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Survey of Various Fuzzy and Uncertain Decision-Making Methods
Takaaki Fujita, Florentin Smarandache
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This paper provides a comprehensive survey of uncertainty-aware multi-criteria decision-making (MCDM) methods, organizing them into a task-oriented taxonomy. It covers problem-level settings, weight elicitation techniques, inter-criteria causality modeling, and various solution procedures including compensatory, distance-to-reference, and outranking frameworks. The authors also introduce new decision-making methods and discuss applications across diverse fields, emphasizing the integration of fuzzy, intuitionistic, neutrosophic, and plithogenic set theories to handle vagueness and incomplete information.
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Takaaki Fujita → authored → Survey of Various Fuzzy and Uncertain Decision-Making Methods
confidence 100% · Survey of Various Fuzzy and Uncertain Decision-Making Methods Takaaki Fujita, Florentin Smarandache
Florentin Smarandache → authored → Survey of Various Fuzzy and Uncertain Decision-Making Methods
confidence 100% · Survey of Various Fuzzy and Uncertain Decision-Making Methods Takaaki Fujita, Florentin Smarandache
Fuzzy Set → isatypeof → Uncertain Set
confidence 95% · To model such uncertainty in a mathematically disciplined manner, many generalized set-theoretic formalisms have been proposed, including Fuzzy Sets
TOPSIS → isatypeof → MCDM
confidence 95% · decision-making itself admits a wide variety of methodologies; for example, TOPSIS
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Abstract
Abstract:Decision-making in real applications is often affected by vagueness, incomplete information, heterogeneous data, and conflicting expert opinions. This survey reviews uncertainty-aware multi-criteria decision-making (MCDM) and organizes the field into a concise, task-oriented taxonomy. We summarize problem-level settings (discrete, group/consensus, dynamic, multi-stage, multi-level, multiagent, and multi-scenario), weight elicitation (subjective and objective schemes under fuzzy/linguistic inputs), and inter-criteria structure and causality modelling. For solution procedures, we contrast compensatory scoring methods, distance-to-reference and compromise approaches, and non-compensatory outranking frameworks for ranking or sorting. We also outline rule/evidence-based and sequential decision models that produce interpretable rules or policies. The survey highlights typical inputs, core computational steps, and primary outputs, and provides guidance on choosing methods according to robustness, interpretability, and data availability. It concludes with open directions on explainable uncertainty integration, stability, and scalability in large-scale and dynamic decision environments.
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Survey of Various Fuzzy and Uncertain Decision-Making Methods Takaaki Fujita, Florentin Smarandache Neutrosophic Scien ce International Association (NSIA) Publis hing House Gallup - Guayaquil United States of America – Ecuador 2026 Editor: Neutrosophic Science International Association (NSIA) Publishing House https://fs.unm.edu/NSIA/ Division of Mathematics and Sciences University of New Mexico 705 Gurley Ave., Gallup Campus NM 87301, United States of America University of Guayaquil Av . Kennedy and Av. Delta “Dr. Salvador Allende” University Campus Guayaquil 090514, Ecuador Peer-Reviewers: Fernando A. F. Ferreira ISCTE Business School, BRU-IUL, University Institute of Lisbon, Avenida das Forças Armadas, 1649-026 Lisbon, Portugal Email: fernando.alberto.ferreira@iscte-iul.pt Julio J. Valdés National Research Council Canada, M-50, 1200 Montreal Road, Ottawa, Ontario K1A 0R6, Canada Email: julio.valdes@nrc-cnrc.gc.ca Tieta Putri College of Engineering, Department of Computer Science and Software Engineering, University of Canterbury, Christchurch, New Zealand Contents in this book The remainder of this book is organized as follows. 1 Introduction7 1.1 Uncertain Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 1.2 Uncertain Decision-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 1.3 Our Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 2 Preliminaries19 2.1 Fuzzy Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Intuitionistic fuzzy set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Neutrosophic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Plithogenic Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Rough Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6 Soft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.7 Uncertain set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.8 Fuzzy Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.9 Uncertain decision-making (UDM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 Problem-level frameworks31 3.1 Fuzzy Multiple-Criteria Decision Making (Fuzzy MCDM) . . . . . . . . . . . . . . . . . . . 31 3.2 Fuzzy MADM (Fuzzy Multi-attribute decision-making) . . . . . . . . . . . . . . . . . . . . . 34 3.3 Fuzzy Group-Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4 Fuzzy dynamic decision-making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.5 Fuzzy Multiple Objective Decision-making . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.6 Fuzzy ethical decision making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.7 Fuzzy Consensus decision-making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.8 Fuzzy Strategic decision making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.9 Fuzzy Multi-Expert Decision-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.10 Fuzzy Multi-Stage Decision-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.11 Fuzzy Multi-Level Decision-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.12 Fuzzy Multi-Agent Decision-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.13 Fuzzy Multi-Scenario Decision-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3 4 Weight elicitation Decision-Methods93 4.1 Fuzzy Analytic hierarchy process (Fuzzy AHP) . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.2 Fuzzy LOPCOW (Fuzzy Linear Optimization for Comprehensive Weight) . . . . . . . . . . 98 4.3 Fuzzy SIWEC (Fuzzy simple weight calculation for criteria) . . . . . . . . . . . . . . . . . . 102 4.4 Fuzzy judgment matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.5 Fuzzy Analytic network process (ANP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.6 Fuzzy Ordinal Priority Approach (OPA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.7 Fuzzy PIPRECIA (Pivot Pairwise Relative Criteria Importance Assessment) . . . . . . . . 118 4.8 Fuzzy SWARA (Fuzzy Stepwise Weight Assessment Ratio Analysis) . . . . . . . . . . . . . 122 4.9 Fuzzy CILOS (Fuzzy Criterion Impact Loss) . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.10 Fuzzy IDOCRIW (Fuzzy Entropy-CILOS integrated objective weighting) . . . . . . . . . . 132 4.11 Fuzzy BWM (Fuzzy Best-Worst Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.12 Fuzzy CRITIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.13 Fuzzy MEREC (Fuzzy MEthod based on the Removal Effects of Criteria) . . . . . . . . . . 146 4.14 Fuzzy FUCOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5 Structure / causality decision-modelling (inter-criteria influence)155 5.1 Fuzzy DEMATEL (Fuzzy Decision Making Trial and Evaluation Laboratory) . . . . . . . . 155 5.2 Fuzzy ISM (Fuzzy Interpretive Structural Modeling) . . . . . . . . . . . . . . . . . . . . . . 159 5.3 Fuzzy MICMAC (cross-impact / driving–dependence analysis) . . . . . . . . . . . . . . . . 165 5.4 Fuzzy Cognitive Map (FCM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6 Compensatory scoring methods on a decision matrix (“weighted-sum” families)175 6.1 Fuzzy COPRAS (Fuzzy Complex Proportional Assessment) . . . . . . . . . . . . . . . . . . 177 6.2 Fuzzy MACBETH (Fuzzy Measuring Attractiveness by a Categorical Based Evaluation Tech- nique) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.3 Fuzzy CoCoSo (Fuzzy Combined Compromise Solution) . . . . . . . . . . . . . . . . . . . . 184 6.4 Fuzzy SAW (Fuzzy Simple Additive Weighting) . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.5 Fuzzy RAFSI (Fuzzy Ranking of Alternatives through Functional mapping of criterion Sub- Intervals into a single Interval) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.6 Fuzzy RATMI (Fuzzy Ranking of Alternatives by Trace-to-Median Index) . . . . . . . . . . 197 6.7 Fuzzy RANCOM (Fuzzy RANking COMparison) . . . . . . . . . . . . . . . . . . . . . . . . 202 6.8 Fuzzy AROMAN (Fuzzy Alternative Ranking Order Method accounting for two-step normal- ization) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.9 Fuzzy MAUT (Fuzzy multi-attribute utility theory) . . . . . . . . . . . . . . . . . . . . . . . 212 6.10 Fuzzy SMART (Fuzzy Simple Multi-Attribute Rating Technique) . . . . . . . . . . . . . . . 216 6.11 Fuzzy REGIME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 6.12 Fuzzy TODIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.13 Fuzzy GRA (Fuzzy Grey Relational Analysis) . . . . . . . . . . . . . . . . . . . . . . . . . . 227 6.14 Fuzzy ARAS (Fuzzy Additive Ratio Assessment) . . . . . . . . . . . . . . . . . . . . . . . . 231 6.15 Fuzzy WASPAS (Fuzzy Weighted Aggregated Sum Product Assessment) . . . . . . . . . . . 235 6.16 Fuzzy MOORA (Fuzzy Multi-Objective Optimization on the basis of Ratio Analysis) . . . . 239 6.17 Fuzzy Preference Selection Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 6.18 FROV (Fuzzy Range of Value method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 6.19 Fuzzy MOOSRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 7 Distance-to-reference / border / compromise-index methods251 7.1 Fuzzy TOPSIS (Fuzzy Technique for Order of Preference by Similarity to Ideal Solution) . 252 7.2 Fuzzy MARCOS (Fuzzy Measurement Alternatives and Ranking according to Compromise Solution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 7.3 Fuzzy CODAS (Fuzzy COmbinative Distance-based ASsessment) . . . . . . . . . . . . . . . 259 7.4 Fuzzy EDAS (Fuzzy Evaluation based on Distance from Average Solution) . . . . . . . . . . 263 7.5 Uncertain VIKOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 7.6 Uncertain MABAC (Multi-attributive border approximation area comparison) . . . . . . . . 270 7.7 Fuzzy MAIRCA (Fuzzy Multi-attributive ideal-real comparative analysis) . . . . . . . . . . 273 8 Outranking (non-compensatory / dominance relations) Decision Methods277 8.1 Fuzzy ELECTRE (Fuzzy Elimination and choice translating reality) . . . . . . . . . . . . . 278 8.2 Fuzzy FlowSort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 8.3 Uncertain PROMETHEE (Preference Ranking Organization METhod for Enrichment of Evaluations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 8.4 Fuzzy QUALIFLEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 8.5 Fuzzy ORESTE (Fuzzy Organization Rangement Et Synthese De Donnees Relationnelles) . 292 9 Rule induction / learning / evidence / sequential decision methods297 9.1 Fuzzy Decision Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 9.2 Fuzzy DRSA (Fuzzy Dominance-based rough approximation) . . . . . . . . . . . . . . . . . 301 9.3 Fuzzy Markov Decision Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 9.4 Fuzzy Evidential Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 10 Other Related Decision Methods313 10.1 Fuzzy Goal programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 10.2 Fuzzy DEA (Fuzzy Data Envelopment Analysis) . . . . . . . . . . . . . . . . . . . . . . . . 318 10.3 Fuzzy Social Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 10.4 Fuzzy decision tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 10.5 Fuzzy decision matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 10.6 Fuzzy Ultrafilter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 10.7 Fuzzy SWOT Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 10.8 Fuzzy Cost-Benefit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 10.9 Fuzzy Decision curve analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 10.10Fuzzy Rational Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 11 Applications of Decision-Making349 11.1 Applications of Computer Science and Engineering . . . . . . . . . . . . . . . . . . . . . . . 349 11.2 Applications of Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 11.3 Applications of Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 11.4 Applications of Business . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 11.5 Applications of Medicine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 11.6 Applications of Social Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 12 Discussions: New Decision-Making Methods353 12.1 Unified Decisional Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 12.2 Unified Uncertain Decisional Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 12.3 Iterated Multi-Criteria Decision-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 12.4 Iterated Multi-Attribute Decision-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 12.5 Iterated Multi-Objective Decision-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 12.6 Analytic SuperHyperNetwork Process (ASHNP) . . . . . . . . . . . . . . . . . . . . . . . . 361 12.7 Analytic Recursive SuperHyperNetwork Process (ARSHNP) . . . . . . . . . . . . . . . . . . 366 12.8 SuperHyperDecision-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 12.9 Uncertain SuperHyperDecision-Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 13 Conclusion373 A Graphic Structure and Uncertain Graphic Structure377 A.1 Graphic structure: a unifying incidence-based model . . . . . . . . . . . . . . . . . . . . . . 377 A.2 Uncertain graphic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 Appendix (List of Tables)380 Appendix (List of Figures)383 Survey of Various Fuzzy and Uncertain Decision-Making Methods Takaaki Fujita 1∗ and Florentin Smarandache 2 1 Independent Researcher, Tokyo, Japan. Email: Takaaki.fujita060@gmail.com 2 University of New Mexico, Gallup Campus, NM 87301, USA. Email: fsmarandache@gmail.com Abstract Decision-making in real applications is often affected by vagueness, incomplete information, heterogeneous data, and conflicting expert opinions. This survey reviews uncertainty-aware multi-criteria decision-making (MCDM) and organizes the field into a concise, task-oriented taxonomy. We summarize problem-level settings (discrete, group/consensus, dynamic, multi-stage, multi-level, multi- agent, and multi-scenario), weight elicitation (subjective and objective schemes under fuzzy/linguistic in- puts), and inter-criteria structure and causality modelling. For solution procedures, we contrast compen- satory scoring methods, distance-to-reference and compromise approaches, and non-compensatory outrank- ing frameworks for ranking or sorting. We also outline rule/evidence-based and sequential decision models that produce interpretable rules or policies. The survey highlights typical inputs, core computational steps, and primary outputs, and provides guidance on choosing methods according to robustness, interpretability, and data availability. It concludes with open directions on explainable uncertainty integration, stability, and scalability in large-scale and dynamic decision environments. Keywords:Fuzzy Set, Neutrosophic Set, Plithogenic Set, MCDM, Decision-Making Chapter 1 Introduction 1.1 Uncertain Set Real-world systems rarely provide perfectly crisp information. Observations may be imprecise, partially reliable, or incomplete. To model such uncertainty in a mathematically disciplined manner, many gener- alized set-theoretic formalisms have been proposed, including Fuzzy Sets [1], Intuitionistic Fuzzy Sets [2], Neutrosophic Sets [3,4], Vague Sets [5], Hesitant Fuzzy Sets [6], Picture Fuzzy Sets [7], Quadripartitioned Neutrosophic Sets [8], PentaPartitioned Neutrosophic Sets [9], Plithogenic Sets [10], HyperFuzzy Sets [11], and HyperNeutrosophic Sets [12]. Applications of fuzzy models and their extensions—outlined in later parts of this book—have been investigated extensively in decision science, chemistry, control, and machine learning [13]. In practice, the appropriate set model depends on (i) the phenomenon being described and (i) how many uncertainty components are needed to represent it faithfully. In the classical fuzzy paradigm, each elementxof a universeXis assigned a single membership grade μ(x)∈[0,1], indicating the extent to whichxbelongs to a given fuzzy set [1]. For reference, the comparison between classical (crisp) sets and fuzzy sets is presented in Table 1.1. Table 1.1: Concise comparison between classical (crisp) sets and fuzzy sets. AspectClassical (Crisp) SetFuzzy Set Membership modelIndicator functionχ A :X→0,1Membership functionμ A :X→[0,1] [1] Decision boundarySharp (in/out)Gradual (degrees of belonging) Expresses uncertainty Not directly (binary membership only) Yes (graded membership captures vagueness) Basic operations∪,∩, c defined via Boolean logicGeneralized via t-norms/t-conorms (e.g., max/min as a common choice) Typical use casesExact categories, deterministic rulesImprecise concepts, soft thresholds, human judgments For an intuitionistic fuzzy set, eachx∈Xis characterized by a pair(μ(x),ν(x))of membership and non-membership values, whereμ,ν:X→[0,1]satisfy 0≤μ(x) +ν(x)≤1, 7 Chapter 1. Introduction [2,14] and the residual quantity1−μ(x)−ν(x)is commonly interpreted as hesitation. A neutrosophic set refines this representation by associating to eachx∈Xa triple (T(x),I(x),F(x)), whereT(x),I(x), andF(x)denote degrees of truth, indeterminacy, and falsity, typically in[0,1]. Unlike the intuitionistic fuzzy constraint, neutrosophic components are not required to sum to1, which enables the encoding of incomplete, inconsistent, or redundant information in a flexible manner [14,15]. 1 Neutrosophy emphasizes the conceptual significance of neutrality and indeterminacy and has stimulated parallel devel- opments in neutrosophic logic, probability, statistics, measure theory, integration, and related formalisms. These tools now appear in a wide variety of scientific and engineering applications [13]. Plithogenic sets further broaden this landscape by modeling each element through its attribute values to- gether with corresponding degrees of appurtenance, and by introducing a contradiction (dissimilarity) func- tion between distinct attribute values [10,17,18]. This added structure supports context-aware aggregation of heterogeneous and potentially conflicting assessments, thereby refining classical fuzzy, intuitionistic fuzzy, and neutrosophic descriptions (see, e.g., [19, 20]). For quick reference, Table 1.2 summarizes the canoni- cal information associated with each element across several representative set extensions (using notation harmonized for this book). Table 1.2: Representative set extensions and the canonical information stored per element. Set TypeCanonical data attached to each element Fuzzy SetMembership mappingμ:X→[0,1]. Intuitionistic Fuzzy Set Membershipμand non-membershipνwithμ(x)+ν(x)≤1; the gap1−μ(x)− ν(x)is interpreted as hesitation. Neutrosophic SetTriple(T,I,F)withT,I,F∈[0,1](truth, indeterminacy, falsity), treated as independent coordinates. Plithogenic SetTuple(P,v,Pv,pdf,pCF)where pdf:P×Pv→[0,1] s encodess-dimensional appurtenance and pCF:Pv×Pv→[0,1] t is a symmetric contradiction map taking values in[0,1] t . As a further generalization, Uncertain Sets are also known [20]. Uncertain sets extend classical sets by assigning graded, multi-component membership information (e.g., truth and indeterminacy) to elements, thereby modeling vagueness and incomplete data in a rigorous way. These concepts are also often studied in conjunction with notions such as grey sets, interval sets, rough sets, near sets, soft sets, and hypersoft sets. 1.2 Uncertain Decision-Making Decision-making selects an action among alternatives by evaluating objectives, criteria, constraints, and uncertainty to achieve desired outcomes. These ideas have been extended by incorporating uncertainty- oriented logics, leading to diverse research directions such as Fuzzy Decision-Making [21,22], Intuitionistic 1 Intuitionistic fuzzy sets do not treat indeterminacy as a primary coordinate, whereas neutrosophic operators incorporate indeterminacy on the same footing as truth-membership and falsehood-nonmembership [14,16]. It is broadly accepted that the neutrosophic set framework subsumes several well-known models, including intuitionistic fuzzy sets, inconsistent intuitionistic fuzzy sets (covering picture fuzzy and ternary fuzzy sets), Pythagorean fuzzy sets, spherical fuzzy sets, andq-rung orthopair fuzzy sets. In a similar spirit, neutrosophication extends a range of decision and uncertainty paradigms, such as regret theory, grey system theory, and three-way decision theory [16]. Chapter 1. Introduction Fuzzy Decision-Making [23, 24], Neutrosophic Decision-Making [25, 26], and Plithogenic Decision-Making [27–29]. Moreover, decision-making itself admits a wide variety of methodologies; for example, TOPSIS [30], AHP [31], ANP [32], CoCoSo, and DEMATEL are well-known representative methods. By combining (i) a chosen uncertain-set paradigm (e.g., fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, or other uncertain sets) with (i) a selected decision-making technique, one can investigate what novel and practically meaningful outcomes emerge. In particular, research explores how efficiently real-world decision processes can be supported or improved in terms of accuracy, robustness, transparency, and computational cost. (i) Uncertain-set paradigm •Fuzzy set •Intuitionistic fuzzy set •Neutrosophic set •Plithogenic set •Other uncertain sets (i) Decision-making tech- nique •AHP / ANP •TOPSIS •CoCoSo •DEMATEL •Other MCDM / GDM methods Combine & Integrate Uncertain Decision-Making (Model + Method) Investigate outcomes •Novel theoretical results (well-definedness, consistency, properties) •Practical performance (accuracy, robustness, interpretability) •Efficiency (computational cost, scalability, decision speed) •Applicability under constraints and incomplete information Figure 1.1: Conceptual diagram: combining an uncertain-set paradigm with a decision-making technique to obtain novel and practically meaningful outcomes. 1.3 Our Contributions In view of the discussion above, the paradigms of decision-making and uncertain sets—including fuzzy sets and neutrosophic sets—are of fundamental importance. In this book, we conduct a comprehensive survey of methods for uncertain decision-making. Our primary objective is to provide a useful reference that supports and informs future research by specialists in this area. In addition, we propose several new decision-making methods. For reference, a compact taxonomy of methods inUncertain Decision Science, classified by task, input structure, and typical outputs, is presented in Table 1.3. Since the table extends across multiple pages, we apologize for any inconvenience. In this book, each of these concepts will be discussed briefly, together with related methods where appropriate. Chapter 1. Introduction Table 1.3: Compact taxonomy of methods inUncertain Decision Science(classified by task, input structure, and typical outputs). Method Pri- mary task Typical inputsPrimary out- puts Notes / when used A. Problem-level frameworks (problem statements / settings) Fuzzy MCDM (gen- eral) [33] Frame- work Alternatives×crite- ria; fuzzy/uncertain evaluations; weights Ranking / best alternative Umbrella viewpoint for most items below. Fuzzy MADM [34] Frame- work (dis- crete) Finite alternatives; attributes;fuzzy numbers / linguistic ratings RankingA common discrete MCDM setting (matrix- based). Fuzzy Group- Decision Making [35] Frame- work (group) Multiple experts; in- dividual fuzzy matri- ces; expert weights Collective ma- trix; ranking Includes aggregation + (optional) consensus stage. Fuzzy Consensus decision- making Con- sensus process Experts’ fuzzy opin- ions; similarity/con- sensus threshold Consensus- reaching+ final choice Explicit iteration until consensus is reached. Fuzzy Multiple Objective Decision- making [36] Opti- mization (multi- objec- tive) Continuous decision vector; multiple (pos- sibly fuzzy) objec- tives/goals Compromise solution Often reduced to amembership- maximization scalariza- tion. Fuzzy ethi- cal decision making Rule/cri- teria decision Ethical dimensions; fuzzy rules or fuzzy scores per action Action choice / ranking Domain-specific MCDM (often fuzzy inference). Fuzzy Strategic decision making Planning decision Projects/measures; goalachievement +cost/resource satisfaction Optimal plan / selection set Often modeled as weighted fuzzy satisfac- tion under constraints. Fuzzy Dynamic Decision- Making (Dynamic /multi- period MCDA) (cf. [37]) Frame- work (dy- namic) Time-indexed eval- uations/weights ( ̃x ij (t),w j (t)); up- date/discount rules; (optional) scenario data Time- aggregated rankingor adaptive pol- icy/plan Models evolving pref- erences/performances across periods; aggre- gates overt= 1,...,T (e.g. discounted, rolling, or state-updated). Continued on next page. Chapter 1. Introduction Method Pri- mary task Typical inputsPrimary out- puts Notes / when used Fuzzy Multi- Expert Decision- Making Frame- work (group / multi- expert) Experts’ fuzzy eval- uations (matrices or preference relations); expert weights; ag- gregation/consensus rule Collective fuzzyas- sessment; ranking/choice General group MCDA: aggregatesmultiple experts’ fuzzy opinions (optionally with consen- sus constraints) into a single decision. Fuzzy Multi-Stage Decision- Making Frame- work (multi- stage) Stage-indexed cri- teria/alternatives; inter-stagecon- straints;fuzzy transition/feedback rules Stage-wise decisions; final plan/policy Sequentialdecisions across stages; earlier choices affect feasible sets and evaluations at later stages (fuzzy un- certainty propagated). Fuzzy Multi-Level Decision- Making Frame- work (hierar- chical / bilevel) Leader–follower ob- jectives/constraints (or hierarchical crite- ria); fuzzy goals and constraints Hierarchi- cal solution (compromise) Models decisions across levels (e.g., bilevel): upper-levelchoices influencelower-level feasible responses under fuzzy preferences/tar- gets. Fuzzy Multi- Agent Decision- Making Frame- work (multi- agent) Agents’utili- ties/preferences (fuzzy); interaction protocol (coopera- tive/competitive); information sharing rules Joint decision, equilibrium, or negotiated outcome Multiple agents with possiblyconflicting fuzzy preferences; so- lution via aggregation, bargaining, or game- theoretic negotiation mechanisms. Fuzzy Multi- Scenario Decision- Making Frame- work (scenario- based) Scenario setΩ; sce- nario weights/prob- abilities (possibly fuzzy); scenario-wise fuzzy performances Robust rank- ing/choice across scenar- ios Evaluatesalterna- tives under multiple scenarios; aggregates scenario-wise fuzzy out- comes (e.g., expected, worst-case, regret) for robust selection. B. Weight elicitation (derive criterion weights) Fuzzy AHP [38] Weight- ing (pair- wise) Fuzzyreciprocal pairwise comparison matrices (hierarchy) Local/global weights; prior- ities Tree/hierarchy; geomet- ric mean / eigenvector variants. Fuzzy ANP [39] Weight- ing (net- work) Pairwisecompar- isons + dependence network; superma- trix Global priori- ties Generalizes AHP to interdependencies (limit supermatrix). Fuzzy BWM [40] Weight- ing (best– worst) Best-to-others and others-to-worst fuzzy ratios Weights (opti- mization) Consistency-controlled, fewer comparisons than AHP. Continued on next page. Chapter 1. Introduction Method Pri- mary task Typical inputsPrimary out- puts Notes / when used Fuzzy CILOS (Criterion Impact Loss) [41] Ob- jective weight- ing Normalized fuzzy decisionmatrix; impact-loss compu- tation per criterion (fuzzy) Objective weights (pos- siblyfuzzy ̃w j ) Assigns larger weights to criteria causing greater information/im- pact loss when omitted or deteriorated (fuzzy loss-based weighting). Fuzzy SWARA [42] Weight- ing (step- wise) Orderedcriteria; fuzzy stepwise im- portance coefficients Weights (se- quential) Fast elicitation when an importance order is known. Fuzzy FU- COM [43] Weight- ing (full consis- tency) Orderedcriteria; consecutive fuzzy priority ratios Weights + de- viation index Optimization enforces ratio + transitivity consistency. Fuzzy PI- PRECIA [44] Weight- ing (pivot/step) Sequential compar- isons to a pivot / neighbor; fuzzy scales Weights (se- quential) Ordinary + inverse pass often used for robust- ness. Fuzzy OPA [45] Weight- ing (ordinal) Experts’ordinal rankings (+ optional fuzzy importances) Weights (max– min model) No pairwise matrices; uses an optimization model. Fuzzy judgment matrix Data structure Fuzzy preference de- grees (often comple- mentary) in[0,1] Inputfor weights/ consistency Used in fuzzy AHP-like pipelines and preference modeling. Fuzzy LOPCOW (Loga- rithmic Percentage Change- driven Objective Weight- ing) [46] Ob- jective weight- ing Fuzzy decision ma- trix (defuzzified); benefit/cost types; normalization Criterion weightsw j (and impor- tance ranking) ComputesPV j = 100|ln(RMS j /σ j )|from normalized data, then normalizesPV j to obtainw j . Fuzzy SIWEC (F-SIWEC) [47] Weight- ing (direct rating; dispersion- adjusted) Experts’ linguistic importance ratings →TFNs; normal- ization; (optional) expert weights Fuzzy(or defuzzified) criterion weights ̃w j (and criteria ranking) Fast weighting without pairwise comparisons; uses dispersion (e.g. standarddeviation) to reflect experts’ disagreement. Continued on next page. Chapter 1. Introduction Method Pri- mary task Typical inputsPrimary out- puts Notes / when used Fuzzy IDOCRIW [48] Ob- jective weight- ing (inte- grated) Normalized fuzzy decisionmatrix (benefit/cost); en- tropydispersion; CILOS impact-loss (fuzzy/defuzzified); aggregation rule Integrated ob- jective weights w (IDOCRIW) j (and criteria ranking) Purelydata-driven weightscombining informationdisper- sion and relative loss; used to reduce sub- jectivity and stabilize weighting when criteria scales/variability differ. C. Structure / causality modelling (inter-criteria influence) Fuzzy DEMA- TEL [49] Causal analysis Fuzzy direct-relation (influence) matrix among criteria Total relation; cause/effect groups Outputs prominence (D+R) and relation (D−R). Fuzzy ISM (Inter- pretive Structural Model- ing) [50] Struc- tural mod- elling (hierar- chy) Expert reachability judgments (binary→ fuzzy); reachability matrix; transitive closure Hierarchical directed influ- ence graph Fuzzy ISM uses graded reachability from lin- guistic terms; fuzziness is propagated through closure. Fuzzy MICMAC (cross- impact / driving– dependence) [51] Influence classifi- cation (ISM)reachabil- ity matrix (bi- nary/fuzzy); cross- impact strengths Driving power /depen- dence indices; clusters (au- tonomous, dependent, linkage, driv- ing) Fuzzy MICMAC com- putes graded driving– dependence strengths and yields robust clus- tering under ambiguity. Fuzzy To- talISM (TISM) Inter- pretable structure model ISM reachability + link-wise rationales (expertexplana- tions) Hierarchy +anno- tated edges (strength + interpretation) Adds explicit justifica- tions to each link while preserving the ISM- derivedhierarchical structure. Fuzzy Cog- nitive Map (FCM) [52] Causal network (dy- namic) Signeddirected causal graph; fuzzy edge weights; initial concept activations Scenario dynamics; steady states /trajecto- ries; influence graph Uses fuzzy causal weights and nonlin- ear updates to model feedback loops and uncertain propagation. D. Compensatory scoring on a decision matrix (“weighted-sum” families) Fuzzy SAW [53] Ranking (addi- tive) Normalized fuzzy decision matrix + weights Scoreand ranking Prototype weighted- sum method; highly interpretable. Fuzzy SMART [54] Ranking (addi- tive) Value scales + swing weights; fuzzy rat- ings possible Utility score; ranking Essentially structured SAW with explicit value scaling. Continued on next page. Chapter 1. Introduction Method Pri- mary task Typical inputsPrimary out- puts Notes / when used Fuzzy MAUT [55] Utility aggrega- tion Single-attribute util- ities; weights; fuzzy outcomes possible Overall utility; ranking Utility-theoretic (often additive/multiplicative forms). Fuzzy MAC- BETH [56] Value scale construc- tion Qualitativepair- wisedifference- of-attractiveness judgments Value func- tions + rank- ing Builds numerical value scales from linguistic categories. Fuzzy CO- PRAS [57] Ranking (bene- fit/cost) Decisionmatrix; weights; benefit/cost partition Utility degree; ranking Separates benefit and cost sums in the scoring. Fuzzy MOORA [58] Ranking (ratio) Normalized decision matrix;weights; benefit–cost split Netscore; ranking Often used with Multi- MOORA extensions. Fuzzy ARAS [59] Ranking (add ideal alt.) Decision matrix + added optimal alter- native; weights Utility ratio; ranking Compares each alter- native to an explicitly added ideal. Fuzzy WAS- PAS [60] Ranking (hybrid) Decisionmatrix; weights; normaliza- tion Integrated score; ranking Combines WSM (sum) and WPM (product). Fuzzy Co- CoSo [61] Ranking (compro- mise) Decisionmatrix; weights; normalized sums/powers Compromise score; ranking Combines additive and multiplicative aggrega- tions. Fuzzy REGIME Pairwise win/loss aggrega- tion Pairwisecompar- isons per criterion; weights Preference ma- trix; ranking Aggregates criterion- wise wins/losses across alternatives. Fuzzy TODIM [62] Prospect- theory ranking Referencepoint; gains/losses; atten- uation parameter; weights Dominance values; rank- ing Captures loss aversion / asymmetric preference. Fuzzy GRA Simi- larity ranking Reference sequence; normalizeddata; distinguishing coeffi- cient Relational grade; ranking Closeness to ideal pat- tern via grey relational grades. Fuzzy Preference Selection Index Ranking (index) Decision matrix; nor- malization (often no explicit weights) Preference in- dex; ranking Weight-light / weight- free variants are com- mon. Fuzzy Range of Value method Ranking (inter- val/value range) Normalized perfor- mances;weights; value ranges Overall value; ranking Emphasizes value inter- vals/ranges in aggrega- tion. Continued on next page. Chapter 1. Introduction Method Pri- mary task Typical inputsPrimary out- puts Notes / when used Fuzzy MOOSRA (Multi- Objective Optimiza- tionon the basis of Simple Ratio Anal- ysis) [63] Ratio- based scoring Normalized fuzzy performances; weights; benefit/cost partition Ratio score; ranking Computes a weighted ratio-type utility from fuzzy normalized values (often benefit-over-cost style) to obtain a com- pensatory ranking. Fuzzy AROMAN (Alterna- tive Rank- ing Order Method Accounting for two-step Normaliza- tion) [64] Ranking (normalization- based scoring) Decisionmatrix; weights; benefit/cost partition; two-step normalization Overall utility score; ranking Performs two-step nor- malization of criterion values, then aggregates weighted normalized performances to obtain a final ranking. Fuzzy RANCOM (e.g., PTF- RANCOM) [65] Weight- ing (expert- judgment based) Criteria;experts’ linguistic importance assessments (fuzzy); (optional)expert weights; aggregation operator Criterion weights (and criteria rank- ing) Aggregates fuzzy im- portance evaluations, scores and ranks cri- teria, then derives normalized weight coef- ficients for subsequent MCDA methods. Fuzzy RAFSI [66] Ranking (func- tional map- ping; rank- reversal resis- tant) Fuzzy decision ma- trix (e.g., TFNs); weights; ideal/anti- idealreference points;mapping interval(n 1 ,n 2 ) Utility score; ranking Maps each criterion sub-interval to a com- mon interval, applies max/min normaliza- tion (viaA,H), then aggregatesweighted scores. Fuzzy RATMI Ranking (trace- to- median index) Fuzzy decision ma- trix; weights; bene- fit/cost types; nor- malization; parame- terv Index scoreE i ; ranking Aggregates weighted normalizedperfor- mances via a trace index and a median-similarity term to obtain a final compromise score. E. Distance-to-reference / border / compromise-index methods Uncertain TOP- SIS [67] Distance to (ideal, nadir) (uncertain/fuzzy) decisionmatrix; weights; metric Closeness coef- ficient; ranking Ranks by distance to FPIS/FNIS (ideal/anti- ideal). Continued on next page. Chapter 1. Introduction Method Pri- mary task Typical inputsPrimary out- puts Notes / when used Uncertain VIKOR [68] Com- promise program- ming Decisionmatrix; weights; best/worst; parameterv Compromise indexQ; solution set Balances group utility and individual regret. Uncertain MABAC [69] Border approxi- mation Decisionmatrix; weights; border area construction Deviation score; ranking Ranks by signed dis- tance from border ap- proximation area. Fuzzy EDAS [70] Distance from average Decisionmatrix; averagesolution; positive/negative distances Appraisal score; ranking Uses distances to the av- erage (not to ideal). Fuzzy CO- DAS [71] Distance (Eu- clid + Taxicab) Decisionmatrix; negative-ideal; dis- tance measures Relative as- sessment; ranking Often uses Euclidean + Manhattan discrimina- tion. Fuzzy MAR- COS [72] Utility vs ideal/anti- ideal Extendedmatrix withideal/anti- ideal; normalization; weights Utility func- tions; ranking Explicitly includes both ideal and anti-ideal ref- erences. Fuzzy MAIRCA (Multi- Attributive Ideal–Real Com- parative Analysis) Ideal– real com- parative ranking Fuzzy decision ma- trix; weights; the- oretical/ideal distri- bution vs observed (real) performance Gap-based score; ranking Compares “theoretical” (ideal)expectations with “real” fuzzy performances;ranks by aggregated fuzzy deviations (gaps). Fuzzy FlowSort Outranking- based sorting Preferencefunc- tions and flows (PROMETHEE); referencepro- files; fuzzy evalu- ations/weights Category as- signment via flows PROMETHEE-flow sorting: compares al- ternatives to limiting profiles and assigns classes using posi- tive/negative/net flow rules (fuzzy inputs). F. Outranking (non-compensatory / dominance relations) Uncertain ELEC- TRE [73] Out- ranking / kernel Concordance– discordance indices; thresholds; weights Outranking graph; kernel set Non-compensatory; yields a dominance relation and best set. Uncertain PROMETHEE [74] Out- ranking flows Preference functions on pairwise differ- ences; weights Positive/neg- ative/net flows Produces complete or partial ranking via flows. Continued on next page. Chapter 1. Introduction Method Pri- mary task Typical inputsPrimary out- puts Notes / when used Fuzzy QUAL- IFLEX [75] Out- ranking (permutation- based ranking) Pairwise (possibly fuzzy) preference re- lations per criterion; criterionweights (optional) Best order- ing (ranking) maximizing concordance Evaluates permutations; aggregates (fuzzy) con- cordance of pairwise preferences to select the most consistent ranking. Fuzzy ORESTE [76] Out- ranking (ordinal- distance ranking) Ordinal ranks / preference intensities (possiblyfuzzy); distance/aggregation rule Ranking (often robust under weak data) Uses (fuzzy) ordinal in- formation and distance- based aggregation when precise cardinal evalua- tions are unavailable. G. Rule induction / learning / evidence / sequential decisions Fuzzy DRSA Dominance- based rules Orderedcriteria; dominance relations; fuzzy boundaries Decision rules; class approxi- mations Produces if–then rules and reduct-like struc- tures. Fuzzy Decision Tree [77] Pre- dictive model Training data; fuzzy splits / fuzzy parti- tions Tree model; predictions Learning-oriented; can support decision recom- mendation. Fuzzy Ev- idential Reasoning Evidence aggrega- tion Belief degrees / ev- idences (often DS- like) + weights Aggregated belief / utility; ranking Combines multiple un- certain evidences sys- tematically. Fuzzy Markov Decision Process Sequen- tial control States,actions, (fuzzy)transi- tions/rewards Optimal pol- icy;value function Multi-stagedecision under uncertainty (dy- namic programming). Chapter 1. Introduction Chapter 2 Preliminaries This chapter collects the basic notation and background used throughout the book. 2.1 Fuzzy Set Fuzzy set theory extends the classical notion of a subset by allowing graded membership, quantified by a value in[0,1][1,78,79]. The core definition is recalled next. Definition 2.1.1(Fuzzy set).[1] LetXbe a nonempty set. Afuzzy setAonXis specified by a membership function μ A :X→[0,1]. Equivalently, one may write A=(x,μ A (x))|x∈X, whereμ A (x)indicates the degree to whichxbelongs toA. A particularly common numeric object in fuzzy decision analysis is thetriangular fuzzy number(TFN) [80,81]. It models an imprecise quantity through a simple piecewise-linear membership profile. Definition 2.1.2(Triangular fuzzy number (TFN)).[82, 83] LetXbe a universe. Atriangular fuzzy number(TFN) ̃xis a fuzzy set onR whose membership functionμ ̃x :R→[0,1]is determined by three real parameters ̃x= (l x ,m x ,u x )∈R 3 , l x ≤m x ≤u x , via μ ̃x (t) = 0,t < l x , t−l x m x −l x , l x ≤t≤m x ,(m x > l x ), u x −t u x −m x , m x ≤t≤u x ,(u x > m x ), 0,t > u x . 19 Chapter 2. Preliminaries Ifl x =m x (resp.m x =u x ), the corresponding middle expression is interpreted in the limiting sense, so that μ ̃x (m x ) = 1and the membership curve remains triangular. If, in addition,0< l x ≤m x ≤u x , then ̃xis called apositiveTFN. Herem x represents the modal (most plausible) value, whilel x andu x serve as lower and upper bounds. 2.2 Intuitionistic fuzzy set Intuitionistic fuzzy sets enrich fuzzy sets by recording both membership and non-membership information, leaving room for an explicit hesitation component [2]. A standard formulation is as follows. Definition 2.2.1(Intuitionistic fuzzy set).[84] LetEbe a nonempty set. Anintuitionistic fuzzy set(IFS) AonEis given by A= 〈x,μ A (x),ν A (x)〉:x∈E , where μ A ,ν A :E−→[0,1] are, respectively, the membership and non-membership functions, and for eachx∈E, 0≤μ A (x) +ν A (x)≤1. The remaining part, π A (x) := 1−μ A (x)−ν A (x), is called thehesitation degreeatx. The classical fuzzy-set situation is recovered whenν A (x) = 1−μ A (x)for allx∈E, equivalentlyπ A (x) = 0 for everyx. As extensions ofintuitionistic fuzzy sets, several related concepts are known, such as the following: • Generalized intuitionistic fuzzy sets [85]: Generalized intuitionistic fuzzy sets assign each element mem- bership and nonmembership degrees in[0,1], constrained by min(μ,ν)≤0.5, extending Atanassov’s μ+ν≤1condition for richer, flexible uncertainty modelling. • Cubic intuitionistic fuzzy sets [86, 87]: Cubic intuitionistic fuzzy sets model each element using an interval valued intuitionistic pair and a crisp intuitionistic pair, preserving membership and nonmem- bership information simultaneously fully. • Bipolar intuitionistic fuzzy sets [88,89]: Bipolar intuitionistic fuzzy sets assign each element positive membership and nonmembership in[0,1]and negative membership and nonmembership in[−1,0], satisfying corresponding sum bounds simultaneously. • Complex intuitionistic fuzzy sets [90, 91]: Complex intuitionistic fuzzy sets assign complex valued membership and nonmembership grades; membership, nonmembership, and their sum remain within the unit circle or unit square always. • Interval-valued intuitionistic fuzzy sets [92,93]: Interval-valued intuitionistic fuzzy sets represent each element by membership and nonmembership intervals in[0,1], with upper bounds satisfyingμ+ν≤1, capturing imprecision explicitly. Chapter 2. Preliminaries 2.3 Neutrosophic Set Neutrosophic sets represent uncertainty by assigning three (in general independent) degrees to each element: truthT, indeterminacyI, and falsityF, typically taken in[0,1][4,14,15,94]. This explicit indeterminacy component provides a flexible generalization of both fuzzy sets and intuitionistic fuzzy sets. Definition 2.3.1(Neutrosophic set).[95,96] LetXbe a nonempty set. Aneutrosophic set(NS)AonX is described by three functions T A :X→[0,1], I A :X→[0,1], F A :X→[0,1], where, for eachx∈X, the valuesT A (x),I A (x), andF A (x)quantify the degrees of truth, indeterminacy, and falsity of the statement “x∈A”, respectively. They satisfy 0≤T A (x) +I A (x) +F A (x)≤3. For reference, an illustrative diagram of a Neutrosophic Set is presented in Figure 2.1. X x∈X Element under evaluation A Neutrosophic description ofx A(x) = ( T A (x), I A (x), F A (x) ) T A (x) 01 truth I A (x) 01 indeterminacy F A (x) 01 falsity 0≤T A (x) +I A (x) +F A (x)≤3 Figure 2.1: Conceptual illustration of a neutrosophic set. For eachx∈X, the membership status is described by the triple ( T A (x),I A (x),F A (x) ) representing truth, indeterminacy, and falsity degrees, respectively. As extensions ofneutrosophic sets, several related concepts are known, such as the following: •Cubic Neutrosophic Sets [97,98]:Represent each element by both an interval neutrosophic triple (T,I,F)and a single-valued triple, capturing range uncertainty plus a representative point simulta- neously for decision tasks. •Bipolar Neutrosophic Sets [99,100]:Assign two neutrosophic triples per element: positive (sup- porting) and negative (opposing) truth–indeterminacy–falsity, modelling pros and cons with uncer- tainty on both sides in evaluations. •m-polar Neutrosophic Sets [101]:Generalize bipolar tompoles; each element hasmneutrosophic triples, one per perspective/agent/state, enabling multi-attitude assessments with indeterminacy and inconsistency across criteria sets too. •Interval-valued Neutrosophic Sets [102,103]:Replace truth, indeterminacy, and falsity degrees with intervals in[0,1], allowing bounded uncertainty about each component instead of single numbers during aggregation and ranking steps. Chapter 2. Preliminaries •SuperHyperNeutrosophic Sets [104–106]:Extends neutrosophic sets to superhyperstructures, assigning truth, indeterminacy, and falsity degrees to higher-order, nested entities and relations. •Complex Neutrosophic Sets [107,108]:Use complex-valued truth, indeterminacy, and falsity grades (typically magnitude and phase), representing oscillatory or periodic information while retain- ing neutrosophic separation of components for signal contexts. •Neutrosophic Offsets [109–111]:Neutrosophic offsets extend neutrosophic sets by allowing truth, indeterminacy, or falsity values to fall below 0 or exceed 1 independently. As subclasses of neutrosophic offsets, concepts such as neutrosophic oversets [112,113] and neutrosophic undersets [3,114] are also known. •Spherical Neutrosophic Sets [115,116]:Constrain single-valued degrees so thatT,I, andFlie in [0,1]and satisfy a spherical norm bound, e.g.,T 2 +I 2 +F 2 ≤1, ensuring feasible triples for decision models. As a further extension, n-HyperSpherical Neutrosophic Sets are also known [117–119]. 2.4 Plithogenic Set Plithogenic set assigns multi-criteria membership vectors to elements, modulated by contradiction degrees between attribute values and dominance levels interactions globally [17,120–122]. Definition 2.4.1(Plithogenic Set).[17, 120] LetPbe a nonempty universe of discourse, and letvbe a (fixed) attribute whose possible values form a nonempty setPv. Fix dimensionss,t∈N. Aplithogenic seton(P,v,Pv)is a quintuple PS= (P, v, Pv, pdf, pCF), where •pdf:P×Pv−→[0,1] s is thedegree of appurtenance function(DAF); forx∈Panda∈Pv,pdf(x,a) is the (possibly vector–valued) membership degree ofxcorresponding to the attribute valuea; •pCF:Pv×Pv−→[0,1] t is thedegree of contradiction function(DCF), satisfying pCF(a,a) = 0, pCF(a,b) =pCF(b,a)for alla,b∈Pv. In plithogenic theory, a (typically fixed)dominant attribute valuea ∗ ∈Pvis chosen, and set–theoretic operations (such as union and intersection) are defined by combining the appurtenance degreespdfwith the contradiction degreespCF(·,a ∗ )in order to model interaction and opposition between different attribute values. Plithogenic sets are known for their ability to generalize a wide variety of concepts. If needed, see [20] for further details. Chapter 2. Preliminaries 2.5 Rough Set Rough set theory models imprecision by approximating a target subset through acertainpart (lower approx- imation) and apossiblepart (upper approximation), constructed from an indiscernibility relation [123–126]. The classical Pawlak approximations are recalled below. Definition 2.5.1(Rough set approximations).[127] LetXbe a nonempty universe, and letR⊆X×X be an equivalence (indiscernibility) relation. Forx∈X, write the equivalence class ofxas [x] R :=y∈X|(x,y)∈R. Given any subsetU⊆X, define: 1.Lower approximation: U:=x∈X|[x] R ⊆U. Elements ofUare those that belong toUwith certainty (their entire class is contained inU). 2.Upper approximation: U:=x∈X|[x] R ∩U6=∅. Elements ofUare those that may belong toU(their class meetsU). The pair(U,U)is the rough-set representation ofU, and it always satisfies U⊆U⊆U. As extensions of rough set theory, various concepts have been studied, such as probabilistic rough sets [128,129] and granular rough sets [130,131]. If needed, see [132] for further details. 2.6 Soft set Soft sets provide a parameter-based description of uncertainty: each parameter selects a subset of the universe, and the family of all such selections forms the model. This framework was introduced by Molodtsov (1999) and has been widely used in decision problems [133,134]. Definition 2.6.1(Soft set).[134] LetUbe a universe set and letEbe a set of parameters. TakeA⊆E and denote byP(U)the power set ofU. A pair(F,A)is called asoft setoverUif F:A→P(U). For each parameter∈A, the subsetF()⊆Uis called the-approximationof(F,A). Thus, a soft set is a parameterized family of subsets ofU. As extensions of soft set theory, a wide variety of concepts have been proposed, such as HyperSoft Sets [135,136], SuperHyperSoft Sets [137,138], and TreeSoft Sets [139,140]. If needed, see, for example, [141] for further details. Chapter 2. Preliminaries 2.7 Uncertain set Anuncertain setassociates with each element a degree taken from a chosen uncertainty model, thereby providing a unifying umbrella for fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, and related frame- works [20,142]. Definition 2.7.1(Uncertain model).[142] LetUdenote the class of alluncertain models. EachM∈U is determined by: • a nonempty set Dom(M)⊆[0,1] k ofadmissible degree tuplesfor some fixed integerk≥1; and • model-specific algebraic or geometric constraints imposed on elements of Dom(M)(for example,μ+ ν≤1in the intuitionistic fuzzy setting, or0≤T+I+F≤3in the neutrosophic setting). Typical instances include: •Fuzzy model:Dom(M) = [0,1]; •Intuitionistic fuzzy model:Dom(M) =(μ,ν)∈[0,1] 2 :μ+ν≤1; •Neutrosophic model:Dom(M) =(T,I,F)∈[0,1] 3 : 0≤T+I+F≤3; •Plithogenic model,and many further extensions. Definition 2.7.2(Uncertain set (U-set)).[142] LetXbe a nonempty universe, and fix an uncertain model Mwith degree-domain Dom(M)⊆[0,1] k . Anuncertain set of typeM(briefly, aU-set) onXis a pair U= (X,μ M ), where μ M :X−→Dom(M) is theuncertainty-degree function(membership map) ofU. Forx∈X, the valueμ M (x)∈Dom(M)encodes the degree(s) to whichxbelongs toU, as prescribed by the modelM. As noted in the remark, various generalizations are possible. For reference, Table 2.1 presents a catalogue of uncertainty-set families (U-Sets) organized by the dimensionkof the degree-domain Dom(M)⊆[0,1] k (cf. [20]). 2.8 Fuzzy Graph A fuzzy graph assigns each vertex and edge a membership degree in[0,1], representing uncertain connectivity and relationship strength as graded, rather than binary, links [143]. As related concepts, Fuzzy Digraph [214], Fuzzy HyperGraph [143], and Fuzzy SuperHyperGraph [215] are also known. Definition 2.8.1(Fuzzy graph).[79] Afuzzy graphon a vertex setVis a pairG= (σ,μ)consisting of: Chapter 2. Preliminaries Table 2.1: A catalogue of uncertainty-set families (U-Sets) by the dimensionkof the degree-domain Dom(M)⊆[0,1] k [20]. knote Representative U-Set model(s) whose degree-domain is a subset of[0,1] k 1Fuzzy Set [1,143]; N-Fuzzy Set [144–146] Shadowed Set [147–149] 2Intuitionistic Fuzzy Set [2,150]; Vague Set [5,151]; Bipolar Fuzzy Set (two-component descrip- tion) [152, 153]; Pythagorean Fuzzy Set [154, 155]; Fermatean fuzzy Set [156, 157]; Variable Fuzzy Set [158–160]; Paraconsistent Fuzzy Set [161,162]; Bifuzzy Set [163,164] 3Single-Valued Neutrosophic Set [94,96]; Picture Fuzzy Set [7,165]; Ternary Fuzzy Set [166]; Hesitant Fuzzy Set [6, 167]; Spherical Fuzzy Set [168, 169]; Tripolar Fuzzy Set (three- component formalisms) [170–172]; Neutrosophic Vague Set [173,174] 4Quadripartitioned Neutrosophic Set [8,175]; Double-Valued Neutrosophic Set [176,177]; Dual Hesitant Fuzzy Set [178,179]; Ambiguous Set [180–182]; Turiyam Neutrosophic Set [183–186] 5Pentapartitioned Neutrosophic Set [187–189]; Triple-Valued Neutrosophic Set [190–193] 6Hexapartitioned Neutrosophic Set [194]; Bipolar Neutrosophic Set [99,195]; Bipolar Picture Fuzzy Sets [196,197]; Quadruple-Valued Neutrosophic Set [192,198] 7Heptapartitioned Neutrosophic Set [199–201]; Quintuple-Valued Neutrosophic Set [192,202, 203] 8Octapartitioned Neutrosophic Set [194]; Bipolar Quadripartitioned Neutrosophic Set [204, 205]; Bipolar Double-valued Neutrosophic Set 9Nonapartitioned Neutrosophic Set [194] n(n≥1) Multi-valued (Fuzzy) Sets [206]; MultiFuzzy Set [207];n-Refined Fuzzy Set [208,209] 2n(n≥1)n-Refined Intuitionistic Fuzzy Set [209]; Multi-Intuitionistic Fuzzy Set [207] 3n(n≥1)n-Refined Neutrosophic Set [209,210]; Multi-Neutrosophic Set [207,211,212] Reading guide.In the U-Set scheme [142], each modelMis specified by a degree-domain Dom(M)⊆[0,1] k and a membership mapμ M :X→Dom(M). The table groups representative families by the ambient dimensionk(i.e., how many numerical components are stored per element). (a) A widely cited viewpoint is that neutrosophic sets provide a unifying umbrella covering several earlier multi-component fuzzy models (and their generalizations); see [16]. (b) Ambiguous sets are commonly presented as subclasses of certain four-component neutrosophic families; see [8,175,182]. (c) Turiyam neutrosophic sets are reported as subclasses of quadripartitioned neutrosophic sets; see [213]. • A vertex membership functionσ:V→[0,1], whereσ(x)gives the degree to whichx∈Vbelongs to the graph. • An edge membership functionμ:V×V→[0,1], which is a fuzzy relation onσ, satisfying μ(x,y)≤σ(x)∧σ(y),∀x,y∈V, where∧denotes the minimum operator. The associatedcrisp graphG ∗ = (σ ∗ ,μ ∗ )is determined by σ ∗ =x∈V|σ(x)>0, μ ∗ =(x,y)∈V×V|μ(x,y)>0. Afuzzy subgraphH= (σ ′ ,μ ′ )ofGis obtained by choosing a subsetX⊆Vand defining • a restricted vertex membershipσ ′ :X→[0,1], • an edge membershipμ ′ :X×X→[0,1]such that μ ′ (x,y)≤σ ′ (x)∧σ ′ (y),∀x,y∈X. Chapter 2. Preliminaries A wide variety of extensions of fuzzy graphs have been studied [4, 216, 217]. These can be represented within the framework of uncertain graphs, which extend uncertain sets to graph structures. We now state the uncertain graph-theoretic notions. Definition 2.8.2(Uncertain graph).[218] LetG= (V,E)be a finite, undirected, loopless graph, and let Mbe an uncertain model with degree-domain Dom(M). Anuncertain graph of typeMis a triple G M = (V,E,μ M ), where μ M :V∪E−→Dom(M) assigns an uncertainty degree in Dom(M)to each vertexv∈Vand each edgee∈E. Optionally, one may impose model-dependent consistency relations between vertex- and edge-degrees (e.g., boundingμ M (e)in terms ofμ M (u)andμ M (v)fore=u,vin fuzzy or intuitionistic fuzzy settings), but such constraints are dictated by the chosen modelMand are not fixed at the level of this general definition. For convenience, Table 2.2 lists representative uncertainty-graph families, organized by the dimensionkof the degree-domain Dom(M)⊆[0,1] k (cf. [20,218]). Table 2.2: A catalogue of uncertainty-graph families (uncertain graphs) by the dimensionkof the degree- domain Dom(M)⊆[0,1] k (cf. [218]). kRepresentative uncertainty-graph type(s)G M = (V,E,μ M )withμ M :V∪E→Dom(M)⊆[0,1] k 1Fuzzy graph;N-graph [219]; shadowed-graph variants [220] 2Intuitionistic fuzzy graph [221, 222]; vague graph [223]; bipolar fuzzy graph [224, 225]; intuitionistic evi- dence graph; variable fuzzy graph; paraconsistent fuzzy graph; bifuzzy graph [226,227] 3Neutrosophic graph [4] (a) ; hesitant fuzzy graph [228, 229]; tripolar fuzzy graph; three-way fuzzy graph; picture fuzzy graph [230, 231]; spherical fuzzy graph [168, 232]; inconsistent intuitionistic fuzzy graph; ternary fuzzy / neutrosophic-fuzzy graph; neutrosophic vague graph 4Quadripartitioned neutrosophic graph [233, 234]; double-valued neutrosophic graph [176]; dual hesitant fuzzy graph [235]; ambiguous graph (b) ; local-neutrosophic graph; support-neutrosophic graph; turiyam neutrosophic graph [185,236] (c) 5Pentapartitioned neutrosophic graph [237]; triple-valued neutrosophic graph [191] 6Hexapartitioned neutrosophic graph; quadruple-valued neutrosophic graph [191] 7Heptapartitioned neutrosophic graph [238]; quintuple-valued neutrosophic graph [191] 8Octapartitioned neutrosophic graph 9Nonapartitioned neutrosophic graph n n-refined fuzzy graph; multi-valued (fuzzy) graphs; multi-fuzzy graphs [239] 2n n-refined intuitionistic fuzzy graph; multi-intuitionistic fuzzy graphs 3n n-refined neutrosophic graph [240]; multi-neutrosophic graphs (a) Neutrosophic graph models are often treated as broad frameworks that can specialize to many degree-based graph formalisms under suitable constraints. (b) Ambiguous-graph models are commonly presented as subclasses of certain quadripartitioned and also double-valued neutro- sophic graph models. (c) Turiyam neutrosophic graphs are reported as subclasses of certain quadripartitioned neutrosophic graph models. 2.9 Uncertain decision-making (UDM) Using uncertain sets, one can defineuncertain decision-making(UDM). The definition is given below. Definition 2.9.1(Uncertain decision-making (UDM) of typeM).LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. Fix anuncertain modelMwith degree-domain Dom(M)⊆[0,1] k (cf. [20,142]). Chapter 2. Preliminaries Anuncertain decision-making instance of typeMis the data D M = ( A,C,X M ,w M ,Agg M ,Score M ) , where: (i)X M = (μ ij )∈Dom(M) m×n is anuncertain evaluation matrix, whereμ ij ∈Dom(M)encodes the uncertain assessment ofA i underC j ; (i)w M = (ω 1 ,...,ω n )∈Dom(M) n (or, optionally,w∈[0,1] n with ∑ j w j = 1) is anuncertain criterion- importance profile; (i)Agg M is a model-dependent aggregation operator which assigns to each alternativeA i an overall uncertain score s i :=Agg M ( (μ i1 ,...,μ in ),(ω 1 ,...,ω n ) ) ∈Dom(M); (iv)Score M :Dom(M)→Ris aranking functional(model-dependent score/defuzzification) inducing a preorder onAby A i M A j ⇐⇒Score M (s i )≥Score M (s j ). AsolutionofD M is any alternative A ? ∈arg max A i ∈A Score M (s i ), together with the ranking induced by Score M . Remark 2.9.2(Specializations).IfMis the fuzzy model Dom(M) = [0,1], then Definition 2.9.1 recovers a standard fuzzy decision-making setting. IfMis intuitionistic fuzzy, neutrosophic, plithogenic, etc., then one obtains the corresponding multi-component uncertain decision-making setting by changing Dom(M) and the operators(Agg M ,Score M ). Theorem 2.9.3(Well-definedness of uncertain decision-making of typeM).LetA=A 1 ,...,A m and C=C 1 ,...,C n be finite. Fix an uncertain modelMwith degree-domainDom(M)⊆[0,1] k . Assume: (A1)the aggregation operator is a total map Agg M :Dom(M) n ×Dom(M) n −→Dom(M); (A2)the ranking functional is a total map Score M :Dom(M)−→R. Then, for every UDM instance D M = ( A,C,X M ,w M ,Agg M ,Score M ) withX M = (μ ij )∈Dom(M) m×n , w M = (ω 1 ,...,ω n )∈Dom(M) n , the following objects are well-defined: Chapter 2. Preliminaries (i)the aggregated uncertain scoress i ∈Dom(M),i= 1,...,m; (i)the induced preference relation M onAgiven by A i M A j ⇐⇒Score M (s i )≥Score M (s j ); (i)the solution setarg max A i ∈A Score M (s i ), which is nonempty. Moreover, M is atotal preorderonA(reflexive, transitive, and total). Proof.(i) Aggregated uncertain scores exist.Fixi∈ 1,...,m. BecauseX M ∈Dom(M) m×n , the row(μ i1 ,...,μ in )belongs to Dom(M) n . Alsow M = (ω 1 ,...,ω n )∈Dom(M) n . By assumption (A1), the value s i :=Agg M ( (μ i1 ,...,μ in ),(ω 1 ,...,ω n ) ) is defined and lies in Dom(M). (i) Scores are real numbers.By (A2), Score M (s i )∈Ris defined for eachi. (i) The relation M is well-defined and a total preorder.DefineA i M A j iff Score M (s i )≥ Score M (s j ). Since≥is a well-defined total preorder onR, it follows immediately that: •Reflexive:Score M (s i )≥Score M (s i )for alli, henceA i M A i . •Transitive:ifA i M A j andA j M A ` , then Score M (s i )≥Score M (s j )≥Score M (s ` ), henceA i M A ` . •Total:for anyi,j, either Score M (s i )≥Score M (s j )or Score M (s j )≥Score M (s i ), henceA i M A j or A j M A i . Therefore M is a total preorder onA. (iv) A maximizer exists.The setScore M (s i ) :i= 1,...,m⊂Ris finite, hence attains its maximum. Therefore, arg max A i ∈A Score M (s i )6=∅, so at least one solution alternativeA ? exists, and the solution set is well-defined. For reference, Table 2.3 presents a catalogue of uncertainty-decision families (UDM). Chapter 2. Preliminaries Table 2.3: A catalogue of uncertainty-decision families (UDM) by the dimensionkof the degree-domain Dom(M)⊆[0,1] k (cf. [20]). knote Representative UDM family/families whose assessments take values in Dom(M)⊆[0,1] k 1Fuzzy decision-making [241] / fuzzy MCDM and MADM (single membership degree);N- fuzzy decision-making (multi-membership collapsed tok=1via aggregation); shadowed de- cision variants 2Intuitionistic fuzzy decision-making (membership/non-membership with constraint) [242, 243]; vague decision-making [244, 245]; bipolar/bi-component fuzzy decision-making [246, 247]; paraconsistent/bifuzzy decision-making 3Single-valued neutrosophic decision-making (truth/indeterminacy/falsity) [248, 249]; hesi- tant fuzzy decision-making [250, 251]; picture-fuzzy decision-making [252]; spherical-fuzzy decision-making [253,254]; tripolar fuzzy decision-making [255]; neutrosophic-vague decision- making [256] 4Quadripartitioned neutrosophic decision-making [257]; double-valued neutrosophic decision- making [258]; dual-hesitant decision-making [259,260]; ambiguous-decision-making; turiyam- neutrosophic decision-making [261] 5Pentapartitioned neutrosophic decision-making [262, 263]; triple-valued neutrosophic decision-making 6Hexapartitioned neutrosophic decision-making; quadruple-valued neutrosophic decision- making 7Heptapartitioned neutrosophic decision-making [199]; quintuple-valued neutrosophic decision-making 8Octapartitioned neutrosophic decision-making 9Nonapartitioned neutrosophic decision-making n(n≥1)n-refined fuzzy decision-making; multi-valued (fuzzy) decision-making; multi-fuzzy decision- making [264] 2n(n≥1)n-refined intuitionistic fuzzy decision-making; multi-intuitionistic fuzzy decision-making 3n(n≥1)n-refined neutrosophic decision-making [265–267]; multi-neutrosophic decision-making Reading guide.This table mirrors the “U-Set catalogue” viewpoint: an uncertain decision family is characterized, at the type level, by the degree-domain Dom(M)⊆[0,1] k used to encode assessments and/or weights, together with model-dependent aggregation and ranking operators(Agg M ,Score M ). Thus, changingkchanges how many numerical components are stored per assessment. Chapter 2. Preliminaries Chapter 3 Problem-level frameworks In this section, we present fundamental decision-making frameworks. 3.1 Fuzzy Multiple-Criteria Decision Making (Fuzzy MCDM) Multiple-Criteria Decision Making evaluates alternatives across several criteria, assigns weights, aggregates scores or outranking relations, and ranks choices systematically overall [268, 269]. Fuzzy MCDM models ratings and weights as fuzzy numbers or linguistic terms, propagates uncertainty through aggregation, then defuzzifies rankings robustly [270,271]. Definition 3.1.1(Fuzzy MCDM problem (evaluation-and-ranking form)).[270,271] LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C k a finite set of criteria. LetFN(R)denote a chosen class of fuzzy numbers onR(e.g. triangular or trapezoidal), and letFN([0,1])denote fuzzy numbers supported in[0,1]. Afuzzy multiple-criteria decision-making (fuzzy MCDM) instanceis the data D= ( A,C, ̃ X, ̃w,Agg,Score ) , where: (i) ̃ X= ( ̃x it )∈FN(R) m×k is thefuzzy decision matrix, where ̃x it represents the (possibly linguistic) performance ofA i under criterionC t ; (i) ̃w= ( ̃w 1 ,..., ̃w k )∈FN([0,1]) k is thefuzzy weight vectorwith ̃w t describing the uncertain importance of criterionC t ; (i)Agg is anaggregation operatorproducing, for each alternativeA i , an overall fuzzy score ̃s i :=Agg ( ( ̃x i1 ,..., ̃x ik ),( ̃w 1 ,..., ̃w k ) ) ∈FN(R); a common choice is the fuzzy weighted average ̃s i = ∑ k t=1 ̃w t ⊗ ̃x it ∑ k t=1 ̃w t , where⊗and/are fuzzy arithmetic operations inFN(R); 31 Chapter 3. Problem-level frameworks (iv)Score:FN(R)→Ris aranking functional(defuzzification/score), and the induced preorder onAis A i A j ⇐⇒Score( ̃s i )≥Score( ̃s j ). Asolutionof the fuzzy MCDM instance is any alternativeA ? ∈Asatisfying A ? ∈arg max A i ∈A Score( ̃s i ), together with the (total preorder) ranking ofAinduced by Score( ̃s i ). Remark 3.1.2(Ideal/anti-ideal (distance-to-reference) fuzzy MCDM).Many fuzzy MCDM methods re- place Score by a distance-to-reference ranking. Given fuzzy positive and negative ideal points (or solutions) I + ,I − in the criterion space, define for eachA i two distancesd + i =d( ̃ x i ,I + )andd − i =d( ̃ x i ,I − )(using a chosen fuzzy distance), and rank by a closeness/approximation index, e.g. Γ i := d − i d + i +d − i ∈[0,1], A i A j ⇐⇒Γ i ≥Γ j . This produces a compromise ranking relative to the ideal and anti-ideal references. Definition 3.1.3(Uncertain Multiple-Criteria Decision Making (UMCDM)).LetA=A 1 ,...,A m be a finite nonempty set of alternatives andC=C 1 ,...,C n a finite nonempty set of criteria. Fix an uncertain modelM. AnUMCDM instance of typeMis the data D M = ( A,C,X M ,w,Agg M ,Score M ) , where: (i)X M = (μ ij )∈Dom(M) m×n is theuncertain evaluation matrix, whereμ ij ∈Dom(M)encodes the (uncertain) evaluation ofA i w.r.t. criterionC j ; (i)w= (w 1 ,...,w n )∈[0,1] n is acriterion weight vectorwith ∑ n j=1 w j = 1(allowing uncertain weights is possible by takingw∈Dom(M) n instead); (i)Agg M is anaggregation operatorproducing an overallM-degree for each alternative: s i :=Agg M ( (μ i1 ,...,μ in ),w ) ∈Dom(M), i= 1,...,m; (iv)Score M :Dom(M)→Ris aranking functional(score/defuzzification), and it induces a preference relation onAby A i M A k ⇐⇒Score M (s i )≥Score M (s k ). Abest alternative(solution) is any A ? ∈arg max 1≤i≤m Score M (s i ). Chapter 3. Problem-level frameworks Theorem 3.1.4(Uncertain-set structure and well-definedness of UMCDM).Let D M = ( A,C,X M ,w,Agg M ,Score M ) be an UMCDM instance as in Definition 3.1.3, whereA,Care finite nonempty andMis an uncertain model. Assume: (A1)Agg M :Dom(M) n ×∆ n →Dom(M)is a total mapping, where∆ n :=w∈[0,1] n : ∑ n j=1 w j = 1; (A2)Score M :Dom(M)→Ris a total mapping. Then: (i)(Uncertain-set structure) The evaluation matrixX M = (μ ij )defines an uncertain set of typeM on the product universeA×Cvia μ:A×C →Dom(M), μ(A i ,C j ) :=μ ij . Hence ( A×C,μ ) is an uncertain set of typeM. Moreover, the aggregated-score maps:A→Dom(M) defined bys(A i ) :=s i also yields an uncertain set ( A,s ) of typeM. (i)(Well-defined ranking) The relation M onAinduced byScore M ◦sis a total preorder (reflexive, transitive, and total). (i)(Existence of a best alternative) The solution setarg max 1≤i≤m Score M (s i )is nonempty. Proof.(i) Uncertain-set structure.BecauseX M ∈Dom(M) m×n , each entryμ ij lies in Dom(M). Define μ(A i ,C j ) :=μ ij onA×C. This is a well-defined functionμ:A×C →Dom(M), so ( A×C,μ ) is an uncertain set of typeM. Next, fixi∈1,...,m. Since(μ i1 ,...,μ in )∈Dom(M) n andw∈∆ n , assumption (A1) implies that s i =Agg M ( (μ i1 ,...,μ in ),w ) ∈Dom(M) exists. Hence the mappings:A →Dom(M),s(A i ) :=s i , is well-defined and ( A,s ) is an uncertain set of typeM. (i) Well-defined ranking and total preorder.By (A2), each Score M (s i )∈Ris defined. Define A i M A k iff Score M (s i )≥Score M (s k ). Since≥onRis reflexive, transitive, and total, the induced relation M inherits these properties: • Reflexive: Score M (s i )≥Score M (s i ), henceA i M A i . Chapter 3. Problem-level frameworks • Transitive: ifA i M A j andA j M A ` , then Score M (s i )≥Score M (s j )≥Score M (s ` ), henceA i M A ` . • Total: for anyi,k, either Score M (s i )≥Score M (s k )or Score M (s k )≥Score M (s i ), henceA i M A k or A k M A i . Thus M is a total preorder. (i) Existence of a best alternative.The setScore M (s i ) :i= 1,...,m⊂Ris finite, hence it attains a maximum. Therefore arg max 1≤i≤m Score M (s i )6=∅, proving existence of at least one best alternative. As a reference, a catalogue of uncertainty-aware MCDM families organized by the dimensionkof the degree-domain is presented in Table 3.1. Table 3.1: A catalogue of uncertainty-aware MCDM families by the dimensionkof the degree-domain Dom(M)⊆[0,1] k . knoteRepresentative uncertainty-aware MCDM model(s) whose degree-domain is a subset of [0,1] k 1Fuzzy MCDM. 2Intuitionistic Fuzzy MCDM [272, 273]; Bipolar Fuzzy MCDM [274, 275]; Pythagorean Fuzzy MCDM [276,277]; Fermatean Fuzzy MCDM [278,279] 3Hesitant Fuzzy MCDM [280, 281]; Picture Fuzzy MCDM [282, 283]; Spherical Fuzzy MCDM [284,285]; (Single-valued) Neutrosophic MCDM [286,287]. 4Quadripartitioned Neutrosophic MCDM [288,289]. n(n≥1)Plithogenic MCDM [290, 291] (vector-valued degrees over attribute values: for Valset(v) =a 1 ,...,a n ,μ(x) = (μ(x,a 1 ),...,μ(x,a n ))∈[0,1] n ; augmented with a contradiction functionc:Valset(v)×Valset(v)→[0,1]on values). Reading guide.Here,kcounts the number of numerical components stored per evaluation (i.e., the ambient dimension of the degree-domain). For hesitant fuzzy information, the classificationk= 3can be realized by a standard vectorization (e.g., an envelope/summary map of a hesitant set of grades into three numbers in[0,1] 3 ). Plithogenic models naturally induce variable dimensionk=nbecause degrees are assigned per attribute–value and|Valset(v)|depends on the chosen value set; the contradiction mapc(·,·)is an additional structure on values. In addition to uncertainty-related extensions, approaches such as Linguistic MCDM [292,293], Behavioral MCDM [294], Rough MCDM [295, 296], Hybrid MCDM [297, 298], Soft MCDM [299, 300], Grey MCDM [301, 302], MCGDM [303, 304], Stratified Multi-Criteria Decision-Making (SMCDM) [305, 306], Z-Number Based MCDM [307,308], Potentially All Pairwise RanKings of all possible Alternatives (PAPRIKA) [309, 310], Aggregated Indices Randomization Method (AIRM) [311, 312], Multi-Attribute Global Inference of Quality (MAGIQ) [313], UTA methods [314], Stochastic Multicriteria Acceptability Analysis (SMAA) [315], Conjoint Value Hierarchy (CVA) [316], and Large-scale GDM [317,318] have also been studied, and further developments and extensions can be expected in the future. 3.2 Fuzzy MADM (Fuzzy Multi-attribute decision-making) MADM evaluates a finite set of alternatives across multiple attributes, weights attributes, aggregates per- formances, and selects the best option [319,320]. Fuzzy MADM expresses attribute ratings and weights as fuzzy numbers or linguistic terms, aggregates fuzzily, then defuzzifies to rank alternatives [321,322]. Chapter 3. Problem-level frameworks Definition 3.2.1(Fuzzy MADM problem (discrete multi-attribute decision model)).[322,323] Let A=A 1 ,...,A m be a finite set of alternatives and C=C 1 ,...,C n a finite set of attributes (criteria). Afuzzy MADM instanceis specified by: (1) afuzzy decision matrix ̃ R= ( ̃r ij )∈FN(R) m×n , ̃r ij is the fuzzy performance rating ofA i w.r.t.C j ; (2) afuzzy weight vector ̃ W= ( ̃w 1 ,..., ̃w n )∈FN(R ≥0 ) n , ̃w j is the fuzzy importance ofC j . HereFN(·)denotes a chosen class of fuzzy numbers (e.g. triangular/trapezoidal). Adecision procedureconsists of two phases: (P1)(Aggregation)For each alternativeA i , compute an aggregated fuzzy utility ̃u i :=Agg ( ̃r i1 ,..., ̃r in ; ̃w 1 ,..., ̃w n ) ∈FN(R), where Agg is a specified fuzzy aggregation operator. A common choice is the fuzzy weighted average (FWA) ̃u i = ⊕ n j=1 ( ̃w j ⊗ ̃r ij ) ⊕ n j=1 ̃w j , with⊕,⊗and/interpreted in the adopted fuzzy arithmetic. (P2)(Ranking)Rank alternatives by a fuzzy ranking rule A i A k ⇐⇒ ̃u i F ̃u k , where F is a fixed preorder on fuzzy numbers (e.g. induced by a defuzzification/score map Score: FN(R)→Rvia ̃u i F ̃u k ⇐⇒Score( ̃u i )≤Score( ̃u k )). A(best) solutionis anyA ? ∈Asuch thatA ? A i for allA i ∈A. We next consider Uncertain MADM, obtained by generalizing the above framework using Uncertain Sets. Definition 3.2.2(Uncertain MADM (UMADM)).LetA=A 1 ,...,A m be a finite nonempty set of alternatives andC=C 1 ,...,C n a finite nonempty set of attributes (criteria). Fix an uncertain modelM with degree-domain Dom(M)⊆[0,1] k . Anuncertain multi-attribute decision-making instance of typeM(UMADM instance) is the data D M = ( A,C,R M ,w,Agg M ,Score M ) , where: Chapter 3. Problem-level frameworks (i)R M = (μ ij )∈Dom(M) m×n is theuncertain decision matrix, whereμ ij ∈Dom(M)encodes the uncertain evaluation of alternativeA i under attributeC j ; (i)w= (w 1 ,...,w n )∈[0,1] n is a (crisp) attribute-weight vector with ∑ n j=1 w j = 1(optionally, one may allow uncertain weightsw∈Dom(M) n ); (i)Agg M is an aggregation operator producing an overallM-degree for each alternative: s i :=Agg M ( (μ i1 ,...,μ in ),w ) ∈Dom(M), i= 1,...,m; (iv)Score M :Dom(M)→Ris a ranking functional inducing the preference relation A i M A ` ⇐⇒Score M (s i )≥Score M (s ` ). Abest alternative(solution) is any A ? ∈arg max 1≤i≤m Score M (s i ). Theorem 3.2.3(Uncertain-set structure and well-definedness of UMADM).Let D M = ( A,C,R M ,w,Agg M ,Score M ) be a UMADM instance in the sense of Definition 3.2.2, withA,Cfinite nonempty andMan uncertain model. Assume: (A1)Agg M :Dom(M) n ×∆ n →Dom(M)is a total mapping, where∆ n :=w∈[0,1] n : ∑ n j=1 w j = 1; (A2)Score M :Dom(M)→Ris a total mapping. Then: (i)(Uncertain-set structure on the evaluation domain) The decision matrixR M = (μ ij )induces an uncertain set of typeMonA×Cvia μ:A×C →Dom(M), μ(A i ,C j ) :=μ ij . Hence ( A×C,μ ) is an uncertain set of typeM. (i)(Uncertain-set structure on alternatives) The aggregated-score assignments:A →Dom(M) defined bys(A i ) :=s i is well-defined and yields an uncertain set ( A,s ) of typeM. (i)(Well-defined preference and existence of a best alternative) The induced relation M on Ais a total preorder (reflexive, transitive, total), and the solution setarg max 1≤i≤m Score M (s i )is nonempty. Chapter 3. Problem-level frameworks Proof.(i) Uncertain-set structure onA×C.BecauseR M ∈Dom(M) m×n , each entryμ ij belongs to Dom(M). Defineμ(A i ,C j ) :=μ ij . This is a well-defined mappingμ:A×C →Dom(M), hence ( A×C,μ ) is an uncertain set of typeM. (i) Aggregated uncertain scores exist and form an uncertain set onA.Fixi∈1,...,m. Since (μ i1 ,...,μ in )∈Dom(M) n andw∈∆ n , assumption (A1) implies that s i =Agg M ( (μ i1 ,...,μ in ),w ) ∈Dom(M) is defined. Therefore the mappings:A→Dom(M),s(A i ) :=s i , is well-defined, and ( A,s ) is an uncertain set of typeM. (i) The induced preference M is well-defined and a total preorder.By (A2), each value Score M (s i )∈Rexists. DefineA i M A ` iff Score M (s i )≥Score M (s ` ). Because≥onRis reflexive, transitive, and total, the induced relation M inherits these properties: • Reflexive: Score M (s i )≥Score M (s i ), henceA i M A i . • Transitive: ifA i M A j andA j M A ` , then Score M (s i )≥Score M (s j )≥Score M (s ` ), henceA i M A ` . • Total: for anyi,`, either Score M (s i )≥Score M (s ` )or Score M (s ` )≥Score M (s i ), henceA i M A ` or A ` M A i . Thus M is a total preorder. (iv) Existence of a best alternative.The setScore M (s i ) :i= 1,...,m⊂Ris finite, hence attains a maximum. Therefore, arg max 1≤i≤m Score M (s i )6=∅, so at least one best alternative exists and the solution set is well-defined. Examples of uncertainty-aware MADM families, grouped by the degree-domain dimensionkof Dom(M)⊆ [0,1] k , are listed in Table 3.2. As related concepts beyond uncertain MADM, Rough MADM [335], Soft MADM [336, 337], Multiple at- tribute group decision-making (MAGDM) [338, 339], Grey MADM [340], Z-number madm [341, 342], and Linguistic MADM [343,344] are also known. Chapter 3. Problem-level frameworks Table 3.2: Related uncertainty-aware MADM families (examples) grouped by the degree-domain dimension kof Dom(M)⊆[0,1] k . knoteRepresentative uncertainty-aware MADM model(s) 1Fuzzy multi-attribute decision making. 2Intuitionistic Fuzzy multi-attribute decision making [324,325]. 3Hesitant Fuzzy multi-attribute decision making [326, 327]; Spherical fuzzy multi- attribute decision making [328, 329]; Picture Fuzzy multi-attribute decision making [330,331]; Neutrosophic multi-attribute decision making [332,333]. n(n≥1)Plithogenic multi-attribute decision making [334] (vector-valued degrees over attribute values, coupled with a contradiction function on values). Note.Thek= 3placement for hesitant fuzzy MADM follows the same convention as in Table 3.1, namely, representing hesitant information via a3-component embedding/summary in[0,1] 3 . Plithogenic MADM yieldsk=nwheren=|Valset(v)| depends on the chosen attribute value set. 3.3 Fuzzy Group-Decision Making Group Decision-Making combines the preferences of multiple decision makers, negotiates trade-offs, measures consensus, and selects an alternative that reflects collective judgments [345,346]. As a concept that may be viewed as roughly opposite to Group Decision-Making, personal decision-making [347, 348] is also known. Fuzzy Group-Decision Making represents opinions by fuzzy numbers or linguistic terms, aggregates them using fuzzy operators, evaluates consensus, and then ranks the alternatives [349,350]. Definition 3.3.1(Fuzzy group decision making (consensus + selection framework)).[351] Afuzzy group decision making(FGDM) problem consists of: X=x 1 ,...,x n (alternatives), E=e 1 ,...,e m (experts/decision makers). Experts express their opinions infuzzyform (often by linguistic terms encoded as fuzzy numbers), and their assessments are aggregated to obtain a collective decision; a consensus stage is typically applied before the final selection stage. (A) Multi-attribute evaluation data (typical MADM-type FGDM).LetC=c 1 ,...,c p be cri- teria. Fix a classFN(R)of fuzzy numbers (e.g. TFNs). Each experte h provides: ̃ R (h) = ( ̃r (h) ij ) ∈FN(R) n×p (fuzzy ratings), ̃w (h) = ( ̃w (h) 1 ,..., ̃w (h) p )∈FN(R ≥0 ) p (fuzzy criterion weights). (B) Group aggregation.Choose aggregation operators Agg R :FN(R) m →FN(R), Agg w :FN(R ≥0 ) m →FN(R ≥0 ), and define the collective fuzzy ratings and weights by ̃r ij :=Agg R ( ̃r (1) ij ,..., ̃r (m) ij ) , ̃w j :=Agg w ( ̃w (1) j ,..., ̃w (m) j ) . (For TFNs, a common choice is componentwise arithmetic mean.) Chapter 3. Problem-level frameworks (C) Selection (collective scoring + ranking).Fix a fuzzy aggregation/scoring operator Agg C across criteria. A standard choice is the fuzzy weighted average (FWA): ̃s i :=Agg C ( ( ̃r i1 ,..., ̃r ip ),( ̃w 1 ,..., ̃w p ) ) = ⊕ p j=1 ( ̃w j ⊗ ̃r ij ) ⊕ p j=1 ̃w j ∈FN(R), where⊕,⊗and/are the adopted fuzzy arithmetic operations. Let Score:FN(R)→Rbe a ranking functional (defuzzification/score). Define the collective preorder onX by x i x k ⇐⇒Score( ̃s i )≥Score( ̃s k ), and select any x ? ∈arg max x i ∈X Score( ̃s i ). (D) Consensus (optional but standard in FGDM).Fix a consensus measure Cons (e.g. built from pairwise similarity of experts’ fuzzy preferences) and a thresholdγ∈[0,1]. Aconsensus stateis reached when Cons≥γ; otherwise, a moderator-guided iterative revision of experts’ opinions may be performed until Cons≥γ, after which the selection step (C) is applied. As an extension of the above framework, we define Uncertain Group-Decision Making. The definition is given below. Definition 3.3.2(Uncertain Group-Decision Making (UGDM) of typeM).LetA=A 1 ,...,A m be a fi- nite nonempty set of alternatives,C=C 1 ,...,C n a finite nonempty set of criteria, andE=E 1 ,...,E r a finite nonempty set of experts (decision makers). Fix an uncertain modelMwith degree-domain Dom(M)⊆ [0,1] k . Anuncertain group decision-making instance of typeM(UGDM instance) is the data G M = ( A,C,E,X M ,w,λ,Agg E ,Agg C ,Score M ,Cons ) , where: (i)X M = ( μ (t) ij ) ∈Dom(M) m×n×r is theexpert-indexed uncertain decision array, whereμ (t) ij ∈Dom(M) encodes expertE t ’s uncertain evaluation ofA i underC j ; (i)w= (w 1 ,...,w n )∈[0,1] n is a criterion-weight vector with ∑ n j=1 w j = 1; (i)λ= (λ 1 ,...,λ r )∈[0,1] r is an expert-weight vector with ∑ r t=1 λ t = 1(optional; if omitted set λ t = 1/r); (iv)Agg E :Dom(M) r →Dom(M)is anexpert-aggregation operatorused entrywise to obtain a collective matrix ̄μ ij :=Agg E ( μ (1) ij ,...,μ (r) ij ;λ ) ∈Dom(M), i= 1,...,m, j= 1,...,n; (we allow Agg E to depend parametrically onλ); Chapter 3. Problem-level frameworks (v) Agg C :Dom(M) n ×∆ n →Dom(M)is acriterion-aggregation operator(with∆ n =w∈[0,1] n : ∑ j w j = 1), producing an overallM-degree s i :=Agg C ( ( ̄μ i1 ,..., ̄μ in ),w ) ∈Dom(M), i= 1,...,m; (vi)Score M :Dom(M)→Ris aranking functionalinducing a preorder onAby A i M A ` ⇐⇒Score M (s i )≥Score M (s ` ); (vii)Cons: [0,1]is a (possibly optional)consensus indexcomputed from the experts’ assessments (e.g. via distances/similarities betweenμ (t) ij and the collective ̄μ ij ), together with a chosen threshold γ∈[0,1]used operationally to accept the group aggregation. AUGDM best alternative(solution) is any A ? ∈arg max 1≤i≤m Score M (s i ). Theorem 3.3.3(Uncertain-set structure and well-definedness of UGDM).LetG M be a UGDM instance, withA,C,Efinite nonempty andMan uncertain model. Assume: (A1)Agg E :Dom(M) r ×∆ r →Dom(M)is a total mapping (where∆ r =λ∈[0,1] r : ∑ t λ t = 1); (A2)Agg C :Dom(M) n ×∆ n →Dom(M)is a total mapping; (A3)Score M :Dom(M)→Ris a total mapping. Then: (i)(Uncertain-set structure on the evaluation universe) The expert-indexed arrayX M induces an uncertain set of typeMon U:=A×C×E via μ:U→Dom(M), μ(A i ,C j ,E t ) :=μ (t) ij . Hence(U,μ)is an uncertain set of typeM. (i)(Collective matrix is well-defined and yields an uncertain set) The entrywise aggregated mapping ̄μ:A×C →Dom(M), ̄μ(A i ,C j ) := ̄μ ij , is well-defined, so(A×C, ̄μ)is an uncertain set of typeM. (i)(Aggregated alternative scores form an uncertain set onA) The mapping s:A→Dom(M), s(A i ) :=s i , is well-defined, so(A,s)is an uncertain set of typeM. Chapter 3. Problem-level frameworks (iv)(Preference relation and solution set are well-defined) The induced relation M onAis a total preorder, and the solution setarg max 1≤i≤m Score M (s i )is nonempty. Proof.(i) Uncertain-set structure onU=A×C×E.BecauseX M ∈Dom(M) m×n×r , every entryμ (t) ij belongs to Dom(M). Defineμ(A i ,C j ,E t ) :=μ (t) ij . This is a well-defined mappingU→Dom(M), hence (U,μ)is an uncertain set of typeM. (i) Collective matrix exists and defines an uncertain set onA×C.Fix(i,j). The expert list (μ (1) ij ,...,μ (r) ij )lies in Dom(M) r andλ∈∆ r . By (A1), the aggregated value ̄μ ij :=Agg E ( μ (1) ij ,...,μ (r) ij ;λ ) ∈Dom(M) exists. Therefore ̄μ:A×C →Dom(M), ̄μ(A i ,C j ) = ̄μ ij is well-defined, and(A×C, ̄μ)is an uncertain set of typeM. (i) Alternative-level aggregation exists and defines an uncertain set onA.Fixi. The criterion tuple( ̄μ i1 ,..., ̄μ in )belongs to Dom(M) n andw∈∆ n . By (A2), the overall score s i :=Agg C ( ( ̄μ i1 ,..., ̄μ in ),w ) ∈Dom(M) exists. Hences:A→Dom(M),s(A i ) =s i is well-defined and(A,s)is an uncertain set of typeM. (iv) Well-defined preference relation and existence of a best alternative.By (A3), each Score M (s i )∈ Rexists. DefineA i M A ` iff Score M (s i )≥Score M (s ` ). Since≥onRis reflexive, transitive, and total, M is a total preorder. Finally, the finite setScore M (s i ) :i= 1,...,m⊂Rattains a maximum, hence arg max 1≤i≤m Score M (s i )6=∅, so at least one best alternative exists and the solution set is well-defined. A catalogue of uncertainty-awaregroup decision-making(GDM) families is presented in Table 3.3. Uncertain Group Decision-Making is not the only perspective; related notions such as rough group decision making [360, 361], soft group decision making [362], grey group decision making [363, 364], multigroup decision making [365, 366], Z-Number group decision making [367, 368], consensus group decision making [369, 370], multi-stage group decision making [371, 372], multi-criteria group decision-making (MCGDM) [373,374], and linguistic group decision making [375,376] are also known. Chapter 3. Problem-level frameworks Table 3.3: A compact catalogue of uncertainty-awaregroup decision-making(GDM) families by the dimen- sionkof the degree-domain Dom(M)⊆[0,1] k . knoteRepresentative GDM family/families whose per-evaluation degree-domain is a subset of[0,1] k 1Fuzzy group decision making: group aggregation/consensus over scalar fuzzy evalua- tions in[0,1](membership grades), typically via expert weights and fuzzy aggregation operators. 2Intuitionistic fuzzy group decision making [352, 353]: group aggregation/consensus over evaluations in[0,1] 2 (membership, non-membership). 3Hesitant fuzzy group decision making [354,355]: group aggregation/consensus where each evaluation is represented in a[0,1] 3 -type degree-domain (per the adopted hesi- tant encoding). 3Neutrosophic group decision making [356, 357]: group aggregation/consensus over evaluations in[0,1] 3 (truth, indeterminacy, falsity). 3Spherical fuzzy group decision making [358,359]: group aggregation/consensus over evaluations in[0,1] 3 satisfyingμ 2 +ν 2 +π 2 ≤1(membership, non-membership, hesitancy). n † non-vector add-on Plithogenic group decision making: scalar degrees on attribute–value pairs (n= |Valset(v)|) coupled with a contradiction functionc(·,·)on values. Reading guide.Herekdenotes the dimension of the degree-domain used for each expert’s evaluation (or preference statement). In GDM, evaluations are collected per expert and then aggregated (optionally with a consensus-reaching stage) into a collective assessment. (†) For plithogenic models, the effective dimension depends on the numbernof admissible values of the chosen attribute, because degrees are attached to attribute–value pairs rather than stored as a fixed-length vector. 3.4 Fuzzy dynamic decision-making Dynamic decision-making chooses actions over time, updating information and preferences, accounting for state transitions, and optimizing cumulative outcomes [377, 378]. Fuzzy dynamic decision-making models evolving states, rewards, or criteria with fuzzy sets, propagates uncertainty through updates, and selects adaptive policies [379,380]. Definition 3.4.1(Fuzzy dynamic decision-making (multi-stage fuzzy MADM)).[379,380] Let A=A 1 ,...,A m andC=C 1 ,...,C n be finite sets of alternatives and criteria. Let T=t 1 ,...,t p be a finite set of decision stages (time periods). Fix a chosen class of fuzzy numbersFN(R)(e.g. triangular/trapezoidal), equipped with fuzzy arithmetic (⊕,⊗, )consistent withFN(R). Afuzzy dynamic decision-making instanceis specified by: (1)Stage-wise fuzzy decision matrices:for each staget k ∈T, a fuzzy decision matrix ̃ R (k) = ( ̃r (k) ij ) ∈FN(R) m×n , ̃r (k) ij ratesA i w.r.t.C j at timet k . Chapter 3. Problem-level frameworks (2)(Optional) Stage-wise fuzzy criterion weights:for each staget k , a fuzzy weight vector ̃ W (k) = ( ̃w (k) 1 ,..., ̃w (k) n )∈FN(R ≥0 ) n . (If weights are time-invariant, write ̃ W (k) = ̃ W.) (3)Stage weights (dynamic importance of time):a weight vector ω= (ω 1 ,...,ω p )∈[0,1] p , p ∑ k=1 ω k = 1, whereω k quantifies the importance of staget k (e.g. giving larger weight to recent stages). (4)Aggregation operators: • a within-stage (across-criteria) fuzzy aggregation map Agg C :FN(R) n ×FN(R ≥0 ) n →FN(R), • a cross-stage (dynamic) fuzzy aggregation map Agg T :FN(R) p ×[0,1] p →FN(R). (5)A ranking functional:a score/defuzzification map Score:FN(R)→R that induces a preorder on fuzzy numbers by ̃x F ̃y⇐⇒Score( ̃x)≤Score( ̃y). Adynamic fuzzy decision procedurecomputes, for each alternativeA i : (P1)Within-stage fuzzy utility.For each staget k , define a stage utility ̃s (k) i :=Agg C ( ̃r (k) i1 ,..., ̃r (k) in ; ̃w (k) 1 ,..., ̃w (k) n ) ∈FN(R). A standard choice is the fuzzy weighted average (FWA): ̃s (k) i = n ⊕ j=1 ( ̃w (k) j ⊗ ̃r (k) ij ) n ⊕ j=1 ̃w (k) j , assuming the denominator is nonzero in the adopted fuzzy arithmetic. (P2)Cross-stage (dynamic) aggregation.Aggregate the stage utilities into an overall dynamic utility: ̃u i :=Agg T ( ̃s (1) i ,..., ̃s (p) i ;ω 1 ,...,ω p ) ∈FN(R). A standard choice is the dynamic fuzzy weighted average (DFWA): ̃u i = p ⊕ k=1 ( ω k ⊗ ̃s (k) i ) , interpretingω k ⊗(·)as scalar multiplication in the fuzzy-number model. Chapter 3. Problem-level frameworks (P3)Ranking and selection.Define the crisp overall scoreU i :=Score( ̃u i )∈Rand rank by A i A ` ⇐⇒U i ≥U ` . A(best) solutionis any A ? ∈arg max A i ∈A Score( ̃u i ). In theuncertain-setviewpoint, the core data at every stage are degrees in a fixed degree-domain Dom(M)⊆ [0,1] k . Definition 3.4.2(Uncertain dynamic decision-making (UDDM) of typeM).Let A=A 1 ,...,A m (alternatives),C=C 1 ,...,C n (criteria), and let T=t 1 ,...,t p (decision stages / time periods) be finite nonempty sets. Fix an uncertain modelMwith degree-domain Dom(M)⊆[0,1] k . Anuncertain dynamic decision-making instance of typeMis the tuple D dyn M = ( A,C,T,(X (1) M ,...,X (p) M ),(w (1) M ,...,w (p) M ),ω,Agg C,M ,Agg T,M ,Score M ) , where: (i)Stage-wise uncertain evaluation matrices:for eachk∈1,...,p, X (k) M = (μ (k) ij )∈Dom(M) m×n , whereμ (k) ij ∈Dom(M)encodes the uncertain assessment ofA i under criterionC j at staget k . (i)Stage-wise uncertain criterion weights (optional but standard):for eachk, w (k) M = (ω (k) 1 ,...,ω (k) n )∈Dom(M) n (or, alternatively,w (k) ∈[0,1] n with ∑ j w (k) j = 1). (i)Stage-importance weights (time aggregation weights):a vector ω= (ω 1 ,...,ω p )∈[0,1] p , p ∑ k=1 ω k = 1. (Thusω k models the relative importance of staget k , e.g. recency weighting.) (iv)Within-stage aggregation (across criteria):a model-dependent total map Agg C,M :Dom(M) n ×Dom(M) n −→Dom(M), which assigns to each alternativeA i at staget k an aggregated uncertain stage-utility s (k) i :=Agg C,M ( (μ (k) i1 ,...,μ (k) in ), w (k) M ) ∈Dom(M). Chapter 3. Problem-level frameworks (v)Cross-stage aggregation (dynamic synthesis):a model-dependent total map Agg T,M :Dom(M) p ×[0,1] p −→Dom(M), which aggregates stage-utilities into an overall uncertain dynamic utility u i :=Agg T,M ( (s (1) i ,...,s (p) i ), ω ) ∈Dom(M). (vi)Ranking functional:a total map Score M :Dom(M)−→R. It induces a preference relation onAby A i dyn M A j ⇐⇒Score M (u i )≥Score M (u j ). The(dynamic) uncertain best-alternative setis arg max A i ∈A Score M (u i ). Theorem 3.4.3(Uncertain-set structure and well-definedness of UDDM).LetD dyn M be an uncertain dy- namic decision-making instance as in Definition 3.4.2. Assume: (A1)Agg C,M :Dom(M) n ×Dom(M) n →Dom(M)is a total map; (A2)Agg T,M :Dom(M) p ×[0,1] p →Dom(M)is a total map; (A3)Score M :Dom(M)→Ris a total map; (A4)ω∈[0,1] p satisfies ∑ p k=1 ω k = 1. Then: (i)The overall dynamic utilitiesu i ∈Dom(M)are well-defined for alli= 1,...,m. (i)The mapping μ dyn M :A→Dom(M), μ dyn M (A i ) :=u i , is well-defined; hence U dyn M := ( A,μ dyn M ) is anuncertain set (U-set) of typeMon the universeA. (i)The induced relation dyn M onAis atotal preorder(reflexive, transitive, and total). (iv)The solution setarg max A i ∈A Score M (u i )is nonempty (so a best alternative exists). Chapter 3. Problem-level frameworks Proof.(i) Stage-utilities exist inDom(M).Fixi∈ 1,...,mandk∈ 1,...,p. BecauseX (k) M ∈ Dom(M) m×n , theith row (μ (k) i1 ,...,μ (k) in )∈Dom(M) n . Also, by definition,w (k) M ∈Dom(M) n (or it is a crisp admissible surrogate; the present theorem covers the Dom(M) n case explicitly). By assumption (A1), the value s (k) i =Agg C,M ( (μ (k) i1 ,...,μ (k) in ),w (k) M ) is well-defined and lies in Dom(M). (i) Dynamic utilities exist inDom(M).For fixedi, the stage-utility vector(s (1) i ,...,s (p) i )lies in Dom(M) p by (i), and the stage-weight vectorωlies in[0,1] p by (A4). By (A2), the overall dynamic utility u i =Agg T,M ( (s (1) i ,...,s (p) i ),ω ) is well-defined and lies in Dom(M), proving (i) of the theorem statement. (i) U-set structure.Defineμ dyn M :A →Dom(M)byμ dyn M (A i ) =u i . By (i) this is a well-defined map into Dom(M), henceU dyn M = (A,μ dyn M )is a U-set of typeMonA, proving (i). (iv) The preference relation is a total preorder.By (A3), each Score M (u i )∈Ris well-defined. DefineA i dyn M A j iff Score M (u i )≥Score M (u j ). Since≥onRis reflexive, transitive, and total, the induced relation onAinherits these properties; thus dyn M is a total preorder, proving (i). (v) Existence of a maximizer.The setScore M (u i ) :i= 1,...,m ⊂Ris finite, hence attains its maximum. Therefore arg max A i ∈A Score M (u i )6=∅, proving (iv). Table 3.4 presents related uncertainty-model variants of Dynamic Fuzzy Decision-Making. Table 3.4: Related uncertainty-model variants of Dynamic Fuzzy Decision-Making. kRelated Dynamic Fuzzy Decision-Making variant(s) 1 Dynamic Fuzzy Decision-Making 2 Dynamic Intuitionistic Fuzzy Decision-Making 3 Dynamic Hesitant Fuzzy Decision-Making 3 Dynamic Spherical Fuzzy Decision-Making 3 Dynamic Neutrosophic Decision-Making nDynamic Plithogenic Decision-Making 3.5 Fuzzy Multiple Objective Decision-making Multiple Objective Decision-making optimizes several conflicting objectives simultaneously, generating Pareto- efficient solutions and selecting a compromise using preferences or weights [381, 382]. Fuzzy Multiple Ob- jective Decision-making represents objectives, goals, or constraints fuzzily, computes satisfaction degrees, aggregates them, and selects a compromise solution [383,384]. Chapter 3. Problem-level frameworks Definition 3.5.1(FMODM as fuzzy multi-objective programming with fuzzy goals).[383,384] LetX⊆R n be a nonempty feasible set and letk≥2be the number of objectives. Infuzzy multi-objective decision- making(FMODM), the objectives remain explicit and (typically) carry weights reflecting their relative significance. Fix a continuoust-normT: [0,1] 2 →[0,1]and itsk-ary extension (still denotedT). LetF(R)be a chosen class of fuzzy numbers onR, and suppose theith objective is a fuzzy-number-valued mapping ̃ f i :X→F(R), i= 1,...,k. Assume for each objectiveitwo fuzzy reference levels are given: ̃m i ∈F(R) (undesired level), ̃ M i ∈F(R) (desired level), together with a metric (distance)D:F(R)×F(R)→R ≥0 . (1) Objective-wise satisfaction (application) functions.Define, for eachi, a satisfaction degree H i :X→[0,1]by H i (x) :=min 1− 1 1 +D ( ̃m i , ̃ f i (x) ) , 1 1 +D ( ̃ M i , ̃ f i (x) ) , or more generally by thet-norm aggregation H i (x) :=T ( 1− 1 1 +D ( ̃m i , ̃ f i (x) ) , 1 1 +D ( ̃ M i , ̃ f i (x) ) ) . Intuitively, largerH i (x)means “far from undesired” and “close to desired” for objectivei. (2) Fuzzy decision (intersection of fuzzy goals).Letw 1 ,...,w k ∈[0,1]be (crisp) importance weights with ∑ k i=1 w i = 1. Define the overall fuzzy decision membership (the intersection of fuzzy goals) by μ D (x) :=T ( H 1 (x) w 1 ,...,H k (x) w k ) , x∈X. (Any other weightedt-norm construction may be used; the key point is that at-norm models the conjunction of goals.) (3) FMODM solution.AnFMODM (compromise) solutionis any x ? ∈arg max x∈X μ D (x). Equivalently, the fuzzy multi-objective problem is reduced to the single-objective scalarization max x∈X T ( H 1 (x),...,H k (x) ) , withH i constructed from desired/undesired levels and a metric on fuzzy numbers. In anuncertainsetting, objective values and/or goal achievements are represented in a fixed degree-domain Dom(M)⊆[0,1] d (fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, etc.), and the multiobjective prob- lem is reduced to aU-seton the feasible set via a model-dependent aggregation and a real-valued score. Chapter 3. Problem-level frameworks Definition 3.5.2(Uncertain multiple objective decision-making (UMODM) of typeM).LetX⊆R n be a nonempty feasible set, and letk≥2. Fix an uncertain modelMwith degree-domain Dom(M)⊆[0,1] d . Anuncertain multiple objective decision-making instance of typeMis the tuple P M = ( X,(f 1 ,...,f k ),(G 1 ,...,G k ), w,Agg M ,Score M ) , where: (i)Objectives:f i :X→R(crisp objectives) fori= 1,...,k. (If objective evaluations are uncertain, replacef i by an uncertain evaluation map ̃ f i :X→Dom(M); the construction below is unchanged by composing withG i .) (i)Objective-to-degree (goal/satisfaction) maps:for eachi, a map G i :R→Dom(M)(or more generallyG i :X→Dom(M)) that converts the objective levelf i (x)into an uncertain degreeμ i (x) :=G i (f i (x))∈Dom(M). (i)Importance weights:a vectorw= (w 1 ,...,w k )∈[0,1] k with ∑ k i=1 w i = 1. (iv)Uncertain goal aggregation:a total map Agg M :Dom(M) k ×[0,1] k −→Dom(M), interpreted as the conjunction/compromise operator over thekuncertain goal degrees. (v)Ranking functional:a total map Score M :Dom(M)→R. For each feasiblex∈X, define theobjective-wise uncertain degree vector μ(x) := ( μ 1 (x),...,μ k (x) ) ∈Dom(M) k , μ i (x) :=G i (f i (x)), and define theoverall uncertain decision degree μ D,M (x) :=Agg M ( μ(x),w ) =Agg M ( (μ 1 (x),...,μ k (x)),(w 1 ,...,w k ) ) ∈Dom(M). The induced preference onXis x M y⇐⇒Score M ( μ D,M (x) ) ≥Score M ( μ D,M (y) ) . AUMODM (compromise) solutionis any x ? ∈arg max x∈X Score M ( μ D,M (x) ) . Theorem 3.5.3(Uncertain-set structure and well-definedness of UMODM).LetP M be a UMODM instance as in Definition 3.5.2. Assume: (A1)X6=∅. Chapter 3. Problem-level frameworks (A2)For eachi,G i :R→Dom(M)is a total map andf i :X→Ris a total map. (A3)Agg M :Dom(M) k ×[0,1] k →Dom(M)is a total map. (A4)Score M :Dom(M)→Ris a total map. (A5)w∈[0,1] k satisfies ∑ k i=1 w i = 1. Then: (i)The mappingμ D,M :X→Dom(M)is well-defined. Consequently, U modm M := (X,μ D,M ) is anuncertain set (U-set) of typeMon the universeX. (i)The induced relation M onXis atotal preorder. (i)IfXis finite, then the UMODM solution setarg max x∈X Score M (μ D,M (x))is nonempty. Proof.(i) Well-definedness and U-set structure.Fixx∈X. For eachi, sincef i is total onX, f i (x)∈Ris well-defined. By (A2),G i is total, henceμ i (x) =G i (f i (x))∈Dom(M)is well-defined. Therefore μ(x) = (μ 1 (x),...,μ k (x))∈Dom(M) k is well-defined. By (A3) and (A5), Agg M (μ(x),w)∈Dom(M)is well-defined, henceμ D,M (x)∈Dom(M)exists for everyx∈X. Thusμ D,M :X→Dom(M)is a well-defined map, andU modm M = (X,μ D,M )is a U-set of typeM. (i) M is a total preorder.By (A4), Score M (μ D,M (x))∈Ris well-defined for eachx∈X. Define x M yiff Score M (μ D,M (x))≥Score M (μ D,M (y)). Since≥onRis reflexive, transitive, and total, the induced relation M inherits these properties and is therefore a total preorder. (i) Existence of a maximizer whenXis finite.IfXis finite, then the real set S:=Score M (μ D,M (x)) :x∈X is finite and thus attains a maximum. Hence arg max x∈X Score M (μ D,M (x))6=∅. Table 3.5 presents related uncertainty-model variants of Multiple Objective Decision-Making (MODM). Chapter 3. Problem-level frameworks Table 3.5: Related uncertainty-model variants of Multiple Objective Decision-Making (MODM). kRelated MODM variant(s) 1 Fuzzy Multiple Objective Decision-Making [385] 2 Intuitionistic Fuzzy Multiple Objective Decision-Making 3 Hesitant Fuzzy Multiple Objective Decision-Making 3 Spherical Fuzzy Multiple Objective Decision-Making 3 Neutrosophic Multiple Objective Decision-Making nPlithogenic Multiple Objective Decision-Making 3.6 Fuzzy ethical decision making Ethical decision making selects actions by comparing moral principles and consequences, balancing rights, duties, and stakeholders to justify choices [386,387]. Fuzzy ethical decision making assigns graded compliance to ethical dimensions, applies fuzzy rules and inference, defuzzifies scores, and selects actions. Definition 3.6.1(Fuzzy ethical decision making (rule-based fuzzy selection)).LetS 6=∅be a set of situations(contexts, states) and letA6=∅be a set of feasibleactions. LetD=d 1 ,...,d k be a finite set of ethical dimensions (e.g. privacy, fairness, transparency), each evaluated on a graded scale in[0,1]. For each dimensiond j , assume anethical compliance degree μ j :S×A−→[0,1],(s,a)7−→μ j (s,a), whereμ j (s,a)is the degree to which actionasatisfies dimensiond j in situations. Fix a continuoust-normTand ans-normSon[0,1](for conjunction/disjunction in fuzzy inference), and fix a finite fuzzy rule baseRwhose rules have the form R:IF(μ 1 isL 1 )AND·AND(μ k isL k )THEN(EisL E ), where eachL j andL E is a linguistic label (e.g.low/medium/high) represented by membership functions on[0,1]. Such conditional rules score ethical behaviour by combining (for example) privacy, fairness, and transparency into anethical decision score. For each(s,a)∈S×A, fuzzy inference (Mamdani-type) produces a fuzzy output ̃ E s,a on[0,1]by μ ̃ E s,a (e) :=S R∈R ( T ( α R (s,a), μ L R E (e) ) ) , whereα R (s,a)∈[0,1]is the firing strength of ruleRcomputed from the antecedent labels viaT, andμ L R E is the membership function of the consequent label ofR. Let Defuzz be a defuzzification functional (e.g. centroid), and define thecrisp ethical score E(s,a) :=Defuzz ( ̃ E s,a ) ∈[0,1]. (Defuzzification is required when non-boolean linguistic values must be converted to actionable numeric outputs.) Afuzzy ethical decisionin situationsis any action a ? ∈arg max a∈A E(s,a). Chapter 3. Problem-level frameworks In anuncertainformulation, each ethical dimension is evaluated in a fixed uncertain degree-domain Dom(M)⊆ [0,1] d (chosen uncertain modelM), and the overall ethical acceptability is represented as anuncertain set on the action space; actions are then ranked via a real-valued score. Definition 3.6.2(Uncertain ethical decision making (UEDM) of typeM).LetS 6=∅be a set of situations (contexts) andA 6=∅a set of feasible actions. LetD=d 1 ,...,d k be a finite set of ethical dimensions (k≥1). Fix an uncertain modelMwith degree-domain Dom(M)⊆[0,1] d . Anuncertain ethical decision making instance of typeMis a tuple E M = ( S,A,D,(μ 1 ,...,μ k ),w,Agg M ,Score M ) , where: (i)Dimension-wise uncertain compliance degrees:for eachj= 1,...,k, a total map μ j :S×A−→Dom(M), whereμ j (s,a)encodes the uncertain degree to which actionasatisfies ethical dimensiond j in situation s. (i)Importance weights:a vectorw= (w 1 ,...,w k )∈[0,1] k with ∑ k j=1 w j = 1. (i)Uncertain ethical aggregation operator:a total map Agg M :Dom(M) k ×[0,1] k −→Dom(M), interpreted as combining thekethical-dimension degrees into a single overall ethical acceptability degree. (iv)Ranking functional:a total map Score M :Dom(M)→R. For each(s,a)∈S×Adefine theoverall uncertain ethical degree μ eth,M (s,a) :=Agg M ( (μ 1 (s,a),...,μ k (s,a)),(w 1 ,...,w k ) ) ∈Dom(M). Thus, for each fixed situations, the mapping μ eth,M (s,·) :A→Dom(M), a7→μ eth,M (s,a), defines anuncertain set (U-set) of typeMon the universeA. AUEDM preferenceat situationsis the relation M,s onAgiven by a M,s b⇐⇒Score M ( μ eth,M (s,a) ) ≥Score M ( μ eth,M (s,b) ) . AUEDM ethical decisionat situationsis any action a ? (s)∈arg max a∈A Score M ( μ eth,M (s,a) ) . Chapter 3. Problem-level frameworks Theorem 3.6.3(Uncertain-set structure and well-definedness of UEDM).LetE M be a UEDM instance as in Definition 3.6.2. Assume: (A1)S 6=∅andA6=∅andDis finite. (A2)Eachμ j :S×A→Dom(M)is a total map. (A3)Agg M :Dom(M) k ×[0,1] k →Dom(M)is a total map. (A4)Score M :Dom(M)→Ris a total map. (A5)w∈[0,1] k satisfies ∑ k j=1 w j = 1. Then: (i)The mappingμ eth,M :S×A→Dom(M)is well-defined. For everys∈S,(A,μ eth,M (s,·))is a U-set of typeMonA. (i)For eachs∈S, the relation M,s is a total preorder onA. (i)IfAis finite, then for everys∈Sthe argmax setarg max a∈A Score M (μ eth,M (s,a))is nonempty. Proof.(i) Well-definedness and U-set structure.Fix(s,a)∈ S ×A. By (A2), each valueμ j (s,a)∈ Dom(M)exists and is uniquely determined, hence thek-tuple(μ 1 (s,a),...,μ k (s,a))∈Dom(M) k is well- defined. By (A3) and (A5), the aggregated degree μ eth,M (s,a) =Agg M ( (μ 1 (s,a),...,μ k (s,a)),w ) is a well-defined element of Dom(M). Thereforeμ eth,M is a well-defined map onS ×A. For fixeds, the functionμ eth,M (s,·) :A→Dom(M)is a membership assignment in the modelM, hence(A,μ eth,M (s,·))is a U-set of typeM. (i) M,s is a total preorder.Fixs∈S. By (A4), the real score Score M (μ eth,M (s,a))∈Ris well-defined for eacha∈ A. Definea M,s biff Score M (μ eth,M (s,a))≥Score M (μ eth,M (s,b)). Since≥onRis reflexive, transitive, and total, the induced relation M,s is a total preorder. (i) Existence of a maximizer for finiteA.IfAis finite, then the finite set S s :=Score M (μ eth,M (s,a)) :a∈A⊆R attains a maximum. Hence the argmax set is nonempty. Table 3.6 presents related uncertainty-model variants of Fuzzy Ethical Decision-Making. Chapter 3. Problem-level frameworks Table 3.6: Related uncertainty-model variants of Fuzzy Ethical Decision-Making. kRelated Fuzzy Ethical Decision-Making variant(s) 1 Fuzzy Ethical Decision-Making 2 Intuitionistic Fuzzy Ethical Decision-Making 3 Hesitant Fuzzy Ethical Decision-Making 3 Spherical Fuzzy Ethical Decision-Making 3 Neutrosophic Ethical Decision-Making nPlithogenic Ethical Decision-Making 3.7 Fuzzy Consensus decision-making Consensus decision-making is a collaborative process in which participants discuss alternatives, reconcile differing views, and seek a broadly acceptable decision supported by the whole group [388, 389]. Fuzzy consensus decision-making is a group process where experts’ fuzzy evaluations are iteratively adjusted toward a collective fuzzy matrix until consensus threshold holds, then alternatives ranked [390,391]. Definition 3.7.1(Fuzzy consensus decision-making (consensus-reaching + selection)).[390,391] Let A=A 1 ,...,A m (alternatives), C=C 1 ,...,C n (criteria), E=e 1 ,...,e p (experts). Fix a classFNof fuzzy numbers (e.g. TFNs), and let each experte k provide a fuzzy decision matrix ̃ R (k) = ( ̃r (k) ij ) ∈FN m×n , ̃r (k) ij is the fuzzy evaluation ofA i underC j . Letw= (w 1 ,...,w p )be expert weights withw k ≥0and ∑ p k=1 w k = 1. (1) Distance, similarity, and consensus index.Assume a normalized distance on fuzzy numbers d:FN×FN→[0,1]. Extend it to decision matrices by D ( ̃ R (k) , ̃ R (`) ) := 1 mn m ∑ i=1 n ∑ j=1 d ( ̃r (k) ij , ̃r (`) ij ) ∈[0,1], and define the corresponding similarity S ( ̃ R (k) , ̃ R (`) ) := 1−D ( ̃ R (k) , ̃ R (`) ) ∈[0,1]. A (weighted)group consensus indexis GCI( ̃ R (1) ,..., ̃ R (p) ) := p ∑ k=1 w k S ( ̃ R (k) , ̃ R c ) ∈[0,1], where ̃ R c is a collective matrix obtained by a fixed fuzzy aggregation operator Agg R (e.g. componentwise weighted average): ̃ R c :=Agg R ( ̃ R (1) ,..., ̃ R (p) ;w ) ∈FN m×n . Chapter 3. Problem-level frameworks (2) Consensus-reaching condition.Fix a consensus thresholdβ∈[0,1]. The experts are said toreach consensusif GCI( ̃ R (1) ,..., ̃ R (p) )≥β. If this fails, a consensus-reaching process iteratively revises experts’ opinions (and possibly their weights) until the threshold is met. Such iterative revision/feedback is the standard viewpoint of consensus reaching in group decision making. (3) A generic fuzzy feedback (opinion-adjustment) step.At iterationt, given current matrices ̃ R (k) (t), compute the collective matrix ̃ R c (t) =Agg R ( ̃ R (1) (t),..., ̃ R (p) (t);w(t)). For each experte k that is requested to adjust, choose anacceptance degreeθ k (t)∈[0,1]and update entrywise by the convex- combination rule ̃r (k) ij (t+ 1) := (1−θ k (t))⊗ ̃r (k) ij (t)⊕θ k (t)⊗ ̃r c ij (t), i= 1,...,m, j= 1,...,n, where⊕,⊗denote the adopted fuzzy arithmetic (e.g. extension-principle addition and scalar multiplica- tion). This expresses “move (to degreeθ k ) toward a trusted/collective opinion”, a common mechanism in consensus-reaching models. Repeat (1)–(3) until GCI≥β. (4) Selection after consensus.Let ̃ R c be the final collective matrix. Fix criterion weights ̃v= ( ̃v 1 ,..., ̃v n ) (crisp or fuzzy) and a final scoring/defuzzification map Score:FN→R. Define the overall fuzzy score of alternativeA i by a chosen operator Agg C , e.g. the fuzzy weighted average: ̃s i :=Agg C ( ̃r c i1 ,..., ̃r c in ; ̃v 1 ,..., ̃v n ) ∈FN, A ? ∈arg max 1≤i≤m Score( ̃s i ). Any suchA ? is called afuzzy consensus decision. Uncertain consensus decision-making generalizes consensus-reaching group decision processes by allowing each assessment to take values in a fixeduncertain degree-domainDom(M)⊆[0,1] d (an uncertain model M). The collective outcome after consensus is represented as anuncertain seton the alternative space, and a real-valued scoring functional induces ranking and selection. Definition 3.7.2(Uncertain consensus decision-making (UCDM) of typeM).Let A=A 1 ,...,A m (alternatives),C=C 1 ,...,C n (criteria),E=e 1 ,...,e p (experts), withm,n,p≥1. Fix an uncertain modelMwith degree-domain Dom(M)⊆[0,1] d . Anuncertain consensus decision-making instance of typeMis a tuple C M = ( A,C,E,(R (k) ) p k=1 , w, d M ,Agg R , β,Up M ,Agg C ,Score M ) , where: Chapter 3. Problem-level frameworks (i)Uncertain evaluation matrices:each experte k provides an uncertain decision matrix R (k) = ( r (k) ij ) ∈Dom(M) m×n , r (k) ij ∈Dom(M)is the uncertain evaluation ofA i underC j . (i)Expert weights:w= (w 1 ,...,w p )∈[0,1] p with ∑ p k=1 w k = 1. (i)Distance and similarity onDom(M):a normalized distance d M :Dom(M)×Dom(M)→[0,1], and its entrywise extension to matrices D M (R (k) ,R (`) ) := 1 mn m ∑ i=1 n ∑ j=1 d M ( r (k) ij ,r (`) ij ) ∈[0,1], with similarityS M (R (k) ,R (`) ) := 1−D M (R (k) ,R (`) ). (iv)Collective aggregation across experts:a total aggregation operator Agg R :Dom(M) p ×[0,1] p →Dom(M) extended entrywise to matrices by R c :=Agg R ( R (1) ,...,R (p) ;w ) ∈Dom(M) m×n , r c ij :=Agg R ( r (1) ij ,...,r (p) ij ;w ) . (v)Consensus index and threshold:a group consensus index GCI M ( R (1) ,...,R (p) ) := p ∑ k=1 w k S M ( R (k) ,R c ) ∈[0,1], and a thresholdβ∈[0,1]. (vi)Opinion-update (feedback) operator:a total update map Up M :Dom(M)×Dom(M)×[0,1]→Dom(M), which updates an expert’s entry toward the collective entry with acceptance degreeθ∈[0,1]. At iterationt, for each requested expertkand each(i,j): r (k) ij (t+ 1) :=Up M ( r (k) ij (t), r c ij (t), θ k (t) ) , θ k (t)∈[0,1]. (vii)Post-consensus selection:a criterion-aggregation operator Agg C :Dom(M) n ×[0,1] n →Dom(M), (using criterion weightsv= (v 1 ,...,v n )∈[0,1] n , ∑ j v j = 1), and a scoring functional Score M :Dom(M)→R. Consensus-reaching phase.Starting from(R (k) (0)) p k=1 , define fort= 0,1,2,...the collective matrix R c (t) =Agg R (R (1) (t),...,R (p) (t);w)and compute GCI M (t) :=GCI M (R (1) (t),...,R (p) (t)). Consensus is reachedat the first timeTsuch that GCI M (T)≥β. Chapter 3. Problem-level frameworks Decision (uncertain-set output) phase.LetR c :=R c (T)be the final collective matrix. For each alternativeA i , define the overall uncertain utility degree u i :=Agg C ( r c i1 ,...,r c in ;v 1 ,...,v n ) ∈Dom(M). Then the mapping μ U :A→Dom(M), μ U (A i ) :=u i , is thecollective uncertain utility set(a U-set of typeM) onA. A final ranking is induced by A i M A ` ⇐⇒Score M (u i )≥Score M (u ` ), and any A ? ∈arg max A i ∈A Score M ( μ U (A i ) ) is called anuncertain consensus decision. Theorem 3.7.3(Uncertain-set structure and well-definedness of UCDM).Consider a UCDM instanceC M as in Definition 3.7.2. Assume: (A1)m,n,p≥1andA,C,Eare finite nonempty. (A2)R (k) ∈Dom(M) m×n for allk(hence each entryr (k) ij ∈Dom(M)exists). (A3)d M is a total mapDom(M)×Dom(M)→[0,1]. (A4)Agg R andAgg C are total maps with codomainDom(M). (A5)Up M is a total map with codomainDom(M), andθ k (t)∈[0,1]for allk,t. (A6)Score M :Dom(M)→Ris a total map; andw,vare normalized weight vectors. Then: (i)For everyt≥0, the matricesR (k) (t)produced by the update rule are well-defined elements of Dom(M) m×n , and the collective matrixR c (t)is well-defined. ConsequentlyGCI M (t)∈[0,1]is well- defined. (i)If consensus is reached at some finite timeT, then the post-consensus utility mappingμ U :A → Dom(M)is a well-defined U-set of typeMonA. (i)The induced relation M onAis a total preorder. (iv)If consensus is reached at some finite timeTandAis finite, then the argmax set defining an uncertain consensus decision is nonempty. Proof.(i) Well-definedness along iterations.Fixt≥0and suppose inductively thatR (k) (t)∈ Dom(M) m×n for allk(true att= 0by (A2)). By (A4), for each(i,j)the value r c ij (t) =Agg R ( r (1) ij (t),...,r (p) ij (t);w ) ∈Dom(M) Chapter 3. Problem-level frameworks is well-defined, henceR c (t)∈Dom(M) m×n is well-defined. By (A5), the update r (k) ij (t+ 1) =Up M ( r (k) ij (t),r c ij (t),θ k (t) ) ∈Dom(M) is well-defined for eachk,i,j, soR (k) (t+ 1)∈Dom(M) m×n for allk. Thus, by induction, allR (k) (t)and R c (t)are well-defined for everyt. Next, by (A3) each entrywise distanced M (r (k) ij (t),r c ij (t))∈[0,1]is well-defined, so the averagesD M (R (k) (t),R c (t))∈ [0,1]and similaritiesS M (R (k) (t),R c (t))∈[0,1]are well-defined. Finally, GCI M (t)is a convex combination of values in[0,1]using weightsw, hence GCI M (t)∈[0,1]is well-defined. (i) U-set structure after consensus.Assume consensus is reached at timeT. ThenR c =R c (T)∈ Dom(M) m×n is well-defined by (i). For each alternativeA i , by (A4) the aggregation u i =Agg C (r c i1 ,...,r c in ;v)∈Dom(M) is well-defined. Thereforeμ U (A i ) :=u i defines a well-defined mapμ U :A →Dom(M), i.e. a U-set of type MonA. (i) M is a total preorder.By (A6), each score Score M (u i )∈Ris well-defined. Since≥onRis reflexive, transitive, and total, the induced relationA i M A ` ⇐⇒Score M (u i )≥Score M (u ` )is a total preorder. (iv) Existence of a maximizer for finiteA.IfAis finite and consensus is reached, then the finite set of real scoresScore M (μ U (A i )) :A i ∈Aattains a maximum. Hence the argmax set is nonempty. Table 3.7 presents related uncertainty-model variants of Fuzzy Consensus Decision-Making. Table 3.7: Related uncertainty-model variants of Fuzzy Consensus Decision-Making. kRelated Fuzzy Consensus Decision-Making variant(s) 1 Fuzzy Consensus Decision-Making 2 Intuitionistic Fuzzy Consensus Decision-Making 3 Hesitant Fuzzy Consensus Decision-Making 3 Spherical Fuzzy Consensus Decision-Making 3 Neutrosophic Consensus Decision-Making nPlithogenic Consensus Decision-Making 3.8 Fuzzy Strategic decision making Strategic decision making selects long-term initiatives under resource constraints, aligning actions with organizational goals, risks, and competitive priorities [392, 393]. Fuzzy strategic decision making scores strategic plans by fuzzy goal achievement and cost satisfaction, aggregates via t-norms, selects best [394,395]. Chapter 3. Problem-level frameworks Definition 3.8.1(Fuzzy strategic decision-making (strategy-map based)).LetM=M 1 ,...,M p be a finite set of strategic measures (projects). Astrategic plan(decision) is a vector x= (x 1 ,...,x p )∈D ⊆0,1 p , wherex ` = 1means that measureM ` is selected, andDencodes feasibility constraints (e.g. resource budgets, precedence constraints). LetG=g 1 ,...,g s be strategic goals (from a formalized strategy map). For each goalg i , fix a set I i =I i1 ,...,I im i of its (resultant) indicators and initial/target values(v 0 ij ,v ? ij )for each indicatorI ij . (1) Strategic criteria (goal-achievement degrees).For eachi, assume the strategy-map formalization provides a mapping that assigns to every planx∈Dadegree of achievement S i (x)∈[0,1], interpreted as the (graded) level at which goalg i is achieved underx. (2) Economic criteria (resource costs and efficiency).LetR=1,...,rbe resource types. Define cost/consumption criteria C u :D →R ≥0 (u∈R), and (optionally) additional economic-efficiency criteria E v :D →R(v= 1,...,q). Thus, the criteria are partitioned into astrategic groupS i s i=1 and aneconomic groupC u r u=1 (and possiblyE v q v=1 ). (3) Fuzzy evaluation and fuzzy selection.Assume each criterion is assessed fuzzily by a satisfaction (membership) degree: μ S i :D →[0,1] (i= 1,...,s), μ C u :D →[0,1] (u= 1,...,r), μ E v :D →[0,1] (v= 1,...,q). LetTbe ak-aryt-norm (conjunction operator), and letw= (w 1 ,...,w s+r+q )be nonnegative importance weights with ∑ w j = 1. Define the overallfuzzy decision membershipof a planxby μ D (x) :=T ( (μ S 1 (x)) w 1 ,...,(μ S s (x)) w s ,(μ C 1 (x)) w s+1 ,...,(μ C r (x)) w s+r , (μ E 1 (x)) w s+r+1 ,...,(μ E q (x)) w s+r+q ) ∈[0,1]. Afuzzy strategic decision(optimal strategic plan) is any x ? ∈arg max x∈D μ D (x). We now define Uncertain Strategic Decision-Making (USDM), obtained by extending this framework using Uncertain Sets. Chapter 3. Problem-level frameworks Definition 3.8.2(Uncertain strategic decision-making (USDM) of typeM).LetM=M 1 ,...,M p be a finite set of strategic measures (projects), and let D ⊆0,1 p be a nonempty feasible set of strategic plans, wherex= (x 1 ,...,x p )∈Dindicates the selected measures. Fix anuncertain modelMwith degree-domain Dom(M)⊆[0,1] d . LetG=g 1 ,...,g s be strategic goals and letR=1,...,rbe resource types. Assume that each plan x∈Dinduces uncertain evaluations (degrees) for: φ i :D →Dom(M) (i= 1,...,s)(goal-achievement degrees), ψ u :D →Dom(M) (u= 1,...,r)(resource-feasibility/cost satisfaction degrees), and optionally additional uncertain criteria η v :D →Dom(M) (v= 1,...,q). Letk:=s+r+qand enumerate these criteria as a single family ζ 1 ,...,ζ k :D →Dom(M), where(ζ 1 ,...,ζ s ) = (φ 1 ,...,φ s ),(ζ s+1 ,...,ζ s+r ) = (ψ 1 ,...,ψ r ), and(ζ s+r+1 ,...,ζ k ) = (η 1 ,...,η q )(if q= 0, omit the last block). (1) Aggregation in the uncertain degree-domain.Choose a total aggregation operator Agg M :Dom(M) k ×∆ k −→Dom(M),∆ k := w∈[0,1] k : k ∑ j=1 w j = 1 , and fix a weight vectorw∈∆ k . (2) Scoring and selection.Fix a total scoring functional Score M :Dom(M)→R. Define the overall uncertain strategic utility of a planx∈Dby U M (x) :=Agg M ( ζ 1 (x),...,ζ k (x);w ) ∈Dom(M), and the induced real score by Score(x) :=Score M (U M (x))∈R. Anuncertain strategic decisionis any plan x ? ∈arg max x∈D Score M ( U M (x) ) . (3) Uncertain-set output (strategic utility set).Define the mapping μ U :D →Dom(M), μ U (x) :=U M (x). ThenU= (D,μ U )is called thestrategic uncertain utility set(a U-set of typeM) induced by the instance. Chapter 3. Problem-level frameworks Theorem 3.8.3(Uncertain-set structure and well-definedness of USDM).Let a USDM instance of typeM be given as in Definition 3.8.2. Assume: (A1)p≥1andD ⊆0,1 p is finite and nonempty. (A2)Dom(M)⊆[0,1] d is nonempty and each criterion mapζ j :D →Dom(M)is total. (A3)Agg M :Dom(M) k ×∆ k →Dom(M)is total andw∈∆ k . (A4)Score M :Dom(M)→Ris total. Then: (i)The mappingμ U :D →Dom(M)defined byμ U (x) =Agg M (ζ 1 (x),...,ζ k (x);w)is well-defined; hence U= (D,μ U )is a well-defined uncertain set (U-set) of typeMonD. (i)The binary relation M onDdefined by x M y⇐⇒Score M (μ U (x))≥Score M (μ U (y)) is a total preorder. (i)The setarg max x∈D Score M (μ U (x))is nonempty; hence an uncertain strategic decision exists. Proof.(i) Uncertain-set structure.Fixx∈ D. By (A2), eachζ j (x)∈Dom(M)is well-defined. By (A3), applying the total map Agg M (·;w)to thek-tuple(ζ 1 (x),...,ζ k (x))yields a unique value μ U (x) =Agg M ( ζ 1 (x),...,ζ k (x);w ) ∈Dom(M). Thusμ U :D →Dom(M)is a well-defined mapping, and thereforeU= (D,μ U )is a U-set of typeM. (i) Total preorder.By (A4), Score M (μ U (x))∈Ris well-defined for allx∈D. Since≥onRis reflexive, transitive, and total, the induced relation M inherits these properties, hence is a total preorder. (i) Existence of an optimizer.BecauseDis finite and nonempty by (A1), the finite set of real numbers Score M (μ U (x)) :x∈Dattains a maximum. Therefore arg max x∈D Score M (μ U (x))6=∅. Table 3.8 presents related uncertainty-model variants of Fuzzy Strategic Decision-Making. Chapter 3. Problem-level frameworks Table 3.8: Related uncertainty-model variants of Fuzzy Strategic Decision-Making. kRelated Fuzzy Strategic Decision-Making variant(s) 1 Fuzzy Strategic Decision-Making 2 Intuitionistic Fuzzy Strategic Decision-Making 3 Hesitant Fuzzy Strategic Decision-Making 3 Spherical Fuzzy Strategic Decision-Making 3 Neutrosophic Strategic Decision-Making nPlithogenic Strategic Decision-Making 3.9 Fuzzy Multi-Expert Decision-Making Multi-expert decision-making aggregates several experts’ evaluations or pairwise preferences, weights ex- perts, measures consensus, and produces a collective ranking or choice [396, 397]. Fuzzy multi-expert decision-making represents experts’ judgments as fuzzy numbers or linguistic terms, aggregates them via fuzzy operators, enforces consensus, and ranks alternatives robustly [398]. Definition 3.9.1(Fuzzy Multi-Expert Decision-Making (FMEDM)).(cf. [399,400]) LetA=A 1 ,...,A m be a finite set of alternatives,K=K 1 ,...,K p a finite set of criteria, andE=E 1 ,...,E n a finite set of experts. (1) Expert contribution factors.Assign each expertE i a contribution factor (weight)c i ∈[0,1]such that n ∑ i=1 c i = 1. (2) Fuzzy evaluations.LetFdenote a chosen class of fuzzy numbers onR. Each expertE i provides a fuzzy evaluation ̃x (i) rj ∈F(r= 1,...,m;j= 1,...,p), interpreted as the (uncertain/vague) performance of alternativeA r under criterionK j . (3) Multi-expert aggregation (group evaluation).Fix fuzzy-number operations⊕(addition) and⊗ (scalar multiplication byc i ). The aggregated (group) fuzzy evaluation is defined entrywise by the weighted fuzzy averaging operator ̃x rj := n ⊕ i=1 ( c i ⊗ ̃x (i) rj ) ,(r= 1,...,m;j= 1,...,p). (4) Aggregation across criteria and decision rule.Letw 1 ,...,w p ≥0be criterion weights with ∑ p j=1 w j = 1(crisp weights; fuzzy weights can be used analogously). Define the fuzzy overall score ofA r by ̃s r := p ⊕ j=1 ( w j ⊗ ̃x rj ) ∈F. Chapter 3. Problem-level frameworks LetΦ :F→Rbe a ranking functional (e.g., a defuzzification map). A (recommended) group decision is any A ? ∈arg max 1≤r≤m Φ( ̃s r ). (Common instantiation: trapezoidal fuzzy numbers).A standardized trapezoidal fuzzy number (STFN) is a quadruple ̃x= (a l ,a m ,a n ,a u )witha l ≤a m ≤a n ≤a u and membership function μ ̃x (t) = t−a l a m −a l , a l ≤t≤a m , 1,a m ≤t≤a n , a u −t a u −a n , a n ≤t≤a u , 0,otherwise. Forc≥0, definec⊗(a l ,a m ,a n ,a u ) := (ca l ,ca m ,ca n ,ca u )and (a l ,a m ,a n ,a u )⊕(b l ,b m ,b n ,b u ) := (a l +b l , a m +b m , a n +b n , a u +b u ), so the multi-expert aggregation in (3) becomes componentwise weighted averaging. Definition 3.9.2(Uncertain multi-expert decision-making (UMEDM) of typeM).Let A=A 1 ,...,A m 6=∅(alternatives),C=C 1 ,...,C n 6=∅(criteria), and E=e 1 ,...,e p 6=∅(experts). Fix anuncertain modelMwith degree-domain Dom(M)⊆[0,1] d (for somed≥1). (1) Expert and criterion weights.Let α= (α 1 ,...,α p )∈∆ p ,∆ p := α∈[0,1] p : p ∑ h=1 α h = 1 , be expert weights, and let w= (w 1 ,...,w n )∈∆ n be criterion weights. (2) Expert-wise uncertain evaluations.Each experte h provides an uncertain decision matrix R (h) = ( r (h) ij ) ∈Dom(M) m×n , r (h) ij ∈Dom(M)evaluatesA i w.r.t.C j by experte h . (3) Aggregation operators inDom(M).Choose total aggregation operators Agg E :Dom(M) p ×∆ p →Dom(M)(across experts), Agg C :Dom(M) n ×∆ n →Dom(M)(across criteria), Chapter 3. Problem-level frameworks and a total scoring functional Score M :Dom(M)→R. (4) Collective evaluation, utility, and selection.Define thecollective uncertain decision matrix R c = (r c ij )∈Dom(M) m×n entrywise by r c ij :=Agg E ( r (1) ij ,...,r (p) ij ; a lpha ) ∈Dom(M), i= 1,...,m, j= 1,...,n. For each alternativeA i , define itsoverall uncertain utility degreeby u i :=Agg C ( r c i1 ,...,r c in ;w ) ∈Dom(M), and its induced real score by U i :=Score M (u i )∈R. A(recommended) uncertain multi-expert decisionis any A ? ∈arg max 1≤i≤m U i =arg max A i ∈A Score M (u i ). (5) Uncertain-set output (multi-expert utility set).Define μ U :A→Dom(M), μ U (A i ) :=u i . ThenU:= (A,μ U )is called themulti-expert uncertain utility set(U-set of typeM) induced by the instance. Theorem 3.9.3(Uncertain-set structure and well-definedness of UMEDM).Let a UMEDM instance of typeMbe given as in Definition 3.9.2. Assume: (A1)A,C,Eare finite and nonempty. (A2)Dom(M)⊆[0,1] d is nonempty. (A3)For every experte h and every(i,j), the evaluation entryr (h) ij ∈Dom(M)is specified (i.e. eachR (h) is a totalm×nmatrix overDom(M)). (A4)α∈∆ p andw∈∆ n . (A5)Agg E andAgg C are total functions with codomainDom(M), andScore M is a total function with codomainR. Then: (i)μ U :A →Dom(M)is well-defined; henceU= (A,μ U )is a well-defined uncertain set (U-set) of type MonA. (i)The binary relation M onAgiven by A i M A ` ⇐⇒Score M ( μ U (A i ) ) ≥Score M ( μ U (A ` ) ) is a total preorder. Chapter 3. Problem-level frameworks (i)The maximizer setarg max A i ∈A Score M (μ U (A i ))is nonempty; in particular, a UMEDM solutionA ? exists. Proof.(i)FixA i ∈ AandC j ∈ C. By (A3), thep-tuple(r (1) ij ,...,r (p) ij )∈Dom(M) p is well-defined. By (A4)–(A5), applying the total map Agg E (·;α)yields a unique value r c ij =Agg E ( r (1) ij ,...,r (p) ij ;α ) ∈Dom(M). ThusR c ∈Dom(M) m×n is well-defined. Again by (A4)–(A5), then-tuple(r c i1 ,...,r c in )∈Dom(M) n is well-defined and u i =Agg C ( r c i1 ,...,r c in ;w ) ∈Dom(M) is uniquely determined. Thereforeμ U (A i ) :=u i is well-defined for eachA i , henceμ U :A →Dom(M)is a well-defined mapping andU= (A,μ U )is a U-set of typeM. (i)By (A5), each score Score M (μ U (A i ))∈Ris well-defined. Since≥onRis reflexive, transitive, and total, the induced relation M is reflexive, transitive, and total; hence it is a total preorder. (i)BecauseAis finite and nonempty by (A1), the finite set of real numbersScore M (μ U (A i )) :A i ∈A attains a maximum. Therefore the argmax set is nonempty, so at least one maximizerA ? exists. Table 3.9 presents related uncertainty-model variants of Fuzzy Multi-Expert Decision-Making. Table 3.9: Related uncertainty-model variants of Fuzzy Multi-Expert Decision-Making. kRelated Fuzzy Multi-Expert Decision-Making variant(s) 1 Fuzzy Multi-Expert Decision-Making 2 Intuitionistic Fuzzy Multi-Expert Decision-Making 3 Hesitant Fuzzy Multi-Expert Decision-Making 3 Spherical Fuzzy Multi-Expert Decision-Making 3 Neutrosophic Multi-Expert Decision-Making [401,402] nPlithogenic Multi-Expert Decision-Making 3.10Fuzzy Multi-Stage Decision-Making Multi-stage decision-making selects actions across sequential stages, where earlier choices affect later feasible sets, costs, and outcomes, optimizing total performance [403,404]. Fuzzy multi-stage decision-making models stage-dependent evaluations, constraints, or transitions with fuzzy sets, propagating uncertainty through stages to choose an adaptive plan [405,406]. Definition 3.10.1(Fuzzy Multi-Stage Decision-Making (FMSDM)).[405,406] Fix a finite horizonm∈N. For each stagei∈ 1,...,m, letS i be a (finite) state set and, for eachs∈S i , letD i (s)be a nonempty (finite) set of admissible decisions. Let the (crisp) transition map be τ i :S i ×D i (s)→S i+1 (i= 1,...,m−1). Assume that each stage-wise cost is given by a fuzzy number: β i :S i ×D i (s)→FN(R), Chapter 3. Problem-level frameworks whereFN(R)denotes the class of fuzzy numbers onR. A (deterministic) policy is a sequenceπ= (π 1 ,...,π m )withπ i (s)∈D i (s). Given an initial states 1 ∈S 1 , the induced trajectory is s i+1 =τ i ( s i ,π i (s i ) ) (i= 1,...,m−1), and the inducedcomposite fuzzy costis the fuzzy number β π (s 1 ) :=β 1 ( s 1 ,π 1 (s 1 ) ) ⊕β 2 ( s 2 ,π 2 (s 2 ) ) ⊕ · ⊕β m ( s m ,π m (s m ) ) , where⊕is theextended (fuzzy) sumdefined byα-cuts: for fuzzy numbersX,Yandα∈(0,1], [X⊕Y] α = [X] α + [Y] α (Minkowski sum of intervals). To compare fuzzy costs under a minimization objective, fix theα-cut dominance preorderonFN(R): for fuzzy numbersX,Y, XY⇐⇒ ∀α∈(0,1] :inf[X] α ≤inf[Y] α and sup[X] α ≤sup[Y] α . WriteX≺YifXYand notYX. A policyπ ? is called(dominance-)optimalats 1 if there is no policyπsuch thatβ π (s 1 )≺β π ? (s 1 ). Equiva- lently, the (possibly set-valued) solution set is the set of-minimal composite costs: Π ? (s 1 ) := π:@π ′ withβ π ′ (s 1 )≺β π (s 1 ) . Definition 3.10.2(Extended minimum cost and fuzzy set of best policies).AssumeΠis a finite set of admissible policies (e.g., all deterministic policies over finiteS i andD i (·)). Define theextended minimum of fuzzy numbers(X k ) N k=1 by Zadeh’s extension principle: μ ̃ min(X 1 ,...,X N ) (u) :=sup u=min(u 1 ,...,u N ) min 1≤k≤N μ X k (u k ), u∈R. Then theminimum (extended) fuzzy costats 1 is β ? (s 1 ) := ̃ min β π (s 1 ) :π∈Π . To associate afuzzy set of best policieswithβ ? (s 1 ), fix any well-defined matching (similarity) index Match: FN(R)×FN(R)→[0,1]. A common choice is theintersection height: Match(X,Y) :=sup u∈R minμ X (u),μ Y (u). Define the fuzzy set ̃ Π ? (s 1 )onΠby μ ̃ Π ? (s 1 ) (π) :=Match ( β π (s 1 ),β ? (s 1 ) ) . In particular, if a unique policyπ ? strictly dominates all others (under≺), then ̃ Π ? (s 1 )collapses to the crisp singletonπ ? . Chapter 3. Problem-level frameworks We now defineUncertain Multi-Stage Decision-Making(UMSDM) by extending stage-wise evaluations from fuzzy quantities to a general uncertain modelMwith degree-domain Dom(M)⊆[0,1] k . Definition 3.10.3(Uncertain Multi-Stage Decision-Making (UMSDM) of typeM).LetMbe an uncertain model with degree-domain Dom(M)⊆[0,1] k for some integerk≥1. Fix a finite horizonm∈N. For each stagei∈1,...,m, letS i be a finite nonempty state set, and for eachs∈S i , letD i (s)be a finite nonempty set of admissible decisions at stagei. Define the feasible state-decision set D i :=(s,d) :s∈S i , d∈D i (s). For eachi= 1,...,m−1, let τ i :D i −→S i+1 be a (crisp) transition map, and for eachi= 1,...,m, let β i :D i −→Dom(M) be theuncertain stage-evaluation map, whereβ i (s,d)represents the uncertain cost, utility, reward, or performance degree associated with taking decisiondin statesat stagei. Assume further that a model-dependent stage-composition operator Comp M :Dom(M) m −→Dom(M) and a model-dependent ranking functional Score M :Dom(M)−→R are fixed. Adeterministic policyis a sequence π= (π 1 ,...,π m ), where each π i :S i −→ ⋃ s∈S i D i (s) satisfiesπ i (s)∈D i (s)for alls∈S i . LetΠ i denote the set of all such stage-idecision rules, and let Π := Π 1 ×·×Π m be the set of all deterministic policies. Given an initial states 1 ∈S 1 and a policyπ∈Π, the induced state trajectory s π 1 ,s π 2 ,...,s π m is defined recursively by s π 1 :=s 1 , s π i+1 :=τ i ( s π i ,π i (s π i ) ) (i= 1,...,m−1). Chapter 3. Problem-level frameworks The induced stage-wise uncertain evaluations are b π i (s 1 ) :=β i ( s π i ,π i (s π i ) ) ∈Dom(M) (i= 1,...,m). Thecomposite uncertain performanceofπats 1 is B π M (s 1 ) :=Comp M ( b π 1 (s 1 ),...,b π m (s 1 ) ) ∈Dom(M), and the associated scalarized performance value is J π M (s 1 ) :=Score M ( B π M (s 1 ) ) ∈R. Under a minimization objective, define the induced policy preorder by π M ρ⇐⇒J π M (s 1 )≤J ρ M (s 1 ). A policyπ ? ∈Πis calledoptimalats 1 if J π ? M (s 1 ) =min π∈Π J π M (s 1 ). Equivalently, the optimal-policy set is Π ? M (s 1 ) :=arg min π∈Π J π M (s 1 ). If a maximization objective is intended instead, one replaces arg min with arg max, or equivalently replaces ≤with≥in the induced preorder. Theorem 3.10.4(Well-definedness of UMSDM of typeM).LetMbe an uncertain model with degree- domainDom(M)⊆[0,1] k , and let U M = ( (S i ) m i=1 ,(D i ) m i=1 ,(τ i ) m−1 i=1 ,(β i ) m i=1 ,Comp M ,Score M ) be a UMSDM instance as in Definition 3.10.3. Assume: (A1) eachS i is finite and nonempty; (A2) eachD i (s)is finite and nonempty for everyi∈1,...,mands∈S i ; (A3) each transition map τ i :D i →S i+1 (i= 1,...,m−1) is a total function; (A4) each stage-evaluation map β i :D i →Dom(M) (i= 1,...,m) is a total function; (A5) the composition operator Comp M :Dom(M) m →Dom(M) is a total function; Chapter 3. Problem-level frameworks (A6) the ranking functional Score M :Dom(M)→R is a total function. Then, for every initial states 1 ∈S 1 , the following statements hold: (i) the policy setΠis finite and nonempty; (i) for every policyπ∈Π, the induced trajectory s π 1 ,s π 2 ,...,s π m is uniquely determined; (i) for every policyπ∈Π, the stage-wise uncertain evaluations b π i (s 1 )∈Dom(M) (i= 1,...,m) and the composite uncertain performance B π M (s 1 )∈Dom(M) are well-defined; (iv) for every policyπ∈Π, the scalar value J π M (s 1 ) =Score M ( B π M (s 1 ) ) ∈R is well-defined; (v) the relation M onΠ, defined by π M ρ⇐⇒J π M (s 1 )≤J ρ M (s 1 ), is a total preorder; (vi) the optimal-policy set Π ? M (s 1 ) =arg min π∈Π J π M (s 1 ) is nonempty. Hence Uncertain Multi-Stage Decision-Making of typeMis well-defined. Proof.First, for each stagei, define Π i :=π i :S i → ⋃ s∈S i D i (s) :π i (s)∈D i (s)for alls∈S i . SinceS i is finite and eachD i (s)is finite and nonempty, the number of such functions is |Π i |= ∏ s∈S i |D i (s)|, Chapter 3. Problem-level frameworks which is a positive finite integer. Therefore eachΠ i is finite and nonempty, and hence Π = Π 1 ×·×Π m is also finite and nonempty. This proves(i). Next, fix an initial states 1 ∈S 1 and a policyπ∈Π. We show that the induced trajectory is uniquely determined. Sets π 1 :=s 1 . Assume inductively thats π i ∈S i has been uniquely defined for somei∈ 1,...,m−1. Becauseπ i (s π i )∈D i (s π i ), the pair ( s π i ,π i (s π i ) ) ∈D i . By assumption(A3), the transition mapτ i is total, so s π i+1 :=τ i ( s π i ,π i (s π i ) ) ∈S i+1 exists and is unique. By induction, the whole trajectory s π 1 ,s π 2 ,...,s π m exists and is unique. Thus(i)holds. For each stagei, since ( s π i ,π i (s π i ) ) ∈D i andβ i :D i →Dom(M)is total by(A4), the value b π i (s 1 ) :=β i ( s π i ,π i (s π i ) ) is well-defined and belongs to Dom(M). Hence them-tuple ( b π 1 (s 1 ),...,b π m (s 1 ) ) ∈Dom(M) m is well-defined. By(A5), the composition operator Comp M is total, so B π M (s 1 ) =Comp M ( b π 1 (s 1 ),...,b π m (s 1 ) ) ∈Dom(M) is well-defined. This proves(i). By(A6), Score M is a total map from Dom(M)intoR. Therefore J π M (s 1 ) :=Score M ( B π M (s 1 ) ) ∈R is well-defined for everyπ∈Π. Thus(iv)holds. Now define, forπ,ρ∈Π, π M ρ⇐⇒J π M (s 1 )≤J ρ M (s 1 ). Since≤onRis reflexive, transitive, and total, the induced relation M onΠis also reflexive, transitive, and total. Hence M is a total preorder. This proves(v). Finally, becauseΠis finite and nonempty by(i), the set J π M (s 1 ) :π∈Π⊆R Chapter 3. Problem-level frameworks is a finite nonempty set of real numbers. Every finite nonempty subset ofRattains its minimum. Therefore there exists at least one policyπ ? ∈Πsuch that J π ? M (s 1 ) =min π∈Π J π M (s 1 ). Equivalently, Π ? M (s 1 ) =arg min π∈Π J π M (s 1 )6=∅. Thus(vi)holds. All required objects therefore exist and are unambiguously defined. Hence UMSDM of typeMis well- defined. Table 3.10 presents related uncertainty-model variants of Fuzzy Multi-Stage Decision-Making. Table 3.10: Related uncertainty-model variants of Fuzzy Multi-Stage Decision-Making. kRelated Fuzzy Multi-Stage Decision-Making variant(s) 1 Fuzzy Multi-Stage Decision-Making 2 Intuitionistic Fuzzy Multi-Stage Decision-Making 3 Hesitant Fuzzy Multi-Stage Decision-Making 3 Spherical Fuzzy Multi-Stage Decision-Making 3 Neutrosophic Multi-Stage Decision-Making nPlithogenic Multi-Stage Decision-Making 3.11Fuzzy Multi-Level Decision-Making Multi-level decision-making models hierarchical leaders and followers, where upper-level decisions constrain lower-level responses, solved as bilevel or hierarchical optimization problems [407–409]. Fuzzy multi-level decision-making introduces fuzzy goals, constraints, or preferences at different hierarchy levels, yielding compromise solutions under imprecise leader–follower information. Definition 3.11.1(FuzzyL-Level Decision-Making).LetL≥2be the number of decision levels. For each level`∈1,...,L, letx ` ∈X ` ⊆R n ` be the decision vector controlled by the`-th level, and set x:= (x 1 ,...,x L )∈X:= L ∏ `=1 X ` . Assume that uncertainty/imprecision is modeled by fuzzy numbers. Let ̃ Rdenote a chosen class of fuzzy numbers onR. For each level`, let the (fuzzy-valued) objective be ̃ f ` :X→ ̃ R. Let the system constraints be given by fuzzy-valued mappings ̃g r :X→ ̃ R, r= 1,...,p. Fix: Chapter 3. Problem-level frameworks • anorder-inducing ranking functionalR: ̃ R→R(e.g., a defuzzification/ranking operator) that will be used to compare fuzzy objectives; • afeasibility satisfaction levelη∈(0,1]. For a fuzzy number ̃z∈ ̃ Rwith membership functionμ ̃z :R→[0,1], define thepossibilitythat ̃z≤0by Pos( ̃z≤0) :=sup t≤0 μ ̃z (t)∈[0,1]. Define theη-feasible set F η := x∈X:Pos( ̃g r (x)≤0)≥ηfor allr= 1,...,p . Define recursively therational-response correspondencesS ` (backward induction): S L (x 1:L−1 ) :=arg min x L ∈X L R ( ̃ f L (x 1:L−1 ,x L ) ) : (x 1:L−1 ,x L )∈F η , S ` (x 1:`−1 ) :=arg min x ` ∈X ` R ( ̃ f ` (x 1:`−1 ,x ` ,x `+1:L ) ) :∃x `+1:L ∈S `+1 (x 1:` ),(x 1:L )∈F η , for`=L−1,L−2,...,1, wherex i:j := (x i ,...,x j ). A vectorx ? = (x ? 1 ,...,x ? L )∈F η is called afuzzyL-level (Stackelberg) solutionif x ? 2:L ∈S 2 (x ? 1 )andx ? 1 ∈S 1 (∅), equivalently, ifx ? is generated by the above backward-induction optimal reactions. Remark 3.11.2(Recovery of classical multilevel programming).If all data are crisp (so ̃ f ` and ̃g r take values inR), Pos( ̃g r (x)≤0)∈0,1, and one takesη= 1withRequal to the identity, then Definition 3.11.1 reduces to the standard (crisp)L-level Stackelberg decision/optimization model. Theorem 3.11.3(Well-definedness / existence under compactness).Assume: 1. eachX ` is nonempty and compact; 2.F η is nonempty and closed inX(hence compact); 3. for each`, the scalarized objectivex7→R( ̃ f ` (x))is continuous onX. Then for every`, the correspondenceS ` is nonempty-valued, and there exists at least one fuzzyL-level solutionx ? ∈F η . Chapter 3. Problem-level frameworks Proof.We argue by backward induction. LevelL.Fixx 1:L−1 such that the feasible slicex L ∈X L : (x 1:L−1 ,x L )∈F η is nonempty. BecauseF η is closed andX L is compact, this slice is compact. Continuity ofx L 7→R( ̃ f L (x 1:L−1 ,x L ))implies the minimum is attained (Weierstrass theorem), henceS L (x 1:L−1 )6=∅. Induction step.AssumeS `+1 is nonempty-valued. Fixx 1:`−1 and consider the set of admissiblex ` ∈X ` for which there existsx `+1:L ∈S `+1 (x 1:` )with(x 1:L )∈F η . Nonemptiness follows from the induction hypothesis together withF η 6=∅. Compactness follows from compactness ofXand closedness ofF η (after projecting onto the relevant coordinates). Sincex7→R( ̃ f ` (x))is continuous onX, the induced minimization over this compact nonempty set attains its minimum, soS ` (x 1:`−1 )6=∅. Applying this to`= 1yields a nonemptyS 1 (∅)and thus a backward-induction chain producing at least onex ? ∈F η . We now defineUncertain Multi-Level Decision-Making(UMLDM) by replacing the level-wise crisp/fuzzy objectives in multi-level decision-making withuncertain-set-valued objectives, in the sense of [20,142]. Definition 3.11.4(UncertainL-Level Decision-Making (UMLDM) of typeM).LetL≥2be the number of hierarchical decision levels, and letMbe an uncertain model with degree-domain Dom(M)⊆[0,1] k for some integerk≥1. For each level`∈1,...,L, letX ` be a nonempty decision set for the`-th decision maker, and put X:= L ∏ `=1 X ` . An element ofXis written as x= (x 1 ,...,x L ), x ` ∈X ` . We write x i:j := (x i ,...,x j ) (1≤i≤j≤L). Assume that the set of globally feasible decisions is a nonempty subset F M ⊆X. For each level`∈1,...,L, let f `,M :X−→Dom(M) be theuncertain objective mapof level`, and let Score `,M :Dom(M)−→R be a totalranking / scalarization mapused to compare uncertain objective values at level`. Thus each pair (X,f `,M )is an uncertain set (U-set) of typeMon the common decision universeX. Chapter 3. Problem-level frameworks For each`∈1,...,Land each feasible prefixx 1:`−1 , define thefeasible continuation set F (`) M (x 1:`−1 ) := x `:L ∈X ` ×·×X L : (x 1:`−1 ,x `:L )∈F M . For convenience, when`= 1, we interpretx 1:0 as the empty prefix∅, so that F (1) M (∅) =F M . Therational-response correspondencesare defined recursively from the lowest level upward. LevelL:for each feasible prefixx 1:L−1 , set R L,M (x 1:L−1 ) :=argmin x L ∈F (L) M (x 1:L−1 ) Score L,M ( f L,M (x 1:L−1 ,x L ) ) . Levels`=L−1,L−2,...,1:for each feasible prefixx 1:`−1 , define theadmissible rational-continuation set A `,M (x 1:`−1 ) := x `:L ∈F (`) M (x 1:`−1 ) :x `+1:L ∈R `+1,M (x 1:` ) , and then define R `,M (x 1:`−1 ) =argmin x `:L ∈A `,M (x 1:`−1 ) Score `,M ( f `,M (x 1:`−1 ,x `:L ) ) . An element x ? = (x ? 1 ,...,x ? L )∈F M is called anuncertainL-level solution(oruncertain Stackelberg solution) of typeMif x ? ∈R 1,M (∅). Equivalently,x ? is obtained by backward induction, where each level`chooses a continuation minimizing its scalarized uncertain objective under the assumption that all lower levels`+ 1,...,Lrespond rationally. Remark 3.11.5(Specializations).IfMis the fuzzy model with Dom(M) = [0,1], then Definition 3.11.4 reduces to a fuzzy multi-level decision-making model. IfMis chosen as an intuitionistic fuzzy, neutrosophic, plithogenic, or other uncertainty model, then one obtains the corresponding uncertainty-aware multi-level decision-making framework by changing the degree-domain Dom(M)and the model-dependent ranking maps Score `,M . Theorem 3.11.6(Well-definedness of UMLDM).Let M= ( L, M,(X ` ) L `=1 , F M ,(f `,M ) L `=1 ,(Score `,M ) L `=1 ) be a UMLDM instance as in Definition 3.11.4. Assume: 1. eachX ` is finite and nonempty; 2.F M ⊆X= ∏ L `=1 X ` is nonempty; Chapter 3. Problem-level frameworks 3. for each`∈1,...,L, the map f `,M :X→Dom(M) is total; 4. for each`∈1,...,L, the map Score `,M :Dom(M)→R is total. Then the following hold. 1. For every`∈1,...,Land every feasible prefixx 1:`−1 arising from some feasible vector inF M , the feasible continuation setF (`) M (x 1:`−1 )is a finite nonempty set. 2. For every`∈ 1,...,Land every such feasible prefixx 1:`−1 , the response setR `,M (x 1:`−1 )is well- defined and nonempty. 3. In particular,R 1,M (∅)6=∅, so the set of uncertainL-level solutions is nonempty. Hence Uncertain Multi-Level Decision-Making of typeMis well-defined. Proof.For each`∈1,...,L, define the set of feasible prefixes of length`−1by P `−1 := x 1:`−1 ∈X 1 ×·×X `−1 :∃x `:L with(x 1:`−1 ,x `:L )∈F M . By convention, P 0 =∅. SinceF M 6=∅, we haveP 0 6=∅. We prove by backward induction on`that for everyx 1:`−1 ∈P `−1 , F (`) M (x 1:`−1 ) is finite and nonempty, and R `,M (x 1:`−1 ) is well-defined and nonempty. Step 1: the last level`=L. Fixx 1:L−1 ∈P L−1 . By definition ofP L−1 , there exists at least onex L ∈X L such that (x 1:L−1 ,x L )∈F M . Hence F (L) M (x 1:L−1 ) =x L ∈X L : (x 1:L−1 ,x L )∈F M is nonempty. SinceX L is finite,F (L) M (x 1:L−1 )is also finite. Chapter 3. Problem-level frameworks Becausef L,M is total, for eachx L ∈F (L) M (x 1:L−1 ), f L,M (x 1:L−1 ,x L )∈Dom(M) is defined. Because Score L,M is total, Score L,M ( f L,M (x 1:L−1 ,x L ) ) ∈R is defined for everyx L ∈F (L) M (x 1:L−1 ). Therefore we are minimizing a real-valued function over a finite nonempty set. Hence the minimum is attained, so R L,M (x 1:L−1 ) =argmin x L ∈F (L) M (x 1:L−1 ) Score L,M ( f L,M (x 1:L−1 ,x L ) ) is well-defined and nonempty. Step 2: induction step. Assume that for some`∈1,...,L−1, the statement has already been proved for level`+ 1; namely, for every feasible prefixx 1:` ∈P ` , F (`+1) M (x 1:` ) is finite and nonempty, and R `+1,M (x 1:` ) is well-defined and nonempty. Now fixx 1:`−1 ∈P `−1 . By definition ofP `−1 , there exists some ̄x `:L ∈X ` ×·×X L such that (x 1:`−1 , ̄x `:L )∈F M . Hence F (`) M (x 1:`−1 ) is nonempty; sinceX ` ×·×X L is finite, it is also finite. Let ̄x ` denote the`-th component of ̄x `:L . Then (x 1:`−1 , ̄x ` , ̄x `+1:L )∈F M , so the prefix (x 1:`−1 , ̄x ` )∈P ` . By the induction hypothesis, R `+1,M (x 1:`−1 , ̄x ` )6=∅. Choose any y `+1:L ∈R `+1,M (x 1:`−1 , ̄x ` ). Chapter 3. Problem-level frameworks Again by the induction hypothesis, y `+1:L ∈F (`+1) M (x 1:`−1 , ̄x ` ), which means (x 1:`−1 , ̄x ` ,y `+1:L )∈F M . Therefore ( ̄x ` ,y `+1:L )∈A `,M (x 1:`−1 ). So the admissible rational-continuation set A `,M (x 1:`−1 ) is nonempty. Since it is a subset of the finite setF (`) M (x 1:`−1 ), it is finite. Now, for every x `:L ∈A `,M (x 1:`−1 ), the point(x 1:`−1 ,x `:L )∈Xis defined, hence by totality off `,M , f `,M (x 1:`−1 ,x `:L )∈Dom(M) is defined; and by totality of Score `,M , Score `,M ( f `,M (x 1:`−1 ,x `:L ) ) ∈R is defined. ThusR `,M (x 1:`−1 )is the argmin of a real-valued function over a finite nonempty set: R `,M (x 1:`−1 ) =argmin x `:L ∈A `,M (x 1:`−1 ) Score `,M ( f `,M (x 1:`−1 ,x `:L ) ) . HenceR `,M (x 1:`−1 )is well-defined and nonempty. This completes the backward induction. Applying the result to`= 1, we obtain that R 1,M (∅)6=∅. Every element ofR 1,M (∅)is an uncertainL-level solution. Therefore the solution set is nonempty, and UMLDM is well-defined. Table 3.11 presents related uncertainty-model variants of Fuzzy Multi-Level Decision-Making. Chapter 3. Problem-level frameworks Table 3.11: Related uncertainty-model variants of Fuzzy Multi-Level Decision-Making. kRelated Fuzzy Multi-Level Decision-Making variant(s) 1 Fuzzy Multi-Level Decision-Making 2 Intuitionistic Fuzzy Multi-Level Decision-Making 3 Hesitant Fuzzy Multi-Level Decision-Making 3 Spherical Fuzzy Multi-Level Decision-Making 3 Neutrosophic Multi-Level Decision-Making nPlithogenic Multi-Level Decision-Making 3.12Fuzzy Multi-Agent Decision-Making Multi-agent decision-making coordinates or analyzes multiple autonomous agents with possibly conflict- ing utilities, seeking joint policies, equilibria, or negotiated agreements under interaction rules [410–412]. Fuzzy multi-agent decision-making represents agents’ utilities, beliefs, or strategies by fuzzy sets, enabling negotiation and coordination under vague preferences and uncertain perceptions. Definition 3.12.1(Fuzzy-number domain).LetFdenote the class of (real) fuzzy numbers, i.e., fuzzy sets ̃aonRwith membership functionμ ̃a :R→[0,1]that are (i) normal, (i) convex, (i) upper semicontinuous, and (iv) have compact support. Forα∈(0,1], theα-cut is the (nonempty) compact interval ̃a α :=x∈R:μ ̃a (x)≥α= [a(α),a(α)]. Define the (Zadeh-extension) sum ̃a⊕ ̃ b∈Fand positive scalar productγ ̃a∈F(γ≥0) byα-cuts: ( ̃a⊕ ̃ b) α = ̃a α + ̃ b α = [a(α) +b(α),a(α) +b(α)],(γ ̃a) α =γ ̃a α . Definition 3.12.2(Fuzzy multi-agent decision-making).Let A=A 1 ,...,A m (alternatives), C=C 1 ,...,C n (criteria), D=1,...,K(agents). AFuzzy Multi-Agent Decision-Making(FMADM) instance is a tuple FMADM= ( A,C,D,( ̃ X (k) ) k∈D ,(w (k) ) k∈D ,λ,Agg,Score,Π ) , where: 1. For each agentk∈D, ̃ X (k) = ( ̃x (k) ij ) m×n ∈F m×n is the fuzzy evaluation matrix, with ̃x (k) ij ∈Fdescribing the (perceived) performance ofA i onC j by agentk. 2. For each agentk∈D, w (k) = (w (k) 1 ,...,w (k) n )∈[0,1] n , n ∑ j=1 w (k) j = 1, is the criterion-weight vector of agentk(crisp weights; fuzzy weights can be handled analogously by replacingw (k) j with ̃w (k) j ∈F). Chapter 3. Problem-level frameworks 3.λ= (λ 1 ,...,λ K )∈[0,1] K is the agent-importance vector with ∑ K k=1 λ k = 1. 4. Agg:F K →Fis an agent-aggregation operator. A canonical choice is the weighted fuzzy mean Agg( ̃a 1 ,..., ̃a K ) := K ⊕ k=1 λ k ̃a k . 5. Score:F→Ris a scoring (defuzzification) functional. A standard example is the centroid score Score( ̃a) = ∫ R xμ ̃a (x)dx ∫ R μ ̃a (x)dx ,assuming ∫ R μ ̃a (x)dx >0. 6.Πis a decision rule producing either a rankingonAor a choice setS ⊆A. Thecollectivefuzzy evaluation matrix is defined entrywise by ̃x ij :=Agg ( ̃x (1) ij ,..., ̃x (K) ij ) ∈F, ̃ X:= ( ̃x ij ) m×n ∈F m×n . A common induced (collective) fuzzy utility ofA i is ̃u i := n ⊕ j=1 w j ̃x ij ∈F,with somew∈[0,1] n , n ∑ j=1 w j = 1(e.g.,w= K ∑ k=1 λ k w (k) ). ThenΠmay return the score-based preorder A i A ` ⇐⇒Score( ̃u i )≤Score( ̃u ` ), or the maximal (choice) setS=A i ∈A:Score( ̃u i ) =max r Score( ̃u r ). Theorem 3.12.3(Well-definedness of the canonical FMADM aggregation).AssumeFis as in Defini- tion 3.12.1. LetAgg( ̃a 1 ,..., ̃a K ) = ⊕ K k=1 λ k ̃a k withλ k ≥0and ∑ K k=1 λ k = 1. Then, for every ( ̃a 1 ,..., ̃a K )∈F K , one hasAgg( ̃a 1 ,..., ̃a K )∈F. Consequently, the collective matrix ̃ Xin Definition 3.12.2 is well-defined inF m×n . Proof.Fixα∈(0,1]. Each ̃a α k is a compact interval inR. By theα-cut definitions, ( K ⊕ k=1 λ k ̃a k ) α = K ∑ k=1 λ k ̃a α k , which is a (nonempty) compact interval since Minkowski sums and nonnegative scalar multiples preserve compact intervals. Standard closure properties for fuzzy numbers imply that the resulting fuzzy set is again normal, convex, upper semicontinuous, and has compact support; hence it lies inF. Applying this entrywise yields ̃ X∈F m×n . We now defineUncertain Multi-Agent Decision-Making(UMADM) by extending fuzzy multi-agent decision- making from fuzzy-number-valued evaluations to a general uncertain modelMwith degree-domain Dom(M)⊆ [0,1] k . Chapter 3. Problem-level frameworks Definition 3.12.4(Uncertain Multi-Agent Decision-Making (UMADM) of typeM).Let A=A 1 ,...,A m (alternatives),C=C 1 ,...,C n (criteria),D=1,...,K(agents), wherem,n,K∈Nandm,n,K≥1. Fix an uncertain modelMwith degree-domain Dom(M)⊆[0,1] k for some integerk≥1. AnUncertain Multi-Agent Decision-Making instance of typeMis a tuple UMADM M = ( A,C,D,(X (k) M ) k∈D ,(w (k) M ) k∈D ,λ,Agg M ,Util M ,Score M ) , where: 1. For each agentk∈D, X (k) M = (μ (k) ij ) m×n ∈Dom(M) m×n is theuncertain evaluation matrixof agentk, whereμ (k) ij ∈Dom(M)denotes the uncertain assessment of alternativeA i under criterionC j given by agentk. 2. For each agentk∈D, w (k) M = (ω (k) 1 ,...,ω (k) n )∈Dom(M) n is theuncertain criterion-importance profileof agentk. (If desired, one may instead use crisp weights w (k) ∈[0,1] n , but here we keep the general uncertain form.) 3. λ= (λ 1 ,...,λ K )∈[0,1] K , K ∑ k=1 λ k = 1, is theagent-importance vector. 4. Agg M :Dom(M) K −→Dom(M) is amodel-dependent agent-aggregation operator, used to combine theKagent-specific assessments of the same entry into a collective uncertain assessment. 5. Util M :Dom(M) n ×Dom(M) n −→Dom(M) is amodel-dependent utility aggregation operator, used to combine the collective criterion-wise assess- ments of an alternative with a collective criterion-weight profile. 6. Score M :Dom(M)−→R is aranking functional(score / scalarization map). Chapter 3. Problem-level frameworks Thecollective uncertain evaluation matrixis defined entrywise by μ ij :=Agg M ( μ (1) ij ,...,μ (K) ij ) ∈Dom(M), X M := (μ ij ) m×n ∈Dom(M) m×n . Similarly, thecollective uncertain criterion-weight profileis defined componentwise by ω j :=Agg M ( ω (1) j ,...,ω (K) j ) ∈Dom(M), w M := (ω 1 ,...,ω n )∈Dom(M) n . For each alternativeA i ∈A, define itscollective uncertain utilityby u i :=Util M ( (μ i1 ,...,μ in ),(ω 1 ,...,ω n ) ) ∈Dom(M). The induced score ofA i is s i :=Score M (u i )∈R. Thecollective preference relation M onAis defined by A i M A j ⇐⇒s i ≥s j . AsolutionofUMADM M is any alternative A ? ∈arg max A i ∈A s i . The corresponding set of optimal alternatives is denoted by A ? M :=arg max A i ∈A s i . Remark 3.12.5(Interpretation).Definition 3.12.4 separates the multi-agent decision process into three layers: 1. agent-wise uncertain assessmentsX (k) M and weight profilesw (k) M ; 2. inter-agent aggregation via Agg M ; 3. alternative-wise utility construction via Util M , followed by ranking through Score M . Thus UMADM is a model-independent framework: onceM, Agg M , Util M , and Score M are specified, one obtains a concrete uncertainty-aware multi-agent decision method. Remark 3.12.6(Specializations).IfMis the fuzzy model with Dom(M) = [0,1], then Definition 3.12.4 reduces to a fuzzy multi-agent decision-making framework. IfMis chosen as an intuitionistic fuzzy, neu- trosophic, hesitant fuzzy, spherical fuzzy, plithogenic, or other uncertainty model, then one obtains the corresponding uncertainty-aware multi-agent decision-making setting by replacing Dom(M), Agg M , Util M , and Score M accordingly. Chapter 3. Problem-level frameworks Theorem 3.12.7(Well-definedness of UMADM of typeM).Let UMADM M = ( A,C,D,(X (k) M ) k∈D ,(w (k) M ) k∈D ,λ,Agg M ,Util M ,Score M ) be a UMADM instance as in Definition 3.12.4. Assume: (A1)A,C, andDare finite nonempty sets; (A2) for eachk∈D, X (k) M = (μ (k) ij )∈Dom(M) m×n andw (k) M = (ω (k) 1 ,...,ω (k) n )∈Dom(M) n ; (A3) Agg M :Dom(M) K →Dom(M) is a total map; (A4) Util M :Dom(M) n ×Dom(M) n →Dom(M) is a total map; (A5) Score M :Dom(M)→R is a total map. Then the following objects are well-defined: (i) the collective uncertain evaluation matrix X M = (μ ij ) m×n ∈Dom(M) m×n ; (i) the collective uncertain criterion-weight profile w M = (ω 1 ,...,ω n )∈Dom(M) n ; (i) the collective uncertain utilities u i ∈Dom(M) (i= 1,...,m); (iv) the real-valued scores s i =Score M (u i )∈R(i= 1,...,m); (v) the preference relation M onA, defined by A i M A j ⇐⇒s i ≥s j , which is a total preorder; Chapter 3. Problem-level frameworks (vi) the optimal set A ? M =arg max A i ∈A s i , which is nonempty. Hence Uncertain Multi-Agent Decision-Making of typeMis well-defined. Proof.By (A1), the setsA=A 1 ,...,A m ,C=C 1 ,...,C n , andD=1,...,Kare finite and nonempty. Step 1: Well-definedness of the collective matrixX M . Fixi∈1,...,mandj∈1,...,n. By (A2), for everyk∈D, μ (k) ij ∈Dom(M). Hence theK-tuple (μ (1) ij ,...,μ (K) ij )∈Dom(M) K . Since Agg M is total by (A3), the value μ ij :=Agg M (μ (1) ij ,...,μ (K) ij ) is defined and belongs to Dom(M). Because this holds for every pair(i,j), the matrix X M = (μ ij ) m×n belongs to Dom(M) m×n . Thus (i) is proved. Step 2: Well-definedness of the collective weight profilew M . Fixj∈1,...,n. By (A2), for everyk∈D, ω (k) j ∈Dom(M). Hence (ω (1) j ,...,ω (K) j )∈Dom(M) K . Again by totality of Agg M , ω j :=Agg M (ω (1) j ,...,ω (K) j ) is defined and belongs to Dom(M). Therefore w M = (ω 1 ,...,ω n )∈Dom(M) n . So (i) holds. Step 3: Well-definedness of the collective uncertain utilitiesu i . Chapter 3. Problem-level frameworks Fixi∈1,...,m. From Step 1, (μ i1 ,...,μ in )∈Dom(M) n , and from Step 2, (ω 1 ,...,ω n )∈Dom(M) n . Hence ( (μ i1 ,...,μ in ),(ω 1 ,...,ω n ) ) ∈Dom(M) n ×Dom(M) n . By totality of Util M in (A4), the value u i =Util M ( (μ i1 ,...,μ in ),(ω 1 ,...,ω n ) ) is defined and belongs to Dom(M). Thus (i) holds. Step 4: Well-definedness of the scoress i . By Step 3,u i ∈Dom(M)for eachi. Since Score M is total by (A5), s i :=Score M (u i )∈R is well-defined for everyi= 1,...,m. Hence (iv) holds. Step 5: The induced preference relation is a total preorder. Define A i M A j ⇐⇒s i ≥s j . Since≥onRis reflexive, transitive, and total, the induced relation M onAis also reflexive, transitive, and total. Therefore M is a total preorder onA. This proves (v). Step 6: Existence of an optimal alternative. The set of scores s 1 ,...,s m ⊂R is finite and nonempty becauseAis finite and nonempty. Every finite nonempty subset ofRhas a maximum. Hence there exists at least one indexi ? ∈1,...,msuch that s i ? =max 1≤i≤m s i . Therefore A ? M =arg max A i ∈A s i 6=∅. So (vi) holds. All required objects are thus well-defined, and the optimal set is nonempty. Hence UMADM of typeMis well-defined. Table 3.12 presents related uncertainty-model variants of Fuzzy Multi-Agent Decision-Making. Chapter 3. Problem-level frameworks Table 3.12: Related uncertainty-model variants of Fuzzy Multi-Agent Decision-Making. kRelated Fuzzy Multi-Agent Decision-Making variant(s) 1 Fuzzy Multi-Agent Decision-Making 2 Intuitionistic Fuzzy Multi-Agent Decision-Making 3 Hesitant Fuzzy Multi-Agent Decision-Making 3 Spherical Fuzzy Multi-Agent Decision-Making 3 Neutrosophic Multi-Agent Decision-Making nPlithogenic Multi-Agent Decision-Making 3.13Fuzzy Multi-Scenario Decision-Making Multi-scenario decision-making evaluates alternatives under several scenarios, aggregates scenario outcomes using probabilities or weights, and selects robust or expected-optimal alternatives [413,414]. Fuzzy multi- scenario decision-making treats scenario performances or scenario weights as fuzzy, aggregating with fuzzy expectation, worst-case, or regret criteria to select robustly. Definition 3.13.1(Fuzzy multi-scenario decision model).Afuzzy multi-scenario decision-making problem is a tuple D= ( X, K,(K j ) m j=1 ,(Y jt ) j,t ,Σ, μ, w, π,Agg C ,Agg S , ) , where: 1.X=x 1 ,...,x n is the finite set of alternatives. 2.K=K 1 ,...,K m is the finite set of (top-level) criteria. 3. For eachj∈ 1,...,m,K j =k j1 ,...,k jT j is the finite set of indicators (subcriteria) defining criterionK j . 4. For each indicatork jt ,Y jt is its value domain (numerical, ordinal, linguistic, etc.). 5.Σ =σ 1 ,...,σ s is a finite set ofscenarios(distinct requirement profiles / evaluation policies). 6.μis a family of scenario-dependent membership mappings (defined precisely in Definition 3.13.2). 7.w= (w 1 ,...,w m )∈[0,1] m is a criterion-weight vector with ∑ m j=1 w j = 1. 8.π= (π 1 ,...,π s )∈[0,1] s is a scenario-weight vector with ∑ s r=1 π r = 1. 9. Agg C is a criterion-aggregation operator (within a scenario). 10. Agg S is a scenario-aggregation operator (across scenarios). 11.is a total preorder on the chosen score space (e.g., on[0,1]or on a class of fuzzy numbers) used for ranking. Definition 3.13.2(Scenario-dependent membership structure).Fixx∈X,j∈1,...,m, and a scenario σ∈Σ. Let the indicator observations forxbe y jt (x)∈Y jt (t= 1,...,T j ). Ascenario-dependent indicator membershipis a mapping μ σ jt :Y jt −→[0,1], y7−→μ σ jt (y), Chapter 3. Problem-level frameworks and the inducedscenario-dependent criterion satisfactionis defined by an indicator-aggregatorΦ σ j as μ σ j (x) := Φ σ j ( μ σ j1 (y j1 (x)),...,μ σ jT j (y jT j (x)) ) ∈[0,1]. Typical choices include: Φ σ j =min (conjunctive / “all indicators must be good”), Φ σ j =max (disjunctive / “any indicator may suffice”), or a weighted mean on[0,1], or a t-norm/t-conorm composition. Scenario semantics (e.g.,obligatory, desirable,not required) are encoded by howμ σ jt andΦ σ j are chosen. Definition 3.13.3(Within-scenario overall score).Assume the score space is[0,1]. For each scenario σ∈Σ, define thewithin-scenario overall score U σ (x) :=Agg C ( (μ σ j (x)) m j=1 ;w ) ∈[0,1]. A canonical (compensatory) choice is the convex combination U σ (x) = m ∑ j=1 w j μ σ j (x). A canonical non-compensatory alternative is the weighted minimum U σ (x) =min 1≤j≤m ( μ σ j (x) ) λ j , λ j >0, or any monotone aggregation on[0,1] m compatible with the intended decision logic. Definition 3.13.4(Across-scenario aggregation and ranking).Define themulti-scenario overall scoreof x∈Xby U(x) :=Agg S ( (U σ r (x)) s r=1 ;π ) . A standard choice is again the convex combination U(x) = s ∑ r=1 π r U σ r (x)∈[0,1]. The induced ranking is x a x b ⇐⇒U(x a )≤U(x b ), and the set of optimal alternatives is arg max x∈X U(x) =x∈X:U(x)≥U(z)∀z∈X. Proposition 3.13.5(Well-definedness).Assume: 1. For everyσ∈Σ,j, andt,μ σ jt :Y jt →[0,1]is well-defined. 2. For everyσandj,Φ σ j : [0,1] T j →[0,1]is well-defined. 3.w∈[0,1] m and ∑ m j=1 w j = 1, andπ∈[0,1] s and ∑ s r=1 π r = 1. 4.Agg C : [0,1] m ×∆ m →[0,1]andAgg S : [0,1] s ×∆ s →[0,1](where∆ m ,∆ s are simplices of weights) are aggregation operators that return values in[0,1]. Chapter 3. Problem-level frameworks Then for everyx∈X, the quantitiesμ σ j (x),U σ (x), andU(x)are well-defined and belong to[0,1]. Hence, Definition 3.13.4 yields a well-defined ranking. Proof.Fixx∈X. By (1), eachμ σ jt (y jt (x))∈[0,1]is defined. By (2),μ σ j (x) = Φ σ j (·)∈[0,1]is defined for eachj. Thus(μ σ j (x)) m j=1 ∈[0,1] m , and by (4) we obtainU σ (x) =Agg C ((μ σ j (x)) j ;w)∈[0,1]. Likewise, (U σ r (x)) s r=1 ∈[0,1] s , and by (4) againU(x) =Agg S ((U σ r (x)) r ;π)∈[0,1]. ThereforeU:X→[0,1]is well-defined, and the preorder induced by≤on[0,1]defines a well-defined ranking onX. We now defineUncertain Multi-Scenario Decision-Making(UMScDM) by extending fuzzy multi-scenario decision-making from fuzzy assessments to a general uncertain modelMwith degree-domain Dom(M)⊆ [0,1] k . Definition 3.13.6(Uncertain Multi-Scenario Decision-Making (UMScDM) of typeM).Let A=A 1 ,...,A m (alternatives),C=C 1 ,...,C n (criteria),Σ =σ 1 ,...,σ s (scenarios), wherem,n,s∈Nandm,n,s≥1. Fix an uncertain modelMwith degree-domain Dom(M)⊆[0,1] k for some integerk≥1. AnUncertain Multi-Scenario Decision-Making instance of typeMis a tuple UMScDM M = ( A,C,Σ,(X (r) M ) s r=1 ,w M ,π,Agg C,M ,Agg S,M ,Score M ) , where: 1. For each scenarior∈1,...,s, X (r) M = (μ (r) ij ) m×n ∈Dom(M) m×n is thescenario-dependent uncertain evaluation matrix, whereμ (r) ij ∈Dom(M)denotes the uncertain assessment of alternativeA i under criterionC j in scenarioσ r . 2. w M = (ω 1 ,...,ω n )∈Dom(M) n is theuncertain criterion-importance profile. 3. π= (π 1 ,...,π s )∈[0,1] s , s ∑ r=1 π r = 1, is thescenario-weight vector. Chapter 3. Problem-level frameworks 4. Agg C,M :Dom(M) n ×Dom(M) n −→Dom(M) is awithin-scenario criterion-aggregation operator. For each scenariorand each alternativeA i , it produces an uncertain scenario-specific utility u (r) i :=Agg C,M ( (μ (r) i1 ,...,μ (r) in ),(ω 1 ,...,ω n ) ) ∈Dom(M). 5. Agg S,M :Dom(M) s ×[0,1] s −→Dom(M) is anacross-scenario aggregation operator. It combines the scenario-specific utilities of an alternative into its overall uncertain utility u i :=Agg S,M ( (u (1) i ,...,u (s) i ),π ) ∈Dom(M). 6. Score M :Dom(M)−→R is aranking functional(score / scalarization map), and the corresponding real-valued score ofA i is s i :=Score M (u i )∈R. The induced preference relation M onAis defined by A i M A j ⇐⇒s i ≥s j . AsolutionofUMScDM M is any alternative A ? ∈arg max A i ∈A s i , and the set of all optimal alternatives is denoted by A ? M :=arg max A i ∈A s i . Remark 3.13.7(Interpretation).Definition 3.13.6 separates the decision process into two aggregation layers: 1.within-scenario aggregation, where criterion-wise uncertain assessments are combined into a scenario- specific uncertain utilityu (r) i ; 2.across-scenario aggregation, where the family(u (1) i ,...,u (s) i )is combined into the overall uncertain utilityu i . Thus UMScDM models decisions in which the performance of each alternative depends on several possible scenarios, while all evaluations remain encoded in a general uncertain modelM. Remark 3.13.8(Specializations).IfMis the fuzzy model with Dom(M) = [0,1], then Definition 3.13.6 reduces to a fuzzy multi-scenario decision-making framework. IfMis chosen as an intuitionistic fuzzy, neutrosophic, hesitant fuzzy, spherical fuzzy, plithogenic, or another uncertainty model, then one obtains the corresponding uncertainty-aware multi-scenario decision-making framework by replacing Dom(M), Agg C,M , Agg S,M , and Score M accordingly. Chapter 3. Problem-level frameworks Theorem 3.13.9(Well-definedness of UMScDM of typeM).Let UMScDM M = ( A,C,Σ,(X (r) M ) s r=1 ,w M ,π,Agg C,M ,Agg S,M ,Score M ) be a UMScDM instance as in Definition 3.13.6. Assume: (A1)A,C, andΣare finite nonempty sets; (A2) for everyr∈1,...,s, X (r) M = (μ (r) ij )∈Dom(M) m×n ; (A3) w M = (ω 1 ,...,ω n )∈Dom(M) n ; (A4) π= (π 1 ,...,π s )∈[0,1] s with s ∑ r=1 π r = 1; (A5) Agg C,M :Dom(M) n ×Dom(M) n →Dom(M) is a total map; (A6) Agg S,M :Dom(M) s ×[0,1] s →Dom(M) is a total map; (A7) Score M :Dom(M)→R is a total map. Then the following objects are well-defined: (i) for every alternativeA i and every scenarioσ r , the scenario-specific uncertain utility u (r) i =Agg C,M ( (μ (r) i1 ,...,μ (r) in ),(ω 1 ,...,ω n ) ) belongs toDom(M); (i) for every alternativeA i , the overall uncertain utility u i =Agg S,M ( (u (1) i ,...,u (s) i ),π ) belongs toDom(M); (i) for every alternativeA i , the real-valued score s i =Score M (u i ) is well-defined; Chapter 3. Problem-level frameworks (iv) the induced preference relation M onA, defined by A i M A j ⇐⇒s i ≥s j , is a total preorder; (v) the optimal set A ? M =arg max A i ∈A s i is nonempty. Hence Uncertain Multi-Scenario Decision-Making of typeMis well-defined. Proof.By (A1), the sets A=A 1 ,...,A m ,C=C 1 ,...,C n ,Σ =σ 1 ,...,σ s are finite and nonempty. Step 1: Well-definedness of the scenario-specific uncertain utilitiesu (r) i . Fixi∈1,...,mandr∈1,...,s. By (A2), thei-th row of the scenario-dependent matrixX (r) M is (μ (r) i1 ,...,μ (r) in )∈Dom(M) n . By (A3), (ω 1 ,...,ω n )∈Dom(M) n . Hence ( (μ (r) i1 ,...,μ (r) in ),(ω 1 ,...,ω n ) ) ∈Dom(M) n ×Dom(M) n . Since Agg C,M is total by (A5), the value u (r) i =Agg C,M ( (μ (r) i1 ,...,μ (r) in ),(ω 1 ,...,ω n ) ) is defined and belongs to Dom(M). This proves (i). Step 2: Well-definedness of the overall uncertain utilitiesu i . Fixi∈1,...,m. By Step 1, for eachr= 1,...,s, u (r) i ∈Dom(M). Therefore (u (1) i ,...,u (s) i )∈Dom(M) s . By (A4), π= (π 1 ,...,π s )∈[0,1] s . Chapter 3. Problem-level frameworks Hence ( (u (1) i ,...,u (s) i ),π ) ∈Dom(M) s ×[0,1] s . Since Agg S,M is total by (A6), the value u i =Agg S,M ( (u (1) i ,...,u (s) i ),π ) is defined and belongs to Dom(M). Thus (i) holds. Step 3: Well-definedness of the scoress i . By Step 2,u i ∈Dom(M)for eachi. Since Score M is total by (A7), s i :=Score M (u i )∈R is well-defined for eachi= 1,...,m. Hence (i) holds. Step 4: The induced preference relation is a total preorder. Define A i M A j ⇐⇒s i ≥s j . Since≥onRis reflexive, transitive, and total, the induced relation M onAis also reflexive, transitive, and total. Therefore M is a total preorder onA. This proves (iv). Step 5: Existence of an optimal alternative. The set of scores s 1 ,...,s m ⊂R is finite and nonempty becauseAis finite and nonempty. Every finite nonempty subset ofRhas a maximum. Hence there exists at least one indexi ? ∈1,...,msuch that s i ? =max 1≤i≤m s i . Therefore A ? M =arg max A i ∈A s i 6=∅. Thus (v) holds. All required objects are therefore defined unambiguously, and the solution set is nonempty. Hence UMScDM of typeMis well-defined. Table 3.13 presents related uncertainty-model variants of Fuzzy Multi-Scenario Decision-Making. Chapter 3. Problem-level frameworks Table 3.13: Related uncertainty-model variants of Fuzzy Multi-Scenario Decision-Making. kRelated Fuzzy Multi-Scenario Decision-Making variant(s) 1 Fuzzy Multi-Scenario Decision-Making 2 Intuitionistic Fuzzy Multi-Scenario Decision-Making 3 Hesitant Fuzzy Multi-Scenario Decision-Making 3 Spherical Fuzzy Multi-Scenario Decision-Making 3 Neutrosophic Multi-Scenario Decision-Making nPlithogenic Multi-Scenario Decision-Making Chapter 3. Problem-level frameworks Chapter 4 Weight elicitation Decision-Methods Weight elicitation estimates criterion importance in decision-making by surveys, pairwise comparisons, trade-off queries, entropy/optimization, or learning, producing normalized weights for aggregation and rank- ing. For reference, a comparison table is presented in Table 4.1. Table 4.1: A concise comparison of fuzzy weight-elicitation methods. MethodTypeTypical inputMain idea / strength Main limitation / note Fuzzy AHPSubjective Fuzzy pairwise com- parison matrix Classical hierarchical weighting with consis- tency checking; intuitive, transparent, and widely used in practice. Requires many pairwise comparisons when the number of criteria be- comes large. Fuzzy LOP- COW ObjectiveFuzzy decision ma- trix Data-driven weighting based on criterion vari- ability or discrimination power. Does not directly in- corporate the decision- maker’s subjective pref- erences. F-SIWECSubjective Expert importance assessments A simple expert-oriented weighting procedure with relatively low elicitation burden. The precise mathemat- ical formulation should be stated explicitly, since variants may differ across studies. Fuzzy judg- ment matrix Founda- tional tool Fuzzy pairwise judg- ments A basic structure for rep- resenting fuzzy preference information used by many weighting approaches. Not a standalone weight- ing method unless cou- pled with a specific derivation rule. Fuzzy ANPSubjective Fuzzy pairwise com- parisons on a net- work Extends AHP to settings with interdependent crite- ria and feedback relations. Model construction and computation are gener- ally more involved than in AHP. Continued on the next page. 93 Chapter 4. Weight elicitation Decision-Methods Table 4.1 (continued). MethodTypeTypical inputMain idea / strength Main limitation / note Fuzzy OPASubjective Ordinal rankings or priority orders Requires less information than full pairwise com- parison and is efficient in elicitation. Uses less detailed prefer- ence information than cardinal comparison- based methods. Fuzzy PI- PRECIA Subjective Sequential relative importance judg- ments Stepwise weighting method with lower burden than full pairwise compar- ison. Final weights may de- pend on the ordering of criteria. Fuzzy SWARA Subjective Ordered criteria and comparative impor- tance coefficients Expert-friendly and easy- to-implement stepwise weighting approach. Sensitive to criterion or- dering and the reliability of expert judgments. Fuzzy CILOS ObjectiveFuzzy decision ma- trix Determines weights through the criterion impact-loss concept with- out requiring direct pref- erence input. Although preference-free, it may be less intuitive for practitioners. Fuzzy IDOCRIW Integrated objective Fuzzy decision ma- trix Integrates multiple objec- tive weighting ideas, often combining CILOS-type and entropy-type informa- tion. Computationally more complex than a sin- gle objective weighting scheme. Fuzzy BWMSubjective Best-to-others and others-to-worst com- parisons Requires fewer compar- isons than AHP and pro- vides a consistency-based framework. Requires clear and reli- able identification of the best and worst criteria. Fuzzy CRITIC ObjectiveFuzzy decision ma- trix Uses both contrast inten- sity and inter-criterion conflict to derive weights. Does not reflect direct expert preferences unless combined with a subjec- tive approach. Fuzzy MEREC ObjectiveFuzzy decision ma- trix Evaluates the effect of removing each criterion on the overall performance structure. Can be sensitive to the selected normalization procedure and per- formance aggregation model. Fuzzy FU- COM Subjective Ranked criteria and comparative priori- ties Achieves weighting with very few comparisons while emphasizing con- sistency. Requires a reliable prior ranking of criteria before weight derivation. 4.1 Fuzzy Analytic hierarchy process (Fuzzy AHP) AHP structures decisions into a hierarchy, uses pairwise comparisons to derive weights, and synthesizes global priorities for ranking [415–417]. Fuzzy AHP replaces crisp comparisons with fuzzy numbers, computes fuzzy weights and defuzzified priorities to rank alternatives under uncertainty [418–420]. Chapter 4. Weight elicitation Decision-Methods Definition 4.1.1(TFNs and FAHP).(cf. [421,422])(0) Positive TFN, arithmetic, and defuzzifica- tion.Apositive triangular fuzzy number(TFN) is ̃x= (l,m,u)∈R 3 >0 with0< l≤m≤u. For positive TFNs ̃x= (l x ,m x ,u x )and ̃y= (l y ,m y ,u y ), define componentwise ̃x⊕ ̃y:= (l x +l y , m x +m y , u x +u y ), ̃x⊗ ̃y:= (l x l y , m x m y , u x u y ), ̃x α := (l α x , m α x , u α x ) (α >0), ̃x −1 := (1/u x ,1/m x ,1/l x ), ̃x ̃y:= ̃x⊗ ̃y −1 . A standard crisp representative (centroid/COA) is COA( ̃x) := l+m+u 3 . (1) Decision hierarchy.Adecision hierarchyis a rooted treeH= (V,root,ch), where each internal node p∈Vhas children ch(p) =e 1 ,...,e n representing criteria/subcriteria, and leaves represent alternatives. (2) Fuzzy pairwise comparisons at a node.Fix an internal nodepwith children ch(p) =e 1 ,...,e n . Afuzzy reciprocal pairwise comparison matrixatpis ̃ A (p) = ( ̃a (p) ij )∈(TFN >0 ) n×n , ̃a (p) i = (1,1,1), ̃a (p) ji = ( ̃a (p) ij ) −1 . (3) Local weights by fuzzy geometric mean.For each rowi, define the fuzzy geometric mean ̃g (p) i := n ⊗ j=1 ̃a (p) ij 1/n , and the normalized local fuzzy weight ̃w (p) i := ̃g (p) i ( n ⊕ k=1 ̃g (p) k ) . Optionally obtain crisp local weights byw (p) i :=COA( ̃w (p) i )and normalize(w (p) 1 ,...,w (p) n )to sum to1. (4) Hierarchical synthesis (global priorities).SetW(root) = 1and propagate priorities down the tree: W(e i ) :=W(p)w (p) i for each edgep→e i . For a leaf (alternative)a,W(a)is its final priority and alternatives are ranked by decreasingW(a). (A fully fuzzy-end variant replaces products by⊗and defuzzifies only at the end.) Remark 4.1.2(Reduction to classical AHP).If ̃a (p) ij = (a (p) ij ,a (p) ij ,a (p) ij )for alli,j,p, then the above reduces to the corresponding crisp AHP geometric-mean weighting and synthesis. By extending AHP using Uncertain Sets, we obtain the following formulation. Chapter 4. Weight elicitation Decision-Methods Definition 4.1.3(Uncertain AHP (UAHP) of typeM).LetH= (V,root,ch)be a finite rooted tree (decision hierarchy), where each internal nodep∈Vhas a finite nonempty set of children ch(p) =e 1 ,...,e n p , representing criteria/subcriteria, and the leavesAlt⊆Vrepresent alternatives. Fix anuncertainty model Mwith degree-domain Dom(M)⊆[0,1] d (d≥1). (1) Local uncertain pairwise comparisons.For each internal nodep∈V, letn p :=|ch(p)|and assume a localuncertain pairwise comparison matrix A (p) = ( a (p) ij ) ∈Dom(M) n p ×n p , subject to the basic AHP normalization constraints in Dom(M): a (p) i =1 M (i= 1,...,n p ), a (p) ji =Inv M ( a (p) ij ) (i6=j), where1 M ∈Dom(M)is the model’s “unit-comparison” element and Inv M :Dom(M)→Dom(M)is a (total) reciprocal/inversion operator in Dom(M). (2) Local priority extraction.Fix a (total) priority-extraction operator Pri M :Dom(M) n×n −→∆ n ,∆ n := x∈R n ≥0 : n ∑ i=1 x i = 1 , which, given a local matrixA (p) , returns acrisplocal priority vector w (p) =Pri M ( A (p) ) ∈∆ n p . (Thusw (p) i is the relative importance of childe i ∈ch(p)w.r.t. nodep.) (3) Hierarchical synthesis (global priorities).Define global prioritiesW:V→[0,1]recursively by W(root) = 1, W(e i ) =W(p)w (p) i for each edgep→e i . For each alternativea∈Alt, its final priority isW(a), and alternatives are ranked by decreasingW(a). (4) UAHP output as an uncertain set over alternatives.LetAlt=A 1 ,...,A m be the leaf set. Define a membership map μ U :Alt→[0,1], μ U (A i ) :=W(A i ). ThenU:= (Alt,μ U )is called theUAHP priority uncertain setinduced by the hierarchy and uncertain comparisons. Theorem 4.1.4(Uncertain-set structure and well-definedness of UAHP).Consider a UAHP instance of typeMas in Definition 4.1.3. Assume: (A1)The hierarchyHis a finite rooted tree and every internal node has finitely many children. Chapter 4. Weight elicitation Decision-Methods (A2)Dom(M)⊆[0,1] d is nonempty,1 M ∈Dom(M)is fixed, andInv M :Dom(M)→Dom(M)is total. (A3)For each internal nodep, the matrixA (p) ∈Dom(M) n p ×n p is fully specified and satisfiesa (p) i =1 M anda (p) ji =Inv M (a (p) ij ). (A4)Pri M is total and satisfiesPri M (A)∈∆ n for everyA∈Dom(M) n×n . Then: (i)The global priority mapW:V→[0,1]is well-defined and satisfies ∑ a∈Alt W(a) = 1. (i)The induced mappingμ U :Alt→[0,1]is well-defined; henceU= (Alt,μ U )is a well-defined uncertain set onAlt. (i)The UAHP solution set arg max A∈Alt μ U (A) is nonempty; therefore at least one optimal alternative exists. Proof.(i) Well-definedness of local weights.Fix an internal nodepwithn p children. By (A3),A (p) is a well-defined element of Dom(M) n p ×n p . By (A4), applying the total map Pri M yields a uniquely determined vector w (p) =Pri M (A (p) )∈∆ n p . In particular,w (p) i ≥0and ∑ n p i=1 w (p) i = 1. (i) Well-definedness ofWand normalization on leaves.DefineW(root) = 1. For any nodev6=root, there is a unique directed path root=v 0 →v 1 → · →v ` =vin a rooted tree. Define recursively W(v t ) =W(v t−1 )w (v t−1 ) ι t , wherev t is theι t -th child ofv t−1 . This gives a unique real valueW(v)≥0, hence Wis well-defined on all nodes. To show ∑ a∈Alt W(a) = 1, proceed by induction on the tree. For each internal nodep, the total weight assigned to its children is ∑ e∈ch(p) W(e) = n p ∑ i=1 W(p)w (p) i =W(p) n p ∑ i=1 w (p) i =W(p). Thus, the massW(p)is conserved when propagated frompto its children. Starting fromW(root) = 1, iterating this conservation down to the leaves implies the sum of leaf weights equals1. (i) Uncertain-set output and existence of an optimum.By (i), eachW(a)∈[0,1]is well-defined for every alternativea∈Alt, soμ U (a) =W(a)defines a well-defined mappingμ U :Alt→[0,1]and hence U= (Alt,μ U )is an uncertain set. BecauseAltis finite and nonempty, the real-valued functionμ U attains its maximum onAlt, so arg max A∈Alt μ U (A)6= ∅. Chapter 4. Weight elicitation Decision-Methods Table 4.2: Related uncertainty-model variants of AHP (classified by the degree-domain dimensionk). kRelated AHP variant(s) 1 Fuzzy AHP [423,424] 2 Intuitionistic Fuzzy AHP [425,426] 2 Pythagorean Fuzzy AHP [427,428] 3 Hesitant Fuzzy AHP [429,430] 3 Spherical Fuzzy AHP [431,432] 3 Neutrosophic AHP [433,434] nPlithogenic AHP [435,436] Related uncertainty-model variants of AHP, classified by the degree-domain dimensionk, are listed in Table 4.2. In addition to Uncertain AHP, related concepts such as Rough AHP [437,438], Soft AHP [439,440], Modified analytic hierarchy process [441,442], TOPSIS-AHP [443,444], Interval AHP [445,446], Grey AHP [447,448], AHP–BOCR [449, 450], Linking Pin AHP [451], AHPSort [452, 453], Stochastic AHP [454, 455], AHP- K [456, 457], Incomplete AHP [458, 459], Monte Carlo analytic hierarchical process (MCAHP) [460, 461], Entropy-weight AHP [462,463], and Linguistic AHP [464,465] have also been studied. 4.2 Fuzzy LOPCOW (Fuzzy Linear Optimization for Comprehensive Weight) LOPCOW computes objective criterion weights from normalized data using logarithmic percentage change of dispersion, emphasizing informative criteria [466,467]. Fuzzy LOPCOW defuzzifies fuzzy ratings, normalizes performances, computes RMS-to-variance log coefficients, and yields objective weights under uncertainty [468,469]. Definition 4.2.1(Fuzzy LOPCOW objective weighting (score-based, well-defined form)).[468, 469] Let A=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. LetFN(R)be a chosen class of fuzzy numbers (e.g., TFNs). Assume a fuzzy decision matrix ̃ X= ( ̃x ij )∈FN(R) m×n . LetC=C + ̇ ∪C − be a partition into benefit (↑) and cost (↓) criteria. (0) Score/defuzzification map.Fix a map (ranking/defuzzification) S:FN(R)→R, and define the induced crisp matrixX= (x ij )∈R m×n by x ij :=S( ̃x ij ). (For TFNs ̃x= (l,m,u), a common choice isS( ̃x) = (l+m+u)/3.) (1) Linear normalization.For each criterionj, set x max j :=max 1≤i≤m x ij , x min j :=min 1≤i≤m x ij . Chapter 4. Weight elicitation Decision-Methods Define the normalized matrixR= (r ij )∈[0,1] m×n by r ij := x max j −x ij x max j −x min j , C j ∈C − (cost), x ij −x min j x max j −x min j , C j ∈C + (benefit), with the convention that ifx max j =x min j thenr ij := 0for alli(zero-variation criterion). (2) Dispersion statistics per criterion.For eachj, define the root-mean-square (RMS) and the standard deviation: RMS j := √ √ √ √ 1 m m ∑ i=1 r 2 ij , ̄r j := 1 m m ∑ i=1 r ij , σ j := √ √ √ √ 1 m m ∑ i=1 (r ij − ̄r j ) 2 . (Soσ j ≥0and RMS j ≥0.) (3) Logarithmic percentage-change coefficient.Define the LOPCOW percentage value of criterionj by PV j := ∣ ∣ ∣ ∣ ln ( RMS j σ j ) ∣ ∣ ∣ ∣ ·100, σ j >0, 0,σ j = 0, where ln denotes the natural logarithm. (4) Objective criterion weights.If ∑ n `=1 PV ` >0, define w j := PV j ∑ n `=1 PV ` ∈[0,1], j= 1,...,n, so that ∑ n j=1 w j = 1. If ∑ n `=1 PV ` = 0(all criteria have zero variation after normalization), setw j := 1/n. The resulting vectorw= (w 1 ,...,w n )is called theFuzzy LOPCOW (objective) weight vectorassociated with( ̃ X,S,C + ,C − ). The definition of Uncertain LOPCOW (ULOPCOW) is given below. Definition 4.2.2(Uncertain LOPCOW (ULOPCOW) objective weighting).LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. Fix an uncertainty modelMwith degree-domain Dom(M)⊆[0,1] d (d≥1). Assume anuncertain decision matrix ̃ X= ( ̃x ij )∈Dom(M) m×n , where ̃x ij is the uncertain evaluation ofA i underC j . LetC=C + ̇ ∪C − be a partition into benefit and cost criteria. (0) Score (crisp representative) map.Fix a total mapping S M :Dom(M)→R, Chapter 4. Weight elicitation Decision-Methods and define the induced crisp matrixX= (x ij )∈R m×n by x ij :=S M ( ̃x ij ). (Examples: for TFNs ̃x= (l,m,u), one may takeS M ( ̃x) = (l+m+u)/3; for interval values ̃x= [`,u], one may takeS M ( ̃x) = (`+u)/2.) (1) Linear normalization (benefit/cost).For each criterionj, set x max j :=max 1≤i≤m x ij , x min j :=min 1≤i≤m x ij . Define the normalized matrixR= (r ij )∈[0,1] m×n by r ij := x max j −x ij x max j −x min j , C j ∈C − (cost), x ij −x min j x max j −x min j , C j ∈C + (benefit), with the convention that ifx max j =x min j thenr ij := 0for alli. (2) Dispersion statistics per criterion.For eachj, define RMS j := √ √ √ √ 1 m m ∑ i=1 r 2 ij , ̄r j := 1 m m ∑ i=1 r ij , σ j := √ √ √ √ 1 m m ∑ i=1 (r ij − ̄r j ) 2 . (3) LOPCOW percentage value.Define PV j := ∣ ∣ ∣ ∣ ln ( RMS j σ j ) ∣ ∣ ∣ ∣ ·100, σ j >0, 0,σ j = 0, j= 1,...,n, where ln denotes the natural logarithm. (4) Objective weight vector.If ∑ n `=1 PV ` >0, define w j := PV j ∑ n `=1 PV ` ∈[0,1], j= 1,...,n, so that ∑ n j=1 w j = 1. If ∑ n `=1 PV ` = 0, setw j := 1/n. The vectorw= (w 1 ,...,w n )is called theUncertain LOPCOW (ULOPCOW) objective weight vectorasso- ciated with( ̃ X,S M ,C + ,C − ). Theorem 4.2.3(Uncertain-set output and well-definedness of ULOPCOW).Let ̃ X∈Dom(M) m×n be given as in Definition 4.2.2, withm,n≥1. Assume: Chapter 4. Weight elicitation Decision-Methods (A1)The score mapS M :Dom(M)→Ris total (defined for every element ofDom(M)). (A2)The benefit/cost partitionC=C + ̇ ∪C − is fixed. Then: (i)The normalized matrixR= (r ij )defined in(4.2.2)is well-defined and satisfies0≤r ij ≤1for alli,j. (i)For eachj,RMS j andσ j are well-defined real numbers and satisfyRMS j ≥0,σ j ≥0. (i)EachPV j is well-defined and satisfiesPV j ≥0. (iv)The weight vectorw= (w 1 ,...,w n )is well-defined and belongs to the probability simplex ∆ n = w∈R n ≥0 : n ∑ j=1 w j = 1 . (v)The mapping μ w :C →[0,1], μ w (C j ) :=w j , defines anuncertain setW:= (C,μ w )(theULOPCOW weight uncertain set). Proof.(i) Normalization is well-defined and bounded.Fixj∈ 1,...,n. By (A1), eachx ij = S M ( ̃x ij )is a real number, hencex max j ,x min j ∈Rare well-defined. Ifx max j =x min j , we setr ij = 0by convention, sor ij is defined. Assumex max j > x min j . For benefit criteria, r ij = x ij −x min j x max j −x min j , and sincex min j ≤x ij ≤x max j , it follows that0≤r ij ≤1. For cost criteria, r ij = x max j −x ij x max j −x min j , and again0≤r ij ≤1by the same bounding. (i) Dispersion statistics exist and are nonnegative.For each fixedj, the valuesr ij m i=1 are real and bounded by (i). Therefore RMS j is the square root of an average of squares of real numbers and is well-defined with RMS j ≥0. Similarly, ̄r j is a finite average andσ j is the square root of an average of squared deviations, hence is well-defined withσ j ≥0. (i) Log-percentage values are well-defined.Ifσ j = 0, thenPV j is defined to be0. Ifσ j >0, then RMS j ≥0. Moreover, ifσ j >0then the columnr ij is not constant, so at least oner ij 6= 0and hence RMS j >0. Thus RMS j /σ j >0and ln(RMS j /σ j )is defined. Taking absolute value and multiplying by100 yieldsPV j ≥0. Chapter 4. Weight elicitation Decision-Methods (iv) Weights form a simplex vector.If ∑ n `=1 PV ` >0, then eachw j is a nonnegative ratio, sow j ≥0 and n ∑ j=1 w j = n ∑ j=1 PV j ∑ n `=1 PV ` = ∑ n j=1 PV j ∑ n `=1 PV ` = 1. If ∑ n `=1 PV ` = 0, thenw j = 1/n≥0and ∑ n j=1 w j = 1. (v) Uncertain-set structure on criteria.By (iv),w j ∈[0,1]for allj. Henceμ w (C j ) :=w j defines a well-defined membership mapμ w :C →[0,1], soW= (C,μ w )is an uncertain set. Related concepts of LOPCOW under uncertainty-aware models are listed in Table 4.3. Table 4.3: Related concepts of LOPCOW under uncertainty-aware models. kRelated LOPCOW concept(s) 2 Intuitionistic Fuzzy LOPCOW 2q-Rung Fuzzy LOPCOW [470,471] 3 Hesitant Fuzzy LOPCOW [472] 3 Spherical Fuzzy LOPCOW [473] 3 Neutrosophic LOPCOW [474] 4.3 Fuzzy SIWEC (Fuzzy simple weight calculation for criteria) SIWEC derives criterion weights from experts’ direct ratings, normalizes scores, aggregates across experts, and outputs a weight vector [475,476]. Fuzzy SIWEC maps linguistic ratings to fuzzy numbers, normalizes, scales by dispersion, aggregates, then defuzzifies or ranks fuzzy weights [47,477]. Definition 4.3.1(Fuzzy SIWEC (F-SIWEC): fuzzy simple weight calculation for criteria).[477,478] Let E=1,...,ebe the set of expert decision makers (EDMs) andC=C 1 ,...,C n the set of criteria. Step 1 (linguistic evaluation→triangular fuzzy numbers).Each EDMi∈Eevaluates each criterion C j using a linguistic term that is mapped to a triangular fuzzy number (TFN) ̃x ij = (x l ij ,x m ij ,x u ij )∈R 3 + , x l ij ≤x m ij ≤x u ij . Collect these ratings into the fuzzy decision matrix ̃ X= [ ̃x ij ] e×n . Step 2 (normalization).For each criterionj, let M j :=max i∈E x u ij , and define the normalized TFN ̃n ij = ( x l ij M j , x m ij M j , x u ij M j ) . Chapter 4. Weight elicitation Decision-Methods Step 3 (standard-deviation scaling).Defuzzify each normalized TFN by n def ij := n l ij + 4n m ij +n u ij 6 , and compute the (sample) standard deviation for each criterionjacross EDMs: stdev j := √ √ √ √ 1 e−1 e ∑ i=1 ( n def ij − ̄n def j ) 2 , ̄n def j := 1 e e ∑ i=1 n def ij . Scale the normalized TFNs by this dispersion: ̃v ij :=stdev j ̃n ij = (stdev j n l ij ,stdev j n m ij ,stdev j n u ij ). Step 4 (aggregation over EDMs).For each criterionj, aggregate scaled TFNs across EDMs: ̃s j := e ⊕ i=1 ̃v ij = (s l j ,s m j ,s u j ), where⊕denotes componentwise addition of TFNs. Step 5 (fuzzy weight vector).Define the fuzzy weight of criterionC j as the TFN ̃w j := (w l j ,w m j ,w u j ) = ( s l j ∑ n k=1 s u k , s m j ∑ n k=1 s m k , s u j ∑ n k=1 s l k ) , j= 1,...,n. Step 6 (optional defuzzification and normalization).A crisp weight can be obtained by w def j := w l j + 4w m j +w u j 6 ∈R + . If required, renormalize: w ∗ j := w def j ∑ n k=1 w def k . The output ̃w= ( ̃w 1 ,..., ̃w n )(orw ∗ ) is called theFuzzy SIWECcriteria-weight vector. Fuzzy SIWEC is extended by using Uncertain Sets. Definition 4.3.2(Uncertain sets and uncertain numbers).LetUbe a nonempty universe. Anuncertain setonUis a mapping μ:U→[0,1], whereμ(u)is interpreted as the degree of uncertainty-membership ofu. Chapter 4. Weight elicitation Decision-Methods Anuncertain numberis an uncertain set ̃xonRwhoseα-cuts [ ̃x] α :=t∈R:μ ̃x (t)≥α(α∈(0,1]) are nonempty compact intervals. Write [ ̃x] α = [x − α ,x + α ] (α∈(0,1]). Denote byUNa chosen class of uncertain numbers closed under the operations below, and UN >0 := ̃x∈UN:supp( ̃x)⊆(0,∞),supp( ̃x) :=t:μ ̃x (t)>0. Define theupper support endpoint(a positive scalar representative) by up( ̃x) :=sup supp( ̃x)∈(0,∞) ( ̃x∈UN >0 ). Definition 4.3.3((Recall) Uncertain-number arithmetic viaα-cuts).Let ̃x, ̃y∈UNandc≥0. Define: [ ̃x⊕ ̃y] α := [x − α +y − α , x + α +y + α ],[c ̃x] α := [cx − α , cx + α ], for allα∈(0,1]. (These are Minkowski sum and scalar multiplication of intervalα-cuts.) Let Score:UN→R ≥0 be a fixedscore(crisp representative) map satisfying: (S1)(Positivity) Score( ̃x)≥0for all ̃x∈UN, and Score( ̃x)>0for all ̃x∈UN >0 ; (S2)(Positive homogeneity) Score(c ̃x) =cScore( ̃x)for allc≥0; (S3)(Additivity) Score( ̃x⊕ ̃y) =Score( ̃x) +Score( ̃y). (For instance, one may take Score( ̃x) = ∫ 1 0 x − α +x + α 2 dαwhen it is finite.) Here, we define Uncertain SIWEC (U-SIWEC), obtained by extending SIWEC using Uncertain Sets. Definition 4.3.4(Uncertain SIWEC (U-SIWEC): uncertain simple integrated weight estimation for crite- ria).LetE=1,...,ebe a finite set of experts andC=C 1 ,...,C n a finite set of criteria. Anuncertain SIWEC instanceis an uncertain evaluation matrix ̃ X= ( ̃x ij )∈(UN >0 ) e×n , ̃x ij is experti’s uncertain rating of criterionC j . Fix up and Score as in Definitions 4.3.2–4.3.3. TheU-SIWEC procedureconstructs criterion weights as follows. Step 1 (normalization by upper support).For each criterionj, set M j :=max i∈E up( ̃x ij )∈(0,∞), Chapter 4. Weight elicitation Decision-Methods and define normalized uncertain ratings ̃n ij := 1 M j ̃x ij ∈UN >0 . Step 2 (crisp representatives).Define n cr ij :=Score( ̃n ij )∈R >0 . Step 3 (dispersion across experts).For each criterionj, compute the mean and (population) standard deviation ̄n cr j := 1 e e ∑ i=1 n cr ij , σ j := √ √ √ √ 1 e e ∑ i=1 ( n cr ij − ̄n cr j ) 2 ∈R ≥0 . Step 4 (dispersion scaling in the uncertain domain).Scale each normalized uncertain rating byσ j : ̃v ij :=σ j ̃n ij ∈UN. Step 5 (aggregation over experts).Aggregate expert information for each criterion by ̃s j := e ⊕ i=1 ̃v ij ∈UN. Step 6 (objective weights).Let S j :=Score( ̃s j )∈R ≥0 , T:= n ∑ k=1 S k ∈R ≥0 . Define thecrisp weight vectorw= (w 1 ,...,w n )by w j := S j T , T >0, 1 n , T= 0. Optionally, define anuncertain weight vector ̃w= ( ̃w 1 ,..., ̃w n )by ̃w j := 1 T ̃s j , T >0, 1 n ̃u, T= 0, where ̃u∈UN >0 is any fixed reference uncertain number (e.g. ̃uwith Score( ̃u) = 1). Chapter 4. Weight elicitation Decision-Methods Theorem 4.3.5(Uncertain-set structure and well-definedness of U-SIWEC).Assume ̃ X∈(UN >0 ) e×n and thatUNis closed under⊕andc (·)forc≥0. AssumeScoresatisfies(S1)–(S3)in Definition 4.3.3. Then the U-SIWEC procedure in Definition 4.3.4 is well-defined and satisfies: (i)Each ̃n ij , ̃v ij , and ̃s j is an uncertain number; hence each is an uncertain set onR. (i)The crisp outputw= (w 1 ,...,w n )is a valid criterion-weight vector: w j ≥0for allj, n ∑ j=1 w j = 1. (i)IfT >0, then the uncertain output satisfies ̃w j ∈UNand Score( ̃w j ) =w j , n ∑ j=1 Score( ̃w j ) = 1. Proof.(i) Closure / uncertain-set structure.Fixj. Since each ̃x ij ∈UN >0 , the upper endpoint up( ̃x ij )is finite and positive, henceM j =max i up( ̃x ij )∈(0,∞)exists. By closure ofUNunder scalar multiplication, ̃n ij = (1/M j ) ̃x ij ∈UN >0 is defined. Nextσ j ∈R ≥0 is defined by a finite sum of real numbersn cr ij =Score( ̃n ij ). Again by closure, ̃v ij =σ j ̃n ij ∈UN, and then ̃s j = ⊕ e i=1 ̃v ij ∈UNby closure under⊕. Each of these objects is, by definition, an uncertain number and therefore an uncertain set onR. (i) Nonnegativity of scores and weights.By (S1),S j =Score( ̃s j )≥0for allj, henceT= ∑ k S k ≥0 andw j ≥0in both branches (T >0andT= 0). (i) Normalization ∑ j w j = 1.IfT >0, then n ∑ j=1 w j = n ∑ j=1 S j T = 1 T n ∑ j=1 S j = T T = 1. IfT= 0, thenw j = 1/ngives ∑ j w j = 1. (iv) Consistency of uncertain weights whenT >0.AssumeT >0and define ̃w j = (1/T) ̃s j . Then ̃w j ∈UNby closure. Using (S2), Score( ̃w j ) =Score ( 1 T ̃s j ) = 1 T Score( ̃s j ) = S j T =w j . Summing yields ∑ j Score( ̃w j ) = ∑ j w j = 1. For reference, related concepts of SIWEC under uncertainty-aware models are listed in Table 4.4. Chapter 4. Weight elicitation Decision-Methods Table 4.4: Related concepts of SIWEC under uncertainty-aware models. kRelated SIWEC concept(s) 1 Fuzzy SIWEC 2 Intuitionistic Fuzzy SIWEC 3 Neutrosophic SIWEC nPlithogenic SIWEC 4.4 Fuzzy judgment matrix Judgment matrix is a pairwise-comparison matrix in which each entrya ij represents the relative preference (or importance) of alternativeX i over alternativeX j ; it is often assumed reciprocal and is used to derive priority weights and to assess consistency [479–481]. A fuzzy judgment matrix is a pairwise-comparison matrix whose entries lie in[0,1]and quantify graded preference degrees, so thata ij >0.5favorsX i over X j ,a ij = 0.5indicates indifference, and (in the complementary case)a ij +a ji = 1[482,483]. Definition 4.4.1(Fuzzy judgment matrix).[482,483] LetX=X 1 ,...,X n be a finite set of alternatives. A matrix A= (a ij )∈R n×n is called afuzzy judgment matrixif 0≤a ij ≤1 (∀i,j∈1,...,n), wherea ij is interpreted as the (graded) preference degree ofX i overX j in a pairwise comparison. A common semantic convention is: a ij >0.5⇒X i is preferred toX j , a ij = 0.5⇒X i andX j are indifferent, a ij <0.5⇒X j is preferred toX i . Definition 4.4.2(Fuzzy complementary judgment matrix).A fuzzy judgment matrixA= (a ij )is called fuzzy complementaryif it satisfies a ij +a ji = 1 (∀i,j∈1,...,n), (which impliesa i = 0.5for alli). Definition 4.4.3(Uncertain preference numbers and arithmetic).Let UN [0,1] := ̃x∈UN:supp( ̃x)⊆[0,1]. For ̃x, ̃y∈UN [0,1] define the(truncated) sumandcomplementbyα-cuts: [ ̃x⊕ T ̃y] α := [ min1,x − α +y − α ,min1,x + α +y + α ] , [1 ̃x] α := [ 1−x + α ,1−x − α ] , α∈(0,1], where[ ̃x] α = [x − α ,x + α ]and[ ̃y] α = [y − α ,y + α ]. (Thus1 ̃xis the pointwise complement of ̃xon[0,1].) Next, we extend the judgment matrix using Uncertain Sets. The definition is given below. Chapter 4. Weight elicitation Decision-Methods Definition 4.4.4(Uncertain judgment matrix).LetX=X 1 ,...,X n be a finite set of alternatives. An uncertain judgment matrix(UJM) onXis a matrix ̃ A= ( ̃a ij )∈ ( UN [0,1] ) n×n , where ̃a ij is interpreted as the uncertain (graded) preference ofX i overX j . A semantic convention is induced by Score: Score( ̃a ij )>0.5⇒X i is preferred toX j ,Score( ̃a ij ) = 0.5⇒X i andX j are indifferent. The UJM is called: (i)uncertain complementaryif ̃a ij =1 ̃a ji (∀i,j), (which implies ̃a i is the “uncertain indifference” around0.5); (i)uncertain reciprocalif all entries are supported in(0,1]and a fixed uncertain-number product⊗is available such that ̃a ij ⊗ ̃a ji = ̃ 1(∀i,j), for a unit uncertain number ̃ 1concentrated at1. Theorem 4.4.5(Uncertain-set structure and well-definedness).Let ̃ A= ( ̃a ij )∈(UN [0,1] ) n×n . Then: (i)Each entry ̃a ij is an uncertain set on[0,1]and, a fortiori, an uncertain set onR. Hence an uncertain judgment matrix is a matrix whose entries carry uncertain-set structure. (i)The complement operator1 (·)in Definition 4.4.3 is well-defined onUN [0,1] and mapsUN [0,1] into itself. (i)If ̃ Ais uncertain complementary, then the complement relation is coherent: for alli,jand allα∈(0,1], [ ̃a ij ] α = [1−a + ji,α ,1−a − ji,α ]=⇒Score( ̃a ij ) = 1−Score( ̃a ji ) wheneverScoreis affine with respect to complements, i.e.Score(1 ̃x) = 1−Score( ̃x). Proof.(i)By definition, ̃a ij ∈UN [0,1] ⊆UNis an uncertain number, hence an uncertain set onR. Therefore each matrix entry carries uncertain-set structure. (i)Fix ̃x∈UN [0,1] and write[ ̃x] α = [x − α ,x + α ]with0≤x − α ≤x + α ≤1. Then Definition 4.4.3 gives [1 ̃x] α = [1−x + α ,1−x − α ]. Since0≤1−x + α ≤1−x − α ≤1, eachα-cut is a nonempty compact interval in[0,1]. Hence1 ̃x∈UN [0,1] , so the operation is well-defined and closed. (i)If ̃ Ais uncertain complementary, then ̃a ij =1 ̃a ji by definition. Assuming Score(1 ̃x) = 1−Score( ̃x), we obtain Score( ̃a ij ) =Score(1 ̃a ji ) = 1−Score( ̃a ji ), which is exactly the coherence of complementary semantics at the score level. Related concepts of judgment matrices under uncertainty-aware models are listed in Table 4.5. Chapter 4. Weight elicitation Decision-Methods Table 4.5: Related concepts of judgment matrices under uncertainty-aware models. kRelated judgment matrix concept(s) 1 Fuzzy Judgment Matrix 2 Intuitionistic Fuzzy Judgment Matrix [426,484] 3 Hesitant Fuzzy Judgment Matrix [485–487] 3 Spherical Fuzzy Judgment Matrix [488,489] 3 Neutrosophic Judgment Matrix [490,491] 4.5 Fuzzy Analytic network process (ANP) ANP evaluates interdependent criteria and alternatives using pairwise comparisons, builds a supermatrix, and derives global priorities via limit supermatrix [492–494]. Fuzzy ANP extends ANP with fuzzy pairwise judgments, forming fuzzy supermatrices and defuzzified limit priorities to handle uncertainty [39,495,496]. Definition 4.5.1(Fuzzy Analytic Network Process (FANP): supermatrix formulation).[497, 498] Let C=C 1 ,...,C m be clusters, whereC i =e i1 ,...,e in i withn i ≥1, and let E:= m ⊔ i=1 C i be the set of all elements. LetD ⊆C×Cbe a directed dependence relation, where(C i ,C j )∈Dmeans that “C i influencesC j ” (allowing inner dependencei=jand outer dependencei6=j). Fix a classFN >0 of positive fuzzy numbers, closed under the operations used below, and fix a ranking (defuzzification) map Score:FN >0 −→R >0 . (0) Fuzzy pairwise comparisons and local priorities.Apositive reciprocal fuzzy pairwise comparison matrixis ̃ A= ( ̃a rs )∈FN n×n >0 , satisfying ̃a r = 1, ̃a sr = ̃a −1 rs (1≤r,s≤n). Itslocal fuzzy priority vector ̃w= ( ̃w 1 ,..., ̃w n )∈FN n >0 is obtained, for example, by the fuzzy geometric mean method: ̃g r := ( n ∏ s=1 ̃a rs ) 1/n , ̃w r := ̃g r /( n ∑ t=1 ̃g t ) , r= 1,...,n, where the product, power, sum, and division are understood in the fuzzy-number sense (e.g. via the extension principle or an equivalent admissible fuzzy arithmetic). The associatedcrisp normalized priority vectoris defined by w ] r := Score( ̃w r ) ∑ n t=1 Score( ̃w t ) , r= 1,...,n. Thenw ] = (w ] 1 ,...,w ] n )∈R n ≥0 and n ∑ r=1 w ] r = 1. Chapter 4. Weight elicitation Decision-Methods (1) Element-to-element influence vectors and unweighted supermatrix.Fix(C i ,C j )∈ Dand a target elemente jk ∈C j . Compare the elements ofC i pairwise with respect to their influence one jk by a positive reciprocal fuzzy comparison matrix ̃ A (i→j|k) ∈FN n i ×n i >0 . Let w ](i→j|k) ∈R n i ≥0 be the corresponding crisp normalized local priority vector obtained from the above procedure. Define the blockW ij ∈R n i ×n j by (W ij ) •k :=w ](i→j|k) (k= 1,...,n j ), and setW ij := 0if(C i ,C j ) /∈D. The resulting block matrix W:= ( W ij ) 1≤i,j≤m ∈R ( ∑ i n i )×( ∑ j n j ) is called theunweighted supermatrix. (2) Cluster weights and weighted (column-stochastic) supermatrix.For each clusterC j , let Γ(j) :=i: (C i ,C j )∈D. Obtain fuzzy cluster-comparison judgments among the influencing clustersC i :i∈Γ(j), compute the corresponding fuzzy cluster-priority vector, and then defuzzify/normalize it to get α (j),] = (α (j),] i ) i∈Γ(j) ∈R |Γ(j)| ≥0 , ∑ i∈Γ(j) α (j),] i = 1. Define the weighted blocks ̄ W ij := α (j),] i W ij , i∈Γ(j), 0,i/∈Γ(j), ̄ W:= ( ̄ W ij ) 1≤i,j≤m . Then ̄ Wis column-stochastic, i.e. each column sums to1. (3) Limit supermatrix and global priorities.If the limit exists, define thelimit supermatrixby W ∞ :=lim t→∞ ̄ W t . If ̄ W t is cyclic with periodN, use the cycle-average W ∞ := 1 N N−1 ∑ t=0 ̄ W t . LetA⊆Ebe the designated set of alternatives (usually a subset of the elements in one cluster). Theglobal priorityof an alternative is read from the corresponding row ofW ∞ (equivalently, from stabilized columns), and the alternatives are ranked in decreasing order of global priority. We now extend Fuzzy ANP using Uncertain Sets. The related definitions are given below. Chapter 4. Weight elicitation Decision-Methods Definition 4.5.2(Uncertain set and uncertain number).LetUbe a nonempty universe. Anuncertain set onUis a mappingμ:U→[0,1]. Anuncertain numberis an uncertain set ̃xonRwhoseα-cuts [ ̃x] α :=t∈R:μ ̃x (t)≥α(α∈(0,1]) are nonempty compact intervals. LetUNbe a fixed class of uncertain numbers and set UN >0 := ̃x∈UN:supp( ̃x)⊆(0,∞). AssumeUN >0 is closed under theuncertain inverse: if[ ̃x] α = [x − α ,x + α ]with0< x − α ≤x + α , define [ ̃x −1 ] α := [ 1 x + α , 1 x − α ] (α∈(0,1]). Fix ascore(crisp representative) map Score:UN >0 →(0,∞), and assume it isreciprocity-compatible: Score( ̃x −1 ) = 1 Score( ̃x) (∀ ̃x∈UN >0 ),Score( ̃ 1) = 1, where ̃ 1is the uncertain number concentrated at1. Definition 4.5.3(Uncertain reciprocal judgment matrix).Letn≥2. Anuncertain reciprocal judgment matrixis ̃ A= ( ̃a rs )∈(UN >0 ) n×n such that ̃a r = ̃ 1(∀r), ̃a sr = ̃a −1 rs (∀r6=s). Its induced crisp reciprocal matrix is A:= (a rs )∈(0,∞) n×n , a rs :=Score( ̃a rs ). Definition 4.5.4(Uncertain ANP (UANP): score-induced supermatrix formulation).LetC=C 1 ,...,C m beclusters, where C i =e i1 ,...,e in i (n i ≥1), E:= m ⊔ i=1 C i is the set of all elements. LetD ⊆C×Cbe a directed dependence relation;(C i ,C j )∈Dmeans “C i influences C j ” (allowingi=j). (0) Local priorities from uncertain pairwise judgments.Fix(C i ,C j )∈ Dand a target element e jk ∈C j . Decision makers provide an uncertain reciprocal judgment matrix ̃ A (i→j|k) ∈(UN >0 ) n i ×n i comparing the elements ofC i with respect to their influence one jk . Let A (i→j|k) := ( Score( ̃a (i→j|k) rs ) ) ∈(0,∞) n i ×n i Chapter 4. Weight elicitation Decision-Methods be the induced crisp reciprocal matrix. Define thelocal priority vectorw (i→j|k) ∈R n i >0 as the normalized Perron vector ofA (i→j|k) : A (i→j|k) w (i→j|k) =λ max w (i→j|k) , n i ∑ r=1 w (i→j|k) r = 1. (1) Unweighted supermatrix.For each(i,j), define the blockW ij ∈R n i ×n j ≥0 by setting itskth column as (W ij ) •k :=w (i→j|k) (k= 1,...,n j ), and setW ij := 0if(C i ,C j ) /∈D. Theunweighted supermatrixis the block matrix W:= ( W ij ) 1≤i,j≤m ∈R N×N ≥0 , N:= m ∑ i=1 n i . (2) Cluster weights and weighted supermatrix.For each target clusterC j , letΓ(j) :=i: (C i ,C j )∈ D. Cluster weights are obtained by uncertain pairwise comparisons among clusters inΓ(j), yielding (after applying Score and Perron normalization) a vector α (j) = (α (j) i ) i∈Γ(j) ∈R |Γ(j)| ≥0 , ∑ i∈Γ(j) α (j) i = 1. Define the weighted blocks ̄ W ij := α (j) i W ij , i∈Γ(j), 0,i/∈Γ(j), ̄ W:= ( ̄ W ij ) 1≤i,j≤m ∈R N×N ≥0 . (3) Limit supermatrix and global priorities.If ̄ Wisprimitive(some power has strictly positive entries), define W ∞ :=lim t→∞ ̄ W t . In general (even if periodic), define the Cesàro limit (always used in practice when cycling occurs): W ∞ :=lim T→∞ 1 T T−1 ∑ t=0 ̄ W t ,whenever the limit exists. LetA⊆Ebe the designated set of alternatives (elements representing alternatives). Theglobal priorityof a∈Ais read from the corresponding row ofW ∞ (equivalently from stabilized columns), and alternatives are ranked by decreasing global priority. Theorem 4.5.5(Uncertain-set structure and well-definedness of UANP).Under the assumptions of Defi- nitions 4.5.2–4.5.4, the UANP construction is well-defined in the following sense. (i)(Uncertain-set structure) Every entry ̃a (i→j|k) rs ∈UN >0 is an uncertain set on(0,∞)(hence onR). Thus each local comparison matrix ̃ A (i→j|k) is a matrix of uncertain sets. Chapter 4. Weight elicitation Decision-Methods (i)(Local priorities are well-defined) For each local uncertain reciprocal judgment matrix ̃ A (i→j|k) , the induced crisp matrixA (i→j|k) = (Score( ̃a (i→j|k) rs ))is a positive reciprocal matrix. Hence its Perron eigenvector exists, is unique up to scaling, and the normalized vectorw (i→j|k) is uniquely determined in the simplex∆ n i :=x∈R n i ≥0 : ∑ r x r = 1. (i)(Weighted supermatrix is column-stochastic) Each column of ̄ Wsums to1, i.e. 1 > ̄ W=1 > , where1is the all-ones vector inR N . (iv)(Limit priorities are well-defined) If ̄ Wis primitive, then the limitW ∞ =lim t→∞ ̄ W t exists. Con- sequently, global priorities read fromW ∞ are well-defined (independent of the iteration index for sufficiently larget). More generally, whenever the Cesàro limit in Definition 4.5.4 exists, it yields a well-defined steady-state priority extraction even in cyclic cases. Proof.(i)Each ̃a rs ∈UN >0 is an uncertain set onR. Hence every local matrix ̃ A (i→j|k) is entrywise an uncertain set. (i)Let ̃ A= ( ̃a rs )be an uncertain reciprocal judgment matrix. Then ̃a r = ̃ 1impliesa r =Score( ̃ 1) = 1. Also ̃a sr = ̃a −1 rs and reciprocity-compatibility give a sr =Score( ̃a sr ) =Score( ̃a −1 rs ) = 1 Score( ̃a rs ) = 1 a rs . ThusA= (a rs )is a positive reciprocal matrix. By the Perron–Frobenius theorem,Ahas a (strictly) positive principal eigenvector, unique up to a positive scalar; normalizing it to sum to1yields a uniquew∈∆ n . (i)Fix a target clusterC j and an elemente jk ∈C j . Thekth column of the blockW ij equalsw (i→j|k) for eachi∈Γ(j), so n i ∑ r=1 (W ij ) rk = n i ∑ r=1 w (i→j|k) r = 1. In the weighted supermatrix, the same column is scaled byα (j) i and summed overi∈Γ(j), giving ∑ i∈Γ(j) n i ∑ r=1 ( ̄ W ij ) rk = ∑ i∈Γ(j) α (j) i n i ∑ r=1 (W ij ) rk = ∑ i∈Γ(j) α (j) i = 1. Hence every column of ̄ Wsums to1, i.e.1 > ̄ W=1 > . (iv)If ̄ Wis primitive and column-stochastic, then ̄ W > is a primitive row-stochastic matrix. By standard Markov-chain (Perron–Frobenius) convergence,( ̄ W > ) t converges ast→ ∞, hence ̄ W t converges as well, andW ∞ :=lim t→∞ ̄ W t exists. Therefore the priorities extracted fromW ∞ are well-defined. If ̄ Wis not primitive, the Cesàro averaging is the standard remedy; whenever the Cesàro limit exists, it yields a unique steady-state projection for priority extraction. Related concepts of ANP under uncertainty-aware models are listed in Table 4.6. As concepts other than Uncertain ANP, DEMATEL-ANP [507,508], BOCR-based ANP [509], Group ANP [510,511], ANP-TOPSIS [510,512], and Rough ANP [513,514] are also known. Chapter 4. Weight elicitation Decision-Methods Table 4.6: Related concepts of ANP under uncertainty-aware models. kRelated ANP concept(s) 2 Intuitionistic Fuzzy ANP [499,500] 2 Pythagorean Fuzzy ANP [501,502] 3 Hesitant Fuzzy ANP [503] 3 Spherical Fuzzy ANP [504] 3 Neutrosophic ANP [505,506] 4.6 Fuzzy Ordinal Priority Approach (OPA) Ordinal Priority Approach (OPA) is a max–min optimization method that converts experts’ ordinal rankings into consistent criterion weights, without pairwise comparisons, yielding a prioritized alternative ordering [515,516]. Fuzzy Ordinal Priority Approach (OPA) extends OPA by representing linguistic importance as triangular fuzzy numbers, solving a fuzzy max–min model to obtain fuzzy weights and rankings [517–519]. Definition 4.6.1(Fuzzy Ordinal Priority Approach (OPA-F): TFN-based formulation).[517–519] LetI be a finite set of experts,Ja finite set of criteria, andKa finite set of alternatives. Putn:=|J|. Let TFN:=(l,m,u)∈R 3 :l≤m≤u be the set of triangular fuzzy numbers (TFNs). We use the componentwise partial order onTFN: (l 1 ,m 1 ,u 1 )(l 2 ,m 2 ,u 2 )⇐⇒l 1 ≥l 2 , m 1 ≥m 2 , u 1 ≥u 2 . (Any fixed TFN ranking/defuzzification rule can be used later for producing a crisp ranking.) Fuzzy linguistic inputs.LetLbe a linguistic scale and letφ:L→TFNmap each term to a TFN. Each experti∈Iprovides anordinal rankingof criteria, represented by a permutation π i :1,...,n→J, whereπ i (1)is the most important criterion under expertiandπ i (n)is the least important. In addition, expertiprovides a linguistic importance assessment for each ranked criterion, encoded as TFNs ̃a i,r :=φ ( ` i,r ) ∈TFN, r= 1,...,n, where` i,r ∈Lis the linguistic term attached to ther-th ranked criterionπ i (r). Decision variables (fuzzy weights).Introduce TFN decision variables ̃ W i,r ∈TFN ≥0 (i∈I, r= 1,...,n), ̃ Z∈TFN, where ̃ W i,r is the (fuzzy) weight assigned by expertito ther-th ranked criterionπ i (r), and ̃ Zis the max–min (bottleneck) objective variable. TFN arithmetic convention.For ̃x= (l x ,m x ,u x )and ̃y= (l y ,m y ,u y )andλ≥0, use ̃x⊕ ̃y:= (l x +l y , m x +m y , u x +u y ), λ ̃x:= (λl x , λm x , λu x ), Chapter 4. Weight elicitation Decision-Methods and the common triangular subtraction approximation ̃x ̃y:= (l x −u y , m x −m y , u x −l y ). (Any other consistent TFN arithmetic can be substituted if preferred.) OPA-F core model (fuzzy max–min program).Fix a (possibly fuzzy) normalization constant ̃ 1∈TFN, typically ̃ 1 = (1,1,1)(or a tolerance form such as(1−ε,1,1 +ε)). TheFuzzy Ordinal Priority Approach (OPA-F)determines fuzzy criterion weights by solving: max ̃ Z(4.1) s.t. ̃a i,r ( ̃ W i,r ̃ W i,r+1 ) ̃ Z,∀i∈I,∀r= 1,...,n−1,(4.2) ̃a i,n ̃ W i,n ̃ Z,∀i∈I,(4.3) ⊕ i∈I n ⊕ r=1 ̃ W i,r = ̃ 1,(4.4) ̃ W i,r = (l w i,r ,m w i,r ,u w i,r ),0≤l w i,r ≤m w i,r ≤u w i,r ,∀i∈I,∀r.(4.5) Aggregated criterion weights and ranking.Define the aggregated fuzzy weight of criterionj∈Jby collecting the ranks wherejappears: ̃ W j := ⊕ i∈I ̃ W i,r(i,j) ,wherer(i,j)is the unique rank withπ i (r(i,j)) =j. A crisp weight vector can be obtained via any defuzzification/ranking map Defuzz:TFN→R ≥0 , e.g. Defuzz(l,m,u) = (l+m+u)/3, followed by normalization: w j := Defuzz( ̃ W j ) ∑ t∈J Defuzz( ̃ W t ) . Reduction to crisp OPA.If all inputs are degenerate TFNs(l,m,u) = (p,p,p)and one takes ̃ 1 = (1,1,1), then (4.1)–(4.5) reduces to a crisp OPA-type max–min linear program. By extending it using Uncertain Sets, we obtain the following formulation. Definition 4.6.2(Uncertainty space, uncertain set, membership, expected value).Anuncertainty spaceis a triple(Γ,L,M)whereΓis a nonempty set,Lis aσ-algebra onΓ, andM:L →[0,1]is anuncertain measure(satisfying, e.g., normality, duality, and subadditivity). Anuncertain setis a measurable mapping ξ: Γ→P(R), such that for every Borel setB⊆R, the events γ∈Γ :ξ(γ)⊆B∈L,γ∈Γ :B⊆ξ(γ)∈L. Chapter 4. Weight elicitation Decision-Methods An uncertain setξis said to have amembership functionμ ξ :R→[0,1]if it represents the uncertain-set membership semantics in the standard way (equivalently, sup x∈R μ ξ (x) = 1). Ifξis a nonempty uncertain set, itsexpected valueis defined by E[ξ] := ∫ ∞ 0 Mξ≥rdr− ∫ 0 −∞ Mξ≤rdr, provided at least one of the two integrals is finite. Definition 4.6.3(Uncertain Ordinal Priority Approach (UOPA)).LetIbe a finite set of experts andJa finite set of criteria, withn:=|J|≥2. Each experti∈Iprovides: 1. anordinal rankingof criteria, represented as a permutationπ i :1,...,n →J(rank1= most important); 2. anuncertain importance profilealong the ranks: for each rankr∈ 1,...,n, an uncertain set (uncertain number) ξ i,r : Γ→P(R) with finite expected valuea i,r :=E[ξ i,r ]∈R. (1) Reduction to a deterministic OPA core via expected values.Define deterministic coefficients a i,r :=E[ξ i,r ], i∈I, r= 1,...,n. Introduce decision variablesw i,r ∈R ≥0 andz∈R, and solve the linear program maxz(4.6) s.t.a i,r ( w i,r −w i,r+1 ) ≥z,∀i∈I,∀r= 1,...,n−1,(4.7) a i,n w i,n ≥z,∀i∈I,(4.8) ∑ i∈I n ∑ r=1 w i,r = 1,(4.9) w i,r ≥0,∀i∈I,∀r.(4.10) (2) Aggregated criterion weights.For each criterionj∈J, define its aggregated (crisp) weight by W j := ∑ i∈I w i,r(i,j) ,wherer(i,j)is the unique rank withπ i (r(i,j)) =j. ThenW j ≥0and ∑ j∈J W j = 1. (3) Output as uncertain sets (degenerate uncertain numbers).Define, for eachj∈J, a degenerate uncertain set (singleton-valued mapping) Ξ j : Γ→P(R),Ξ j (γ) :=W j . The familyΞ j j∈J is called theUOPA uncertain weight vector. Chapter 4. Weight elicitation Decision-Methods Theorem 4.6.4(Uncertain-set structure and well-definedness of UOPA).Assume in Definition 4.6.3 that, for alli∈Iandr∈ 1,...,n, the uncertain setξ i,r has a finite expected valuea i,r =E[ξ i,r ]and that a i,r ≥0. Then: 1. the coefficientsa i,r in(4.6)–(4.10)are well-defined real numbers, so the optimization model is well-posed as a finite-dimensional linear program; 2. the feasible region of(4.6)–(4.10)is nonempty and compact, and the objective is bounded above; hence an optimal solution exists; 3. the UOPA outputΞ j j∈J defined in Definition 4.6.3(3) forms a family of uncertain sets (on the same uncertainty space), i.e., the method yields anuncertain-set structuredweight vector. Proof.(1) Well-defined coefficients.By assumption, eachξ i,r has finite expected valueE[ξ i,r ]in the sense of Definition 4.6.2. Thereforea i,r ∈Ris uniquely determined, and (4.6)–(4.10) is a deterministic linear program with finitely many variables and linear constraints. (2) Nonemptiness (feasibility).LetN:=|I|n. Setw i,r := 1/Nfor alli,r, and setz:= 0. Then (4.9) and (4.10) hold. Moreover, sincea i,r ≥0andw i,r −w i,r+1 = 0, the left-hand side of (4.7) equals0, and the left-hand side of (4.8) equalsa i,n /N≥0. Hence all constraints hold withz= 0. So the feasible set is nonempty. (3) Compactness and boundedness.From (4.9) and (4.10), the vector(w i,r )lies in a standard simplex, hence the feasible set projected towis closed and bounded. The full feasible set in(w,z)is also closed: it is an intersection of closed halfspaces. Also, for every feasible(w,z)and everyi∈I, (4.8) impliesz≤a i,n w i,n ≤a i,n ·1 =a i,n because0≤w i,n ≤1 under (4.9)–(4.10). Hencezis bounded above by max i∈I a i,n <∞. Therefore the linear objective maxzis bounded above on the feasible region. (4) Existence of an optimizer.A bounded linear functional attains its maximum on a nonempty compact polyhedron. Thus (4.6)–(4.10) has at least one optimal solution. (5) Uncertain-set structured output.For eachj∈J, the mappingΞ j (γ) =W j is a constant (singleton-valued) set-valued function, hence measurable with respect toL, and therefore is an uncertain set in the sense of Definition 4.6.2. Consequently,Ξ j j∈J is a family of uncertain sets representing the UOPA weight vector. Related concepts of Ordinal Priority Approach (OPA) under uncertainty-aware models are listed in Ta- ble 4.7. Related concepts other than Uncertain OPA include the interval ordinal priority approach [516, 522], the grey ordinal priority approach [523,524], and the rough ordinal priority approach [525,526]. Chapter 4. Weight elicitation Decision-Methods Table 4.7: Related concepts of Ordinal Priority Approach (OPA) under uncertainty-aware models. kRelated OPA concept(s) 1 Fuzzy Ordinal Priority Approach (OPA) 2 Intuitionistic Fuzzy Ordinal Priority Approach (OPA) 3 Neutrosophic Ordinal Priority Approach (OPA) [520,521] 4.7 Fuzzy PIPRECIA (Pivot Pairwise Relative Criteria Importance Assessment) PIPRECIA is an MCDM weighting method using a pivot criterion and sequential pairwise comparisons to derive criteria importance coefficients efficiently [527, 528]. Fuzzy PIPRECIA extends PIPRECIA by expressing judgments with fuzzy numbers, handling vagueness and uncertainty while computing robust criteria weights consistently [44,529]. Definition 4.7.1(TFN-based fuzzy PIPRECIA (ordinary + inverse)).[44,529] LetC 1 ,...,C n be criteria written in a fixed order (the order isnotrequired to be sorted by importance). LetR≥1decision makers (DMs) provide linguistic judgments encoded aspositive triangular fuzzy numbers(TFNs), using two TFN- valued scales: a “>1” scale (often called the 1–2 scale) and a “<1” scale (often called the 0–1 scale). (Concrete TFN tables for these two scales are commonly given in the literature.) Assume TFN arithmetic(⊗, )and TFN powers are available (e.g., componentwise in the standard fuzzy- AHP convention). Step 1 (ordinary consecutive comparisons).For eachj= 2,...,nand each DMr∈ 1,...,R, let ̃s (r) j ∈TFN >0 encode the relative importance ofC j versusC j−1 : ̃s (r) j > ̃ 1ifC j is judged more important thanC j−1 , = ̃ 1ifC j is judged equally important asC j−1 , < ̃ 1ifC j is judged less important thanC j−1 , ̃ 1 := (1,1,1), where>/<are interpreted at the linguistic-scale level. Step 2 (aggregation by geometric mean).Aggregate the DMs’ TFNs by the (TFN) geometric mean: ̃s j :=GM ( ̃s (1) j ,..., ̃s (R) j ) := ( R ⊗ r=1 ̃s (r) j ) 1/R , j= 2,...,n, as commonly required in fuzzy PIPRECIA implementations. Step 3 (coefficients).Define TFN coefficients ̃ k j by ̃ k 1 := ̃ 1, ̃ k j := ̃ 2 ̃s j (j= 2,...,n), ̃ 2 := (2,2,2), i.e., if ̃s j = (l j ,m j ,u j )then ̃ k j := (2−u j ,2−m j ,2−l j )(to preserve the TFN order). This corresponds to the “subtract from2” rule in fuzzy PIPRECIA. Chapter 4. Weight elicitation Decision-Methods Step 4 (recursive fuzzy weights).Define ̃q j by ̃q 1 := ̃ 1, ̃q j := ̃q j−1 ̃ k j (j= 2,...,n). Step 5 (relative weights from the ordinary pass).Define the (ordinary-pass) fuzzy relative weights by ̃w j := ̃q j ( n ⊕ t=1 ̃q t ) , j= 1,...,n. Step 6–9 (inverse pass).Repeat Steps 1–5in reverse directionby comparingC j withC j+1 (starting from the penultimate criterion) to obtain inverse-pass TFNs ̃w ′ 1 ,..., ̃w ′ n . Step 10 (defuzzification and final criterion weights).Let Defuzz:TFN >0 →R >0 be a fixed defuzzi- fication map. A widely used choice (often reported as DFV) is the weighted mean Defuzz(l,m,u) := l+ 4m+u 6 . Defuzzify and average the ordinary and inverse weights: w j :=Defuzz( ̃w j ), w ′ j :=Defuzz( ̃w ′ j ), w final j := w j +w ′ j 2 , j= 1,...,n. This “defuzzify then average” rule is the standard finalization step in fuzzy PIPRECIA. Finally, normalize w final j ← w final j ∑ n t=1 w final t so that n ∑ j=1 w final j = 1, and use ( w final 1 ,...,w final n ) as the criterion-weight vector. As an extension based on Uncertain Sets, the definition of Uncertain PIPRECIA (U-PIPRECIA) is given below. Definition 4.7.2(Uncertain PIPRECIA (U-PIPRECIA): expected-value realization).LetC 1 ,...,C n be criteria listed in a fixed order (n≥2), and letD=1,...,Rbe a finite set of decision makers (DMs). Work on a fixed uncertainty space(Γ,L,M). For eachj= 2,...,nand each DMr∈ D, assume the DM provides anuncertain relative-importance assessment Ξ (r) j : Γ→P(R), interpreted as the (uncertain) importance ratio ofC j versusC j−1 . Assume: Ξ (r) j (γ)⊆R >0 (∀γ∈Γ),E[Ξ (r) j ]<∞. Chapter 4. Weight elicitation Decision-Methods Define the corresponding deterministic coefficients by expected values: s (r) j :=E[Ξ (r) j ]∈R >0 , j= 2,...,n, r∈D. Step 1 (DM aggregation).Aggregate DMs by the geometric mean (positive scalar aggregation): s j := ( R ∏ r=1 s (r) j ) 1/R ∈R >0 , j= 2,...,n, and sets 1 := 1. Step 2 (PIPRECIA coefficients).Define k 1 := 1, k j := 2−s j , j= 2,...,n. (These are the crisp counterparts of the standard “subtract from2” rule.) Step 3 (recursive importance sequence).Define q 1 := 1, q j := q j−1 k j (j= 2,...,n), wheneverk j >0. Step 4 (normalized criterion weights).LetQ:= ∑ n t=1 q t (assumeQ >0). Define w j := q j Q , j= 1,...,n. The vectorw= (w 1 ,...,w n )is called theU-PIPRECIA (ordinary-pass) weight vector. Step 5 (optional inverse pass and finalization).Repeat Steps 1–4 in reverse order by comparingC j withC j+1 to obtainw ′ 1 ,...,w ′ n , and set w final j := w j +w ′ j 2 , w final j ← w final j ∑ n t=1 w final t . Definition 4.7.3(Uncertain-set output of U-PIPRECIA).Letw= (w 1 ,...,w n )be the (ordinary-pass or final) weight vector obtained from Definition 4.7.2. Define, for eachj= 1,...,n, a degenerate uncertain set (singleton-valued mapping) W j : Γ→P(R), W j (γ) :=w j . ThenW= (W 1 ,...,W n )is called theU-PIPRECIA uncertain weight vector. Theorem 4.7.4(Uncertain-set structure and well-definedness of U-PIPRECIA).Assume the setting of Definition 4.7.2 and suppose: (A1)(Finite positive expectations) For allj= 2,...,nandr∈1,...,R,s (r) j =E[Ξ (r) j ]∈(0,2). Chapter 4. Weight elicitation Decision-Methods Then: (i)s j ∈(0,2)and hencek j = 2−s j ∈(0,2)for allj≥2; (i)the recursionq 1 = 1,q j =q j−1 /k j is well-defined and yieldsq j >0for allj; (i)Q= ∑ n t=1 q t satisfiesQ∈(0,∞), and thereforew j =q j /Qis well-defined withw j >0and ∑ n j=1 w j = 1; (iv)the output family(W j ) n j=1 in Definition 4.7.3 is a family of uncertain sets on(Γ,L,M)(=an uncertain- set structure). The same conclusions hold for the inverse-pass and the final averaged weights whenever (A1) holds for the inverse comparisons as well. Proof.(i)Fixj∈2,...,n. By (A1), eachs (r) j ∈(0,2), hence ∏ R r=1 s (r) j ∈(0,2 R )and therefore s j = ( R ∏ r=1 s (r) j ) 1/R ∈(0,2). Consequentlyk j = 2−s j ∈(0,2). Alsok 1 = 1by definition. (i)Sincek j >0for allj≥2, division byk j is valid. Inductively,q 1 = 1>0and ifq j−1 >0then q j =q j−1 /k j >0. Thus allq j are well-defined and positive. (i)Because eachq t >0andn <∞, the sumQ= ∑ n t=1 q t is finite and strictly positive. Hence each w j =q j /Qis well-defined and positive. Moreover, n ∑ j=1 w j = n ∑ j=1 q j Q = 1 Q n ∑ j=1 q j = Q Q = 1. (iv)For each fixedj, the mappingW j (γ) =w j is constant (singleton-valued), hence measurable with respect toLand therefore qualifies as an uncertain set. Thus(W j ) n j=1 is an uncertain-set structured output. The inverse pass is the same construction applied to the reverse ordering, so the same argument applies, and the final averaging/renormalization is an ordinary algebraic post-processing that preserves well-definedness whenever both passes are well-defined. For reference, related concepts of PIPRECIA under uncertainty-aware models are listed in Table 4.8. As related PIPRECIA-based methods other than Uncertain PIPRECIA, PIPRECIA-S [536, 537], Grey PIPRECIA [538,539], and Rough PIPRECIA [540,541] are also known. Chapter 4. Weight elicitation Decision-Methods Table 4.8: Related concepts of PIPRECIA under uncertainty-aware models. kRelated PIPRECIA concept(s) 2 Intuitionistic Fuzzy PIPRECIA [530] 2 Fermatean Fuzzy PIPRECIA [531] 3 Picture Fuzzy PIPRECIA [532] 3 Hesitant Fuzzy PIPRECIA [533] 3 Neutrosophic PIPRECIA [534,535] 4.8 Fuzzy SWARA (Fuzzy Stepwise Weight Assessment Ratio Analysis) SWARA ranks criteria by importance, elicits stepwise comparative coefficients between successive criteria, computes recalculated weights sequentially, and normalizes them for use in scoring models later [542,543]. Fuzzy SWARA uses linguistic terms for stepwise importance ratios, converts them to fuzzy numbers, prop- agates fuzziness through weight calculation, and defuzzifies final weights for ranking [544–546]. Definition 4.8.1(Fuzzy SWARA (TFN-based step-wise weighting)).[544–546] LetC=C 1 ,...,C n be a finite set of criteria. Assume that the decision makers (DMs) provide aconsensus ordering C 1 C 2 ·C n (from the most important to the least important criterion). (0) Triangular fuzzy numbers and arithmetic.Let TFN >0 :=(l,m,u)∈R 3 : 0< l≤m≤u be the set of positive triangular fuzzy numbers (TFNs). For ̃x= (l x ,m x ,u x )and ̃y= (l y ,m y ,u y )inTFN >0 , define ̃x⊕ ̃y:= (l x +l y , m x +m y , u x +u y ), ̃y −1 := ( 1 u y , 1 m y , 1 l y ) , ̃x ̃y:= ̃x⊗ ̃y −1 = ( l x u y , m x m y , u x l y ) , ̃ 1 := (1,1,1). (Thus is well-defined onTFN >0 .) (1) Fuzzy stepwise comparative importance.For eachj= 2,...,n, let ̃ b j = (l j ,m j ,u j )∈TFN >0 be thefuzzy comparative importanceofC j−1 relative toC j (elicited from linguistic judgments encoded as TFNs). No ̃ b 1 is needed. (2) Fuzzy coefficients.Define TFN coefficients ̃e j by ̃e 1 := ̃ 1, ̃e j := ̃ b j ⊕ ̃ 1 (j= 2,...,n). (3) Recalculated fuzzy weights (unnormalized).Define ̃ f j ∈TFN >0 recursively by ̃ f 1 := ̃ 1, ̃ f j := ̃ f j−1 ̃e j (j= 2,...,n). Chapter 4. Weight elicitation Decision-Methods (4) Normalized fuzzy weights.Let ̃ F:= ⊕ n t=1 ̃ f t . Thefuzzy SWARA weightofC j is ̃w j := ̃ f j ̃ F= (w ` j ,w m j ,w u j )∈TFN >0 , j= 1,...,n. (5) Defuzzification (crisp weights).A commonly used crisp weight is obtained by centroid (center-of- area) defuzzification: w j := w ` j +w m j +w u j 3 , j= 1,...,n. Optionally, normalizewagain byw j ←w j / ∑ n t=1 w t so that ∑ n j=1 w j = 1. The resulting vector(w 1 ,...,w n )is called theFuzzy SWARA (defuzzified) criterion-weight vector. Using Uncertain Sets, we define Uncertain SWARA (U-SWARA) as follows. Definition 4.8.2(Uncertain SWARA (U-SWARA): expected-value realization).LetC=C 1 ,...,C n be a finite set of criteria,n≥2. Assume a fixedconsensus importance order C 1 C 2 ·C n (from most to least important). Work on a fixed uncertainty space(Γ,L,M). For each stepj= 2,...,n, decision makers provide anuncertain stepwise comparative-importance coefficient (the “relative importance ofC j−1 overC j ”) Ξ j : Γ→P(R), assumed to satisfy Ξ j (γ)⊆R ≥0 (∀γ∈Γ), s j :=E[Ξ j ]<∞. (Thuss j is the crisp stepwise coefficient obtained by expected-value reduction of uncertainty.) Define theSWARA recalculation factorsby k 1 := 1, k j := 1 +s j (j= 2,...,n). Define theunnormalized sequential weightsrecursively by q 1 := 1, q j := q j−1 k j (j= 2,...,n), wheneverk j >0. LetQ:= ∑ n t=1 q t (assumeQ >0), and define thenormalized SWARA weightsby w j := q j Q , j= 1,...,n. The vectorw= (w 1 ,...,w n )is called theU-SWARA criterion-weight vector. Chapter 4. Weight elicitation Decision-Methods Definition 4.8.3(Uncertain-set output of U-SWARA).Letw= (w 1 ,...,w n )be obtained from Defini- tion 4.8.2. Define degenerate uncertain sets (singleton-valued mappings) W j : Γ→P(R), W j (γ) :=w j , j= 1,...,n. ThenW= (W 1 ,...,W n )is called theU-SWARA uncertain weight vector. Theorem 4.8.4(Uncertain-set structure and well-definedness of U-SWARA).Assume the setting of Defi- nition 4.8.2 and suppose: (A1)(Nonnegative finite expectations)s j =E[Ξ j ]∈[0,∞)for allj= 2,...,n. Then: (i)k j = 1 +s j ∈[1,∞)for allj= 2,...,n, hencek j >0; (i)the recursionq 1 = 1,q j =q j−1 /k j is well-defined and yieldsq j >0for allj; (i)Q= ∑ n t=1 q t satisfiesQ∈(0,∞), and thusw j =q j /Qis well-defined withw j >0and ∑ n j=1 w j = 1; (iv)the family(W j ) n j=1 in Definition 4.8.3 forms an uncertain-set structured output on(Γ,L,M). Proof.(i)By (A1), for eachj≥2one hass j ≥0, hencek j = 1 +s j ≥1>0. (i)Sincek j >0, division byk j is valid. Inductivelyq 1 = 1>0; ifq j−1 >0thenq j =q j−1 /k j >0. Thus allq j exist and are positive. (i)Becausen <∞and eachq t >0, the sumQ= ∑ n t=1 q t is finite and strictly positive. Hencew j =q j /Q is well-defined and positive. Moreover, n ∑ j=1 w j = n ∑ j=1 q j Q = 1 Q n ∑ j=1 q j = Q Q = 1. (iv)For eachj, the mapW j (γ) =w j is constant (singleton-valued), hence measurable; therefore it is an uncertain set. Consequently(W j ) n j=1 is an uncertain-set structured output. Related uncertainty-model variants of SWARA, classified by the degree-domain dimensionk, are listed in Table 4.9. As related concepts other than Uncertain SWARA, Rough SWARA [561], Soft SWARA [562], SWARA- TOPSIS [563,564], SWARA-AHP [565,566], and Grey SWARA [567,568] are also known. Chapter 4. Weight elicitation Decision-Methods Table 4.9: Related uncertainty-model variants of SWARA (classified by the degree-domain dimensionk). kRelated SWARA variant(s) 1 Fuzzy SWARA 2 Intuitionistic Fuzzy SWARA [547,548] 2 Pythagorean Fuzzy SWARA [549,550] 2 Fermatean Fuzzy SWARA [551,552] 3 Hesitant Fuzzy SWARA [553,554] 3 Spherical Fuzzy SWARA [555,556] 3 Neutrosophic SWARA [557,558] n Plithogenic SWARA [559,560] 4.9 Fuzzy CILOS (Fuzzy Criterion Impact Loss) CILOS derives objective criterion weights by quantifying information or impact loss when each criterion deteriorates, emphasizing criteria that most affect overall performance [569, 570]. Fuzzy CILOS extends CILOS using fuzzy numbers for performances, computes fuzzy impact-loss measures per criterion, and produces objective weights robust under imprecision [571]. Definition 4.9.1(Fuzzy CILOS (Criterion Impact Loss) weighting).[571] Let A=A 1 ,...,A m andC=C 1 ,...,C n be the sets of alternatives and criteria, respectively. Assume that each criterionC j is either ofbenefit type orcost type. LetFN >0 be a fixed class of positive fuzzy numbers, and let ̃ X= ( ̃x ij ) m×n ∈FN m×n >0 be the fuzzy decision matrix, where ̃x ij is the fuzzy performance of alternativeA i under criterionC j . Fix a total ranking / defuzzification map Score:FN >0 →R >0 . Define the associated positive crisp score matrix S= (s ij ) m×n , s ij :=Score( ̃x ij )>0. Step 1: Benefit-type transformation.Define the transformed matrixX= (x ij ) m×n ∈R m×n >0 by x ij := s ij ,ifC j is a benefit criterion, min 1≤r≤m s rj s ij ,ifC j is a cost criterion. Thus, after transformation, every criterion is treated as a benefit criterion. Step 2: Criterion-wise optima.For each criterionj∈1,...,n, define x ? j :=max 1≤i≤m x ij . Chapter 4. Weight elicitation Decision-Methods Choose an index ι j ∈arg max 1≤i≤m x ij . Step 3: Square matrix of criterion-optimal alternatives.Define the matrixA= (a ij ) n×n by a ij :=x ι i j , i,j= 1,...,n. Hence, thei-th row ofAis the transformed performance vector of an alternative that is optimal for criterion C i . In particular, a i =x ? i . Step 4: Relative-loss matrix.Define the matrixP= (p ij ) n×n by p ij := x ? j −a ij x ? j , i6=j, 0,i=j. Thusp ij ∈[0,1]represents the relative loss of criterionC j when the alternative that is best for criterionC i is selected. Step 5: CILOS balance matrix.DefineF= (f ij ) n×n by f ij := p ij ,i6=j, − n ∑ r=1 r6=j p rj , i=j. Equivalently, F= − ∑ r6=1 p r1 p 12 ·p 1n p 21 − ∑ r6=2 p r2 ·p 2n . . . . . . . . . . . . p n1 p n2 · − ∑ r6=n p rn . Step 6: Criterion-significance vector and weights.A nonzero vector q= (q 1 ,...,q n ) > ∈R n ≥0 \0 satisfying Fq= 0 is called acriterion-significance vectorof the fuzzy CILOS model. The correspondingFuzzy CILOS weight vectoris w= (w 1 ,...,w n ) > , w j := q j ∑ n `=1 q ` , j= 1,...,n. Then w j ≥0and n ∑ j=1 w j = 1. Chapter 4. Weight elicitation Decision-Methods We now defineUncertain CILOS(UCILOS) by extending the criterion-impact-loss idea to a general uncer- tain modelM. Definition 4.9.2(Uncertain CILOS (UCILOS) of typeM).Let A=A 1 ,...,A m andC=C 1 ,...,C n be the sets of alternatives and criteria, respectively, wherem,n∈Nandm,n≥1. Fix an uncertain modelMwith degree-domain Dom(M)⊆[0,1] k for some integerk≥1. Assume that each criterionC j is designated either as abenefit criterionor as acost criterion. Let X M = (μ ij ) m×n ∈Dom(M) m×n be an uncertain decision matrix, whereμ ij ∈Dom(M)is the uncertain performance of alternativeA i under criterionC j . Assume further that a total score map Score M :Dom(M)−→R >0 is fixed. Define the associated positive score matrix S= (s ij ) m×n ∈R m×n >0 , s ij :=Score M (μ ij ). Step 1: Benefit-type transformation.DefineB= (b ij ) m×n ∈R m×n >0 by b ij := s ij ,ifC j is a benefit criterion, min 1≤r≤m s rj s ij ,ifC j is a cost criterion. Thus every criterion is transformed into benefit form. Step 2: Column normalization.For each criterionj, let β j := m ∑ r=1 b rj . Define the normalized matrix Z= (z ij ) m×n ∈R m×n >0 by z ij := b ij β j . Chapter 4. Weight elicitation Decision-Methods Step 3: Criterion-wise optima.For eachj∈1,...,n, define z ? j :=max 1≤i≤m z ij , and choose any index ι j ∈arg max 1≤i≤m z ij . Step 4: Square matrix of criterion-optimal alternatives.Define the square matrix A= (a ij ) n×n by a ij :=z ι i j , i,j= 1,...,n. Hence thei-th row ofAis the normalized criterion profile of an alternative that is optimal for criterionC i . In particular, a i =z ? i . Step 5: Relative-loss matrix.DefineP= (p ij ) n×n by p ij := 0,i=j, z ? j −a ij z ? j , i6=j. Thusp ij is the relative loss of criterionC j when the alternative that is optimal for criterionC i is selected. Step 6: Total loss and criterion significance.For each criterionj, define its total impact loss by L j := n ∑ i=1 i6=j p ij . Define the corresponding criterion-significance score by q j := 1 1 +L j , j= 1,...,n. Step 7: UCILOS weight vector.TheUncertain CILOS weight vectoris w= (w 1 ,...,w n ) > ∈[0,1] n , w j := q j ∑ n `=1 q ` , j= 1,...,n. Remark 4.9.3.Definition 4.10.2 is an objective weighting construction. It generalizes the criterion-impact- loss idea by first converting each uncertain entryμ ij ∈Dom(M)into a positive representative score and then applying a loss-based criterion-weighting mechanism to the induced score matrix. Theorem 4.9.4(Well-definedness of UCILOS).Let U CILOS = ( A,C,M,X M ,Score M ) be a UCILOS instance as in Definition 4.10.2. Assume: Chapter 4. Weight elicitation Decision-Methods (A1)AandCare finite nonempty sets; (A2) X M = (μ ij ) m×n ∈Dom(M) m×n ; (A3) each criterionC j is designated either as benefit type or as cost type; (A4) Score M :Dom(M)→R >0 is a total map. Then all quantities appearing in Definition 4.10.2 are well-defined. More precisely: (i) the score matrixS= (s ij )∈R m×n >0 is well-defined; (i) the benefit-type matrixB= (b ij )∈R m×n >0 is well-defined; (i) the normalized matrixZ= (z ij )∈R m×n >0 is well-defined, and for eachj, m ∑ i=1 z ij = 1; (iv) for eachj, the optimal valuez ? j and at least one maximizerι j ∈arg max i z ij exist; (v) the square matrixA= (a ij )and the relative-loss matrixP= (p ij )are well-defined, with 0≤p ij ≤1 (1≤i,j≤n); (vi) the total lossesL j and significance scoresq j are well-defined and satisfy L j ∈[0,n−1], q j ∈ [ 1 n ,1 ] ; (vii) the weight vectorw= (w 1 ,...,w n ) > is well-defined and satisfies w j >0for allj,and n ∑ j=1 w j = 1. Hence Uncertain CILOS of typeMis well-defined. Proof.By (A2), for everyi∈1,...,mandj∈1,...,n, μ ij ∈Dom(M). Since Score M is total by (A4), each value s ij :=Score M (μ ij ) Chapter 4. Weight elicitation Decision-Methods is well-defined and belongs toR >0 . Therefore the score matrix S= (s ij ) m×n ∈R m×n >0 is well-defined. This proves (i). Next, defineB= (b ij )as in Definition 4.10.2. IfC j is a benefit criterion, then b ij =s ij >0. IfC j is a cost criterion, then b ij = min 1≤r≤m s rj s ij . Because the finite sets 1j ,...,s mj ⊂R >0 has a positive minimum ands ij >0, this quotient is well-defined and strictly positive. Hence everyb ij ∈R >0 , soB∈R m×n >0 . This proves (i). For eachj, define β j := m ∑ r=1 b rj . Since eachb rj >0andm≥1, we haveβ j >0. Therefore z ij := b ij β j is well-defined and positive. ThusZ= (z ij )∈R m×n >0 is well-defined. Moreover, m ∑ i=1 z ij = m ∑ i=1 b ij β j = 1 β j m ∑ i=1 b ij = 1. This proves (i). Fixj∈1,...,n. Since the finite nonempty setz 1j ,...,z mj ⊂R >0 has a maximum, the number z ? j :=max 1≤i≤m z ij exists, and the argmax set arg max 1≤i≤m z ij is nonempty. Hence at least one indexι j can be chosen. This proves (iv). Now define a ij :=z ι i j . Since eachι i ∈1,...,m, everya ij is well-defined and belongs toR >0 . ThereforeA= (a ij )is well-defined. For the relative-loss matrix, first note thatz ? j >0. Also, sincez ? j is the maximum in columnj, 0< a ij ≤z ? j . Chapter 4. Weight elicitation Decision-Methods Hence, fori6=j, p ij = z ? j −a ij z ? j is well-defined and satisfies0≤p ij ≤1. By definitionp j = 0. ThereforeP= (p ij )is well-defined and all its entries lie in[0,1]. This proves (v). For eachj, L j := n ∑ i=1 i6=j p ij is a finite sum ofn−1numbers from[0,1]. Therefore 0≤L j ≤n−1. Consequently, q j := 1 1 +L j is well-defined and satisfies 1 n ≤q j ≤1. This proves (vi). Finally, because eachq j >0, the denominator n ∑ `=1 q ` is strictly positive. Hence w j := q j ∑ n `=1 q ` is well-defined for everyj, andw j >0. Moreover, n ∑ j=1 w j = n ∑ j=1 q j ∑ n `=1 q ` = ∑ n j=1 q j ∑ n `=1 q ` = 1. Thuswis a well-defined normalized weight vector. This proves (vii). Therefore every step of the UCILOS construction is mathematically well-defined. Related concepts of CILOS under uncertainty-aware models are listed in Table 4.10. Table 4.10: Related concepts of CILOS under uncertainty-aware models. kRelated CILOS concept(s) 1 Fuzzy CILOS 2 Intuitionistic Fuzzy CILOS 3 Neutrosophic CILOS Chapter 4. Weight elicitation Decision-Methods 4.10Fuzzy IDOCRIW (Fuzzy Entropy-CILOS integrated objective weighting) IDOCRIW objectively derives criterion weights by integrating entropy dispersion and CILOS impact-loss indices from a normalized decision matrix data alone [572,573]. Fuzzy IDOCRIW extends IDOCRIW using fuzzy numbers for performances, computing entropy and loss measures fuzzily, then defuzzifying weights robustly overall [48,574]. Definition 4.10.1(Fuzzy IDOCRIW weights (Entropy–CILOS integrated objective weighting)).[48,574] LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. Let ̃ X= ( ̃x ij )∈F m×n be a fuzzy decision matrix, whereFdenotes the chosen class of fuzzy evaluations (e.g., TFNs, IVIFNs, picture-fuzzy numbers, etc.). Assume ascore/defuzzificationmap s:F→R >0 that induces comparisons and yields strictly positive scalars d ij :=s( ̃x ij ) (i= 1,...,m, j= 1,...,n). Partition the criteria into benefit and cost types:C=BtC, whereBare beneficial andCare non-beneficial criteria. Form a benefit-oriented (positive) matrix ̂ D= ( ̂ d ij )by ̂ d ij := d ij ,j∈B, min 1≤i≤m d ij d ij , j∈C, for alli,j. (Thus all criteria are converted to the “larger-is-better” direction.) (1) Entropy objective weights.Define the column-normalized proportions p ij := ̂ d ij ∑ m r=1 ̂ d rj , i= 1,...,m, j= 1,...,n, and adopt the convention0ln0 := 0. Letk:= 1/ln(m). The entropy of criterionjis E j :=−k m ∑ i=1 p ij ln(p ij ), its divergence is∆ j := 1−E j , and the entropy weight vector is w (E) j := ∆ j ∑ n `=1 ∆ ` , j= 1,...,n. (2) CILOS objective weights.For each criterionj, pick an index s j ∈arg max 1≤i≤m ̂ d ij (one of the best-performing alternatives under criterionj). Construct the square matrixB= (b ij )∈R n×n >0 by b ij := ̂ d s i j (i,j= 1,...,n), Chapter 4. Weight elicitation Decision-Methods so thatb i =max 1≤r≤m ̂ d ri . Define the relative impact-loss matrixR= (r ij )∈[0,1] n×n by r ij := b j −b ij b j , r j := 0 (i,j= 1,...,n). Form the (column-Laplacian) weight-system matrixF= (f ij )∈R n×n : f ij := r ij ,i6=j, − n ∑ r=1 r rj , i=j. ACILOS weight vectoris any vectorw (C) ∈R n ≥0 \0satisfying F w (C) = 0, n ∑ j=1 w (C) j = 1. (Under standard irreducibility/nondegeneracy conditions onR, such a normalized nonnegative null-vector is unique.) (3) IDOCRIW aggregation (integrated objective weights).Thefuzzy IDOCRIWweight vector is defined by the multiplicative fusion w (IDOCRIW) j := w (E) j w (C) j ∑ n `=1 w (E) ` w (C) ` , j= 1,...,n. Thenw (IDOCRIW) ∈R n ≥0 and ∑ n j=1 w (IDOCRIW) j = 1. We now defineUncertain CILOS(UCILOS) by extending the criterion-impact-loss idea to a general uncer- tain modelM. Definition 4.10.2(Uncertain CILOS (UCILOS) of typeM).Let A=A 1 ,...,A m andC=C 1 ,...,C n be the sets of alternatives and criteria, respectively, wherem,n∈Nandm,n≥1. Fix an uncertain modelMwith degree-domain Dom(M)⊆[0,1] k for some integerk≥1. Assume that each criterionC j is designated either as abenefit criterionor as acost criterion. Let X M = (μ ij ) m×n ∈Dom(M) m×n be an uncertain decision matrix, whereμ ij ∈Dom(M)is the uncertain performance of alternativeA i under criterionC j . Chapter 4. Weight elicitation Decision-Methods Assume further that a total score map Score M :Dom(M)−→R >0 is fixed. Define the associated positive score matrix S= (s ij ) m×n ∈R m×n >0 , s ij :=Score M (μ ij ). Step 1: Benefit-type transformation.DefineB= (b ij ) m×n ∈R m×n >0 by b ij := s ij ,ifC j is a benefit criterion, min 1≤r≤m s rj s ij ,ifC j is a cost criterion. Thus every criterion is transformed into benefit form. Step 2: Column normalization.For each criterionj, let β j := m ∑ r=1 b rj . Define the normalized matrix Z= (z ij ) m×n ∈R m×n >0 by z ij := b ij β j . Step 3: Criterion-wise optima.For eachj∈1,...,n, define z ? j :=max 1≤i≤m z ij , and choose any index ι j ∈arg max 1≤i≤m z ij . Step 4: Square matrix of criterion-optimal alternatives.Define the square matrix A= (a ij ) n×n by a ij :=z ι i j , i,j= 1,...,n. Hence thei-th row ofAis the normalized criterion profile of an alternative that is optimal for criterionC i . In particular, a i =z ? i . Chapter 4. Weight elicitation Decision-Methods Step 5: Relative-loss matrix.DefineP= (p ij ) n×n by p ij := 0,i=j, z ? j −a ij z ? j , i6=j. Thusp ij is the relative loss of criterionC j when the alternative that is optimal for criterionC i is selected. Step 6: Total loss and criterion significance.For each criterionj, define its total impact loss by L j := n ∑ i=1 i6=j p ij . Define the corresponding criterion-significance score by q j := 1 1 +L j , j= 1,...,n. Step 7: UCILOS weight vector.TheUncertain CILOS weight vectoris w= (w 1 ,...,w n ) > ∈[0,1] n , w j := q j ∑ n `=1 q ` , j= 1,...,n. Remark 4.10.3.Definition 4.10.2 is an objective weighting construction. It generalizes the criterion- impact-loss idea by first converting each uncertain entryμ ij ∈Dom(M)into a positive representative score and then applying a loss-based criterion-weighting mechanism to the induced score matrix. Theorem 4.10.4(Well-definedness of UCILOS).Let U CILOS = ( A,C,M,X M ,Score M ) be a UCILOS instance as in Definition 4.10.2. Assume: (A1)AandCare finite nonempty sets; (A2) X M = (μ ij ) m×n ∈Dom(M) m×n ; (A3) each criterionC j is designated either as benefit type or as cost type; (A4) Score M :Dom(M)→R >0 is a total map. Then all quantities appearing in Definition 4.10.2 are well-defined. More precisely: (i) the score matrixS= (s ij )∈R m×n >0 is well-defined; Chapter 4. Weight elicitation Decision-Methods (i) the benefit-type matrixB= (b ij )∈R m×n >0 is well-defined; (i) the normalized matrixZ= (z ij )∈R m×n >0 is well-defined, and for eachj, m ∑ i=1 z ij = 1; (iv) for eachj, the optimal valuez ? j and at least one maximizerι j ∈arg max i z ij exist; (v) the square matrixA= (a ij )and the relative-loss matrixP= (p ij )are well-defined, with 0≤p ij ≤1 (1≤i,j≤n); (vi) the total lossesL j and significance scoresq j are well-defined and satisfy L j ∈[0,n−1], q j ∈ [ 1 n ,1 ] ; (vii) the weight vectorw= (w 1 ,...,w n ) > is well-defined and satisfies w j >0for allj,and n ∑ j=1 w j = 1. Hence Uncertain CILOS of typeMis well-defined. Proof.By (A2), for everyi∈1,...,mandj∈1,...,n, μ ij ∈Dom(M). Since Score M is total by (A4), each value s ij :=Score M (μ ij ) is well-defined and belongs toR >0 . Therefore the score matrix S= (s ij ) m×n ∈R m×n >0 is well-defined. This proves (i). Next, defineB= (b ij )as in Definition 4.10.2. IfC j is a benefit criterion, then b ij =s ij >0. IfC j is a cost criterion, then b ij = min 1≤r≤m s rj s ij . Because the finite sets 1j ,...,s mj ⊂R >0 has a positive minimum ands ij >0, this quotient is well-defined and strictly positive. Hence everyb ij ∈R >0 , soB∈R m×n >0 . This proves (i). Chapter 4. Weight elicitation Decision-Methods For eachj, define β j := m ∑ r=1 b rj . Since eachb rj >0andm≥1, we haveβ j >0. Therefore z ij := b ij β j is well-defined and positive. ThusZ= (z ij )∈R m×n >0 is well-defined. Moreover, m ∑ i=1 z ij = m ∑ i=1 b ij β j = 1 β j m ∑ i=1 b ij = 1. This proves (i). Fixj∈1,...,n. Since the finite nonempty setz 1j ,...,z mj ⊂R >0 has a maximum, the number z ? j :=max 1≤i≤m z ij exists, and the argmax set arg max 1≤i≤m z ij is nonempty. Hence at least one indexι j can be chosen. This proves (iv). Now define a ij :=z ι i j . Since eachι i ∈1,...,m, everya ij is well-defined and belongs toR >0 . ThereforeA= (a ij )is well-defined. For the relative-loss matrix, first note thatz ? j >0. Also, sincez ? j is the maximum in columnj, 0< a ij ≤z ? j . Hence, fori6=j, p ij = z ? j −a ij z ? j is well-defined and satisfies0≤p ij ≤1. By definitionp j = 0. ThereforeP= (p ij )is well-defined and all its entries lie in[0,1]. This proves (v). For eachj, L j := n ∑ i=1 i6=j p ij is a finite sum ofn−1numbers from[0,1]. Therefore 0≤L j ≤n−1. Consequently, q j := 1 1 +L j Chapter 4. Weight elicitation Decision-Methods is well-defined and satisfies 1 n ≤q j ≤1. This proves (vi). Finally, because eachq j >0, the denominator n ∑ `=1 q ` is strictly positive. Hence w j := q j ∑ n `=1 q ` is well-defined for everyj, andw j >0. Moreover, n ∑ j=1 w j = n ∑ j=1 q j ∑ n `=1 q ` = ∑ n j=1 q j ∑ n `=1 q ` = 1. Thuswis a well-defined normalized weight vector. This proves (vii). Therefore every step of the UCILOS construction is mathematically well-defined. 4.11Fuzzy BWM (Fuzzy Best-Worst Method) BWM selects the best and worst criteria, compares best-to-others and others-to-worst, then solves a con- strained optimization to obtain consistent criterion weights for MCDA ranking tasks [575, 576]. Fuzzy BWM replaces pairwise comparison ratios with fuzzy numbers, solves for fuzzy weights under consistency constraints, and defuzzifies or ranks by possibility to prioritize criteria [577–579]. Definition 4.11.1(TFN-based Fuzzy Best–Worst Method (FBWM)).[577–579] LetC=c 1 ,...,c n be a set of decision criteria(n≥2). A triangular fuzzy number (TFN) is denoted by ̃a= (a l ,a m ,a u )with 0< a l ≤a m ≤a u . Step 1 (Best and worst criteria).A decision-maker selects: c B ∈C(best/most important), c W ∈C(worst/least important). Step 2 (Fuzzy pairwise-comparison vectors).Using a linguistic scale mapped to TFNs, the decision- maker provides: • theFuzzy Best-to-Othersvector ̃ A B = ( ̃a B1 , ̃a B2 ,..., ̃a Bn ) , ̃a Bj = (a l Bj ,a m Bj ,a u Bj ), where ̃a Bj encodes the degree to whichc B is preferred overc j ; Chapter 4. Weight elicitation Decision-Methods • theFuzzy Others-to-Worstvector ̃ A W = ( ̃a 1W , ̃a 2W ,..., ̃a nW ) , ̃a jW = (a l jW ,a m jW ,a u jW ), where ̃a jW encodes the degree to whichc j is preferred overc W . As neutral self-comparisons, set ̃a B = ̃a W W = (1,1,1). Step 3 (Deriving fuzzy weights by linear programming).The FBWM output is a TFN weight for each criterion ̃w j = (w l j ,w m j ,w u j ) (j= 1,...,n), together with a satisfaction degreeβ∈[0,1]. Lett∈ l,m,udenote the TFN component (lower/mid- dle/upper). Introduce tolerance parametersd t j >0andq t j >0. Define the following linear program: maxβ s.t.0≤w t B −a t Bj w t j ≤d t j ,β≤1− w t B −a t Bj w t j d t j ,∀j,∀t∈l,m,u, −q t j ≤w t j −a t jW w t W ≤0,β≤1 + w t j −a t jW w t W q t j ,∀j,∀t∈l,m,u, n ∑ i=1 w m i = 1, w u j + n ∑ i=1 i6=j w l i ≤1, w l j + n ∑ i=1 i6=j w u i ≥1,∀j, 0≤w l j ≤w m j ≤w u j ,∀j, 0≤β≤1. Any optimal solution ̃w j n j=1 is called theFBWM fuzzy weight vectorassociated with( ̃ A B , ̃ A W ). Uncertain BWM of type (M) is defined as follows. Definition 4.11.2(Admissible comparison scale for an uncertain model).LetMbe an uncertain model with degree-domain Dom(M)⊆[0,1] k (nonempty). Anadmissible (ratio) comparison scaleforMis a map κ M :Dom(M)−→[1,Λ] for some fixed constantΛ≥1. The valueκ M (d)is interpreted as acrisp preference intensityinduced by the uncertainty-degree tupled∈Dom(M). Remark.The choice ofκ M is model-dependent (e.g., a score/defuzzification plus rescaling) and should be specified by the decision analyst. Chapter 4. Weight elicitation Decision-Methods Definition 4.11.3(Uncertain BWM of typeM(U-BWM)).LetC=c 1 ,...,c n be a set of criteria with n≥2. Fix an uncertain modelMwith Dom(M)6=∅and an admissible scaleκ M as in Definition 4.11.2. Step 1 (Best and worst criteria).Select c B ∈C(best / most important), c W ∈C(worst / least important). Step 2 (Uncertain comparison vectors).Provide the following two U-sets (uncertain vectors) of type M: •Best-to-Othersuncertain preferences A (M) B :C →Dom(M), c j 7→a (M) Bj , wherea (M) Bj encodes how stronglyc B is preferred overc j ; •Others-to-Worstuncertain preferences A (M) W :C →Dom(M), c j 7→a (M) jW , wherea (M) jW encodes how stronglyc j is preferred overc W . Set neutral self-comparisonsa (M) B =a (M) W W so that κ M ( a (M) B ) =κ M ( a (M) W W ) = 1. Step 3 (Crisp intensities induced byM).Define induced (crisp) ratio intensities ˆa Bj :=κ M ( a (M) Bj ) ∈[1,Λ],ˆa jW :=κ M ( a (M) jW ) ∈[1,Λ] (j= 1,...,n). Step 4 (Weight derivation via a minimax linear program).TheU-BWM weight vectoris any optimal solutionw= (w 1 ,...,w n )of the linear program minξ s.t.−ξ≤w B −ˆa Bj w j ≤ξ, j= 1,...,n, −ξ≤w j −ˆa jW w W ≤ξ, j= 1,...,n, n ∑ j=1 w j = 1, w j ≥0 (j= 1,...,n), ξ≥0. The optimal valueξ ? is interpreted as a (model-induced)consistency deviation. Theorem 4.11.4(Well-definedness of U-BWM).Under the assumptions of Definition 4.11.3 (in particular, Dom(M)6=∅andκ M :Dom(M)→[1,Λ]exists), the U-BWM optimization problem is well-defined: Chapter 4. Weight elicitation Decision-Methods (i) the feasible region is nonempty; (i) an optimal solution(w ? ,ξ ? )exists; (i) every optimal weight vector satisfiesw ? ∈∆ n−1 :=w∈R n ≥0 : ∑ n j=1 w j = 1. Proof.(i) Since Dom(M)6=∅, the uncertain comparison vectorsA (M) B ,A (M) W can be chosen (and are given) with values in Dom(M), hence the induced intensitiesˆa Bj ,ˆa jW are well-defined real numbers in[1,Λ]. Consider the uniform weightsw j := 1/nfor allj; then ∑ n j=1 w j = 1andw j ≥0hold. Define ξ 0 :=max max 1≤j≤n ∣ ∣ w B −ˆa Bj w j ∣ ∣ ,max 1≤j≤n ∣ ∣ w j −ˆa jW w W ∣ ∣ ≥0. With(w,ξ) = ( 1 n 1,ξ 0 )all linear inequalities in Definition 4.11.3 are satisfied, so the feasible region is nonempty. (i) The feasible set is described by finitely many linear equalities/inequalities, hence it is a closed polyhedron. Moreover, the constraints ∑ n j=1 w j = 1andw j ≥0imply0≤w j ≤1for allj. For any feasiblew, choosing ξ=max max j |w B −ˆa Bj w j |,max j |w j −ˆa jW w W | always yields a feasible pair(w,ξ), so the objectiveξis bounded below by0. Because the simplex constraint makes thew-coordinates bounded and the inequalities forceξto be at least the maximum of finitely many continuous expressions, the minimum of the linear objective over this nonempty closed feasible set is attained. Equivalently (and standard in linear programming), a feasible LP with objective bounded below admits an optimal solution. (i) By construction, every feasible (hence every optimal) solution satisfies the simplex constraints ∑ n j=1 w j = 1andw j ≥0, i.e.,w∈∆ n−1 . Related concepts of BWM under uncertainty-aware models are listed in Table 4.11. Table 4.11: Related concepts of BWM under uncertainty-aware models. kRelated BWM concept(s) 2 Intuitionistic Fuzzy BWM [580,581] 2 Bipolar Fuzzy BWM [582] 2 Pythagorean Fuzzy BWM [40] 3 Spherical Fuzzy BWM [583,584] 3 Hesitant Fuzzy BWM [585,586] 3 Picture fuzzy BWM (cf. [587]) 3 Neutrosophic BWM [588–590] nPlithogenic BWM As extension- or related concepts other than Uncertain BWM, BWAHP [581], Bayesian BWM [591], Rough BWM [592,593], BWM-TOPSIS [594,595], Z-number BWM [596], Interval BWM [597], Robust BWM [598], Belief-based BWM [599], and Grey BWM [600,601] are also known. Chapter 4. Weight elicitation Decision-Methods 4.12Fuzzy CRITIC CRITIC is an objective weighting method that assigns criterion weights using data dispersion and inter- criterion conflict, emphasizing informative, discriminative, and nonredundant criteria [602, 603]. Fuzzy CRITIC computes objective criterion weights from fuzzy ratings using dispersion and intercriteria correla- tion, emphasizing informative, nonredundant criteria in scoring [604–606]. Definition 4.12.1(Fuzzy CRITIC weighting).[604–606] LetA=A 1 ,...,A m be a set of alternatives andC=C 1 ,...,C n a set of criteria. Assume a fuzzy decision matrix ̃ X= ( ̃x ij ) m×n , ̃x ij ∈F, whereFis a chosen family of fuzzy evaluations (e.g., TFNs, trapezoidal fuzzy numbers, interval-valued fuzzy numbers, etc.). Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Fix adefuzzification/scoremap S:F→R, which converts fuzzy evaluations into real numbers (e.g., for a TFN ̃x= (`,m,u)one may takeS( ̃x) = (`+m+u)/3). Define the crisp matrixX= (x ij )by x ij :=S( ̃x ij ) (i= 1,...,m;j= 1,...,n). (1) Min–max normalization.For each criterionC j , define x min j :=min 1≤i≤m x ij , x max j :=max 1≤i≤m x ij , and assumex max j 6=x min j for at least onej. The normalized matrixR= (r ij )is defined by r ij := x ij −x min j x max j −x min j , C j ∈C ben , x max j −x ij x max j −x min j , C j ∈C cost . Thenr ij ∈[0,1]. (2) Dispersion of each criterion.Let ̄r j := 1 m m ∑ i=1 r ij , σ j := √ √ √ √ 1 m−1 m ∑ i=1 (r ij − ̄r j ) 2 . (3) Inter-criteria correlation.Forj,k∈1,...,ndefine Pearson correlation ρ jk := ∑ m i=1 (r ij − ̄r j )(r ik − ̄r k ) √ ∑ m i=1 (r ij − ̄r j ) 2 √ ∑ m i=1 (r ik − ̄r k ) 2 , Chapter 4. Weight elicitation Decision-Methods whenever the denominator is nonzero (otherwise setρ jk := 0). (4) Information content and weights.Define the CRITIC information content of criterionC j as Γ j :=σ j n ∑ k=1 ( 1−ρ jk ) , and (when ∑ n j=1 Γ j >0) define fuzzy-CRITIC objective weights by w j := Γ j ∑ n `=1 Γ ` , j= 1,...,n. The resulting vectorw= (w 1 ,...,w n ) > is called theFuzzy CRITICweight vector associated with( ̃ X,S). Proposition 4.12.2(Well-definedness and simplex property).In Definition 4.12.1, assume ∑ n j=1 Γ j >0. Thenw j ≥0for alljand ∑ n j=1 w j = 1. Proof.For eachj,σ j ≥0by definition. Also,ρ jk ≤1, hence1−ρ jk ≥0, so ∑ n k=1 (1−ρ jk )≥0and therefore Γ j ≥0. With ∑ n j=1 Γ j >0, the formulaw j = Γ j / ∑ n `=1 Γ ` yieldsw j ≥0and ∑ n j=1 w j = 1immediately. Using Uncertain Sets, we define Uncertain CRITIC weighting of typeMas follows. Definition 4.12.3(Admissible scoring map for an uncertain model).LetMbe an uncertain model with nonempty degree-domain Dom(M)⊆[0,1] k . Anadmissible scoreforMis a map S M :Dom(M)−→R such thatS M (d)is finite for everyd∈Dom(M). (Examples include defuzzification/score functions for fuzzy degrees, or scalarization maps for multi-component degrees such as intuitionistic/neutrosophic/plithogenic tuples.) Definition 4.12.4(Uncertain CRITIC weighting of typeM).LetA=A 1 ,...,A m be alternatives and C=C 1 ,...,C n criteria withm≥2andn≥1. Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M as in Definition 4.12.3. Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M), wherex (M) ij encodes the uncertainty-degree evaluation of alternativeA i under criterionC j . Step 0 (Crisp projection via scoring).Define the real-valued matrixX= (x ij )by x ij :=S M ( x (M) ij ) ∈R. Chapter 4. Weight elicitation Decision-Methods Step 1 (Min–max normalization).For each criterionC j , set x min j :=min 1≤i≤m x ij , x max j :=max 1≤i≤m x ij . Define the normalized matrixR= (r ij )by r ij := x ij −x min j x max j −x min j ,ifC j ∈C ben andx max j > x min j , x max j −x ij x max j −x min j ,ifC j ∈C cost andx max j > x min j , 0,ifx max j =x min j . (Thus degenerate criteria with zero range are handled byr ij = 0.) Step 2 (Dispersion).Let ̄r j := 1 m m ∑ i=1 r ij , σ j := √ √ √ √ 1 m−1 m ∑ i=1 (r ij − ̄r j ) 2 (j= 1,...,n). Step 3 (Inter-criteria correlation).Forj,k∈1,...,ndefine ρ jk := ∑ m i=1 (r ij − ̄r j )(r ik − ̄r k ) √ ∑ m i=1 (r ij − ̄r j ) 2 √ ∑ m i=1 (r ik − ̄r k ) 2 ,ifσ j >0andσ k >0, 0,otherwise. Step 4 (CRITIC information and objective weights).Define the information content ofC j by Γ j :=σ j n ∑ k=1 ( 1−ρ jk ) , j= 1,...,n. If ∑ n j=1 Γ j >0, define theUncertain CRITICweights by w j := Γ j ∑ n `=1 Γ ` , j= 1,...,n. The vectorw= (w 1 ,...,w n ) > is called theUncertain CRITIC weight vector of typeMassociated with (X (M) ,S M ). Theorem 4.12.5(Well-definedness of Uncertain CRITIC).In Definition 4.12.4, assumem≥2,Dom(M)6= ∅, andS M is admissible. Then: •the matricesXandRare well-defined (all entries are finite real numbers); •for everyj,k,ρ jk is well-defined and satisfies−1≤ρ jk ≤1; Chapter 4. Weight elicitation Decision-Methods •Γ j ≥0for allj; •if ∑ n j=1 Γ j >0, then the resulting weight vector satisfies w j ≥0 (j= 1,...,n), n ∑ j=1 w j = 1, i.e.,w∈∆ n−1 :=w∈R n ≥0 : ∑ n j=1 w j = 1. Proof.SinceS M :Dom(M)→Ris admissible, eachx ij =S M (x (M) ij )is a finite real number, henceX is well-defined. For each fixedj,x min j andx max j exist because the index set1,...,mis finite. The normalization rule definesr ij either by a ratio with denominatorx max j −x min j >0or by the constant value 0whenx max j =x min j . ThereforeRis well-defined and has finite real entries. Next,σ j ≥0holds by definition of a square root of a nonnegative quantity. Ifσ j = 0orσ k = 0, Definition 4.12.4 setsρ jk = 0, hence it is well-defined. Assumeσ j >0andσ k >0. Let u i :=r ij − ̄r j , v i :=r ik − ̄r k (i= 1,...,m). Then the numerator ofρ jk equals ∑ m i=1 u i v i , and the denominator equals √ ∑ m i=1 u 2 i √ ∑ m i=1 v 2 i , which is positive underσ j >0andσ k >0. By the Cauchy–Schwarz inequality, ∣ ∣ m ∑ i=1 u i v i ∣ ∣ ≤ √ √ √ √ m ∑ i=1 u 2 i √ √ √ √ m ∑ i=1 v 2 i , hence|ρ jk |≤1, so−1≤ρ jk ≤1. For eachj,σ j ≥0and1−ρ jk ≥0becauseρ jk ≤1. Therefore ∑ n k=1 (1−ρ jk )≥0and thusΓ j = σ j ∑ n k=1 (1−ρ jk )≥0. Finally, if ∑ n j=1 Γ j >0, thenw j = Γ j / ∑ n `=1 Γ ` is well-defined, satisfiesw j ≥0, and n ∑ j=1 w j = ∑ n j=1 Γ j ∑ n `=1 Γ ` = 1. Hencew∈∆ n−1 . Related concepts of CRITIC under uncertainty-aware models are listed in Table 4.12. As extensions of CRITIC other than Uncertain CRITIC, several related concepts are also known, including Rough CRITIC [623], Modified CRITIC [624,625], and CRITID [626]. Chapter 4. Weight elicitation Decision-Methods Table 4.12: Related concepts of CRITIC under uncertainty-aware models. kRelated CRITIC concept(s) 2 Intuitionistic Fuzzy CRITIC [607,608] 2 Pythagorean Fuzzy CRITIC [609,610] 2 Fermatean Fuzzy CRITIC [611,612] 3 Picture Fuzzy CRITIC [613,614] 3 Spherical Fuzzy CRITIC [615–617] 3 Neutrosophic CRITIC [618–620] nPlithogenic CRITIC [621,622] 4.13Fuzzy MEREC (Fuzzy MEthod based on the Removal Effects of Criteria) The Method based on the Removal Effects of Criteria (MEREC) is an objective weighting method that determines criterion importance by measuring how the overall performance changes when each criterion is removed [627]. As related methods, approaches such as ITARA are also known [628,629]. Fuzzy MEREC derives objective weights by evaluating the change in fuzzy overall performance caused by removing each criterion, assigning larger weights to criteria with greater impact [630,631]. Definition 4.13.1(TFN-based Fuzzy MEREC (MEREC-F)).[630,631] LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. LetB ⊆ CandK ⊆ Cdenote the sets of benefit-type and cost-type criteria, respectively, withB∪K=CandB∩K=∅. Assume the (positive) triangular fuzzy decision matrix ̃ X= ( ̃x ij ) m×n , ̃x ij = (l ij ,m ij ,u ij ),0< l ij ≤m ij ≤u ij (i= 1,...,m;j= 1,...,n). Step 1 (Fuzzy normalization).Define the normalized TFNs ̃r ij = (r l ij ,r m ij ,r u ij )by ̃r ij = ( l ij /u ? j , m ij /u ? j , u ij /u ? j ) , C j ∈B, ( l − j /u ij , l − j /m ij , l − j /l ij ) , C j ∈K, where u ? j :=max 1≤i≤m u ij >0, l − j :=min 1≤i≤m l ij >0. Then ̃r ij is a positive TFN and (componentwise) lies in(0,1]. Step 2 (Scalarization / defuzzification).Convert ̃r ij to a scalarη ij ∈(0,1]by a fixed defuzzification mapδ:TFN >0 →(0,∞). A standard choice (used widely for TFNs) is η ij :=δ( ̃r ij ) = r l ij + 4r m ij +r u ij 6 . (If numerical zeros may occur in practice, one may replaceη ij by maxη ij ,εwith a fixed tinyε >0so that ln(η ij )is well-defined.) Step 3 (Overall performance with all criteria).Define, for each alternativeA i , S i :=ln 1 + 1 n n ∑ j=1 ∣ ∣ ln(η ij ) ∣ ∣ . Chapter 4. Weight elicitation Decision-Methods Step 4 (Leave-one-criterion-out performance).For each criterionC j and each alternativeA i , define S (−j) i :=ln 1 + 1 n n ∑ k=1 k6=j ∣ ∣ ln(η ik ) ∣ ∣ . Step 5 (Removal effect).For each criterionC j , define its removal effect (impact) by V j := m ∑ i=1 ∣ ∣ S (−j) i −S i ∣ ∣ . Step 6 (Objective criterion weights).If ∑ n j=1 V j >0, define the (crisp) MEREC-F weights by w j := V j ∑ n k=1 V k , j= 1,...,n. Theorem 4.13.2(Well-definedness of MEREC-F weights).Under the assumptions of Definition (MEREC- F), if ∑ n j=1 V j >0, then w j ≥0 (j= 1,...,n),and n ∑ j=1 w j = 1. Moreover, ∑ n j=1 V j = 0holds if and only ifS (−j) i =S i for alli,j, i.e., removing any single criterion never changes the overall performance values. Proof.By construction, eachV j is a finite sum of absolute values, henceV j ≥0. If ∑ n j=1 V j >0, thenw j is well-defined and nonnegative, and n ∑ j=1 w j = n ∑ j=1 V j ∑ n k=1 V k = ∑ n j=1 V j ∑ n k=1 V k = 1. Finally, ∑ n j=1 V j = 0holds exactly when everyV j = 0, equivalently|S (−j) i −S i |= 0for alli,j, i.e.,S (−j) i =S i for alli,j. Uncertain MEREC is defined as follows. Definition 4.13.3(Uncertain MEREC weighting of typeM).LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria, withm≥1andn≥2. LetB ⊆ CandK ⊆ C denote the sets of benefit-type and cost-type criteria, respectively, withB∪K=CandB∩K=∅. Fix an uncertain modelMwith Dom(M)6=∅and an admissible positive scoreS M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M) (i= 1,...,m;j= 1,...,n). Chapter 4. Weight elicitation Decision-Methods Step 0 (Crisp projection).Define the positive real matrixX= (x ij )by x ij :=S M ( x (M) ij ) ∈(0,∞). Step 1 (Positive normalization).For each criterionC j , define x ? j :=max 1≤i≤m x ij >0, x − j :=min 1≤i≤m x ij >0. Define the normalized scalarsη ij ∈(0,1]by η ij := x ij x ? j , C j ∈B, x − j x ij , C j ∈K. (Optional numerical stabilization).Fix a tiny constantε∈(0,1)and replaceη ij by η ε ij :=maxη ij ,ε∈[ε,1]. For notational simplicity, writeη ij for the stabilized values in what follows. Step 2 (Overall performance with all criteria).For each alternativeA i , define S i :=ln 1 + 1 n n ∑ j=1 ∣ ∣ ln(η ij ) ∣ ∣ . Step 3 (Leave-one-criterion-out performance).For eachj∈ 1,...,nand eachi∈ 1,...,m, define S (−j) i :=ln 1 + 1 n n ∑ k=1 k6=j ∣ ∣ ln(η ik ) ∣ ∣ . Step 4 (Removal effect).For each criterionC j , define its removal effect by V j := m ∑ i=1 ∣ ∣ S (−j) i −S i ∣ ∣ . Step 5 (Objective weights).If ∑ n j=1 V j >0, define theUncertain MEREC weightsby w j := V j ∑ n k=1 V k , j= 1,...,n. The vectorw= (w 1 ,...,w n ) > is called theUncertain MERECweight vector of typeMassociated with (X (M) ,S M ). Chapter 4. Weight elicitation Decision-Methods Theorem 4.13.4(Well-definedness of Uncertain MEREC).Under the assumptions of Definition 4.13.3: 1. All quantitiesη ij ,S i ,S (−j) i , andV j are well-defined and finite real numbers, withη ij ∈(0,1]and V j ≥0. 2. If ∑ n j=1 V j >0, then the weights are well-defined and satisfy w j ≥0 (j= 1,...,n), n ∑ j=1 w j = 1, i.e.,w∈∆ n−1 :=w∈R n ≥0 : ∑ n j=1 w j = 1. 3. Moreover, ∑ n j=1 V j = 0holds if and only ifS (−j) i =S i for alliandj, i.e., removing any single criterion never changes the performance values. Proof.(1) By admissibility ofS M , eachx ij =S M (x (M) ij )is strictly positive and finite. Hence for eachj, the extremax ? j andx − j exist (finite index set) and are positive. IfC j ∈ Bthenη ij =x ij /x ? j ∈(0,1]; ifC j ∈ K thenη ij =x − j /x ij ∈(0,1]. After optional stabilization,η ij ∈[ε,1]. Therefore ln(η ij )is defined and finite, so|ln(η ij )|is finite. Consequently, the sums inS i andS (−j) i are finite, their arguments are>1, and thus S i andS (−j) i are finite real numbers. Finally, eachV j is a finite sum of absolute values, henceV j ≥0and finite. (2) If ∑ n j=1 V j >0, then eachw j =V j / ∑ n k=1 V k is well-defined and nonnegative. Moreover, n ∑ j=1 w j = n ∑ j=1 V j ∑ n k=1 V k = ∑ n j=1 V j ∑ n k=1 V k = 1, sow∈∆ n−1 . (3) SinceV j ≥0for allj, one has ∑ n j=1 V j = 0if and only ifV j = 0for allj. ButV j = ∑ m i=1 |S (−j) i −S i |= 0 holds if and only if|S (−j) i −S i |= 0for alli, equivalentlyS (−j) i =S i for alli. Related concepts of MEREC under uncertainty-aware models are listed in Table 4.13. Table 4.13: Related concepts of MEREC under uncertainty-aware models. kRelated MEREC concept(s) 2 Intuitionistic Fuzzy MEREC [632] 2 Pythagorean Fuzzy MEREC [633,634] 3 Spherical Fuzzy MEREC [630,635] 3 Neutrosophic MEREC [636,637] Chapter 4. Weight elicitation Decision-Methods 4.14Fuzzy FUCOM FUCOM orders criteria, elicits comparative priorities between consecutive criteria, and finds weights by minimizing deviation from these ratios while enforcing full consistency conditions in practice [638–640]. Fuzzy FUCOM expresses comparative priorities as fuzzy ratios, solves for fuzzy weights under consistency constraints, and applies defuzzification/possibility measures to obtain a stable ranking result [641,642]. Definition 4.14.1(Fuzzy FUCOM (FUCOM-F) with triangular fuzzy numbers).[641, 642] LetC= C 1 ,...,C n be a finite set of criteria. Assume that decision makers (DMs) provide animportance ranking C j(1) <C j(2) <·<C j(n) , wherej(·)is a permutation of1,...,nand ties are allowed. (0) Triangular fuzzy numbers (TFNs) and basic operations.Let TFN >0 :=(l,m,u)∈R 3 : 0< l≤m≤u. For ̃x= (l x ,m x ,u x )and ̃y= (l y ,m y ,u y )inTFN >0 define ̃x⊕ ̃y:= (l x +l y , m x +m y , u x +u y ), ̃x⊗ ̃y:= (l x l y , m x m y , u x u y ), ̃y −1 := ( 1 u y , 1 m y , 1 l y ) , ̃x ̃y:= ̃x⊗ ̃y −1 = ( l x u y , m x m y , u x l y ) , ̃ 1 := (1,1,1). We use the componentwise preorder on TFNs: ̃a ̃ b⇐⇒a ` ≤b ` , a m ≤b m , a u ≤b u for ̃a= (a ` ,a m ,a u ), ̃ b= (b ` ,b m ,b u ), and componentwise absolute value| ̃a|:= (|a ` |,|a m |,|a u |). (1) Fuzzy comparative priorities between consecutive ranks.For each rankk= 1,...,n−1, DMs elicit a TFN ̃φ k/(k+1) ∈TFN >0 , interpreted as the comparative priority ofC j(k) overC j(k+1) . (2) Unknown fuzzy weights.The FUCOM-F goal is to determine fuzzy criterion weights ̃w j(k) = (w ` j(k) ,w m j(k) ,w u j(k) )∈TFN ≥0 (k= 1,...,n), with the usual TFN feasibility0≤w ` j(k) ≤w m j(k) ≤w u j(k) . (3) FUCOM-F consistency conditions (ideal equalities).The two FUCOM consistency requirements are: (C1)(Ratio consistency)For eachk= 1,...,n−1, ̃w j(k) ̃w j(k+1) = ̃φ k/(k+1) equivalently ̃w j(k) = ̃w j(k+1) ⊗ ̃φ k/(k+1) . Chapter 4. Weight elicitation Decision-Methods (C2)(Transitivity)For eachk= 1,...,n−2, ̃w j(k) ̃w j(k+2) = ̃φ k/(k+1) ⊗ ̃φ (k+1)/(k+2) equivalently ̃w j(k) = ̃w j(k+2) ⊗ ̃φ k/(k+1) ⊗ ̃φ (k+1)/(k+2) . In practice, exact equality may not hold; FUCOM-F therefore minimizes the deviation from full consistency. (4) FUCOM-F optimization model (minimizing deviation from maximum consistency).Let ξ≥0be a (crisp) deviation variable (DMC). FUCOM-F determines the fuzzy weights by solving: minξ(4.11) s.t. ∣ ∣ ̃w j(k) − ̃w j(k+1) ⊗ ̃φ k/(k+1) ∣ ∣ (ξ,ξ,ξ),k= 1,...,n−1,(4.12) ∣ ∣ ̃w j(k) − ̃w j(k+2) ⊗ ̃φ k/(k+1) ⊗ ̃φ (k+1)/(k+2) ∣ ∣ (ξ,ξ,ξ),k= 1,...,n−2,(4.13) n ⊕ k=1 ̃w j(k) = ̃ 1,(4.14) 0≤w ` j(k) ≤w m j(k) ≤w u j(k) ,k= 1,...,n.(4.15) Ifξ= 0is attainable, the obtained weights satisfy full (fuzzy) consistency. (5) Defuzzification (optional; crisp weights).A common defuzzification for ̃w= (w ` ,w m ,w u )is the graded mean integration representation (GMIR): GMIR( ̃w) := w ` + 4w m +w u 6 . Define crisp weights and (optionally) renormalize: w i :=GMIR( ̃w i ), w i ← w i ∑ n t=1 w t so that n ∑ i=1 w i = 1. The resulting(w 1 ,...,w n )is called theFUCOM-F (defuzzified) criterion-weight vector. Using Uncertain Sets, we define Uncertain FUCOM (U-FUCOM) as follows. Definition 4.14.2(Uncertain FUCOM (U-FUCOM): expected-value realization).LetC=C 1 ,...,C n be a finite set of criteria,n≥2. Assume decision makers provide animportance order C j(1) <C j(2) <·<C j(n) , wherej(·)is a permutation of1,...,n(ties allowed). Work on a fixed uncertainty space(Γ,L,M). (1) Uncertain comparative priorities.For each consecutive rankk= 1,...,n−1, let Φ k : Γ→P(R) be anuncertain comparative-priority coefficientrepresenting the relative importance ofC j(k) overC j(k+1) . Assume: Φ k (γ)⊆R >0 (∀γ∈Γ), φ k :=E[Φ k ]∈(0,∞). Chapter 4. Weight elicitation Decision-Methods Define alsoφ k,k+2 :=φ k φ k+1 fork= 1,...,n−2. (2) Decision variables (crisp weights) and deviation.Letw j(1) ,...,w j(n) ∈R ≥0 be unknown criterion weights and letξ∈R ≥0 be the deviation-from-full-consistency variable. (3) U-FUCOM optimization model.U-FUCOM determines(w,ξ)by solving the linear program: minξ(4.16) s.t.−ξ≤w j(k) −φ k w j(k+1) ≤ξ,k= 1,...,n−1,(4.17) −ξ≤w j(k) −φ k,k+2 w j(k+2) ≤ξ,k= 1,...,n−2,(4.18) n ∑ k=1 w j(k) = 1,(4.19) w j(k) ≥0 (k= 1,...,n), ξ≥0.(4.20) Any optimal solutionw ? = (w ? 1 ,...,w ? n )(with indices permuted back fromj(·)) is called aU-FUCOM weight vector. If the optimal value satisfiesξ ? = 0, the obtained weights are (fully) consistent with the expected ratiosφ k and transitivity productsφ k φ k+1 . Definition 4.14.3(Uncertain-set output of U-FUCOM).Letw ? be any U-FUCOM optimal weight vector from Definition 4.14.2. Define singleton-valued uncertain sets W i : Γ→P(R), W i (γ) :=w ? i , i= 1,...,n. ThenW= (W 1 ,...,W n )is called theU-FUCOM uncertain weight vector. Theorem 4.14.4(Uncertain-set structure and well-definedness of U-FUCOM).Assume the setting of Definition 4.14.2 and suppose: (A1)(Positive finite expected ratios)φ k =E[Φ k ]∈(0,∞)for allk= 1,...,n−1. Then: (i)the feasible set of(4.16)–(4.20)is nonempty; (i)an optimal solution(w ? ,ξ ? )exists; (i)every optimal weight vector satisfiesw ? i ≥0and ∑ n i=1 w ? i = 1; (iv)the outputW= (W 1 ,...,W n )in Definition 4.14.3 is an uncertain-set structured output on(Γ,L,M). Proof.(i) Feasibility.Consider the uniform weight vector ̄wdefined by ̄w i := 1/nfor alli. Let B:=max 1,max 1≤k≤n−1 φ k ,max 1≤k≤n−2 φ k φ k+1 ∈(0,∞). Chapter 4. Weight elicitation Decision-Methods For anyk, ∣ ∣ ̄w j(k) −φ k ̄w j(k+1) ∣ ∣ ≤ ̄w j(k) +φ k ̄w j(k+1) ≤ 1 n + φ k n ≤ 1 +B n , and similarly ∣ ∣ ̄w j(k) −(φ k φ k+1 ) ̄w j(k+2) ∣ ∣ ≤ 1 n + φ k φ k+1 n ≤ 1 +B n . Hence( ̄w, ̄ ξ)with ̄ ξ:= (1 +B)/nsatisfies all constraints, so the feasible set is nonempty. (i) Existence of an optimizer.All feasiblewlie in the standard simplex ∆ :=w∈R n :w i ≥0, n ∑ i=1 w i = 1, which is compact. Moreover, part (i) shows there exists a feasible point withξ= ̄ ξ, hence the optimal value satisfies0≤ξ ? ≤ ̄ ξ. Therefore it suffices to minimize the continuous function(w,ξ)7→ξover the compact set F ∩ ( ∆×[0, ̄ ξ] ) , whereFis the feasible set; compactness follows because all constraints in (4.17)–(4.20) are closed linear inequalities/equalities. By the Weierstrass theorem, an optimizer(w ? ,ξ ? )exists. (i) Normalization and nonnegativity.These are enforced directly by (4.19)–(4.20). (iv) Uncertain-set structure.For eachi, the mapW i (γ) =w ? i is constant (singleton-valued), hence measurable, and thus an uncertain set. ThereforeW= (W 1 ,...,W n )is an uncertain-set structured output on(Γ,L,M). Chapter 4. Weight elicitation Decision-Methods Chapter 5 Structure / causality decision-modelling (inter-criteria influence) Structure/causality modelling methods (e.g., DEMATEL, ISM, MICMAC, FCM) elicit inter-criteria influ- ence matrices, derive causal graphs, identify driving/dependent factors, and prioritize criteria accordingly. For convenience, a concise comparison of the four structure/causality decision-modeling frameworks consid- ered in this chapter is presented in Table 5.1. Table 5.1: A concise comparison of four structure/causality decision-modeling frameworks. Method Primary purposeTypical input / core mecha- nism Typical output / role in analysis DEMATEL To analyze the strength and di- rection of inter-criteria causal in- fluence. Direct-influence matrix; normal- ization and total-relation matrix computation; cause–effect anal- ysis via prominence and relation indices. Cause/effect grouping, influence intensity, and overall centrality of criteria. ISMTo derive a hierarchical struc- tural model from pairwise influ- ence or reachability judgments. Reachability-based relation ma- trix; transitive closure; level par- titioning of factors. A layered directed hierarchy showing which factors are foun- dational, intermediate, or top- level. MICMAC To classify factors according to driving power and dependence power. Reachability matrix (often ob- tained from ISM); row/column aggregation and quadrant-based classification. Partition of factors into au- tonomous, dependent, linkage, and driving classes. FCMTo model feedback-rich causal systems and simulate their dy- namic behavior over time. Signed weighted causal graph (or weight matrix) together with an activation/update rule. Dynamic trajectories, equilib- rium states, cycles, and scenario- based system behavior under feedback. 5.1 Fuzzy DEMATEL (Fuzzy Decision Making Trial and Evaluation Laboratory) DEMATEL analyzes causal relationships among criteria using direct influence matrices, computes total influence, and separates factors into cause and effect groups [643–645]. Fuzzy DEMATEL models uncertain causal influences among criteria using fuzzy direct-relation matrices, derives total relations, and identifies cause–effect groups via prominence (D+R) and relation (D−R) [154,646,647]. 155 Chapter 5. Structure / causality decision-modelling (inter-criteria influence) Definition 5.1.1(TFN-based fuzzy DEMATEL data and outputs).[154,646,647] LetC=C 1 ,...,C n be the set of criteria. LetTFN:=(l,m,u)∈R 3 :l≤m≤u. (1) Linguistic-to-fuzzy encoding and expert aggregation.LetLbe a linguistic scale and letφ:L→ [0,1] 3 map each linguistic term`to a TFNφ(`) = (l ` ,m ` ,u ` )(e.g. a 5-level scale). AssumeP≥1experts provide judgments` (p) ij ∈Lfor the influenceC i →C j . Define the individual fuzzy direct-relation matrices ̃ X (p) = ( ̃x (p) ij ) n×n , ̃x (p) ij :=φ(` (p) ij ), ̃x (p) i = (0,0,0), and their componentwise mean (aggregation) ̃ X= ( ̃x ij ) n×n , ̃x ij := 1 P P ∑ p=1 ̃x (p) ij . (2) Defuzzification, normalization, and total relation.For a TFN ̃x= (l,m,u), define BNP defuzzi- fication by BNP( ̃x) := l+m+u 3 . Defuzzify ̃ Xentrywise to obtain the crisp direct-relation matrix X= (x ij ), x ij :=BNP( ̃x ij ). Let s:=max 1≤i≤n n ∑ j=1 x ij , N:= 1 s X. Assumeρ(N)<1. The total-relation matrix is T:=N(I−N) −1 = ∞ ∑ k=1 N k . (3) Dispatch/receive and cause–effect indices.Let1∈R n be the all-ones vector and define D:=T1, R:=T T 1. For each criterionC i , define Prominence i :=D i +R i ,Relation i :=D i −R i . Then Prominence i measures overall centrality, while Relation i >0(resp.<0) indicates a net cause (resp. net effect). Definition 5.1.2(Uncertain DEMATEL (U-DEMATEL): expected-value realization).LetC=C 1 ,...,C n be a finite set of criteria withn≥2, and fix an uncertainty space(Γ,L,M). (1) Uncertain direct-relation matrix.Anuncertain direct influence assessmentis ann×nmatrix ̃ X= ( ̃ X ij ) ∈US(R ≥0 ) n×n , ̃ X i ≡0, Chapter 5. Structure / causality decision-modelling (inter-criteria influence) where each entry ̃ X ij : Γ→ P(R ≥0 )is an uncertain number encoding the (nonnegative) direct influence C i →C j . Assume the expected values exist and define thecrisp expected direct-relation matrix X= (x ij )∈R n×n ≥0 , x ij :=E[ ̃ X ij ]<∞. (2) Normalization.Let s:=max 1≤i≤n n ∑ j=1 x ij . Assumes >0and define the normalized matrix N:= 1 s X. (3) Total-relation matrix.Define thetotal-relation matrixby the Neumann series T:= ∞ ∑ k=1 N k , whenever the series converges. (Equivalently, when it converges one may writeT=N(I−N) −1 .) (4) Prominence and relation indices.Let1∈R n be the all-ones vector and set D:=T1, R:=T T 1. For each criterionC i , define Prominence i :=D i +R i ,Relation i :=D i −R i . Then Relation i >0indicates a net cause and Relation i <0indicates a net effect (with respect to the expected influence network). Using Uncertain Sets, we define U-DEMATEL as follows. Definition 5.1.3(Uncertain-set output of U-DEMATEL).Let(Prominence i ,Relation i ) n i=1 be the indices produced by Definition 5.1.2. Define singleton-valued uncertain sets Π i : Γ→P(R),Π i (γ) :=Prominence i ,R i : Γ→P(R),R i (γ) :=Relation i . The collection ( (Π i ,R i ) ) n i=1 is called theU-DEMATEL uncertain index family. Theorem 5.1.4(Uncertain-set structure and well-definedness of U-DEMATEL).In the setting of Defini- tion 5.1.2, assume: (A1)(Finite expectations)x ij =E[ ̃ X ij ]<∞for alli,j, andX6= 0(sos >0). (A2)(Spectral-radius condition)ρ(N)<1, whereρ(·)denotes the spectral radius. Chapter 5. Structure / causality decision-modelling (inter-criteria influence) Then: (i)the seriesT= ∑ ∞ k=1 N k converges (entrywise and in any matrix norm); (i)(I−N)is invertible andT=N(I−N) −1 ; (i)the vectorsD=T1andR=T T 1are well-defined and finite; (iv)the indicesProminence i andRelation i are well-defined real numbers; (v)the outputs in Definition 5.1.3 form an uncertain-set structured output on(Γ,L,M). Proof.(i)Under (A2), the matrixNsatisfiesρ(N)<1. A standard result in linear algebra implies that for any induced matrix norm‖·‖there existsk 0 such that‖N k 0 ‖<1; hence ∑ k≥1 N k converges absolutely in that norm. Therefore the Neumann series ∞ ∑ k=0 N k converges, and so doesT= ∑ ∞ k=1 N k . (i)Since ∑ ∞ k=0 N k converges, the Neumann-series identity holds: (I−N) ( ∞ ∑ k=0 N k ) =I= ( ∞ ∑ k=0 N k ) (I−N), so(I−N)is invertible and (I−N) −1 = ∞ ∑ k=0 N k . Multiplying byNgives T= ∞ ∑ k=1 N k =N ∞ ∑ k=0 N k =N(I−N) −1 . (i)By (i),Thas finite real entries, henceD=T1andR=T T 1are finite vectors inR n . (iv)Each Prominence i =D i +R i and Relation i =D i −R i is therefore a finite real number. (v)For eachi,Π i (γ) =Prominence i andR i (γ) =Relation i are constant singleton-valued maps, hence measurable and therefore uncertain sets. This yields an uncertain-set structured output on(Γ,L,M). Related concepts of DEMATEL under uncertainty-aware models are listed in Table 5.2. Uncertain DEMATEL has also been extended in other directions; for example, Rough DEMATEL [666, 667], AHP-DEMATEL [668,669], TOPSIS-DEMATEL [670,671], Interval DEMATEL [672,673], Linguistic DEMATEL [674,675], and Grey DEMATEL [676,677] are also known as related variants. Chapter 5. Structure / causality decision-modelling (inter-criteria influence) Table 5.2: Related concepts of DEMATEL under uncertainty-aware models. kRelated DEMATEL concept(s) Representative references 1 Fuzzy DEMATEL[648,649] 2 Intuitionistic Fuzzy DEMATEL[650–652] 2 Bipolar Fuzzy DEMATEL[653,654] 2 Pythagorean fuzzy DEMATEL[655,656] 2 Fermatean fuzzy DEMATEL[657–659] 3 Picture Fuzzy DEMATEL[660] 3 Hesitant Fuzzy DEMATEL[661] 3 Spherical Fuzzy DEMATEL[662,663] 3 Neutrosophic DEMATEL[664,665] 5.2 Fuzzy ISM (Fuzzy Interpretive Structural Modeling) Interpretive Structural Modeling structures complex factors by expert judgments on pairwise reachability, builds a transitive reachability matrix, and derives a hierarchical directed graph [678–680]. Fuzzy ISM replaces binary reachability with graded membership degrees from linguistic judgments, propagates fuzziness through transitive closure, and yields a hierarchical influence graph [681, 682]. As an extension, concepts such as Fuzzy Total Interpretive Structural Modeling have also been studied [683,684]. Definition 5.2.1(Fuzzy Interpretive Structural Modeling (Fuzzy ISM)).[681,682] LetF=f 1 ,...,f n be a finite set of factors (system variables), withn≥2. Step 1 (Directional judgments: SSIM symbols).For each unordered pairi,jwithi6=j, an expert specifies thedirectionof influence using s ij ∈V,A,X,O, interpreted as: V:f i influencesf j , A:f j influencesf i , X:f i andf j influence each other, O:f i andf j are unrelated. (Equivalently, one may store these symbols in a self-structural interaction matrix (SSIM).) Step 2 (Fuzzy strength of influence).Fix a familyFof fuzzy degrees representing influence strengths, together with a mapping from linguistic terms toF. Typical choices include: •F= [0,1](direct fuzzy degrees), or •F=TFN [0,1] (triangular fuzzy numbers within[0,1]). For each pairi,jwithi6=j, assign a fuzzy influence strength ̃a ij ∈F Chapter 5. Structure / causality decision-modelling (inter-criteria influence) corresponding to the expert’s linguistic assessment of the influence magnitude. Step 3 (Fuzzy adjacency / AFSSIM).Define the (possibly aggregated) fuzzy adjacency matrix ̃ A= ( ̃a → ij ) n×n ∈F n×n by setting ̃a → i := 1 F (full self-reachability) and, fori6=j, ( ̃a → ij , ̃a → ji ) := ( ̃a ij ,0 F ), s ij =V, (0 F , ̃a ij ), s ij =A, ( ̃a ij , ̃a ij ), s ij =X, (0 F ,0 F ), s ij =O, where0 F and1 F denote the null and unit elements inF. If multiple experts provide matrices ̃ A (1) ,..., ̃ A (p) , their aggregation (AFSSIM) is obtained by a fixed aggregation operator Agg:F p →Fapplied entrywise: ̃ A ij :=Agg ( ̃ A (1) ij ,..., ̃ A (p) ij ) . Step 4 (Fuzzy reachability matrix via fuzzy transitive closure).AssumeFsupports a fuzzy con- junction∧and disjunction∨(e.g., min/max on[0,1]). For matrices overF, define the max–min product ◦by ( ̃ P◦ ̃ Q) ik := n ∨ j=1 ( ̃ P ij ∧ ̃ Q jk ) . Let ̃ A (1) := ̃ Aand ̃ A (t+1) := ̃ A (t) ◦ ̃ A. Thefuzzy reachability matrix(FRM) is defined as the fuzzy transitive closure ̃ R:= ̃ A (1) ∨ ̃ A (2) ∨ · ∨ ̃ A (n) . Step 5 (Defuzzification and thresholding: DFRM).Fix a crisp score/defuzzification map Score:F→ [0,1]and a thresholdθ∈(0,1]. Define thedefuzzified (binary) reachability matrixD= (d ij )∈0,1 n×n by d ij := 1,Score( ̃ R ij )≥θ, 0,Score( ̃ R ij )< θ. Step 6 (Level partitioning and ISM hierarchy).For eachi, define the reachability set, antecedent set, and intersection set: Reach(i) :=f j ∈F |d ij = 1,Ante(i) :=f j ∈F |d ji = 1,Inter(i) :=Reach(i)∩Ante(i). Thetop levelconsists of allf i satisfying Reach(i) =Inter(i). Remove these factors and repeat the same construction on the remaining set until all factors are assigned to levels. The resulting layered digraph (with edgesf i →f j whend ij = 1) is called theFuzzy ISM modelinduced by the expert judgments and the chosen fuzzy/defuzzification settings. Proposition 5.2.2(Basic well-posedness and termination).AssumeFis finite andFadmits∧,∨so that max–min composition is defined. Then: •the fuzzy transitive closure ̃ Rin Definition 5.2.1 is well-defined; Chapter 5. Structure / causality decision-modelling (inter-criteria influence) •for any score mapScoreand thresholdθ∈(0,1], the binary matrixDis well-defined; •the level partitioning procedure terminates after at mostniterations and assigns every factor to exactly one level. Proof.All entries of ̃ Alie inF, and ̃ A (t) is obtained from finitely many∧,∨operations, hence is well-defined; so is ̃ Ras a finite join. Applying Score and thresholding yields a binary entry for each pair(i,j), soD is well-defined. At each level-partition iteration, at least one factor is selected for the current top level (a standard property of reachability/antecedent-based partitioning on finite sets), hence the remaining set strictly decreases. Therefore the procedure terminates in at mostnsteps, and disjointness of removed sets implies each factor is assigned to exactly one level. We now generalize this framework by allowing pairwise influences to take values in an arbitrary uncertain modelM. Definition 5.2.3(Uncertain ISM of typeM).Let F=f 1 ,...,f n be a finite nonempty set of factors, wheren≥1. Fix an uncertain modelMwith degree-domain Dom(M)⊆[0,1] k for some integerk≥1. Assume that the direct influence structure is represented by anuncertain influence relation R M :F ×F −→Dom(M), whereR M (f i ,f j )expresses the uncertain degree to which factorf i directly influences factorf j . Fix further: • a total score map Score M :Dom(M)−→[0,1], which converts uncertain influence values into scalar influence strengths; • a threshold θ∈(0,1]. Step 1 (Scored adjacency matrix).Define the scored adjacency matrix A= (a ij ) n×n ∈[0,1] n×n by a ij := 1,i=j, Score M ( R M (f i ,f j ) ) , i6=j. Chapter 5. Structure / causality decision-modelling (inter-criteria influence) Thus self-reachability is fixed at1, while off-diagonal entries are induced from the uncertain relation. Step 2 (Max–min reachability closure).For matricesP,Q∈[0,1] n×n , define their max–min product P◦Qby (P◦Q) ik :=max 1≤j≤n minP ij ,Q jk . Define recursively A [1] :=A, A [t+1] :=A [t] ◦A(t≥1). Theuncertain reachability matrixis defined by ̃ R:=A [1] ∨A [2] ∨·∨A [n] , where∨denotes the entrywise maximum. Writing ̃ R= (r ij ) n×n , eachr ij ∈[0,1]represents the aggregated uncertain reachability strength fromf i tof j . Step 3 (Binary reachability matrix).Define the binary reachability matrix D= (d ij ) n×n ∈0,1 n×n by thresholding: d ij := 1, r ij ≥θ, 0, r ij < θ. Step 4 (Level partitioning).Set U 1 :=F. For each iterationt≥1, providedU t 6=∅, define for everyf i ∈U t : Reach t (i) :=f j ∈U t :d ij = 1, Ante t (i) :=f j ∈U t :d ji = 1, Inter t (i) :=Reach t (i)∩Ante t (i). Thet-th level is L t :=f i ∈U t :Reach t (i) =Inter t (i). IfU t 6=∅, remove the identified level and define U t+1 :=U t t . Repeat until the remaining set becomes empty. The ordered family of levels (L 1 ,L 2 ,...,L T ) together with the directed graph onFwhose edges are given by f i →f j ⇐⇒d ij = 1 is called theUncertain ISM model of typeMinduced by (F,R M ,Score M ,θ). Chapter 5. Structure / causality decision-modelling (inter-criteria influence) Theorem 5.2.4(Well-definedness of Uncertain ISM).Assume the data in Definition 5.2.3 satisfy: (A1)F=f 1 ,...,f n is finite and nonempty; (A2)R M :F ×F →Dom(M)is a total map; (A3)Score M :Dom(M)→[0,1]is a total map; (A4)θ∈(0,1]. Then: (i) the scored adjacency matrixA, all powersA [t] (t= 1,...,n), and the uncertain reachability matrix ̃ R are well-defined; (i) the binary reachability matrixDis well-defined, reflexive, and transitive; (i) for every iterationtwithU t 6=∅, the levelL t is well-defined and nonempty; (iv) the level-partitioning procedure terminates after at mostniterations; (v) the obtained levelsL 1 ,...,L T are pairwise disjoint and satisfy F=L 1 tL 2 t·tL T . Hence Uncertain ISM of typeMis well-defined. Proof.(i) Well-definedness ofA,A [t] , and ̃ R.For each pair(i,j), the valueR M (f i ,f j )∈Dom(M)is defined by (A2), and Score M ( R M (f i ,f j ) ) ∈[0,1] is defined by (A3). Therefore every entrya ij ofAis well-defined, so A∈[0,1] n×n . Now letP,Q∈[0,1] n×n . Since minP ij ,Q jk ∈[0,1]for everyj, and the maximum of finitely many numbers in[0,1]again lies in[0,1], the entry (P◦Q) ik =max 1≤j≤n minP ij ,Q jk is well-defined and belongs to[0,1]. HenceP◦Q∈[0,1] n×n . By induction, eachA [t] ∈[0,1] n×n is well-defined fort= 1,...,n. Since ̃ Ris the entrywise maximum of finitely many matricesA [1] ,...,A [n] , it is also well-defined and belongs to[0,1] n×n . Chapter 5. Structure / causality decision-modelling (inter-criteria influence) (i) Well-definedness, reflexivity, and transitivity ofD.Because everyr ij ∈[0,1]is well-defined and θ∈(0,1], each entryd ij ∈0,1is well-defined by thresholding. ThusD∈0,1 n×n . Sincea i = 1for everyi, we also haver i ≥1, hencer i = 1. Becauseθ≤1, it follows thatd i = 1for alli. ThereforeDis reflexive. We next show transitivity. For a directed path π= (i=i 0 ,i 1 ,...,i m =j) in1,...,n, define its strength by str(π) :=min 0≤q<m a i q i q+1 . By construction of ̃ R=A [1] ∨·∨A [n] , the entryr ij is the maximum strength of all simple directed paths fromitoj; it suffices to consider simple paths because removing a cycle does not decrease the minimum edge value along the path, and any simple path has length at mostn−1. Now supposed ij = 1andd jk = 1. Then r ij ≥θ, r jk ≥θ. Choose simple pathsπ ij fromitojandπ jk fromjtokwhose strengths arer ij andr jk , respectively. Concatenating them gives a path fromitok; after removing repeated cycles, one obtains a simple path fromitokwhose strength is at least minr ij ,r jk ≥θ. Hence r ik ≥minr ij ,r jk ≥θ, sod ik = 1. ThereforeDis transitive. (i) Nonemptiness of each levelL t .Fix an iterationtwithU t 6=∅. Restrict the relation represented byDtoU t . SinceDis reflexive and transitive, the restricted relation is also reflexive and transitive. Define an equivalence relation∼ t onU t by f i ∼ t f j ⇐⇒d ij = 1andd ji = 1. LetU t /∼ t be the set of equivalence classes, and define a relation t on these classes by [f i ] t [f j ]⇐⇒d ij = 1. BecauseDis reflexive and transitive, t is a partial order on the finite setU t /∼ t . Hence it has at least one maximal class; choose one and denote it by[f i ? ]. Take anyf j ∈Reach t (i ? ). Thend i ? j = 1, so [f i ? ] t [f j ]. By maximality of[f i ? ], this implies[f j ] = [f i ? ]. Therefored ji ? = 1, i.e., f j ∈Ante t (i ? ). Chapter 5. Structure / causality decision-modelling (inter-criteria influence) Thus Reach t (i ? )⊆Ante t (i ? ). Consequently, Reach t (i ? ) =Reach t (i ? )∩Ante t (i ? ) =Inter t (i ? ). Hencef i ? ∈L t , soL t 6=∅. (iv) Termination.WheneverU t 6=∅, part (i) shows thatL t 6=∅. Therefore U t+1 =U t t is a strict subset ofU t . Since|U 1 |=n, after at mostniterations the remaining set becomes empty. Hence the procedure terminates. (v) Disjointness and covering property.By construction, L t ⊆U t andU t+1 =U t t . Therefore the setsL 1 ,L 2 ,...,L T are pairwise disjoint. Since the procedure stops exactly when no factors remain, every factor is removed at some step, whence F=L 1 tL 2 t·tL T . All required objects are therefore well-defined, and the hierarchical decomposition exists and terminates. For reference, related concepts of Interpretive Structural Modeling under uncertainty-aware models are listed in Table 5.3. Table 5.3: Related concepts of Interpretive Structural Modeling under uncertainty-aware models. kRelated ISM concept(s) 1 Fuzzy Interpretive Structural Modeling 2 Intuitionistic Fuzzy Interpretive Structural Modeling 3 Neutrosophic Interpretive Structural Modeling [685–687] 5.3 Fuzzy MICMAC (cross-impact / driving–dependence analysis) MICMAC analyzes a binary reachability matrix to compute driving power and dependence, classifying factors into autonomous, dependent, linkage, and driving clusters [688, 689]. Fuzzy MICMAC computes driving and dependence strengths from fuzzy reachability values, classifies criteria using graded impacts, and supports robust clustering under ambiguous judgments [690–692]. Chapter 5. Structure / causality decision-modelling (inter-criteria influence) Definition 5.3.1(Fuzzy MICMAC (driving–dependence analysis)).[693–695] LetF=f 1 ,...,f n be a finite set of system factors (n≥2). Assume that afuzzy reachability matrix(FRM) has been obtained (typically after fuzzy transitive closure in Fuzzy ISM), ̃ R= ( ̃r ij ) n×n , ̃r ij ∈F, whereFis a chosen fuzzy evaluation family (e.g., TFNs in[0,1]), introduced to handle uncertainty in expert judgments. Step 1 (Defuzzification / scoring).Fix a score map Score:F→[0,1]and define the crisp reachability matrix R= ( r ij ) n×n , r ij :=Score( ̃r ij )∈[0,1]. Step 2 (Thresholding: DFRM).Fix a thresholdθ∈(0,1]and define thedefuzzified (binary) reachability matrix(DFRM) D= ( d ij ) n×n ∈0,1 n×n , d ij :=1[r ij ≥θ]. (Thresholding assigns1when the value is at least the cutoff and0otherwise.) Step 3 (Driving power and dependence power).Define thedriving poweranddependence powerof each factorf i by DRP(i) := n ∑ j=1 d ij ,DEP(i) := n ∑ j=1 d ji . Row sums measure how many factors are reachable (directly or indirectly), while column sums measure how strongly a factor is influenced by others. Step 4 (Quadrant-based MICMAC classification).Choose cutoffs(τ D ,τ P )for “high/low” driving and dependence; a common choice is τ D := 1 n n ∑ i=1 DRP(i), τ P := 1 n n ∑ i=1 DEP(i). Classify each factorf i into one of the four MICMAC clusters (autonomous, dependent, linkage, independent) according to its(DRP(i),DEP(i))location. Formally, F aut :=f i :DRP(i)< τ D and DEP(i)< τ P , F dep :=f i :DRP(i)< τ D and DEP(i)≥τ P , F lin :=f i :DRP(i)≥τ D and DEP(i)≥τ P , F ind :=f i :DRP(i)≥τ D and DEP(i)< τ P . The resulting partition is called theFuzzy MICMAC classificationinduced by( ̃ R,Score,θ,τ D ,τ P ). (Driv- ing/dependence-based grouping is the standard purpose of fuzzy MICMAC in ISM-based studies.) Theorem 5.3.2(Well-definedness of Fuzzy MICMAC).Under Definition 5.3.1, assume: •Fis finite withn≥2; Chapter 5. Structure / causality decision-modelling (inter-criteria influence) • Score:F→[0,1]is well-defined onF; •θ∈(0,1]and(τ D ,τ P )∈R 2 are fixed. Then: 1. The matricesRandDare well-defined, withR∈[0,1] n×n andD∈0,1 n×n . 2. For eachi,DRP(i)andDEP(i)are well-defined integers satisfying 0≤DRP(i)≤n,0≤DEP(i)≤n. 3. The four setsF aut ,F dep ,F lin ,F ind form a partition ofF, i.e., they are pairwise disjoint and their union equalsF. Proof.(1) Since Score maps every ̃r ij ∈Finto[0,1], each entryr ij is a well-defined real number in[0,1]. Withθ∈(0,1], the indicatord ij =1[r ij ≥θ]is well-defined and lies in0,1for every(i,j). HenceRand Dare well-defined matrices of the stated types. (2) Because eachd ij ∈0,1and there are exactlynterms in each sum, DRP(i) = ∑ n j=1 d ij and DEP(i) = ∑ n j=1 d ji are well-defined integers with bounds0≤DRP(i)≤nand0≤DEP(i)≤n. (3) Fix anyf i ∈ F. Exactly one of the two inequalities DRP(i)< τ D or DRP(i)≥τ D holds, and exactly one of DEP(i)< τ P or DEP(i)≥τ P holds. Therefore,f i belongs to exactly one of the four sets defined in Step 4. This showsF aut ∪F dep ∪F lin ∪F ind =F. Moreover, nof i can satisfy two incompatible pairs of inequalities simultaneously, so the four sets are pairwise disjoint. Thus they form a partition ofF. We now extend this idea to a general uncertain modelM. Definition 5.3.3(Uncertain MICMAC of typeM).Let F=f 1 ,...,f n be a finite nonempty set of factors, wheren≥1. Fix an uncertain modelMwith degree-domain Dom(M)⊆[0,1] k for some integerk≥1. Assume that anuncertain reachability matrixof typeMis given: ̃ R M = (ρ ij ) n×n ∈Dom(M) n×n , whereρ ij ∈Dom(M)represents the uncertain reachability degree from factorf i to factorf j . Typically, ̃ R M is obtained from an Uncertain ISM procedure. Fix further: Chapter 5. Structure / causality decision-modelling (inter-criteria influence) • a total score map Score M :Dom(M)−→[0,1], • a threshold θ∈(0,1]. Step 1 (Scored reachability matrix).Define the scored reachability matrix R= (r ij ) n×n ∈[0,1] n×n by r ij :=Score M (ρ ij ), i,j= 1,...,n. Step 2 (Binary reachability matrix).Define the binary reachability matrix D= (d ij ) n×n ∈0,1 n×n by thresholding: d ij := 1, r ij ≥θ, 0, r ij < θ. Step 3 (Driving power and dependence power).For each factorf i ∈F, define itsdriving powerand dependence powerby Drv M (i) := n ∑ j=1 d ij ,Dep M (i) := n ∑ j=1 d ji . Thus Drv M (i)is the row sum ofD, and Dep M (i)is the column sum ofD. Step 4 (MICMAC cutoffs).Define the average driving-power and dependence-power cutoffs by τ drv := 1 n n ∑ i=1 Drv M (i), τ dep := 1 n n ∑ i=1 Dep M (i). Step 5 (Quadrant-based classification).Define the four MICMAC classes by F aut := f i ∈F:Drv M (i)< τ drv and Dep M (i)< τ dep , F dep := f i ∈F:Drv M (i)< τ drv and Dep M (i)≥τ dep , F lin := f i ∈F:Drv M (i)≥τ drv and Dep M (i)≥τ dep , F drv := f i ∈F:Drv M (i)≥τ drv and Dep M (i)< τ dep . These are called, respectively, theautonomous,dependent,linkage, anddrivingclasses. The quadruple ( F aut ,F dep ,F lin ,F drv ) is called theUncertain MICMAC classification of typeMinduced by ( ̃ R M ,Score M ,θ). Chapter 5. Structure / causality decision-modelling (inter-criteria influence) Remark 5.3.4.Definition 5.3.3 is model-independent. The uncertain modelMonly enters through the degree-domain Dom(M)and the score map Score M , which converts uncertain reachability values into scalar strengths in[0,1]. Theorem 5.3.5(Well-definedness of Uncertain MICMAC).Under Definition 5.3.3, assume: (A1)F=f 1 ,...,f n is finite and nonempty; (A2) ̃ R M = (ρ ij ) n×n ∈Dom(M) n×n ; (A3)Score M :Dom(M)→[0,1]is a total map; (A4)θ∈(0,1]. Then: (i) the scored matrixR= (r ij )∈[0,1] n×n and the binary matrixD= (d ij )∈0,1 n×n are well-defined; (i) for everyi∈1,...,n, the quantitiesDrv M (i)andDep M (i)are well-defined integers satisfying 0≤Drv M (i)≤n,0≤Dep M (i)≤n; (i) the cutoffsτ drv andτ dep are well-defined real numbers; (iv) the four classes F aut ,F dep ,F lin ,F drv are well-defined and form a partition ofF, that is, they are pairwise disjoint and their union isF. Hence Uncertain MICMAC of typeMis well-defined. Proof.(i) Well-definedness ofRandD.By (A2), each entryρ ij belongs to Dom(M). Since Score M is total by (A3), the value r ij :=Score M (ρ ij ) is well-defined and belongs to[0,1]for everyi,j. Therefore R= (r ij ) n×n ∈[0,1] n×n is well-defined. Becauseθ∈(0,1], for eachr ij ∈[0,1]exactly one of the inequalities r ij ≥θorr ij < θ holds. Hence eachd ij is well-defined and belongs to0,1. Therefore D= (d ij ) n×n ∈0,1 n×n Chapter 5. Structure / causality decision-modelling (inter-criteria influence) is well-defined. (i) Well-definedness of driving and dependence powers.For each fixedi, the sums Drv M (i) = n ∑ j=1 d ij ,Dep M (i) = n ∑ j=1 d ji contain exactlynterms, each equal to0or1. Therefore both are well-defined integers, and clearly 0≤Drv M (i)≤n,0≤Dep M (i)≤n. (i) Well-definedness of the cutoffs.Since each Drv M (i)and Dep M (i)is a real number, the averages τ drv = 1 n n ∑ i=1 Drv M (i), τ dep = 1 n n ∑ i=1 Dep M (i) are well-defined real numbers. (iv) Partition property of the four classes.Fix any factorf i ∈F. Exactly one of the two relations Drv M (i)< τ drv orDrv M (i)≥τ drv holds, and exactly one of the two relations Dep M (i)< τ dep orDep M (i)≥τ dep holds. Hence exactly one of the following four combinations is true: Drv M (i)< τ drv and Dep M (i)< τ dep , Drv M (i)< τ drv and Dep M (i)≥τ dep , Drv M (i)≥τ drv and Dep M (i)≥τ dep , Drv M (i)≥τ drv and Dep M (i)< τ dep . Thereforef i belongs to exactly one of F aut ,F dep ,F lin ,F drv . This shows both that the union of the four classes isFand that they are pairwise disjoint. Hence the four classes form a partition ofF, and Uncertain MICMAC is well-defined. Related concepts of MICMAC under uncertainty-aware models are listed in Table 5.4. Chapter 5. Structure / causality decision-modelling (inter-criteria influence) Table 5.4: Related concepts of MICMAC under uncertainty-aware models. kRelated MICMAC concept(s) 1 Fuzzy MICMAC 2 Intuitionistic Fuzzy MICMAC 3 Neutrosophic MICMAC [696,697] 5.4 Fuzzy Cognitive Map (FCM) A cognitive map is a signed directed graph of causal concepts; updating node states via weighted sums simulates system behavior and feedback loops [698, 699]. Fuzzy Cognitive Maps use fuzzy causal weights and fuzzy concept activations, apply nonlinear update functions, and model uncertain feedback dynamics for scenario evaluation [700–702]. Definition 5.4.1(Fuzzy Cognitive Map (FCM)).[700, 701] LetN≥1and letC=C 1 ,...,C N be a finite set ofconcepts(states/variables/features of the modeled system). AFuzzy Cognitive Map (FCM)is a weighted directed graph (equivalently, a weight matrix) FCM= (C,W,f), where: (i)W= (w ji )∈[−1,1] N×N is thecausal connection (adjacency) matrixwithw i = 0. The entryw ji represents the signed strength of the causal influenceC j →C i : w ji >0(positive causality), w ji <0(negative causality), w ji = 0(no direct relation). (i)f:R→Iis anactivation (transfer) functionmapping to a bounded intervalI= [0,1](unipolar) or I= [−1,1](bipolar), typically monotone and continuous (e.g. sigmoid). FCM state and reasoning dynamics.Thestateof the FCM at discrete timek∈N 0 is a vector A(k) = (A 1 (k),...,A N (k))∈I N , whereA i (k)is the activation level of conceptC i . Given an initial stateA(0)∈I N , the standard synchronous FCM update rule is A i (k+ 1) =f A i (k) + N ∑ j=1 j6=i A j (k)w ji , i= 1,...,N.(5.1) Equivalently, with componentwise application off, A(k+ 1) =f ( A(k) +A(k)W ) . (Variants omit the self-memory termA i (k)and useA(k+ 1) =f(A(k)W).) Common choice off(unipolar sigmoid).A widely used activation function is the logistic sigmoid f λ (x) = 1 1 +e −λx , λ >0, Chapter 5. Structure / causality decision-modelling (inter-criteria influence) which ensuresA i (k)∈[0,1]for allk. Outcome.Iterating (5.1) generates a trajectory inI N that may converge to a fixed point, enter a limit cycle, or exhibit more complex attractors depending onA(0),W, andf. Below, we present the Uncertain Cognitive Map obtained by extending the framework using Uncertain Sets. Definition 5.4.2(Uncertain Cognitive Map (UCM): expected-weight dynamics).LetN≥1and letC= C 1 ,...,C N be a finite set of concepts. Fix an uncertainty space(Γ,L,M)and a bounded intervalI= [0,1] (unipolar) orI= [−1,1](bipolar). (1) Uncertain causal weights.Anuncertain cognitive mapis specified by a matrix of uncertain numbers ̃ W= ( ̃w ji ) ∈US([−1,1]) N×N , ̃w i ≡0, where ̃w ji encodes the uncertain signed causal influenceC j →C i . Assume the expectations exist and define theexpected weight matrix W:= (w ji )∈[−1,1] N×N , w ji :=E[ ̃w ji ]. (2) State space and activation function.Letf:R→Ibe an activation (transfer) function such that f(R)⊆I. (Examples include the unipolar logistic sigmoid or a bipolar tanh-type map.) (3) Expected-value UCM dynamics.AUCM stateat discrete timek∈N 0 is a vector A(k) = (A 1 (k),...,A N (k))∈I N . GivenA(0)∈I N , define the synchronous update rule by A i (k+ 1) =f A i (k) + N ∑ j=1 j6=i A j (k)w ji , i= 1,...,N.(5.2) Equivalently, A(k+ 1) =f ( A(k) +A(k)W ) , wherefis applied componentwise. The tripleUCM:= (C, ̃ W,f)is called anUncertain Cognitive Map. Definition 5.4.3(Uncertain-set output induced by a UCM).LetA(0)∈I N be fixed and let(A(k)) k≥0 be generated by (5.2). For eachk∈N 0 and each concepti, define the singleton-valued uncertain set A i,k : Γ→P(I),A i,k (γ) :=A i (k). Then ( A i,k ) k∈N 0 i=1,...,N is called theuncertain-set state familyinduced by the UCM (via expected-value real- ization). Chapter 5. Structure / causality decision-modelling (inter-criteria influence) Theorem 5.4.4(Uncertain-set structure and well-definedness of UCM dynamics).In the setting of Defi- nition 5.4.2, assume: (A1)(Finite expectations)E[ ̃w ji ]exists and is finite for alli,j. (A2)(Bounded transfer)f(R)⊆I. Then, for every initial stateA(0)∈I N : (i)the recursion(5.2)defines a unique sequence(A(k)) k≥0 inI N ; (i)eachA i (k)∈Iis well-defined for alliand allk; (i)the family in Definition 5.4.3 is an uncertain-set structured output on(Γ,L,M). Proof.FixA(0)∈I N . Step 1 (existence of the expected weight matrix).By (A1), eachw ji :=E[ ̃w ji ]exists as a real number, henceW= (w ji )is a well-defined real matrix. Step 2 (one-step update is well-defined).AssumeA(k)∈I N . For eachi, define the real input u i (k) :=A i (k) + ∑ j6=i A j (k)w ji ∈R, which is well-defined because it is a finite sum of real products. Then (A2) impliesA i (k+1) :=f(u i (k))∈I. ThereforeA(k+ 1)∈I N . Step 3 (existence and uniqueness for all times).Step 2 shows that the mapF:I N →I N defined by F(A) =f(A+AW)(componentwisef) is a well-defined function. Hence the recursionA(k+ 1) =F(A(k)) produces a unique sequence(A(k)) k≥0 inI N by deterministic iteration of a function. Step 4 (uncertain-set structured output).For each(i,k),A i,k (γ) =A i (k)is a constant singleton- valued set map, hence it is an uncertain set on(Γ,L,M). This proves (i). As related concepts beyond Uncertain Cognitive Maps, Grey Cognitive Maps [717, 718], Rough Cogni- tive Maps [719, 720], Evidential Cognitive Maps [721, 722], Dynamic Cognitive Maps [723, 724], Granular Cognitive Maps [725, 726], Cognitive HyperMaps [727–729], Bipolar Cognitive Maps [730, 731], Linguistic Cognitive Maps [732], and Probabilistic Cognitive Maps [733,734] are also well known. Chapter 5. Structure / causality decision-modelling (inter-criteria influence) Table 5.5: A compact catalogue of Cognitive Map variants by the uncertainty encoding (indexed by a convenient degree-domain dimensionk). kCognitive Map variantRemark (how uncertainty is represented) 1Fuzzy Cognitive Maps (FCM) [703,704] Causal influences (edge weights) and/or concept activations are mod- eled by single graded degrees (typically normalized to[0,1]when needed) and updated by nonlinear dynamical rules. 2Intuitionistic Fuzzy Cognitive Maps [705,706] Each influence is encoded by a two-component degree (e.g., member- ship and non-membership, with implicit hesitation), yielding a richer uncertainty description than scalar weights. 2Pythagorean Fuzzy Cognitive Maps (PFCM) [707,708] Pythagorean Fuzzy Cognitive Maps (PFCM) (degrees(μ,ν)∈[0,1] 2 withμ 2 +ν 2 ≤1; hesitationπ= √ 1−μ 2 −ν 2 is derived). 3Hesitant Fuzzy Cognitive Maps [709,710] Each influence is assessed by multiple plausible degrees (a finite hes- itation set); for catalogue purposes this is often summarized into a fixed-lengthk=3encoding (e.g., min/representative/max). 3Neutrosophic Cognitive Maps [711–714] Each influence carries truth/indeterminacy/falsity components in [0,1] 3 , enabling explicit representation of indeterminacy and incon- sistency in causal strengths. 3n (n≥ 1) Refined Neutrosophic Cognitive Maps [704,714] Refined Neutrosophic Cognitive Maps (R-NCMs): degrees in [0,1] 3n , typically(T 1 ,...,T n , I 1 ,...,I n , F 1 ,...,F n )for each con- cept/edge. nPlithogenic Cognitive Maps [715,716] Influences are attribute–value based with scalar appurtenance degrees on attribute–value pairs, coupled with a contradiction function on values; aggregation uses plithogenic operators. Chapter 6 Compensatory scoring methods on a decision ma- trix (“weighted-sum” families) Compensatory scoring aggregates weighted, normalized criterion performances into overall utility scores, allowing trade-offs where strong performance compensates weak criteria, producing rankings via sums/prod- ucts. For convenience, a concise comparison of the representative decision-matrix methods discussed in this chap- ter is presented in Table 6.1. Table 6.1: A concise comparison of representative decision-matrix methods discussed in this chapter. Method Primary role Core mechanismTypical final quantity Short note COPRASUtility rankingUses weighted normalized per- formances, separates benefit and cost sums, and computes rela- tive significance. Relative significance and utility degree Explicit benefit– cost decomposi- tion. MAC- BETH Value-scale con- struction and ranking Converts qualitative pairwise attractiveness judgments into a numerical value scale, usually via linear programming. Cardinal value scale / overall scores Strongly judgment- driven and scale- oriented. CoCoSoCompromise ranking Combines weighted-sum and weighted-product logics into a compromise aggregation. Combined compro- mise score Hybrid SAW– WPM style method. SAWUtility rankingAggregates normalized crite- rion performances by a simple weighted sum. Weighted-sum util- ity Canonical com- pensatory base- line. RAFSIUtility rankingMaps criterion values to a com- mon interval using ideal and anti-ideal anchors, then aggre- gates them. RAFSI scoreEmphasizes common-interval functional map- ping. Continued on the next page. 175 Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Table 6.1 (continued). Method Primary role Core mechanismTypical final quantity Short note RATMIUtility rankingCombines a trace-based per- formance term with a median- similarity term after weighting. Trace-to-median index Matrix-geometry flavored ranking rule. RANCOM Criteria weight- ing Ranks criteria, builds a compar- ison matrix from rank relations, and normalizes row sums. Criterion-weight vector Mainly a weight- ing tool rather than a direct alternative- ranking method. AROMAN Utility rankingUses two-step normalization, separates benefit and cost ef- fects, and applies an exponential preference score. Ranking indexDesigned to bal- ance two normal- ization views. MAUTUtility rankingApplies single-attribute utility functions and combines them into an overall multi-attribute utility. Overall utilityPreference- theoretic and utility-based. SMARTUtility rankingUses value functions and swing- type weights, then computes a simple weighted additive score. SMART scoreTransparent and easy to imple- ment. REGIMEPairwise ranking Compares alternatives pairwise on each criterion, aggregates weighted win–loss indicators, and forms net dominance. Net guide indexComparison- based rather than purely di- rect scoring. TODIMDominance ranking Uses prospect-theoretic gains and losses relative to a reference and aggregates dominance val- ues. Overall dominance value Captures asym- metric treatment of losses. GRASimilarity rank- ing Measures closeness to an ideal or reference sequence via grey rela- tional coefficients and grades. Grey relational grade Reference- similarity based. ARASUtility ratio ranking Aggregates weighted normalized performances relative to an ex- plicitly defined ideal alternative. Utility degreeIdeal-alternative baseline is cen- tral. WASPASCompromise ranking Combines weighted-sum and weighted-product utilities through a mixing parameter. Integrated WAS- PAS utility Robust WSM– WPM hybrid. MOORARatio rankingUses vector normalization and computes a weighted benefit- minus-cost score. MOORA scoreSimple ratio- analysis family. PSIObjective weighting and ranking Derives criterion weights from dispersion of normalized data and then computes a preference index. Preference selection index More data-driven than subjective weighting meth- ods. ROVInterval utility ranking Forms pessimistic and optimistic weighted utilities and ranks by an attitude-dependent score. Value interval / attitude score Naturally sup- ports optimistic– pessimistic evalu- ation. Continued on the next page. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Table 6.1 (continued). Method Primary role Core mechanismTypical final quantity Short note MOOSRA Ratio rankingComputes a weighted benefit-to- cost ratio after normalization. MOOSRA scoreDirect bene- fit/cost ratio form. Note.Although all methods are discussed in the same general decision-matrix context, they are not identical in role. In particular, RANCOM is primarily a criterion-weighting method, while MACBETH is fundamentally a value-scale construction framework that can subsequently support ranking. 6.1 Fuzzy COPRAS (Fuzzy Complex Proportional Assessment) COPRAS ranks alternatives using weighted normalized criteria, separately aggregating benefit and cost sums to compute relative significance and utility [618, 735, 736]. Fuzzy COPRAS models ratings and weights as fuzzy numbers, defuzzifies or compares them, then computes COPRAS significance and utility under uncertainty [737,738]. Definition 6.1.1(Fuzzy COPRAS (COPRAS-F)).[737,738] LetA=A 1 ,...,A m be alternatives and C=C 1 ,...,C n criteria. Partition criteria into benefit and cost types C=C + ̇ ∪C − , J + :=j:C j ∈C + , J − :=j:C j ∈C − . Assume a TFN decision matrix ̃ D= ( ̃x ij )∈(TFN) m×n and TFN criterion weights ̃ w= ( ̃w 1 ,..., ̃w n )∈ (TFN) n . (0) BNP defuzzification.For a TFN ̃x= (l,m,u)define BNP( ̃x) := l+m+u 3 . (1) Defuzzify and (optionally) normalize weights.Set x ij :=BNP( ̃x ij ), q j :=BNP( ̃w j ), and (optionally) normalizeqbyq j ←q j / ∑ n t=1 q t so that ∑ n j=1 q j = 1. (2) Column-sum normalization and weighting.Define n ij := x ij ∑ m p=1 x pj ,ˆx ij :=q j n ij . (3) Benefit/cost sums.For each alternativeA i , set P i := ∑ j∈J + ˆx ij , R i := ∑ j∈J − ˆx ij , R min :=min 1≤i≤m R i . Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (4) Relative significance and utility degree.Define Q i :=P i + R min ∑ m t=1 R t R i ∑ m t=1 ( R min R t ) , Q max :=max 1≤i≤m Q i , N i := Q i Q max ×100%. The final ranking is obtained by sortingQ i (equivalentlyN i ) in descending order. Using Uncertain Sets, we define Uncertain COPRAS (U-COPRAS) as follows. Definition 6.1.2(Uncertain COPRAS (U-COPRAS): expected-score formulation).LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. Partition criteria into benefit and cost types C=C + ̇ ∪C − , J + :=j:C j ∈C + , J − :=j:C j ∈C − . Fix an uncertainty space(Γ,L,M). (1) Uncertain decision matrix and uncertain weights.Anuncertain COPRAS instanceconsists of: • anuncertain decision matrix ̃ X= ( ̃x ij )∈US(R ≥0 ) m×n , where ̃x ij is the uncertain performance ofA i under criterionC j ; • anuncertain weight vector ̃w= ( ̃w 1 ,..., ̃w n )∈US(R ≥0 ) n , where ̃w j is the uncertain importance of criterionC j . Assume all expectations exist and define theexpected (crisp) scores x ij :=E[ ̃x ij ]∈R ≥0 , q j :=E[ ̃w j ]∈R ≥0 . (2) Normalized expected weights.If ∑ n t=1 q t >0, set ̄q j := q j ∑ n t=1 q t ∈[0,1] (j= 1,...,n), so that ∑ n j=1 ̄q j = 1. If ∑ n t=1 q t = 0, set ̄q j := 1/nfor allj. (3) Column-sum normalization and weighting (on expected data).For each criterionj, let s j := m ∑ p=1 x pj . Assumes j >0for allj(non-degenerate criteria). Define n ij := x ij s j ∈[0,1],ˆx ij := ̄q j n ij ∈[0,1]. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (4) Benefit/cost aggregated sums.For each alternativeA i , define P i := ∑ j∈J + ˆx ij , R i := ∑ j∈J − ˆx ij . IfJ − 6=∅, also define R min :=min 1≤i≤m R i . (5) Relative significance and utility degree.IfJ − =∅, setQ i :=P i . IfJ − 6=∅andR i >0for alli, define Q i :=P i + R min ∑ m t=1 R t R i ∑ m t=1 ( R min R t ) . LetQ max :=max 1≤i≤m Q i and, ifQ max >0, define the utility percentage N i := Q i Q max ×100%. The U-COPRAS ranking is obtained by sortingQ i (equivalentlyN i ) in descending order. Definition 6.1.3(Uncertain-set output induced by U-COPRAS).Under Definition 6.1.2, define the singleton- valued uncertain sets Q i : Γ→P(R),Q i (γ) :=Q i , and (whenQ max >0) the singleton-valued uncertain sets N i : Γ→P([0,100]),N i (γ) :=N i . Then(Q i ) m i=1 (and(N i ) m i=1 ) is called theuncertain-set structured COPRAS output(via expected-score realization). Theorem 6.1.4(Uncertain-set structure and well-definedness of U-COPRAS).In the setting of Defini- tion 6.1.2, assume: (A1)(Finite expectations)E[ ̃x ij ]andE[ ̃w j ]exist and are finite for alli,j. (A2)(Non-degenerate columns)s j = ∑ m p=1 x pj >0for allj. (A3)(Cost-sum positivity when needed) EitherJ − =∅, or elseR i >0for alli. (A4)(Nontrivial overall score)Q max >0wheneverN i is formed. Then: (i)all quantities in Steps(2)–(5)of Definition 6.1.2 are well-defined real numbers; (i)the induced ranking preorderA i A k ⇐⇒Q i ≥Q k is well-defined onA; (i)the outputs(Q i )and(N i )in Definition 6.1.3 constitute uncertain sets on(Γ,L,M). Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Proof.Assumption (A1) guarantees thatx ij =E[ ̃x ij ]andq j =E[ ̃w j ]exist as finite real numbers, hence Steps (1)–(2) are well-defined. If ∑ t q t >0, the normalization ̄q j =q j / ∑ t q t is a real number in[0,1]; if ∑ t q t = 0, the fallback ̄q j = 1/nis a real number and still satisfies ∑ j ̄q j = 1. By (A2), eachs j >0, son ij =x ij /s j is well-defined and lies in[0,1]becausex ij ≥0andx ij ≤s j . Then ˆx ij = ̄q j n ij is well-defined and nonnegative, soP i = ∑ j∈J + ˆx ij andR i = ∑ j∈J − ˆx ij are well-defined finite sums. IfJ − =∅, thenQ i =P i is well-defined. Otherwise, by (A3) we haveR i >0and alsoR t >0for everyt, hence each fractionR min /R t is well-defined and positive, so ∑ m t=1 (R min /R t )>0. Therefore the COPRAS expression forQ i is well-defined for alli. The ranking preorderA i A k ⇐⇒Q i ≥Q k is well-defined because(Q i )are real numbers. Finally, when Q max >0(assumption (A4)), eachN i = (Q i /Q max )·100is a well-defined real number in[0,100]. The set mapsQ i (γ) =Q i andN i (γ) =N i are singleton-valued uncertain sets on(Γ,L,M), proving (i). For reference, COPRAS and representative uncertainty-aware variants are listed in Table 6.2. Table 6.2: COPRAS and representative uncertainty-aware variants (organized by the degree-domain dimen- sionk). kRepresentative COPRAS-type model(s) whose evaluation degrees live in a subset of[0,1] k 1Fuzzy COPRAS. 2Intuitionistic Fuzzy COPRAS [57,739]; Pythagorean fuzzy COPRAS [740,741]. 3Hesitant Fuzzy COPRAS [742,743]; Spherical fuzzy COPRAS [744,745]; Picture Fuzzy COPRAS [746, 747]; Neutrosophic COPRAS [748,749]. Note.Herekdenotes the ambient dimension of the degree-domain used to encode each criterion evaluation (and, if applicable, each weight). 6.2 Fuzzy MACBETH (Fuzzy Measuring Attractiveness by a Categorical Based Eval- uation Technique) MACBETH elicits qualitative pairwise judgments of attractiveness differences, converts them into numerical value scales via linear programming, and aggregates weighted scores to rank alternatives consistently [750, 751]. Fuzzy MACBETH replaces crisp categories with fuzzy linguistic terms or fuzzy numbers, propagating imprecision through the scale-construction model to obtain fuzzy scores and robust rankings [752–754]. Definition 6.2.1(Fuzzy MACBETH (triangular fuzzy semantic scale + F-LP-MACBETH)).[752–754] LetA=a 1 ,...,a n be a finite set ofevaluation elements(actions, alternatives, or performance levels) that are totally preordered by a decision maker (P= strict preference,I= indifference). Assumea + ∈Ais the most attractive element anda − ∈Ais the least attractive element. (1) Semantic categories and their triangular fuzzy encoding.Let the semantic categories be C 0 ,C 1 ,...,C 6 , whereC 0 denotesindifferenceandC 1 ,...,C 6 denote increasingdifferences of attractive- ness. Associate each categoryC k with a triangular fuzzy number (TFN) ̃ A k = (` k ,m k ,u k )∈R 3 ≥0 , ` k ≤m k ≤u k , Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) given by the standard Fuzzy-MACBETH scale ̃ A 0 = (0,0,0), ̃ A 1 = (1,1,2), ̃ A 2 = (1,2,3), ̃ A 3 = (2,3,4), ̃ A 4 = (3,4,5), ̃ A 5 = (4,5,6), ̃ A 6 = (5,6,6). (Equivalently: fork= 2,3,4,5, take` k =k−1,m k =k,u k =k+ 1; fork= 1,` 1 =m 1 = 1,u 1 = 2; for k= 6,` 6 = 5,m 6 =u 6 = 6.) (2) Fuzzified judgment matrix.For each ordered pair(x,y)∈A×AwithxPy, the decision maker assigns a categoryc(x,y)∈1,...,6(or0for indifference). Define the fuzzified judgment as ̃ d(x,y) := ̃ A c(x,y) . (3) TFN arithmetic and TFN order.For TFNs ̃p= (p 1 ,p 2 ,p 3 )and ̃q= (q 1 ,q 2 ,q 3 )define ̃p⊕ ̃q= (p 1 +q 1 , p 2 +q 2 , p 3 +q 3 ), ̃p ̃q= (p 1 −q 1 , p 2 −q 2 , p 3 −q 3 ). Use the componentwise preorderon TFNs: (` 1 ,m 1 ,u 1 )(` 2 ,m 2 ,u 2 )⇐⇒` 1 ≥` 2 , m 1 ≥m 2 , u 1 ≥u 2 . Let COA(`,m,u) := (`+m+u)/3be the centroid defuzzification map. (4) F-LP-MACBETH (fuzzy linear program) and fuzzy basic scale.Decision variables are TFNs ̃v(x) = (v L x ,v M x ,v U x )for allx∈A(thepre-cardinal / basic fuzzy scale). Compute ̃vby solving the following optimization problem: min COA ( ̃v(a + ) ̃v(a − ) ) subject to the constraints (Origin) ̃v(a − ) = (0,0,0), (Indifference) ̃v(x) ̃v(y) = (0,0,0)∀(x,y)∈C 0 , (Positive separation) ̃v(x) ̃v(y)(1,1,2)∀(x,y)∈C k , k∈1,...,6, (Cardinal consistency across categories) ̃v(x) ̃v(y) ̃v(w) ̃v(z) ∀(x,y)∈C k ,∀(w,z)∈C k ′ , k > k ′ , k,k ′ ∈1,...,6. Any feasible optimizer ̃vis called aFuzzy-MACBETH basic (pre-cardinal) scale. (5) Defuzzification and cardinalization.Define the basic crisp scalev x :=COA( ̃v(x))for eachx∈A. To obtain acardinalscaleE x , anchor two reference levels: choose a “neutral” levelx neu and a “good” level x good and impose E x neu = 0, E x good = 100. Then define the affine transformation E x =αv x +β, with(α,β)determined by the two anchoring equations above. The resulting mappingx7→E x is called the (fuzzy) MACBETH cardinal value function. Using Uncertain Sets, we present Uncertain MACBETH (U-MACBETH) as an extension of the original framework below. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Definition 6.2.2(Uncertain MACBETH (U-MACBETH): expected-interval LP scale construction).Let A=a 1 ,...,a N be a finite set of evaluation elements (actions/alternatives/levels). Assume a total preorder onAwith strict partPand indifference partI: xPy⇐⇒(xyandy6x), xIy⇐⇒(xyandyx). Leta + ∈Abe a most attractive element anda − ∈Aa least attractive element (witha + Pa − ). (1) Semantic categories and uncertain numerical bounds.Let the MACBETH semantic categories beC 0 ,C 1 ,...,C 6 , whereC 0 denotesindifferenceandC 1 ,...,C 6 denote increasingdifferences of attractive- ness. Fix an uncertainty space(Γ,L,M). For each categoryk∈ 0,...,6, assign anuncertain interval bound pair ( ̃ L k , ̃ U k )∈US(R ≥0 )×US(R ≥0 ), interpreted as (uncertain) lower/upper bounds on a value difference belonging to categoryC k . Assume the expected bounds exist and put L k :=E[ ̃ L k ], U k :=E[ ̃ U k ] (k= 0,...,6), with 0 =L 0 =U 0 ,0≤L k ≤U k <∞(k≥1), U k ≤L k+1 (k= 0,...,5) (category separation on expected bounds). (2) Uncertain MACBETH judgment matrix.For each ordered pair(x,y)∈A×AwithxPy, the decision maker provides a category label c(x,y)∈1,...,6. For indifference pairs(x,y)∈I, setc(x,y) := 0. This yields a (category-valued) judgment matrixc: A×A→0,...,6. (3) Expected-interval linear program (basic value scale).Decision variables are real valuesv(x)∈R for allx∈A. Define the feasible set by the expected-interval constraints: (Anchor)v(a − ) = 0, (Indifference)v(x)−v(y) = 0∀(x,y)∈I, (Category bounds)L c(x,y) ≤v(x)−v(y)≤U c(x,y) ∀(x,y)∈A×AwithxPy, (Monotonicity)v(x)≥v(y)∀x,y∈Awithxy. Among all feasiblev, define abasic MACBETH scaleby solving minv(a + )subject to the above constraints.(6.1) Any optimizerv ? of (6.1) is called anU-MACBETH basic value scale. (4) Cardinalization (0–100 scale) and ranking.Ifv ? (a + )>0, define the cardinal value function E:A→[0,100]by E(x) := 100· v ? (x)−v ? (a − ) v ? (a + )−v ? (a − ) = 100· v ? (x) v ? (a + ) (x∈A). Rank elements by the induced preorder x U-MACBETH y⇐⇒E(x)≥E(y). Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Definition 6.2.3(Uncertain-set structured U-MACBETH output).Under Definition 6.2.2, fix an optimizer v ? of (6.1) (and the inducedEwhenv ? (a + )>0). Define singleton-valued uncertain sets V x : Γ→P(R),V x (γ) :=v ? (x), and, whenv ? (a + )>0, E x : Γ→P([0,100]),E x (γ) :=E(x). Then(V x ) x∈A (and(E x ) x∈A ) is called theuncertain-set structured U-MACBETH output(via expected- interval realization). Theorem 6.2.4(Uncertain-set structure and well-definedness of U-MACBETH).In the setting of Defini- tion 6.2.2, assume: (A1)(Finite expectations)E[ ̃ L k ]andE[ ̃ U k ]exist and are finite for allk= 0,...,6. (A2)(Separated expected category bounds)0 =L 0 =U 0 ,0≤L k ≤U k , andU k ≤L k+1 fork= 0,...,5. (A3)(Feasibility) The constraint set of(6.1)is nonempty. (A4)(Nontrivial top anchor) Every feasiblevsatisfiesv(a + )>0(equivalently, the constraints force a strictly positive separation betweena + anda − ). Then: (i)the LP(6.1)is well-defined and attains an optimal solutionv ? ; (i)the cardinal valuesE(x)are well-defined real numbers in[0,100]for allx∈A; (i)the ranking preorder U-MACBETH is well-defined onA; (iv)the mapsV x andE x in Definition 6.2.3 are uncertain sets on(Γ,L,M). Proof.By (A1) the expected boundsL k ,U k are finite real numbers. Hence all constraints in Definition 6.2.2 are ordinary linear equalities/inequalities, so (6.1) is a standard linear program. By (A3) the feasible region is nonempty. The anchor constraintv(a − ) = 0and the monotonicity constraints ensurev(a + )≥0for every feasiblevbecausea + a − . Therefore the objective minv(a + )is bounded below by0. Since the feasible region is a closed polyhedron and the objective is continuous linear, the minimum is attained at some optimizerv ? , proving (i). By (A4) one hasv ? (a + )>0, so the normalizationE(x) = 100v ? (x)/v ? (a + )is well-defined for everyx∈A. Moreover,v ? (a − ) = 0and monotonicity implies0≤v ? (x)≤v ? (a + )for allx∈A(becausea + xa − holds in a total preorder). HenceE(x)∈[0,100], proving (i). The preorder U-MACBETH is well-defined because it compares real numbersE(x), proving (i). Finally,V x (γ) =v ? (x)andE x (γ) =E(x)are singleton-valued set maps fromΓintoP(R)(resp. P([0,100])), hence they are uncertain sets. This proves (iv). Related concepts of MACBETH under uncertainty-aware models are listed in Table 6.3. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Table 6.3: Related concepts of MACBETH under uncertainty-aware models. kRelated MACBETH concept(s) 1 Fuzzy MACBETH 2 Intuitionistic Fuzzy MACBETH 3 Neutrosophic MACBETH 6.3 Fuzzy CoCoSo (Fuzzy Combined Compromise Solution) CoCoSo combines SAW and WEP aggregation to compute compromise scores, producing stable, robust rankings even after alternative changes small perturbations [755, 756]. Fuzzy CoCoSo expresses perfor- mances and preferences by triangular fuzzy numbers, normalizes them, aggregates fuzzy sums/powers, then defuzzifies final scores consistently [641,757]. Definition 6.3.1(Fuzzy CoCoSo (TFN-based Combined Compromise Solution)).[641, 757] LetA= A 1 ,...,A m be alternatives andC=C 1 ,...,C n criteria. Partition criteria into benefit and cost sets C=C + ̇ ∪C − . LetTFN ≥0 :=(l,m,u)∈R 3 ≥0 :l≤m≤u. (0) TFN arithmetic (nonnegative convention).For ̃x= (l x ,m x ,u x ), ̃y= (l y ,m y ,u y )∈TFN ≥0 and c >0define ̃x⊕ ̃y:= (l x +l y , m x +m y , u x +u y ), ̃x⊗ ̃y:= (l x l y , m x m y , u x u y ), c ̃x:= (cl x , cm x , cu x ), ̃x c:= ( l x c , m x c , u x c ) . Forw≥0, define the (scalar) TFN power by ̃x w := (l w x , m w x , u w x ), (with the standard convention0 0 := 1if needed). Fix a defuzzification/score map (centroid) Defuzz(l,m,u) := l+m+u 3 . (1) Input data (fuzzy decision matrix + weights).Assume a TFN decision matrix ̃ Y= ( ̃y ij )∈(TFN ≥0 ) m×n , ̃y ij = (l ij ,m ij ,u ij ), and a (crisp) weight vectorw= (w 1 ,...,w n )withw j ≥0and ∑ n j=1 w j = 1. (If TFN weights are given, one may first defuzzify and renormalize them to obtain suchw.) (2) Normalization (fuzzy lift of CoCoSo (7)–(8)).For each criterionj, set the global lower/upper scalars y j :=min 1≤i≤m l ij ,y j :=max 1≤i≤m u ij ,∆ j :=y j −y j >0. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Define the normalized TFNs ̃ t ij ∈TFN ≥0 by ̃ t ij := ( l ij −y j ∆ j , m ij −y j ∆ j , u ij −y j ∆ j ) , C j ∈C + (benefit), ( y j −u ij ∆ j , y j −m ij ∆ j , y j −l ij ∆ j ) , C j ∈C − (cost). Then0≤ ̃ t ij ≤(1,1,1)componentwise and the construction reduces to the crisp normalization when ̃y ij = (y ij ,y ij ,y ij ). (3) Fuzzy comparability sequences (lift of CoCoSo (9)–(10)).Define ̃ G i := n ⊗ j=1 ( ̃ t ij ) w j ∈TFN ≥0 , ̃ H i := n ⊕ j=1 ( w j ̃ t ij ) ∈TFN ≥0 . (4) Three fuzzy appraisal scores (lift of CoCoSo (11)–(13)).Let S GH := m ∑ p=1 Defuzz ( ̃ G p ⊕ ̃ H p ) >0, G min :=min p Defuzz( ̃ G p ), H min :=min p Defuzz( ̃ H p ), G max :=max p Defuzz( ̃ G p ), H max :=max p Defuzz( ̃ H p ). Define ̃as (α) i := ( ̃ G i ⊕ ̃ H i ) S GH , ̃as (β) i := ( ̃ H i H min ) ⊕ ( ̃ G i G min ) , and for a parameterλ∈[0,1](oftenλ= 0.5), ̃as (γ) i := ( λ ̃ H i ⊕(1−λ) ̃ G i ) ( λH max + (1−λ)G max ) . (5) Final fuzzy CoCoSo score (lift of CoCoSo (14)) and ranking.Define the final TFN score ̃as i := ( ̃as (α) i ⊗ ̃as (β) i ⊗ ̃as (γ) i ) 1/3 ⊕ 1 3 ( ̃as (α) i ⊕ ̃as (β) i ⊕ ̃as (γ) i ) . DefuzzifyAS i :=Defuzz( ̃as i )∈R ≥0 and rank alternatives by A p FCoCoSo A q ⇐⇒AS p ≥AS q . AnyA ? ∈arg max 1≤i≤m AS i is called aFuzzy CoCoSo(compromise) solution. Using Uncertain Sets, we define Uncertain CoCoSo as follows. Definition 6.3.2(Uncertainty space, uncertain variable, and induced uncertain set).Anuncertainty space is a triple(Γ,L,M)whereΓ6=∅,Lis aσ-algebra onΓ, andM:L →[0,1]is an uncertain measure. A (real-valued)uncertain variableis anL-measurable mapξ: Γ→R. Any uncertain variableξinduces a singleton-valueduncertain set X ξ : Γ→P(R),X ξ (γ) :=ξ(γ). WhenE[ξ]∈Rexists (finite), it is used as a crisp score. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Definition 6.3.3(Uncertain CoCoSo (U-CoCoSo): scenario-wise CoCoSo with expected-score decision). LetA=A 1 ,...,A m be alternatives andC=C 1 ,...,C n criteria, partitioned as C=C + ̇ ∪C − (benefit/cost), J + :=j:C j ∈C + , J − :=j:C j ∈C − . Fix an uncertainty space(Γ,L,M). (1) Uncertain input data.Anuncertain CoCoSo instanceconsists of • anuncertain decision matrixΞ = (ξ ij )∈UV m×n ≥0 , where eachξ ij : Γ→R ≥0 is an uncertain variable representing the performance ofA i underC j ; • anuncertain weight vectorΩ = (ω 1 ,...,ω n )∈UV n ≥0 , whereω j : Γ→R ≥0 is an uncertain variable representing the importance ofC j ; • a parameterλ∈[0,1](oftenλ= 1 2 ). HereUV ≥0 denotes the class of nonnegative uncertain variables on(Γ,L,M). (2) Scenario-wise normalized weights.For eachγ∈Γ, define W(γ) := n ∑ t=1 ω t (γ), w j (γ) := ω j (γ) W(γ) (j= 1,...,n), so that ∑ n j=1 w j (γ) = 1wheneverW(γ)>0. (3) Scenario-wise normalization of performances.For each criterionjand scenarioγ, set ξ j (γ) :=min 1≤i≤m ξ ij (γ),ξ j (γ) :=max 1≤i≤m ξ ij (γ),∆ j (γ) :=ξ j (γ)−ξ j (γ). Assuming∆ j (γ)>0, define the normalized scorest ij (γ)∈[0,1]by t ij (γ) := ξ ij (γ)−ξ j (γ) ∆ j (γ) , j∈J + (benefit), ξ j (γ)−ξ ij (γ) ∆ j (γ) , j∈J − (cost). (4) CoCoSo comparability sequences (SAW-like and WEP-like).Define, for each alternativeA i and scenarioγ, H i (γ) := n ∑ j=1 w j (γ)t ij (γ), G i (γ) := n ∏ j=1 ( t ij (γ) ) w j (γ) , with the convention0 0 := 1when it occurs. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (5) Three appraisal scores.Let S GH (γ) := m ∑ p=1 ( G p (γ) +H p (γ) ) , H min (γ) :=min p H p (γ), G min (γ) :=min p G p (γ), H max (γ) :=max p H p (γ), G max (γ) :=max p G p (γ). Assuming the denominators below are positive, define as (α) i (γ) := G i (γ) +H i (γ) S GH (γ) , as (β) i (γ) := H i (γ) H min (γ) + G i (γ) G min (γ) , as (γ) i (γ) := λH i (γ) + (1−λ)G i (γ) λH max (γ) + (1−λ)G max (γ) . (6) Final U-CoCoSo score and decision rule.Define the final scenario-wise score AS i (γ) := ( as (α) i (γ)as (β) i (γ)as (γ) i (γ) ) 1/3 + as (α) i (γ) +as (β) i (γ) +as (γ) i (γ) 3 . ThenAS i : Γ→R ≥0 is an uncertain variable (under the well-definedness conditions below), and it induces an uncertain set AS i (γ) :=AS i (γ)⊆R ≥0 . If the expectation exists, define the crisp utility U i :=E[AS i ]∈R, rank alternatives byA p U-CoCoSo A q ⇐⇒U p ≥U q , and select any A ? ∈arg max 1≤i≤m U i . Theorem 6.3.4(Uncertain-set structure and well-definedness of U-CoCoSo).In Definition 6.3.3, assume: (A1)Eachξ ij andω j is anL-measurable mapΓ→R ≥0 . (A2)Weight normalization is valid:W(γ) = ∑ n t=1 ω t (γ)>0for allγ∈Γ. (A3)Nondegenerate criteria ranges:∆ j (γ)>0for allj= 1,...,nand allγ∈Γ. (A4)Positive denominators for appraisal scores: S GH (γ)>0, H min (γ)>0, G min (γ)>0, λH max (γ) + (1−λ)G max (γ)>0 (∀γ∈Γ). (A5)Finite expected scores:E[AS i ]∈Rexists for alli= 1,...,m. Then: Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (i)all quantities in Steps(2)–(6)of Definition 6.3.3 are well-defined for everyγ∈Γ; (i)eachAS i : Γ→R ≥0 is an uncertain variable, henceAS i (γ) =AS i (γ)is an uncertain set on (Γ,L,M); (i)the ranking preorder U-CoCoSo induced byU i =E[AS i ]is well-defined onA, andarg max i U i is nonempty. Proof.Fix anyγ∈Γ. By (A2),W(γ)>0, so each normalized weightw j (γ) =ω j (γ)/W(γ)is a well-defined real number and ∑ j w j (γ) = 1. By (A3),∆ j (γ)>0, so each normalized performancet ij (γ)is well-defined. Moreover, by construction, t ij (γ)∈[0,1]. Hence the weighted sumH i (γ) = ∑ j w j (γ)t ij (γ)is well-defined and belongs to[0,1]. AlsoG i (γ) = ∏ j (t ij (γ)) w j (γ) is well-defined as a finite product of nonnegative reals (using0 0 := 1when needed), so G i (γ)∈[0,1]. Assumption (A4) guarantees that every denominator appearing inas (α) i (γ),as (β) i (γ),as (γ) i (γ)is strictly positive, so these appraisal scores are well-defined real numbers. ConsequentlyAS i (γ)is well-defined. To show measurability, note that sums, products, scalar division by a strictly positive measurable function, and pointwise min/max preserveL-measurability. Under (A1)–(A4), each mappingγ7→AS i (γ)is obtained from(ξ ij )and(ω j )by finitely many such operations, thereforeAS i is anL-measurable mapΓ→R ≥0 , i.e., an uncertain variable. This proves (i) and (i). Finally, by (A5) eachU i =E[AS i ]is a finite real number, so the preorderA p U-CoCoSo A q ⇐⇒U p ≥U q is well-defined. SinceAis finite, max i U i exists and arg max i U i is nonempty, proving (i). CoCoSo and representative uncertainty-aware variants are listed in Table 6.4. Table 6.4: CoCoSo and representative uncertainty-aware variants (organized by the degree-domain dimen- sionk). kRepresentative CoCoSo-type model(s) whose evaluation degrees live in a subset of[0,1] k 1Fuzzy CoCoSo. 2Intuitionistic Fuzzy CoCoSo [758, 759]; Pythagorean fuzzy Cocoso [760, 761]; Fermatean fuzzy CoCoSo [762]. 3Picture Fuzzy CoCoSo [763]; Spherical fuzzy Cocoso [764,765]; Neutrosophic CoCoSo [766,767]. nPlithogenic CoCoSo [768] Note.Herekdenotes the ambient dimension of the degree-domain used to encode each criterion evaluation (and, if applicable, each weight). In this convention, picture-fuzzy and (single-valued) neutrosophic evaluations are treated as three-component degrees, hencek= 3. As concepts other than Uncertain CoCoSo, Rough CoCoSo [769], Grey CoCoSo [770,771], SWARA-CoCoSo [772,773], and Extended CoCoSo [774,775] are also known. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) 6.4 Fuzzy SAW (Fuzzy Simple Additive Weighting) Classical SAW normalizes performance scores, multiplies each criterion by its weight, sums weighted values, and ranks alternatives by maximum total score under given criteria [776,777]. Fuzzy SAW normalizes fuzzy performance ratings, multiplies by criterion weights, sums weighted fuzzy scores, defuzzifies them, and ranks alternatives by highest overall utility final index [778–780]. Definition 6.4.1(Fuzzy SAW (FSAW) with triangular fuzzy numbers).[778–780] LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. Assume each criterionC j is either abenefitcriterion (↑; larger is better) or acostcriterion (↓; smaller is better). (0) Triangular fuzzy numbers (TFNs) and arithmetic.Let TFN ≥0 :=(l,m,u)∈R 3 : 0≤l≤m≤u. For ̃x= (l x ,m x ,u x ), ̃y= (l y ,m y ,u y )∈TFN ≥0 andα≥0, define ̃x⊕ ̃y:= (l x +l y , m x +m y , u x +u y ), α ̃x:= (αl x , αm x , αu x ). (1) Fuzzy decision matrix.ATFN-based fuzzy decision matrixis ̃ X= ( ̃x ij )∈(TFN ≥0 ) m×n , ̃x ij = (a ij ,b ij ,c ij ), where ̃x ij represents the (possibly linguistic) performance ofA i underC j . IfP≥1decision makers provide TFNs ̃x (p) ij = (a (p) ij ,b (p) ij ,c (p) ij ), a standard aggregation is the TFN geometric mean ̃x ij := ( P ∏ p=1 a (p) ij ) 1/P , ( P ∏ p=1 b (p) ij ) 1/P , ( P ∏ p=1 c (p) ij ) 1/P . (Geometric-mean aggregation for the fuzzy MCDM matrix is commonly used.) (2) Criterion weights.Letw= (w 1 ,...,w n )be a (crisp) weight vector with w j ≥0, n ∑ j=1 w j = 1. (Weights can be obtained, for example, by defuzzifying and normalizing linguistic judgments.) (3) Normalization of TFNs.To map heterogeneous criteria to comparable scales and keep normalized TFNs in[0,1], a standard linear scale transformation for benefit criteria is: ̃r ij := ( a ij c ∗ j , b ij c ∗ j , c ij c ∗ j ) , c ∗ j :=max 1≤i≤m c ij ,(C j :↑). This normalization is widely used in fuzzy SAW implementations. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) For cost criteria, one common dual form (after ensuring positivity) is: ̃r ij := ( a − j c ij , a − j b ij , a − j a ij ) , a − j :=min 1≤i≤m a ij ,(C j :↓), which is the TFN analogue of reciprocal cost normalization. (4) Weighted normalized matrix and fuzzy SAW score.Define the weighted normalized TFN entries by ̃v ij :=w j ̃r ij = (w j r ` ij , w j r m ij , w j r u ij ), and the total fuzzy score of alternativeA i by the TFN sum ̃s i := n ⊕ j=1 ̃v ij ∈TFN ≥0 . Equivalently, in matrix form this is the fuzzy weighted summation ̃s= ̃ Rw(the SAW aggregation of normalized ratings with weights). (5) Defuzzification and ranking.Convert ̃s i = (s ` i ,s m i ,s u i )to a crisp score via (signed-distance / centroid) defuzzification: Score(A i ) :=Defuzz( ̃s i ) := s ` i +s m i +s u i 3 . Then rank alternatives by descending score: A p FSAW A q ⇐⇒Score(A p )≥Score(A q ). (Using(a+b+c)/3as a defuzzification rule for TFNs is standard in fuzzy SAW practice.) (6) Reduction to crisp SAW.If every TFN is degenerate ̃x ij = (x ij ,x ij ,x ij ), then ̃r ij and ̃s i are degenerate as well, and the above procedure reduces to the classical (crisp) SAW method. Using Uncertain Sets, we define Uncertain SAW (U-SAW) as follows. Definition 6.4.2(Uncertain SAW (U-SAW): scenario-wise SAW with expected-score decision).LetA= A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria, partitioned as C=C + ̇ ∪C − (benefit/cost), J + :=j:C j ∈C + , J − :=j:C j ∈C − . Fix an uncertainty space(Γ,L,M). (1) Uncertain input data.Anuncertain SAW instanceconsists of: • anuncertain decision matrix Ξ = (ξ ij )∈UV m×n , ξ ij : Γ→R ≥0 , whereξ ij (γ)is the (nonnegative) performance ofA i underC j in scenarioγ; Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) • anuncertain criterion-weight vector Ω = (ω 1 ,...,ω n )∈(UV ≥0 ) n , ω j : Γ→R ≥0 . (2) Scenario-wise weight normalization.For each scenarioγ∈Γ, define W(γ) := n ∑ t=1 ω t (γ), w j (γ) := ω j (γ) W(γ) (j= 1,...,n), so that ∑ n j=1 w j (γ) = 1wheneverW(γ)>0. (3) Scenario-wise performance normalization.For each criterionjand scenarioγ, define ξ j (γ) :=min 1≤i≤m ξ ij (γ),ξ j (γ) :=max 1≤i≤m ξ ij (γ). Assumingξ j (γ)>0for benefit criteria andξ j (γ)>0for cost criteria, define normalized scoresr ij (γ)∈[0,1] by r ij (γ) := ξ ij (γ) ξ j (γ) , j∈J + (benefit), ξ j (γ) ξ ij (γ) , j∈J − (cost). (4) Scenario-wise SAW aggregation.Define, for each alternativeA i and scenarioγ, S i (γ) := n ∑ j=1 w j (γ)r ij (γ)∈[0,1]. ThenS i : Γ→[0,1]is theuncertain SAW score(a scenario-wise SAW utility). (5) Uncertain-set structure and decision rule.EachS i induces an uncertain set S i (γ) :=S i (γ)⊆[0,1]. IfE[S i ]exists and is finite, define the crisp utility U i :=E[S i ]∈R, rank alternatives by A p U-SAW A q ⇐⇒U p ≥U q , and select any A ? ∈arg max 1≤i≤m U i . Theorem 6.4.3(Uncertain-set structure and well-definedness of U-SAW).In Definition 6.4.2, assume: (A1)(Measurability) Eachξ ij andω j isL-measurable. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (A2)(Positive total weight)W(γ) = ∑ n t=1 ω t (γ)>0for allγ∈Γ. (A3)(Normalization denominators) For allγ∈Γ: ξ j (γ)>0∀j∈J + , ξ j (γ)>0∀j∈J − , ξ ij (γ)>0∀(i,j)∈1,...,m×J − . (A4)(Finite expectations)E[S i ]exists and is finite for alli= 1,...,m. Then: (i)the normalized weightsw j (γ)and normalized scoresr ij (γ)are well-defined for everyγ∈Γ, and r ij (γ)∈[0,1]; (i)eachS i : Γ→[0,1]is an uncertain variable, henceS i (γ) =S i (γ)is an uncertain set on(Γ,L,M); (i)the preorder U-SAW induced byU i =E[S i ]is well-defined, and sinceAis finite,arg max i U i is nonempty. Proof.Fixγ∈Γ. By (A2),W(γ)>0, hence each normalized weightw j (γ) =ω j (γ)/W(γ)is a well-defined real number and ∑ n j=1 w j (γ) = 1. Forj∈J + , (A3) givesξ j (γ)>0, sor ij (γ) =ξ ij (γ)/ξ j (γ)is well-defined and lies in[0,1]because0≤ ξ ij (γ)≤ξ j (γ)by definition of the maximum. Forj∈J − , (A3) ensuresξ ij (γ)>0andξ j (γ)>0, so r ij (γ) =ξ j (γ)/ξ ij (γ)is well-defined and belongs to[0,1]sinceξ j (γ)≤ξ ij (γ). Therefore, for eachithe SAW scoreS i (γ) = ∑ n j=1 w j (γ)r ij (γ)is well-defined and lies in[0,1], as a convex combination of numbers in[0,1]. To prove thatS i is an uncertain variable, observe that under (A1) the mapsξ ij andω j are measurable; pointwise min/max of finitely many measurable functions is measurable; sums/products and division by a strictly positive measurable function preserve measurability. Henceγ7→w j (γ),γ7→r ij (γ), and consequently γ7→S i (γ)are allL-measurable. ThusS i is an uncertain variableΓ→[0,1], and the induced singleton mappingS i (γ) =S i (γ)is an uncertain set. This proves (i) and (i). Finally, (A4) yields finite realsU i =E[S i ]. HenceA p U-SAW A q ⇐⇒U p ≥U q defines a well-defined preorder onA. SinceAis finite, max i U i exists and arg max i U i is nonempty. This proves (i). Related concepts of SAW under uncertainty-aware models are listed in Table 6.5. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Table 6.5: Related concepts of SAW under uncertainty-aware models. kRelated SAW concept(s) 1 Fuzzy SAW 2 Intuitionistic Fuzzy SAW [781,782] 3 Neutrosophic SAW [783,784] 6.5 Fuzzy RAFSI (Fuzzy Ranking of Alternatives through Functional mapping of cri- terion Sub-Intervals into a single Interval) RAFSI maps criterion values to a common interval using ideal and anti-ideal points, then aggregates weighted normalized scores [785, 786]. Fuzzy RAFSI uses fuzzy numbers for evaluations, maps them to a common interval, normalizes benefit/cost, aggregates fuzzy weighted scores [787,788]. Definition 6.5.1(TFN-based Fuzzy RAFSI).[787,788] LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. Partition criteria into benefit (max) and cost (min) types: C=C + ̇ ∪C − . Assume evaluations are given by triangular fuzzy numbers (TFNs) ̃x ij = (l ij ,m ij ,u ij ),0≤l ij ≤m ij ≤u ij , forming the fuzzy decision matrix ̃ X= ( ̃x ij )∈(TFN ≥0 ) m×n . Letw= (w 1 ,...,w n )∈[0,1] n be criterion weights with ∑ n j=1 w j = 1. TFN arithmetic (standard positive convention).For TFNs ̃a= (a 1 ,a 2 ,a 3 ), ̃ b= (b 1 ,b 2 ,b 3 )andα≥0, define ̃a⊕ ̃ b:= (a 1 +b 1 , a 2 +b 2 , a 3 +b 3 ), α ̃a:= (αa 1 , αa 2 , αa 3 ), and (for strictly positive TFNs) the reciprocal ̃a −1 := ( 1 a 3 , 1 a 2 , 1 a 1 ) . Fix the centroid score (defuzzification) map COA(l,m,u) := l+m+u 3 . Step 1 (reference points: ideal/anti-ideal).For each criterionC j , define crisp reference points (via COA-ordering): a I j := max 1≤i≤m COA( ̃x ij ), C j ∈C + , min 1≤i≤m COA( ̃x ij ), C j ∈C − , a N j := min 1≤i≤m COA( ̃x ij ), C j ∈C + , max 1≤i≤m COA( ̃x ij ), C j ∈C − . (Equivalently, a decision-maker may specifya I j ,a N j externally.) Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Step 2 (functional mapping to a common interval).Fix numbersn 1 >0andn 2 > n 1 (oftenn 1 = 1, n 2 = 6). Define the affine map f j (x) := n 2 −n 1 a I j −a N j x+ a I j n 1 −a N j n 2 a I j −a N j (a I j 6=a N j ), and set the standardized TFN ̃s ij :=f j ( ̃x ij ) := ( f j (l ij ), f j (m ij ), f j (u ij ) ) ∈TFN >0 . (Ifa I j =a N j , set ̃s ij := (n 1 ,n 1 ,n 1 )for alli.) Step 3 (arithmetic and harmonic means of the common interval).Define A:= n 1 +n 2 2 , H:= 2 1 n 1 + 1 n 2 . Step 4 (RAFSI normalization).Define the normalized TFNs ̂ ̃s ij by ̂ ̃s ij := 1 2A ̃s ij , C j ∈C + , H 2 ̃s −1 ij , C j ∈C − . Step 5 (criteria function / overall score and ranking).For each alternativeA i , define its overall fuzzy RAFSI score ̃ V(A i ) := n ⊕ j=1 ( w j ̂ ̃s ij ) ∈TFN >0 . DefuzzifyV i :=COA( ̃ V(A i ))and rank alternatives by A p F-RAFSI A q ⇐⇒V p ≥V q . Any maximizerA ? ∈arg max i V i is called aFuzzy RAFSI best alternative. Using Uncertain Sets, we define Uncertain RAFSI (U-RAFSI) as follows. Definition 6.5.2(Uncertainty space, uncertain variable, and induced uncertain set).Anuncertainty space is a triple(Γ,L,M)whereΓ6=∅,Lis aσ-algebra onΓ, andM:L →[0,1]is an uncertain measure. A (real-valued)uncertain variableis anL-measurable mapξ: Γ→R. Every uncertain variableξinduces a singleton-valueduncertain set X ξ : Γ→P(R),X ξ (γ) :=ξ(γ). Whenever it exists and is finite,E[ξ]∈Rdenotes the expected value ofξ. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Definition 6.5.3(Uncertain RAFSI (U-RAFSI): scenario-wise RAFSI with expected-score decision).Let A=A 1 ,...,A m be alternatives andC=C 1 ,...,C n criteria, partitioned into benefit/cost sets C=C + ̇ ∪C − , J + :=j:C j ∈C + , J − :=j:C j ∈C − . Fix an uncertainty space(Γ,L,M)and fixed constants0< n 1 < n 2 . Set the arithmetic and harmonic anchors A:= n 1 +n 2 2 , H:= 2 1 n 1 + 1 n 2 . (1) Uncertain evaluations and weights.Anuncertain RAFSI instanceis specified by: • anuncertain decision matrix Ξ = (ξ ij )∈UV m×n , ξ ij : Γ→R, • anuncertain criterion-weight vector Ω = (ω 1 ,...,ω n )∈(UV ≥0 ) n , ω j : Γ→R ≥0 . (2) Scenario-wise weight normalization.For each scenarioγ∈Γ, define W(γ) := n ∑ t=1 ω t (γ), w j (γ) := ω j (γ) W(γ) (j= 1,...,n). (3) Scenario-wise ideal/anti-ideal reference points.For each criterionjand scenarioγ, define the reference scalars a I j (γ) := max 1≤i≤m ξ ij (γ), j∈J + , min 1≤i≤m ξ ij (γ), j∈J − , a N j (γ) := min 1≤i≤m ξ ij (γ), j∈J + , max 1≤i≤m ξ ij (γ), j∈J − . (Thusa I j (γ)is the scenario-wise ideal anda N j (γ)is the scenario-wise anti-ideal.) (4) Scenario-wise functional mapping to the common interval[n 1 ,n 2 ].Whenevera I j (γ)6=a N j (γ), define the affine map f j,γ (x) := n 2 −n 1 a I j (γ)−a N j (γ) x+ a I j (γ)n 1 −a N j (γ)n 2 a I j (γ)−a N j (γ) . Define the standardized (mapped) performance s ij (γ) := f j,γ ( ξ ij (γ) ) , a I j (γ)6=a N j (γ), n 1 ,a I j (γ) =a N j (γ), (i= 1,...,m;j= 1,...,n). (Sos ij (γ)∈[n 1 ,n 2 ]whenevera I j (γ)6=a N j (γ).) Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (5) RAFSI normalization (benefit/cost).Define normalized valueŝs ij (γ)by ̂s ij (γ) := s ij (γ) 2A ,j∈J + , H 2 · 1 s ij (γ) , j∈J − . (6) Scenario-wise RAFSI score and uncertain-set output.For each alternativeA i , define its (scenario-wise) RAFSI score V i (γ) := n ∑ j=1 w j (γ)̂s ij (γ)∈R. ThenV i : Γ→Ris an uncertain variable, and it induces an uncertain set V i (γ) :=V i (γ). (7) Expected-score ranking and selection.IfE[V i ]exists and is finite, define U i :=E[V i ]∈R, A p U-RAFSI A q ⇐⇒U p ≥U q , and select any A ? ∈arg max 1≤i≤m U i . Theorem 6.5.4(Uncertain-set structure and well-definedness of U-RAFSI).In Definition 6.5.3, assume: (A1)(Measurability) Eachξ ij andω j isL-measurable. (A2)(Positive total weight)W(γ) = ∑ n t=1 ω t (γ)>0for allγ∈Γ. (A3)(Common-interval anchors)0< n 1 < n 2 . (A4)(Cost positivity after mapping) For everyγ∈Γandj∈J − , the mapped values satisfys ij (γ)>0for alli(in particular this holds ifn 1 >0). (A5)(Finite expectations)E[V i ]exists and is finite for alli= 1,...,m. Then: (i)for everyγ∈Γ, the quantitiesw j (γ),a I j (γ),a N j (γ),s ij (γ),̂s ij (γ), andV i (γ)are well-defined real numbers; (i)eachV i is an uncertain variable, henceV i (γ) =V i (γ)is an uncertain set; (i)the preorder U-RAFSI is well-defined andarg max i E[V i ]6=∅. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Proof.Fixγ∈Γ. (i) By (A2),W(γ)>0, hencew j (γ) =ω j (γ)/W(γ)is well-defined and ∑ n j=1 w j (γ) = 1. Sincem <∞, the pointwise maxima/minima ofξ ij (γ) m i=1 exist inR, soa I j (γ)anda N j (γ)are well-defined reals. Ifa I j (γ)6=a N j (γ), thenf j,γ is a well-defined affine map ands ij (γ) =f j,γ (ξ ij (γ))is a real number. If a I j (γ) =a N j (γ), we sets ij (γ) =n 1 , which is well-defined by (A3). Thuss ij (γ)is well-defined in all cases. Moreover, fora I j (γ)6=a N j (γ)one hasf j,γ (a N j (γ)) =n 2 andf j,γ (a I j (γ)) =n 1 , sos ij (γ)∈[n 1 ,n 2 ]whenever ξ ij (γ)∈[a N j (γ),a I j (γ)], which holds by definition of max/min. Hences ij (γ)≥n 1 >0. In particular, for j∈J − we haves ij (γ)>0, so the reciprocal1/s ij (γ)in the cost normalization is well-defined. Therefore ̂s ij (γ)is well-defined for alli,j, and so isV i (γ) = ∑ n j=1 w j (γ)̂s ij (γ). (i) Under (A1), eachξ ij andω j is measurable. Finite sums preserve measurability, and pointwise min/max of finitely many measurable functions is measurable. On the setγ:a I j (γ)6=a N j (γ),s ij (γ)is obtained from measurable functions by arithmetic operations and division by a nonzero measurable function, hence is measurable there; on the complementary set it is the constantn 1 , hence measurable. Thuss ij is measurable on all ofΓ. Since1/s ij is measurable wherevers ij >0(and (A4) guarantees this for cost criteria), it follows that̂s ij is measurable. Consequently eachV i isL-measurable, i.e., an uncertain variable, and V i (γ) =V i (γ)is an uncertain set. (i) By (A5) eachU i =E[V i ]is a finite real number, soA p U-RAFSI A q ⇐⇒U p ≥U q defines a well-defined preorder. SinceAis finite, max i U i exists and arg max i U i is nonempty. Related concepts of RAFSI under uncertainty-aware models are listed in Table 6.6. Table 6.6: Related concepts of RAFSI under uncertainty-aware models. kRelated RAFSI concept(s) 1 Fuzzy RAFSI 2 Intuitionistic Fuzzy RAFSI 2 Fermatean Fuzzy RAFSI [789] 3 Neutrosophic RAFSI [790] 6.6 Fuzzy RATMI (Fuzzy Ranking of Alternatives by Trace-to-Median Index) RATMI is a matrix-based MCDM ranking method that normalizes criterion values, applies criterion weights, and produces a single trace-to-median index by combining a trace-based performance term with a median- similarity term [791, 792]. Related concepts, such as RAMS-RATMI [793, 794], are also known. Fuzzy RATMI extends RATMI to settings where evaluations are fuzzy or linguistic; the fuzzy assessments are first converted to comparable scalar scores (e.g., by defuzzification), and then the same trace-to-median aggregation is used to rank alternatives under uncertainty [795]. Definition 6.6.1(Fuzzy RATMI (FRATMI): Fuzzy Ranking of Alternatives by Trace-to-Median Index). [795] LetA=A 1 ,...,A m be alternatives andC=C 1 ,...,C n criteria (m,n≥2). Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Assume apositivefuzzy decision matrix ̃ X= ( ̃x ij ) m×n , ̃x ij ∈F >0 , whereF >0 is a chosen family of positive fuzzy evaluations (e.g., TFNs, trapezoidal fuzzy numbers, interval- valued fuzzy numbers). Step 0 (Score/defuzzification).Fix a score mapS:F >0 →(0,∞)and define a crisp matrix x ij :=S( ̃x ij )>0 (i= 1,...,m;j= 1,...,n). (Example for TFN ̃x= (`,m,u):S( ̃x) = (`+m+u)/3.) Step 1 (Normalization).For each criterionC j , define r ij := x ij max 1≤i≤m x ij , C j ∈C ben , min 1≤i≤m x ij x ij , C j ∈C cost . Thenr ij ∈(0,1]. Step 2 (Weights and weighted normalization).Letw= (w 1 ,...,w n ) > be criterion weights with w j ≥0, n ∑ j=1 w j = 1. Define the weighted normalized matrixU= (u ij )by u ij :=w j r ij (i= 1,...,m;j= 1,...,n). Step 3 (Optimal alternative components).Define the componentwise optimal vectorQ= (q 1 ,...,q n ) by q j :=max 1≤i≤m u ij (j= 1,...,n). Letk:=|C ben |andh:=|C cost |(k+h=n), and decomposeQ=Q max ∪Q min according toC ben andC cost . Step 4 (Magnitudes).Define the magnitudes of the benefit- and cost-components: Q k := √ ∑ C j ∈C ben q 2 j , Q h := √ ∑ C j ∈C cost q 2 j . For each alternativeA i , define U ik := √ ∑ C j ∈C ben u 2 ij , U ih := √ ∑ C j ∈C cost u 2 ij . Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Step 5 (Trace index: MCRAT component).Define diagonal matrices F:= ( Q k 0 0Q h ) , G i := ( U ik 0 0U ih ) , T i :=F G i . Define the trace score tr i :=tr(T i ) =Q k U ik +Q h U ih . Step 6 (Median similarity: RAMS component).Define the median lengths M:= √ Q 2 k +Q 2 h 2 , M i := √ U 2 ik +U 2 ih 2 , and the median similarity MS i := M i M . Step 7 (Trace-to-median index: RATMI).Letv∈[0,1](typicallyv= 0.5). Define tr ∗ :=min 1≤i≤m tr i ,tr − :=max 1≤i≤m tr i ,MS ∗ :=min 1≤i≤m MS i ,MS − :=max 1≤i≤m MS i . Assume tr − >tr ∗ and MS − >MS ∗ . The(fuzzy) RATMI indexofA i is E i :=v tr i −tr ∗ tr − −tr ∗ + (1−v) MS i −MS ∗ MS − −MS ∗ . Ranking rule:rank alternatives in descending order ofE i . We now present Uncertain RATMI of typeM, obtained by extending the framework using Uncertain Sets. Definition 6.6.2(Uncertain RATMI of typeM(U-RATMI)).LetA=A 1 ,...,A m be alternatives and C=C 1 ,...,C n criteria, withm,n≥2. Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Fix an uncertain modelMwith Dom(M)6=∅and an admissible positive scoreS M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M) (i= 1,...,m;j= 1,...,n). Step 0 (Crisp projection).Define the positive real matrixX= (x ij )by x ij :=S M ( x (M) ij ) ∈(0,∞). Step 1 (Normalization).For each criterionC j , define x max j :=max 1≤i≤m x ij , x min j :=min 1≤i≤m x ij . Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Thenx max j >0andx min j >0. Define r ij := x ij x max j , C j ∈C ben , x min j x ij , C j ∈C cost . (i= 1,...,m;j= 1,...,n). Hencer ij ∈(0,1]. Step 2 (Weights and weighted normalization).Letw= (w 1 ,...,w n ) > be criterion weights satisfying w j ≥0, n ∑ j=1 w j = 1. Define the weighted normalized matrixU= (u ij )by u ij :=w j r ij (i= 1,...,m;j= 1,...,n). Step 3 (Componentwise optimal vector).Define q j :=max 1≤i≤m u ij (j= 1,...,n), and setQ= (q 1 ,...,q n ). Step 4 (Benefit/cost magnitudes).Letk:=|C ben |andh:=|C cost |(k+h=n). Define Q k := √ ∑ C j ∈C ben q 2 j , Q h := √ ∑ C j ∈C cost q 2 j , and for each alternativeA i , U ik := √ ∑ C j ∈C ben u 2 ij , U ih := √ ∑ C j ∈C cost u 2 ij . (Empty sums are understood as0.) Step 5 (Trace index component).Define diagonal matrices F:= ( Q k 0 0Q h ) , G i := ( U ik 0 0U ih ) , T i :=F G i , and the trace score tr i :=tr(T i ) =Q k U ik +Q h U ih . Step 6 (Median similarity component).Define M:= √ Q 2 k +Q 2 h 2 , M i := √ U 2 ik +U 2 ih 2 ,MS i := M i M . Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Step 7 (Trace-to-median index).Letv∈[0,1]. Define tr ∗ :=min 1≤i≤m tr i ,tr − :=max 1≤i≤m tr i ,MS ∗ :=min 1≤i≤m MS i ,MS − :=max 1≤i≤m MS i , and the ranges∆ tr :=tr − −tr ∗ ,∆ MS :=MS − −MS ∗ . Define normalized components T i := tr i −tr ∗ ∆ tr ,∆ tr >0, 0,∆ tr = 0, S i := MS i −MS ∗ ∆ MS ,∆ MS >0, 0,∆ MS = 0. TheUncertain RATMI indexofA i is E i :=v T i + (1−v)S i . Ranking rule:rank alternatives in descending order ofE i . Theorem 6.6.3(Well-definedness and boundedness of U-RATMI).Under the assumptions of Defini- tion 6.6.2 (in particular,m,n≥2,Dom(M)6=∅, andS M :Dom(M)→(0,∞)), all quantities in the U-RATMI procedure are well-defined and finite. Moreover, for each alternativeA i , 0≤T i ≤1,0≤S i ≤1,0≤E i ≤1. Hence the ranking rule in Definition 6.6.2 is always well-defined. Proof.SinceS M maps Dom(M)into(0,∞), eachx ij >0is finite. Because1,...,mis finite,x max j and x min j exist and satisfyx max j ≥x ij ≥x min j >0. Thus, for benefit criteria,r ij =x ij /x max j ∈(0,1], and for cost criteria,r ij =x min j /x ij ∈(0,1]. Withw j ≥0and ∑ j w j = 1, eachu ij =w j r ij is finite and lies in[0,1]. Therefore eachq j =max i u ij exists and is finite. The quantitiesQ k ,Q h ,U ik ,U ih are square roots of finite sums of squares, hence finite and nonnegative. At least one weight is positive, so at least oneq j >0, which impliesQ 2 k +Q 2 h = ∑ n j=1 q 2 j >0, henceM >0and MS i =M i /Mis well-defined and finite. The extrema tr ∗ ,tr − ,MS ∗ ,MS − exist becausemis finite. If∆ tr >0, thenT i is the standard min–max normalization of tr i , hence0≤T i ≤1; if∆ tr = 0, the definition setsT i = 0, so still0≤T i ≤1. The same argument gives0≤S i ≤1. Finally, withv∈[0,1],E i =vT i + (1−v)S i is a convex combination of two numbers in[0,1], so0≤E i ≤1. Therefore the indexE i exists for everyi, and sorting byE i defines a valid ranking. Related concepts of RATMI under uncertainty-aware models are listed in Table 6.7. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Table 6.7: Related concepts of RATMI under uncertainty-aware models. kRelated RATMI concept(s) 1 Fuzzy RATMI 2 Intuitionistic Fuzzy RATMI 2 Fermatean Fuzzy RATMI 3 Neutrosophic RATMI 6.7 Fuzzy RANCOM (Fuzzy RANking COMparison) RANCOM ranks criteria by importance judgments, builds a comparison matrix from ranks, sums rows, and normalizes to weights [796, 797]. Fuzzy RANCOM encodes linguistic importance as fuzzy numbers, aggregates experts, scores and ranks criteria, then derives normalized weights [65,798]. Definition 6.7.1(Fuzzy RANCOM (PTF-RANCOM) for subjective criteria weighting).[65, 798] Let C=C 1 ,...,C n be a finite set of criteria andE=E 1 ,...,E r a finite set of experts. Fix an integer q≥1. (A) Polytopic fuzzy numbers (PTFNs).Apolytopic fuzzy number (PTFN)is a triple z=〈α,η,ζ〉∈[0,1] 3 satisfyingα q +η q +ζ q ≤1, whereαis the positive-membership degree,ηis the neutral-membership degree, andζis the negative- membership degree. (B) Score functional.Forz=〈α,η,ζ〉, define the (crisp) score Score(z) := 1 +α q +η q −ζ q 3 ∈R. (C) PTF weighted aggregation (PTFWA).Given PTFNsz 1 =〈α 1 ,η 1 ,ζ 1 〉,...,z r =〈α r ,η r ,ζ r 〉and expert weightsλ= (λ 1 ,...,λ r )withλ k ≥0and ∑ r k=1 λ k = 1, define PTFWA λ (z 1 ,...,z r ) := 〈( 1− r ∏ k=1 (1−α q k ) λ k ) 1/q , r ∏ k=1 η λ k k , r ∏ k=1 ζ λ k k 〉 . (D) Input data of Fuzzy RANCOM.Each expertE k provides, for every criterionC j , a PTFN impor- tance assessment ι jk =〈α jk ,η jk ,ζ jk 〉(j= 1,...,n;k= 1,...,r), typically obtained by encoding linguistic terms into PTFNs. (E) PTF-RANCOM procedure and output weights. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Step 1.(Aggregate experts)For each criterionC j , compute its integrated PTF importance ι j :=PTFWA λ (ι j1 ,...,ι jr )∈[0,1] 3 . Step 2.(Score and rank)Computes j :=Score(ι j )and define the rank position ς j := 1 +|t∈1,...,n:s t > s j |∈1,...,n, so that smallerς j means higher importance (ties allowed). Step 3.(Ranking comparison matrix)DefineB= (b gj )∈[0,1] n×n by b gj := 1, ς g < ς j , 0.5, ς g =ς j , 0, ς g > ς j , (g,j= 1,...,n). Step 4.(Row sums)Define the row-sum vectorh∈R n ≥0 by h j := n ∑ g=1 b jg (j= 1,...,n). Step 5.(Normalized criterion weights)Define the criterion weights w j := h j ∑ n t=1 h t ∈[0,1], j= 1,...,n, so that ∑ n j=1 w j = 1. The vectorw= (w 1 ,...,w n )is called the(PTF-)fuzzy RANCOM weight vectorofC. Definition 6.7.2(Uncertain RANCOM of typeM).LetC=C 1 ,...,C n be a finite set of criteria (n≥2) andE=E 1 ,...,E r a finite set of experts (r≥1). Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M . Input (expert uncertain importance assessments).Each expertE k provides, for every criterionC j , an uncertain importance degree ι jk ∈Dom(M) (j= 1,...,n;k= 1,...,r), interpreted as the importance ofC j under the uncertainty modelM. Step 1 (Expert aggregation).Letλ= (λ 1 ,...,λ r )be expert weights withλ k ≥0and ∑ r k=1 λ k = 1. Fix an aggregation operator Agg M :Dom(M) r →Dom(M), and compute the aggregated importance for each criterion: ι j :=Agg M (ι j1 ,...,ι jr )∈Dom(M), j= 1,...,n. (Whenr= 1, take Agg M as the identity.) Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Step 2 (Score and induced ranking).Compute crisp scores s j :=S M (ι j )∈R, j= 1,...,n, and define the rank position (ties allowed) by ς j := 1 + ∣ ∣ t∈1,...,n:s t > s j ∣ ∣ ∈1,...,n. Thus smallerς j means higher importance. Step 3 (Ranking comparison matrix).DefineB= (b gj )∈[0,1] n×n by b gj := 1, ς g < ς j , 0.5, ς g =ς j , 0, ς g > ς j , (g,j= 1,...,n). Step 4 (Row sums).Define h j := n ∑ g=1 b jg (j= 1,...,n). Step 5 (Normalized criterion weights).Define the Uncertain RANCOM weights by w j := h j ∑ n t=1 h t (j= 1,...,n), provided that ∑ n t=1 h t >0. The vectorw= (w 1 ,...,w n ) > is called theUncertain RANCOM weight vector of typeM. Theorem 6.7.3(Well-definedness of Uncertain RANCOM).Under Definition 6.7.2, assume: •n≥2,r≥1, andDom(M)6=∅; • Agg M :Dom(M) r →Dom(M)is well-defined onDom(M) r ; •S M :Dom(M)→Ris admissible. Then: (i) The aggregated importancesι j and scoress j are well-defined for allj. (i) The rank positionsς j are well-defined integers in1,...,n. (i) The matrixBand the row-sumsh j are well-defined, with 0≤b gj ≤1,0≤h j ≤n(g,j= 1,...,n). Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (iv) ∑ n t=1 h t >0always holds, hence the weightsw j are well-defined and satisfy w j ≥0 (j= 1,...,n), n ∑ j=1 w j = 1, i.e.,w∈∆ n−1 :=w∈R n ≥0 : ∑ n j=1 w j = 1. Proof.(i) Since eachι jk ∈Dom(M)and Agg M maps Dom(M) r into Dom(M), eachι j =Agg M (ι j1 ,...,ι jr ) is well-defined and belongs to Dom(M). Admissibility ofS M implies eachs j =S M (ι j )is a finite real number. (i) For each fixedj, the sett:s t > s j is a subset of the finite set1,...,n, so its cardinality is a well-defined integer between0andn−1. Henceς j = 1 +|t:s t > s j |lies in1,...,n. (i) For any pair(g,j), exactly one of the three relationsς g < ς j ,ς g =ς j , orς g > ς j holds, sob gj is well-defined and belongs to0,0.5,1 ⊂[0,1]. Therefore each row sumh j = ∑ n g=1 b jg is well-defined and satisfies0≤h j ≤n. (iv) Consider the diagonal entriesb j . Sinceς j =ς j , we haveb j = 0.5for allj. Thus n ∑ t=1 h t = n ∑ t=1 n ∑ g=1 b tg = n ∑ t=1 b t + n ∑ t=1 n ∑ g=1 g6=t b tg ≥ n ∑ t=1 b t = n 2 >0. Hence the denominator inw j =h j / ∑ n t=1 h t is strictly positive, sow j is well-defined. Sinceh j ≥0, we have w j ≥0for allj, and n ∑ j=1 w j = ∑ n j=1 h j ∑ n t=1 h t = 1. Thereforew∈∆ n−1 . Related concepts of RANCOM under uncertainty-aware models are listed in Table 6.8. Table 6.8: Related concepts of RANCOM under uncertainty-aware models. kRelated RANCOM concept(s) 1 Fuzzy RANCOM 2 Intuitionistic Fuzzy RANCOM 2 Fermatean Fuzzy RANCOM 3 Neutrosophic RANCOM 6.8 Fuzzy AROMAN (Fuzzy Alternative Ranking Order Method accounting for two- step normalization) AROMAN ranks alternatives using two-step normalization, separates benefit and cost effects, then computes an exponential preference score [799, 800]. Fuzzy AROMAN replaces crisp ratings with fuzzy numbers, performs two-step fuzzy normalization, aggregates weighted effects, and defuzzifies rankings [801]. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Definition 6.8.1(Fuzzy AROMAN (Alternative Ranking Order Method accounting for two-step normal- ization)).[801] LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. Partition the criteria into benefit- and cost-type indices J max ̇ ∪J min =1,...,n, wherej∈J max means “larger is better” andj∈J min means “smaller is better”. Assume evaluations are given as triangular fuzzy numbers (TFNs) ̃x ij = (l ij ,m ij ,u ij )∈TFN ≥0 , i= 1,...,m, j= 1,...,n. IfK≥1experts provide TFNs ̃x (k) ij , set the aggregated fuzzy decision matrix ̃x ij := 1 K ( ̃x (1) ij ⊕·⊕ ̃x (K) ij ) . TFN arithmetic convention (componentwise, positive case).For TFNs ̃a= (a 1 ,a 2 ,a 3 ), ̃ b= (b 1 ,b 2 ,b 3 )andα≥0, define ̃a⊕ ̃ b:= (a 1 +b 1 ,a 2 +b 2 ,a 3 +b 3 ), α ̃a:= (αa 1 ,αa 2 ,αa 3 ). If all TFNs involved are nonnegative, define the (approximate) product and power by ̃a⊗ ̃ b:= (a 1 b 1 ,a 2 b 2 ,a 3 b 3 ), ̃a p := (a p 1 ,a p 2 ,a p 3 ) (p >0), and for a monotone functionf:R ≥0 →R ≥0 (e.g.f=exp), f( ̃a) := (f(a 1 ),f(a 2 ),f(a 3 )). Step 1 (Normalization No. 1: min–max).For each criterionj, define fuzzy extrema across alternatives by any fixed TFN preorder (e.g. by a score/defuzzification map; see below): ̃x min j :=min 1≤i≤m ̃x ij , ̃x max j :=max 1≤i≤m ̃x ij . Set the first normalized TFNs ̃ t ij by ̃ t ij := ( ̃x ij ̃x min j ) ( ̃x max j ̃x min j ), j∈J max , ( ̃x max j ̃x ij ) ( ̃x max j ̃x min j ), j∈J min , where , denote the chosen TFN subtraction/division rules (any consistent TFN arithmetic may be used). Step 2 (Normalization No. 2: vector-type).Define the second normalized TFNs ̃ t ∗ ij by ̃ t ∗ ij := ̃x ij √ √ √ √ m ⊕ p=1 ( ̃x pj ⊗ ̃x pj ) , j∈J max , ̃x −1 ij √ √ √ √ m ⊕ p=1 ( ̃x −1 pj ⊗ ̃x −1 pj ) , j∈J min , Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) where ̃x −1 is the TFN reciprocal (defined whenever the TFN is strictly positive). Step 3 (Aggregated two-step normalization).Fixβ∈[0,1]and set ̃ t norm ij :=β ̃ t ij ⊕(1−β) ̃ t ∗ ij . Step 4 (Weighting).Letw= (w 1 ,...,w n )be nonnegative criterion weights with ∑ n j=1 w j = 1. Define the weighted normalized TFNs ̂ ̃ t ij :=w j ̃ t norm ij . Step 5 (Separate aggregation by criterion type).Define the fuzzy sums ̃ L i := ⊕ j∈J min ̂ ̃ t ij , ̃ A i := ⊕ j∈J max ̂ ̃ t ij . Step 6 (Type-balance exponent and final fuzzy ranking index).Set the type-balance coefficient λ:= ∑ j∈J min w j ∈[0,1], and define ̂ ̃ L i := ̃ L λ i , ̂ ̃ A i := ̃ A 1−λ i , ̃ R i :=exp ( ̂ ̃ A i ̂ ̃ L i ) . Step 7 (Defuzzification and ordering).Fix a TFN score/defuzzification map, e.g. the centroid score Score(l,m,u) := l+m+u 3 . TheFuzzy AROMAN rankingis the preorder onAgiven by A p FAROMAN A q ⇐⇒Score( ̃ R p )≥Score( ̃ R q ). Any maximizer of Score( ̃ R i )is called aFuzzy AROMAN best alternative. We now defineUncertain AROMAN(U-AROMAN) by extending AROMAN to a general uncertain model M, using a total score map to convert uncertain evaluations into positive real values and then performing the two-step normalization on the induced score matrix. Definition 6.8.2(Uncertain AROMAN of typeM).Let A=A 1 ,...,A m andC=C 1 ,...,C n be the sets of alternatives and criteria, respectively, wherem,n∈N,m≥1, andn≥2. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Fix an uncertain modelMwith degree-domain Dom(M)⊆[0,1] k for some integerk≥1. Let X M = (μ ij ) m×n ∈Dom(M) m×n be an uncertain decision matrix, whereμ ij ∈Dom(M)is the uncertain evaluation of alternativeA i under criterionC j . Partition the criterion index set as J max ̇ ∪J min =1,...,n, whereJ max is the set of benefit criteria andJ min is the set of cost criteria. Fix: • a total score map Score M :Dom(M)−→R >0 , • a parameter β∈[0,1], • a weight vector w= (w 1 ,...,w n )∈(0,1) n , n ∑ j=1 w j = 1. Define the positive score matrix S= (s ij ) m×n ∈R m×n >0 , s ij :=Score M (μ ij ). For each criterionj, define s min j :=min 1≤i≤m s ij , s max j :=max 1≤i≤m s ij , δ j :=s max j −s min j . Assumeδ j >0for allj. Step 1 (Normalization No. 1: min–max normalization).Define t (1) ij := s ij −s min j δ j , j∈J max , s max j −s ij δ j , j∈J min . Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Step 2 (Normalization No. 2: vector-type normalization).For each criterionj, define ν j := √ √ √ √ m ∑ r=1 s 2 rj , η j := √ √ √ √ m ∑ r=1 s −2 rj . Then define t (2) ij := s ij ν j , j∈J max , s −1 ij η j , j∈J min . Step 3 (Aggregated two-step normalization).Define t ij :=β t (1) ij + (1−β)t (2) ij . Step 4 (Weighting).Define u ij :=w j t ij , i= 1,...,m, j= 1,...,n. Step 5 (Separate aggregation by criterion type).Define L i := ∑ j∈J min u ij , A i := ∑ j∈J max u ij , i= 1,...,m. Step 6 (Type-balance coefficient and final ranking index).Define λ:= ∑ j∈J min w j . Assume 0< λ <1. TheUncertain AROMAN ranking indexof alternativeA i is R i :=exp ( A 1−λ i −L λ i ) , i= 1,...,m. Step 7 (Ranking).The U-AROMAN preference relation onAis defined by A p UAROMAN A q ⇐⇒R p ≥R q . Any alternative A ? ∈arg max 1≤i≤m R i is called aU-AROMAN best alternative. Theorem 6.8.3(Well-definedness of Uncertain AROMAN).Under Definition 6.8.2, assume: Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (A1)AandCare finite nonempty sets; (A2)X M = (μ ij )∈Dom(M) m×n ; (A3)Score M :Dom(M)→R >0 is a total map; (A4)J max ̇ ∪J min =1,...,n, with bothJ max 6=∅andJ min 6=∅; (A5) for everyj∈1,...,n, δ j =s max j −s min j >0; (A6)w= (w 1 ,...,w n )∈(0,1) n satisfies ∑ n j=1 w j = 1; (A7)β∈[0,1]. Then the following hold: (i) the matricesS= (s ij ),T (1) = (t (1) ij ),T (2) = (t (2) ij ),T= (t ij ), andU= (u ij )are well-defined; (i) for alli,j, 0≤t (1) ij ≤1,0< t (2) ij ≤1,0≤t ij ≤1,0≤u ij ≤w j ; (i) for everyi, 0≤L i ≤λ,0≤A i ≤1−λ; (iv) the coefficientλsatisfies0< λ <1, each ranking indexR i is a well-defined positive real number, and the relation UAROMAN is a total preorder onA; (v) the set arg max 1≤i≤m R i is nonempty. Hence Uncertain AROMAN of typeMis well-defined. Proof.By (A2), eachμ ij ∈Dom(M). Since Score M is total by (A3), s ij :=Score M (μ ij ) is well-defined and belongs toR >0 . ThereforeS= (s ij )∈R m×n >0 is well-defined. For each fixed criterionj, the sets 1j ,...,s mj ⊂R >0 is finite and nonempty, sos min j ands max j are well- defined. By (A5),δ j >0, hence the min–max normalized values t (1) ij = s ij −s min j δ j , j∈J max , s max j −s ij δ j , j∈J min Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) are well-defined. Sinces min j ≤s ij ≤s max j , it follows that 0≤t (1) ij ≤1. Next, because eachs rj >0, the sums m ∑ r=1 s 2 rj and m ∑ r=1 s −2 rj are strictly positive, soν j >0andη j >0. Hence t (2) ij = s ij ν j , j∈J max , s −1 ij η j , j∈J min is well-defined for everyi,j. Moreover, ν 2 j = m ∑ r=1 s 2 rj ≥s 2 ij =⇒0< s ij ν j ≤1, and similarly η 2 j = m ∑ r=1 s −2 rj ≥s −2 ij =⇒0< s −1 ij η j ≤1. Thus 0< t (2) ij ≤1. Sinceβ∈[0,1], the aggregated normalized value t ij =βt (1) ij + (1−β)t (2) ij is a convex combination of numbers in[0,1], hence 0≤t ij ≤1. Becausew j >0by (A6), the weighted value u ij =w j t ij is well-defined and satisfies 0≤u ij ≤w j . This proves (i) and (i). Now define L i = ∑ j∈J min u ij , A i = ∑ j∈J max u ij . Since0≤u ij ≤w j , we obtain 0≤L i ≤ ∑ j∈J min w j =λ,0≤A i ≤ ∑ j∈J max w j = 1−λ. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) This proves (i). BecauseJ min 6=∅,J max 6=∅, and everyw j >0, we have 0< λ= ∑ j∈J min w j <1. Hence the exponentsλand1−λare strictly positive. SinceL i ,A i ≥0, the powersL λ i andA 1−λ i are well-defined nonnegative real numbers. Therefore R i =exp ( A 1−λ i −L λ i ) is well-defined and strictly positive for everyi. The relation A p UAROMAN A q ⇐⇒R p ≥R q is induced by the usual order onR, so it is reflexive, transitive, and total. Hence it is a total preorder on A. This proves (iv). Finally, sinceAis finite and nonempty, the finite set R 1 ,...,R m ⊂R >0 attains a maximum. Therefore arg max 1≤i≤m R i 6=∅. This proves (v), and hence U-AROMAN is well-defined. Related concepts of AROMAN under uncertainty-aware models are listed in Table 6.9. Table 6.9: Related concepts of AROMAN under uncertainty-aware models. kRelated AROMAN concept(s) 1 Fuzzy AROMAN 2 Intuitionistic Fuzzy AROMAN 2 Fermatean Fuzzy AROMAN 3 Neutrosophic AROMAN 6.9 Fuzzy MAUT (Fuzzy multi-attribute utility theory) MAUT (multi-attribute utility theory) models preferences with single-attribute utility functions and trade- off weights, combining them (often additively) into an overall utility used to choose the maximum-utility alternative option [802–804]. Fuzzy MAUT represents utilities, weights, or outcomes as fuzzy sets/num- bers, computing fuzzy expected utilities and then defuzzifying or using dominance to select under vague assessments [805–807]. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Definition 6.9.1(Fuzzy MAUT (fuzzy multi-attribute utility theory)).(cf. [808,809]) LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C K a finite set of attributes (criteria). (0) Fuzzy numbers and defuzzification.Fix a classFN(R)of fuzzy numbers onR(e.g. triangular or trapezoidal), and letFN([0,1])denote fuzzy numbers supported in[0,1]. Fix a defuzzification (score) functional Defuzz:FN([0,1])→[0,1] (e.g. centroid/COA), used only for final ranking. (1) Fuzzy performance ratings.Afuzzy MAUT instancespecifies, for each alternativeA i and attribute C k , a fuzzy rating ̃x ik ∈FN(R), i= 1,...,m, k= 1,...,K, interpreted as the uncertain value of attributeC k for alternativeA i . (2) Attribute utility functions.For each attributeC k , fix a (normalized) utility function u k :R→[0,1]. Itsfuzzy extensionmaps fuzzy ratings to fuzzy utilities: ̃u ik :=u k ( ̃x ik )∈FN([0,1]), defined by Zadeh’s extension principle. Equivalently, inα-cut form: if( ̃x ik ) α = [x L ik (α),x U ik (α)], then ( ̃u ik ) α = [ min t∈[x L ik (α),x U ik (α)] u k (t),max t∈[x L ik (α),x U ik (α)] u k (t) ] (α∈(0,1]). (In particular, ifu k is monotone increasing, then( ̃u ik ) α = [u k (x L ik (α)),u k (x U ik (α))].) (3) Weights.Either use crisp weightsw= (w 1 ,...,w K )withw k ≥0and ∑ K k=1 w k = 1, or use fuzzy weights ̃w k ∈FN([0,1])(with a chosen normalization rule). (4) Overall fuzzy utility (additive MAUT aggregation).Using fuzzy arithmetic onFN([0,1]), define the overall fuzzy utility ofA i by ̃ U(A i ) := K ⊕ k=1 ( w k ̃u ik ) ,(crisp weights), K ⊕ k=1 ( ̃w k ⊗ ̃u ik ) ,(fuzzy weights), i= 1,...,m, where⊕is fuzzy addition, is scalar multiplication, and⊗is fuzzy multiplication (or another chosen t-norm-like weighting operation). (5) Ranking and solution.Defuzzify the overall utilities: U i :=Defuzz ( ̃ U(A i ) ) ∈[0,1], i= 1,...,m, Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) and define the induced preorder A i FMAUT A j ⇐⇒U i ≥U j . Any maximizer A ? ∈arg max A i ∈A U i is called aFuzzy MAUT(best-utility) solution. (Reduction to classical MAUT).If each fuzzy rating is degenerate ̃x ik = (x ik ,x ik ,x ik )(or the crisp singleton), and weights are crisp, then ̃u ik becomes crispu k (x ik )and the model reduces to U(A i ) = K ∑ k=1 w k u k (x ik ). Using Uncertain Sets, we define Uncertain MAUT (U-MAUT) as follows. Definition 6.9.2(Uncertain MAUT (U-MAUT): multi-attribute utility with uncertain inputs).LetA= A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C K a finite set of attributes (criteria). Fix an uncertainty space(Γ,L,M). (1) Uncertain performance ratings (attribute outcomes).For each alternativeA i and attributeC k , the attribute outcome is modeled by an uncertain variable x ik : Γ→R, i= 1,...,m, k= 1,...,K. Equivalently, eachx ik induces an uncertain setX ik (γ) :=x ik (γ). (2) Single-attribute utility functions and uncertain utilities.For each attributeC k , fix a utility function u k :R→[0,1]. Define theuncertain utilityof alternativeA i under attributeC k by composition: U ik (γ) := (u k ◦x ik )(γ) =u k ( x ik (γ) ) ∈[0,1]. ThusU ik : Γ→[0,1]is an uncertain variable and induces the uncertain setU ik (γ) :=U ik (γ). (3) Attribute weights (crisp or uncertain).Either: (Wc)crisp weights:w= (w 1 ,...,w K )∈[0,1] K with ∑ K k=1 w k = 1; or (Wu)uncertain weights:uncertain variablesw k : Γ→[0,1]such that K ∑ k=1 w k (γ) = 1for allγ∈Γ. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (4) Overall uncertain utility (additive MAUT aggregation).Define the overall utility ofA i by U i (γ) := K ∑ k=1 w k U ik (γ),under (Wc), K ∑ k=1 w k (γ)U ik (γ),under (Wu), γ∈Γ. EachU i : Γ→[0,1]induces an uncertain set U(A i )(γ) :=U i (γ)⊆[0,1]. (5) Decision rule (expected-utility ranking).IfE[U i ]exists for alli, define the (crisp) MAUT ranking by A p U-MAUT A q ⇐⇒E[U p ]≥E[U q ], and select any A ? ∈arg max A i ∈A E[U i ]. Theorem 6.9.3(Uncertain-set structure and well-definedness of U-MAUT).In Definition 6.9.2, assume: (A1)(Measurability of outcomes) Eachx ik : Γ→RisL-measurable. (A2)(Utility regularity) Eachu k :R→[0,1]is Borel-measurable. (A3)(Weights) Either(Wc)holds, or(Wu)holds with eachw k L-measurable. Then: (i)For alli,k,U ik =u k ◦x ik is an uncertain variable taking values in[0,1], henceU ik (γ) =U ik (γ)is an uncertain set. (i)For alli, the aggregated utilityU i is an uncertain variable taking values in[0,1], henceU(A i )(γ) = U i (γ)is an uncertain set. (i)If, additionally,E[U i ]exists for alli(for example, if the underlying expectation is defined for all bounded uncertain variables), then the expected-utility ranking in Definition 6.9.2 is well-defined. Proof.(i) Fixi,k. By (A1)x ik isL-measurable, and by (A2)u k is Borel-measurable. Therefore the compositionU ik =u k ◦x ik isL-measurable, hence an uncertain variable. Sinceu k (R)⊆[0,1], one has U ik (γ)∈[0,1]for allγ. ThusU ik (γ) =U ik (γ)defines a singleton-valued uncertain set. (i) Fixi. Under (Wc),U i (γ) = ∑ K k=1 w k U ik (γ)is a finite linear combination ofL-measurable functions, henceL-measurable. Under (Wu), each productw k (γ)U ik (γ)isL-measurable (product of measurable func- tions), and the finite sum is measurable; henceU i is an uncertain variable. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Moreover, in both casesw k (γ)≥0and ∑ K k=1 w k (γ) = 1(with the crisp case interpreted as constant functions), and0≤U ik (γ)≤1. Hence 0≤U i (γ) = K ∑ k=1 w k (γ)U ik (γ)≤ K ∑ k=1 w k (γ)·1 = 1, soU i (γ)∈[0,1]for allγ. ThusU(A i )(γ) =U i (γ)is a well-defined uncertain set. (i) IfE[U i ]exists for alli, then eachE[U i ]is a finite real number, so the preorderA p U-MAUT A q ⇐⇒ E[U p ]≥E[U q ]is well-defined and the argmax set is nonempty becauseAis finite. Related concepts of MAUT under uncertainty-aware models are listed in Table 6.10. Table 6.10: Related concepts of MAUT under uncertainty-aware models. kRelated MAUT concept(s) 1 Fuzzy MAUT 2 Intuitionistic Fuzzy MAUT 2 Fermatean Fuzzy MAUT 3 Neutrosophic MAUT 6.10Fuzzy SMART (Fuzzy Simple Multi-Attribute Rating Technique) SMART (Simple Multi-Attribute Rating Technique) assigns each criterion a value scale, elicits swing weights for importance, and computes a simple weighted-sum score for each alternative to rank them overall [810, 811]. Fuzzy SMART uses linguistic ratings and fuzzy weights (e.g., triangular numbers), aggregates via fuzzy arithmetic, and derives rankings by defuzzification or comparing fuzzy scores directly [812,813]. Definition 6.10.1(Fuzzy SMART (fuzzy Simple Multi-Attribute Rating Technique)).[812,813] LetA= A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. Partition criteria into benefit and cost sets C=C + ̇ ∪C − . LetP=1,...,pbe the set of decision makers (DMs). (0) TFNs, fuzzy arithmetic, and defuzzification.LetTFN:=(l,m,u)∈R 3 :l≤m≤ube the set of triangular fuzzy numbers (TFNs). For ̃x= (l x ,m x ,u x ), ̃y= (l y ,m y ,u y )∈TFNandα≥0, define ̃x⊕ ̃y:= (l x +l y , m x +m y , u x +u y ), α ̃x:= (αl x , αm x , αu x ). Fix a defuzzification map (e.g. centroid/COA) Defuzz(l,m,u) := l+m+u 3 . (When TFNs are used, centroid-type defuzzification is standard in fuzzy SMART implementations.) Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (1) Fuzzy criterion weights (group aggregation).Each DMt∈Passigns a TFN weight ̃w (t) j ∈TFN ≥0 to criterionC j . Define the aggregated fuzzy weight by the TFN mean ̃w j := 1 p p ⊕ t=1 ̃w (t) j ∈TFN ≥0 , j= 1,...,n. Defuzzify and normalize to obtain crisp SMART weights ˆw j :=Defuzz( ̃w j ), w j := ˆw j ∑ n r=1 ˆw r , j= 1,...,n, so thatw j ≥0and ∑ n j=1 w j = 1. (2) Fuzzy ratings and aggregation.Each DMt∈Pprovides a TFN rating ̃x (t) ij ∈TFN ≥0 forA i under C j . Aggregate by the TFN mean: ̃x ij := 1 p p ⊕ t=1 ̃x (t) ij ∈TFN ≥0 , i= 1,...,m, j= 1,...,n. Thus ̃ X= ( ̃x ij )is the aggregated fuzzy decision matrix. (3) Normalization (benefit/cost).Choose a crisp ranking of TFNs via Defuzz and define, for each criterionj, x j :=min 1≤i≤m Defuzz( ̃x ij ),x j :=max 1≤i≤m Defuzz( ̃x ij ),∆ j :=x j −x j >0. Define the (crisp) normalized ratingsr ij ∈[0,1]by r ij := Defuzz( ̃x ij )−x j ∆ j , C j ∈C + , x j −Defuzz( ̃x ij ) ∆ j , C j ∈C − . (Equivalently, one may normalize directly at the TFN level and then defuzzify; both variants appear in fuzzy SMART practice.) (4) SMART aggregation (simple additive weighting).The SMART score of alternativeA i is S i := n ∑ j=1 w j r ij , i= 1,...,m, which is the classical SAW-type synthesis used by SMART. (5) Ranking and solution.Rank alternatives by descendingS i : A p FSMART A q ⇐⇒S p ≥S q , and any maximizer A ? ∈arg max 1≤i≤m S i is called aFuzzy SMARTsolution. (Reduction to classical SMART).If all TFNs are degenerate (crisp) andp= 1, then ̃x ij and ̃w j reduce to crisp values, and the above procedure reduces to the standard SMART/SAW scoring formula. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Using Uncertain Sets, we define Uncertain SMART (U-SMART) as follows. Definition 6.10.2(Uncertain SMART (U-SMART): uncertain Simple Multi-Attribute Rating Technique). LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. Fix an uncertainty space(Γ,L,M). (1) Uncertain raw ratings (attribute performances).For each alternativeA i and criterionC j , the (raw) performance is modeled by an uncertain variable X ij : Γ→R, i= 1,...,m, j= 1,...,n. Thus eachX ij induces the uncertain setX ij (γ) :=X ij (γ). (2) Value functions and uncertain value scores.For each criterionC j , fix a Borel-measurablevalue function v j :R→[0,1]. Define the uncertain value score V ij (γ) := (v j ◦X ij )(γ) =v j ( X ij (γ) ) ∈[0,1]. ThenV ij : Γ→[0,1]is an uncertain variable inducingV ij (γ) :=V ij (γ). (3) Swing weights (crisp or uncertain).Either: (Wc)crisp weights:w= (w 1 ,...,w n )∈[0,1] n with ∑ n j=1 w j = 1; or (Wu)uncertain weights:uncertain variablesW j : Γ→[0,1]such that n ∑ j=1 W j (γ) = 1for allγ∈Γ. (4) Overall uncertain SMART score (simple weighted sum).Define the overall SMART score of alternativeA i by S i (γ) := n ∑ j=1 w j V ij (γ),under (Wc), n ∑ j=1 W j (γ)V ij (γ),under (Wu), γ∈Γ. EachS i : Γ→[0,1]induces the uncertain set S(A i )(γ) :=S i (γ)⊆[0,1]. (5) Decision rule (expected SMART score).IfE[S i ]exists for alli, define the ranking A p U-SMART A q ⇐⇒E[S p ]≥E[S q ], and select any maximizer A ? ∈arg max A i ∈A E[S i ]. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Theorem 6.10.3(Uncertain-set structure and well-definedness of U-SMART).In Definition 6.10.2, as- sume: (A1)EachX ij : Γ→RisL-measurable. (A2)Each value functionv j :R→[0,1]is Borel-measurable. (A3)Either(Wc)holds, or(Wu)holds and eachW j isL-measurable. Then: (i)For alli,j,V ij =v j ◦X ij is an uncertain variable with values in[0,1], soV ij (γ) =V ij (γ)is an uncertain set. (i)For alli, the aggregated scoreS i is an uncertain variable with values in[0,1], soS(A i )(γ) =S i (γ) is an uncertain set. (i)If, additionally,E[S i ]exists for alli(e.g., if expectation is defined for all bounded uncertain variables), then the expected-score ranking in Definition 6.10.2 is well-defined andarg max A i ∈A E[S i ]6=∅. Proof.(i) Fixi,j. By (A1)X ij isL-measurable, and by (A2)v j is Borel-measurable. HenceV ij =v j ◦X ij isL-measurable, i.e., an uncertain variable. Sincev j (R)⊆[0,1],V ij (γ)∈[0,1]for allγ. Therefore V ij (γ) =V ij (γ)is a well-defined uncertain set. (i) Fixi. Under (Wc),S i (γ) = ∑ n j=1 w j V ij (γ)is a finite linear combination of measurable functions, hence measurable. Under (Wu), each productW j (γ)V ij (γ)is measurable and the finite sum remains measurable; henceS i is an uncertain variable. Moreover, in both cases the weights are nonnegative and sum to1(pointwise in the uncertain-weight case), and0≤V ij (γ)≤1. Thus 0≤S i (γ) = n ∑ j=1 weight j (γ)V ij (γ)≤ n ∑ j=1 weight j (γ)·1 = 1, soS i (γ)∈[0,1]for allγ. HenceS(A i )(γ) =S i (γ)is an uncertain set. (i) If eachE[S i ]exists and is finite, then the relationA p U-SMART A q ⇐⇒E[S p ]≥E[S q ]is a well-defined preorder. SinceAis finite, max A i ∈A E[S i ]is attained, so the argmax set is nonempty. Related concepts of SMART under uncertainty-aware models are listed in Table 6.11. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Table 6.11: Related concepts of SMART under uncertainty-aware models. kRelated SMART concept(s) 1 Fuzzy SMART 2 Intuitionistic Fuzzy SMART 2 Fermatean Fuzzy SMART 3 Neutrosophic SMART 6.11Fuzzy REGIME REGIME compares alternatives pairwise on each criterion, records win/lose/tie signs, multiplies by weights, and sums to build a preference matrix for ranking across all criteria. Fuzzy REGIME replaces crisp perfor- mances and win/lose signs with fuzzy comparisons, yielding fuzzy preference intensities; aggregation and defuzzification produce rankings that reflect measurement uncertainty better [814]. Definition 6.11.1(Fuzzy REGIME (F-REGIME) for MADM).[814] LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. For each criterionC j , lets j ∈ +1,−1 indicate its type (s j = +1for a benefit criterion;s j =−1for a cost criterion). Letw= (w 1 ,...,w n )be criterion weights with w j ≥0, n ∑ j=1 w j = 1. Assume afuzzy decision matrix ̃ X= ( ̃x ij ) m×n with ̃x ij ∈F, where F denotes a chosen class of fuzzy numbers onR(e.g., normal and convex fuzzy numbers). (1) Fuzzy pairwise comparator.For each criterionC j , fix a mapping κ j :F×F−→[−1,1] satisfying the following minimal axioms: (K1)(Anti-symmetry)κ j ( ̃a, ̃ b) =−κ j ( ̃ b, ̃a)for all ̃a, ̃ b∈F. (K2)(Crisp consistency) for crispa,b∈R(embedded in F),κ j (a,b) =sgn(a−b). (2) REGIME identifier (criterionwise).For each ordered pair(A i ,A k )withi6=kand each criterion C j , define the (possibly fuzzy/graded) REGIME identifier R (j) ik :=s j κ j ( ̃x ij , ̃x kj )∈[−1,1]. Collect these into theREGIME vector R(A i ,A k ) := ( R (1) ik ,...,R (n) ik ) ∈[−1,1] n . StackingR(A i ,A k )over all ordered pairs(i,k),i6=k, yields theREGIME matrix. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (3) Guide index (pairwise aggregation).Define the guide index ofA i overA k by the weighted sum GI ik := n ∑ j=1 w j R (j) ik ∈[−1,1]. (4) Net guide index and ranking.Define the net guide index of each alternative by NGI(A i ) := m ∑ k=1 k6=i GI ik . The F-REGIME ranking is obtained by sorting alternatives in descending order ofNGI(A i )(ties may be broken by a secondary rule, e.g., average rank under individual criteria or a chosen defuzzification tie-break). Proposition 6.11.2(Basic well-definedness).Under Theorem 6.11.1 and (K1), for alli6=k, R (j) ik =−R (j) ki (∀j),GI ik =−GI ki . HenceNGI(A i )is well-defined and measures the net pairwise advantage ofA i over the remaining alternatives under the chosen fuzzy comparator. Proof.By definition,R (j) ik =s j κ j ( ̃x ij , ̃x kj ) =−s j κ j ( ̃x kj , ̃x ij ) =−R (j) ki using (K1). Summing with weights yieldsGI ik =−GI ki . The formula forNGI(A i )is a finite sum, hence well-defined. Using Uncertain Sets, we define Uncertain REGIME as follows. Definition 6.11.3(Uncertain REGIME of typeM(U-REGIME)).LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria, withm≥2andn≥1. For each criterionC j , lets j ∈+1,−1indicate its type (s j = +1for benefit;s j =−1for cost). Letw= (w 1 ,...,w n )be weights with w j ≥0, n ∑ j=1 w j = 1. Fix an uncertain modelMwith nonempty degree-domain Dom(M)⊆[0,1] k . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M), wherex (M) ij encodes the (uncertain) performance of alternativeA i on criterionC j . (1) Criterionwise uncertain comparator.For each criterionC j , fix a mapping κ (M) j :Dom(M)×Dom(M)−→[−1,1] satisfying the minimal axioms: (U1)(Anti-symmetry)κ (M) j (a,b) =−κ (M) j (b,a)for alla,b∈Dom(M). Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (U2)(Neutrality on equality)κ (M) j (a,a) = 0for alla∈Dom(M). (WhenMis fuzzy and Dom(M) = [0,1],κ (M) j may be a fuzzy/possibilistic comparison intensity; otherM admit analogous score-based or possibility-based comparators.) (2) REGIME identifiers (criterionwise).For each ordered pair(A i ,A k )withi6=kand each criterion C j , define R (j) ik :=s j κ (M) j ( x (M) ij , x (M) kj ) ∈[−1,1]. Collect them into theREGIME vector R(A i ,A k ) := ( R (1) ik ,...,R (n) ik ) ∈[−1,1] n . (3) Guide index (pairwise aggregation).Define the pairwise guide index ofA i overA k by GI ik := n ∑ j=1 w j R (j) ik ∈[−1,1]. (4) Net guide index and ranking.Define the net guide index of each alternative by NGI(A i ) := m ∑ k=1 k6=i GI ik ∈R. The U-REGIME ranking is obtained by sorting alternatives in descending order ofNGI(A i )(ties may be handled by any fixed secondary rule). Theorem 6.11.4(Well-definedness and anti-symmetry of U-REGIME).Under Definition 6.11.3, the U- REGIME quantities are well-defined and satisfy: (i) For alli6=kand allj,R (j) ik andGI ik exist and lie in[−1,1]. (i) For alli6=kand allj, R (j) ik =−R (j) ki ,GI ik =−GI ki . (i) For eachi,NGI(A i )is a well-defined finite real number. Proof.(i) Sincex (M) ij ,x (M) kj ∈Dom(M)andκ (M) j maps Dom(M)×Dom(M)into[−1,1], the valueκ (M) j (x (M) ij ,x (M) kj ) is well-defined in[−1,1]. Multiplying bys j ∈ ±1preserves the range, hence eachR (j) ik ∈[−1,1]exists. The weighted sum definingGI ik is finite and is a convex combination of numbers in[−1,1], soGI ik ∈[−1,1]. (i) By anti-symmetry (U1), R (j) ki =s j κ (M) j (x (M) kj ,x (M) ij ) =−s j κ (M) j (x (M) ij ,x (M) kj ) =−R (j) ik . Multiplying by weights and summing overjyieldsGI ki =−GI ik . (i) For fixedi,NGI(A i ) = ∑ k6=i GI ik is a finite sum of finite real numbers (at mostm−1terms), hence is well-defined and finite. As related concepts, Spherical fuzzy REGIME [815] and Neutrosophic REGIME [816] are also known. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) 6.12Fuzzy TODIM TODIM ranks by prospect theory: it computes dominance from gains and losses relative to a reference, uses an attenuation parameter for losses, and aggregates dominance [817–819]. Fuzzy TODIM evaluates criteria values as fuzzy numbers, computes fuzzy dominance for gains and losses, and obtains a final ordering via defuzzification or fuzzy preference relations [820]. Definition 6.12.1(TFN-based Fuzzy TODIM).LetA=A 1 ,...,A m be a finite set of alternatives and C=C 1 ,...,C n a finite set of criteria. PartitionC=C ben ∪C cost into benefit- and cost-type criteria. (1) Triangular fuzzy evaluations.Assume a TFN decision matrix ̃ X= ( ̃x ij ) m×n , ̃x ij = (` ij , μ ij , u ij ),0< ` ij ≤μ ij ≤u ij , where ̃x ij is the (linguistic-to-fuzzy) assessment ofA i w.r.t.C j . If multiple decision-makerse= 1,...,kprovide TFNs ̃x (e) ij , use the componentwise average aggregation ̃x ij := 1 k k ∑ e=1 ̃x (e) ij = ( 1 k k ∑ e=1 ` (e) ij , 1 k k ∑ e=1 μ (e) ij , 1 k k ∑ e=1 u (e) ij ) . (2) TFN weights and reference criterion.Let TFN criterion weights be ̃w j = (` w j ,μ w j ,u w j ),0< ` w j ≤μ w j ≤u w j , j= 1,...,n, (aggregated from multiple experts in the same way if needed). Fix a score/defuzzification maps:TFN >0 →R >0 , e.g. s(`,μ,u) := `+ 2μ+u 4 . Define the normalized crisp weights w j := s( ̃w j ) ∑ n t=1 s( ̃w t ) , j= 1,...,n. Let the reference criterion be any maximizer r∈arg max 1≤j≤n w j , and define relative weights (projection to the reference criterion) w jr := w j w r , j= 1,...,n, W r := n ∑ t=1 w tr . (3) Normalized TFN performances.Define normalized TFN performanceŝx ij (larger is better) as follows. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) ForC j ∈C ben , set ̃x max j := ( max i ` ij ,max i μ ij ,max i u ij ) ,̂x ij := ̃x ij ̃x max j . ForC j ∈C cost , set ̃x min j := ( min i ` ij ,min i μ ij ,min i u ij ) ,̂x ij := ̃x min j ̃x ij . Here TFN division is the standard positive-TFN operation (` 1 ,μ 1 ,u 1 ) (` 2 ,μ 2 ,u 2 ) := ( ` 1 u 2 , μ 1 μ 2 , u 1 ` 2 ) . (4) TFN distance (vertex method).For TFNs ̃a= (` a ,μ a ,u a )and ̃ b= (` b ,μ b ,u b )define d( ̃a, ̃ b) := √ 1 3 [ (` a −` b ) 2 + (μ a −μ b ) 2 + (u a −u b ) 2 ] . (5) Gain/Loss matrices (pairwise comparisons).Use the scores(·)to compare TFNs: write ̃x ̃yiff s( ̃x)≥s( ̃y). For each criterionC j and each pair(i,k), define gain and loss: G j ik , L j ik ∈R, i,k= 1,...,m, j= 1,...,n. IfC j ∈C ben (benefit), set G j ik := d(̂x ij ,̂x kj ), ̃x ij ̃x kj , 0, ̃x ij ≺ ̃x kj , L j ik := 0, ̃x ij ̃x kj , −d(̂x ij ,̂x kj ), ̃x ij ≺ ̃x kj . IfC j ∈C cost (cost), set G j ik := 0, ̃x ij ̃x kj , d(̂x ij ,̂x kj ), ̃x ij ≺ ̃x kj , L j ik := −d(̂x ij ,̂x kj ), ̃x ij ̃x kj , 0, ̃x ij ≺ ̃x kj . (ThusG j ik ≥0andL j ik ≤0always, andG j i =L j i = 0.) (6) Dominance degree per criterion (prospect-type asymmetry).Fix a loss attenuation parameter θ >0. Define (square-root form) dominance contributions: Φ j(+) ik := √ G j ik w jr W r ,Φ j(−) ik :=− 1 θ √ (−L j ik )W r w jr ,Φ j ik := Φ j(+) ik + Φ j(−) ik . LetΦ j := (Φ j ik ) m×m be the dominance matrix for criterionC j . (7) Overall dominance and overall value.Define the overall dominance matrix∆ = (δ ik ) m×m by δ ik := n ∑ j=1 Φ j ik . Define the overall value of alternativeA i by normalizing row-sums: S i := m ∑ k=1 δ ik ,Ξ(A i ) := S i −min 1≤t≤m S t max 1≤t≤m S t −min 1≤t≤m S t ∈[0,1]. Rank alternatives by decreasingΞ(A i )(ties allowed). Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Proposition 6.12.2(Well-definedness).Under the standing assumptions ̃x ij ∈TFN >0 and ̃w j ∈TFN >0 , and withθ >0, the quantitiesG j ik ,L j ik ,Φ j ik ,δ ik ,Ξ(A i )are well-defined real numbers and0≤Ξ(A i )≤1for alli. Proof.The vertex distanced(·,·)is real and nonnegative by construction. Hence eachG j ik is either0or a nonnegative real, and eachL j ik is either0or a nonpositive real. Sincew j >0andw r =max j w j >0, all ratiosw jr andW r are positive, soΦ j(+) ik andΦ j(−) ik are real. Thereforeδ ik andS i are real. Finally,Ξ(A i )is the standard affine normalization ofS i m i=1 , hence lies in[0,1]whenever the denominator is nonzero; if all S i are equal, defineΞ(A i ) := 0for alli(or any constant in[0,1]). Using Uncertain Sets, we define Uncertain TODIM as follows. Definition 6.12.3(Admissible distance on an uncertain model).LetMbe an uncertain model. An admissible distanceis a map d M :Dom(M)×Dom(M)−→[0,∞) satisfying: (i)d M (a,b) =d M (b,a), (i)d M (a,a) = 0, and (i)d M (a,b)<∞for alla,b∈Dom(M). Definition 6.12.4(Uncertain TODIM of typeM(U-TODIM)).LetA=A 1 ,...,A m be alternatives andC=C 1 ,...,C n criteria, withm≥2andn≥1. PartitionC=C ben ̇ ∪ C cost into benefit and cost criteria. Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M), anduncertain criterion weights w (M) j ∈Dom(M) (j= 1,...,n). Fix an admissible scoreS M and an admissible distanced M . Step 1 (Crisp weights and reference criterion).Define positive crisp weights by ω j := exp(S M (w (M) j )) ∑ n t=1 exp(S M (w (M) t )) ∈(0,1), n ∑ j=1 ω j = 1. Choose a reference criterion r∈arg max 1≤j≤n ω j , and define relative weights ω jr := ω j ω r (j= 1,...,n), W r := n ∑ t=1 ω tr . Step 2 (Normalization to a benefit orientation).For each criterionC j , define the criterionwise best and worst degrees in Dom(M)by x max j ∈arg max 1≤i≤m S M (x (M) ij ), x min j ∈arg min 1≤i≤m S M (x (M) ij ). Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Define the normalized distance-to-best performance ∆ (j) i := d M ( x (M) ij , x max j ) , C j ∈C ben , d M ( x (M) ij , x min j ) , C j ∈C cost , (i= 1,...,m). (Thus, for each criterion, smaller∆ (j) i means better.) Step 3 (Pairwise gains/losses).For each criterionC j and pair(i,k)define G j ik :=max0,∆ (j) k −∆ (j) i ≥0, L j ik :=−max0,∆ (j) i −∆ (j) k ≤0. HenceG j ik measures gain ofA i overA k , whileL j ik is the (negative) loss. Step 4 (Prospect-type dominance per criterion).Fix a loss attenuation parameterθ >0. Define dominance contributions Φ j(+) ik := √ G j ik ω jr W r ,Φ j(−) ik :=− 1 θ √ (−L j ik )W r ω jr ,Φ j ik := Φ j(+) ik + Φ j(−) ik . LetΦ j := (Φ j ik ) m×m be the dominance matrix for criterionC j . Step 5 (Overall dominance and final value).Define the overall dominance matrix∆ = (δ ik )by δ ik := n ∑ j=1 Φ j ik . Let S i := m ∑ k=1 δ ik . Define the final TODIM value by min–max normalization: Ξ(A i ) := S i −min 1≤t≤m S t max 1≤t≤m S t −min 1≤t≤m S t ,maxS t >minS t , 0,maxS t =minS t , and rank alternatives by decreasingΞ(A i ). Theorem 6.12.5(Well-definedness of U-TODIM).Under Definition 6.12.4, assumeDom(M)6=∅,S M is admissible,d M is an admissible distance, andθ >0. Then: (i) All quantities in U-TODIM are well-defined finite real numbers. (i) The crisp weights satisfyω j ∈(0,1)and ∑ n j=1 ω j = 1, and thusω jr >0andW r >0. (i) For eachi, the final value satisfies0≤Ξ(A i )≤1. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Proof.(i) Admissibility ofS M impliesS M (w (M) j )∈R, hence exp(S M (w (M) j ))∈(0,∞)is finite. Therefore the normalization definingω j yields well-defined positive reals with sum1. (i) Sincermaximizesω j , we haveω r >0, hence each ratioω jr =ω j /ω r is well-defined and positive, and W r = ∑ t ω tr >0. (i) Becausemis finite andS M (x (M) ij )are finite, the argmax/argmin sets used to choosex max j ,x min j are nonempty, so the selections exist. By admissibility ofd M , each∆ (j) i is a finite nonnegative real. ThusG j ik andL j ik are well-defined finite reals withG j ik ≥0andL j ik ≤0. Withθ >0,ω jr >0, andW r >0, the square-root expressions inΦ j(+) ik andΦ j(−) ik are well-defined finite reals, soΦ j ik and henceδ ik andS i are finite real numbers. Finally,Ξ(A i )is obtained by standard min–max normalization. If maxS t >minS t , then0≤Ξ(A i )≤1for alli; if maxS t =minS t , the definition setsΞ(A i ) = 0. Related concepts of TODIM under uncertainty-aware models are listed in Table 6.12. Table 6.12: Related concepts of TODIM under uncertainty-aware models. kRelated TODIM concept(s) 2 Intuitionistic Fuzzy TODIM [821] 2 Pythagorean Fuzzy TODIM [822–824] 2 Fermatean Fuzzy TODIM [825,826] 3 Hesitant Fuzzy TODIM [827,828] 3 Picture Fuzzy TODIM [829] 3 Spherical Fuzzy TODIM [830,831] 3 Neutrosophic TODIM [832–835] nPlithogenic TODIM [836] As related concepts to Uncertain TODIM, several extensions are also known, including Rough TODIM [837, 838], Linguistic TODIM [839, 840], Grey TODIM [841], TODIM-VIKOR [826, 842], and Extended TODIM [843,844]. 6.13Fuzzy GRA (Fuzzy Grey Relational Analysis) Classical GRA (Grey Relational Analysis) sets a reference sequence, computes grey relational coefficients from absolute deviations, aggregates coefficients with weights, and ranks alternatives by highest relational grade overall [845,846]. Fuzzy GRA normalizes fuzzy criterion ratings, defines an ideal reference sequence, measures fuzzy deviations, computes grey relational coefficients and weighted grades, then ranks alternatives by similarity [847,848]. Definition 6.13.1(TFN-based Fuzzy Grey Relational Analysis (Fuzzy GRA)).[849, 850] LetA= A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. Assume a partition into benefit and cost criteria C=C + ̇ ∪C − , J + :=j:C j ∈C + , J − :=j:C j ∈C − . Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) LetTFN >0 :=(l,m,u)∈R 3 >0 :l≤m≤udenote positive triangular fuzzy numbers (TFNs). Suppose the fuzzy decision matrix is ̃ X= ( ̃x ij )∈(TFN >0 ) m×n , ̃x ij = (l ij ,m ij ,u ij ), and letw= (w 1 ,...,w n )∈[0,1] n be criterion weights with ∑ n j=1 w j = 1. Step 1 (Normalization to a comparable fuzzy scale).For each criterionj, define the positive scalars u max j :=max 1≤i≤m u ij , l min j :=min 1≤i≤m l ij . Define the normalized TFNs ̃r ij ∈(TFN >0 )by ̃r ij := ( l ij u max j , m ij u max j , u ij u max j ) , j∈J + (benefit), ( l min j u ij , l min j m ij , l min j l ij ) , j∈J − (cost). (Thus, larger performance yields larger normalized values for both benefit and cost criteria.) Step 2 (Reference/ideal sequence).Define the fuzzy reference value for each criterion by the TFN ̃r ∗ j := (1,1,1)∈TFN >0 , j= 1,...,n, and the corresponding reference sequence ̃ R ∗ := ( ̃r ∗ 1 ,..., ̃r ∗ n ). Step 3 (Distance to the reference in the fuzzy domain).Fix a metric (distance)D:TFN >0 × TFN >0 →R ≥0 . A standard choice is the vertex-distance: D ( (l 1 ,m 1 ,u 1 ),(l 2 ,m 2 ,u 2 ) ) := √ (l 1 −l 2 ) 2 + (m 1 −m 2 ) 2 + (u 1 −u 2 ) 2 3 . Define the deviation (distance to the reference) by ∆ ij :=D ( ̃r ij , ̃r ∗ j ) ∈R ≥0 , i= 1,...,m, j= 1,...,n. Let ∆ min :=min 1≤i≤m min 1≤j≤n ∆ ij ,∆ max :=max 1≤i≤m max 1≤j≤n ∆ ij . Step 4 (Grey relational coefficients).Fix the distinguishing coefficientρ∈(0,1]. If∆ max = 0(all alternatives coincide with the reference), defineγ ij := 1for alli,j. Otherwise define the fuzzy grey relational coefficient by γ ij := ∆ min +ρ∆ max ∆ ij +ρ∆ max ∈(0,1], i= 1,...,m, j= 1,...,n. Step 5 (Grey relational grades and ranking).Define the grey relational grade (overall similarity) of alternativeA i by the weighted sum Γ i := n ∑ j=1 w j γ ij ∈[0,1], i= 1,...,m. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Rank alternatives by descendingΓ i : A p FGRA A q ⇐⇒Γ p ≥Γ q . Any maximizer A ? ∈arg max 1≤i≤m Γ i is called aFuzzy GRA best alternative. The definition of Uncertain GRA of typeMis given below. Definition 6.13.2(Uncertain GRA of typeM(U-GRA)).LetA=A 1 ,...,A m be alternatives and C=C 1 ,...,C n criteria, withm,n≥2. Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M), and criterion weightsw= (w 1 ,...,w n )satisfyingw j ≥0and ∑ n j=1 w j = 1. Fix an uncertain modelMwith Dom(M)6=∅, together with an admissible scoreS M and an admissible distanced M . Fix a distinguishing coefficientρ∈(0,1]. Step 1 (Crisp projection).Definey ij :=S M (x (M) ij )∈Rand form the real matrixY= (y ij ). Step 2 (Min–max normalization to a benefit orientation).For each criterionC j , define y min j :=min 1≤i≤m y ij , y max j :=max 1≤i≤m y ij . Define a normalized scalar matrixR= (r ij )by r ij := y ij −y min j y max j −y min j , C j ∈C ben andy max j > y min j , y max j −y ij y max j −y min j , C j ∈C cost andy max j > y min j , 0,y max j =y min j , (i= 1,...,m;j= 1,...,n), so thatr ij ∈[0,1]and larger values always indicate better performance. Step 3 (Reference sequence).Define the reference (ideal) sequence by r ∗ j := 1 (j= 1,...,n). Step 4 (Deviation to the reference).Define ∆ ij := ∣ ∣ r ∗ j −r ij ∣ ∣ =|1−r ij |∈[0,1],∆ min :=min i,j ∆ ij ,∆ max :=max i,j ∆ ij . Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Step 5 (Grey relational coefficients).If∆ max = 0, setγ ij := 1for alli,j. Otherwise define γ ij := ∆ min +ρ∆ max ∆ ij +ρ∆ max ∈(0,1], i= 1,...,m, j= 1,...,n. Step 6 (Grey relational grades and ranking).Define the grey relational grade of alternativeA i by Γ i := n ∑ j=1 w j γ ij ∈[0,1], i= 1,...,m, and rank by descendingΓ i . (Optional) Uncertainty-aware distance refinement.Instead of∆ ij =|1−r ij |, one may define an uncertainty-aware deviation usingd M , e.g., ∆ (M) ij :=d M ( x (M) ij , x (M)∗ j ) , wherex (M)∗ j is a chosen reference degree in Dom(M)(e.g., anS M -maximizer). The remaining steps are then applied with∆ (M) ij in place of∆ ij . Theorem 6.13.3(Well-definedness of U-GRA).Under Definition 6.13.2, assumem,n≥2,Dom(M)6=∅, S M is admissible, andρ∈(0,1]. Then: (i) The normalized valuesr ij are well-defined and satisfy0≤r ij ≤1. (i) The deviations satisfy0≤∆ ij ≤1, hence∆ min ,∆ max exist. (i) The grey relational coefficientsγ ij are well-defined and satisfy0< γ ij ≤1when∆ max >0, andγ ij = 1 when∆ max = 0. (iv) The grey relational grades satisfy0≤Γ i ≤1for alli, and the ranking byΓ i is well-defined. Proof.(i) By admissibility ofS M , eachy ij =S M (x (M) ij )is a finite real number, so for eachjthe extrema y min j andy max j exist (finite index set). Ify max j > y min j , the stated min–max formula definesr ij . Because y ij ∈[y min j ,y max j ], it follows thatr ij ∈[0,1]for both benefit and cost cases. Ify max j =y min j , the definition setsr ij = 0, sor ij ∈[0,1]always. (i) Sincer ∗ j = 1andr ij ∈[0,1], we have∆ ij =|1−r ij |∈[0,1]. Thus∆ min and∆ max exist as minima/max- ima over a finite set. (i) If∆ max = 0, then all∆ ij = 0and the definition setsγ ij = 1. If∆ max >0, thenρ∆ max >0and the denominator∆ ij +ρ∆ max ≥ρ∆ max >0, soγ ij is well-defined. Moreover,∆ ij ≥∆ min implies γ ij = ∆ min +ρ∆ max ∆ ij +ρ∆ max ≤1, and positivity of the numerator yieldsγ ij >0. (iv) Sincew j ≥0, ∑ j w j = 1, and0< γ ij ≤1, the weighted sum satisfies0≤Γ i ≤1. Therefore sorting by Γ i defines a valid ranking (ties allowed). Related concepts of GRA under uncertainty-aware models are listed in Table 6.13. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Table 6.13: Related concepts of GRA under uncertainty-aware models. kRelated GRA concept(s) 1 Fuzzy GRA 2 Intuitionistic Fuzzy GRA [851] 2 Fermatean Fuzzy GRA 3 Hesitant Fuzzy GRA [852,853] 3 Neutrosophic GRA [854,855] 6.14Fuzzy ARAS (Fuzzy Additive Ratio Assessment) ARAS ranks alternatives by normalized weighted criteria sums, comparing each option to an optimal al- ternative through utility ratios [856, 857]. Fuzzy ARAS represents ratings and weights as fuzzy numbers, normalizes them, computes fuzzy optimality sums, defuzzifies, and ranks by utility [858–860]. Definition 6.14.1(TFN-based Fuzzy ARAS (FARAS)).[858–860] LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. Partition criteria into benefit and cost sets: J + ̇ ∪J − =1,...,n, wherej∈J + means “larger is better” andj∈J − means “smaller is better”. Assume the (fuzzy) performance ofA i underC j is apositive TFN ̃a ij = (l ij ,m ij ,u ij ) (0< l ij ≤m ij ≤u ij ), i= 1,...,m, j= 1,...,n, and letw= (w 1 ,...,w n )be acrispweight vector withw j ≥0and ∑ n j=1 w j = 1. TFN arithmetic (standard positive convention).For positive TFNs ̃x= (l x ,m x ,u x )and ̃y= (l y ,m y ,u y )define ̃x⊕ ̃y:= (l x +l y , m x +m y , u x +u y ), ̃x −1 := ( 1 u x , 1 m x , 1 l x ) , ̃x ̃y:= ̃x⊗ ̃y −1 , r ̃x:= (rl x , rm x , ru x ) (r≥0). Step 0 (add the optimal/ideal alternative).IntroduceA 0 and define, for each criterionj, ̃a 0j := max 1≤i≤m ̃a ij , j∈J + , min 1≤i≤m ̃a ij , j∈J − , where max/min may be taken w.r.t. any fixed total preorder on TFNs (e.g. via a defuzzification-based score). Form the extended decision matrix ̃ A= ( ̃a ij ) (m+1)×n (i= 0,1,...,m;j= 1,...,n). Step 1 (normalization).For eachj∈J + (benefit), set ̃a ∗ ij := ̃a ij m ⊕ p=0 ̃a pj , i= 0,1,...,m. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) For eachj∈J − (cost), apply the two-stage transform: ̃ b ij := ̃a −1 ij , ̃a ∗ ij := ̃ b ij m ⊕ p=0 ̃ b pj , i= 0,1,...,m. Let ̃ A ∗ = ( ̃a ∗ ij )denote the normalized TFN matrix. Step 2 (weighted normalized matrix).Define ̃ ˆa ij :=w j ̃a ∗ ij , i= 0,1,...,m, j= 1,...,n, and write ̃ ˆ A= ( ̃ ˆa ij ). Step 3 (optimality function and defuzzification).For eachi= 0,1,...,m, define the (fuzzy) opti- mality value ̃ S i := n ⊕ j=1 ̃ ˆa ij . If ̃ S i = (S ` i ,S m i ,S u i ), define the crisp score by the COA/centroid rule: S i :=Defuzz( ̃ S i ) := S ` i +S m i +S u i 3 . Step 4 (utility degree and ranking).Define, for each real alternativeA i (i= 1,...,m), the utility degree K i := S i S 0 ∈[0,1]. The FARAS ranking is the total preorder onAgiven by A p FARAS A q ⇐⇒K p ≥K q . The definition of Uncertain ARAS (U-ARAS), which is an extension based on Uncertain Sets, is given below. Definition 6.14.2(Uncertain ARAS (U-ARAS): additive ratio assessment under uncertainty).LetA= A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. LetJ + ̇ ∪J − = 1,...,nbe the partition into benefit (J + ) and cost (J − ) criteria. Fix an uncertainty space(Γ,L,M). (0) Uncertain performance matrix.For each(i,j), letX ij : Γ→(0,∞)be an uncertain variable representing the (positive) performance ofA i under criterionC j . Denote the induced uncertain set by X ij (γ) :=X ij (γ). (1) Crisp criterion weights (ARAS standard).Letw= (w 1 ,...,w n )∈[0,1] n satisfy w j ≥0, n ∑ j=1 w j = 1. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (2) Construct the optimal (ideal) alternative.IntroduceA 0 and define, for eachj, X 0j (γ) := max 1≤i≤m X ij (γ), j∈J + , min 1≤i≤m X ij (γ), j∈J − , γ∈Γ. This yields uncertain variablesX 0j : Γ→(0,∞)and uncertain setsX 0j (γ) :=X 0j (γ). (3) ARAS normalization (scenario-wise).Define normalized uncertain variablesR ij : Γ→[0,1]as follows. • Forj∈J + (benefit), R ij (γ) := X ij (γ) ∑ m p=0 X pj (γ) , i= 0,1,...,m. • Forj∈J − (cost), use the reciprocal transform: Y ij (γ) := 1 X ij (γ) , R ij (γ) := Y ij (γ) ∑ m p=0 Y pj (γ) , i= 0,1,...,m. LetR ij (γ) :=R ij (γ)be the induced uncertain sets. (4) Weighted normalized values and additive optimality function.Define V ij (γ) :=w j R ij (γ)∈[0,1], S i (γ) := n ∑ j=1 V ij (γ)∈[0,1], i= 0,1,...,m, and the induced uncertain setsS(A i )(γ) :=S i (γ). (5) Utility degree and decision rule.AssumeS 0 (γ)>0for allγ∈Γand define the (scenario-wise) utility degree K i (γ) := S i (γ) S 0 (γ) ∈[0,1], i= 1,...,m, with induced uncertain setsK(A i )(γ) :=K i (γ). IfE[K i ]exists for alli, rank by expected utility: A p U-ARAS A q ⇐⇒E[K p ]≥E[K q ], and select any maximizer A ? ∈arg max A i ∈A E[K i ]. Theorem 6.14.3(Uncertain-set structure and well-definedness of U-ARAS).In Definition 6.14.2, assume: (A1)For alli,j,X ij : Γ→(0,∞)isL-measurable. (A2)w j ≥0and ∑ n j=1 w j = 1. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (A3)For allγand allj, ∑ m p=0 X pj (γ)>0(automatic from positivity), and forj∈J − also ∑ m p=0 1/X pj (γ)> 0(automatic from positivity). Then: (i)For eachj, the ideal-performance mapX 0j is anL-measurable uncertain variable, henceX 0j is an uncertain set. (i)For alli,j, the normalized valuesR ij areL-measurable uncertain variables taking values in[0,1], and S i areL-measurable uncertain variables taking values in[0,1]; henceR ij andS(A i )are uncertain sets. (i)If, additionally,S 0 (γ)>0for allγ, then eachK i is anL-measurable uncertain variable with values in[0,1], andK(A i )is an uncertain set. (iv)If, further,E[K i ]exists for alli, then the expected-utility ranking U-ARAS is well-defined and arg max A i ∈A E[K i ]6=∅. Proof.(i) Fixj. Forj∈J + ,X 0j (γ) =max 1≤i≤m X ij (γ). The pointwise maximum of finitely many measurable functions is measurable; henceX 0j isL-measurable. Forj∈J − ,X 0j (γ) =min 1≤i≤m X ij (γ) and the pointwise minimum of finitely many measurable functions is measurable. Positivity ofX ij implies X 0j (γ)∈(0,∞). (i) Fix(i,j). The denominators in the normalization are strictly positive by (A3). Forj∈J + ,R ij is a ratio of measurable functions with positive denominator, hence measurable. Moreover,0≤R ij (γ)≤1because X ij (γ)≤ ∑ m p=0 X pj (γ). Forj∈J − , the reciprocalY ij = 1/X ij is measurable (composition withx7→1/x on(0,∞)), and againR ij =Y ij / ∑ m p=0 Y pj is measurable with0≤R ij ≤1. Thus eachR ij (γ) =R ij (γ)is an uncertain set. NowV ij (γ) =w j R ij (γ)is measurable and lies in[0,1]sincew j ∈[0,1]andR ij ∈[0,1]. ThereforeS i = ∑ n j=1 V ij is a finite sum of measurable functions, hence measurable, and satisfies0≤S i (γ)≤ ∑ n j=1 w j = 1 by (A2). HenceS(A i )is an uncertain set. (i) IfS 0 (γ)>0for allγ, thenK i =S i /S 0 is a ratio of measurable functions with positive denominator, hence measurable. Also0≤K i (γ)≤1becauseS 0 (γ)≥max 0≤p≤m S p (γ)≥S i (γ)holds in standard ARAS settings whereA 0 is constructed as an ideal; thusK i (γ)∈[0,1]andK(A i )is an uncertain set. (iv) IfE[K i ]exists for alli, then the rankingA p U-ARAS A q ⇐⇒E[K p ]≥E[K q ]is well-defined. SinceA is finite, the maximum expected utility is attained, so the argmax set is nonempty. Related concepts of ARAS under uncertainty-aware models are listed in Table 6.14. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Table 6.14: Related concepts of ARAS under uncertainty-aware models. kRelated ARAS concept(s) 1 Fuzzy ARAS 2 Intuitionistic Fuzzy ARAS [861] 3 Picture fuzzy ARAS [862,863] 3 Spherical Fuzzy ARAS [864,865] 3 Neutrosophic ARAS [99,866] 6.15Fuzzy WASPAS (Fuzzy Weighted Aggregated Sum Product Assessment) WASPAS ranks alternatives by combining weighted sum and weighted product models after normalization, producing an integrated utility score [867,868]. Fuzzy WASPAS uses fuzzy ratings and weights, normalizes them, computes fuzzy WSM and WPM utilities, defuzzifies, and ranks [869,870]. Definition 6.15.1(TFN-based Fuzzy WASPAS (WASPAS-F)).[871, 872] LetA=A 1 ,...,A m be alternatives andC=C 1 ,...,C n criteria. Assume a TFN decision matrix and TFN weights ̃ X= ( ̃x ij ) m×n , ̃x ij ∈TFN >0 , ̃ w= ( ̃w 1 ,..., ̃w n ), ̃w j ∈TFN >0 , and (optionally) ∑ n j=1 ̃w j = (1,1,1)in the chosen TFN arithmetic. LetJ + ⊆1,...,nbe benefit criteria indices andJ − ⊆1,...,ncost criteria indices, withJ + ̇ ∪J − =1,...,n. Fix a total preorderon TFNs (used to compute max and min over finite TFN-sets). Forj∈ 1,...,n define ̃x max j :=max 1≤i≤m ̃x ij , ̃x min j :=min 1≤i≤m ̃x ij , where max,min are taken w.r.t.. Step 1 (Normalization). Define the normalized fuzzy matrix ̃ ̄ X= ( ̃ ̄x ij )by ̃ ̄x ij := ̃x ij ̃x max j , j∈J + , ̃x min j ̃x ij , j∈J − , (1≤i≤m,1≤j≤n). Step 2 (WSM part). Define the weighted normalized TFNs for the weighted-sum model (WSM) by ̃ ˆx (q) ij := ̃ ̄x ij ⊗ ̃w j , ̃ Q i := n ⊕ j=1 ̃ ˆx (q) ij , i= 1,...,m. Step 3 (WPM part). Assume (as typical after normalization) that ̃ ̄x ij = (a,b,c)satisfies0< a≤b≤c≤1 and that ̃w j = (α,β,γ)satisfies0< α≤β≤γ≤1. Use the standard TFN power rule (a,b,c) (α,β,γ) := ( a γ , b β , c α ) . Define the weighted normalized TFNs for the weighted-product model (WPM) by ̃ ˆx (p) ij := ( ̃ ̄x ij ) ̃w j , ̃ P i := n ⊗ j=1 ̃ ˆx (p) ij , i= 1,...,m. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Step 4 (Defuzzification). Let COA(l,m,u) := l+m+u 3 (centroid/center-of-area). Define crisp scores Q i :=COA( ̃ Q i ), P i :=COA( ̃ P i ). Step 5 (Integrated utility and ranking). Fixλ∈[0,1](or set it by the balancing rule below) and define K i :=λQ i + (1−λ)P i , i= 1,...,m. A commonly used balancing choice is λ:= ∑ m i=1 P i ∑ m i=1 Q i + ∑ m i=1 P i . The fuzzy WASPAS ranking is obtained by sortingK i in descending order (largerK i means a better alternative). Definition 6.15.2(Uncertain WASPAS (U-WASPAS): WSM–WPM compromise under uncertainty).Let A=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. LetJ + ̇ ∪J − = 1,...,nbe the partition into benefit (J + ) and cost (J − ) criteria. Fix an uncertainty space(Γ,L,M). (0) Uncertain performance matrix (positive data).For each(i,j), let X ij : Γ→(0,∞) be an uncertain variable representing the performance ofA i under criterionC j . Write the induced uncertain set asX ij (γ) :=X ij (γ). (1) Criterion weights and WASPAS mixing parameter.Letw= (w 1 ,...,w n )∈[0,1] n satisfy w j ≥0, n ∑ j=1 w j = 1, and fixλ∈[0,1](the WSM–WPM compromise parameter). (2) Normalization (scenario-wise).For each criterionjand scenarioγ∈Γ, define the normalizing scalars M j (γ) :=max 1≤i≤m X ij (γ), m j (γ) :=min 1≤i≤m X ij (γ). Define the normalized uncertain variablesR ij : Γ→(0,1]by R ij (γ) := X ij (γ) M j (γ) , j∈J + (benefit), m j (γ) X ij (γ) , j∈J − (cost). LetR ij (γ) :=R ij (γ)be the induced uncertain sets. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (3) WSM part (weighted sum model).Define the WSM utility (uncertain variable) ofA i by Q i (γ) := n ∑ j=1 w j R ij (γ)∈[0,1],Q(A i )(γ) :=Q i (γ). (4) WPM part (weighted product model).Define the WPM utility (uncertain variable) ofA i by P i (γ) := n ∏ j=1 ( R ij (γ) ) w j ∈(0,1],P(A i )(γ) :=P i (γ). (Herex7→x w j is taken on(0,1].) (5) Integrated WASPAS utility and decision rule.Define the integrated utility (uncertain variable) K i (γ) :=λQ i (γ) + (1−λ)P i (γ)∈[0,1],K(A i )(γ) :=K i (γ). IfE[K i ]exists for alli, rank alternatives by expected utility: A p U-WASPAS A q ⇐⇒E[K p ]≥E[K q ], and select any maximizer A ? ∈arg max A i ∈A E[K i ]. U-WASPAS, which is an extension based on Uncertain Sets, is defined as follows. Theorem 6.15.3(Uncertain-set structure and well-definedness of U-WASPAS).In Definition 6.15.2, as- sume: (A1)For alli,j,X ij : Γ→(0,∞)isL-measurable. (A2)w j ≥0and ∑ n j=1 w j = 1, andλ∈[0,1]. Then: (i)For eachj, the mapsM j (γ) =max i X ij (γ)andm j (γ) =min i X ij (γ)areL-measurable uncertain variables with values in(0,∞). (i)For alli,j, the normalized valuesR ij areL-measurable uncertain variables taking values in(0,1]; henceR ij are uncertain sets. (i)For eachi,Q i andP i areL-measurable uncertain variables withQ i (γ)∈[0,1]andP i (γ)∈(0,1]for allγ; henceQ(A i )andP(A i )are uncertain sets. (iv)For eachi,K i is anL-measurable uncertain variable with values in[0,1], henceK(A i )is an uncertain set. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (v)IfE[K i ]exists for alli, then the expected-utility ranking U-WASPAS is well-defined andarg max A i ∈A E[K i ]6= ∅. Proof.(i) For fixedj,M j andm j are pointwise maximum/minimum of finitely many measurable maps X 1j ,...,X mj , hence measurable. Positivity of eachX ij impliesM j ,m j ∈(0,∞). (i) Fix(i,j). Forj∈J + , R ij = X ij M j is measurable as a quotient of measurable functions with strictly positive denominator. Moreover0< R ij (γ)≤1sinceX ij (γ)≤M j (γ). Forj∈J − , R ij = m j X ij is measurable with0< R ij (γ)≤1sincem j (γ)≤X ij (γ). (i) SinceR ij are measurable andw j are constants,w j R ij are measurable, and the finite sumQ i = ∑ j w j R ij is measurable. Also0≤Q i (γ)≤ ∑ j w j = 1. ForP i = ∏ j (R ij ) w j : for eachj, the mapx7→x w j is continuous on(0,1], hence(R ij ) w j is measurable; the finite product is measurable. SinceR ij (γ)∈(0,1], one hasP i (γ)∈(0,1]. (iv)K i =λQ i +(1−λ)P i is a linear combination of measurable maps, hence measurable. BecauseQ i ∈[0,1], P i ∈(0,1], andλ∈[0,1], it follows thatK i (γ)∈[0,1]. (v) IfE[K i ]exists for alli, then the relationA p U-WASPAS A q ⇐⇒E[K p ]≥E[K q ]is well-defined. Since Ais finite, the maximum expected value is attained, so the argmax set is nonempty. Related concepts of WASPAS under uncertainty-aware models are listed in Table 6.15. Table 6.15: Related concepts of WASPAS under uncertainty-aware models. kRelated WASPAS concept(s) 1 Fuzzy WASPAS 2 Intuitionistic Fuzzy WASPAS [873,874] 2 Bipolar fuzzy WASPAS [875,876] 2 Pythagorean Fuzzy WASPAS [877,878] 2 Fermatean fuzzy WASPAS [879,880] 3 Picture Fuzzy WASPAS [881,882] 3 Neutrosophic WASPAS [883,884] 3 Spherical Fuzzy WASPAS [885,886] 3 Pythagorean neutrosophic WASPAS [887] 3 Hesitant Fuzzy WASPAS [888,889] In addition to Uncertain WASPAS, related variants such as Rough WASPAS [593, 890] and Extended WASPAS [891] are also known. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) 6.16Fuzzy MOORA (Fuzzy Multi-Objective Optimization on the basis of Ratio Anal- ysis) MOORA normalizes criterion values, applies criterion weights, and then ranks alternatives using the ag- gregatedbenefit-minus-costscore across multiple objectives [892, 893]. As an extension of MOORA, the Multi-MOORA approach is also well known [893–895]. Fuzzy MOORA generalizes this procedure by repre- senting ratings and/or weights as fuzzy numbers, performing fuzzy normalization and weighting, and finally ranking alternatives by defuzzifying the resultingbenefit-minus-costfuzzy scores [896,897]. Definition 6.16.1(TFN-based Fuzzy MOORA).[898, 899] LetA=A 1 ,...,A m be alternatives and C=C 1 ,...,C n criteria. Assume a partition into benefit and cost criteria C=C + ̇ ∪C − ,C + =C 1 ,...,C g ,C − =C g+1 ,...,C n , and a crisp weight vectorw= (w 1 ,...,w n )withw j ≥0and ∑ n j=1 w j = 1. Let ̃ X= ( ̃x ij )∈(TFN) m×n be the TFN decision matrix with ̃x ij = (x ` ij ,x m ij ,x u ij ). (0) TFN arithmetic used.All operations below are componentwise: (x ` ,x m ,x u )⊕(y ` ,y m ,y u ) = (x ` +y ` , x m +y m , x u +y u ), α (x ` ,x m ,x u ) = (αx ` , αx m , αx u ) (α≥0), and we use the standard triangular approximation for subtraction: (x ` ,x m ,x u ) (y ` ,y m ,y u ) = (x ` −y ` , x m −y m , x u −y u ). (1) Normalization.For each criterionj, set d j := √ √ √ √ m ∑ i=1 ( (x ` ij ) 2 + (x m ij ) 2 + (x u ij ) 2 ) >0, ̃x ∗ ij := ̃x ij /d j . (2) Weighting.Define ̃v ij :=w j ̃x ∗ ij and collect ̃ V= ( ̃v ij ). (3) Ratio score (benefit minus cost).For each alternativeA i , define the fuzzy MOORA score ̃y i := g ⊕ j=1 ̃v ij n ⊕ j=g+1 ̃v ij = (y ` i ,y m i ,y u i ). (4) Defuzzification and ranking.Define BNP( ̃y i ) := (y ` i +y m i +y u i )/3. The final ranking is the preorder onAgiven by A p A q ⇐⇒BNP( ̃y p )≥BNP( ̃y q ). Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Using Uncertain Sets, we define Uncertain MOORA of typeM(U-MOORA) as follows. Definition 6.16.2(Uncertain MOORA of typeM(U-MOORA)).LetA=A 1 ,...,A m be alternatives andC=C 1 ,...,C n criteria withm≥2andn≥1. Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M), andcriterion weightsw= (w 1 ,...,w n )withw j ≥0and ∑ n j=1 w j = 1. Step 0 (Crisp projection).Define the real-valued matrixY= (y ij )by y ij :=S M ( x (M) ij ) ∈R. Step 1 (Vector normalization).For each criterionC j , define d j := √ √ √ √ m ∑ i=1 y 2 ij ≥0. Define normalized performancesr ij by r ij := y ij d j , d j >0, 0, d j = 0, (i= 1,...,m;j= 1,...,n). Step 2 (Weighting).Define the weighted normalized performances v ij :=w j r ij (i= 1,...,m;j= 1,...,n). Step 3 (MOORA ratio score: benefit minus cost).For each alternativeA i , define the U-MOORA score Ψ i := ∑ C j ∈C ben v ij − ∑ C j ∈C cost v ij . Ranking rule: A p UMOORA A q ⇐⇒Ψ p ≥Ψ q . Theorem 6.16.3(Well-definedness of U-MOORA).Under Definition 6.16.2, assumeS M is admissible and m≥2. Then: Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) (i) All quantitiesy ij ,d j ,r ij ,v ij , andΨ i are well-defined finite real numbers. (i) Ifd j >0thenr ij =y ij /d j is well-defined; ifd j = 0thenr ij = 0is well-defined by definition. (i) The ranking relation induced byΨ i is well-defined (ties allowed). Proof.(i) SinceS M is admissible, eachy ij =S M (x (M) ij )is finite. Hence eachy 2 ij is finite and nonnegative, sod j = √ ∑ m i=1 y 2 ij is well-defined and finite. (i) Ifd j >0, division byd j is valid andr ij is finite. Ifd j = 0, then ∑ m i=1 y 2 ij = 0impliesy ij = 0for alli; settingr ij = 0is consistent and yields a finite value. (i) Sincew j ≥0andr ij are finite,v ij =w j r ij is finite. The sums definingΨ i are finite sums of finite reals, henceΨ i is finite. Therefore, sorting alternatives by the real scoresΨ i defines a well-defined preorder. Related concepts of MOORA under uncertainty-aware models are listed in Table 6.16. Table 6.16: Related concepts of MOORA under uncertainty-aware models. kRelated MOORA concept(s) 2 Intuitionistic Fuzzy MOORA [900,901] 2 Pythagorean Fuzzy MOORA [898,902] 2 Fermatean Fuzzy MOORA [903,904] 3 Neutrosophic MOORA [905,906] As related concepts to Uncertain MOORA, several variants are known, including AHP–MOORA [907,908], TOPSIS–MOORA [909,910], and Grey MOORA [911,912]. 6.17Fuzzy Preference Selection Index Classical PSI derives objective criterion weights from dispersion of normalized performances, avoiding sub- jective weighting, computes a preference index for each alternative, and ranks by it [913, 914]. Fuzzy PSI estimates criterion weights from fuzzy dispersion of normalized ratings, avoiding subjective weighting, com- putes preference selection indices, defuzzifies them, and ranks alternatives by priority [915,916]. Definition 6.17.1(Fuzzy Preference Selection Index (FPSI)).[917] LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria, partitioned into benefit criteriaC b and cost criteriaC c . Assume a fuzzy decision matrix ̃ X= ( ̃x ij ) m×n , ̃x ij ∈FN >0 (i= 1,...,m, j= 1,...,n), whereFN >0 denotes the class of positive fuzzy numbers onRwith compactα-cuts. For ̃z∈FN >0 , write its α-cut as[ ̃z] α = [z − α ,z + α ]forα∈[0,1]. (Fuzzy arithmetic viaα-cuts).For ̃u, ̃v∈FNandλ∈R, define (extension principle / interval arithmetic) [ ̃u⊕ ̃v] α = [u − α +v − α , u + α +v + α ],[ ̃u ̃v] α = [u − α −v + α , u + α −v − α ], Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) [λ ̃u] α = [λu − α , λu + α ], λ≥0, [λu + α , λu − α ], λ <0, [ ̃u⊗ ̃v] α = [ minu − α v − α ,u − α v + α ,u + α v − α ,u + α v + α ,maxu − α v − α ,u − α v + α ,u + α v − α ,u + α v + α ] , and, provided0 /∈[ ̃v] α for allα, [ ̃u ̃v] α = [ minu − α /v − α ,u − α /v + α ,u + α /v − α ,u + α /v + α ,maxu − α /v − α ,u − α /v + α ,u + α /v − α ,u + α /v + α ] . Define the fuzzy square by ̃z 2 := ̃z⊗ ̃z. Step 1 (Normalization).For eachj, define c + j :=max 1≤i≤m sup(supp( ̃x ij )) (j∈C b ), a − j :=min 1≤i≤m inf(supp( ̃x ij )) (j∈C c ), and set the normalized fuzzy rating ̃ R ij ∈FN >0 by ̃ R ij := ̃x ij c + j , j∈C b , a − j ̃x ij , j∈C c . This is the fuzzy analogue of the classical PSI normalizationsR ij =x ij /x max j (benefit) andR ij =x min j /x ij (cost). Step 2 (Fuzzy preference variation).Let ̃ R j be the fuzzy mean of criterionj: ̃ R j := 1 m m ⊕ i=1 ̃ R ij . Choose a scaling constantκ >0(common choices:κ= 1as in the basic PSI variance-analogy, orκ=m−1 to mimic sample-variance scaling). Define the fuzzy preference variation ̃ PV j := 1 κ m ⊕ i=1 ( ̃ R ij ̃ R j ) 2 . This generalizes the classical PSI stepPV j = ∑ m i=1 (R ij −R j ) 2 and its interval-valued fuzzy counterpart. Step 3 (Fuzzy deviation and overall preference weights).Define the fuzzy deviation ̃ δ j := 1 ̃ PV j , and assume ⊕ n j=1 ̃ δ j is strictly positive (in the sense that0is not contained in any of itsα-cuts), so that fuzzy division is well-defined. Then define the (data-driven) fuzzy overall preference weight of criterionjby ̃w j := ̃ δ j ( n ⊕ k=1 ̃ δ k ) , j= 1,...,n. This matches the classical PSI constructionδ j = 1−PV j andw j =δ j / ∑ k δ k , and its interval-valued fuzzy analogue. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Step 4 (Fuzzy preference selection index).For each alternativeA i , define its fuzzy preference selection index ̃ I i := n ⊕ j=1 ( ̃ R ij ⊗ ̃w j ) , i= 1,...,m, which generalizesI i = ∑ n j=1 (R ij w j )and its interval-valued fuzzy version. Step 5 (Ranking).Fix a score (ranking) functionalS:FN→R(e.g., centroid-based or expected-value- based). Rank alternatives by decreasingS( ̃ I i ); the best alternative is any maximizer ofS( ̃ I i ). Using Uncertain Sets, we define Uncertain Preference Selection Index of typeM(U-PSI) as follows. Definition 6.17.2(Uncertain Preference Selection Index of typeM(U-PSI)).LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria, withm≥2andn≥1. Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Fix an uncertain modelMwith Dom(M)6=∅and an admissible positive scoreS M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M) (i= 1,...,m;j= 1,...,n). Step 0 (Crisp projection).Define a positive real matrixY= (y ij )by y ij :=S M ( x (M) ij ) ∈(0,∞). Step 1 (Normalization).For each criterionC j , define y max j :=max 1≤i≤m y ij >0, y min j :=min 1≤i≤m y ij >0. Define normalized performancesR= (r ij )by r ij := y ij y max j , C j ∈C ben , y min j y ij , C j ∈C cost . (i= 1,...,m;j= 1,...,n). Thenr ij ∈(0,1]. Step 2 (Preference variation / dispersion).For each criterionj, define the mean ̄r j := 1 m m ∑ i=1 r ij , and the (scaled) preference variation PV j := 1 m m ∑ i=1 (r ij − ̄r j ) 2 . Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Step 3 (Deviation and objective criterion weights).Define the deviation δ j := 1−PV j , j= 1,...,n, and the objective weights w j := δ j ∑ n t=1 δ t , j= 1,...,n. Step 4 (Uncertain PSI index and ranking).For each alternativeA i , define its Preference Selection Index I i := n ∑ j=1 w j r ij . Rank alternatives in descending order ofI i (ties allowed). Theorem 6.17.3(Well-definedness and simplex property of U-PSI).Under Definition 6.17.2 (in particular, m≥2,Dom(M)6=∅, andS M :Dom(M)→(0,∞)), the U-PSI procedure is well-defined. More precisely: (i) The normalization producesr ij ∈(0,1]for alli,j. (i) For eachj, one has0≤PV j ≤ 1 4 , henceδ j = 1−PV j ∈[ 3 4 ,1]. In particular, ∑ n t=1 δ t >0, so the weightsw j are well-defined. (i) The weight vectorw= (w 1 ,...,w n ) > satisfiesw j ≥0and ∑ n j=1 w j = 1(i.e.,w∈∆ n−1 ). (iv) For each alternativeA i , the indexI i is well-defined and satisfies0< I i ≤1. Proof.(i) By admissibility ofS M , eachy ij >0is finite. Hence for eachj, the extremay max j andy min j exist and are strictly positive (finite index set). ForC j ∈ C ben ,r ij =y ij /y max j ∈(0,1]. ForC j ∈ C cost , r ij =y min j /y ij ∈(0,1]. Thusr ij ∈(0,1]for alli,j. (i) Fixj. Sincer ij ∈[0,1], the sample values have range at most1. By Popoviciu’s inequality on variances, the average squared deviation satisfies PV j = 1 m m ∑ i=1 (r ij − ̄r j ) 2 ≤ (1−0) 2 4 = 1 4 , and clearlyPV j ≥0. Henceδ j = 1−PV j ∈[ 3 4 ,1]. Therefore n ∑ t=1 δ t ≥n· 3 4 >0, sow j =δ j / ∑ t δ t is well-defined. (i) Since eachδ j ≥0, one hasw j ≥0, and n ∑ j=1 w j = ∑ n j=1 δ j ∑ n t=1 δ t = 1. (iv) Becausew j ≥0, ∑ j w j = 1, andr ij ∈(0,1], the weighted sumI i = ∑ j w j r ij is finite. Moreover, I i ≤ ∑ j w j ·1 = 1. Since at least onew j >0and allr ij >0, one hasI i >0. Related concepts of Preference Selection Index under uncertainty-aware models are listed in Table 6.17. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Table 6.17: Related concepts of Preference Selection Index under uncertainty-aware models. kRelated Preference Selection Index concept(s) 1 Fuzzy Preference Selection Index 2 Intuitionistic Fuzzy Preference Selection Index 3 Neutrosophic Preference Selection Index 6.18FROV (Fuzzy Range of Value method) Classical ROV calculates each alternative’s value interval from weighted normalized criteria, then ranks using pessimistic, optimistic, or midpoint scores reflecting decision attitude under uncertainty settings [918,919]. Fuzzy Range of Value (ROV) computes each alternative’s fuzzy utility interval from weighted normalized fuzzy ratings, defuzzifies bounds, and ranks using optimistic/pessimistic or midrange score [920,921]. Definition 6.18.1(TFN-based Fuzzy Range of Value (FROV) method).[920,921] LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria, partitioned into benefit and cost criteria: C=C + ̇ ∪C − . Assume the fuzzy decision matrix consists of (nonnegative) triangular fuzzy numbers (TFNs) ̃ X= ( ̃x ij )∈(TFN ≥0 ) m×n , ̃x ij = (l ij ,m ij ,u ij ),0≤l ij ≤m ij ≤u ij , and letw= (w 1 ,...,w n )∈[0,1] n be criterion weights with ∑ n j=1 w j = 1. Fix an attitude parameter λ∈[0,1](oftenλ= 1 2 ). Step 1 (fuzzy linear normalization to[0,1]).For each criterionC j , define the global lower and upper anchors x j :=min 1≤i≤m l ij ,x j :=max 1≤i≤m u ij ,∆ j :=x j −x j . Assume∆ j >0for allj. Define the normalized TFN ̃r ij = (r ` ij ,r m ij ,r u ij )∈TFN [0,1] by ̃r ij := ( l ij −x j ∆ j , m ij −x j ∆ j , u ij −x j ∆ j ) , C j ∈C + (benefit), ( x j −u ij ∆ j , x j −m ij ∆ j , x j −l ij ∆ j ) , C j ∈C − (cost). (Thus, componentwise,0≤r ` ij ≤r m ij ≤r u ij ≤1.) Step 2 (pessimistic/optimistic overall values = “range of value”).Define, for each alternativeA i , the pessimistic and optimistic aggregated utilities V − i := n ∑ j=1 w j r ` ij , V + i := n ∑ j=1 w j r u ij , and the inducedvalue range ROV(A i ) := [ V − i , V + i ] ⊆[0,1]. Equivalently, one may form the TFN aggregated utility ̃ V i := n ∑ j=1 w j ̃r ij = ( n ∑ j=1 w j r ` ij , n ∑ j=1 w j r m ij , n ∑ j=1 w j r u ij ) = (V − i ,V m i ,V + i ), Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) whose support endpoints coincide with the range ROV(A i ). Step 3 (crisp ROV score and ranking).Define the (attitude-dependent) crisp score Score FROV (A i ) := (1−λ)V − i +λV + i ∈[0,1]. The FROV ranking is the preorder onAgiven by A p FROV A q ⇐⇒Score FROV (A p )≥Score FROV (A q ). Any maximizer A ? ∈arg max A i ∈A Score FROV (A i ) is called aFROV best alternative. Using Uncertain Sets, we define Uncertain Range of Value method of typeM(U-ROV) as follows. Definition 6.18.2(Uncertain Range of Value method of typeM(U-ROV)).LetA=A 1 ,...,A m be alternatives andC=C 1 ,...,C n criteria, withm,n≥2. Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Fix an uncertain modelMwith Dom(M)6=∅and an admissible positive scoreS M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M) (i= 1,...,m;j= 1,...,n). Letw= (w 1 ,...,w n )be criterion weights withw j ≥0and ∑ n j=1 w j = 1. Fix an attitude parameter λ∈[0,1]. Step 0 (Crisp projection).Define the positive real matrixY= (y ij )by y ij :=S M ( x (M) ij ) ∈(0,∞). Step 1 (Linear normalization to[0,1]).For each criterionC j , define anchors y j :=min 1≤i≤m y ij ,y j :=max 1≤i≤m y ij ,∆ j :=y j −y j ≥0. Define normalized valuesr ij ∈[0,1]by r ij := y ij −y j ∆ j , C j ∈C ben and∆ j >0, y j −y ij ∆ j , C j ∈C cost and∆ j >0, 0,∆ j = 0. (Thus, largerr ij is always better; degenerate criteria with∆ j = 0contribute0.) Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Step 2 (Value range: pessimistic and optimistic utilities).Define the pessimistic and optimistic aggregated utilities for eachA i by V − i := n ∑ j=1 w j r ij , V + i := n ∑ j=1 w j r ij , afterapplying different attitudes to uncertainty. To explicitly model uncertainty, introduce a criterionwise admissible interval radiusε ij ∈[0,r ij ]and define r − ij :=r ij −ε ij ∈[0,1], r + ij :=r ij +ε ij ∈[0,1], then set V − i := n ∑ j=1 w j r − ij , V + i := n ∑ j=1 w j r + ij . Theuncertain range of valueofA i is ROV(A i ) := [ V − i , V + i ] ⊆[0,1]. Step 3 (Attitude-dependent crisp score and ranking).Define the attitude score Score UROV (A i ) := (1−λ)V − i +λV + i ∈[0,1]. Rank alternatives by decreasing Score UROV (A i ). Theorem 6.18.3(Well-definedness of U-ROV).Under Definition 6.18.2, assumem,n≥2,Dom(M)6=∅, andS M :Dom(M)→(0,∞)is admissible. Chooseε ij ∈[0,r ij ]so thatr + ij =r ij +ε ij ≤1for alli,j. Then: (i) All normalized valuesr ij and boundsr − ij ,r + ij are well-defined and lie in[0,1]. (i) For eachi,0≤V − i ≤V + i ≤1andROV(A i )is a well-defined interval in[0,1]. (i) For eachi,0≤Score UROV (A i )≤1; hence the ranking rule is well-defined. Proof.(i) SinceS M is admissible, eachy ij >0is finite, hencey j ,y j exist and are finite. If∆ j >0, the linear formulas yieldr ij ∈[0,1]; if∆ j = 0,r ij = 0by definition. Withε ij ∈[0,r ij ]andr + ij ≤1, we obtain 0≤r − ij ≤r ij ≤r + ij ≤1. (i) Sincew j ≥0, ∑ j w j = 1, andr − ij ,r + ij ∈[0,1], the weighted sums satisfy0≤V − i ≤V + i ≤1. Hence ROV(A i ) = [V − i ,V + i ]is a well-defined interval contained in[0,1]. (i) Because Score UROV (A i )is a convex combination ofV − i andV + i withλ∈[0,1], it follows that Score UROV (A i )∈[0,1]. Therefore sorting by these scores defines a valid ranking (ties allowed). Related concepts of Range of Value under uncertainty-aware models are listed in Table 6.18. Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Table 6.18: Related concepts of Range of Value under uncertainty-aware models. kRelated Range of Value concept(s) 1 Fuzzy Range of Value 2 Intuitionistic Fuzzy Range of Value 3 Neutrosophic Range of Value 6.19Fuzzy MOOSRA MOOSRA normalizes criterion values, applies criterion weights, computes a benefit-to-cost ratio based on weighted sums, and ranks alternatives in descending order [922, 923]. As an extension, variants such as MULTIMOOSRAL have also been studied [924, 925]. Fuzzy MOOSRA replaces crisp criterion values with fuzzy numbers, propagates uncertainty throughα-cut operations or defuzzification, and then ranks alternatives based on the resulting fuzzy ratios in a robust manner [926,927]. Definition 6.19.1(MOOSRA score (crisp baseline)).[922, 923] LetA=A 1 ,...,A m be the set of alternatives andC=C 1 ,...,C n the set of criteria. LetJ + ⊆ 1,...,nbe the index set ofbenefit (to be maximized) criteria andJ − ⊆ 1,...,nthe index set ofcost(to be minimized) criteria, with J + ∪J − =1,...,nandJ + ∩J − =∅. Letw j >0be criterion weights with ∑ n j=1 w j = 1. Given a (crisp) decision matrixX= (x ij )∈R m×n + , define the normalized matrixH= (h ij )by h ij := x ij √ ∑ m p=1 x 2 pj (i= 1,...,m, j= 1,...,n), and define the MOOSRA performance score of alternativeA i by Y i (X) := ∑ j∈J + w j h ij ∑ j∈J − w j h ij , i= 1,...,m, assuming the denominator is positive. Alternatives are ranked in descending order ofY i . Definition 6.19.2(Fuzzy MOOSRA (F-MOOSRA) via the extension principle /α-cuts).Let ̃ X= ( ̃x ij ) be afuzzy decision matrixwhere each entry ̃x ij is a nonnegative fuzzy number (e.g. triangular/trapezoidal) and( ̃x ij ) α = [x L ij (α),x U ij (α)]⊆R + denotes itsα-cut forα∈(0,1]. Define the feasible set of crisp realizations at levelαby D α := X= (x ij )∈R m×n + ∣ ∣ ∣ x ij ∈( ̃x ij ) α for alli,j . LetY i (·)be the (crisp) MOOSRA score from the previous definition. Thefuzzy MOOSRA scoreof alternative A i is the fuzzy number ̃ Y i whoseα-cuts are ( ̃ Y i ) α = [ Y L i (α), Y U i (α) ] , Y L i (α) :=inf X∈D α Y i (X), Y U i (α) :=sup X∈D α Y i (X), provided ∑ j∈J − w j h ij (X)>0for allX∈ D α . Equivalently, ̃ Y i is obtained from ̃ Xby Zadeh’s extension principle applied to the mappingX7→Y i (X). Finally,Fuzzy MOOSRAranks alternatives by a fixed fuzzy-number comparison rule applied to ̃ Y i (e.g. centroid defuzzification, possibility/necessity dominance, orα-level interval dominance). Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Using Uncertain Sets, we define Uncertain MOOSRA of typeM(U-MOOSRA) as follows. Definition 6.19.3(Uncertain MOOSRA of typeM(U-MOOSRA)).LetA=A 1 ,...,A m be alternatives andC=C 1 ,...,C n criteria withm≥2andn≥2. LetJ + ⊆1,...,nbe the set of benefit criteria and J − ⊆1,...,nthe set of cost criteria, withJ + ∪J − =1,...,nandJ + ∩J − =∅, and assumeJ − 6=∅. Fix an uncertain modelMwith Dom(M)6=∅and an admissible positive scoreS M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M) (i= 1,...,m;j= 1,...,n), and criterion weightsw= (w 1 ,...,w n )withw j >0and ∑ n j=1 w j = 1. (If uncertain weights are provided, one may first score them and normalize to obtain suchw j .) Step 0 (Crisp projection).Define a positive real matrixX= (x ij )by x ij :=S M ( x (M) ij ) ∈(0,∞). Step 1 (Vector normalization).For each criterionj, define d j := √ √ √ √ m ∑ p=1 x 2 pj >0, h ij := x ij d j ∈(0,1] (i= 1,...,m). Step 2 (Weighted sums for benefit and cost parts).For each alternativeA i , define N i := ∑ j∈J + w j h ij ≥0, D i := ∑ j∈J − w j h ij >0, and the U-MOOSRA score Y i := N i D i ∈[0,∞). Ranking rule:rank alternatives in descending order ofY i . Theorem 6.19.4(Well-definedness and positivity of U-MOOSRA).Under Definition 6.19.3, assumem≥ 2,J − 6=∅,Dom(M)6=∅, andS M :Dom(M)→(0,∞)is admissible. Then: (i)d j >0for everyj, hence allh ij are well-defined and satisfyh ij ∈(0,1]. (i) For everyi, one hasD i >0, henceY i is well-defined and satisfiesY i ≥0. (i) The ranking rule induced byY i is well-defined (ties allowed). Chapter 6. Compensatory scoring methods on a decision matrix (“weighted-sum” families) Proof.(i) SinceS M maps into(0,∞), eachx pj >0. Therefore ∑ m p=1 x 2 pj >0and henced j = √ ∑ m p=1 x 2 pj > 0. Thush ij =x ij /d j is well-defined and strictly positive. Moreover,x ij ≤d j becaused j ≥ √ x 2 ij =x ij , henceh ij ≤1. (i) BecauseJ − 6=∅,w j >0for allj, andh ij >0for alli,j, the sumD i = ∑ j∈J − w j h ij is a sum of at least one strictly positive term, henceD i >0. ThereforeY i =N i /D i is well-defined and nonnegative. (i) EachY i is a finite real number, so sorting alternatives byY i yields a well-defined preorder. Related concepts of MOOSRA under uncertainty-aware models are listed in Table 6.19. Table 6.19: Related concepts of MOOSRA under uncertainty-aware models. kRelated MOOSRA concept(s) 1 Fuzzy MOOSRA 2 Intuitionistic Fuzzy MOOSRA 3 Neutrosophic MOOSRA Chapter 7 Distance-to-reference / border / compromise-index methods Distance-to-reference, border-based, and compromise-index methods rank alternatives by first normalizing the criteria, then measuring distances to reference entities (e.g., ideal/anti-ideal points, boundary profiles, or border regions), and finally aggregating these distances into a single compromise score that balances closeness, separation, and trade-offs across criteria. For convenience, a concise comparison of the repre- sentative distance-to-reference, border-based, and compromise-index methods discussed in this chapter is presented in Table 7.1. Table 7.1: A concise comparison of representative distance-to-reference, border-based, and compromise- index methods. Method Reference entity Core mechanismTypical final quantity Ranking direction TOPSISPositive ideal solu- tion and negative ideal solution Normalizes and weights the decision matrix, computes separation from the ideal and anti-ideal points, and evaluates relative closeness. Closeness coeffi- cient Larger is better MARCOSIdeal and anti-ideal alternatives Extends the decision matrix by adding ideal and anti-ideal rows, normalizes toward the ideal, and computes utility degrees relative to both reference alternatives. MARCOS utilityLarger is better CODASNegative ideal solu- tion Measures Euclidean and taxicab distances from the negative ideal, then forms a relative assessment matrix using a threshold-based cor- rection. Assessment scoreLarger is better Continued on the next page. 251 Chapter 7. Distance-to-reference / border / compromise-index methods Table 7.1 (continued). Method Reference entity Core mechanismTypical final quantity Ranking direction EDASAverage solutionComputes positive and negative distances from the average crite- rion value, aggregates them with weights, and forms an appraisal score. Appraisal scoreLarger is better VIKORBest and worst cri- terion values Uses normalized regret from the best value, aggregates group utility and individual regret, and combines them into a compromise index. Compromise index Q i Smaller is better MABACBorder approxima- tion area Normalizes and weights perfor- mances, constructs a border area by criterion, and sums signed devia- tions from that border. Border-deviation score Larger is better MAIRCATheoretical vs. real distribution of cri- terion importance Compares a theoretical distribu- tion matrix with the real weighted- normalized distribution and aggre- gates the resulting gaps. Overall gap / cri- terion function Smaller is better Note.Although all of these methods belong to the same broad family of reference-based MCDM techniques, they differ in the type of reference they use: TOPSIS and VIKOR are ideal-solution based, MARCOS uses both ideal and anti-ideal alternatives, CODAS uses a negative ideal, EDAS uses an average solution, MABAC uses a border approximation area, and MAIRCA compares theoretical and real distributions. 7.1 Fuzzy TOPSIS (Fuzzy Technique for Order of Preference by Similarity to Ideal Solution) TOPSIS ranks alternatives by distances to ideal and anti-ideal solutions after normalization and weighting, selecting the option with highest relative closeness coefficient [928,929]. Fuzzy TOPSIS ranks alternatives under uncertainty using fuzzy ratings and weights, defines fuzzy ideal and anti-ideal solutions, computes distances, and selects highest closeness coefficient [930,931]. Definition 7.1.1(TFN-based Fuzzy TOPSIS).[419, 932] LetA=A 1 ,...,A m be alternatives and C=C 1 ,...,C n criteria, partitioned into benefit and cost types C=C + ̇ ∪C − . AssumeK≥1decision makers provide TFN ratings ̃x (k) ij = (a (k) ij ,b (k) ij ,c (k) ij )and TFN weights ̃w (k) j = (w (k) j1 ,w (k) j2 ,w (k) j3 ). (0) Vertex distance.For TFNs ̃x= (a 1 ,b 1 ,c 1 )and ̃y= (a 2 ,b 2 ,c 2 ), set d( ̃x, ̃y) := √ (a 1 −a 2 ) 2 + (b 1 −b 2 ) 2 + (c 1 −c 2 ) 2 3 . Chapter 7. Distance-to-reference / border / compromise-index methods (1) Group aggregation (min–mean–max).Define aggregated TFNs by ̃x ij := ( min k a (k) ij , 1 K K ∑ k=1 b (k) ij ,max k c (k) ij ) , ̃w j := ( min k w (k) j1 , 1 K K ∑ k=1 w (k) j2 ,max k w (k) j3 ) . Let ̃ X= ( ̃x ij )and ̃w= ( ̃w 1 ,..., ̃w n ). (2) Normalization.Define ̃ R= ( ̃r ij )by ̃r ij := ( a ij /c ∗ j , b ij /c ∗ j , c ij /c ∗ j ) , C j ∈C + , c ∗ j :=max 1≤i≤m c ij , ( a − j /c ij , a − j /b ij , a − j /a ij ) , C j ∈C − , a − j :=min 1≤i≤m a ij . (3) Weighting.Define the weighted normalized matrix ̃ V= ( ̃v ij )by ̃v ij := ̃r ij ⊗ ̃w j = (r ij1 w j1 , r ij2 w j2 , r ij3 w j3 ), and write ̃v ij = (v ij1 ,v ij2 ,v ij3 ). (4) FPIS/FNIS.Define (as crisp TFNs) v ∗ j :=max 1≤i≤m v ij3 , v − j :=min 1≤i≤m v ij1 , ̃v ∗ j := (v ∗ j ,v ∗ j ,v ∗ j ), ̃v − j := (v − j ,v − j ,v − j ), and setA ∗ := ( ̃v ∗ 1 ,..., ̃v ∗ n ),A − := ( ̃v − 1 ,..., ̃v − n ). (5) Separation and closeness coefficient.For each alternativeA i , define d ∗ i := n ∑ j=1 d( ̃v ij , ̃v ∗ j ), d − i := n ∑ j=1 d( ̃v ij , ̃v − j ), C i := d − i d − i +d ∗ i ∈[0,1]. (6) Ranking.The fuzzy TOPSIS ranking is the total preorder onAinduced byCC i : A p A q ⇐⇒C p ≥C q , i.e., rank alternatives by decreasingCC i (ties allowed). Using Uncertain Sets, we define Uncertain TOPSIS of typeM(U-TOPSIS) as follows. Chapter 7. Distance-to-reference / border / compromise-index methods Definition 7.1.2(Uncertain TOPSIS of typeM(U-TOPSIS)).LetA=A 1 ,...,A m be alternatives and C=C 1 ,...,C n criteria withm,n≥2. Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Fix an uncertain modelMwith Dom(M)6=∅, together with an admissible positive scoreS M and an admissible distanced M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M) (i= 1,...,m;j= 1,...,n). Letw= (w 1 ,...,w n )be criterion weights withw j >0and ∑ n j=1 w j = 1. (If uncertain weights are provided, first score them viaS M and normalize to obtain suchw j .) Step 0 (Crisp projection).Definey ij :=S M (x (M) ij )>0and formY= (y ij )∈(0,∞) m×n . Step 1 (Vector normalization).For each criterionj, define d j := √ √ √ √ m ∑ i=1 y 2 ij >0, r ij := y ij d j ∈(0,1] (i= 1,...,m). Step 2 (Weighted normalized matrix).Define v ij :=w j r ij ∈(0,1] (i= 1,...,m;j= 1,...,n). Step 3 (Ideal and anti-ideal solutions).Define the (crisp) positive ideal solution (PIS) and negative ideal solution (NIS) by v ∗ j := max 1≤i≤m v ij , C j ∈C ben , min 1≤i≤m v ij , C j ∈C cost , v − j := min 1≤i≤m v ij , C j ∈C ben , max 1≤i≤m v ij , C j ∈C cost . LetV ∗ = (v ∗ 1 ,...,v ∗ n )andV − = (v − 1 ,...,v − n ). Step 4 (Separation measures).For each alternativeA i , define the separations D ∗ i := n ∑ j=1 d ( v ij ,v ∗ j ) , D − i := n ∑ j=1 d ( v ij ,v − j ) , wheredis any metric onR(e.g.,d(a,b) =|a−b|). (Equivalently, one may used M directly on Dom(M)by definingD ∗ i := ∑ j d M (x (M) ij ,x (M)∗ j )with suitablex (M)∗ j .) Step 5 (Closeness coefficient and ranking).Define the closeness coefficient C i := D − i D − i +D ∗ i , D − i +D ∗ i >0, 0,D − i +D ∗ i = 0, i= 1,...,m, and rank alternatives by decreasingCC i (ties allowed). Chapter 7. Distance-to-reference / border / compromise-index methods Theorem 7.1.3(Well-definedness and boundedness of U-TOPSIS).Under Definition 7.1.2, assumem,n≥ 2,Dom(M)6=∅,S M :Dom(M)→(0,∞)is admissible, andw j >0with ∑ j w j = 1. Then: (i)d j >0for every criterionj, hencer ij andv ij are well-defined. (i)V ∗ andV − exist and are well-defined. (i) For eachi,D ∗ i ≥0,D − i ≥0, andCC i is well-defined with0≤C i ≤1. (iv) The ranking induced byCC i is well-defined (ties allowed). Proof.(i) SinceS M is admissible and maps into(0,∞), eachy ij >0. Thus ∑ m i=1 y 2 ij >0andd j = √ ∑ i y 2 ij >0, sor ij =y ij /d j is well-defined. Moreover,y ij ≤d j impliesr ij ≤1, hencer ij ∈(0,1]. With w j >0,v ij =w j r ij is well-defined. (i) For eachj, the setsv 1j ,...,v mj are finite, so the maxima and minima in the definitions ofv ∗ j andv − j exist. HenceV ∗ andV − are well-defined. (i) Each separation measure is a finite sum of nonnegative distances, henceD ∗ i ,D − i ≥0and finite. If D − i +D ∗ i >0, thenCC i =D − i /(D − i +D ∗ i )is well-defined and lies in[0,1]. IfD − i +D ∗ i = 0, the definition setsCC i = 0, which also lies in[0,1]. (iv) Since eachCC i is a real number, sorting alternatives byCC i defines a well-defined preorder. Related concepts of TOPSIS under uncertainty-aware models are listed in Table 7.2. Table 7.2: Related concepts of TOPSIS under uncertainty-aware models. kRelated TOPSIS concept(s) 2 Intuitionistic Fuzzy TOPSIS [933–935] 2 Vague TOPSIS [936] 2 Bipolar fuzzy TOPSIS [937,938] 2 Pythagorean Fuzzy TOPSIS [939,940] 2 Fermatean Fuzzy TOPSIS [941,942] 3 Hesitant Fuzzy TOPSIS [943,944] 3 Picture Fuzzy TOPSIS [945,946] 3 Spherical Fuzzy TOPSIS [947,948] 3 Neutrosophic TOPSIS [949–951] 4 Quadripartitioned Neutrosophic TOPSIS [204,952] 6 Bipolar neutrosophic TOPSIS [953,954] nPlithogenic TOPSIS [955,956] As related concepts beyond Uncertain TOPSIS, several extensions are also known, including Rough TOPSIS [957, 958], Grey TOPSIS [959, 960], Soft TOPSIS [961], HyperSoft TOPSIS [962, 963], SuperHyperSoft TOPSIS [964], Group-TOPSIS [965,966], Interval TOPSIS [967,968], OWA-TOPSIS [969,970], Qualitative TOPSIS [971,972], TOPSIS-AHP [973,974], and Linguistic TOPSIS [975,976]. Chapter 7. Distance-to-reference / border / compromise-index methods 7.2 Fuzzy MARCOS (Fuzzy Measurement Alternatives and Ranking according to Compromise Solution) Classical MARCOS adds ideal and antiideal alternatives, normalizes the decision matrix, applies criterion weights, computes utility degrees relative to ideals, and ranks accordingly for choice [636, 977]. Fuzzy MARCOS appends ideal and antiideal alternatives, normalizes fuzzy matrix, computes weighted sums, derives utility degrees relative to ideals, defuzzifies, ranks all options consistently overall [978,979]. Definition 7.2.1(TFN-based Fuzzy MARCOS (Measurement Alternatives and Ranking according to Com- promise Solution)).[978,979] LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. LetC + (benefit) andC − (cost) be a partition ofC. Assume the performance ratings arepositive triangular fuzzy numbers (TFNs) ̃x ij = (x ` ij ,x m ij ,x u ij ),0< x ` ij ≤x m ij ≤x u ij , i= 1,...,m, j= 1,...,n, and letw= (w 1 ,...,w n )be a (crisp) weight vector withw j ≥0and ∑ n j=1 w j = 1. (0) TFN arithmetic conventions.For TFNs ̃a= (a ` ,a m ,a u )and ̃ b= (b ` ,b m ,b u )(positive), define ̃a⊕ ̃ b:= (a ` +b ` , a m +b m , a u +b u ), r ̃a:= (ra ` , ra m , ra u ) (r≥0), ̃a⊗ ̃ b:= (a ` b ` , a m b m , a u b u ), ̃a ̃ b:= ( a ` b u , a m b m , a u b ` ) , so that division preserves the TFN order when all components are positive. To take max/min over a finite TFN-set, fix any total preorder on TFNs; a standard choice is the defuzzifi- cation score Score TFN ( ̃a) := a ` + 4a m +a u 6 . Then ̃a ̃ b⇐⇒Score TFN ( ̃a)≤Score TFN ( ̃ b). Step 1 (initial fuzzy decision matrix).Let ̃ X:= ( ̃x ij )∈(TFN >0 ) m×n . Step 2 (anti-ideal and ideal solutions; matrix expansion).For each criterionC j , define the anti-ideal (worst) and ideal (best) TFNs: ̃x AAI j := min 1≤i≤m ̃x ij , C j ∈C + , max 1≤i≤m ̃x ij , C j ∈C − , ̃x AI j := max 1≤i≤m ̃x ij , C j ∈C + , min 1≤i≤m ̃x ij , C j ∈C − . Introduce two additional alternativesA 0 :=AAI andA m+1 :=AI and form theextendedmatrix ̃ X ext = ( ̃x ext ij )∈(TFN >0 ) (m+2)×n , where ̃x ext 0j := ̃x AAI j , ̃x ext m+1,j := ̃x AI j , and ̃x ext ij := ̃x ij fori= 1,...,m. Chapter 7. Distance-to-reference / border / compromise-index methods Step 3 (normalization).Define the normalized TFNs ̃n ij by ̃n ij := ̃x ext ij ̃x AI j , C j ∈C + , ̃x AI j ̃x ext ij , C j ∈C − , i= 0,1,...,m+ 1, j= 1,...,n. Collect ̃ N:= ( ̃n ij ). Step 4 (weighting).Define the weighted normalized TFNs ̃v ij :=w j ̃n ij , ̃ V:= ( ̃v ij ). Step 5 (row aggregation).For each extended alternativeA i (i= 0,1,...,m+ 1), define the aggregated TFN ̃ S i := n ⊕ j=1 ̃v ij . Step 6 (degrees of usefulness w.r.t. AAI and AI).Define ̃ K − i := ̃ S i ̃ S 0 , ̃ K + i := ̃ S i ̃ S m+1 , i= 1,...,m. Step 7 (auxiliary fuzzy sum and maximal reference).Set ̃ T i := ̃ K − i ⊕ ̃ K + i (i= 1,...,m), and define the componentwise maximal TFN ̃ D:= (d ` ,d m ,d u ) := ( max i t ` i ,max i t m i ,max i t u i ) ,where ̃ T i = (t ` i ,t m i ,t u i ). Defuzzify ̃ Dby the graded mean: d def :=Defuzz( ̃ D) := d ` + 4d m +d u 6 . Step 8 (utility functions relative to AAI and AI).Define ̃ f( ̃ K − i ) := 1 d def ̃ K − i , ̃ f( ̃ K + i ) := 1 d def ̃ K + i . Step 9 (final utility and ranking).Defuzzify (componentwise TFNs) by Defuzz(`,m,u) = (`+4m+u)/6 and put K − i :=Defuzz( ̃ K − i ), K + i :=Defuzz( ̃ K + i ), f − i :=Defuzz ( ̃ f( ̃ K − i ) ) , f + i :=Defuzz ( ̃ f( ̃ K + i ) ) . Then the (crisp) final MARCOS utility is f(K i ) := K − i +K + i 1 + 1−f + i f + i + 1−f − i f − i . Rank alternatives by decreasingf(K i ): A p FMARCOS A q ⇐⇒f(K p )≥f(K q ). Chapter 7. Distance-to-reference / border / compromise-index methods Referring to the above, we define Uncertain MARCOS of typeM(U-MARCOS) as follows. Definition 7.2.2(Uncertain MARCOS of typeM(U-MARCOS)).LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria, withm,n≥2. Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Fix an uncertain modelMwith Dom(M)6=∅and an admissible positive scoreS M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M) (i= 1,...,m;j= 1,...,n), and acrispweight vectorw= (w 1 ,...,w n )satisfying w j >0, n ∑ j=1 w j = 1. Introduce two auxiliary alternatives: A 0 :=AAI (anti-ideal), A m+1 :=AI (ideal). Step 0 (Crisp projection).Definey ij :=S M (x (M) ij )∈(0,∞)and setY= (y ij )∈(0,∞) m×n . Step 1 (Ideal / anti-ideal components and matrix extension).For each criterionC j , define y AI j := max 1≤i≤m y ij , C j ∈C ben , min 1≤i≤m y ij , C j ∈C cost , y AAI j := min 1≤i≤m y ij , C j ∈C ben , max 1≤i≤m y ij , C j ∈C cost . Define the extended matrixY ext = (y ext ij )∈(0,∞) (m+2)×n by y ext 0j :=y AAI j , y ext m+1,j :=y AI j , y ext ij :=y ij (i= 1,...,m). Step 2 (Normalization toward the ideal).Fori= 0,1,...,m+ 1, define n ij := y ext ij y AI j , C j ∈C ben , y AI j y ext ij , C j ∈C cost . Thenn ij ∈(0,1]andn m+1,j = 1. Step 3 (Weighting and aggregation).Define v ij :=w j n ij , S i := n ∑ j=1 v ij (i= 0,1,...,m+ 1). Chapter 7. Distance-to-reference / border / compromise-index methods Step 4 (Utility degrees).For each real alternativeA i (i= 1,...,m), define K − i := S i S 0 , K + i := S i S m+1 , T i :=K − i +K + i . Step 5 (Final utility function and ranking).LetD:=max 1≤i≤m T i and define f − i := K − i D , f + i := K + i D . Define the MARCOS utility U i := K − i +K + i 1 + 1−f + i f + i + 1−f − i f − i , i= 1,...,m, and rank alternatives by decreasingU i . Theorem 7.2.3(Well-definedness of U-MARCOS).Under Definition 7.2.2, assumeS M is admissible and w j >0for allj. Then all quantitiesn ij ,S i ,K − i ,K + i ,T i ,D,f − i ,f + i ,U i are well-defined and finite. Moreover, S 0 >0,S m+1 = ∑ n j=1 w j >0,D >0, andU i ≥0for alli. Proof.Admissibility ofS M impliesy ij >0finite, so the extremay AI j andy AAI j exist and are positive. Hence the normalization in Step 2 has positive denominators, andn ij ∈(0,1]is well-defined. Sincew j >0, eachv ij =w j n ij >0, soS i = ∑ j v ij >0for alli, in particularS 0 >0. Becausen m+1,j = 1, we haveS m+1 = ∑ n j=1 w j >0. ThusK − i =S i /S 0 andK + i =S i /S m+1 are well-defined and positive, hence T i =K − i +K + i >0and thereforeD=max i T i >0. Consequentlyf − i ,f + i ∈(0,1]and the denominators inU i are strictly positive, soU i is well-defined and finite. Finally,U i is a ratio of positive terms, hence U i ≥0. Related concepts of MARCOS under uncertainty-aware models are listed in Table 7.3. Table 7.3: Related concepts of MARCOS under uncertainty-aware models. kRelated MARCOS concept(s) 1 Fuzzy MARCOS 2 Intuitionistic Fuzzy MARCOS [979,980] 3 Hesitant Fuzzy MARCOS [981] 3 Picture Fuzzy MARCOS [982,983] 3 Spherical Fuzzy MARCOS [984,985] 3 Neutrosophic MARCOS [636,986] 7.3 Fuzzy CODAS (Fuzzy COmbinative Distance-based ASsessment) Classical CODAS defines a negative ideal solution, computes Euclidean and taxicab distances for each alternative, applies a threshold, and ranks by larger distances overall performance [987,988]. Fuzzy CODAS defines fuzzy negative ideal solution, computes Euclidean and taxicab distances of each alternative, applies a threshold, defuzzifies, and ranks; higher distance means better [989,990]. Chapter 7. Distance-to-reference / border / compromise-index methods Definition 7.3.1(Fuzzy CODAS (COmbinative Distance-based ASsessment) — trapezoidal-fuzzy, group MCDM).[989, 990]Input data.LetA=A 1 ,...,A m be alternatives,C=C 1 ,...,C n criteria, and letQ=1,...,qbe the set of decision makers (DMs). Partition criteria into benefit and cost sets C=C + ̇ ∪C − . Assume all ratings and weights arepositive trapezoidal fuzzy numbers ̃x (`) ij = (x (`) ij,1 ,x (`) ij,2 ,x (`) ij,3 ,x (`) ij,4 )∈TrFN >0 , ̃w (`) j ∈TrFN >0 , where`∈Q,i∈ 1,...,m,j∈ 1,...,n. Fix: (i) trapezoidal-fuzzy arithmetic(⊕, ,⊗, )onTrFN >0 (componentwise as standard), (i) a defuzzification/ranking mapD:TrFN >0 →Rused only to take max/min over finite fuzzy sets, and (i) a threshold parameterθ≥0. Step 1 (aggregate the fuzzy decision matrices).Define the averaged fuzzy decision matrix ̃ X= ( ̃x ij ) by ̃x ij := 1 q q ⊕ `=1 ̃x (`) ij . Step 2 (aggregate the fuzzy weights).Define the averaged fuzzy weight vector ̃w= ( ̃w 1 ,..., ̃w n )by ̃w j := 1 q q ⊕ `=1 ̃w (`) j . Step 3 (normalization).For each criterionj, choose a fuzzy “maximum” ̃x max j ∈arg max 1≤i≤m D( ̃x ij ). Define the normalized fuzzy matrix ̃ N= ( ̃n ij )by ̃n ij := ̃x ij ̃x max j , C j ∈C + (benefit), ̃x max j ̃x ij , C j ∈C − (cost). Step 4 (weighted normalized matrix).Define ̃ R= ( ̃r ij )by ̃r ij := ̃w j ⊗ ̃n ij . Step 5 (fuzzy negative-ideal solution).For eachj, define ̃ns j ∈arg min 1≤i≤m D( ̃r ij ), ̃ NS:= ( ̃ns 1 ,..., ̃ns n ). Chapter 7. Distance-to-reference / border / compromise-index methods Step 6 (distances from the fuzzy negative-ideal solution).For trapezoidal fuzzy numbers ̃a= (a 1 ,a 2 ,a 3 ,a 4 )and ̃ b= (b 1 ,b 2 ,b 3 ,b 4 )define: d E ( ̃a, ̃ b) := √ (a 1 −b 1 ) 2 + 2(a 2 −b 2 ) 2 + 2(a 3 −b 3 ) 2 + (a 4 −b 4 ) 2 6 , d H ( ̃a, ̃ b) := |a 1 −b 1 |+ 2|a 2 −b 2 |+ 2|a 3 −b 3 |+|a 4 −b 4 | 6 . Then for each alternativeA i define the (combinative) distance components ED i := n ∑ j=1 d E ( ̃r ij , ̃ns j ), HD i := n ∑ j=1 d H ( ̃r ij , ̃ns j ). Step 7 (relative assessment matrix).Define the threshold functiont:R→0,1by t(x) := 1,|x|≥θ, 0,|x|< θ. Fori,k∈1,...,mdefine the relative assessment entry p ik := (ED i −ED k ) +t(ED i −ED k ) (HD i −HD k ), and collectRA= (p ik )∈R m×m . Step 8 (assessment score).For eachi, define the CODAS assessment score AS i := m ∑ k=1 p ik . Step 9 (ranking / solution).TheFuzzy CODASranking is the total preorder onAinduced byAS: A i FCODAS A k ⇐⇒AS i ≥AS k . A (best) solution is any A ? ∈arg max A i ∈A AS i . Using Uncertain Sets, we define Uncertain CODAS of typeM(U-CODAS) as follows. Definition 7.3.2(Uncertain CODAS of typeM(U-CODAS)).LetA=A 1 ,...,A m be alternatives and C=C 1 ,...,C n criteria withm≥2andn≥1. Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Fix an uncertain modelMwith Dom(M)6=∅and an admissible positive scoreS M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M) (i= 1,...,m;j= 1,...,n), Chapter 7. Distance-to-reference / border / compromise-index methods and criterion weightsw= (w 1 ,...,w n )withw j >0and ∑ n j=1 w j = 1. Fix a threshold parameterθ≥0. Step 0 (Crisp projection).Definey ij :=S M (x (M) ij )∈(0,∞)and setY= (y ij )∈(0,∞) m×n . Step 1 (Normalization).For each criterionj, define y max j :=max 1≤i≤m y ij >0. Define normalized performancesn ij ∈(0,1]by n ij := y ij y max j , C j ∈C ben , y max j y ij , C j ∈C cost . Step 2 (Weighted normalized matrix).Define weighted normalized values r ij :=w j n ij (i= 1,...,m;j= 1,...,n). Step 3 (Negative-ideal solution).Define the negative-ideal component for each criterion by ns j :=min 1≤i≤m r ij ,NS:= (ns 1 ,...,ns n ). Step 4 (Distances from the negative-ideal).For each alternativeA i , define the Euclidean and taxicab distances from NS: ED i := √ √ √ √ n ∑ j=1 (r ij −ns j ) 2 , HD i := n ∑ j=1 |r ij −ns j |. Step 5 (Relative assessment matrix).Define the threshold indicatort:R→0,1by t(x) := 1,|x|≥θ, 0,|x|< θ, and for each pair(i,k)define p ik := (ED i −ED k ) +t(ED i −ED k ) (HD i −HD k ). LetRA= (p ik )∈R m×m . Step 6 (Assessment score and ranking).Define AS i := m ∑ k=1 p ik (i= 1,...,m), and rank alternatives by decreasingAS i . Chapter 7. Distance-to-reference / border / compromise-index methods Theorem 7.3.3(Well-definedness of U-CODAS).Under Definition 7.3.2, assumeS M :Dom(M)→(0,∞) is admissible andw j >0for allj. Then all quantitiesn ij ,r ij ,ns j ,ED i ,HD i ,p ik , andAS i are well-defined finite real numbers. Proof.SinceS M maps into(0,∞), eachy ij >0is finite; hencey max j =max i y ij >0exists. Thereforen ij in Step 1 is well-defined and lies in(0,1]for both benefit and cost criteria. Becausew j >0,r ij =w j n ij is well-defined and positive, so eachns j =min i r ij exists and is finite. Thusr ij −ns j are finite reals, soED i andHD i are finite and satisfyED i ≥0,HD i ≥0. Givenθ≥0,t(·) is well-defined and takes values in0,1, so eachp ik is a finite real. Finally,AS i = ∑ m k=1 p ik is a finite sum, hence well-defined and finite. Related concepts of CODAS under uncertainty-aware models are listed in Table 7.4. Table 7.4: Related concepts of CODAS under uncertainty-aware models. kRelated CODAS concept(s) 2 Intuitionistic Fuzzy CODAS [991,992] 2 Pythagorean Fuzzy CODAS [993,994] 3 Neutrosophic CODAS [995,996] 3 Picture Fuzzy CODAS [997,998] 3 Hesitant Fuzzy CODAS [999,1000] 3 Spherical Fuzzy CODAS [1001,1002] 7.4 Fuzzy EDAS (Fuzzy Evaluation based on Distance from Average Solution) EDAS (evaluation based on distance from average solution) ranks alternatives by positive and negative distances from average solution, aggregates weighted distances, and computes appraisal scores [1003,1004]. Fuzzy EDAS uses fuzzy ratings and weights, computes fuzzy distances from average solution, defuzzifies, and ranks by appraisal scores [1005]. Definition 7.4.1(Fuzzy EDAS (evaluation based on distance from average solution)).[1005] LetA= A 1 ,...,A m be alternatives andC=C 1 ,...,C n criteria. Assume a trapezoidal-fuzzy decision matrix and trapezoidal-fuzzy criterion weights ̃ X= ( ̃x ij ) m×n , ̃x ij = (x ij1 ,x ij2 ,x ij3 ,x ij4 ), ̃ W= ( ̃w 1 ,..., ̃w n ), ̃w j = (w j1 ,w j2 ,w j3 ,w j4 ), where all entries are trapezoidal fuzzy numbers. Fix a nonnegative scalar “distance” functiond:R×R→ R ≥0 (e.g. a chosenL 1 -metric at the scalar level), and extend it componentwise to trapezoidal fuzzy numbers by d( ̃a, ̃ b) := ( d(a 1 ,b 1 ),d(a 2 ,b 2 ),d(a 3 ,b 3 ),d(a 4 ,b 4 ) ) , ̃a= (a 1 ,a 2 ,a 3 ,a 4 ), ̃ b= (b 1 ,b 2 ,b 3 ,b 4 ). (0) Defuzzification and theψ-operator.For ̃a= (a 1 ,a 2 ,a 3 ,a 4 )define the graded-mean defuzzification Defuzz( ̃a) := a 1 +a 4 2 + (a 2 −a 1 −a 4 +a 3 ), Chapter 7. Distance-to-reference / border / compromise-index methods and defineψ(“maximum with zero”) by ψ( ̃a) := ̃a,Defuzz( ̃a)>0, ̃ 0,Defuzz( ̃a)≤0, ̃ 0 := (0,0,0,0). (1) Average (reference) solution.For each criterionj, define the average trapezoidal fuzzy value ̃ AV j := 1 m m ⊕ i=1 ̃x ij , where⊕denotes trapezoidal-fuzzy addition and scalar multiplication is componentwise. (2) Positive/negative distances from the average.Define trapezoidal-fuzzy matrices ̃ PDA= ( ̃ pda ij ) and ̃ NDA= ( ̃ nda ij )by ̃ pda ij :=ψ ( d( ̃x ij , ̃ AV j ) ) , ̃ nda ij :=ψ ( d( ̃ AV j , ̃x ij ) ) . (3) Weighted sums.For each alternativeA i , set ̃sp i := n ⊕ j=1 ( ̃w j ⊗ ̃ pda ij ) , ̃sn i := n ⊕ j=1 ( ̃w j ⊗ ̃ nda ij ) , where⊗is trapezoidal-fuzzy multiplication. (4) Normalization.Write ̃sp i = (sp i1 ,sp i2 ,sp i3 ,sp i4 )and ̃sn i = (sn i1 ,sn i2 ,sn i3 ,sn i4 ). Define normalized (real) values ̃ NSP i := sp i1 max 1≤t≤m Defuzz( ̃sp t ) , ̃ NSN i := 1− sn i1 max 1≤t≤m Defuzz( ̃sn t ) . (5) Appraisal score and ranking.Define the appraisal score AS i := ̃ NSP i + ̃ NSN i 2 ∈[0,1], and rank alternatives in descending order ofAS i (largerAS i indicates a better alternative). Using Uncertain Sets, we define Uncertain EDAS of typeM(U-EDAS) as follows. Chapter 7. Distance-to-reference / border / compromise-index methods Definition 7.4.2(Uncertain EDAS of typeM(U-EDAS)).LetA=A 1 ,...,A m be alternatives and C=C 1 ,...,C n criteria, withm,n≥2. Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Fix an uncertain modelMwith Dom(M)6=∅and an admissible positive scoreS M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M) (i= 1,...,m;j= 1,...,n), and criterion weightsw= (w 1 ,...,w n )with w j ≥0, n ∑ j=1 w j = 1. Step 0 (Crisp projection).Define the positive real matrixY= (y ij )by y ij :=S M ( x (M) ij ) ∈(0,∞). Step 1 (Average solution).For each criterionC j , define the average (reference) value AV j := 1 m m ∑ i=1 y ij ∈(0,∞). Step 2 (Positive/negative distances from the average).For eachi,j, define the positive distance from average (PDA) and negative distance from average (NDA) by PDA ij := max0, y ij −AV j AV j , C j ∈C ben , max0, AV j −y ij AV j , C j ∈C cost , NDA ij := max0, AV j −y ij AV j , C j ∈C ben , max0, y ij −AV j AV j , C j ∈C cost . ThusPDA ij ≥0andNDA ij ≥0. Step 3 (Weighted sums).For each alternativeA i , define SP i := n ∑ j=1 w j PDA ij ≥0, SN i := n ∑ j=1 w j NDA ij ≥0. Step 4 (Normalization).Let SP max :=max 1≤i≤m SP i , SN max :=max 1≤i≤m SN i . Chapter 7. Distance-to-reference / border / compromise-index methods Define normalized values NSP i := SP i SP max , SP max >0, 0,SP max = 0, NSN i := 1− SN i SN max , SN max >0, 1,SN max = 0. Step 5 (Appraisal score and ranking).Define the appraisal score AS i := NSP i +NSN i 2 ∈[0,1], and rank alternatives in descending order ofAS i (ties allowed). Theorem 7.4.3(Well-definedness and boundedness of U-EDAS).Under Definition 7.4.2 (in particular, m,n≥2andS M :Dom(M)→(0,∞)), all quantities in U-EDAS are well-defined and finite. Moreover, 0≤PDA ij ,NDA ij <∞,0≤SP i ,SN i <∞,0≤NSP i ,NSN i ,AS i ≤1. Hence the ranking rule induced byAS i is well-defined. Proof.SinceS M maps into(0,∞), eachy ij >0is finite, and thusAV j = 1 m ∑ i y ij >0is finite. Therefore the ratios in the definitions ofPDA ij andNDA ij have strictly positive denominators and finite numerators, soPDA ij andNDA ij are finite and nonnegative. Becausew j ≥0and ∑ j w j = 1,SP i andSN i are finite nonnegative weighted sums, soSP max and SN max exist (finite maxima over a finite set) and satisfySP max ≥0,SN max ≥0. IfSP max >0, then NSP i =SP i /SP max ∈[0,1]; ifSP max = 0, the definition setsNSP i = 0. Similarly, ifSN max >0, then SN i /SN max ∈[0,1]soNSN i = 1−SN i /SN max ∈[0,1]; ifSN max = 0, the definition setsNSN i = 1∈[0,1]. Finally,AS i is the average of two numbers in[0,1], henceAS i ∈[0,1]. Thus sorting alternatives byAS i yields a well-defined preorder. Related concepts of EDAS under uncertainty-aware models are listed in Table 7.5. Table 7.5: Related concepts of EDAS under uncertainty-aware models. kRelated EDAS concept(s) 2 Intuitionistic Fuzzy EDAS [70,1006] 3 Hesitant Fuzzy EDAS [1007,1008] 3 Spherical Fuzzy EDAS [1009,1010] 3 Neutrosophic EDAS [1011,1012] 7.5 Uncertain VIKOR VIKOR ranks alternatives by distances to ideal and anti-ideal points, computing group utility and individual regret, then a compromise index balancing both [1013, 1014]. Fuzzy VIKOR ranks alternatives under uncertainty using fuzzy ideal and nadir solutions, aggregated utility and regret measures, then a compromise index balancing group benefit and individual regret [892,1015,1016]. Chapter 7. Distance-to-reference / border / compromise-index methods Definition 7.5.1(TFN-based Fuzzy VIKOR).[892, 1015, 1016] LetA=A 1 ,...,A J be alternatives andC=C 1 ,...,C n criteria. LetI b andI c be the index sets of benefit and cost criteria, withI b ̇ ∪I c = 1,...,n. Assume TFN performances ̃ f ij = (l ij ,m ij ,r ij )and TFN weights ̃w i = (l w i ,m w i ,r w i ). (0) Conventions (defuzzification and max/min).Use the “2nd weighted mean” defuzzification Crisp(l,m,r) := l+ 2m+r 4 , and compare TFNs by ̃x ̃y⇐⇒Crisp( ̃x)≤Crisp( ̃y). All TFN additions/scalar divisions below are componentwise; division is only by positive scalars. (1) Fuzzy ideal and nadir values.For each criterionidefine ̃ f ? i := max 1≤j≤J ̃ f ij , i∈I b , min 1≤j≤J ̃ f ij , i∈I c , ̃ f i := min 1≤j≤J ̃ f ij , i∈I b , max 1≤j≤J ̃ f ij , i∈I c , where max,min are taken w.r.t.. Write ̃ f ? i = (l ? i ,m ? i ,r ? i )and ̃ f i = (l i ,m i ,r i ). (2) Normalized fuzzy deviations.Define the (crisp) ranges ∆ i := r ? i −l i , i∈I b , r i −l ? i , i∈I c , (∆ i >0), and for each alternativeA j , ̃ d ij := ( ̃ f ? i − ̃ f ij )/∆ i , i∈I b , ( ̃ f ij − ̃ f ? i )/∆ i , i∈I c . (3) Group utility and individual regret (fuzzy).For each alternativeA j define TFNs ̃ S j := n ∑ i=1 ̃w i ̃ d ij , ̃ R j :=max 1≤i≤n ( ̃w i ̃ d ij ) , where products/sums are componentwise and max is w.r.t.. (4) Compromise index (fuzzy).Let ̃ S:=min 1≤j≤J ̃ S j , ̃ R:=min 1≤j≤J ̃ R j , and set S max :=max 1≤j≤J r( ̃ S j ), R max :=max 1≤j≤J r( ̃ R j ), wherer(l,m,r) :=randl(l,m,r) :=ldenote right/left endpoints. Fixv∈[0,1]. Define ̃ Q j :=v ̃ S j − ̃ S S max −l( ̃ S) + (1−v) ̃ R j − ̃ R R max −l( ̃ R) . Chapter 7. Distance-to-reference / border / compromise-index methods (5) Ranking and compromise solution (crisp).Defuzzify S j :=Crisp( ̃ S j ), R j :=Crisp( ̃ R j ), Q j :=Crisp( ̃ Q j ), and rank alternatives increasingly byQ j . LetA (1) ,A (2) be the first two in theQ-ranking andA (J) the last. Set DQ:= 1 J−1 ,Adv:= Q(A (2) )−Q(A (1) ) Q(A (J) )−Q(A (1) ) . ThenA (1) is the (single) compromise solution if Adv≥DQandA (1) is also best bySand/or byR. If the stability condition fails, proposeA (1) ,A (2) . If the advantage condition fails, proposeA (1) ,...,A (M) whereMis the largest index withQ(A (M) )−Q(A (1) )< DQ. Using Uncertain Sets, we define Uncertain VIKOR of typeM(U-VIKOR) as follows. Definition 7.5.2(Uncertain VIKOR of typeM(U-VIKOR)).LetA=A 1 ,...,A m be alternatives and C=C 1 ,...,C n criteria, withm,n≥2. Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M), and criterion weightsw= (w 1 ,...,w n )with w j >0, n ∑ j=1 w j = 1. Fix the VIKOR compromise parameterv∈[0,1]. Step 0 (Crisp projection).Definey ij :=S M (x (M) ij )∈R. Step 1 (Best and worst values per criterion).For each criterionC j , define the best and worst (projected) values f ∗ j := max 1≤i≤m y ij , C j ∈C ben , min 1≤i≤m y ij , C j ∈C cost , f − j := min 1≤i≤m y ij , C j ∈C ben , max 1≤i≤m y ij , C j ∈C cost . Define the range∆ j :=f ∗ j −f − j ≥0. Step 2 (Normalized regret per criterion).For each alternativeA i and criterionC j , define d ij := f ∗ j −y ij ∆ j , C j ∈C ben and∆ j >0, y ij −f ∗ j ∆ j , C j ∈C cost and∆ j >0, 0,∆ j = 0, Chapter 7. Distance-to-reference / border / compromise-index methods so thatd ij ∈[0,1]and smaller is better (less regret). Step 3 (Group utility and individual regret measures).Define S i := n ∑ j=1 w j d ij , R i :=max 1≤j≤n (w j d ij ). Step 4 (Compromise index).Let S ∗ :=min 1≤i≤m S i , S − :=max 1≤i≤m S i , R ∗ :=min 1≤i≤m R i , R − :=max 1≤i≤m R i . Define the VIKOR compromise index Q i := v S i −S ∗ S − −S ∗ + (1−v) R i −R ∗ R − −R ∗ , S − > S ∗ andR − > R ∗ , S i −S ∗ S − −S ∗ ,S − > S ∗ andR − =R ∗ , R i −R ∗ R − −R ∗ ,S − =S ∗ andR − > R ∗ , 0,S − =S ∗ andR − =R ∗ . Rank alternatives in ascending order ofQ i (smallerQ i is better). (Optional) VIKOR compromise solution set.LetA (1) ,A (2) ,...,A (m) be theQ-sorted alternatives and set DQ:= 1 m−1 . One may apply the standard VIKOR acceptable-advantage and acceptable-stability rules on(S i ,R i ,Q i )to output either a single compromise solution or a small compromise set. Theorem 7.5.3(Well-definedness of U-VIKOR).Under Definition 7.5.2, assumem,n≥2,Dom(M)6=∅, S M is admissible, andw j >0with ∑ n j=1 w j = 1. Then: (i) For alli,j, the quantitiesf ∗ j ,f − j ,∆ j , andd ij are well-defined, with0≤d ij ≤1. (i) For eachi,S i andR i are well-defined finite real numbers satisfying0≤S i ≤1and0≤R i ≤1. (i) The compromise indexQ i is well-defined and satisfies0≤Q i ≤1for alli. (iv) The ranking induced byQ i is well-defined (ties allowed). Proof.(i) SinceS M is admissible, eachy ij is finite. Because the setsy 1j ,...,y mj are finite,f ∗ j andf − j exist for eachj, hence∆ j =f ∗ j −f − j ≥0is well-defined. If∆ j >0, thend ij is defined by a ratio of finite numbers; if∆ j = 0,d ij = 0by definition. In either case, one has0≤d ij ≤1. Chapter 7. Distance-to-reference / border / compromise-index methods (i) Since0≤d ij ≤1and ∑ j w j = 1,S i = ∑ j w j d ij is a convex combination, hence0≤S i ≤1. Also, 0≤w j d ij ≤w j , soR i =max j (w j d ij )exists and satisfies0≤R i ≤max j w j ≤1. (i) The extremaS ∗ ,S − ,R ∗ ,R − exist becausemis finite. Each case in the definition ofQ i avoids division by zero by switching to a single normalized component or settingQ i = 0when both ranges vanish. Whenever a fraction is used, its numerator lies between0and the corresponding denominator, hence the fraction lies in[0,1]. ThereforeQ i ∈[0,1]. (iv) Since eachQ i is a real number, sorting alternatives byQ i defines a well-defined preorder. Related concepts of VIKOR under uncertainty-aware models are listed in Table 7.6. Table 7.6: Related concepts of VIKOR under uncertainty-aware models. kRelated VIKOR concept(s) 2 Intuitionistic Fuzzy VIKOR [1017,1018] 2 Pythagorean Fuzzy VIKOR [1019,1020] 2 Fermatean Fuzzy VIKOR [1021] 2 Bipolar fuzzy VIKOR [1022,1023] 3 Hesitant Fuzzy VIKOR [1024,1025] 3 Spherical Fuzzy VIKOR [1026–1028] 3 Neutrosophic VIKOR [1029,1030] nPlithogenic VIKOR [1031] Beyond Uncertain VIKOR, several related variants are also known, such as Grey VIKOR [1032], Group VIKOR [1033,1034], Z-VIKOR [1035,1036], VIKORSORT [1037,1038], Rough VIKOR [1039,1040], TOP- SIS–VIKOR [1041,1042], and Soft VIKOR [1043]. 7.6 Uncertain MABAC (Multi-attributive border approximation area comparison) MABAC ranks alternatives by distances from a border approximation area across criteria, then sums devi- ations to obtain final scores [1044–1046]. Fuzzy MABAC uses fuzzy ratings and weights, computes a fuzzy border approximation area, and ranks alternatives by defuzzified summed deviations [1047–1049]. Definition 7.6.1(TFN-based Fuzzy MABAC).[1047–1049] LetA=A 1 ,...,A m be alternatives and C=C 1 ,...,C n criteria. Partition criteria into benefit and cost types C=C + ̇ ∪C − , and fix a crisp weight vectorw= (w 1 ,...,w n )withw j ≥0and ∑ n j=1 w j = 1. Let ̃ X= ( ̃x ij )∈(TFN) m×n be the TFN decision matrix, where ̃x ij = (x ` ij ,x m ij ,x r ij ). (0) TFN arithmetic used.All operations are componentwise unless stated otherwise: ̃x⊕ ̃y= (x ` +y ` , x m +y m , x r +y r ), α ̃x= (αx ` , αx m , αx r ) (α≥0), ̃x⊗ ̃y= (x ` y ` , x m y m , x r y r )(nonnegative TFNs), ̃x p = ( (x ` ) p ,(x m ) p ,(x r ) p ) (p >0), Chapter 7. Distance-to-reference / border / compromise-index methods and we use the common triangular subtraction approximation ̃x ̃y= (x ` −y r , x m −y m , x r −y ` ). (1) Normalization.For each criterionj, define x + j :=max 1≤i≤m x r ij , x − j :=min 1≤i≤m x ` ij , and the normalized TFNs ̃ t ij := ̃x ij x − j x + j −x − j , C j ∈C + , ̃x ij x + j x − j −x + j , C j ∈C − , where division is by a positive scalar. (2) Weighting.Define ̃v ij :=w j ̃ t ij and collect ̃ V= ( ̃v ij ). (3) Border approximation area (BAA).For each criterionj, define the border approximation TFN ̃g j := ( m ∏ i=1 ̃v ij ) 1/m , ̃ G:= ( ̃g 1 ,..., ̃g n ). (4) Signed deviations from the border and overall score.Set ̃q ij := ̃v ij ̃g j , ̃ S i := n ⊕ j=1 ̃q ij . (Heuristically, ̃q ij >0indicatesA i lies in the upper approximation area for criterionj, while ̃q ij <0indicates the lower area; a crisp sign test can be implemented after defuzzification.) (5) Defuzzification and ranking.For ̃ S i = (s ` i ,s m i ,s r i ), define Defuzz( ̃ S i ) := s ` i +s m i +s r i 3 . Rank alternatives by Defuzz( ̃ S i )in descending order: A p A q ⇐⇒Defuzz( ̃ S p )≥Defuzz( ̃ S q ). The definition of Uncertain MABAC of typeM(U-MABAC) is given below. Chapter 7. Distance-to-reference / border / compromise-index methods Definition 7.6.2(Uncertain MABAC of typeM(U-MABAC)).LetA=A 1 ,...,A m be alternatives andC=C 1 ,...,C n criteria withm,n≥2. Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M) (i= 1,...,m;j= 1,...,n), and criterion weightsw= (w 1 ,...,w n )withw j ≥0and ∑ n j=1 w j = 1. Step 0 (Crisp projection).Define a real matrixY= (y ij )by y ij :=S M ( x (M) ij ) ∈R. Step 1 (Linear normalization to[0,1]).For each criterionj, define y min j :=min 1≤i≤m y ij , y max j :=max 1≤i≤m y ij ,∆ j :=y max j −y min j ≥0. Define normalized valuest ij ∈[0,1]by t ij := y ij −y min j ∆ j , C j ∈C ben and∆ j >0, y max j −y ij ∆ j , C j ∈C cost and∆ j >0, 0,∆ j = 0. (Thus largert ij always indicates better performance.) Step 2 (Weighting).Define the weighted normalized matrixV= (v ij )by v ij :=w j t ij (i= 1,...,m;j= 1,...,n). Step 3 (Border approximation area).For each criterionj, define the border approximation value (geometric mean): g j := ( m ∏ i=1 v ij ) 1/m ≥0. LetG:= (g 1 ,...,g n ). Step 4 (Signed deviations and overall score).Define deviations q ij :=v ij −g j , S i := n ∑ j=1 q ij . Rank alternatives by decreasingS i . Chapter 7. Distance-to-reference / border / compromise-index methods Theorem 7.6.3(Well-definedness of U-MABAC).Under Definition 7.6.2, assumem,n≥2,Dom(M)6=∅, andS M is admissible. Then all quantitiest ij ,v ij ,g j ,q ij ,S i are well-defined finite real numbers. Proof.BecauseS M is admissible, eachy ij is finite; hence for eachjthe extremay min j ,y max j exist and are finite, so∆ j ≥0is well-defined. If∆ j >0, the normalization formulas definet ij ∈[0,1]; if∆ j = 0, the definition setst ij = 0, sot ij is well-defined in all cases. Sincew j ≥0andt ij ∈[0,1], eachv ij =w j t ij is finite and nonnegative. Therefore the product ∏ m i=1 v ij is well-defined and nonnegative, so the geometric meang j = ( ∏ m i=1 v ij ) 1/m is well-defined and finite (and equals0if any factor is0). Consequently,q ij =v ij −g j andS i = ∑ n j=1 q ij are finite sums/differences of finite reals, hence well-defined. Related concepts of MABAC are listed in Table 7.7. Table 7.7: Related concepts of MABAC under uncertainty-aware models. kRelated MABAC concept(s) 2 Intuitionistic Fuzzy MABAC [1050,1051] 3 Spherical Fuzzy MABAC [43,1052] 3 Hesitant Fuzzy MABAC [1053,1054] 3 Picture Fuzzy MABAC [1055,1056] 3 Neutrosophic MABAC [1057,1058] As a related concept, Rough MABAC [1059,1060] is also known. 7.7 Fuzzy MAIRCA (Fuzzy Multi-attributive ideal-real comparative analysis) MAIRCA ranks alternatives by comparing theoretical ideal distribution of criterion importance with real performances, aggregating deviations to score [1061, 1062]. Fuzzy MAIRCA represents performances and possibly weights as fuzzy numbers, computes fuzzy ideal–real gaps, defuzzifies aggregated deviations for ranking [1063,1064]. Definition 7.7.1(Fuzzy MAIRCA (TFN-based ideal–real comparative analysis)).[1063, 1064] LetA= A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. LetJ + andJ − be the index sets of benefit and cost criteria, withJ + ̇ ∪J − =1,...,n. LetTFN ≥0 :=(l,m,u)∈R 3 ≥0 :l≤m≤u. Assume a triangular-fuzzy decision matrix ̃ X= ( ̃x ij )∈(TFN ≥0 ) m×n , ̃x ij = (l ij ,m ij ,u ij ), and a (crisp) criterion-weight vectorw= (w 1 ,...,w n )withw j ≥0and ∑ n j=1 w j = 1. Fix a probability vectorp= (p 1 ,...,p m )withp i ≥0and ∑ m i=1 p i = 1(typicallyp i = 1/mfor alli). (0) TFN arithmetic and defuzzification.For TFNs ̃a= (a ` ,a m ,a u )and ̃ b= (b ` ,b m ,b u )andα≥0, define ̃a⊕ ̃ b:= (a ` +b ` , a m +b m , a u +b u ), α ̃a:= (αa ` , αa m , αa u ), Chapter 7. Distance-to-reference / border / compromise-index methods and the standard TFN subtraction approximation ̃a ̃ b:= (a ` −b u , a m −b m , a u −b ` ). Fix a defuzzification functional Defuzz:TFN ≥0 →R ≥0 (e.g. centroid) Defuzz(l,m,u) := l+m+u 3 . Step 1 (matrix of theoretical weights).Define the theoretical-weight matrixT (p) = ( ̃ t (p) ij )∈(TFN ≥0 ) m×n by ̃ t (p) ij := (p i w j , p i w j , p i w j ), i= 1,...,m, j= 1,...,n. (This is the “theoretical share” of criterionjallocated to alternativei.) Step 2 (criterion-wise reference bounds).For each criterionj, set the crisp bounds from TFN end- points x − j :=min 1≤i≤m l ij , x + j :=max 1≤i≤m u ij ,∆ j :=x + j −x − j >0. Step 3 (fuzzy normalization factors).Define normalized TFNs ̃r ij ∈TFN ≥0 by ̃r ij := ( l ij −x − j ∆ j , m ij −x − j ∆ j , u ij −x − j ∆ j ) , j∈J + , ( x + j −u ij ∆ j , x + j −m ij ∆ j , x + j −l ij ∆ j ) , j∈J − . (Thus ̃r ij encodes the normalized “real performance” on[0,1]; for cost criteria the order is reversed to keep l≤m≤u.) Step 4 (matrix of real weights).Define the real-weight matrixT (r) = ( ̃ t (r) ij )∈(TFN ≥0 ) m×n by ̃ t (r) ij := (p i w j ) ̃r ij , i= 1,...,m, j= 1,...,n. Step 5 (gap matrix).Define the fuzzy gap matrix ̃ G= ( ̃g ij )by ̃g ij := ̃ t (p) ij ̃ t (r) ij , i= 1,...,m, j= 1,...,n. Step 6 (criterion function and ranking).For each alternativeA i , define its fuzzy criterion function ̃ Q i := n ⊕ j=1 ̃g ij ∈TFN ≥0 , Q i :=Defuzz( ̃ Q i )∈R ≥0 . TheFuzzy MAIRCA rankingis the preorder onAgiven by A i FMAIRCA A k ⇐⇒Q i ≤Q k , i.e.,smallerQ i means a smaller ideal–real gap and hence a better alternative. A best alternative is any A ? ∈arg min A i ∈A Q i . Chapter 7. Distance-to-reference / border / compromise-index methods The definition of Uncertain MAIRCA of typeM(U-MAIRCA) is given below. Definition 7.7.2(Uncertain MAIRCA of typeM(U-MAIRCA)).LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria, withm,n≥2. LetJ + andJ − be the index sets of benefit and cost criteria, withJ + ̇ ∪J − =1,...,n. Fix an uncertain modelMwith Dom(M)6=∅and an admissible positive scoreS M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M) (i= 1,...,m;j= 1,...,n). Letw= (w 1 ,...,w n )be criterion weights withw j ≥0and ∑ n j=1 w j = 1. Fix a probability vector p= (p 1 ,...,p m )withp i ≥0and ∑ m i=1 p i = 1(typicallyp i = 1/m). Step 0 (Crisp projection).Define the positive real matrixY= (y ij )by y ij :=S M ( x (M) ij ) ∈(0,∞). Step 1 (Theoretical weight distribution).Define the theoretical distribution matrixT (p) = (t (p) ij )∈ R m×n ≥0 by t (p) ij :=p i w j , i= 1,...,m, j= 1,...,n. Step 2 (Min–max normalization of real performances).For each criterionj, define y min j :=min 1≤i≤m y ij , y max j :=max 1≤i≤m y ij ,∆ j :=y max j −y min j ≥0. Define normalized valuesr ij ∈[0,1]by r ij := y ij −y min j ∆ j , j∈J + and∆ j >0, y max j −y ij ∆ j , j∈J − and∆ j >0, 0,∆ j = 0. (Thus largerr ij always means better; degenerate criteria with∆ j = 0contribute0.) Step 3 (Real distribution matrix).Define the real distribution matrixT (r) = (t (r) ij )by t (r) ij :=t (p) ij r ij =p i w j r ij . Step 4 (Gap matrix and criterion function).Define the gap matrixG= (g ij )by g ij :=t (p) ij −t (r) ij =p i w j (1−r ij )≥0. Chapter 7. Distance-to-reference / border / compromise-index methods For each alternativeA i , define its criterion function (overall gap) Q i := n ∑ j=1 g ij = n ∑ j=1 p i w j (1−r ij )≥0. Ranking rule:rank alternatives by ascendingQ i (smaller gap is better). Theorem 7.7.3(Well-definedness of U-MAIRCA).Under Definition 7.7.2, assumem,n≥2,Dom(M)6=∅, andS M :Dom(M)→(0,∞)is admissible. Then: (i) All quantitiesy ij ,t (p) ij ,r ij ,t (r) ij ,g ij , andQ i are well-defined and finite. (i) For alli,j, one has0≤r ij ≤1, henceg ij ≥0andQ i ≥0. (i) The ranking induced byQ i is well-defined (ties allowed). Proof.(i) Admissibility ofS M impliesy ij >0finite. Sincep i ,w j are finite and nonnegative,t (p) ij =p i w j is finite. For eachj,y min j ,y max j exist (finite index set) and∆ j ≥0is finite. If∆ j >0, the min–max formulas definer ij ; if∆ j = 0, the definition setsr ij = 0. Thusr ij is well-defined in all cases and finite. Then t (r) ij =t (p) ij r ij andg ij =t (p) ij −t (r) ij are finite, henceQ i = ∑ j g ij is finite. (i) When∆ j >0, min–max normalization yieldsr ij ∈[0,1]; when∆ j = 0,r ij = 0∈[0,1]. Therefore 1−r ij ∈[0,1], sog ij =p i w j (1−r ij )≥0andQ i ≥0. (i) EachQ i is a real number, so sorting alternatives byQ i defines a well-defined preorder. Related concepts of MAIRCA under uncertainty-aware models are listed in Table 7.8. Table 7.8: Related concepts of MAIRCA under uncertainty-aware models. kRelated MAIRCA concept(s) 1 Fuzzy MAIRCA 2 Intuitionistic Fuzzy MAIRCA [1065,1066] 3 Neutrosophic MAIRCA Chapter 8 Outranking (non-compensatory / dominance rela- tions) Decision Methods Outranking methods compare alternatives pairwise using concordance/discordance with thresholds and veto, yielding non-compensatory dominance relations where strong drawbacks cannot be offset by advantages. For convenience, a concise comparison of the representative outranking methods discussed in this chapter is presented in Table 8.1. Table 8.1: A concise comparison of representative outranking decision methods. Method Core comparison basis Main mechanismPrimary output Decision form ELEC- TRE Pairwise comparison of alternatives over concordant and dis- cordant criteria Builds concordance and discor- dance indices, applies thresh- olds (and possibly veto logic), and derives an outranking rela- tion. Outranking relation and kernel Partial dom- inance / choice set FlowSort Comparison of each alternative with or- dered limiting refer- ence profiles Computes preference flows against boundary profiles and assigns the alternative to the category whose profile interval contains its flow. Category as- signment Sorting / ordered clas- sification PROMETHEE I Pairwise criterionwise preference degrees between alternatives Aggregates preference functions with criterion weights and com- putes positive, negative, and net outranking flows. Net flowComplete ranking QUAL- IFLEX Pairwise concordance of alternatives under each possible permu- tation Evaluates all admissible per- mutations, aggregates weighted concordance/discordance over pairs, and selects the permuta- tion with maximal comprehen- sive concordance. Best permuta- tion score Complete ranking Continued on the next page. 277 Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods Table 8.1 (continued). Method Core comparison basis Main mechanismPrimary output Decision form ORESTE Ordinal positions and distance-based prefer- ence intensities Uses ordinal ranks of perfor- mances and criterion impor- tance, computes distance-based scores and preference inten- sities, and derives weak and PIR-type relations. Distance score and PIR rela- tions Weak rank- ing / out- ranking structure Note.All of these methods belong to the outranking family, but they differ in their decision logic. ELECTRE emphasizes non-compensatory dominance through concordance/discordance thresholds, FlowSort is primarily a sorting procedure based on reference profiles, PROMETHEE I yields a complete ranking through net preference flows, QUALIFLEX selects the best global permutation of alternatives, and ORESTE is especially suited to ordinal-information settings with weak preference, indifference, and incomparability-type structures. 8.1 Fuzzy ELECTRE (Fuzzy Elimination and choice translating reality) ELECTRE is an outranking MCDM method that builds concordance/discordance indices, applies thresholds and veto, constructs an outranking relation, and eliminates dominated alternatives [1067, 1068]. Fuzzy ELECTRE is an outranking MCDM method using fuzzy evaluations to compute concordance/discordance and identify alternatives that dominate others under uncertainty [1069,1070]. Definition 8.1.1(TFN-based Fuzzy ELECTRE).[1071,1072] LetA=A 1 ,...,A m be alternatives and C=C 1 ,...,C n criteria. LetBandKbe the index sets of benefit and cost criteria, withB ̇ ∪K= 1,...,n. Assume a crisp decision matrixX= (x ij )∈R m×n >0 and TFN criterion weights ̃w j = (l j ,m j ,u j ). (0) Conventions.For TFNs, use componentwise addition and scalar multiplication: (l,m,u)⊕(l ′ ,m ′ ,u ′ ) = (l+l ′ , m+m ′ , u+u ′ ), α (l,m,u) = (αl, αm, αu) (α≥0). (1) Aggregate and normalize fuzzy weights (three components).IfKdecision makers provide numeric importance scoresy jk for criterionj, set ̃w j = (l j ,m j ,u j ) := ( min k y jk , 1 K K ∑ k=1 y jk ,max k y jk ) . Normalize each component reciprocally: w j1 := (1/l j ) ∑ n t=1 (1/l t ) , w j2 := (1/m j ) ∑ n t=1 (1/m t ) , w j3 := (1/u j ) ∑ n t=1 (1/u t ) . Letw (z) = (w 1z ,...,w nz )forz= 1,2,3. Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods (2) Normalize performances and build three weighted matrices.Define the normalized matrix R= (r ij )by r ij := x ij √ ∑ m p=1 x 2 pj ,j∈B, (1/x ij ) √ ∑ m p=1 (1/x pj ) 2 , j∈K, and forz∈1,2,3define v (z) ij :=r ij w jz , V (z) = (v (z) ij ) m×n . (3) Concordance/discordance sets and indices.Forp6=qandz∈1,2,3define C (z) (p,q) :=j:v (z) pj ≥v (z) qj , D (z) (p,q) :=1,...,n (z) (p,q), C (z) pq := ∑ j∈C (z) (p,q) w jz , D (z) pq := ∑ j∈D (z) (p,q) |v (z) pj −v (z) qj | ∑ n j=1 |v (z) pj −v (z) qj | . Defuzzify by summing components: C ∗ pq := 3 ∑ z=1 C (z) pq , D ∗ pq := 3 ∑ z=1 D (z) pq . (4) Thresholds and outranking.Let ̄ C:= 1 m(m−1) m ∑ p,q=1 p6=q C ∗ pq , ̄ D:= 1 m(m−1) m ∑ p,q=1 p6=q D ∗ pq . Define the fuzzy ELECTRE outranking relation by A p < FE A q ⇐⇒ ( C ∗ pq ≥ ̄ C ) ∧ ( D ∗ pq ≤ ̄ D ) ,(p6=q). (5) Dominance matrix and kernel (best set).Define e pq :=1[C ∗ pq ≥ ̄ C], f pq :=1[D ∗ pq ≤ ̄ D], T:=E◦F, so thatt pq = 1iffA p < FE A q . The kernel is K:=A p ∈A:@q6=pwitht qp = 1. Using an Uncertain Set as the underlying extension framework, we define Uncertain ELECTRE of typeM (U-ELECTRE) as follows. Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods Definition 8.1.2(Uncertain ELECTRE of typeM(U-ELECTRE)).LetA=A 1 ,...,A m be alternatives andC=C 1 ,...,C n criteria, withm≥2andn≥1. LetBandKbe index sets of benefit and cost criteria withB ̇ ∪K=1,...,n. Fix an uncertain modelMwith Dom(M)6=∅and an admissible score S M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M), and criterion weightsw= (w 1 ,...,w n )withw j ≥0and ∑ n j=1 w j = 1. (If uncertain weights are provided, score and normalize them to obtain suchw j .) Step 0 (Crisp projection and performance orientation).Definey ij :=S M (x (M) ij )∈R. Convert all criteria to a “larger is better” orientation by defining z ij := y ij , j∈B(benefit), −y ij , j∈K(cost). LetZ= (z ij ). Step 1 (Normalization).For each criterionj, define d j := √ √ √ √ m ∑ p=1 z 2 pj ≥0, r ij := z ij d j , d j >0, 0, d j = 0, and the weighted normalized matrix v ij :=w j r ij . Step 2 (Concordance and discordance sets).For each ordered pair(p,q)withp6=q, define C(p,q) :=j:v pj ≥v qj ,D(p,q) :=1,...,n (p,q). Step 3 (Concordance index).Define the concordance index C pq := ∑ j∈C(p,q) w j ∈[0,1]. Step 4 (Discordance index).Define the discordance index by D pq := max j∈D(p,q) |v pj −v qj | max 1≤j≤n |v pj −v qj | ,max 1≤j≤n |v pj −v qj |>0, 0,max 1≤j≤n |v pj −v qj |= 0, (p6=q). Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods (ThusD pq ∈[0,1].) Step 5 (Thresholds and outranking relation).Define average thresholds ̄ C:= 1 m(m−1) m ∑ p,q=1 p6=q C pq , ̄ D:= 1 m(m−1) m ∑ p,q=1 p6=q D pq . Define the outranking relation< UE by A p < UE A q ⇐⇒ ( C pq ≥ ̄ C ) ∧ ( D pq ≤ ̄ D ) ,(p6=q). Step 6 (Kernel / best set).Lett pq :=1[A p < UE A q ]and define the kernel K:=A p ∈A:@q6=pwitht qp = 1. Theorem 8.1.3(Well-definedness of U-ELECTRE).Under Definition 8.1.2, assumem≥2,Dom(M)6=∅, andS M is admissible. Then all quantitiesz ij ,d j ,r ij ,v ij , the setsC(p,q),D(p,q), the indicesC pq ,D pq , the thresholds ̄ C, ̄ D, and the kernelKare well-defined. Moreover, 0≤C pq ≤1,0≤D pq ≤1. Proof.SinceS M is admissible, eachy ij is finite, hencez ij is finite. Thus eachd j = √ ∑ m p=1 z 2 pj is well-defined and finite, andr ij is well-defined by case distinction (d j >0yields a valid division;d j = 0setsr ij = 0). Thenv ij =w j r ij is well-defined. For each pair(p,q), the setC(p,q)is determined by finitely many comparisonsv pj ≥v qj , hence is well- defined, and so isD(p,q). Becausew j ≥0and ∑ j w j = 1, the concordance indexC pq = ∑ j∈C(p,q) w j is well-defined and lies in[0,1]. For discordance, the quantities|v pj −v qj |are finite. If their maximum overjis positive, then the ratio definingD pq is valid and lies in[0,1]because the numerator is a maximum over a subset of indices. If the maximum is0, thenv pj =v qj for alljand the definition setsD pq = 0∈[0,1]. HenceD pq is always well-defined and in[0,1]. The thresholds ̄ Cand ̄ Dare averages over finitely many well-defined indices, hence well-defined. Therefore the outranking relation is well-defined, as is the kernelKdefined by a finite quantification overq6=p. Related concepts of ELECTRE under uncertainty-aware models are listed in Table 8.2. In addition to Uncertain ELECTRE, several other extensions are also known, such as ELECTRE TRI [1084,1085], Rough ELECTRE [1086,1087], Extended ELECTRE [1088,1089], ELECTRE IS [1090], Group- ELECTRE [1091, 1092], ELECTRE TRI-nC [1093, 1094], Grey ELECTRE [1095], and Soft ELECTRE [1096,1097]. Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods Table 8.2: Related concepts of ELECTRE under uncertainty-aware models. kRelated ELECTRE concept(s) 2 Intuitionistic Fuzzy ELECTRE [1073,1074] 2 Pythagorean Fuzzy ELECTRE [1075] 2 Fermatean Fuzzy ELECTRE [1076,1077] 3 Neutrosophic ELECTRE [1078–1080] 3 Spherical Fuzzy ELECTRE [1081] 3 Picture Fuzzy ELECTRE [1082] 3 Hesitant Fuzzy ELECTRE [1083] 8.2 Fuzzy FlowSort FlowSort classifies alternatives into ordered categories using PROMETHEE outranking flows versus ref- erence profiles, applying assignment rules by flows [1098, 1099]. Fuzzy FlowSort extends FlowSort to fuzzy evaluations and weights, computing fuzzy outranking degrees, defuzzified flows, then assigning cate- gories [1100,1101]. Definition 8.2.1(Fuzzy FlowSort (F-FlowSort) with limiting profiles).[1100,1101] LetA=a 1 ,...,a m be a finite set of alternatives andG=g 1 ,...,g n a finite set of criteria. AssumeK≥2ordered categories C=C 1 C 2 ·C K , described by(K+ 1)limiting reference profiles R=r 1 ,...,r K+1 (intended order:r 1 r 2 ·r K+1 ), where categoryC k is bounded (in performance) betweenr k (upper) andr k+1 (lower). Fuzzy performance data.For each criteriong j and eachx∈A∪R, the evaluation is a triangular fuzzy number (TFN) ̃g j (x) = (` j (x), m j (x), u j (x)), ` j (x)≤m j (x)≤u j (x), and the criterion weight may be crisp (w j ∈[0,1]) or fuzzy ( ̃w j ), with the standard normalization condition ∑ n j=1 w j = 1(or an agreed defuzzified normalization if ̃w j are used). Fuzzy preference modelling.For each criterionj, fix a (PROMETHEE-type) unicriterion preference function P j :R→[0,1] (e.g., usual, U-shape, V-shape, level, Gaussian), together with any thresholds (indifference, preference) needed by that type. Forx,y∈A∪Rdefine thefuzzy deviation ̃ ∆ j (x,y) := ̃g j (x) ̃g j (y) (using TFN subtraction), and define thefuzzy unicriterion preference ̃ P j (x,y)by applyingP j to ̃ ∆ j (x,y) via the chosen TFN calculus (e.g., interval/vertex evaluation or LR-form). Aggregated (fuzzy) outranking degree.Forx,y∈A∪Rdefine the aggregated fuzzy preference ̃π(x,y) := n ∑ j=1 w j ⊗ ̃ P j (x,y), Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods where⊗is scalar–TFN multiplication (or TFN multiplication if ̃w j are used). Choose a defuzzification operator Def:TFN→R (e.g., Yager-type, centroid, mean of maxima) and set thecrispoutranking degree π d (x,y) :=Def ( ̃π(x,y) ) ∈[0,1]. Local comparison set and PROMETHEE-like flows.For each alternativea i ∈Adefine the local set R ∗ i :=a i ∪R. For anyx∈R ∗ i define itspositive,negative, andnetflows (withinR ∗ i ) by φ + i (x) := 1 |R ∗ i |−1 ∑ y∈R ∗ i y6=x π d (x,y), φ − i (x) := 1 |R ∗ i |−1 ∑ y∈R ∗ i y6=x π d (y,x), φ i (x) :=φ + i (x)−φ − i (x). Category assignment rules.For eacha i ∈Aand eachk∈1,...,Kdefine: (R + ) (positive-flow rule)C φ + (a i ) =C k if φ + i (r k )> φ + i (a i )≥φ + i (r k+1 ). (R − ) (negative-flow rule)C φ − (a i ) =C k if φ − i (r k )≤φ − i (a i )< φ − i (r k+1 ). (R)(net-flow rule)C φ (a i ) =C k if φ i (r k )> φ i (a i )≥φ i (r k+1 ). AFuzzy FlowSort classificationis any mappingσ:A → Cobtained by choosing one of the rules (R + ), (R − ), (R), or by a deterministic fusion policy, e.g. σ(a i ) =C k ∗ wherek ∗ =Mode k + ,k − ,k withC φ + (a i ) =C k + ,C φ − (a i ) =C k − ,C φ (a i ) =C k , and a fixed tie-breaking convention (e.g., choose the worst index among ties). Proposition 8.2.2(Basic well-definedness).Assumeπ d (x,y)∈[0,1]for all relevant(x,y)and letN i := |R ∗ i |. Thenφ + i (x),φ − i (x)∈[0,1]andφ i (x)∈[−1,1]for allx∈ R ∗ i . Moreover, if for a fixed rule (R + ) (resp. (R − ), (R)) the profile flows satisfy φ + i (r 1 )> φ + i (r 2 )>·> φ + i (r K+1 )(resp. similarly forφ − i orφ i ), then the corresponding assignment produces a unique category for eacha i . Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods Proof.Sinceπ d (·,·)∈[0,1], each flow is an average of numbers in[0,1], hence lies in[0,1]. The net flow is a difference of two[0,1]numbers, hence in[−1,1]. Strict monotonicity of profile flows yields a partition of the real line into disjoint half-open intervals, so exactly oneksatisfies the relevant inequality. Using an Uncertain Set as the underlying extension framework, we define Uncertain FlowSort of typeM (U-FlowSort) as follows. Definition 8.2.3(Uncertain FlowSort of typeM(U-FlowSort) with limiting profiles).LetA=a 1 ,...,a m be a finite set of alternatives andG=g 1 ,...,g n a finite set of criteria, withm≥1andn≥1. FixK≥2 ordered categories C=C 1 C 2 ·C K , described by(K+ 1)limiting reference profiles R=r 1 ,...,r K+1 (intended order:r 1 r 2 ·r K+1 ), where categoryC k is bounded byr k (upper) andr k+1 (lower). Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M . Assume uncertain evaluations x (M) j (x)∈Dom(M)for allx∈A∪R, j= 1,...,n, and crisp criterion weightsw= (w 1 ,...,w n )with w j ≥0, n ∑ j=1 w j = 1. Step 1 (Crisp projection).Define crisp performances y j (x) :=S M ( x (M) j (x) ) ∈R(x∈A∪R, j= 1,...,n). Step 2 (Unicriterion preference functions).For each criterionj, fix a PROMETHEE-type preference function P j :R→[0,1], possibly with thresholds (indifference/preference) implicit in its definition. Forx,y∈ A∪Rdefine the unicriterion preference degree π j (x,y) :=P j ( y j (x)−y j (y) ) ∈[0,1]. Step 3 (Aggregated preference index).Define the aggregated outranking (preference) index π(x,y) := n ∑ j=1 w j π j (x,y)∈[0,1], x,y∈A∪R. Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods Step 4 (Local comparison set and flows).For each alternativea i ∈Adefine the local set S i :=a i ∪R, N i :=|S i |=K+ 2. For anyx∈S i define the positive, negative, and net flows: φ + i (x) := 1 N i −1 ∑ y∈S i y6=x π(x,y), φ − i (x) := 1 N i −1 ∑ y∈S i y6=x π(y,x), φ i (x) :=φ + i (x)−φ − i (x). Step 5 (Assignment rules).For eacha i ∈A, assign a category using one of the following deterministic rules: (R + ) Positive-flow rule:choose the uniquek∈1,...,Ksuch that φ + i (r k )> φ + i (a i )≥φ + i (r k+1 ), and setσ(a i ) :=C k . (R − ) Negative-flow rule:choose the uniquek∈1,...,Ksuch that φ − i (r k )≤φ − i (a i )< φ − i (r k+1 ), and setσ(a i ) :=C k . (R) Net-flow rule:choose the uniquek∈1,...,Ksuch that φ i (r k )> φ i (a i )≥φ i (r k+1 ), and setσ(a i ) :=C k . If uniqueness fails due to ties, adopt any fixed tie-breaking rule (e.g. choose the worst category among eligible ones). The resulting mappingσ:A→Cis called anUncertain FlowSort classification. Theorem 8.2.4(Well-definedness of U-FlowSort).Under Definition 8.2.3, assumeS M is admissible and each preference functionP j mapsRinto[0,1]. Then: (i) The unicriterion indicesπ j (x,y)and aggregated indexπ(x,y)are well-defined and lie in[0,1]. (i) For eachiand eachx∈S i , the flows satisfy 0≤φ + i (x)≤1,0≤φ − i (x)≤1,−1≤φ i (x)≤1. (i) The assignment mappingσ:A→Cis well-defined once a deterministic rule (R + ), (R − ), or (R) and a tie-breaking convention are fixed. Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods (iv) If, for a chosen rule, the corresponding profile flows are strictly decreasing (e.g.φ + i (r 1 )>·> φ + i (r K+1 )for (R + )), then eacha i is assigned to auniquecategory without tie-breaking. Proof.(i) SinceS M is admissible, eachy j (x)is finite, hence each differencey j (x)−y j (y)is finite. By assumptionP j :R→[0,1], soπ j (x,y)is well-defined in[0,1]. Becausew j ≥0and ∑ j w j = 1, the aggregated indexπ(x,y) = ∑ j w j π j (x,y)is a convex combination, hence lies in[0,1]and is well-defined. (i) For fixediandx∈ S i ,φ + i (x)is an average ofN i −1numbers in[0,1], hence lies in[0,1]. The same holds forφ − i (x). Thereforeφ i (x) =φ + i (x)−φ − i (x)∈[−1,1]. (i) With flows well-defined, each assignment rule compares finitely many real numbers. A deterministic rule plus a fixed tie-break produces a unique category, henceσis well-defined. (iv) Under strict decrease of the profile flows, the real line is partitioned into disjoint half-open intervals [ φ(r k+1 ), φ(r k ) ) (or the analogous form), so exactly oneksatisfies the defining inequalities. Related concepts of FlowSort under uncertainty-aware models are listed in Table 8.3. Table 8.3: Related concepts of FlowSort under uncertainty-aware models. kRelated FlowSort concept(s) 1 Fuzzy FlowSort 2 Intuitionistic Fuzzy FlowSort 3 Neutrosophic FlowSort 8.3 Uncertain PROMETHEE (Preference Ranking Organization METhod for Enrich- ment of Evaluations) PROMETHEE ranks alternatives using preference functions on pairwise criterion differences, aggregates weighted preferences, and computes positive, negative, and net outranking flows [1102–1104]. Fuzzy PROMETHEE models ratings and/or weights as fuzzy numbers, defuzzifies or compares them, then applies PROMETHEE preference aggregation and flow calculations under uncertainty [1105–1107]. Definition 8.3.1(Fuzzy PROMETHEE I (TFN inputs + Yager defuzzification)).(cf. [1105–1107]) Let A=a 1 ,...,a m be alternatives and letC=1,...,nbe criteria. Assume each criterionj∈Cis either abenefitcriterion (↑) or acostcriterion (↓). Let ̃ f j (a)∈R 3 be the (triangular) fuzzy performance of alternativea∈Aon criterionj, and let ̃w j ∈R 3 be the (triangular) fuzzy weight of criterionj. Fix preference functions (generalized criteria) p j :R→[0,1] (j= 1,...,n), which convert a performance deviation into a preference degree. (1) Yager defuzzification.For a TFN ̃x= (l,m,u), define its Yager magnitude (center/centroid score) by Yag( ̃x) := l+m+u 3 . Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods Define defuzzified performances g j (a) := Yag ( ̃ f j (a) ) , jis benefit (↑), −Yag ( ̃ f j (a) ) , jis cost (↓), and defuzzified weightsˆw j :=Yag( ̃w j ). Normalize the weights (optional but standard) by w j := ˆw j ∑ n t=1 ˆw t (j= 1,...,n). (2) Pairwise preferences.Fora,b∈A, define the deviation on criterionjby d j (a,b) :=g j (a)−g j (b), the unicriterion preference degree by P j (a,b) :=p j ( d j (a,b) ) ∈[0,1], and the aggregated (global) preference index by π(a,b) := n ∑ j=1 w j P j (a,b)∈[0,1]. (3) Outranking flows and complete ranking (PROMETHEE I).Define the positive, negative, and net flows for eacha∈Aby φ + (a) := 1 m−1 ∑ b∈A b6=a π(a,b), φ − (a) := 1 m−1 ∑ b∈A b6=a π(b,a), φ(a) :=φ + (a)−φ − (a). TheFuzzy PROMETHEE IIranking is the complete preorder onAinduced byφ: ab⇐⇒φ(a)≥φ(b). Using an Uncertain Set as the underlying extension framework, we define Uncertain PROMETHEE I of typeM(U-PROMETHEE) as follows. Definition 8.3.2(Uncertain PROMETHEE I of typeM(U-PROMETHEE)).LetA=a 1 ,...,a m be a finite set of alternatives withm≥2, and letC=1,...,nbe criteria. Partition criteria into benefit and cost setsB,K⊆CwithB ̇ ∪K=C. Fix an uncertain modelMwith Dom(M)6=∅and an admissible score S M . Assume uncertain performancesx (M) j (a)∈Dom(M)for eacha∈Aand criterionj∈C. Letw= (w 1 ,...,w n )be criterion weights with w j ≥0, n ∑ j=1 w j = 1. Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods Fix PROMETHEE preference functions (generalized criteria) p j :R→[0,1] (j= 1,...,n), which convert a criterion deviation into a preference degree. Step 1 (Crisp projection and orientation).For eacha∈Aandj∈C, define g j (a) := S M ( x (M) j (a) ) , j∈B(benefit), −S M ( x (M) j (a) ) , j∈K(cost), so that largerg j (a)always means better. Step 2 (Pairwise unicriterion preferences).Fora,b∈Aandj∈C, define d j (a,b) :=g j (a)−g j (b)∈R, P j (a,b) :=p j ( d j (a,b) ) ∈[0,1]. Step 3 (Aggregated preference index).Define the global preference index π(a,b) := n ∑ j=1 w j P j (a,b)∈[0,1], a,b∈A. Step 4 (PROMETHEE I flows).For eacha∈A, define the positive, negative, and net flows φ + (a) := 1 m−1 ∑ b∈A b6=a π(a,b), φ − (a) := 1 m−1 ∑ b∈A b6=a π(b,a), φ(a) :=φ + (a)−φ − (a). Ranking rule (PROMETHEE I).Define the complete preorder onAby a UP b⇐⇒φ(a)≥φ(b). Theorem 8.3.3(Well-definedness and bounds of U-PROMETHEE).Under Definition 8.3.2, assumem≥ 2,Dom(M)6=∅,S M is admissible, and eachp j mapsRinto[0,1]. Then: (i) For alla,b∈A,π(a,b)is well-defined and lies in[0,1]. (i) For alla∈A, the flows satisfy 0≤φ + (a)≤1,0≤φ − (a)≤1,−1≤φ(a)≤1. (i) The ranking relation UP is well-defined (ties allowed). Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods Proof.(i) Admissibility ofS M implies eachg j (a)is finite, hence each deviationd j (a,b)is finite. Since p j :R→[0,1],P j (a,b)is well-defined in[0,1]. Withw j ≥0and ∑ j w j = 1,π(a,b) = ∑ j w j P j (a,b)is a convex combination, hence lies in[0,1]. (i) For fixeda,φ + (a)is the average ofm−1numbers in[0,1], henceφ + (a)∈[0,1]. Similarly,φ − (a)∈[0,1]. Thereforeφ(a) =φ + (a)−φ − (a)∈[−1,1]. (i) Since eachφ(a)is a real number, sorting alternatives byφ(a)defines a complete preorder (ties allowed), hence UP is well-defined. For reference, related concepts of PROMETHEE under uncertainty-aware models are listed in Table 8.4. Table 8.4: Related concepts of PROMETHEE under uncertainty-aware models. kRelated PROMETHEE concept(s) 1 Fuzzy PROMETHEE 2 Intuitionistic Fuzzy PROMETHEE [1106,1108] 2 Bipolar fuzzy PROMETHEE [1109,1110] 2 Pythagorean fuzzy PROMETHEE [1111,1112] 2 Fermatean fuzzy PROMETHEE [1113,1114] 3 Picture Fuzzy PROMETHEE [1115,1116] 3 Hesitant Fuzzy PROMETHEE [503,1117] 3 Spherical Fuzzy PROMETHEE [1118,1119] 3 Neutrosophic PROMETHEE [1120,1121] Linguistic PROMETHEE [1122,1123], Soft PROMETHEE [1112,1124], Grey PROMETHEE [1125,1126], Group-PROMETHEE [1127, 1128], Extended PROMETHEE [1129, 1130], PROMETHEE-GAIA [1113, 1131], SMAA-PROMETHEE [1132,1133], PROMETHEE-MD-2T [1134], and Rough PROMETHEE [1135, 1136] are also known as related variants. 8.4 Fuzzy QUALIFLEX QUALIFLEX ranks alternatives by evaluating all permutations, aggregating concordance of pairwise cri- terion preferences, selecting best ordering [1137, 1138]. Fuzzy QUALIFLEX encodes preferences with fuzzy numbers or linguistic terms, aggregates fuzzy concordance across permutations, yielding robust rank- ing [1139,1140]. Definition 8.4.1(Fuzzy QUALIFLEX (permutation-based outranking)).[1139,1140] LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria. LetFbe a chosen family of fuzzy evaluations (e.g. triangular/trapezoidal fuzzy numbers, hesitant fuzzy numbers, etc.). Assume a fuzzy decision matrix ̃ X= ( ̃x ij )∈F m×n , ̃x ij ∈Fis the evaluation ofA i underC j . Letw= (w 1 ,...,w n )∈[0,1] n be criterion weights with ∑ n j=1 w j = 1. For each criterionC j , fix: • anidealfuzzy value ̃u + j ∈F(best attainable onC j ; for cost criteria, choose the minimum-type ideal); Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods • a real-valuedcloseness/score functional r j :F→R, r j ( ̃x) :=d j ( ̃x, ̃u + j ), whered j is a distance (or dissimilarity) onF. (Thus, smallerr j ( ̃x)means “closer to ideal” on criterionC j .) LetS m denote the set of all permutations of1,...,m. Eachπ∈S m represents a candidate complete ranking P π = ( A π(1) ,A π(2) ,...,A π(m) ) ,whereA π(p) is ranked not worse thanA π(q) ifp < q. For any ordered pair(p,q)with1≤p < q≤m, define thecriterion-wise concordance/discordance indexby φ π j ( A π(p) ,A π(q) ) :=r j ( ̃x π(q)j ) −r j ( ̃x π(p)j ) . Hence: φ π j >0⇒concordance (the higher-ranked alternative is closer to ideal), φ π j = 0⇒ex aequo, φ π j <0⇒discordance. Aggregate across criteria (weighted) for each pair: φ π ( A π(p) ,A π(q) ) := n ∑ j=1 w j φ π j ( A π(p) ,A π(q) ) . Define thecomprehensive concordance/discordance indexof the permutationP π by Φ(π) := ∑ 1≤p<q≤m φ π ( A π(p) ,A π(q) ) . AFuzzy QUALIFLEX solutionis any permutationπ ? ∈S m achieving π ? ∈arg max π∈S m Φ(π), and the induced ranking A π ? (1) A π ? (2) ·A π ? (m) . Using an Uncertain Set, we define Uncertain QUALIFLEX of typeM(U-QUALIFLEX) as follows. Definition 8.4.2(Uncertain QUALIFLEX of typeM(U-QUALIFLEX)).LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria, withm≥2andn≥1. Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M . Assume anuncertain decision matrix X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M), and criterion weightsw= (w 1 ,...,w n )withw j ≥0and ∑ n j=1 w j = 1. Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods Step 0 (Crisp projection and orientation).Definey ij :=S M (x (M) ij )∈Rand convert all criteria to a “larger is better” form: z ij := y ij , C j ∈C ben , −y ij , C j ∈C cost . LetZ= (z ij )∈R m×n . Permutation set.LetS m be the set of all permutations of1,...,m. Eachπ∈S m encodes a candidate complete ranking P π = ( A π(1) ,A π(2) ,...,A π(m) ) ,wherep < q⇒A π(p) is ranked not worse thanA π(q) . Step 1 (Pairwise concordance contribution under a permutation).For any permutationπ∈S m and any ordered pair(p,q)with1≤p < q≤m, define the criterionwise concordance contribution φ π j ( A π(p) ,A π(q) ) :=z π(p)j −z π(q)j ∈R, j= 1,...,n. Thusφ π j >0means the higher-ranked alternative is better on criterionj. Step 2 (Weighted pairwise concordance index).Define the weighted pairwise concordance index φ π ( A π(p) ,A π(q) ) := n ∑ j=1 w j φ π j ( A π(p) ,A π(q) ) ∈R. Step 3 (Comprehensive concordance index of a permutation).Define the comprehensive concor- dance of permutationπby Φ(π) := ∑ 1≤p<q≤m φ π ( A π(p) ,A π(q) ) ∈R. Step 4 (Optimal permutation and ranking).Any maximizer π ? ∈arg max π∈S m Φ(π) is called anUncertain QUALIFLEX solution, and the induced ranking is A π ? (1) UQF A π ? (2) UQF · UQF A π ? (m) . Theorem 8.4.3(Well-definedness of U-QUALIFLEX).Under Definition 8.4.2, assumem≥2,Dom(M)6= ∅, andS M is admissible. Then: (i) For everyπ∈S m , the valueΦ(π)is a well-defined finite real number. (i) The maximizer setarg max π∈S m Φ(π)is nonempty, hence an optimal permutationπ ? exists. (i) The induced ranking byπ ? is well-defined (ties correspond to multiple maximizers). Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods Proof.(i) Admissibility ofS M implies eachy ij is finite; hence each oriented valuez ij is finite. For anyπ and any(p,q), eachφ π j =z π(p)j −z π(q)j is a finite real. Sincew j ≥0and ∑ j w j = 1,φ π = ∑ j w j φ π j is finite, andΦ(π)is a finite sum over the ( m 2 ) pairs, hence finite. (i) The setS m is finite with|S m |=m!. Therefore the finite setΦ(π) :π∈S m ⊂Rattains a maximum, so arg max π∈S m Φ(π)6=∅. (i) Any maximizerπ ? defines an ordering of the alternatives, hence a ranking relation. If multiple maxi- mizers exist, multiple optimal rankings are possible; selecting anyπ ? yields a well-defined ranking. As related concepts, QUALIFLEX variants under uncertainty-aware models are listed in Table 8.5. Table 8.5: Related concepts of QUALIFLEX under uncertainty-aware models. kRelated QUALIFLEX concept(s) 2 Intuitionistic Fuzzy QUALIFLEX [1141] 3 Hesitant Fuzzy QUALIFLEX [1139,1140] 3 Neutrosophic QUALIFLEX [1142–1144] 8.5 Fuzzy ORESTE (Fuzzy Organization Rangement Et Synthese De Donnees Rela- tionnelles) ORESTE derives rankings from ordinal information using preference intensities and distance-based aggre- gation, suitable when precise data unavailable [1145, 1146]. Fuzzy ORESTE represents ordinal judgments as fuzzy ranks, aggregates fuzzy preference distances, producing rankings under vague, imprecise assess- ments [76,1147,1148]. Definition 8.5.1(Fuzzy ORESTE (distance-based outranking)).[76,1147,1148] LetA=a 1 ,...,a m be a finite set of alternatives andC=c 1 ,...,c n a finite set of criteria, partitioned asC=C ben tC cost (benefit vs. cost). LetFdenote a chosen family of fuzzy numbers onR(e.g. triangular, trapezoidal, or general normal convex fuzzy numbers). Afuzzy decision matrixis ̃ X= ( ̃x ij )∈F m×n , ̃x ij ∈Fencodes the (fuzzy) performance ofa i onc j . Afuzzy importance profileis ̃w= ( ̃w 1 ,..., ̃w n )∈F n , ̃w j ∈Fencodes the (fuzzy) importance ofc j . Assume: (A1)A ranking functional Rank:F→Rthat induces a total preorder onF(used to define max/min of fuzzy numbers by comparing Rank values). (A2)A (normalized) distanced F :F×F→[0,1](any metric/pseudometric suitable forF, e.g. viaα-cuts). Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods Step 1 (Ideal points).Define the criterion-wise fuzzy ideal value ̃x + j ∈Fby ̃x + j := arg max i∈1,...,m Rank( ̃x ij ), c j ∈C ben , arg min i∈1,...,m Rank( ̃x ij ), c j ∈C cost . Also define the most important fuzzy weight ̃w + :=arg max j∈1,...,n Rank( ̃w j ). Step 2 (Distances / coordinates).For each(i,j)set d ij :=d F ( ̃x ij , ̃x + j ) ∈[0,1], d j :=d F ( ̃w j , ̃w + ) ∈[0,1]. Interpret(d ij ,d j )as the coordinate of the pair “(a i ,c j )” relative to the ideal. Step 3 (Global preference score).Fix a tradeoff parameterξ∈[0,1]. Theglobal preference scoreofa i w.r.t.c j is D ij := ( ξ d 2 ij + (1−ξ)d 2 j ) 1/2 ∈[0,1], wheresmallerD ij means closer to the ideal and thus better. Theaverage preference scoreofa i is D i := 1 n n ∑ j=1 D ij ∈[0,1]. Theweak ranking(preorder) is induced byD i : a i a k ⇐⇒D i < D k , a i ∼a k ⇐⇒D i =D k . Step 4 (Preference intensities).For paired comparison(a i ,a k )define the criterion-wise preference intensity T j (a i ,a k ) :=maxD kj −D ij ,0∈[0,1], the average preference intensity T(a i ,a k ) := 1 n n ∑ j=1 T j (a i ,a k )∈[0,1], and the net preference intensity ∆T(a i ,a k ) :=T(a i ,a k )−T(a k ,a i )∈[−1,1]. Step 5 (P/I/R test and strong ranking).Fix thresholdsμ∈(0,1](preference threshold) andσ∈(0,1] (indifference bound). Define the PIR relation betweena i anda k by: a i I a k ⇐⇒ ∣ ∣ ∆T(a i ,a k ) ∣ ∣ < μandT(a i ,a k )< σandT(a k ,a i )< σ, Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods a i Ra k ⇐⇒ ∣ ∣ ∆T(a i ,a k ) ∣ ∣ < μand maxT(a i ,a k ),T(a k ,a i )≥σ, a i P a k ⇐⇒ ∣ ∣ ∆T(a i ,a k ) ∣ ∣ ≥μand∆T(a i ,a k )>0. (The case ∣ ∣ ∆T(a i ,a k ) ∣ ∣ ≥μand∆T(a i ,a k )<0yieldsa k Pa i .) AFuzzy ORESTE outputis the weak ranking induced by(D i ) m i=1 together with the PIR structure(P,I,R) onA×A. Abestalternative (distance-to-ideal optimum) is any a ? ∈arg min a i ∈A D i . Using an Uncertain Set, we define Uncertain ORESTE of typeM(U-ORESTE) as follows. Definition 8.5.2(Uncertain ORESTE of typeM(U-ORESTE)).LetA=a 1 ,...,a m be a finite set of alternatives andC=c 1 ,...,c n a finite set of criteria, withm,n≥2. Partition criteria into benefit and cost sets: C=C ben ̇ ∪C cost . Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M . Assume uncertain performancesx (M) ij ∈Dom(M)for each(i,j)and uncertain criterion-importance degrees w (M) j ∈Dom(M)for eachj. Fix an admissible scoreS M and define crisp performances and importance scores y ij := S M (x (M) ij ), c j ∈C ben , −S M (x (M) ij ), c j ∈C cost , β j :=S M (w (M) j ). Thus largery ij is better and largerβ j means more important. Step 1 (Ordinal ranks from projected scores).Define the performance rank of alternativea i under criterionc j by r ij := 1 + ∣ ∣ t∈1,...,m:y tj > y ij ∣ ∣ ∈1,...,m, and define the importance rank of criterionc j by s j := 1 + ∣ ∣ t∈1,...,n:β t > β j ∣ ∣ ∈1,...,n. (Thus smallerr ij and smallers j indicate higher desirability/importance; ties allowed.) Step 2 (Besson–rank distance / global preference coordinates).Fix a tradeoff parameterξ∈[0,1]. Define the ORESTE global distance score D ij := √ ξ ( r ij m ) 2 + (1−ξ) ( s j n ) 2 ∈ [ 0, √ ξ+ (1−ξ) ] = [0,1], and the average distance (global score) of alternativea i : D i := 1 n n ∑ j=1 D ij ∈[0,1]. Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods Step 3 (Weak ranking).Define the weak preorder onAby a i UO a k ⇐⇒D i ≤D k (smaller distance is better). Step 4 (Preference intensity and PIR structure).Fora i ,a k ∈ Adefine the criterionwise preference intensity T j (a i ,a k ) :=maxD kj −D ij ,0∈[0,1], the average preference intensity T(a i ,a k ) := 1 n n ∑ j=1 T j (a i ,a k )∈[0,1], and the net intensity ∆T(a i ,a k ) :=T(a i ,a k )−T(a k ,a i )∈[−1,1]. Fix thresholdsμ∈(0,1]andσ∈(0,1]and define relationsP,I,Rby a i I a k ⇐⇒ |∆T(a i ,a k )|< μandT(a i ,a k )< σandT(a k ,a i )< σ, a i Ra k ⇐⇒ |∆T(a i ,a k )|< μand maxT(a i ,a k ),T(a k ,a i )≥σ, a i P a k ⇐⇒ |∆T(a i ,a k )|≥μand∆T(a i ,a k )>0, (witha k Pa i when|∆T(a i ,a k )|≥μand∆T(a i ,a k )<0). Theorem 8.5.3(Well-definedness of U-ORESTE).Under Definition 8.5.2, assumem,n≥2,Dom(M)6=∅, andS M is admissible. Then: (i) The ranksr ij ∈1,...,mands j ∈1,...,nare well-defined (ties allowed). (i)D ij andD i are well-defined and satisfy0≤D ij ≤1and0≤D i ≤1. (i) For alla i ,a k ,T j (a i ,a k ),T(a i ,a k )∈[0,1]and∆T(a i ,a k )∈[−1,1]are well-defined. (iv) The weak ranking UO and the PIR relations(P,I,R)are well-defined. Proof.(i) Since ally ij andβ j are finite real numbers, the setst:y tj > y ij andt:β t > β j are finite and have well-defined cardinalities. Thereforer ij = 1 +|t:y tj > y ij | ∈ 1,...,mands j = 1 +|t: β t > β j |∈1,...,n. (i) Because1≤r ij ≤mand1≤s j ≤n, one has0< r ij /m≤1and0< s j /n≤1. Withξ∈[0,1], the expression under the square root inD ij lies in[0,1], henceD ij ∈[0,1]. ThenD i is an average ofnnumbers in[0,1], soD i ∈[0,1]. (i) SinceD ij ∈[0,1], we haveD kj −D ij ∈[−1,1], henceT j (a i ,a k ) =maxD kj −D ij ,0∈[0,1]. Averaging yieldsT(a i ,a k )∈[0,1]. Therefore∆T(a i ,a k )is a difference of two numbers in[0,1], so it lies in[−1,1]. (iv) The weak ranking uses comparisons of real numbersD i , hence is well-defined. The relationsP,I,Rare defined by finite inequalities involving∆TandTand fixed thresholdsμ,σ, so they are well-defined. As related concepts, ORESTE variants under uncertainty-aware models are listed in Table 8.6. Chapter 8. Outranking (non-compensatory / dominance relations) Decision Methods Table 8.6: Related concepts of ORESTE under uncertainty-aware models. kRelated ORESTE concept(s) 2 Intuitionistic Fuzzy ORESTE [1149] 3 Hesitant Fuzzy ORESTE [1150] 3 Spherical Fuzzy ORESTE [1151] 3 Neutrosophic ORESTE [1152,1153] Chapter 9 Rule induction / learning / evidence / sequential decision methods This section describes decision-making methods from the perspectives of rule induction, learning, evidence aggregation, and sequential decision-making. For convenience, the main concepts introduced in this chapter are compared in Table 9.1. Table 9.1: Concise comparison of representative rule-induction, learning, evidence-aggregation, and sequential-decision methods. Method Type Basic input structure Core mechanismPrimary output Decision role Decision Tree FuzzyFuzzy-valued at- tributes, fuzzy partitions, or soft split functions Recursively partitions the instance space by fuzzy tests and prop- agates memberships through branches. Predicted class, value, or leaf-based membership Rule induc- tion / classi- fication Decision Tree Uncer- tain Uncertain-valued features scored by an admissible map intoR Converts uncertain fea- tures to crisp scores, then performs hard or soft tree splitting on measurable tests. Deterministic prediction or aggregated soft output Rule induc- tion / classi- fication DRSAFuzzyOrdered decision classes and fuzzy dominance rela- tions on criteria Builds fuzzy lower and upper approximations of class unions through fuzzy dominance and fuzzy quantifiers. Fuzzy approx- imations and graded deci- sion rules Preference rule induc- tion DRSAUncer- tain Uncertain-valued decision table pro- jected to crisp cri- terion scores Defines crisp dominance cones from scored un- certain data and com- putes lower/upper ap- proximations of class unions. Approxi- mation sys- tem and dominance- based rules Preference rule induc- tion Continued on the next page. 297 Chapter 9. Rule induction / learning / evidence / sequential decision methods Table 9.1 (continued). Method Type Basic input structure Core mechanismPrimary output Decision role Markov De- cision Pro- cess FuzzyStates, actions, fuzzy transitions, and fuzzy rewards Propagates fuzzy states and evaluates dis- counted fuzzy returns, typically through fuzzy Bellman-type updates. Fuzzy value function and policy Sequential decision- making Markov De- cision Pro- cess Uncer- tain Uncertain transi- tion degrees and uncertain reward degrees Scores and normalizes uncertain transitions into probabilities, scores rewards, and solves the induced crisp MDP. Value function and optimal policy Sequential decision- making Evidential Reasoning FuzzyFuzzy belief rules, matching degrees, and belief distribu- tions over evalua- tion grades Activates rules, con- structs evidence masses, and aggregates them by ER/Dempster–Shafer style combination. Aggregated belief distribu- tion Evidence ag- gregation / assessment Evidential Reasoning Uncer- tain Uncertain belief degrees and un- certain reliabilities scored into[0,1] Converts uncertain evi- dence into crisp masses and combines them se- quentially under ER aggregation formulas. Synthesized belief distri- bution with incompleteness degree Evidence ag- gregation / assessment Note.Decision trees are primarily local partition-based predictive models; DRSA is an approximation- and dominance-based rule-induction framework; MDPs are sequential optimization models under state transitions; and evidential reasoning is an evidence-fusion framework over evaluation grades. In the uncertain versions, the common structural principle is to start from uncertain-valued inputs in Dom(M), project them through an admissible score, and then apply a well-defined crisp decision mechanism. 9.1 Fuzzy Decision Tree A decision tree is a rule-based model splitting data by feature tests, forming branches to predict classes or values [1154, 1155]. A fuzzy decision tree replaces crisp splits with fuzzy membership functions, enabling soft rules, smoother boundaries, and robustness to noise [1156–1158]. Definition 9.1.1(Fuzzy decision tree (attribute-value / fuzzy-partition form)).[1156–1158] LetX6=∅ be a finite set of training objects, and let F(X) :=A:X→[0,1] denote the family of all fuzzy subsets ofX. ForA∈F(X), define its (sigma-count) fuzzy cardinality by M(A) := ∑ x∈X A(x). Fixm≥1attributes. For each attributei∈ 1,...,m, letX i ⊆ F(X)be the (finite) set of fuzzy attribute-values (linguistic values), and assume|X i | ≥1. Fix at-normT(e.g.T=min) and define the fuzzy intersection (A∧ T B)(x) :=T ( A(x),B(x) ) (x∈X). Chapter 9. Rule induction / learning / evidence / sequential decision methods Afuzzy decision treeis a rooted directed treeTwhose nodes are fuzzy subsets ofX(i.e. elements ofF(X)) such that: (a) Every nodeNofTsatisfiesN∈F(X). (b) For every non-leaf nodeN, there exists an attribute indexi∈1,...,mfor which the set of children ofNis exactly ch(N) =A∧ T N:A∈X i ⊆F(X). (c) Each leaf is assigned one or more class labels (classification decisions). Definition 9.1.2(Soft/complete fuzzy decision tree (membership-propagation semantics)).LetUbe a universe of objects (instances). Asoft (fuzzy) decision treefor a (crisp or fuzzy) target classCconsists of a rooted directed tree whose nodesvcorrespond to fuzzy subsetsS v ⊆Uwith membershipμ v :U→[0,1], and: • each internal nodevselects an attributeaand adiscriminator(soft split)σ v :U→[0,1](e.g. a piecewise-linear function governed by a cut-pointαand widthβ), and for a binary split defines the childrenv L ,v R by μ v L (o) :=μ v (o)σ v (o), μ v R (o) :=μ v (o) ( 1−σ v (o) ) (o∈U). • each leaf`carries a numeric labelL ` ∈R(oftenL ` ∈[0,1]for class-membership prediction). For an input objecto∈U, the tree output (estimated class-membership) is the leaf-aggregation ̂μ C (o) := ∑ `∈Leaves μ ` (o)L ` ∑ `∈Leaves μ ` (o) , with the usual convention that the denominator is positive (e.g.μ root (o) = 1). Using an Uncertain Set, we define an Uncertain decision tree (U-tree) as follows. Definition 9.1.3(Uncertain decision tree (U-tree) as a measurable partition rule).Let(Ω,F)be a mea- surable space of instances and letYbe a label space (e.g.Y=1,...,Kfor classification orY=Rfor regression). Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M . Assumeduncertain-valued features X j : Ω→Dom(M) (j= 1,...,d), and letS=S v v∈V be the node set of a finite rooted directed treeTwith rootr. Anuncertain decision treeis specified by: • a finite rooted treeT= (V,E); Chapter 9. Rule induction / learning / evidence / sequential decision methods • for each internal nodev∈Vasplit ruledetermined by an indexj(v)∈ 1,...,dand a measurable setB v ⊆R, producing two childrenv L ,v R and the measurable tests o∈S v L ⇐⇒S M (X j(v) (o))∈B v , o∈S v R ⇐⇒S M (X j(v) (o)) /∈B v ; • for each leaf`a prediction valueh(`)∈Y. The induced predictorH: Ω→Yis H(o) :=h(`(o)), where`(o)is the unique leaf reached by starting at the root and applying the tests along the unique root-to-leaf path. Definition 9.1.4(Soft uncertain decision tree (membership-propagation form)).Let(Ω,F)be a measurable space. Fix an uncertain modelMand an admissible scoreS M . Asoft uncertain decision treeis a finite rooted treeT= (V,E)with: • a root membershipμ r : Ω→[0,1](typicallyμ r ≡1); • for each internal nodev, a feature indexj(v)and a measurablesoft splitfunction σ v : Ω→[0,1], σ v (o) = Σ v ( S M (X j(v) (o)) ) for some measurableΣ v :R→[0,1](e.g. a sigmoid or triangular membership around a cut-point); with childrenv L ,v R defined by the membership propagation μ v L (o) :=μ v (o)σ v (o), μ v R (o) :=μ v (o) ( 1−σ v (o) ) ; • for each leaf`, a numeric labelL ` ∈R(or a class-probability vector in∆ K−1 ). The tree output is the leaf aggregation ̂ H(o) := ∑ `∈Leaves μ ` (o)L ` ∑ `∈Leaves μ ` (o) , ∑ ` μ ` (o)>0, 0, ∑ ` μ ` (o) = 0, (with0replaced by any fixed default output when the denominator is0). Theorem 9.1.5(Well-definedness of uncertain decision trees).In Definitions 9.1.3 and 9.1.4, assume: (i)S M is admissible (finite everywhere); (i) all setsB v ⊆Rand all functionsΣ v :R→[0,1]are measurable; (i)Tis finite. Then: Chapter 9. Rule induction / learning / evidence / sequential decision methods (a) The hard uncertain decision tree predictorH: Ω→Yis well-defined. (b) In the soft uncertain decision tree, each node membershipμ v is well-defined and satisfies0≤μ v ≤1. (c) For everyo∈Ω, the soft output ̂ H(o)is well-defined as a real number under the stated denominator convention. Proof.(a) SinceTis a rooted tree, each instanceofollows a unique path from the root to a leaf by repeatedly applying the binary test “S M (X j(v) (o))∈B v ”. BecauseS M (X j(v) (o))is a finite real andB v is a fixed set, the test outcome is defined at each internal node. Finiteness ofTimplies the path terminates at a leaf`(o) in finitely many steps. ThusH(o) =h(`(o))is defined for allo. (b) By induction on depth. At the root,μ r (o)∈[0,1]. Supposeμ v (o)∈[0,1]for some nodev. Sinceσ v (o) = Σ v (S M (X j(v) (o)))∈[0,1], we haveμ v L (o) =μ v (o)σ v (o)∈[0,1]andμ v R (o) =μ v (o)(1−σ v (o))∈[0,1]. Thus all node memberships are well-defined and bounded in[0,1]. (c) For eacho, the set of leaves is finite and eachμ ` (o)∈[0,1], so the sums ∑ ` μ ` (o)and ∑ ` μ ` (o)L ` are finite real numbers. If the denominator is positive, the ratio is well-defined; if it is0, the definition assigns a fixed default value, so ̂ H(o)is well-defined for allo. Table 9.2: Related concepts of decision trees under uncertainty-aware models. kRelated decision tree concept(s) 2 Intuitionistic Fuzzy Decision Trees [1159] 3 Hesitant Fuzzy Decision Trees [1160,1161] 3 Neutrosophic Decision Trees [1162,1163] Besides Uncertain Decision Trees, several other related decision-tree concepts are also known, including Boosted Decision Trees [1164,1165], Decision Hypertrees [1166], Rough Decision Trees [1167,1168], Directed Decision Trees [1169,1170], Complete Decision Trees [1171,1172], Phonetic Decision Trees [1173,1174], and Multi-Criteria Decision Trees [1175,1176]. 9.2 Fuzzy DRSA (Fuzzy Dominance-based rough approximation) Dominance-based rough set is a rough-set approach for preference-ordered data that uses dominance rela- tions to approximate decision classes in multicriteria decision analysis [1177–1179]. DRSA uses dominance relations on ordered criteria to approximate decision classes, inducing if–then decision rules and identifying reducts without requiring globally fixed cutoffs in advance [1180, 1181]. Fuzzy DRSA softens dominance and class boundaries using membership grades, producing fuzzy lower/upper approximations and graded decision rules that better capture vague preferences in practice [1182,1183]. Definition 9.2.1(Fuzzy DRSA (fuzzy extensions of the dominance-based rough set approach)).[1182] Let S=〈U, Q∪d, V, f〉 be a (finite) decision table, whereUis the universe of objects,Q=q 1 ,...,q m is the set of condition criteria,dis the decision criterion, andf:U×(Q∪d)→Vis the information function. Assume that the decision attribute induces an ordered family of decision classes Cl 1 ,...,Cl k ,with higher index meaning a (weakly) better class. Chapter 9. Rule induction / learning / evidence / sequential decision methods Fix fuzzy logic connectives: at-normT, at-conormS, and an implicatorI(typically theR-implicator of a left-continuoust-norm). AFuzzy DRSA modelconsists of the following ingredients. (1) Fuzzy (criterion-wise) ordering relations and fuzzy dominance.For each criterionq∈Q, let R q :U×U→[0,1] be a fuzzy preorder (reflexive andT-transitive). For a nonempty set of criteriaP⊆Q, define thefuzzy dominance relation D P (u,v) :=T q∈P ( R q (u,v) ) ∈[0,1], u,v∈U. (ThusD P (u,v)is the degree to which “uis at least as good asvw.r.t. all criteria inP”.) WriteD:=D P whenPis fixed. (2) Fuzzy decision classes and fuzzy class unions.In fuzzy DRSA, each class is allowed to be a fuzzy set Cl t :U→[0,1],Cl t (u)is the credibility thatubelongs to class Cl t . Define thefuzzy upwardandfuzzy downwardunions (one standard choice) by Cl ≥ t (u) :=max s≥t Cl s (u),Cl ≤ t (u) :=max s≤t Cl s (u), u∈U, t= 1,...,k. (3) Fuzzified quantifiers.Let qua ∀ and qua ∃ be fuzzy quantifiers used to aggregate truth degrees over U. Two canonical choices are: (qua ∀ ,qua ∃ ) = (T,S)(finite-Uaggregation) or(qua ∀ ,qua ∃ ) = (inf,sup)(general). If(qua ∀ ,qua ∃ ) = (T,S), interpret qua ∀ (a v ;v∈U) =T v∈U a v ,qua ∃ (a v ;v∈U) =S v∈U a v . (4) Fuzzy DRSA lower/upper approximations.For eacht∈1,...,k, define fuzzy lower and upper approximations of the unions as fuzzy sets onUwith membership degrees in[0,1]: apr qua ∀ ,I D ( Cl ≥ t ) (u) :=qua ∀ ( I ( D(v,u),Cl ≥ t (v) ) ;v∈U ) , apr qua ∃ ,T D ( Cl ≥ t ) (u) :=qua ∃ ( T ( D(u,v),Cl ≥ t (v) ) ;v∈U ) , apr qua ∀ ,I D ( Cl ≤ t ) (u) :=qua ∀ ( I ( D(u,v),Cl ≤ t (v) ) ;v∈U ) , apr qua ∃ ,T D ( Cl ≤ t ) (u) :=qua ∃ ( T ( D(v,u),Cl ≤ t (v) ) ;v∈U ) . We define Uncertain DRSA of typeM(U-DRSA) as follows. Chapter 9. Rule induction / learning / evidence / sequential decision methods Definition 9.2.2(Uncertain DRSA of typeM(U-DRSA)).Let DT=〈U, Q∪d, V, f〉 be a finite decision table, whereUis a finite universe of objects,Q=q 1 ,...,q m is the set of condition criteria,dis the decision criterion, andf:U×(Q∪d)→Vis the information function. Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M . (A) Uncertain-valued criteria and their crisp projections.Assume that each condition criterion q∈Qand the decision criteriondtake values in Dom(M): f(u,q)∈Dom(M) (q∈Q), f(u,d)∈Dom(M) (u∈U). Define the projected (crisp) criterion values g q (u) :=S M ( f(u,q) ) ∈R(q∈Q), g d (u) :=S M ( f(u,d) ) ∈R(u∈U). (B) Ordered decision classes and unions.Assume the decision attribute induces an ordered family of decision classes Cl 1 ,...,Cl K ⊆U,with higher index meaning a (weakly) better class. Define the upward and downward unions Cl ≥t := K ⋃ s=t Cl s ,Cl ≤t := t ⋃ s=1 Cl s , t= 1,...,K. (C) Dominance relations on criteria.For any nonempty subsetP⊆Q, define the (crisp) dominance relation D P :=(u,v)∈U×U:g q (u)≥g q (v)∀q∈P. Equivalently,(u,v)∈D P means thatudominatesvon all criteria inP. Foru∈Udefine the dominance cones D + P (u) :=v∈U: (v,u)∈D P (objects dominatingu), D − P (u) :=v∈U: (u,v)∈D P (objects dominated byu). (D) DRSA lower/upper approximations of unions.For eacht∈1,...,Kdefine: P ( Cl ≥t ) :=u∈U:D + P (u)⊆Cl ≥t ,P ( Cl ≥t ) :=u∈U:D − P (u)∩Cl ≥t 6=∅, P ( Cl ≤t ) :=u∈U:D − P (u)⊆Cl ≤t ,P ( Cl ≤t ) :=u∈U:D + P (u)∩Cl ≤t 6=∅. The quadruple of approximations ( P(Cl ≥t ),P(Cl ≥t ),P(Cl ≤t ),P(Cl ≤t ) ) K t=1 is called theUncertain DRSA approximation systemassociated with(DT,M,S M ,P). Chapter 9. Rule induction / learning / evidence / sequential decision methods Theorem 9.2.3(Well-definedness of U-DRSA).Under Definition 9.2.2, assumeUis finite,Dom(M)6=∅, andS M is admissible. Then, for any nonemptyP⊆Qand anyt∈1,...,K: (i) The dominance relationD P ⊆U×Uis well-defined and is a preorder (reflexive and transitive). (i) The dominance conesD + P (u)andD − P (u)are well-defined subsets ofUfor allu∈U. (i) The setsP(Cl ≥t ),P(Cl ≥t ),P(Cl ≤t ), andP(Cl ≤t )are well-defined subsets ofU. Proof.(i) SinceS M is admissible, eachg q (u) =S M (f(u,q))is a finite real number, so the comparisons g q (u)≥g q (v)are well-defined. HenceD P is well-defined. Reflexivity holds becauseg q (u)≥g q (u)for all q∈P, so(u,u)∈D P . Transitivity: if(u,v)∈D P and(v,w)∈D P , then for eachq∈P,g q (u)≥g q (v)≥ g q (w), hence(u,w)∈D P . ThusD P is a preorder. (i) GivenD P , the conesD + P (u) =v: (v,u)∈D P andD − P (u) =v: (u,v)∈D P are defined by set comprehension over the finite universeU, hence are well-defined subsets ofU. (i) Each approximation set is defined using standard set operations (subset test, intersection, emptiness) applied to the well-defined setsD + P (u),D − P (u)and unions Cl ≥t ,Cl ≤t , hence is well-defined. 9.3 Fuzzy Markov Decision Process A Markov decision process models sequential decisions with states, actions, transition probabilities, and rewards; dynamic programming (value/policy iteration) yields an optimal policy maximizing expected return [1184,1185]. A fuzzy MDP represents transitions, rewards, or states with fuzzy sets when probabilities are imprecise; fuzzy Bellman updates compute fuzzy value functions and robust policies [1186,1187]. Definition 9.3.1(Fuzzy Markov Decision Process (FMDP)).[1186, 1187] Let(S,Σ S )be a (finite or measurable) state space and(A,Σ A )an action space. For eachs∈S, letA(s)⊆Abe the (nonempty) feasible action set. Fix a discount factorγ∈[0,1). Afuzzy Markov decision process(in the possibilistic/max–min sense) is a tuple M= ( S,A(s) s∈S , ̃ P, ̃r,γ ) , where: 1.Fuzzy transition kernel (possibility kernel).For each(s,a)∈S×A(s), the mapping ̃ P(·|s,a) :S→[0,1], s ′ 7→ ̃ P(s ′ |s,a), is anormalfuzzy set onS(i.e. sup s ′ ∈S ̃ P(s ′ |s,a) = 1). It induces a possibility measure on(S,Σ S )by Π(B|s,a) :=sup s ′ ∈B ̃ P(s ′ |s,a), B∈Σ S . Chapter 9. Rule induction / learning / evidence / sequential decision methods 2.Fuzzy reward.For each(s,a)∈S×A(s), the immediate reward is a fuzzy set ̃r(s,a)∈F(R), often restricted to fuzzy numbers (normal, upper semicontinuous, and with bounded support). 3.Policies.A (deterministic)stationary policyis a mapf:S→Awithf(s)∈A(s). More generally, a (possibly randomized) history-dependent policy is a familyπ=π t t≥0 withπ t (·|h t )a probability distribution overA(s t )given a historyh t = (s 0 ,a 0 ,...,s t ). Possibilistic state evolution (fuzzy state / possibility distribution).A fuzzy state (possibility distribution) is ̃x∈F(S), i.e. a map ̃x:S→[0,1]with sup s∈S ̃x(s) = 1. Given a chosen actiona t ∈A(·)at timet, the next fuzzy state is defined by the max–min composition ̃x t+1 (s ′ ) = ∨ s∈S ( ̃x t (s)∧ ̃ P(s ′ |s,a t ) ) , s ′ ∈S, where∨=max and∧=min. Discounted fuzzy return.Let⊕and denote fuzzy addition and scalar multiplication (e.g. defined via Zadeh’s extension principle; equivalently, viaα-cuts when ̃r(s,a)are fuzzy numbers): ( ̃u⊕ ̃v) α = ( ̃u) α + ( ̃v) α ,(c ̃u) α =c( ̃u) α (c≥0). For a fixed policyπand initial fuzzy state ̃x 0 , define the discounted total fuzzy reward (formally) by ̃ G π ( ̃x 0 ) := ∞ ⊕ t=0 γ t ̃r(s t ,a t ), where(s t ,a t )are generated underπand the transition kernel ̃ P. (For bounded rewards andγ <1, this series is well-defined under standard choices of fuzzy arithmetic, typically by checking convergence onα-cuts.) The definition of Uncertain Markov decision process of typeM(U-MDP) is given below. Definition 9.3.2(Uncertain Markov decision process of typeM(U-MDP)).LetSbe a finite nonempty state set andAa finite nonempty action set. For eachs∈S, letA(s)⊆Abe a nonempty feasible action set. Fix a discount factorγ∈[0,1). Fix an uncertain modelMwith Dom(M)6=∅and admissible scoresS P M ,S R M . Anuncertain Markov decision process of typeMis a tuple M M = ( S,A(s) s∈S , P M , R M , γ;S P M ,S R M ) , where: •Uncertain transition degrees:P M :S×A×S→Dom(M)assigns an uncertainty degree tuple P M (s ′ |s,a)∈Dom(M)to each(s,a,s ′ )witha∈A(s). Chapter 9. Rule induction / learning / evidence / sequential decision methods •Uncertain reward degrees:R M :S×A→Dom(M)assigns an uncertainty degree tupleR M (s,a)∈ Dom(M)to each(s,a)witha∈A(s). Induced (crisp) MDP via scoring and normalization.Define the (strictly positive) transition scores α(s ′ |s,a) :=S P M ( P M (s ′ |s,a) ) ∈(0,∞), a∈A(s), and the normalizing constant Z(s,a) := ∑ t∈S α(t|s,a)∈(0,∞). Define the induced transition probabilities p(s ′ |s,a) := α(s ′ |s,a) Z(s,a) ∈(0,1), ∑ s ′ ∈S p(s ′ |s,a) = 1. Define the induced immediate reward r(s,a) :=S R M ( R M (s,a) ) ∈R. The induced (crisp) MDP is ( S,A(s),p,r,γ ) . Policies and value functions.A (deterministic) stationary policy is a mapπ:S→Awithπ(s)∈A(s). For suchπ, define the value functionV π :S→Rby the discounted expectation V π (s) :=E π [ ∞ ∑ t=0 γ t r(S t ,π(S t )) ∣ ∣ ∣ S 0 =s ] , where(S t ) t≥0 evolves according top(·|s,π(s)). The optimal value function is V ? (s) :=sup π V π (s), and an optimal policyπ ? satisfiesV π ? =V ? . Proposition 9.3.3(Probability well-definedness).Under Definition 9.3.2, for every(s,a)witha∈A(s), the induced transition probabilitiesp(·|s,a)are well-defined and satisfy p(s ′ |s,a)≥0, ∑ s ′ ∈S p(s ′ |s,a) = 1. Proof.α(s ′ |s,a) =S P M (P M (s ′ |s,a))>0for everys ′ . HenceZ(s,a) = ∑ t∈S α(t|s,a)>0, sop(s ′ |s,a) = α(s ′ |s,a)/Z(s,a)is well-defined and nonnegative. Summing overs ′ yields ∑ s ′ p(s ′ |s,a) =Z(s,a)/Z(s,a) = 1. Theorem 9.3.4(Well-definedness of discounted values and Bellman optimality).Assume the setting of Definition 9.3.2 and suppose the induced rewards are bounded: there existsR max <∞such that|r(s,a)|≤ R max for all feasible(s,a). Then: Chapter 9. Rule induction / learning / evidence / sequential decision methods (i) For every stationary policyπ, the value functionV π is well-defined and bounded: |V π (s)|≤ R max 1−γ (s∈S). (i) The Bellman expectation operatorT π :R S →R S , (T π V)(s) :=r(s,π(s)) +γ ∑ s ′ ∈S p(s ′ |s,π(s))V(s ′ ), is aγ-contraction in the sup norm‖·‖ ∞ and has a unique fixed point equal toV π . (i) The Bellman optimality operatorT ? :R S →R S , (T ? V)(s) :=max a∈A(s) [ r(s,a) +γ ∑ s ′ ∈S p(s ′ |s,a)V(s ′ ) ] , is also aγ-contraction and has a unique fixed pointV ? . Moreover, there exists an optimal deterministic stationary policyπ ? attaining the maximizers inT ? . Proof.(i) Since|r(S t ,π(S t ))|≤R max almost surely, the discounted series is absolutely bounded by ∞ ∑ t=0 γ t R max = R max 1−γ , so the expectation definingV π (s)is finite and the bound holds. (i) For anyV,W∈R S and anys∈S, |(T π V)(s)−(T π W)(s)|=γ ∣ ∣ ∣ ∣ ∣ ∑ s ′ p(s ′ |s,π(s)) (V(s ′ )−W(s ′ )) ∣ ∣ ∣ ∣ ∣ ≤γ ∑ s ′ p(s ′ |s,π(s))‖V−W‖ ∞ =γ‖V−W‖ ∞ . Taking sup s gives‖T π V−T π W‖ ∞ ≤γ‖V−W‖ ∞ , soT π is a contraction. By the Banach fixed-point theorem on the complete metric space(R S ,‖·‖ ∞ ),T π has a unique fixed point. Standard MDP arguments show that this fixed point equals the discounted value functionV π . (i) For anyV,Wand anys, |(T ? V)(s)−(T ? W)(s)|≤max a∈A(s) γ ∣ ∣ ∣ ∣ ∣ ∑ s ′ p(s ′ |s,a)(V(s ′ )−W(s ′ )) ∣ ∣ ∣ ∣ ∣ ≤γ‖V−W‖ ∞ , hence‖T ? V−T ? W‖ ∞ ≤γ‖V−W‖ ∞ , soT ? is a contraction and has a unique fixed pointV ? . SinceA(s) is finite, the maximum in(T ? V)(s)is attained for eachs; choosing an argmax action defines a deterministic stationary policyπ ? , and the fixed-point identityV ? =T ? V ? implies optimality. Related concepts of Markov decision processes under uncertainty-aware models are listed in Table 9.3. Chapter 9. Rule induction / learning / evidence / sequential decision methods Table 9.3: Related concepts of Markov decision processes under uncertainty-aware models. kRelated Markov decision process concept(s) 1 Fuzzy Markov Decision Processes 3 Neutrosophic Markov Decision Processes [1188,1189] 9.4 Fuzzy Evidential Reasoning Classical Evidential Reasoning assigns belief degrees to evaluation grades, combines multiple criteria using evidential reasoning aggregation based on Dempster Shafer calculus, then computes expected utilities [1190, 1191]. Fuzzy Evidential Reasoning represents each criterion as fuzzy belief degrees over evaluation grades, combines evidence across criteria using ER rules, then defuzzifies utilities for ranking [1192,1193]. Definition 9.4.1(Fuzzy Evidential Reasoning (Fuzzy ER)).[1194,1195] LetΘ =H 1 ,...,H N be a finite set ofevaluation grades(hypotheses). LetX i be the domain of thei-th antecedent attribute (i= 1,...,M). For eachi, fix a linguistic term setA i =A i,1 ,...,A i,J i , where eachA i,j is represented by a fuzzy set on X i with membership functionμ i,j :X i →[0,1]. Abelief-rule base(fuzzy rule base with belief structure) is a collectionR=R k L k=1 of rules of the form R k :IFx 1 isA k 1 ∧ · ∧x M isA k M THEN (H j ,β k,j ) N j=1 , whereA k i ∈A i is the antecedent linguistic term used in rulekfor attributei, and thebelief degreessatisfy β k,j ∈[0,1], N ∑ j=1 β k,j ≤1 (k= 1,...,L). (The inequality allowsincompletenessof the consequent belief assignment.) Given an observation (possibly fuzzy) ̃x i ∈ F(X i )for each attributei, define thematching degreebetween ̃x i and the antecedent termA k i by the Max–Min similarity α k i :=max t∈X i min ( μ ̃x i (t),μ A k i (t) ) ∈[0,1], i= 1,...,M, k= 1,...,L. (If the input is crispx i ∈X i , one may takeμ ̃x i (t) =1 x i (t), yieldingα k i =μ A k i (x i ).) Letω k ≥0denote an optionalintrinsic rule weight(importance of rulek). Define theactivation weight (normalized firing strength) of rulekby θ k := ω k ∏ M i=1 α k i ∑ L `=1 ω ` ∏ M i=1 α ` i so thatθ k ≥0, L ∑ k=1 θ k = 1. Evidence-mass construction.For each rulek, define basic probability masses onΘ∪Dby m k (H j ) :=θ k β k,j (j= 1,...,N), Chapter 9. Rule induction / learning / evidence / sequential decision methods m k (D) := 1− N ∑ j=1 m k (H j ) = 1−θ k N ∑ j=1 β k,j , and split the unassigned mass as m k (D) := 1−θ k , ̃m k (D) :=θ k ( 1− N ∑ j=1 β k,j ) ,so thatm k (D) =m k (D) + ̃m k (D). Evidential reasoning aggregation.Initializem (1) (·) =m 1 (·). Fork= 1,...,L−1, combinem (k) and m k+1 by m (k+1) (H j ) =K k+1 ( m (k) (H j )m k+1 (H j ) +m (k) (H j )m k+1 (D) +m (k) (D)m k+1 (H j ) ) , ̃m (k+1) (D) =K k+1 ( ̃m (k) (D) ̃m k+1 (D) + ̃m (k) (D)m k+1 (D) +m (k) (D) ̃m k+1 (D) ) , m (k+1) (D) =K k+1 m (k) (D)m k+1 (D), m (k+1) (D) = ̃m (k+1) (D) +m (k+1) (D), where the normalization factorK k+1 is K k+1 := ( 1− N ∑ j=1 N ∑ t=1 t6=j m (k) (H j )m k+1 (H t ) ) −1 . Fuzzy ER output (belief distribution).TheFuzzy ERsynthesis result is the belief distributionβ= (β 1 ,...,β N ,β D )defined by β j := m (L) (H j ) 1−m (L) (D) (j= 1,...,N), β D := ̃m (L) (D) 1−m (L) (D) . Hereβ D quantifies the remaining (normalized) incompleteness after aggregation. The mapping ER: (θ k ,β k,1 ,...,β k,N ) L k=1 7−→(β 1 ,...,β N ,β D ) is calledfuzzy evidential reasoning. Using an Uncertain Set, we define Uncertain Evidential Reasoning as follows. Definition 9.4.2(Uncertain Evidential Reasoning of typeM(U-ER)).LetΘ =H 1 ,...,H N be a finite set of evaluation grades (hypotheses), withN≥2. LetL≥1be the number of evidence sources (e.g. criteria, rules, experts). Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M :Dom(M)→[0,1]. For each evidence source`∈1,...,Lassume: • uncertain belief degreesb (M) `j ∈Dom(M)for gradesH j (j= 1,...,N), Chapter 9. Rule induction / learning / evidence / sequential decision methods • an uncertain reliability/activation degreeθ (M) ` ∈Dom(M). Define their crisp projections β `j :=S M ( b (M) `j ) ∈[0,1], θ ` :=S M ( θ (M) ` ) ∈[0,1]. (A) Consistency (incomplete belief allowed).Assume for each`, N ∑ j=1 β `j ≤1. Define the incompleteness (ignorance) for source`: β `D := 1− N ∑ j=1 β `j ∈[0,1]. (B) Basic probability masses.For each`, define a basic probability assignment (BPA) onΘ∪Dby m ` (H j ) :=θ ` β `j (j= 1,...,N), m ` (D) := 1− N ∑ j=1 m ` (H j ) = 1−θ ` N ∑ j=1 β `j . Splitm ` (D)into m ` (D) := 1−θ ` , ̃m ` (D) :=θ ` β `D , m ` (D) =m ` (D) + ̃m ` (D). (C) ER aggregation (sequential combination).Setm (1) :=m 1 ,m (1) :=m 1 , ̃m (1) := ̃m 1 . For k= 1,...,L−1, combine(m (k) ,m (k) , ̃m (k) )with(m k+1 ,m k+1 , ̃m k+1 )by K k+1 := 1− N ∑ j=1 N ∑ t=1 t6=j m (k) (H j )m k+1 (H t ) −1 , m (k+1) (H j ) :=K k+1 ( m (k) (H j )m k+1 (H j ) +m (k) (H j )m k+1 (D) +m (k) (D)m k+1 (H j ) ) , m (k+1) (D) :=K k+1 m (k) (D)m k+1 (D), ̃m (k+1) (D) :=K k+1 ( ̃m (k) (D) ̃m k+1 (D) + ̃m (k) (D)m k+1 (D) +m (k) (D) ̃m k+1 (D) ) , m (k+1) (D) :=m (k+1) (D) + ̃m (k+1) (D). (D) Final belief distribution.Let the aggregated masses bem (L) with splitm (L) (D), ̃m (L) (D). Define the normalized belief degrees β j := m (L) (H j ) 1−m (L) (D) (j= 1,...,N), β D := ̃m (L) (D) 1−m (L) (D) . The vector(β 1 ,...,β N ,β D )is called theU-ER synthesis result. Chapter 9. Rule induction / learning / evidence / sequential decision methods Theorem 9.4.3(Well-definedness of U-ER).Under Definition 9.4.2, assume: (i)S M :Dom(M)→[0,1]is admissible; (i) for every`, ∑ N j=1 β `j ≤1(soβ `D ∈[0,1]); (i) for each aggregation stepk→k+1, theconflictsatisfies κ k+1 := N ∑ j=1 N ∑ t=1 t6=j m (k) (H j )m k+1 (H t )<1; (iv) the final normalization denominator is positive: 1−m (L) (D)>0. Then: (a) All massesm ` (H j ),m ` (D),m ` (D), ̃m ` (D)are well-defined in[0,1]. (b) Each intermediate factorK k+1 is well-defined and finite, and all aggregated masses remain in[0,1]. (c) The final belief degreesβ j ,β D are well-defined, satisfy0≤β j ,β D ≤1, and N ∑ j=1 β j +β D = 1. Proof.(a) Sinceθ ` ,β `j ∈[0,1],m ` (H j ) =θ ` β `j ∈[0,1]. Alsom ` (D) = 1−θ ` ∑ j β `j ∈[0,1]because ∑ j β `j ≤1. The split satisfiesm ` (D) = 1−θ ` ∈[0,1]and ̃m ` (D) =θ ` β `D ∈[0,1]. (b) By assumption,κ k+1 <1, henceK k+1 = (1−κ k+1 ) −1 is well-defined and finite. Each update formula is a finite sum/product of nonnegative terms multiplied byK k+1 , hence defines finite nonnegative masses. (These are the standard ER/Dempster-style normalized combinations, which preserve total mass1on Θ∪Dunder the same condition.) (c) Since1−m (L) (D)>0, the normalization definingβ j andβ D is valid. Nonnegativity follows from (b). Finally, N ∑ j=1 β j +β D = ∑ N j=1 m (L) (H j ) + ̃m (L) (D) 1−m (L) (D) = 1−m (L) (D) 1−m (L) (D) = 1, becausem (L) (D) =m (L) (D) + ̃m (L) (D)and the total mass onΘ∪Dequals1. Related concepts of evidential reasoning under uncertainty-aware models are listed in Table 9.4. Chapter 9. Rule induction / learning / evidence / sequential decision methods Table 9.4: Related concepts of evidential reasoning under uncertainty-aware models. kRelated evidential reasoning concept(s) 2 Intuitionistic Fuzzy Evidential Reasoning [1196,1197] 2 Pythagorean Fuzzy Evidential Reasoning [1198,1199] 3 Hesitant Fuzzy Evidential Reasoning [1200] 3 Spherical Fuzzy Evidential Reasoning [1201] 3 Picture Fuzzy Evidential Reasoning [1202] 3 Neutrosophic Evidential Reasoning [1203,1204] nPlithogenic Evidential Reasoning [1205] Chapter 10 Other Related Decision Methods In this section, we describe other decision-making methods and related tools. For convenience, the main concepts introduced in this chapter are compared in Table 10.1. Table 10.1: Concise comparison of representative other related decision methods. MethodType Basic input structure Core mechanismPrimary out- put Deci- sion role Goal Program- ming FuzzyFeasible set, goal functions, aspi- ration levels, and fuzzy tolerances Defines goal-satisfaction memberships and max- imizes the common sat- isfaction level over all goals Compromise so- lution and satis- faction level Goal- based opti- mization Goal Program- ming Uncer- tain Feasible set, real- valued goals, and uncertain tar- gets/tolerances scored into crisp values Projects uncertain tar- gets and tolerances, builds membership-style satisfaction functions, and solves a max–min compromise model U-GP compro- mise solution Goal- based opti- mization Data Envelop- ment Analysis FuzzyDMUs with fuzzy inputs and fuzzy outputs Extends DEA efficiency analysis through fuzzy arithmetic orα-cuts around the empirical frontier Fuzzy efficiency score or interval efficiency Effi- ciency evalua- tion Data Envelop- ment Analysis Uncer- tain DMUs with uncer- tain input/output degrees scored into positive crisp data Converts uncertain ob- servations into crisp DEA data and solves the induced CCR-type efficiency program Crisp efficiency scores Effi- ciency evalua- tion Social Choice FuzzyIndividuals, alter- natives, and fuzzy preference relations Aggregates graded pref- erences into a collective fuzzy preference and se- lects a socially preferred alternative Collective choice or collective pref- erence Col- lective decision- making Continued on the next page. 313 Chapter 10. Other Related Decision Methods Table 10.1 (continued). MethodType Basic input structure Core mechanismPrimary out- put Deci- sion role Social Choice Uncer- tain Uncertain pref- erence relations scored into[0,1] and an aggregation rule Scores uncertain pair- wise preferences, ag- gregates them into a collective relation, and applies a score-based social choice rule Chosen alterna- tive Col- lective decision- making Decision Table FuzzyFuzzy condition states, action states, and rule- table entries Matches inputs to fuzzy rule columns and selects or aggregates actions by graded firing strengths Action config- uration or rule output Rule- based decision support Decision Table Uncer- tain Uncertain condi- tion states and crisp or uncertain action values Scores uncertain con- dition states, evaluates rule firing by at-norm, and selects an action via deterministic tie- breaking Action configura- tion Rule- based decision support Decision Ma- trix FuzzyAlternatives, fuzzy states/events, and fuzzy evaluations Propagates fuzzy pay- offs through fuzzy arith- metic orα-cut compu- tations to obtain aggre- gate characteristics Fuzzy expected evaluation and related indices Matrix- based assess- ment Decision Ma- trix Uncer- tain Alternatives, crite- ria, and uncertain evaluation entries Scores uncertain entries into a crisp matrix and supports normalization, weighting, and aggrega- tion operations Scored decision matrix Matrix- based assess- ment UltrafilterFuzzyFuzzy acceptance degrees on subsets of a universe Assigns graded accep- tance to subsets while preserving monotonic- ity, intersection, and ultrafilter-type di- chotomy Fuzzy ultrafil- ter / accepted family Decisive- choice structure UltrafilterUncer- tain Uncertain accep- tance degrees on subsets with an admissible score Converts uncertain subset-acceptance into scored acceptance satis- fying ultrafilter axioms at the crisp level Uncertain ul- trafilter and induced crisp ultrafilter Decisive- choice structure SWOT Analy- sis FuzzyInternal and ex- ternal factors rep- resented by fuzzy numbers or mem- berships Aggregates internal– external factor pairs, assigns strategy quad- rants, and prioritizes strategies by closeness or scoring rules Prioritized SWOT strate- gies Strategic planning SWOT Analy- sis Uncer- tain Internal and exter- nal factors assessed by uncertain de- grees and impor- tance weights Scores uncertain factors, assigns SO/ST/WO/WT quad- rants, and ranks factor pairs by normalized in- teraction strength Prioritized anno- tated strategies Strategic planning Continued on the next page. Chapter 10. Other Related Decision Methods Table 10.1 (continued). MethodType Basic input structure Core mechanismPrimary out- put Deci- sion role Cost–Benefit Analysis FuzzyFuzzy benefit and cost cashflows, pos- sibly fuzzy discount rates Computes fuzzy present worths and fuzzy benefit–cost measures through fuzzy arith- metic orα-cuts Fuzzy NPV or fuzzy BCR Eco- nomic evalua- tion Cost–Benefit Analysis Uncer- tain Uncertain benefits, costs, and discount- rate degrees scored into crisp quanti- ties Converts uncertain cashflows into crisp present worths and eval- uates projects by NPV or BCR NPV, BCR, and project ranking Eco- nomic evalua- tion Decision Curve Analysis FuzzyBinary outcomes, fuzzy predicted risks, thresholds, and harm function Uses threshold- exceedance degrees from fuzzy risks to compute fuzzy net benefit across thresholds Fuzzy decision curve and op- timal strategy set Threshold- based model evalua- tion Decision Curve Analysis Uncer- tain Binary outcomes, uncertain predicted risks, threshold- exceedance func- tional, and harm function Evaluates uncertain risk exceedance at each threshold and computes net benefit for compet- ing strategies Uncertain deci- sion curve and optimal strategy set Threshold- based model evalua- tion Rational Choice FuzzyAlternatives and a fuzzy binary prefer- ence relation Selects alternatives whose preference degree against every feasible competitor exceeds a fixed threshold Fuzzy choice correspondence Preference- based choice Rational Choice Uncer- tain Alternatives, un- certain binary pref- erence relation, threshold func- tional, and cutoff level Scores uncertain pair- wise preference com- parisons through a threshold functional and induces a choice correspondence Uncertain choice correspondence Preference- based choice Note.Goal programming and cost–benefit analysis are optimization- and evaluation-oriented methods; data envelopment analysis focuses on relative efficiency; social choice and rational choice aggregate or compare preferences; decision tables and decision matrices provide structured rule or data representations; ultrafilters formalize decisive acceptance structures; SWOT supports strategic prioritization; and decision curve analysis evaluates threshold-based decision strategies. In the uncertain variants, the common construction is to begin with uncertain-valued inputs in Dom(M), project them through an admissible score or threshold functional, and then apply a well-defined crisp decision mechanism. 10.1Fuzzy Goal programming Goal programming is an optimization approach that seeks a compromise solution by minimizing weighted deviations from multiple target goals under given constraints [1206,1207]. Fuzzy goal programming extends this idea by representing goals and constraints as fuzzy sets and then minimizing degrees of dissatisfaction, enabling robust optimization when aspiration levels or requirements are imprecise [1208,1209]. Chapter 10. Other Related Decision Methods Definition 10.1.1(Fuzzy goal programming (membership-based GP)).[1208, 1209] LetX⊆R n be a nonempty feasible set and let f i :X→R(i= 1,...,p) be objective (goal) functions with aspiration levels (targets)g i ∈R. Assume each goal has an admissible (tolerance) deviation∆ i >0. Define thegoal-satisfaction membership functionμ i :X→[0,1]by the (symmetric) triangular form μ i (x) := 0,f i (x)≤g i −∆ i , f i (x)−(g i −∆ i ) ∆ i , g i −∆ i ≤f i (x)≤g i , (g i + ∆ i )−f i (x) ∆ i , g i ≤f i (x)≤g i + ∆ i , 0,f i (x)≥g i + ∆ i . (Other one-sided or nonlinear membership profiles can be used, depending on the decision-maker’s prefer- ence.) Thefuzzy goal programming (FGP) modelselects a compromise solution by maximizing the common satis- faction level: max x∈X, λ∈[0,1] λsubject toλ≤μ i (x) (i= 1,...,p). Any optimizerx ? is called anFGP solution. Using an Uncertain Set, we define Uncertain goal programming as follows. Definition 10.1.2(Uncertain goal programming of typeM(U-GP)).LetX⊆R n be a nonempty feasible set, and let f i :X→R(i= 1,...,p) be goal functions withp≥1. Fix aspiration (target) degrees g (M) i ∈Dom(M) (i= 1,...,p), and (strictly positive) tolerance degrees ∆ (M) i ∈Dom(M) (i= 1,...,p), together with an uncertain modelMand admissible scores S M :Dom(M)→R, S + M :Dom(M)→(0,∞). Define the projected targets and tolerances g i :=S M (g (M) i )∈R,∆ i :=S + M (∆ (M) i )∈(0,∞). Define the (membership-style) goal satisfaction for eachiby the triangular profile μ i (x) :=max 0,1− |f i (x)−g i | ∆ i ∈[0,1], x∈X. Chapter 10. Other Related Decision Methods (Thusμ i (x) = 1at the targetf i (x) =g i and decreases linearly to0at distance∆ i .) Theuncertain goal programming problemis max x∈X, λ∈[0,1] λsubject toλ≤μ i (x) (i= 1,...,p). Any optimizerx ? is called aU-GP compromise solution. Theorem 10.1.3(Well-definedness and feasibility of U-GP).Under Definition 10.1.2, assume: (i)X6=∅; (i)S M is admissible andS + M is admissible positive (i.e.∆ i >0for alli); (i) eachf i is well-defined as a real-valued function onX. Then: (a) Each membership functionμ i :X→[0,1]is well-defined. (b) The U-GP feasible set is nonempty (hence the optimization problem is well-defined). (c) Any optimal value satisfies0≤λ ? ≤1. Proof.(a) Fixiandx∈X. Sincef i (x),g i ∈R, the quantity|f i (x)−g i |is finite. Because∆ i >0, the ratio|f i (x)−g i |/∆ i is well-defined and nonnegative. Hence1−|f i (x)−g i |/∆ i is a real number, and taking max0,·yieldsμ i (x)≥0. Moreover, since|f i (x)−g i |/∆ i ≥0, we have1−|f i (x)−g i |/∆ i ≤1, soμ i (x)≤1. Thusμ i (x)∈[0,1]andμ i is well-defined. (b) BecauseX6=∅, choose any ̄x∈X. By (a),μ i ( ̄x)∈[0,1]for alli. Let ̄ λ:= 0. Then ̄ λ≤μ i ( ̄x)holds for everyi, and ̄ λ∈[0,1]. Thus( ̄x, ̄ λ)is feasible, so the feasible set is nonempty. (c) By constructionλ∈[0,1]is enforced, hence any optimum satisfies0≤λ ? ≤1. Related concepts of goal programming under uncertainty-aware models are listed in Table 10.2. As related concepts other than Uncertain Goal Programming, Multi-Goal Programming [1218, 1219], In- teractive Goal Programming [1220, 1221], Meta-Goal Programming [1222, 1223], Weighted Goal Program- ming [1224,1225], Lexicographic Goal Programming [1226,1227], Chebyshev Goal Programming [1228,1229], Group-Goal Programming [1230], Fractional Goal Programming [1231, 1232], and Multi-Choice Goal Pro- gramming [1233,1234] are also known. Chapter 10. Other Related Decision Methods Table 10.2: Related concepts of goal programming under uncertainty-aware models. kRelated goal programming concept(s) 1 Fuzzy Goal Programming 2 Intuitionistic Fuzzy Goal Programming [1210,1211] 2 Pythagorean Fuzzy Goal Programming [1212,1213] 3 Neutrosophic Goal Programming [1214,1215] 3 Picture Fuzzy Goal Programming [1216] 3 Spherical Fuzzy Goal Programming [1217] 10.2Fuzzy DEA (Fuzzy Data Envelopment Analysis) DEA evaluates relative efficiency of decision-making units using linear programming, comparing multiple inputs and outputs against an empirical frontier simultaneously [1235, 1236]. Fuzzy DEA extends DEA by modeling inputs, outputs, or data uncertainty as fuzzy numbers, yielding possibilistic efficiency ranges robustly overall [1237,1238]. Definition 10.2.1(Fuzzy Data Envelopment Analysis (Fuzzy DEA / FDEA)).[1239,1240] Fix integers n≥1(DMUs),m≥1(inputs), ands≥1(outputs). LetJ:=1,...,nbe the index set of decision making units (DMUs). For eachj∈J, DMUjconsumesminputs and producessoutputs. Fuzzy observations.For every inputi∈ 1,...,mand DMUj∈ J, the input is a fuzzy number ̃x ij ∈F(R + ). For every outputr∈1,...,sand DMUj∈J, the output is a fuzzy number ̃y rj ∈F(R + ). (Thus the classical crisp data(x ij ,y rj )are replaced by fuzzy input/output variables.) Underlying crisp DEA map (CCR, input-oriented).Given anycrisp realizationX= (x ij )∈R m×n + andY= (y rj )∈R s×n + , define the (input-oriented) CCR efficiency of DMUp∈Jas θ p (X,Y) :=min θ,λ θ s.t. n ∑ j=1 λ j x ij ≤θx ip (i= 1,...,m), n ∑ j=1 λ j y rj ≥y rp (r= 1,...,s), λ j ≥0 (j= 1,...,n), whereλ= (λ 1 ,...,λ n )are the intensity variables. (Other standard DEA variants such as BCC/VRS are obtained by adding ∑ n j=1 λ j = 1.) Fuzzy efficiency (extension principle).Thefuzzy CCR efficiencyof DMUpis the fuzzy number ̃ θ p ∈ F((0,1])induced from the fuzzy data ̃ X= ( ̃x ij )and ̃ Y= ( ̃y rj )by Zadeh’s extension principle, i.e. ̃ θ p :=Ext(θ p )( ̃ X, ̃ Y). Equivalently (and most commonly in FDEA practice), ̃ θ p is characterized byα-cuts: for eachα∈[0,1], define theα-cut intervals ( ̃x ij ) α = [x L ij (α),x U ij (α)],( ̃y rj ) α = [y L rj (α),y U rj (α)], Chapter 10. Other Related Decision Methods and the corresponding set of admissible crisp datasets D α := ( ∏ i,j [x L ij (α),x U ij (α)] ) × ( ∏ r,j [y L rj (α),y U rj (α)] ) . Then theα-cut of the fuzzy efficiency is the interval ( ̃ θ p ) α = [ θ L p (α),θ U p (α) ] , θ L p (α) :=inf (X,Y)∈D α θ p (X,Y), θ U p (α) :=sup (X,Y)∈D α θ p (X,Y). Hence FDEA yields, for eachα, aninterval efficiency scorerather than a single crisp score, by transforming a fuzzy DEA model into crisp linear programs onα-cut intervals. Finally, the membership function of ̃ θ p can be recovered fromα-cuts via μ ̃ θ p (t) =sup α∈[0,1] :t∈[θ L p (α),θ U p (α)] . Fuzzy DEA (as a method).The collection ̃ θ p p∈J is called theFuzzy DEA evaluationof thenDMUs. Ranking/selection is then performed by a chosen fuzzy-number ranking/defuzzification rule applied to ̃ θ p (or by comparing theα-level intervals). Using an Uncertain Set,the definition of Uncertain DEA of type M is given below. Definition 10.2.2(Uncertain DEA of typeM(U-DEA), CCR input-oriented).Fix integersn≥1(DMUs), m≥1(inputs), ands≥1(outputs). LetJ:=1,...,nindex the DMUs. Fix an uncertain modelM with Dom(M)6=∅and an admissible positive scoreS M . Uncertain observations.For each inputi= 1,...,mand DMUj∈ J, assume an uncertain input degreex (M) ij ∈Dom(M); for each outputr= 1,...,sand DMUj∈J, assume an uncertain output degree y (M) rj ∈Dom(M). Define the induced positive crisp data by scoring: x ij :=S M (x (M) ij )∈(0,∞), y rj :=S M (y (M) rj )∈(0,∞). CCR efficiency (envelopment form).For a fixed DMUp∈J, define its (input-oriented) CCR efficiency as the optimal value θ p :=min θ,λ θ subject to n ∑ j=1 λ j x ij ≤θx ip (i= 1,...,m), n ∑ j=1 λ j y rj ≥y rp (r= 1,...,s), λ j ≥0 (j= 1,...,n), θ≥0. The collectionθ p p∈J is called theU-DEA (CCR) efficiency evaluation. (For the BCC/VRS model add ∑ n j=1 λ j = 1.) Chapter 10. Other Related Decision Methods Theorem 10.2.3(Well-definedness and feasibility of U-DEA).Under Definition 10.2.2, assumeS M is admissible positive and hence all induced data satisfyx ij >0andy rj >0. Then for eachp∈J: (i) The CCR feasible set is nonempty, soθ p is well-defined as an extended real number. (i) The optimal value satisfies0≤θ p ≤1. (i) In particular,θ p is a finite real number (hence U-DEA is well-defined). Proof.(i) Feasibility: chooseλ p := 1andλ j := 0forj6=p, and setθ:= 1. Then for each inputi, n ∑ j=1 λ j x ij =x ip ≤1·x ip , and for each outputr, n ∑ j=1 λ j y rj =y rp ≥y rp . Alsoλ j ≥0andθ≥0. Hence the feasible set is nonempty. (i) Sinceθ≥0is a constraint,θ p ≥0. Because(θ,λ) = (1,e p )is feasible, the minimum satisfiesθ p ≤1. (i) Combining (i) and (i), the optimal value exists and obeys0≤θ p ≤1, so it is finite and real. For reference, related concepts of data envelopment analysis under uncertainty-aware models are listed in Table 10.3. Table 10.3: Related concepts of data envelopment analysis under uncertainty-aware models. kRelated data envelopment analysis concept(s) 2 Intuitionistic Fuzzy Data Envelopment Analysis [1241,1242] 2 Bipolar Fuzzy Data Envelopment Analysis [1243,1244] 2 Pythagorean Fuzzy Data Envelopment Analysis [1245] 3 Hesitant Fuzzy Data Envelopment Analysis [1246] 3 Spherical Fuzzy Data Envelopment Analysis [1247,1248] 3 Neutrosophic Data Envelopment Analysis [1249–1251] nPlithogenic Data Envelopment Analysis [1252–1254] Moreover, besides Uncertain Data Envelopment Analysis, a very large number of other related concepts are also known, including Two-Stage Data Envelopment Analysis [1255,1256], Three-Stage Data Envelopment Analysis [1257, 1258], Four-Stage Data Envelopment Analysis [1259], Global Data Envelopment Analysis [1260,1261], Rough Data Envelopment Analysis [1262,1263], Grey Data Envelopment Analysis [1264,1265], and Multi-Criteria Data Envelopment Analysis [1266,1267]. Chapter 10. Other Related Decision Methods 10.3Fuzzy Social Choice Social choice aggregates individual preferences into collective decisions, using voting or welfare rules while addressing fairness, strategy, and consistency [1268]. Fuzzy social choice uses graded preference relations, aggregates them into a collective fuzzy relation, then selects an alternative via defuzzification [1269–1271]. Definition 10.3.1(Fuzzy social choice (via a fuzzy social choice function)).(cf. [1269–1271]) LetX= x 1 ,...,x m be a finite set of alternatives with|X|≥3, and letN=1,...,nbe a finite set of individuals withn≥2. Fix a nonempty domainTof admissiblefuzzy preference relationsonX. (i) Individual fuzzy preference relations.For eachi∈N, an individual preference is a fuzzy binary relation R i :X×X−→[0,1], whereR i (x,y)is interpreted as the degree to which “xis at least as preferred asy”. Apreference profileis then-tuple R N := (R 1 ,...,R n )∈T n . (One common choice is to takeT=H, the set offuzzy orderings, i.e. relations that are reflexive, connected, and max–min transitive.) (i) Fuzzy social choice function.Afuzzy social choice function(FSCF) is a mapping f:T n −→X that assigns asinglecollective choicef(R N )∈Xto each profileR N of individuals’ fuzzy preference relations. The resulting collective alternativef(R N )is called thefuzzy social choice. Definition 10.3.2(Uncertain preference relations of typeM).LetXbe a finite set of alternatives with |X|≥3and letMbe an uncertain model. Anuncertain preference relation of typeMonXis a mapping R M :X×X−→Dom(M), whereR M (x,y)encodes the (uncertain) degree to which “xis at least as preferred asy”. Given an admissible scoreS M :Dom(M)→[0,1], its inducedscored preference relationis r(x,y) :=S M ( R M (x,y) ) ∈[0,1]. We define Uncertain social choice function of typeM(U-SCF) as follows. Definition 10.3.3(Uncertain social choice function of typeM(U-SCF)).LetXbe a finite set of al- ternatives with|X| ≥3and letN=1,...,nbe voters withn≥2. Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M :Dom(M)→[0,1]. LetT M be a nonempty domain of admissible uncertain preference relations onX. Apreference profileis an n-tuple R:= (R (1) M ,...,R (n) M )∈(T M ) n . Step 1 (Aggregate uncertain preferences to a collective scored relation).Fix an aggregation operator Agg: [0,1] n →[0,1] Chapter 10. Other Related Decision Methods (e.g. weighted mean, OWA, min/max, or any monotone aggregator). For each pair(x,y)∈X×X, define the collective scored relation r ∗ (x,y) :=Agg ( S M (R (1) M (x,y)),...,S M (R (n) M (x,y)) ) ∈[0,1]. Step 2 (Social choice by a score-based rule).Define thecollective dominance scoreof an alternative x∈Xby Score(x) := ∑ y∈X\x r ∗ (x,y)∈[0,m−1]. Anuncertain social choice function(U-SCF) is the mapping f: (T M ) n −→X, f(R)∈arg max x∈X Score(x), with a fixed tie-breaking convention if the argmax is not unique. The chosen alternativef(R)is called the uncertain social choice(U-social choice). Theorem 10.3.4(Well-definedness of U-social choice).Under Definition 10.3.3, assume: (i)T M 6=∅andDom(M)6=∅; (i)S M :Dom(M)→[0,1]is admissible; (i)Agg: [0,1] n →[0,1]is well-defined; (iv) a deterministic tie-breaking rule is fixed. Then the mappingf: (T M ) n →Xis well-defined. Proof.Fix a profileR∈(T M ) n . For each(x,y)and each voteri,R (i) M (x,y)∈Dom(M)by definition, soS M (R (i) M (x,y))∈[0,1]is well-defined. Hence then-tuple of scored inputs to Agg lies in[0,1] n , so r ∗ (x,y) =Agg(·)∈[0,1]is well-defined for every(x,y)∈X×X. Therefore, for eachx∈X, Score(x) = ∑ y6=x r ∗ (x,y)is a finite sum of numbers in[0,1], hence a well-defined real number. SinceXis finite, the setScore(x) :x∈Xattains a maximum, so arg max x∈X Score(x)6=∅. With a fixed tie-breaking convention,f(R)is uniquely determined. Thusfis well-defined on(T M ) n . 10.4Fuzzy decision tables Decision tables map combinations of condition states to actions, enabling transparent rule-based decisions, consistency checks, and maintenance in complex systems [1272]. Fuzzy decision tables extend decision tables with fuzzy sets and linguistic terms, handling uncertainty via graded matching and t-norm aggregation [1273,1274]. Chapter 10. Other Related Decision Methods Definition 10.4.1(Fuzzy decision table (FDT)).[1273, 1274] Letc≥1anda≥1. For each condition indexi∈1,...,cletCS i be acondition subjectwithcondition domainCD i (universe of discourse). For each action indexj∈1,...,aletAS j be anaction subjectwithaction domainAD j . (1) Fuzzy condition states.For eachi, fix a finite nonempty set of linguisticcondition states CT i = ̃ C i1 ,..., ̃ C in i , n i ≥1, where each ̃ C ik is a fuzzy set onCD i , i.e. ̃ C ik :CD i →[0,1]. WriteCT:=CT 1 ,...,CT c . (2) Fuzzy action states (two standard forms). Form 1 (crisp execution flags, fuzzy only in conditions).For eachj, let AV j :=true(x),false(−),nil(·), and define the action-configuration spaceAV:= ∏ a j=1 AV j . Form 2 (fuzzy/linguistic action values allowed).For eachj, fix a finite nonempty set ofaction states AT j = ̃ A j1 ,..., ̃ A jm j , m j ≥1, where each ̃ A j` is a fuzzy set onAD j : ̃ A j` :AD j →[0,1]. Equivalently, one may define an action-value set AV j := ̃a| ̃a∈F(AD j )or a designated linguistic subset of it, and again setAV:= ∏ a j=1 AV j . (3) Fuzzy decision table as a single-hit function.Afuzzy decision tableis a (single-valued) mapping F:CT 1 ×·×CT c −→AV such thateach possible condition-state combination is mapped into exactly one action configuration. Equiv- alently, viewing condition/action combinations as entries, an FDT is a function from the condition-entry space to the action-entry space: F:SPACE(C)→SPACE(A), whereSPACE(C) :=CT 1 ×·×CT c andSPACE(A) :=AV(Form 1 or Form 2). (4) Rule (column) interpretation.Each column of the table corresponds to a rule IFCS 1 is ̃ C 1k 1 ∧ · ∧CS c is ̃ C ck c THEN(AS 1 ,...,AS a )takes configurationF( ̃ C 1k 1 ,..., ̃ C ck c ), where the antecedent is evaluated by a chosent-norm (e.g. min) when consulting the table. Chapter 10. Other Related Decision Methods Using an Uncertain Set, we define Uncertain decision table of typeM(U-DT) as follows. Definition 10.4.2(Uncertain decision table of typeM(U-DT)).Letc≥1anda≥1. Subjects and domains.For each condition indexi∈ 1,...,c, letCS i be a condition subject with a (nonempty) condition domainCD i . For each action indexj∈1,...,a, letAS j be an action subject with an action domainAD j . Uncertain condition states (linguistic states).Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M . For each conditioni, fix a finite nonempty set of linguistic condition states CT i =C (M) i1 ,...,C (M) in i , n i ≥1, where each state is an uncertain set (degree map) onCD i : C (M) ik :CD i →Dom(M). Its induced (scored) membership is μ ik (t) :=S M ( C (M) ik (t) ) ∈[0,1], t∈CD i . Action values.Choose either of the following common forms. Form 1 (crisp execution flags).For eachj, let AV j :=true,false,nil, AV:= a ∏ j=1 AV j . Form 2 (uncertain/linguistic action values).For eachj, fix a finite nonempty set of action states AT j =A (M) j1 ,...,A (M) jm j , m j ≥1, where eachA (M) j` :AD j →Dom(M)is an uncertain set onAD j . SetAV j :=AT j andAV:= ∏ a j=1 AV j . Decision table as a single-hit mapping.Define the condition-entry space SPACE(C) :=CT 1 ×·×CT c . Anuncertain decision table(U-DT) is a mapping F:SPACE(C)−→AV, so that each condition-state combination is mapped to exactly one action configuration. Chapter 10. Other Related Decision Methods Consultation semantics (graded matching).Fix at-normT: [0,1] c →[0,1](e.g. min or product). For an observed inputo= (t 1 ,...,t c )∈CD 1 ×·×CD c and a rule-entry(C (M) 1k 1 ,...,C (M) ck c )∈SPACE(C), define its firing strength by α k 1 ,...,k c (o) :=T ( μ 1k 1 (t 1 ),...,μ ck c (t c ) ) ∈[0,1]. A standard deterministic selection rule is to choose a maximizer (k ? 1 ,...,k ? c )∈arg max (k 1 ,...,k c ) α k 1 ,...,k c (o), with a fixed tie-breaking convention, and output the action Act(o) :=F ( C (M) 1k ? 1 ,...,C (M) ck ? c ) ∈AV. Theorem 10.4.3(Well-definedness of U-DT and its consultation rule).Under Definition 10.4.2, assume: (i)Dom(M)6=∅andS M :Dom(M)→[0,1]is admissible; (i) eachCT i is finite and nonempty, andAVis nonempty; (i) thet-normTis a well-defined map[0,1] c →[0,1]; (iv) a deterministic tie-breaking convention is fixed forarg max. Then: (a) The mappingF:SPACE(C)→AVis well-defined as a function (single-hit table). (b) For every observationo∈CD 1 ×·×CD c , all firing strengthsα k 1 ,...,k c (o)are well-defined in[0,1]. (c) The consultation outputAct(o)∈AVis well-defined for every observationo. Proof.(a) By definition,SPACE(C)is a Cartesian product of nonempty sets, hence nonempty, andFis assumed to be a single-valued mapping intoAV. (b) Fixo= (t 1 ,...,t c ). For any rule-entry(k 1 ,...,k c ), eachμ ik i (t i ) =S M (C (M) ik i (t i ))∈[0,1]is well-defined by admissibility ofS M . SinceTmaps[0,1] c to[0,1],α k 1 ,...,k c (o)∈[0,1]is well-defined. (c) The set of all indices(k 1 ,...,k c )is finite because eachCT i is finite; hence the maximum of the finite set of valuesα k 1 ,...,k c (o)is attained, so arg max is nonempty. With a fixed tie-breaking rule, a unique maximizer is selected, and then Act(o)is defined by applyingFto that entry. Related concepts of decision tables under uncertainty-aware models are listed in Table 10.4. Chapter 10. Other Related Decision Methods Table 10.4: Related concepts of decision tables under uncertainty-aware models. kRelated decision table concept(s) 1 Fuzzy Decision Tables 2 Intuitionistic Fuzzy Decision Tables [1275,1276] 3 Neutrosophic Decision Tables [1277,1278] 10.5Fuzzy decision matrices Decision matrices tabulate alternatives versus criteria or states, record payoffs or ratings, and support se- lection via scoring, weighting, dominance checks, or expected-value calculations [1279,1280]. Fuzzy decision matrices replace payoffs or ratings with fuzzy numbers or fuzzy events, propagate uncertainty through fuzzy arithmetic orα-cuts, and rank alternatives robustly [1281–1283]. Definition 10.5.1(Fuzzy decision matrix (with fuzzy states of the world and fuzzy evaluations)).Let (Ω,A,P)be a probability space. LetX=x 1 ,...,x n be a finite set of alternatives and letm≥1. (1) Fuzzy states of the world.Forj= 1,...,m, letS j be afuzzy eventonΩ, i.e. a fuzzy setS j ∈F(Ω) with membership functionμ S j : Ω→[0,1]whoseα-cuts(S j ) α =ω∈Ω :μ S j (ω)≥αbelong toA(for all α∈(0,1]). Assume that(S 1 ,...,S m )forms afuzzy partitionofΩ: m ∑ j=1 μ S j (ω) = 1 (∀ω∈Ω). (2) Zadeh probabilities of fuzzy states.Define the (Zadeh) probability of a fuzzy eventAby P Z (A) := ∫ Ω μ A (ω)dP(ω), and set p Zj :=P Z (S j ) (j= 1,...,m). (3) Fuzzy evaluations.For each alternativex i and each fuzzy stateS j , letH i,j be a fuzzy number representing the evaluation (consequence) of choosingx i whenS j occurs. ThusH i,j ∈F N (R)and itsα-cut is an interval (H i,j ) α = [ h L i,j (α), h U i,j (α) ] (α∈(0,1]). (4) The fuzzy decision matrix.Thefuzzy decision matrixis the data structure D F = ( X,(S 1 ,...,S m ),(p Z1 ,...,p Zm ),(H i,j ) 1≤i≤n,1≤j≤m ) , often displayed as a table whose columns are the fuzzy statesS j (with probabilitiesp Zj ) and whose entries are the fuzzy evaluationsH i,j . Chapter 10. Other Related Decision Methods (5) Induced fuzzy characteristics (expected value and variance).Using fuzzy arithmetic (e.g. via α-cuts), thefuzzy expected evaluationofx i is (F)EH i := m ∑ j=1 p Zj ·H i,j , so that for eachα∈(0,1], ( (F)EH i ) α = m ∑ j=1 p Zj h L i,j (α), m ∑ j=1 p Zj h U i,j (α) . A commonly used (but dependency-ignoring) fuzzy variance is (F)var t (H i ) := m ∑ j=1 p Zj ( H i,j −(F)EH i ) 2 . A dependency-respecting variance can be defined byα-cuts through optimization: forα∈(0,1], (F)var s (H i ) α = [ var s,L i (α),var s,U i (α) ] , where var s,L i (α) =min m ∑ j=1 p Zj ( h i,j − m ∑ k=1 p Zk h i,k ) 2 ∣ ∣ ∣ ∣ ∣ ∣ h i,j ∈(H i,j ) α , j= 1,...,m , var s,U i (α) =max m ∑ j=1 p Zj ( h i,j − m ∑ k=1 p Zk h i,k ) 2 ∣ ∣ ∣ ∣ ∣ ∣ h i,j ∈(H i,j ) α , j= 1,...,m . Definition 10.5.2(Uncertain decision matrix of typeM(U-DM)).LetA=A 1 ,...,A m be a finite set of alternatives andC=C 1 ,...,C n a finite set of criteria, withm,n≥1. Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M . Anuncertain decision matrix of typeMis a triple ̃ X (M) = ( A,C,X (M) ) , X (M) = ( x (M) ij ) m×n , x (M) ij ∈Dom(M), wherex (M) ij encodes the uncertain evaluation (performance/payoff) of alternativeA i under criterionC j . The inducedscored (crisp) decision matrixis X= ( x ij ) m×n , x ij :=S M ( x (M) ij ) ∈R. IfS M is admissible positive, thenX∈(0,∞) m×n . Definition 10.5.3(Basic transformations on a U-DM).Let ̃ X (M) = (A,C,X (M) )be a U-DM and let X=S M (X (M) )be its scored matrix. • Aweight vectoris anyw= (w 1 ,...,w n )withw j ≥0and ∑ n j=1 w j = 1. Chapter 10. Other Related Decision Methods • Abenefit/cost orientationis a partitionC=C ben ̇ ∪C cost , often converted to “larger is better” by replacingx ij with−x ij onC cost . • Anormalized matrixR= (r ij )is any transformationR=N(X)such thatr ij is finite (common choices: min–max, vector normalization, ratio normalization). Theorem 10.5.4(Well-definedness of uncertain decision matrices).Under Definition 10.5.2, assume Dom(M)6=∅andS M is admissible. Then: (i) The scored decision matrixX= (S M (x (M) ij ))is well-defined and belongs toR m×n . (i) Any finite aggregation of entries ofX(e.g. weighted sums ∑ j w j x ij or pairwise differencesx ij −x kj ) is well-defined as a real number. (i) IfS M is admissible positive, then all ratio-based normalizations of the form r ij = x ij max p x pj , r ij = min p x pj x ij , r ij = x ij √ ∑ p x 2 pj are well-defined (denominators are strictly positive). Proof.(i) Sincex (M) ij ∈Dom(M)andS M is admissible, eachx ij =S M (x (M) ij )is a finite real number. Thus X∈R m×n is well-defined. (i) Finite sums and differences of real numbers are well-defined, so any finite aggregation (including weighted sums withw j ≥0and ∑ j w j = 1) is well-defined. (i) IfS M is admissible positive, thenx ij >0for alli,j. Hence max p x pj >0, min p x pj >0, and ∑ p x 2 pj >0 for each fixedj, so all displayed ratio-based normalizations are well-defined. 10.6Fuzzy Ultrafilter An ultrafilter is a maximal filter on a set: for every subset, either it or its complement belongs to the ultrafilter, never both [1284]. An ultrafilter relates to decision-making by modeling decisive coalitions in collective choice: under Arrow-type axioms, the sets able to determine social outcomes form an ultrafilter. Fuzzy ultrafilters assign each subset a membership degree in[0,1], preserving monotonicity and intersection, with maximal top-level acceptance (cf. [1285–1287]). Definition 10.6.1(Filter and Ultrafilter).(cf. [1284,1288,1289]) LetXbe a nonempty set. A(classical) filteronXis a nonempty familyF ⊆P(X)such that 1.∅/∈FandX∈F; 2. (Upward closed) ifA∈FandA⊆B⊆X, thenB∈F; Chapter 10. Other Related Decision Methods 3. (Closed under finite intersections) ifA,B∈F, thenA∩B∈F. A(classical) ultrafilteronXis a filterUonXwhich is maximal under inclusion: wheneverFis a filter on XwithU ⊆F, one hasF=U. Equivalently, a filterUonXis an ultrafilter if and only if it satisfies theultrafilter dichotomy: ∀A⊆X, A∈UorX ∈U. Definition 10.6.2(Principal Ultrafilter).LetXbe a nonempty set and fix a pointx∈X. Theprincipal ultrafilter atxis the family U x :=A⊆X:x∈A. Equivalently,U x is the unique ultrafilter onXsatisfying A∈U x ⇐⇒x∈A(∀A⊆X). ThusU x expresses the policy “select the specific elementxthroughout,” because every set is accepted exactly when it containsx. Definition 10.6.3(Fuzzy Ultrafilter on a Set).(cf. [1285–1287]) LetXbe a nonempty set. Afuzzy ultrafilteronXis a map ̃ U:P(X)−→[0,1] satisfying, for allA,B⊆X: 1. (Normalization) ̃ U(X) = 1and ̃ U(∅) = 0. 2. (Monotonicity) IfA⊆B, then ̃ U(A)≤ ̃ U(B). 3. (Meet as intersection) ̃ U(A∩B) =min ̃ U(A), ̃ U(B). 4. (Ultrafilter maximality) max ̃ U(A), ̃ U(X )= 1. Definition 10.6.4(Principal Fuzzy Ultrafilter).[1290] LetXbe a nonempty set and fixx∈X. The principal fuzzy ultrafilter atxis the fuzzy ultrafilter ̃ U x :P(X)−→[0,1], ̃ U x (A) := 1, x∈A, 0, x/∈A. In particular, ̃ U x (A) = 1⇐⇒x∈A, so ̃ U x encodes “a fixed elementxis (fully) selected” in a fuzzy-valued format. Definition 10.6.5(Uncertain ultrafilter of typeM).LetXbe a nonempty set and fix an uncertain model Mwith Dom(M)6=∅. Fix an admissible scoreS M :Dom(M)→[0,1]. Anuncertain ultrafilter of typeM(relative toS M )is a mapping U M :P(X)−→Dom(M) whosescored acceptancemap u:P(X)→[0,1], u(A) :=S M ( U M (A) ) satisfies, for allA,B⊆X: Chapter 10. Other Related Decision Methods (U1)(Normalization)u(X) = 1andu(∅) = 0. (U2)(Monotonicity)A⊆B⇒u(A)≤u(B). (U3)(Meet = intersection)u(A∩B) =minu(A),u(B). (U4)(Ultrafilter dichotomy at level1)maxu(A),u(X )= 1. Theorem 10.6.6(Well-definedness and induced crisp ultrafilter).LetU M :P(X)→Dom(M)satisfy Definition 10.6.5. Then: (i)The scored acceptance mapu=S M ◦U M is well-defined and satisfiesu(A)∈[0,1]for allA⊆X. (i)The family F 1 :=A⊆X:u(A) = 1 is a (classical) ultrafilter onX. Proof.(i) SinceS M is admissible,S M (d)∈[0,1]for everyd∈Dom(M); henceu(A) =S M (U M (A))∈[0,1] for allA. (i) First,∅/∈ F 1 becauseu(∅) = 0by (U1), andX∈ F 1 becauseu(X) = 1. IfA∈ F 1 andA⊆B, then u(B)≥u(A) = 1by (U2), henceu(B) = 1andB∈F 1 (upward closed). IfA,B∈F 1 , then by (U3), u(A∩B) =minu(A),u(B)=min1,1= 1, soA∩B∈F 1 (closed under finite intersections). ThusF 1 is a filter. To show maximality, letA⊆X. By (U4), maxu(A),u(X )= 1, so eitheru(A) = 1oru(X ) = 1, i.e., A∈F 1 orX ∈F 1 . HenceF 1 satisfies the ultrafilter dichotomy, and therefore is a classical ultrafilter. Definition 10.6.7(Principal uncertain ultrafilter).LetXbe a nonempty set, fixx∈X, and fix an uncertain modelMwith an admissible scoreS M . Assume there exist two degree tuplesd 1 ,d 0 ∈Dom(M) such that S M (d 1 ) = 1, S M (d 0 ) = 0. DefineU M,x :P(X)→Dom(M)by U M,x (A) := d 1 , x∈A, d 0 , x/∈A. ThenU M,x is called theprincipal uncertain ultrafilter atx. Proposition 10.6.8(Well-definedness of the principal uncertain ultrafilter).Under Definition 10.6.7,U M,x is an uncertain ultrafilter of typeM. Proof.Letu(A) =S M (U M,x (A))∈ 0,1. Thenu(X) = 1,u(∅) = 0. IfA⊆Bandu(A) = 1then x∈A⊆Bsou(B) = 1; otherwiseu(A) = 0≤u(B), proving monotonicity. Alsou(A∩B) = 1iffx∈A andx∈B, i.e. iffu(A) =u(B) = 1, sou(A∩B) =minu(A),u(B). Finally, for anyA, eitherx∈Aor x∈X , hence maxu(A),u(X )= 1. Thus (U1)–(U4) hold. Related concepts of ultrafilters under uncertainty-aware models are listed in Table 10.5. Besides Uncertain Ultrafilters, several other related concepts are also known, including Weak Ultrafilters [1299,1300], Soft Ultrafilters [1288,1289], and HyperSoft Ultrafilters [1301]. Chapter 10. Other Related Decision Methods Table 10.5: Related concepts of ultrafilters under uncertainty-aware models. kRelated ultrafilter concept(s) 2 Intuitionistic Fuzzy Ultrafilter [1291–1293] 3 Neutrosophic Ultrafilter [1294–1297] 3 Picture Fuzzy Ultrafilter [1298] nPlithogenic Ultrafilter [1297] 10.7Fuzzy SWOT Analysis SWOT analysis identifies strengths, weaknesses, opportunities, and threats, organizes internal-external fac- tors, and derives strategies for planning decisions [1302, 1303]. Fuzzy SWOT analysis represents SWOT factors with fuzzy numbers or linguistic terms, aggregates uncertainties, and prioritizes strategies using fuzzy ranking [1304, 1305]. SWOT supports decision-making by structuring internal and external factors, generating strategic alternatives, prioritizing options with weights/scores, and justifying choices transpar- ently under uncertainty. Definition 10.7.1(Fuzzy SWOT Analysis (FSWOT)).[1304, 1305] LetI=I 1 ,...,I n i be the set of internal factorsandE=E 1 ,...,E n e the set ofexternal factors. Fix the SWOT coordinate scale X:= [−10,10] (internal axis),Y:= [−10,10] (external axis), wherex <0representsweakness,x >0strength, andy <0threat,y >0opportunity. AFuzzy SWOT analysisis a procedure that maps the factor sets(I,E)into a prioritized list of strategies by the following mathematical components. (1) Fuzzification of factors.Each internal factorI u is represented by a membership functionμ I u :X → [0,1], and each external factorE v byμ E v :Y →[0,1]. A common parametrization is the triangular form I u ≡(x p u ,x m u ,x o u )andE v ≡(y p v ,y m v ,y o v ), with μ I u (x) =tri(x;x p u ,x m u ,x o u ), μ E v (y) =tri(y;y p v ,y m v ,y o v ), where fora < b < c, tri(t;a,b,c) := 0,t < a, t−a b−a , a≤t≤b, c−t c−b , b≤t≤c, 0,t > c. (2) Aggregation surface (fuzzy SWOT surface).Fix a t-normT: [0,1] 2 →[0,1](oftenT=min). For each pair(I u ,E v )define the aggregated membership surface μ u,v :X ×Y →[0,1], μ u,v (x,y) :=T ( μ I u (x),μ E v (y) ) . (3)α-cut defuzzification and fuzzy area.For a chosenα∈(0,1], define theα-cut region (fuzzy area) A u,v (α) :=(x,y)∈X ×Y:μ u,v (x,y)≥α. Chapter 10. Other Related Decision Methods This set encodes the interaction betweenI u andE v at uncertainty levelα. (4) Quadrant assignment (SO/ST/WO/WT).Let Q SO =(x,y) :x≥0, y≥0, Q ST =(x,y) :x≥0, y≤0, Q W O =(x,y) :x≤0, y≥0, Q W T =(x,y) :x≤0, y≤0. IfA u,v (α)intersects multiple quadrants, define its quadrant by maximal area: quad u,v (α)∈argmax Q∈Q SO ,Q ST ,Q W O ,Q W T Area ( A u,v (α)∩Q ) . (5) Prioritization via ideal-point closeness (one standard choice).AssumeA u,v (α)has positive area. Let its centroid (center of gravity) be (x u,v (α),y u,v (α)) := 1 Area(A u,v (α)) ∫ A u,v (α) (x,y)dxdy. Define the positive and negative ideal points z + = (10,10), z − = (−10,−10), the Euclidean distances d + u,v (α) := ∥ ∥ (x u,v (α),y u,v (α))−z + ∥ ∥ 2 , d − u,v (α) := ∥ ∥ (x u,v (α),y u,v (α))−z − ∥ ∥ 2 , and thecloseness coefficient c u,v (α) := d − u,v (α) d − u,v (α) +d + u,v (α) ∈[0,1]. Larger c u,v (α)indicates higher priority for the pair(I u ,E v )at levelα. (6) Strategy extraction.A (crisp) strategy candidate is an ordered triple σ= (I u ,E v ,quad u,v (α)), interpreted as a strategy generated from the internal factorI u and external factorE v according to the chosen quadrant. The prioritized strategy list at levelαis obtained by sorting candidates by c u,v (α)(or another chosen ranking functional). (7) Multi-αsynthesis (optional).Forα 1 ,...,α K ∈(0,1]and weightsw k ≥0with ∑ K k=1 w k = 1, define a final score Score u,v := K ∑ k=1 w k c u,v (α k ), and rank strategies by Score u,v . Chapter 10. Other Related Decision Methods Definition 10.7.2(Uncertain SWOT analysis of typeM(U-SWOT)).LetI=I 1 ,...,I n i be internal factors andE=E 1 ,...,E n e external factors. Fix an uncertain modelMwith Dom(M)6=∅and an admissible scoreS M . Assume each internal factorI u and external factorE v is assessed by uncertain degrees ι (M) u ∈Dom(M) (u= 1,...,n i ), (M) v ∈Dom(M) (v= 1,...,n e ). Let the internal and external signed scores be x u :=S M (ι (M) u )∈R, y v :=S M ( (M) v )∈R. Interpretation (convention): x u >0 :strength, x u <0 :weakness, y v >0 :opportunity, y v <0 :threat. (If a different sign convention is preferred, replacex u ,y v by affine rescalings.) Fix a nonnegativeimportance weightfor each factor: α u ≥0 (u= 1,...,n i ), β v ≥0 (v= 1,...,n e ), and, for normalization convenience, assume ∑ u α u = 1and ∑ v β v = 1. (1) Strategy candidates (pairing).Each pair(I u ,E v )generates a candidate strategy token σ uv := (u,v). (2) Quadrant (SO/ST/WO/WT) assignment.Define quad(u,v) := SO, x u ≥0, y v ≥0, ST, x u ≥0, y v <0, WO, x u <0, y v ≥0, WT, x u <0, y v <0. (3) Pair priority score.Define the (nonnegative) interaction magnitude κ uv :=α u β v |x u ||y v | ∈R ≥0 . Normalize to obtain a comparable priority score: Score uv := κ uv n i ∑ p=1 n e ∑ q=1 κ pq , ∑ p,q κ pq >0, 1 n i n e , ∑ p,q κ pq = 0, Score uv ∈[0,1]. (4) U-SWOT output.The U-SWOT output is the prioritized list of annotated strategies ( σ uv ,quad(u,v),Score uv ) u,v sorted by decreasing Score uv (ties allowed). Chapter 10. Other Related Decision Methods Theorem 10.7.3(Well-definedness of U-SWOT).Under Definition 10.7.2, assumeDom(M)6=∅andS M is admissible. Then: (i) The signed scoresx u ,y v are well-defined real numbers. (i) For every(u,v), the quadrant labelquad(u,v)∈SO,ST,WO,WTis well-defined. (i) The scoresκ uv ≥0andScore uv ∈[0,1]are well-defined, and n i ∑ u=1 n e ∑ v=1 Score uv = 1. (iv) Consequently, the prioritized strategy list (sorted byScore uv ) is well-defined. Proof.(i) Sinceι (M) u , (M) v ∈Dom(M)andS M is admissible,x u =S M (ι (M) u )andy v =S M ( (M) v )are finite real numbers. (i) Each quadrant condition is a Boolean combination of comparisons of real numbers (x u ≥0,y v ≥0etc.), so exactly one of the four cases applies; thus quad(u,v)is well-defined. (i) By assumptionα u ,β v ≥0, soκ uv =α u β v |x u ||y v | ≥0is well-defined. If ∑ p,q κ pq >0, then Score uv = κ uv / ∑ p,q κ pq is well-defined and lies in[0,1], and summing over(u,v)yields1. If ∑ p,q κ pq = 0, the definition sets Score uv = 1/(n i n e ), which also lies in[0,1]and sums to1. (iv) The set of candidates is finite, hence sorting by the real scores Score uv yields a well-defined ranking (with ties allowed). Related concepts of SWOT under uncertainty-aware models are listed in Table 10.6. Table 10.6: Related concepts of SWOT under uncertainty-aware models. kRelated SWOT concept(s) 1 Fuzzy SWOT 2 Intuitionistic Fuzzy SWOT [1306] 3 Hesitant Fuzzy SWOT [1307,1308] 3 Neutrosophic SWOT [1309–1311] As extensions other than Uncertain SWOT Analysis, several related concepts have been proposed, including the TOWS Matrix [1312, 1313], SWOT-AHP [1314, 1315], ANP-SWOT [1316, 1317], Quantified SWOT [1318,1319], SWOT-QSPM [1320,1321], and SOAR [1322,1323]. Chapter 10. Other Related Decision Methods 10.8Fuzzy Cost-Benefit Analysis Cost-benefit analysis compares monetized benefits and costs over time, discounts cashflows, computes net present value, and guides project selection decisions [1324,1325]. Fuzzy cost-benefit analysis models uncer- tain benefits, costs, or discount rates as fuzzy numbers, aggregates viaα-cuts, ranks alternatives robustly overall [1326–1328]. Definition 10.8.1(Fuzzy Cost–Benefit Analysis (Fuzzy CBA)).[1326–1328] Fix a finite planning horizon T∈Nand letR + := [0,∞). LetF(R + )denote the set of fuzzy numbers onR + (normal, convex, upper semicontinuous, compact support). Consider a project (or design alternative)x. For each timet= 0,1,...,T, model the (uncertain) benefit and cost cash-flows by fuzzy numbers ̃ B t (x)∈F(R + ), ̃ C t (x)∈F(R + ). Let the discount rate be either crispr∈[0,∞)or fuzzy ̃r∈ F([0,∞)). Define the (crisp) discount factor d t (r) := (1 +r) −t and, whenris fuzzy, treatd t ( ̃r)via the extension principle (equivalently,α-cuts). (1) Fuzzy present worth of benefits and costs.Using fuzzy arithmetic (e.g. via Zadeh’s extension principle; equivalently viaα-cuts), define the fuzzy present worths ̃ PW B (x) := T ⊕ t=0 ̃ B t (x) d t (r), ̃ PW C (x) := T ⊕ t=0 ̃ C t (x) d t (r), whenris crisp; and replaced t (r)byd t ( ̃r)whenris fuzzy. Here⊕and denote fuzzy addition and scalar multiplication. α-cut characterization (crispr).Ifris crisp and all cash-flows are nonnegative fuzzy numbers, then for eachα∈(0,1], writing( ̃ B t (x)) α = [B L t (α;x),B U t (α;x)]and( ̃ C t (x)) α = [C L t (α;x),C U t (α;x)], ( ̃ PW B (x) ) α = [ T ∑ t=0 B L t (α;x) (1 +r) t , T ∑ t=0 B U t (α;x) (1 +r) t ] , ( ̃ PW C (x) ) α = [ T ∑ t=0 C L t (α;x) (1 +r) t , T ∑ t=0 C U t (α;x) (1 +r) t ] . (2) Fuzzy benefit–cost ratio (fuzzy CBF/BCR).Assume ̃ PW C (x)is strictly positive in the sense that inf( ̃ PW C (x)) α >0for allα∈(0,1]. Define thefuzzy benefit–cost ratio(also called fuzzy cost–benefit function) ̃ BCR(x) := ̃ PW B (x) ̃ PW C (x), where is fuzzy division. With nonnegativeα-cut intervals( ̃ PW B (x)) α = [PW L B (α;x),PW U B (α;x)]and ( ̃ PW C (x)) α = [PW L C (α;x),PW U C (α;x)]andPW L C (α;x)>0, one obtains ( ̃ BCR(x) ) α = [ PW L B (α;x) PW U C (α;x) , PW U B (α;x) PW L C (α;x) ] . Classically, BCR>1indicates cost-effectiveness; fuzzy CBA keeps this rule but interprets it through a fuzzy comparison/ranking. Chapter 10. Other Related Decision Methods (3) Decision rule (one mathematically standard option).Let ̃ Zbe a fuzzy number onRand define the possibility and necessity of an event ̃ Z≥z 0 by Π( ̃ Z≥z 0 ) :=sup z≥z 0 μ ̃ Z (z), N( ̃ Z≥z 0 ) := 1−sup z<z 0 μ ̃ Z (z). For a confidence levelη∈[0,1], accept projectxif N ( ̃ BCR(x)≥1 ) ≥η(conservative),orΠ ( ̃ BCR(x)≥1 ) ≥η(optimistic). For ranking multiple alternatives, compare ̃ BCR(x)by any fixed fuzzy-number ranking functional (e.g. centroid defuzzification). (4) Incremental fuzzy CBA for mutually exclusive alternatives.Given two alternativesx(chal- lenger) andy(defender), define incremental fuzzy present worths ∆ ̃ PW B := ̃ PW B (x) ̃ PW B (y),∆ ̃ PW C := ̃ PW C (x) ̃ PW C (y), and the incremental ratio ∆ ̃ BCR:= ∆ ̃ PW B ∆ ̃ PW C . Then selectxoveryif (in the chosen fuzzy-comparison sense)∆ ̃ BCR>1; iterate this pairwise elimination to obtain a final choice. Definition 10.8.2(Uncertain cost–benefit analysis of typeM(U-CBA)).Fix a finite planning horizon T∈Nand a finite set of projects (alternatives)X. Fix an uncertain modelMwith Dom(M)6=∅and admissible scoresS $ M ,S + M . For each projectx∈Xand each timet= 0,1,...,T, assume uncertain benefit and cost degrees B (M) t (x)∈Dom(M), C (M) t (x)∈Dom(M), and assume an uncertain discount-rate degreer (M) ∈Dom(M). Define the induced (crisp) nonnegative cashflows and discount rate by scoring: B t (x) :=S $ M ( B (M) t (x) ) ∈[0,∞), C t (x) :=S $ M ( C (M) t (x) ) ∈[0,∞), r:=S + M (r (M) )−1∈[0,∞). (Equivalently, one may defineS r M :Dom(M)→[0,∞)directly; the above form enforces1 +r >0.) Define the discount factor d t (r) := (1 +r) −t ∈(0,1] (t= 0,...,T). (1) Present worths.Define the present worth (PW) of benefits and costs: PW B (x) := T ∑ t=0 B t (x)d t (r),PW C (x) := T ∑ t=0 C t (x)d t (r). Chapter 10. Other Related Decision Methods (2) Net present value and benefit–cost ratio.Define NPV(x) :=PW B (x)−PW C (x)∈R, and, provided PW C (x)>0, define the benefit–cost ratio BCR(x) := PW B (x) PW C (x) ∈[0,∞). (3) Decision rules (two standard options). •Feasibility/acceptance:acceptxif NPV(x)≥0(or equivalently BCR(x)≥1when PW C (x)>0). •Ranking:rank projects by decreasing NPV(x)(or decreasing BCR(x)). Theorem 10.8.3(Well-definedness of U-CBA).Under Definition 10.8.2, assume: (i)S $ M is admissible monetary (finite onDom(M)) andS + M is admissible positive; (i)T <∞andXis finite. Then for every projectx∈X: (a)PW B (x)andPW C (x)are well-defined finite nonnegative real numbers. (b)NPV(x)is well-defined finite real. (c) IfPW C (x)>0, thenBCR(x)is well-defined and finite. Moreover, if there exists at least onexwithPW C (x)>0, then ranking byBCRis well-defined (up to ties) on that subset. Proof.(a) For eacht,B t (x),C t (x)∈[0,∞)by admissibility ofS $ M . Alsor≥0implies1 +r >0and henced t (r) = (1 +r) −t ∈(0,1]is well-defined. Therefore each termB t (x)d t (r)andC t (x)d t (r)is finite and nonnegative. SinceTis finite, the sums defining PW B (x)and PW C (x)are finite and nonnegative. (b) NPV(x)is the difference of two finite reals, hence finite. (c) If PW C (x)>0, then division by PW C (x)is valid and yields a finite real BCR(x)≥0. Finally, ranking by BCR on the subsetx:PW C (x)>0is well-defined because it compares real numbers (and ties are allowed). Related concepts of cost-benefit analysis under uncertainty-aware models are listed in Table 10.7. Chapter 10. Other Related Decision Methods Table 10.7: Related concepts of cost-benefit analysis under uncertainty-aware models. kRelated cost-benefit analysis concept(s) 1 Fuzzy Cost-Benefit Analysis 2 Intuitionistic Fuzzy Cost-Benefit Analysis 3 Neutrosophic Cost-Benefit Analysis (cf. [1329]) 3 Spherical Fuzzy Cost-Benefit Analysis [1330,1331] 10.9Fuzzy Decision curve analysis Decision curve analysis evaluates prediction models across threshold probabilities by net benefit, balancing true positives against false positives to identify clinically useful decision strategies [1332, 1333]. Fuzzy decision curve analysis extends decision curve analysis by using fuzzy risk estimates or thresholds, modeling uncertainty in predictions while preserving net-benefit-based strategy comparison. Let FN [0,1] denote a fixed class of fuzzy numbers supported on the interval[0,1]. Thus, each ̃p∈FN [0,1] is a fuzzy set on[0,1]with membership functionμ ̃p : [0,1]→[0,1], and is interpreted as a fuzzy predicted risk / fuzzy probability. Definition 10.9.1(Fuzzy Decision Curve Analysis (FDCA)).Let Ω =1,...,N be a finite set of cases (patients, objects, or decision units), and let y i ∈0,1(i∈Ω) denote the true binary outcome, wherey i = 1means “event/disease present” andy i = 0means “event/dis- ease absent.” Assume that a given prediction strategySprovides, for each casei∈Ω, a fuzzy predicted risk ̃p (S) i ∈FN [0,1] . Fix a threshold probability t∈(0,1), and let H: [0,1)→[0,∞) be a prescribedtest-harm function(possiblyH≡0). For each casei, define thefuzzy treatment-recommendation degreeat thresholdtby r (S) i (t) := Π ( ̃p (S) i ≥t ) :=sup u∈[t,1] μ ̃p (S) i (u)∈[0,1]. Hencer (S) i (t)is the possibility degree that the fuzzy predicted risk of caseireaches or exceedst. Chapter 10. Other Related Decision Methods Define the corresponding fuzzy-weighted true-positive and false-positive counts by TP (S) F (t) := N ∑ i=1 y i r (S) i (t),FP (S) F (t) := N ∑ i=1 (1−y i )r (S) i (t). Thefuzzy net benefitof strategySat thresholdtis then defined by NB (S) F (t) := 1 N TP (S) F (t)− 1 N FP (S) F (t) t 1−t −H(t). The mapping NB (S) F : (0,1)→R, t7−→NB (S) F (t), is called thefuzzy decision curveof strategyS. IfSis a finite family of competing strategies, then theFDCA-optimal strategy setat thresholdtis Opt F (t) :=arg max S∈S NB (S) F (t). A strategyS 1 is said todominateS 2 on a threshold regionT⊂(0,1)if NB (S 1 ) F (t)≥NB (S 2 ) F (t)for allt∈T, with strict inequality for at least onet∈T. Proposition 10.9.2(Well-definedness of FDCA).Assume that, for a fixed strategyS, ̃p (S) i ∈FN [0,1] (i= 1,...,N), thatt∈(0,1), and thatH(t)<∞. Then: 1.r (S) i (t)∈[0,1]is well-defined for everyi; 2.TP (S) F (t)andFP (S) F (t)are well-defined real numbers satisfying 0≤TP (S) F (t)≤N,0≤FP (S) F (t)≤N; 3.NB (S) F (t)∈Ris well-defined; 4. ifSis finite and nonempty, thenOpt F (t)6=∅. Proof.Since each ̃p (S) i is a fuzzy number on[0,1], its membership functionμ ̃p (S) i : [0,1]→[0,1]is well- defined. Therefore r (S) i (t) =sup u∈[t,1] μ ̃p (S) i (u) is well-defined and belongs to[0,1], proving (1). Chapter 10. Other Related Decision Methods Becausey i ∈0,1and0≤r (S) i (t)≤1, each summand in TP (S) F (t) = N ∑ i=1 y i r (S) i (t) and FP (S) F (t) = N ∑ i=1 (1−y i )r (S) i (t) lies in[0,1]. Hence both sums are well-defined real numbers in[0,N], proving (2). Sincet∈(0,1), the factort/(1−t)is a finite real number. Together withH(t)<∞, this implies that NB (S) F (t) = 1 N TP (S) F (t)− 1 N FP (S) F (t) t 1−t −H(t) is a well-defined real number. Thus (3) holds. Finally, ifSis finite and nonempty, then the set NB (S) F (t) :S∈S ⊂R is finite and nonempty, and therefore attains its maximum. Hence Opt F (t) =arg max S∈S NB (S) F (t)6=∅. This proves (4). Proposition 10.9.3(Reduction to classical DCA).Assume that each fuzzy predicted risk degenerates to a crisp singleton: ̃p (S) i =χ p (S) i , p (S) i ∈[0,1]. Then r (S) i (t) =1 [ p (S) i ≥t ] , and consequently TP (S) F (t) =#i:y i = 1, p (S) i ≥t, FP (S) F (t) =#i:y i = 0, p (S) i ≥t. HenceNB (S) F (t)reduces exactly to the usual decision-curve net benefit NB (S) (t) = TP (S) (t) N − FP (S) (t) N t 1−t −H(t). Proof.If ̃p (S) i =χ p (S) i , then μ ̃p (S) i (u) = 1, u=p (S) i , 0, u6=p (S) i . Therefore r (S) i (t) =sup u∈[t,1] μ ̃p (S) i (u) = 1, p (S) i ≥t, 0, p (S) i < t, which is exactly1[p (S) i ≥t]. The remaining statements follow immediately by substitution into the definitions of TP (S) F (t), FP (S) F (t), and NB (S) F (t). Chapter 10. Other Related Decision Methods We now defineUncertain Decision Curve Analysis(UDCA) by extending fuzzy / probabilistic decision curve analysis to a general uncertain modelM. Definition 10.9.4(Uncertain Decision Curve Analysis (UDCA) of typeM).LetMbe an uncertain model with degree-domain Dom(M)⊆[0,1] k for some integerk≥1. Let Ω =1,...,N be a finite set of cases, and let y i ∈0,1(i= 1,...,N) denote the true binary outcome, wherey i = 1means that the event/disease is present andy i = 0means that it is absent. LetSbe a finite nonempty family of candidate decision strategies. For each strategyS∈S, assume that each casei∈Ωis assigned anuncertain predicted risk ρ (S) i ∈Dom(M). Fix a threshold probability t∈(0,1), and let H: (0,1)→[0,∞) be a prescribedtest-harm function. Assume further that a model-dependentthreshold-exceedance functional Exc M :Dom(M)×(0,1)−→[0,1] is given. Ford∈Dom(M)andt∈(0,1), the value Exc M (d,t) is interpreted as the degree to which the uncertain predicted riskdreaches or exceeds the treatment threshold t. For each strategyS∈S, casei∈Ω, and thresholdt∈(0,1), define theuncertain treatment-recommendation degree r (S) i,M (t) :=Exc M (ρ (S) i ,t)∈[0,1]. Define theuncertain true-positive countanduncertain false-positive countofSat thresholdtby TP (S) M (t) := N ∑ i=1 y i r (S) i,M (t),FP (S) M (t) := N ∑ i=1 (1−y i )r (S) i,M (t). Chapter 10. Other Related Decision Methods Theuncertain net benefitof strategySat thresholdtis defined by NB (S) M (t) := 1 N TP (S) M (t)− 1 N FP (S) M (t) t 1−t −H(t). The mapping NB (S) M : (0,1)→R, t7−→NB (S) M (t), is called theuncertain decision curveof strategyS. The induced preference relation among strategies at thresholdtis S 1 M,t S 2 ⇐⇒NB (S 1 ) M (t)≥NB (S 2 ) M (t). TheUDCA-optimal strategy setat thresholdtis Opt M (t) :=arg max S∈S NB (S) M (t). A strategyS 1 is said todominateS 2 on a threshold regionT⊂(0,1)if NB (S 1 ) M (t)≥NB (S 2 ) M (t)for allt∈T, with strict inequality for at least onet∈T. Theorem 10.9.5(Well-definedness of UDCA).LetMbe an uncertain model, and let U DCA = ( Ω,(y i ) N i=1 ,S,(ρ (S) i ) i,S ,Exc M , H ) be a UDCA instance as in Definition 10.9.4. Assume: (A1)N≥1,Ω =1,...,N, andy i ∈0,1for alli∈Ω; (A2)Sis finite and nonempty; (A3) for everyS∈Sandi∈Ω, ρ (S) i ∈Dom(M); (A4) Exc M :Dom(M)×(0,1)→[0,1] is a total map; (A5) H: (0,1)→[0,∞) is a total map. Then, for every thresholdt∈(0,1)and every strategyS∈S, the following objects are well-defined: Chapter 10. Other Related Decision Methods (i) the uncertain treatment-recommendation degrees r (S) i,M (t) =Exc M (ρ (S) i ,t)∈[0,1] (i= 1,...,N); (i) the uncertain true-positive and false-positive counts TP (S) M (t),FP (S) M (t)∈[0,N]; (i) the uncertain net benefit NB (S) M (t)∈R; (iv) the induced preference relation M,t onS, which is a total preorder; (v) the optimal strategy set Opt M (t) =arg max S∈S NB (S) M (t), which is nonempty. Hence Uncertain Decision Curve Analysis of typeMis well-defined. Proof.Fixt∈(0,1)andS∈S. By (A3), for eachi∈Ω, ρ (S) i ∈Dom(M). Sincet∈(0,1)and Exc M is total by (A4), the value r (S) i,M (t) =Exc M (ρ (S) i ,t) is well-defined and belongs to[0,1]. This proves (i). Next, becausey i ∈0,1by (A1) andr (S) i,M (t)∈[0,1], each summand in TP (S) M (t) = N ∑ i=1 y i r (S) i,M (t) and FP (S) M (t) = N ∑ i=1 (1−y i )r (S) i,M (t) belongs to[0,1]. Hence both sums are well-defined real numbers, and moreover 0≤TP (S) M (t)≤N,0≤FP (S) M (t)≤N. Thus (i) holds. Sincet∈(0,1), the quantity t 1−t Chapter 10. Other Related Decision Methods is a finite real number. Also,H(t)∈[0,∞)is defined by (A5). Therefore NB (S) M (t) = 1 N TP (S) M (t)− 1 N FP (S) M (t) t 1−t −H(t) is a well-defined real number. This proves (i). Now define S 1 M,t S 2 ⇐⇒NB (S 1 ) M (t)≥NB (S 2 ) M (t). Because≥onRis reflexive, transitive, and total, the induced relation M,t onSis also reflexive, transitive, and total. Hence it is a total preorder. This proves (iv). Finally, sinceSis finite and nonempty by (A2), the set NB (S) M (t) :S∈S ⊂R is finite and nonempty, so it attains a maximum. Therefore Opt M (t) =arg max S∈S NB (S) M (t)6=∅. This proves (v), and hence UDCA is well-defined. Proposition 10.9.6(Reduction to classical DCA).Assume thatMis the crisp probabilistic model with Dom(M) = [0,1], and let the threshold-exceedance functional be Exc M (p,t) :=1[p≥t] (p∈[0,1], t∈(0,1)). Then, for every strategyS, we have r (S) i,M (t) =1[ρ (S) i ≥t], and consequently TP (S) M (t) =#i:y i = 1, ρ (S) i ≥t, FP (S) M (t) =#i:y i = 0, ρ (S) i ≥t. HenceNB (S) M (t)reduces exactly to the usual classical decision-curve net benefit formula. Proof.Immediate from the definition of Exc M and substitution into the formulas of Definition 10.9.4. 10.10Fuzzy Rational Choice Rational choice selects the most preferred feasible alternative according to a consistent preference relation, assuming decision makers compare options coherently and choose utility-maximizing outcomes [1334,1335]. Fuzzy rational choice extends rational choice by using fuzzy preference degrees, allowing vague or partial comparisons, and selecting alternatives that sufficiently dominate competitors [1336–1338]. Chapter 10. Other Related Decision Methods Definition 10.10.1(Fuzzy Rational Choice).[1339,1340] LetXbe a finite nonempty set of alternatives, and let P ∗ (X) :=A⊆X:A6=∅ denote the family of all nonempty subsets ofX. Afuzzy preference relationonXis a mapping r:X×X−→[0,1], wherer(a,b)represents the degree to whichais at least as good asb. For eacha∈X, define its1/2-upper contour set by R 1/2 (a) :=x∈X:r(a,x)≥ 1 2 . Thechoice correspondence induced byris the mapping C r :P ∗ (X)−→P(X) given by C r (A) :=a∈A:A⊆R 1/2 (a)=a∈A:r(a,x)≥ 1 2 for allx∈A, A∈P ∗ (X). A choice function C:P ∗ (X)−→P(X) is called afuzzy rational choice functionif there exists a fuzzy preference relationr:X×X→[0,1]such that C(A) =C r (A)for allA∈P ∗ (X). In this case,ris said torationalizeC. We now defineUncertain Rational Choiceby extending fuzzy rational choice from the fuzzy degree-domain [0,1]to a general uncertain modelMwith degree-domain Dom(M)⊆[0,1] k . Definition 10.10.2(Uncertain binary preference relation of typeM).LetXbe a finite nonempty set of alternatives, and letMbe an uncertain model with degree-domain Dom(M)⊆[0,1] k for some integerk≥1. Anuncertain binary preference relation of typeMonXis a mapping R M :X×X−→Dom(M). Fora,b∈X, the value R M (a,b)∈Dom(M) encodes the uncertain degree to whichais at least as good asb, according to the modelM. Chapter 10. Other Related Decision Methods Definition 10.10.3(Threshold functional and induced upper contour set).LetR M :X×X→Dom(M) be an uncertain binary preference relation. Fix a total map Θ M :Dom(M)−→[0,1] and a threshold level λ∈(0,1]. For eacha∈X, define the(Θ M ,λ)-upper contour setofaby U M,Θ,λ (a) := x∈X: Θ M (R M (a,x))≥λ . Thusx∈U M,Θ,λ (a)means that, after evaluating the uncertain comparisonR M (a,x)through the threshold functionalΘ M , the alternativeais judged to dominatexat level at leastλ. Definition 10.10.4(Choice correspondence induced by an uncertain preference relation).Let P ∗ (X) :=A⊆X:A6=∅ be the family of all nonempty subsets ofX. Under the assumptions of Definition 10.10.3, define C M,Θ,λ :P ∗ (X)−→P(X) by C M,Θ,λ (A) :=a∈A:A⊆U M,Θ,λ (a)(A∈P ∗ (X)). Equivalently, C M,Θ,λ (A) = a∈A: Θ M (R M (a,x))≥λfor allx∈A . This set is called theuncertain choice correspondence induced byR M . Definition 10.10.5(Uncertain rational choice function).A mapping C:P ∗ (X)−→P(X) is called anuncertain rational choice function of typeMif there exist • an uncertain binary preference relation R M :X×X→Dom(M), • a total threshold functional Θ M :Dom(M)→[0,1], • and a threshold levelλ∈(0,1], such that C(A) =C M,Θ,λ (A)for allA∈P ∗ (X). In this case, we say thatCisrationalizedby the triple (R M ,Θ M ,λ). Chapter 10. Other Related Decision Methods Theorem 10.10.6(Well-definedness of uncertain rational choice).LetXbe a finite nonempty set, letM be an uncertain model, and let R M :X×X−→Dom(M) be an uncertain binary preference relation. Assume that Θ M :Dom(M)−→[0,1] is a total map and thatλ∈(0,1]. Then the induced mapping C M,Θ,λ :P ∗ (X)−→P(X) defined by C M,Θ,λ (A) = a∈A: Θ M (R M (a,x))≥λfor allx∈A is well-defined. More precisely, for everyA∈P ∗ (X), 1.C M,Θ,λ (A)⊆A; 2.C M,Θ,λ (A)is uniquely determined; 3.C M,Θ,λ (A)∈P(X). HenceC M,Θ,λ is a well-defined set-valued choice correspondence onX. Moreover, if the followingexistence conditionholds: ∀A∈P ∗ (X)∃a∈A∀x∈A,Θ M (R M (a,x))≥λ, then C M,Θ,λ (A)6=∅for allA∈P ∗ (X), soC M,Θ,λ is a genuine choice function in the usual nonempty-valued sense. Proof.Fix anyA∈ P ∗ (X). SinceA⊆XandA6=∅, every elementa∈Aand everyx∈Abelong toX. Therefore, because R M :X×X→Dom(M), the quantity R M (a,x)∈Dom(M) is defined for every pair(a,x)∈A×A. SinceΘ M is total on Dom(M), the value Θ M (R M (a,x))∈[0,1] is defined for every(a,x)∈A×A. Becauseλ∈(0,1], the statement Θ M (R M (a,x))≥λ Chapter 10. Other Related Decision Methods has a definite truth value for every such pair. Hence the set a∈A: Θ M (R M (a,x))≥λfor allx∈A is well-defined by separation from the finite setA. By construction, every element of this set belongs toA, so C M,Θ,λ (A)⊆A. This proves (1). The defining condition depends only on the already fixed data A, R M ,Θ M , λ, so the resulting subsetC M,Θ,λ (A)is uniquely determined. This proves (2). SinceC M,Θ,λ (A)⊆A⊆X, it follows that C M,Θ,λ (A)∈P(X). This proves (3). ThereforeC M,Θ,λ is a well-defined mapping fromP ∗ (X)intoP(X). For the final statement, assume the existence condition ∀A∈P ∗ (X)∃a∈A∀x∈A,Θ M (R M (a,x))≥λ. FixA∈P ∗ (X). Then there exists somea ? ∈Asuch that Θ M (R M (a ? ,x))≥λfor allx∈A. By definition ofC M,Θ,λ (A), this implies a ? ∈C M,Θ,λ (A). HenceC M,Θ,λ (A)6=∅. SinceAwas arbitrary, the conclusion holds for all nonempty feasible setsA. Proposition 10.10.7(Reduction to fuzzy rational choice).LetMbe the fuzzy model with Dom(M) = [0,1]. Take Θ M (u) :=u(u∈[0,1]), and set λ= 1 2 . Then, for every fuzzy binary relation r:X×X→[0,1], the induced uncertain choice correspondence becomes C M,Θ,λ (A) =a∈A:r(a,x)≥ 1 2 for allx∈A, which is exactly the standard1/2-cut fuzzy rational choice rule. Proof.Under Dom(M) = [0,1]andΘ M (u) =u, we have Θ M (R M (a,x))≥λ⇐⇒r(a,x)≥ 1 2 . Substituting this into Definition 10.10.4 yields the stated formula. Chapter 11 Applications of Decision-Making In this chapter, we briefly review application areas of decision-making. 11.1Applications of Computer Science and Engineering This section presents representative applications of decision-making in computer science and engineering. • Software requirements prioritization and release planning: Software requirements prioritization and release planning rank features by value, cost, risk, and dependencies to choose optimal release contents (cf. [1341]). Analyses have been conducted using Fuzzy MADM [1342,1343] and multi-person decision- making [1344]. • IT service management (ITSM) improvement prioritization: ITSM improvement prioritization ranks process and tool initiatives by incident reduction, customer impact, effort, risk, compliance, and ROI (cf. [1345]). Analyses have been conducted using fuzzy sets and neutrosophic sets [1346,1347]. • Cybersecurity control prioritization and risk-treatment planning: Ranks security controls by risk reduction, cost, and feasibility, guiding treatment plans, budgeting, and implementation sequencing against evolving threats effectively (cf. [1348]). Analyses have been conducted using fuzzy decision- making [1349,1350] and neutrosophic decision-making [1351,1352]. • Cloud migration and system architecture alternative selection: Evaluates cloud migration and archi- tecture options by cost, latency, security, scalability, compliance, and operational complexity to choose best-fit design. Analyses have been conducted using fuzzy logic [1353]. 11.2Applications of Manufacturing This section presents representative applications of decision-making in Manufacturing. 349 Chapter 11. Applications of Decision-Making • Production planning and assembly line balancing: Production planning and assembly line balancing schedule tasks and assign workloads to stations, minimizing cycle time, costs, and bottlenecks (cf. [1354]). Cases have been analyzed using fuzzy decision-making [1355,1356] and neutrosophic decision- making [1357,1358]. • Maintenance strategy selection (preventive, predictive, condition-based): Maintenance strategy selec- tion chooses preventive, predictive, or corrective policies by balancing reliability, downtime, cost, risk, and resource constraints (cf. [1359]). Cases have been analyzed using fuzzy MCDM [1360,1361]. • Prioritization in power grids and industrial assets: Prioritization in power grids and industrial as- sets ranks maintenance and investment actions by reliability impact, risk, cost, urgency, and safety constraints. Cases have been analyzed using fuzzy MCDM [1362]. • Project selection and portfolio optimization (including R&D portfolios): Project selection and portfolio optimization choose and balance projects under budget and risk constraints to maximize strategic value and returns (cf. [1363]). Analyses have been conducted using a variety of methods, including Fuzzy MCDM [1364,1365], Fuzzy MULTIMOORA [1366], Fuzzy TOPSIS [1367–1369], Neutrosophic MCDM [1370], Neutrosophic AHP [1371], and neutrosophic TOPSIS [1372]. 11.3Applications of Finance This section presents representative applications of decision-making in Finance. • Credit scoring and loan approval (financial decision-making): Credit scoring and loan approval as- sess borrower risk using financial and behavioral indicators, assigning ratings to decide credit limits and approvals (cf. [1373]). Analyses have been conducted using fuzzy decision-making [1374, 1375], intuitionistic fuzzy decision-making [1376], and neutrosophic decision-making [1377,1378]. 11.4Applications of Business This section presents representative applications of decision-making in Business. • Supplier selection and procurement evaluation: Supplier selection and procurement evaluation ranks vendors by cost, quality, delivery, risk, sustainability, and strategic fit, using multi-criteria analysis to support sourcing decisions (cf. [1379]). A wide range of analytical approaches has been investigated, including Fuzzy MCDM [1380,1381], Fuzzy AHP [1382], Fuzzy TOPSIS [1383,1384], and Neutrosophic TOPSIS [1385–1387]. • Logistics hub / warehouse location selection: Logistics hub / warehouse location selection chooses optimal facility sites by balancing cost, service coverage, demand proximity, capacity, risk, and sus- tainability objectives under multiple criteria (cf. [1388]). Studies have explored analyses using methods such as Fuzzy MCDM [1389], Fuzzy TOPSIS [1390], and Fuzzy BWM [1391]. • Transportation mode and route selection in supply chains: Chooses transport modes and routes by balancing cost, transit time, reliability, capacity, emissions, and risk across the supply chain. It has been studied using fuzzy MCDM approaches [1392,1393] and fuzzy AHP methods [1394–1396], among others. Chapter 11. Applications of Decision-Making • Human resource selection, performance evaluation, and assignment decisions: Selects and assigns personnel by evaluating skills, experience, performance metrics, availability, and fit, balancing fairness, cost, and organizational objectives under uncertainty. Analyses using methods such as fuzzy AHP have been reported [1397,1398]. • Marketing strategy selection (budget allocation and campaign prioritization): Selects marketing strate- gies by comparing channels and campaigns on ROI, reach, timing, risk, and budget constraints to prioritize spending. Analyses have been conducted using fuzzy ANP [1399,1400], fuzzy AHP [1401], and fuzzy DEMATEL [1402]. 11.5Applications of Medicine This section presents representative applications of decision-making in Medicine. • Medical Diagnosis: Interprets symptoms, history, tests, and imaging to infer disease likelihoods, con- firm causes, and guide treatment decisions under uncertainty (cf. [1403]). Several studies have been conducted within frameworks such as fuzzy decision-making [1404,1405], intuitionistic fuzzy decision- making [87,1406], and neutrosophic decision-making [1407,1408]. • Treatment option selection and clinical decision support: Ranks treatment alternatives using pa- tient data and criteria, supporting clinicians with transparent recommendations under uncertainty, constraints, and trade-offs. Analyses using methods such as fuzzy PROMETHEE have been re- ported [1409]. • Bed/ICU allocation and surge planning under uncertain demand and length of stay: Allocates beds and ICU capacity by forecasting uncertain demand and stay lengths, optimizing staffing, resources, triage, and surge responses (cf. [1410]). Analyses using fuzzy MCDM [1411] and fuzzy AHP [1412,1413] have been reported. 11.6Applications of Social Science This section presents representative applications of decision-making in Social Science. • Environmental impact assessment and mitigation option selection: Assesses environmental impacts and selects mitigation options by comparing severity, likelihood, cost, compliance, and stakeholder priorities under uncertainty. Analyses have been conducted using methods such as fuzzy AHP [1414]. • Renewable energy site selection (wind/solar) and technology choice: Selects wind/solar sites by eval- uating resource potential, grid access, land constraints, environmental impacts, costs, and uncertainty (cf. [1415]). It has been analyzed using fuzzy MCDM methods [1416,1417]. • Weather prediction: Forecasts future weather by combining observations and models to estimate tem- perature, precipitation, wind, and uncertainty over time and space (cf. [1418]). It has been analyzed using fuzzy MCDM methods [1419] and neutrosophic MCDM approaches [1420]. • Smartphone selection: Chooses smartphones by comparing price, performance, camera, battery, dis- play, software support, durability, and user preferences under constraints. It has been studied using fuzzy MCDM [1421], fuzzy WASPAS [1422], fuzzy AHP [31], and fuzzy TOPSIS [1423], among others. Chapter 11. Applications of Decision-Making Chapter 12 Discussions: New Decision-Making Methods In this chapter, we examine several new decision-making methods. For convenience, the main concepts introduced in this chapter are compared in Table 12.1. Table 12.1: Concise comparison of the new decision-making methods introduced in this chapter. MethodTypeBasic structure Core mechanismPrimary out- put Decision role Unified Deci- sional Structure (UDS) Crisp meta- framework Alternatives, cri- teria, evaluation domains, decision data, weights, pre- processing, kernel, and decision map Processes raw evaluation data throughPrep→ Ker→Decto produce an admissible decision object Score, ranking, choice set, or sorting/classi- fication Method- unifying decision architec- ture Unified Un- certain Deci- sional Structure (UUDS) Uncer- tain meta- framework Alternatives, criteria, uncer- tainty algebras, uncertain evalu- ations, weights, and representation- invariant operators Applies uncertainty- aware preprocessing, kernel construction, and decision mapping in a representation- independent way Representation- invariant score, ranking, choice set, or sorting/classi- fication Uncertainty- unifying decision architec- ture Iterated Multi-Criteria Decision-Making (I-MCDM) Hierarchi- cal scoring method Criterion tree, leaf scores, and internal MCDM scoring op- erators with local weights Recursively aggregates child criterion scores up- ward until a root score is obtained Final score vector and in- duced ranking Hierar- chical MCDM Iterated Multi- Attribute Decision-Making (I-MADM) Hierar- chical at- tribute aggregation Attribute tree, atomic attribute scores, and internal MADM operators with local weights Recursively combines lower-level attribute scores into higher-level attribute scores Final root score and ranking Hierar- chical MADM Iterated Multi- Objective Decision-Making (I-MODM) Hierarchi- cal opti- mization Feasible set, ob- jective tree, leaf objectives, aggrega- tion operators, and a solver Recursively aggregates objectives into a final scalar objective and then solves it Final scalar objective and solution set Hierar- chical MODM Continued on the next page. 353 Chapter 12. Discussions: New Decision-Making Methods Table 12.1 (continued). MethodTypeBasic structure Core mechanismPrimary out- put Decision role Analytic Hyper- Network Process (AHNP) Hypernet- work prior- itization Directed hyper- network, pairwise- comparison matri- ces, local priori- ties, and a hyper- supermatrix Computes Perron prior- ity vectors on hyperarcs and derives global pri- orities from a stationary distribution Global priority vector and ranking Network- based pri- oritization Analytic Su- perHyperNet- work Process (ASHNP) Superhy- pernetwork prioritiza- tion Directedn- superhypernetwork, pairwise- comparison matri- ces on supernodes, and a superhyper- supermatrix Extends AHNP from ordinary nodes ton- supernodes and com- putes stationary global priorities Global priority vector and ranking on supernodes Hierar- chical network prioritiza- tion Analytic Recur- sive SuperHyper- Network Process (ARSHNP) Recursive superhy- pernetwork prioritiza- tion Directed recursive superhypernetwork, support-based pair- wise matrices, and a recursive hyper- supermatrix Flattens recursive struc- tures through supports, computes local priorities, and extracts global pri- orities by a stationary vector Global priority vector and ranking Recursive hierarchi- cal priori- tization SuperHyperDecision- Making (SHDM) Reachability- based decision system Working universe inP n (D), super- hypercombination rule, seed set, and preorder Generates the least reachable closed set by iterative consequence expansion and selects maximal reachable states Set of maxi- mal reachable decision states Reachability- based choice Uncertain SuperHyperDecision- Making (USHDM) Uncertain reachability- based decision system SHDM structure together with an uncertainty struc- ture and uncertain valuation map Combines reachable- state generation with uncertainty-valued eval- uation and score-induced maximality Set of maximal uncertain deci- sion outcomes Uncertainty- aware reachability- based choice Note.UDS and UUDS are general decision architectures rather than single concrete methods. I-MCDM, I-MADM, and I-MODM are recursive hierarchical aggregation schemes on trees of criteria, attributes, or objectives. AHNP, ASHNP, and ARSHNP are network-based prioritization models using stationary supermatrix ideas on increasingly richer structures. SHDM and USHDM are closure- and reachability-based decision systems, where the final decision is obtained from maximal reachable states, with uncertainty incorporated in the latter case. 12.1Unified Decisional Structure The Unified Decisional Structure is a framework that maps decision data through preprocessing, a decision kernel, and a decision map to produce scores, rankings, choices, or classifications across methods. Definition 12.1.1(Decision tasks and admissible outputs).LetAbe a finite nonempty set of alternatives. Adecision outputonAis one of the following objects: 1. (Scoring) a mapu:A→R; 2. (Ranking) a total preorderonA(reflexive, transitive, and total); Chapter 12. Discussions: New Decision-Making Methods 3. (Choice) a nonempty subsetA ? ⊆A; 4. (Sorting / classification) a mapσ:A → Kinto a finite ordered set of categoriesK=K 1 K 2 ·K L . Denote byO(A)the (disjoint) collection of all such admissible outputs. Definition 12.1.2(Unified Decisional Structure (UDS)).AUnified Decisional Structure(UDS) is a tuple UDS:= ( A,C,(E j ) n j=1 , X, w,Prep,Ker,Dec ) satisfying: 1. (Alternatives and criteria)A=A 1 ,...,A m andC=C 1 ,...,C n are finite nonempty sets. 2. (Evaluation domains) For each criterionC j ,E j is a nonempty set (e.g.R, an interval scale, an ordinal scale). 3. (Decision data) Theevaluation mapis a total function X:A×C → n ⊔ j=1 E j , X(A i ,C j ) =:x ij ∈E j . 4. (Weights) Theweight vectorisw= (w 1 ,...,w n )∈[0,1] n with n ∑ j=1 w j = 1. 5. (Preprocessing operator)Prepis a specification of maps (possibly depending onwand/or on the full matrix(x ij )) producing aprocessed representationPrep(X,w)∈ D, whereDis a fixed data space. This step may include normalization, benefit/cost conversion, scaling, reference-profile construction, etc. 6. (Decision kernel)Ker:D → Zis a total function that maps processed data to anevidence object z∈ Z(e.g. a utility vector, a distance-to-ideal vector, an outranking matrix, a dominance matrix, flows, etc.). 7. (Decision map)Dec:Z →O(A)is a total function that converts evidence into an admissible output (score/ranking/choice/sorting). The induced decision procedure is the composite map DM UDS :=Dec◦Ker◦Prep. Theorem 12.1.3(Well-definedness of UDS).EveryUDSin Theorem 12.1.2 induces a unique decision procedure DM UDS : (X,w)7−→DM UDS (X,w)∈O(A), i.e. the output exists and is uniquely determined by(X,w). Proof.By Theorem 12.1.2,Prepis a total mapping from inputs(X,w)toD,Keris a total mapping from DtoZ, andDecis a total mapping fromZtoO(A). Therefore the compositionDec◦Ker◦Prepis a total function from(X,w)toO(A). Function values are unique, hence the induced output is unique. Chapter 12. Discussions: New Decision-Making Methods 12.2Unified Uncertain Decisional Structure Unified Uncertain Decisional Structure extends UDS with uncertainty algebras and representation-invariant operators, yielding method-agnostic decisions from fuzzy/rough/grey/neutrosophic evaluations. Definition 12.2.1(Uncertainty algebra with representation equivalence).Anuncertainty algebrais a tuple U= (U,∼,ω α α∈I ,sc) where: 1.Uis a nonempty set of uncertain values (e.g. fuzzy numbers, rough intervals, grey numbers, neutro- sophic triples, intuitionistic pairs, linguistic terms, etc.). 2.∼is an equivalence relation onUencodingrepresentational equality(e.g. two different parameteriza- tions describing the same membership function). 3. Eachω α is a (possibly multi-ary) operation onUused in the decision pipeline, andcompatibilityholds: ifu k ∼u ′ k for all arguments, then ω α (u 1 ,...,u p )∼ω α (u ′ 1 ,...,u ′ p )(whenever both sides are defined). 4. sc:U→R q is ascore (crispification) mapthat is constant on equivalence classes: u∼v=⇒sc(u) =sc(v). Definition 12.2.2(Unified Uncertain Decisional Structure (UUDS)).AUnified Uncertain Decisional Struc- ture(UUDS) is a tuple UUDS:= ( A,C,(U j ) n j=1 , ̃ X, w, ̃ Prep, ̃ Ker, ̃ Dec ) such that: 1.A=A 1 ,...,A m andC=C 1 ,...,C n are finite nonempty. 2. For eachj,U j = (U j ,∼ j ,ω α,j α∈I j ,sc j )is an uncertainty algebra in the sense of Theorem 12.2.1. 3. The uncertain evaluation map is total: ̃ X:A×C → n ⊔ j=1 U j , ̃ X(A i ,C j ) =: ̃x ij ∈U j . 4. Weights satisfyw j ∈[0,1]and ∑ n j=1 w j = 1. 5. The preprocessing operator ̃ Prepmaps( ̃ X,w)to a processed space ̃ D, and isrepresentation-invariant: if ̃x ij ∼ j ̃x ′ ij for all(i,j), then ̃ Prep( ̃ X,w) = ̃ Prep( ̃ X ′ ,w). 6. The kernel ̃ Ker: ̃ D → ̃ Zis total and representation-invariant (with respect to the induced equivalence on ̃ Dcoming from the∼ j -compatibility of all used operations). 7. The decision map ̃ Dec: ̃ Z →O(A)is total and depends on uncertain objects only via class-invariant quantities (typically via score maps sc j or their aggregates). Chapter 12. Discussions: New Decision-Making Methods The induced uncertain decision procedure is DM UUDS := ̃ Dec◦ ̃ Ker◦ ̃ Prep. Theorem 12.2.3(Well-definedness of UUDS).LetUUDSbe as in Theorem 12.2.2. Then: 1. The outputDM UUDS ( ̃ X,w)exists and is unique for every input( ̃ X,w). 2. The output isrepresentation-independent: if ̃ Xand ̃ X ′ satisfy ̃x ij ∼ j ̃x ′ ij for all(i,j), then DM UUDS ( ̃ X,w) =DM UUDS ( ̃ X ′ ,w). Proof.(1) By Theorem 12.2.2, ̃ Prepis total from inputs( ̃ X,w)to ̃ D, ̃ Keris total from ̃ Dto ̃ Z, and ̃ Decis total from ̃ ZtoO(A). Hence the composite map exists and is a function; therefore its value is unique. (2) Assume ̃x ij ∼ j ̃x ′ ij entrywise. By the representation-invariance requirement in Theorem 12.2.2(5), ̃ Prep( ̃ X,w) = ̃ Prep( ̃ X ′ ,w). Applying the representation-invariant kernel Theorem 12.2.2(6) yields ̃ Ker ( ̃ Prep( ̃ X,w) ) = ̃ Ker ( ̃ Prep( ̃ X ′ ,w) ) . Finally, ̃ Decdepends only on class-invariant quantities (e.g. score maps constant on∼ j -classes), so applying ̃ Decpreserves equality and givesDM UUDS ( ̃ X,w) =DM UUDS ( ̃ X ′ ,w). 12.3Iterated Multi-Criteria Decision-Making Iterated Multi-Criteria Decision-Making recursively composes MCDM scoring operators across a hierarchical criterion tree, aggregating leaf criterion scores into a final root score ranking alternatives. Definition 12.3.1(MCDM scoring operator).LetA=A 1 ,...,A m be a finite nonempty set of alterna- tives and letk∈N. A(crisp) MCDM scoring operatorof aritykis a total function M:R m×k ×∆ k −→R m , where∆ k :=w∈[0,1] k : ∑ k `=1 w ` = 1is the simplex of weights. Given a performance matrixX∈R m×k and weightsw∈∆ k , the vectorM(X,w) = (u(A 1 ),...,u(A m )) > is interpreted as ascoreonA. (Examples: weighted sum, TOPSIS-score, VIKOR-index, MOORA-score, TODIM-value, PROMETHEE net-flow, etc., provided they output a real-valued score for each alternative.) Definition 12.3.2(Criterion tree).Acriterion treeis a finite rooted treeT= (V,E,r). For a nodev∈V, let Ch(v)⊆Vbe the (finite) set of children ofv. A node is aleafif Ch(v) =∅, and aninternal node otherwise. Theheightht(T)is the maximum number of edges on a directed path from the root to a leaf. Chapter 12. Discussions: New Decision-Making Methods Definition 12.3.3(Iterated Multi-Criteria Decision-Making (I-MCDM)).LetA=A 1 ,...,A m be alter- natives and letT= (V,E,r)be a criterion tree. AnIterated Multi-Criteria Decision-Making(I-MCDM) instance is a collection I-MCDM:= ( A, T,u ` `∈L ,M v ,w (v) v∈V ) , where: 1.L⊆Vis the set of leaves. Each leaf`∈Lis equipped with abase (atomic) score u ` :A→R. (Equivalently,u ` may be obtained from raw measurements/linguistic assessments by any fixed pre- processing and crispification; the present definition only requires the resulting mapA→R.) 2. Each internal nodev∈V v :=|Ch(v)|is equipped with M v :R m×k v ×∆ k v →R m (an MCDM scoring operator) and a weight vectorw (v) ∈∆ k v over its children. Recursive semantics (“multi-criteria of multi-criteria of·”).Define, for every nodev∈V, a score functionu v :A→Rrecursively by: (S1)Ifvis a leaf, setu v :=u v given above (base score). (S2)Ifvis internal, enumerate its children as Ch(v) =c 1 ,...,c k v and form thechild-score matrix X (v) ∈R m×k v by X (v) i` :=u c ` (A i ) (i= 1,...,m, `= 1,...,k v ). Then define the parent score vector by ( u v (A 1 ),...,u v (A m ) ) > :=M v ( X (v) , w (v) ) . Thefinal I-MCDM scoreisu r :A →Rat the root. A final ranking can be induced byA i A k ⇐⇒ u r (A i )≥u r (A k ). Theorem 12.3.4(Well-definedness of I-MCDM).LetI-MCDMbe as in Theorem 12.3.3. Assume thatT is finite and everyM v is a total function on its stated domain. Then: 1. For every nodev∈V, the recursively defined mapu v :A→Rexists and is unique. 2. In particular, the final scoreu r and the induced ranking are uniquely determined by the instance data. Chapter 12. Discussions: New Decision-Making Methods Proof.BecauseTis a finite rooted tree, nodes admit a well-founded order by depth-from-leaves (equivalently, by decreasing distance to leaves). Proceed by induction on this order. Base step (leaves).Ifvis a leaf,u v is given as part of the instance, hence exists and is unique. Inductive step (internal nodes).Assume that for all childrenc∈Ch(v), the functionsu c exist and are unique. Then the child-score matrixX (v) is uniquely determined, since each entryX (v) i` =u c ` (A i )is uniquely determined. BecauseM v is total andw (v) ∈∆ k v is fixed, the vectorM v (X (v) ,w (v) )∈R m exists and is unique, hence defines a unique functionu v . By finiteness, every node is reached after finitely many steps, so allu v exist and are unique. Applying this to the rootryields existence and uniqueness ofu r and thus of the induced ranking. 12.4Iterated Multi-Attribute Decision-Making Iterated Multi-Attribute Decision-Making composes MADM scoring operators along an attribute hierarchy, recursively aggregating lower-level attribute scores into a root score for ranking alternatives. Definition 12.4.1(Attribute/objective tree).Afinite rooted treeT= (V,E,r)is used to represent nested attributesorobjectives. For each nodev∈V, let Ch(v)be its (finite) set of children. Leaves are nodes with Ch(v) =∅. Definition 12.4.2(MADM scoring operator).LetA=A 1 ,...,A m be a finite nonempty set of alterna- tives and letk∈N. AMADM scoring operatorof aritykis a total function M:R m×k ×∆ k −→R m ,∆ k := w∈[0,1] k : k ∑ `=1 w ` = 1 . Given an attribute-performance matrixX∈R m×k and weightsw∈∆ k ,M(X,w)returns one real score per alternative. Definition 12.4.3(Iterated Multi-Attribute Decision-Making (I-MADM)).LetA=A 1 ,...,A m be alternatives and letT= (V,E,r)be anattribute tree. AnIterated Multi-Attribute Decision-Making(I- MADM) instance is a collection I-MADM:= ( A, T,u ` `∈L ,M v ,w (v) v∈V ) , whereL⊆Vis the set of leaves, such that: 1. Each leaf`∈L(an atomic attribute) has a base score u ` :A→R. 2. Each internal nodev∈V v :=|Ch(v)|has an MADM scoring operator M v :R m×k v ×∆ k v →R m and a weight vectorw (v) ∈∆ k v on its children. Chapter 12. Discussions: New Decision-Making Methods Recursive semantics.Define scoresu v :A→Rfor all nodesv∈Vrecursively: 1. Ifvis a leaf,u v is given (base attribute score). 2. Ifvis internal with Ch(v) =c 1 ,...,c k v , formX (v) ∈R m×k v byX (v) i` =u c ` (A i )and set (u v (A 1 ),...,u v (A m )) > :=M v (X (v) ,w (v) ). The final I-MADM score isu r at the root, yielding a ranking byA i A k ⇐⇒u r (A i )≥u r (A k ). Theorem 12.4.4(Well-definedness of I-MADM).Under Theorem 12.4.3, ifTis finite and eachM v is total, then for every nodev∈Vthe scoreu v exists and is unique; in particular,u r is unique. Proof.Proceed by induction from leaves to the root. Leaf scores are given. Assuming child scoresu c are uniquely determined, the matrixX (v) is uniquely determined. Totality ofM v impliesM v (X (v) ,w (v) )∈R m exists uniquely, hence definesu v uniquely. Finiteness ofTensures termination at the root. 12.5Iterated Multi-Objective Decision-Making Iterated Multi-Objective Decision-Making composes objective-aggregation operators along a hierarchical objective tree, recursively producing a final scalar objective, then solving it to obtain optimal decisions. Definition 12.5.1(MODM objective-aggregation operator).LetX ⊆R d be a nonempty feasible decision set. AMODM objective-aggregation operatorof aritykis a total function G: (R X ) k ×∆ k −→R X , where(R X ) k denotesk-tuples of real-valued functions onX. Given objective functionsf 1 ,...,f k :X →R and weightsw∈∆ k ,G(f 1 ,...,f k ;w)returns an aggregated objectiveF:X →R(to be minimized or maximized as specified). Definition 12.5.2(Iterated Multi-Objective Decision-Making (I-MODM)).LetX ⊆R d be a nonempty feasible set and letT= (V,E,r)be anobjective tree. AnIterated Multi-Objective Decision-Making(I- MODM) instance is a collection I-MODM:= ( X, T,f ` `∈L ,G v ,w (v) v∈V ,Solve ) , whereLis the leaf set, such that: 1. Each leaf`∈Lhas an atomic objectivef ` :X →R. 2. Each internal nodevwithk v :=|Ch(v)|has an objective-aggregation operator G v : (R X ) k v ×∆ k v →R X and weightsw (v) ∈∆ k v . Chapter 12. Discussions: New Decision-Making Methods 3.Solveis a fixed optimization rule that maps a scalar objectiveF:X →Rto a nonempty solution set X ? (F)⊆X(e.g. argmin or argmax), whenever such a set exists. Recursive semantics.Define aggregated objectivesF v :X →Rfor allv∈Vrecursively: 1. Ifvis a leaf, setF v :=f v . 2. Ifvis internal with Ch(v) =c 1 ,...,c k v , set F v :=G v (F c 1 ,...,F c k v ;w (v) ). The final scalar objective isF r at the root, and the I-MODM output is the solution set X ? :=Solve(F r )⊆X. Theorem 12.5.3(Well-definedness of I-MODM).Under Theorem 12.5.2, ifTis finite, eachG v is total, andSolve(F r )is defined (i.e. returns a nonempty set), then: 1. each aggregated objectiveF v exists and is unique for allv∈V; 2. the final objectiveF r is unique; hence the output solution setX ? is uniquely determined. Proof.Induct from leaves upward. Leaves give objectivesF v uniquely. Assuming objectivesF c are uniquely defined for all childrenc∈Ch(v), totality ofG v gives a unique functionF v =G v (F c 1 ,...,F c k v ;w (v) ). Finiteness ofTensures the recursion terminates at the root, yielding a uniqueF r . Applying the fixed rule Solveto this uniqueF r yields a uniquely determined output set. 12.6Analytic SuperHyperNetwork Process (ASHNP) In this section, we consider extending the Analytic Network Process (ANP) by using hypergraphs and super- hypergraphs. Hypergraph generalizes graphs: vertices with hyperedges as nonempty vertex subsets, enabling single edges to connect multiple vertices, modeling higher-order relations [1424–1426]. Hypernetwork is a (weighted) hypergraph: nodes and hyperedges with a weight/attribute function on hyperedges, represent- ing multi-node interactions with strengths or confidences [1427].n-SuperHyperGraph uses vertices drawn from then-th iterated powerset of a base set; edges are nonempty families of such supervertices, capturing hierarchy [215, 1428–1432]. Related concepts, such as directed superhypergraphs [1433, 1434] and Meta- SuperHyperGraph [1435–1437] are also known.n-SuperHyperNetwork is a weightedn-SuperHyperGraph: supernodes from then-th powerset and weighted hyperedges over them, encoding hierarchical interactions and strengths [1438,1439]. We also discuss in the Appendix a concept that can generalize arbitrary struc- tures such as graphs and hypergraphs; please refer to it as needed. Analytic HyperNetwork Process (AHNP) extends ANP to directed hypernetworks; pairwise comparisons yield local priorities, assembled into a hyper-supermatrix whose stationary distribution provides global node weights. ASHNP generalizes AHNP to n-supernodes from iterated powersets; superhyperarcs encode hierarchical interactions; eigenvector priorities build a superhyper-supermatrix, producing global importance rankings. Chapter 12. Discussions: New Decision-Making Methods Definition 12.6.1(Iterated powerset and iterated nonempty powerset).(cf. [1440]) LetHbe a set and let P(H)denote its (ordinary) powerset. (1) Iterated powerset. Define the iterated powerset operatorP n (H)forn∈N 0 by P 0 (H) :=H,P n+1 (H) :=P ( P n (H) ) (n∈N 0 ). (2) Iterated nonempty powerset. LetP ∗ (X) :=P(X)\∅be the nonempty powerset of a setX. Define P n ∗ (H)forn∈N 0 by P 0 ∗ (H) :=H,P n+1 ∗ (H) :=P ∗ ( P n ∗ (H) ) =P ( P n ∗ (H) ) \∅(n∈N 0 ). Definition 12.6.2(Hypergraph [1425,1441]).A(finite) hypergraphis an ordered pairH= (V(H),E(H)) where •V(H)is a finite nonempty set ofvertices; •E(H)⊆P(V(H))\∅is a finite set ofhyperedges. Thus each hyperedgee∈E(H)is a nonempty subset ofV(H)(allowing one edge to connect any number of vertices). Definition 12.6.3(n-SuperHyperGraph [215,1442]).LetV 0 be a finite nonemptybasevertex set and let n∈N 0 . Using the iterated powerset from the previous definition, ann-SuperHyperGraph overV 0 is a pair SHG (n) = (V,E) such that ∅6=V⊆P n (V 0 )andE⊆P(V)\∅. Elements ofVare calledn-supervertices, and elements ofEare calledn-superedges. Special cases. Forn= 0, one recovers an ordinary hypergraph on a subset of the base set:V⊆V 0 and E⊆P(V)\∅. Forn= 1, vertices are subsets ofV 0 and edges are nonempty families of such subsets. Remark. SinceE⊆ P(V)⊆ P(P n (V 0 )) =P n+1 (V 0 ), everyn-superedge can be viewed (canonically) as an element of the(n+ 1)-st iterated powerset ofV 0 . Definition 12.6.4(Hypernetwork [1443,1444]).A(weighted) hypernetworkis a triple N= (V,E,w) where •Vis a finite nonempty set ofnodes; •E ⊆P(V)\∅is a finite set ofhyperedges; Chapter 12. Discussions: New Decision-Making Methods •w:E →R ≥0 is aweight/attributefunction on hyperedges (omitwfor the unweighted case). Adirected hypernetworkcan be modeled by replacingEwith a set of ordered pairs E ⊆ ( P(V)\∅ ) × ( P(V)\∅ ) , where each directed hyperedge is of the form(T,H)(tail/head), or by any equivalent head–tail partition formalism. Optional node and hyperedge labelings` V :V→L V and` E :E →L E may be added to encode types. Definition 12.6.5(n-SuperHypernetwork [1444–1446]).LetV 0 be a finite nonempty base node set and let n∈N 0 . Ann-superhypernetworkis a triple N (n) = (V,E,w) where •V⊆P n (V 0 )is a finite nonempty set ofn-supernodes; •E ⊆P(V)\∅is a finite set ofn-superhyperedges; •w:E →R ≥0 is an optional weight function. Equivalently, ann-superhypernetwork is ann-SuperHyperGraph endowed with a hyperedge-weight function. Moreover,E ⊆P(V)⊆P n+1 (V 0 )holds automatically. Definition 12.6.6(Positive reciprocal pairwise-comparison matrix).A matrixM= (m pq )∈R k×k >0 is called positive reciprocalif m p = 1, m pq = 1 m qp (p6=q). Lemma 12.6.7(Perron priority vector).LetM∈R k×k >0 be a positive matrix (in particular, any matrix in Theorem 12.6.6 is positive). Then there exists an eigenvalueλ max >0and an eigenvectorv∈R k >0 such thatMv=λ max v. Moreover,vis unique up to a positive scalar factor. Hence the normalized vector p(M) := v 1 > v ∈∆ k ,∆ k :=x∈R k ≥0 :1 > x= 1, is uniquely determined byM. Proof.This is a standard consequence of the Perron–Frobenius theorem for positive matrices: a positive matrix has a unique (up to scaling) positive eigenvector associated with its spectral radius. Normalization by1 > vfixes the scaling uniquely. Lemma 12.6.8(Unique stationary distribution for primitive column-stochastic matrices).LetW∈R m×m ≥0 becolumn-stochastic, i.e.1 > W=1 > , and assumeWisprimitive(there existst∈Nsuch thatW t has all entries strictly positive). Then there exists a unique vectorπ∈∆ m such that Wπ=π, and for everyx∈∆ m one hasW k x→πask→∞. Chapter 12. Discussions: New Decision-Making Methods Proof.Primitivity implies, by Perron–Frobenius, that eigenvalue1is simple and has a unique positive right eigenvector. Normalizing it to sum to1yields the uniqueπ∈∆ m . ConvergenceW k x→πfor allx∈∆ m follows from the Perron decomposition for primitive stochastic matrices. Definition 12.6.9(Directed hypernetwork).Adirected hypernetworkis a triple H= (V,E,γ), whereVis a finite nonempty set of nodes,Eis a finite set ofdirected hyperarcs e= (T e ,H e ),∅6=T e ⊆V,∅6=H e ⊆V, andγ:E →R >0 is an optional hyperarc weight (omitγif unweighted). Intuitively, the tail setT e influences the head setH e . Definition 12.6.10(Analytic HyperNetwork Process (AHNP)).LetA=A 1 ,...,A m be a finite set of decision elements (criteria, subcriteria, alternatives, or any mix) represented as nodes of a directed hypernetworkH= (V,E,γ)withV=A. AnAnalytic HyperNetwork Process (AHNP)instance consists of: 1. A directed hypernetworkH= (V,E,γ)onV=A. 2. For each hyperarce= (T e ,H e )∈Eand each head nodeh∈H e , a positive reciprocal matrix M (e,h) ∈R |T e |×|T e | >0 whose entries compare the relative influence of tail nodes inT e with respect tothe head nodeh. 3. A fixed rule to assign a nonnegativearc–head mixing weightα e,h ≥0for each(e,h)withh∈H e . (For example,α e,h may be set proportional toγ(e)and then normalized per head node.) Local priorities.For each(e,h), define thelocal priority vector p (e,h) :=p ( M (e,h) ) ∈∆ |T e | as in Theorem 12.6.7. Indexp (e,h) by tail nodes inT e . Global influence (hyper-supermatrix).Define the raw influence matrixS= (s ih ) i,h∈V ∈R m×m ≥0 by s ih := ∑ e=(T e ,H e )∈E h∈H e , i∈T e α e,h p (e,h) (i), wherep (e,h) (i)denotes the component corresponding toi∈T e . To obtain a column-stochastichyper-supermatrixW= (w ih ), set for each columnh: w ih := s ih ∑ u∈V s uh ,if ∑ u∈V s uh >0, δ ih ,if ∑ u∈V s uh = 0, Chapter 12. Discussions: New Decision-Making Methods whereδ ih is the Kronecker delta (a self-loop fallback to avoid a zero column). Final priorities and output.Define the AHNP priority vectorπ∈∆ m as any solution of Wπ=π. If a distinguished subsetV alt ⊆Vrepresents alternatives, rank alternatives by descending values ofπ restricted toV alt (renormalize onV alt if desired). Theorem 12.6.11(Well-definedness of AHNP).Assume an AHNP instance as in Theorem 12.6.10. Then: 1. All local priority vectorsp (e,h) are well-defined and unique. 2. The hyper-supermatrixWis well-defined and column-stochastic. 3. IfWis primitive, then the stationary priority vectorπ∈∆ m exists, is unique, andW k x→πfor everyx∈∆ m . Proof.(1) EachM (e,h) is positive reciprocal, hence positive; apply Theorem 12.6.7. (2) The entriess ih are finite sums of nonnegative terms, hence well-defined and nonnegative. Column normalization definesw ih ≥0and ensures ∑ i w ih = 1for eachh; if the raw column sum is0, the fallback columnδ ih also sums to1. Thus1 > W=1 > . (3) IfWis primitive, apply Theorem 12.6.8 to obtain existence/uniqueness ofπand convergence of powers. Definition 12.6.12(Analytic SuperHyperNetwork Process (ASHNP)).Fix a finite nonempty base node setV 0 and an integern∈N 0 . LetV⊆P n (V 0 )be a finite nonempty set ofn-supernodes. LetH (n) = (V,E,γ) be a directed hypernetwork on the supernode setV(in the sense of Theorem 12.6.9), whose hyperarcs are of the forme= (T e ,H e )with∅6=T e ,H e ⊆V. AnAnalytic SuperHyperNetwork Process (ASHNP)instance consists of: 1. The directed superhypernetworkH (n) = (V,E,γ). 2. For eache= (T e ,H e )∈Eand each head supernodeh∈H e , a positive reciprocal matrix M (e,h) ∈R |T e |×|T e | >0 comparing supernodes inT e with respect tothe head supernodeh. 3. Nonnegative mixing weightsα e,h ≥0assigned to each(e,h)withh∈H e . Chapter 12. Discussions: New Decision-Making Methods Computation.Define local prioritiesp (e,h) :=p(M (e,h) ), build the raw influence matrixS= (s ih ) i,h∈V by s ih := ∑ e=(T e ,H e )∈E h∈H e , i∈T e α e,h p (e,h) (i), and column-normalize it (with the same zero-column fallback) to obtain a column-stochastic superhyper- supermatrixWon the index setV. Define the ASHNP priority vectorπ∈∆ |V| byWπ=πand rank the designated alternative supernodes accordingly. Theorem 12.6.13(Well-definedness of ASHNP).Assume an ASHNP instance as in Theorem 12.6.12. Then: 1. All local priority vectorsp (e,h) are well-defined and unique. 2. The superhyper-supermatrixWis well-defined and column-stochastic. 3. IfWis primitive, then the stationary priority vectorπ∈∆ |V| exists, is unique, andW k x→πfor everyx∈∆ |V| . Proof.Identical to the proof of Theorem 12.6.11, since the construction depends only on the finiteness of the node set and positivity/reciprocity of the pairwise-comparison matrices. The fact thatV⊆ P n (V 0 ) changes only the interpretation of nodes, not the algebra. 12.7Analytic Recursive SuperHyperNetwork Process (ARSHNP) Recursive SuperHyperGraph allows supervertices from an iterated powerset, while edges are recursive set- objects drawn from a depth-k powerset universe, capturing nested, multi-level hyper-relations without mem- bership cycles [1447–1449]. Recursive SuperHyperNetwork is a weighted recursive SuperHyperGraph: it attaches nonnegative weights/attributes to recursive superhyperedges, enabling quantified strengths or con- fidences for hierarchical, nested interactions among supernodes. Analytic Recursive SuperHyperNetwork Process (ARSHNP) extends AHNP to recursive superhypernetworks: pairwise comparisons on supports yield local priorities, assembled into a column-stochastic supermatrix whose stationary vector provides global node importances. Definition 12.7.1(Atomization convention).Throughout, the node setVis treated as a set ofatoms (urelements): elements ofVare not decomposed by the membership relation∈in the recursive constructions below. Equivalently, one may replaceVby a tagged copy At(V) :=(0,v)|v∈V, and work overAt(V). In either case, all recursive set-constructions are builtoverthese atoms. Definition 12.7.2(Depth-kpowerset universe).(cf. [1450,1451]) LetVbe a nonempty set of atoms and letk∈N 0 . Define(S i ) i≥0 by S 0 :=V, S i :=P ( i−1 ⋃ j=0 S j ) (i≥1), Chapter 12. Discussions: New Decision-Making Methods and define theobject universe up to depthkby Obj(V,k) := k ⋃ i=0 S i . Thedepth-kpowerset universegenerated byVis 2 V,k :=P ( Obj(V,k) ) . Lemma 12.7.3(Finite depth and canonical support).LetVbe a set of atoms andk∈N 0 . Every x∈Obj(V,k)admits a finitestructural depth d(x)∈0,1,...,kand a well-definedsupport Supp(x)⊆V. (i) Atoms.Forv∈V, define d(v) := 0,Supp(v) :=v. (i) Set-objects.Forx∈Obj(V,k) (soxis a set of objects), define d(x) := 1 +max ( d(y)|y∈x∪0 ) ,Supp(x) := ⋃ y∈x Supp(y). Thend(x)≤k. Moreover,Supp(x)6=∅wheneverx6=∅. Proof.By construction,S 0 =Vconsists of atoms, hence has depth0. Ifx∈S i for somei≥1, then x⊆ ⋃ i−1 j=0 S j , so everyy∈xlies inS j(y) for somej(y)≤i−1. Thus the recursion is well-founded, and d(y) is defined for ally∈xbefore defining d(x). Induction on the leastiwithx∈S i yields d(x)≤i≤k. If x6=∅, it contains somey, and Supp(y)6=∅; hence Supp(x) = ⋃ y∈x Supp(y)6=∅. Definition 12.7.4(Support of a recursive set).For anyX∈2 V,k (soX⊆Obj(V,k)), define its support by Supp(X) := ⋃ x∈X Supp(x)⊆V, where Supp(x)is as in Theorem 12.7.3. Definition 12.7.5(k-recursive hypergraph).(cf. [1450,1451]) LetVbe a finite nonempty set of atoms and letk∈N 0 . Ak-recursive hypergraphis a pair H= (V,E) such that E⊆2 V,k \∅, where2 V,k is the depth-kpowerset universe from Theorem 12.7.2. Fork= 0, one has2 V,0 =P(V), hence E⊆P(V)\∅andHis an ordinary hypergraph. Definition 12.7.6((n,k)-recursive SuperHyperGraph).LetV 0 be a finite nonempty base set and let n,k∈N 0 . Define iterated powersets by P 0 (V 0 ) :=V 0 ,P n+1 (V 0 ) :=P ( P n (V 0 ) ) . A(n,k)-recursive SuperHyperGraphis a pair RSHG (n,k) = (V,E) such that ∅6=V⊆P n (V 0 )(finite), E⊆2 V,k \∅(finite), where2 V,k is generated fromVas in Theorem 12.7.2. In addition,Vis interpreted as a set of atoms in the sense of Theorem 12.7.1. Chapter 12. Discussions: New Decision-Making Methods Definition 12.7.7((n,k)-Recursive SuperHyperNetwork).LetV 0 be a finite nonempty base set and let n,k∈N 0 . Let RSHG (n,k) = (V,E)be an(n,k)-recursive SuperHyperGraph as in Theorem 12.7.6. A (n,k)-recursive SuperHyperNetworkis a triple N (n,k) = (V,E,w) wherew:E→R ≥0 is an optional weight/attribute function (omitwif unweighted). Theorem 12.7.8(Well-definedness of recursive SuperHyperNetworks).LetN (n,k) = (V,E,w)be as in Theorem 12.7.7. Then: 1. For every recursive superhyperedgee∈E, the supportSupp(e)⊆Vis well-defined. 2. Theflattened(ordinary) weighted hypernetwork N:= (V,E,w) is well-defined, where E:=Supp(e)|e∈E⊆P(V)\∅, w(F) := ∑ e∈E Supp(e)=F w(e) (F∈E). Proof.(1) Sincee∈2 V,k , one hase⊆Obj(V,k), hence Supp(x)is defined for eachx∈eby Theorem 12.7.3. Thus Supp(e) = ⋃ x∈e Supp(x)is well-defined and is a subset ofV. Ife6=∅, then by Theorem 12.7.3 the support of some element is nonempty, so Supp(e)6=∅. (2) Therefore each Supp(e)lies inP(V)\∅, soEis well-defined and finite. For eachF∈E,w(F)is a finite sum of nonnegative reals, hence well-defined. Definition 12.7.9(Positive reciprocal matrix and Perron priority).Letr∈N. A matrixM= (m pq )∈R r×r >0 ispositive reciprocalif m p = 1, m pq = 1 m qp (p6=q). Let∆ r :=x∈R r ≥0 :1 > x= 1. For suchM, letp(M)∈∆ r denote the normalized Perron (principal) right eigenvector; it exists and is unique after normalization (Perron–Frobenius). Definition 12.7.10(Directed recursive superhypernetwork).Adirected recursive superhypernetworkis a triple −→ N (n,k) = (V, −→ E,γ), whereVis as in Theorem 12.7.7, −→ E⊆ ( 2 V,k \∅ ) × ( 2 V,k \∅ ) is a finite set ofdirected recursive superhyperarcse= (T e ,H e ), andγ: −→ E→R >0 is an optional arc weight. Define thetail supportandhead supportby T e :=Supp(T e )⊆V, H e :=Supp(H e )⊆V, which are nonempty by Theorem 12.7.3 and Theorem 12.7.4. Chapter 12. Discussions: New Decision-Making Methods Definition 12.7.11(Analytic Recursive SuperHyperNetwork Process (ARSHNP)).Let −→ N (n,k) = (V, −→ E,γ) be a directed recursive superhypernetwork. AnARSHNPinstance consists of: 1. For each arce= (T e ,H e )∈ −→ Eand each head nodeh∈H e , a positive reciprocal pairwise-comparison matrix M (e,h) ∈R |T e |×|T e | >0 comparing tail nodes inT e with respect tohead nodeh. 2. Nonnegative mixing weightsα e,h ≥0for each(e,h)withh∈H e (e.g., proportional toγ(e)and normalized per head node). Local priorities.For each(e,h), definep (e,h) :=p(M (e,h) )∈∆ |T e | and index its components by nodes in T e . Recursive hyper-supermatrix.DefineS= (s ih ) i,h∈V ∈R |V|×|V| ≥0 by s ih := ∑ e∈ −→ E h∈H e , i∈T e α e,h p (e,h) (i). Column-normalizeSto obtain a column-stochastic matrixW= (w ih ): w ih := s ih ∑ u∈V s uh ,if ∑ u∈V s uh >0, δ ih ,if ∑ u∈V s uh = 0, whereδ ih is the Kronecker delta. Global priorities.Anyπ∈∆ |V| satisfying Wπ=π is called anARSHNP priority vector. If a distinguished subsetV alt ⊆Vrepresents alternatives, rank them by the corresponding components ofπ(renormalizing onV alt if desired). Theorem 12.7.12(Well-definedness of ARSHNP).Assume an ARSHNP instance as in Theorem 12.7.11. Then: 1. Tail/head supportsT e ,H e are well-defined nonempty subsets ofVfor all arcse∈ −→ E. 2. Each local priority vectorp (e,h) is well-defined and unique. 3. The matrixWis well-defined, nonnegative, and column-stochastic. 4. There exists at least oneπ∈∆ |V| such thatWπ=π. If, in addition,Wis primitive, then thisπis unique and strictly positive. Chapter 12. Discussions: New Decision-Making Methods Proof.(1) SinceT e ,H e ∈2 V,k \∅, supports Supp(T e ),Supp(H e )are defined by Theorem 12.7.4. Nonempti- ness follows fromT e ,H e 6=∅and Theorem 12.7.3. (2) EachM (e,h) is positive reciprocal, hence positive. By Perron–Frobenius, it has a unique normalized Perron vector, sop (e,h) is well-defined and unique. (3) Eachs ih is a finite sum of nonnegative reals, hence well-defined ands ih ≥0. Column-normalization gives w ih ≥0and ∑ i∈V w ih = 1for eachh; the fallback columnδ ih also sums to1. ThusWis column-stochastic. (4) SinceWis column-stochastic,‖W‖ 1 = 1, hence its spectral radius satisfiesρ(W)≤1. Also1 > W=1 > implies1is an eigenvalue ofW > , hence1is an eigenvalue ofW. Thereforeρ(W) = 1, and by Perron– Frobenius for nonnegative matrices,Wadmits a nonnegative right eigenvector for eigenvalue1. Normalizing it to sum to1yieldsπ∈∆ |V| withWπ=π. IfWis primitive, the Perron eigenvector is unique up to scaling and strictly positive; normalization makes it unique and strictly positive. 12.8SuperHyperDecision-Making SuperHyperDecision-Making defines hierarchical decision states drawn from an n-th powerset, generates reachable sets via set-valued combination rules, and selects maximal outcomes under a preorder. Definition 12.8.1(SuperHyperdecision-making system (leveln)).Fix an integern≥1and a nonempty setDof atomic decision items. LetΩ⊆P n (D)be a nonemptyworking universe(typically finite), and fix an arityk≥1. Alevel-nsuperhypercombinationis a set-valued map ?: Ω k −→P(Ω)\∅. Given a nonempty seedS 0 ⊆Ω, define theimmediate-consequence operatorT ? :P(Ω)→P(Ω)by T ? (S) :=S∪ ⋃ ?(X 1 ,...,X k ) ∣ ∣ X 1 ,...,X k ∈S , S⊆Ω, and define thereachable (least?-closed) setcontainingS 0 by S ∗ := ∞ ⋃ t=0 T t ? (S 0 ), T 0 ? (S 0 ) :=S 0 , T t+1 ? (S 0 ) :=T ? ( T t ? (S 0 ) ) . Letbe a preorder onΩ(reflexive and transitive). The set ofSuperHyperdecision outcomesis Opt ?, (S 0 ) :=Max (S ∗ ) := X∈S ∗ ∣ ∣ ∀Y∈S ∗ :XY . Theorem 12.8.2(Well-definedness of SuperHyperdecision-making).Assume Theorem 12.8.1 and, in ad- dition, thatΩis finite. Then: 1.T ? :P(Ω)→P(Ω)is well-defined and monotone (i.e.S⊆S ′ ⇒T ? (S)⊆T ? (S ′ )). Chapter 12. Discussions: New Decision-Making Methods 2. The reachable setS ∗ exists, is nonempty, satisfiesS 0 ⊆S ∗ ⊆Ω, and is theleastT ? -closed subset of ΩcontainingS 0 : T ? (S ∗ ) =S ∗ ,(∀S⊆Ω) ( S 0 ⊆S&T ? (S)⊆S ) ⇒S ∗ ⊆S. Moreover the chain stabilizes after finitely many steps: there existsN≤|Ω|such thatT N ? (S 0 ) =S ∗ . 3. For every preorderonΩ, the outcome setOpt ?, (S 0 )is well-defined and nonempty. Proof.(1) LetS⊆Ω. For any(X 1 ,...,X k )∈S k , the value?(X 1 ,...,X k )is a nonempty subset ofΩby definition, hence ⋃ ?(X 1 ,...,X k )|X 1 ,...,X k ∈S⊆Ω. ThereforeT ? (S)⊆Ω, soT ? is well-defined. IfS⊆S ′ , thenS k ⊆(S ′ ) k , hence the union taken overS k is contained in the union taken over(S ′ ) k , givingT ? (S)⊆T ? (S ′ ). (2) The sequenceS 0 ⊆T ? (S 0 )⊆T 2 ? (S 0 )⊆·is increasing by (1), soS ∗ is well-defined as its union and is nonempty because it containsS 0 . ClearlyS ∗ ⊆Ωsince each iterate is a subset ofΩ. To showT ? (S ∗ ) =S ∗ , use monotonicity:T ? (T t ? (S 0 )) =T t+1 ? (S 0 )⊆S ∗ for allt, soT ? (S ∗ )⊆S ∗ . Conversely,S ∗ ⊆T ? (S ∗ )because T ? is extensive (S⊆T ? (S)by definition), hence equality. Minimality follows because anyT ? -closedS containingS 0 contains all iteratesT t ? (S 0 )by induction, hence contains their unionS ∗ . IfΩis finite, the increasing chain stabilizes in at most|Ω|strict growth steps, soT N ? (S 0 ) =S ∗ for someN≤|Ω|. (3) SinceΩis finite,S ∗ is finite and nonempty. Every preorder on a finite nonempty set admits at least one maximal element. Hence Max (S ∗ )is nonempty and well-defined. 12.9Uncertain SuperHyperDecision-Making Uncertain SuperHyperDecision-Making extends this framework by valuing reachable states with uncertain degrees and score-induced preorders, yielding representation-invariant maximal outcomes under uncertainty. Definition 12.9.1(Uncertain-model-induced uncertainty structure).LetMbe an uncertain model with degree-domain Dom(M)⊆[0,1] k . Fix either: (i) a preorder M on Dom(M), or (i) a score map sc M :Dom(M)→Sinto a preordered set(S, S ). Define an uncertainty structure U M = (U,∼,S, S ,sc) by U:=Dom(M),∼:=equality onU, and: • If (i) is chosen, setS:=U, S := M , and sc:=id U . Chapter 12. Discussions: New Decision-Making Methods • If (i) is chosen, set sc:=sc M . Then any U-setU= (X,μ M )induces a preorder onXvia xy⇐⇒sc(μ M (x)) S sc(μ M (y)). Definition 12.9.2(Uncertain SuperHyperdecision-making (USHDM)).AnUncertain SuperHyperdecision- making(USHDM) instance is a tuple USHDM:= ( D,n,Ω,k,?,S 0 ,U,Val ) where(D,n,Ω,k,?,S 0 )satisfy Theorem 12.8.1,U= (U,∼,S, S )is an uncertainty structure, and Val: Ω→Uis an uncertain evaluation map. The reachable setS ∗ is computed exactly as in Theorem 12.8.1, and theuncertain superhyperdecision outcomesare defined by Opt unc ?,Val (S 0 ) :=Max Val (S ∗ ), where Val is the induced preorder. Theorem 12.9.3(Well-definedness and representation invariance of USHDM).Assume Theorem 12.9.2 and thatΩis finite. Then: 1. The setOpt unc ?,Val (S 0 )is well-defined and nonempty. 2. (Representation invariance) IfVal,Val ′ : Ω→UsatisfyVal(X)∼Val ′ (X)for allX∈Ω, then Opt unc ?,Val (S 0 ) =Opt unc ?,Val ′ (S 0 ). Proof.(1) By Theorem 12.8.2(2), the reachable setS ∗ is a finite nonempty subset ofΩ. Val is a preorder (pullback of a preorder), so by the same finiteness argument as in Theorem 12.8.2(3), Max Val (S ∗ )exists and is nonempty. (2) If Val(X)∼Val ′ (X)for allX, then sc(Val(X)) =sc(Val ′ (X))by class-invariance of sc. Hence Val and Val ′ coincide onΩ, so their maximal sets over the sameS ∗ coincide, proving the claim. Chapter 13 Conclusion In this book, we conducted a comprehensive survey of methods for uncertain decision-making, with partic- ular attention to the roles of uncertainty modeling, weight elicitation, structural and causal analysis, com- pensatory and reference-based ranking schemes, outranking procedures, and related rule-based or sequential decision frameworks. We also discussed how diverse uncertainty paradigms—including fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, rough, soft, and other generalized structures—can be incorporated into decision-making processes in a systematic manner. Overall, the survey shows that uncertain decision-making is not a single method, but a broad methodological family in which the representation of uncertainty and the choice of decision mechanism must be designed in a mutually consistent way. In particular, the selection of a suitable framework depends on the problem structure, the type of available information, the interaction among criteria, and the decision context, such as individual, group, dynamic, multi-stage, or multi-scenario settings. We expect that future work will advance in several directions. First, further computational and experi- mental studies are needed in order to compare the stability, robustness, interpretability, and computational complexity of different uncertain decision-making methods under common benchmark settings. Second, more case-study–driven investigations should be developed in practical domains such as engineering design, manufacturing, finance, medicine, logistics, and social decision analysis, so that the theoretical advan- tages of these methods can be evaluated in realistic decision environments. Third, promising research directions include deeper integration with machine learning, intelligent optimization, expert systems, and broader decision-support frameworks, especially in settings where uncertain, incomplete, inconsistent, or multi-source information must be processed. More generally, future studies may also focus on the development of unified mathematical frameworks capable of comparing, generalizing, and connecting existing uncertain decision-making methods. Such work may help clarify the relationships among current models and may support the design of new methods that are both theoretically well-founded and practically applicable. 373 Chapter 13. Conclusion Disclaimer Funding This study was conducted without any financial support from external organizations or grants. Acknowledgments We would like to express our sincere gratitude to everyone who provided valuable insights, support, and encouragement throughout this research. We also extend our thanks to the readers for their interest and to the authors of the referenced works, whose scholarly contributions have greatly influenced this study. Lastly, we are deeply grateful to the publishers and reviewers who facilitated the dissemination of this work. Data Availability Since this research is purely theoretical and mathematical, no empirical data or computational analysis was utilized. Researchers are encouraged to expand upon these findings with data-oriented or experimental approaches in future studies. Ethical Statement As this study does not involve experiments with human participants or animals, no ethical approval was required. Conflicts of Interest The authors declare that they have no conflicts of interest related to the content or publication of this book. Code Availability No code or software was developed for this study. 375 Chapter 13. Conclusion Clinical Trial This study did not involve any clinical trials. Consent to Participate Not applicable. Use of Generative AI and AI-Assisted Tools I use generative AI and AI-assisted tools for tasks such as English grammar checking, and I do not employ them in any way that violates ethical standards. Disclaimer (Others) This work presents theoretical ideas and frameworks that have not yet been empirically validated. Readers are encouraged to explore practical applications and further refine these concepts. Although care has been taken to ensure accuracy and appropriate citations, any errors or oversights are unintentional. The perspectives and interpretations expressed herein are solely those of the authors and do not necessarily reflect the viewpoints of their affiliated institutions. Appendix A Graphic Structure and Uncertain Graphic Struc- ture In this appendix, we briefly examinegraphic structuresanduncertain graphic structures. A graphic structure is a unified incidence-based framework consisting of a vertex set and a collection of edge objects, where each edge is associated with a nonempty finite set of incident vertices. An uncertain graphic structure is a graphic structure equipped with an uncertainty-degree map that assigns to every vertex and edge a degree tuple from a chosen uncertainty model, enabling graded and multi-component descriptions. A.1 Graphic structure: a unifying incidence-based model The definition of a graphic structure is given below. Definition A.1.1(Iterated universe (for nested / super-hyper entities)).LetX 0 be a nonempty set (“atomic” entities). Define iterated power sets by P (0) (X 0 ) :=X 0 ,P (i+1) (X 0 ) :=P ( P (i) (X 0 ) ) (i≥0), and the depth-nuniverse by Ω (n) (X 0 ) := n ⋃ i=0 P (i) (X 0 ). ThusΩ (0) (X 0 ) =X 0 (ordinary vertices), whileΩ (n) (X 0 )(n≥1) contains set-valued and nested entities, enabling super/higher-order vertex types. Definition A.1.2(Graphic structure of depthn).Fix a base setX 0 6=∅and an integern≥0. Agraphic structure of depthnoverX 0 is a tuple GS= ( V,E,Inc,Tail,Head ) satisfying: •Vertex set:V⊆Ω (n) (X 0 )is a nonempty set of vertices (entities). Elements ofVmay be atomic (∈X 0 ) or nested (set-valued), depending onn. 377 Appendix A. Graphic Structure and Uncertain Graphic Structure •Edge set:Eis a (possibly empty) set of edge-objects, disjoint fromV. •Incidence:Inc:E→ P + fin (V)assigns to each edgee∈Eits (finite, nonempty) incident vertex set Inc(e)⊆V. •Optional direction data:Tail,Head:E→P fin (V)are maps such that, for everye∈E, Tail(e)∩Head(e) =∅,Tail(e)∪Head(e) =Inc(e). (If direction is not needed, one may set Tail(e) =Head(e) =∅for alle.) Remark A.1.3(Standard graph notions as special cases).Within Definition A.1.2, many familiar objects are recovered by imposing additional constraints: •(Undirected) simple graph:n= 0, Inc(e)has cardinality2for alle∈E, and Tail=Head=∅. •Directed graph:n= 0,|Inc(e)|= 2and|Tail(e)|=|Head(e)|= 1for alle(so each edge has a unique tail and head). •Hypergraph:n= 0, Inc(e)∈P + fin (V)arbitrary (no restriction to size2). •Directed hypergraph:n= 0, Tail(e),Head(e)nonempty (typically), disjoint, and Tail(e)∪Head(e) = Inc(e). •SuperHyperGraph (nested vertices):n≥1withV⊆Ω (n) (X 0 ), allowing vertices that are sets (or sets of sets, etc.); edges are still incidence sets of such vertices via Inc. •r-regular graph (property):in the undirected graph case, define deg(v) := ∣ ∣ e∈E:v∈Inc(e) ∣ ∣ (v∈V), and require deg(v) =rfor allv∈V. Hence Definition A.1.2 can be viewed as a common “incidence calculus” for graph-like objects. A.2 Uncertain graphic structure The definition of an uncertain graphic structure is given below. Throughout this subsection, letMbe an uncertain modelwith nonempty degree-domain Dom(M)⊆[0,1] k (as defined previously in the uncertain-set framework). Definition A.2.1(Uncertain graphic structure of typeM).LetGS= (V,E,Inc,Tail,Head)be a graphic structure. Anuncertain graphic structure of typeMis a pair UGS M = ( GS,μ M ) , where μ M :V∪E−→Dom(M) is an uncertainty-degree map assigning a degree tuple in Dom(M)to every vertex and every edge. Optional (model-dependent) consistency.Depending on the intended specialization (fuzzy, intuition- istic fuzzy, neutrosophic, plithogenic, etc.), one may additionally impose constraints linking edge-degrees to incident vertex-degrees. For example, in the fuzzy case Dom(M) = [0,1], a common constraint is μ M (e)≤minμ M (u),μ M (v)for an undirected edgeeincident tou,v. Such constraints are not fixed at the level of this general definition; they are selected according to the chosen modelMand the application semantics. Appendix A. Graphic Structure and Uncertain Graphic Structure Remark A.2.2(Underlying crisp structure and reduction to uncertain sets).There is an obviousforgetful projection π:UGS M = ( (V,E,Inc,Tail,Head),μ M ) 7−→(V,E,Inc,Tail,Head), which discards uncertainty degrees and returns the underlying graphic structure. IfE=∅, thenUGS M reduces to an uncertain set onVvia the restrictionμ M | V :V→Dom(M). Theorem A.2.3(Well-definedness and existence).Fix a depthn≥0, a base setX 0 6=∅, and an uncertain modelMwithDom(M)6=∅. (i) For every graphic structureGSof depthnoverX 0 , there exists an uncertain graphic structureUGS M = (GS,μ M )of typeM. (i) The definition ofUGS M iswell-defined: namely, for any choice of mapμ M :V∪E→Dom(M), the pair ( GS,μ M ) is a valid uncertain graphic structure of typeM. (i) Every classical graph-like object (graph, directed graph, hypergraph, superhypergraph, etc.) can be em- bedded as a special case ofUGS M by choosingGSwith the corresponding constraints from Remark A.1.3 and takingμ M to be constant (or crisp-valued, when appropriate). Proof.(i) Since Dom(M)6=∅, choose any fixed degree tupled 0 ∈Dom(M). Defineμ M :V∪E→Dom(M) by the constant mapμ M (z) :=d 0 for allz∈V∪E. Thenμ M is a well-typed function with codomain Dom(M), soUGS M = (GS,μ M )satisfies Definition A.2.1. (i) Letμ M :V∪E→Dom(M)be any function. Definition A.2.1 requires only thatGSis a graphic structure and thatμ M maps each vertex/edge into the degree-domain Dom(M). Both hold by assumption, hence ( GS,μ M ) is a valid uncertain graphic structure. (i) Take any classical structure (e.g., a graph, directed graph, hypergraph, or a nested-vertex superhyper- graph). SelectnandX 0 so that its vertex objects lie inΩ (n) (X 0 ), and encode it as a graphic structureGS by imposing the appropriate constraints from Remark A.1.3 on Inc,Tail,Head. Finally, apply (i) to obtain UGS M = (GS,μ M )(e.g., using a constantμ M ), which realizes the classical object as a special case within the uncertain graphic-structure framework. Appendix (List of Tables) 1.1 Concise comparison between classical (crisp) sets and fuzzy sets. . . . . . . . . . . . . . . .7 1.2 Representative set extensions and the canonical information stored per element. . . . . . . .8 1.3 Compact taxonomy of methods inUncertain Decision Science(classified by task, input struc- ture, and typical outputs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 A catalogue of uncertainty-set families (U-Sets) by the dimensionkof the degree-domain Dom(M)⊆[0,1] k [20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 A catalogue of uncertainty-graph families (uncertain graphs) by the dimensionkof the degree- domain Dom(M)⊆[0,1] k (cf. [218]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 A catalogue of uncertainty-decision families (UDM) by the dimensionkof the degree-domain Dom(M)⊆[0,1] k (cf. [20]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1 A catalogue of uncertainty-aware MCDM families by the dimensionkof the degree-domain Dom(M)⊆[0,1] k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Related uncertainty-aware MADM families (examples) grouped by the degree-domain dimen- sionkof Dom(M)⊆[0,1] k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3 A compact catalogue of uncertainty-awaregroup decision-making(GDM) families by the dimensionkof the degree-domain Dom(M)⊆[0,1] k . . . . . . . . . . . . . . . . . . . . . . . 42 3.4 Related uncertainty-model variants of Dynamic Fuzzy Decision-Making. . . . . . . . . . . . 46 3.5 Related uncertainty-model variants of Multiple Objective Decision-Making (MODM). . . . 50 3.6 Related uncertainty-model variants of Fuzzy Ethical Decision-Making. . . . . . . . . . . . . 53 3.7 Related uncertainty-model variants of Fuzzy Consensus Decision-Making. . . . . . . . . . . 57 3.8 Related uncertainty-model variants of Fuzzy Strategic Decision-Making. . . . . . . . . . . . 61 3.9 Related uncertainty-model variants of Fuzzy Multi-Expert Decision-Making. . . . . . . . . . 64 3.10 Related uncertainty-model variants of Fuzzy Multi-Stage Decision-Making. . . . . . . . . . 70 3.11 Related uncertainty-model variants of Fuzzy Multi-Level Decision-Making. . . . . . . . . . . 77 3.12 Related uncertainty-model variants of Fuzzy Multi-Agent Decision-Making. . . . . . . . . . 84 3.13 Related uncertainty-model variants of Fuzzy Multi-Scenario Decision-Making. . . . . . . . . 91 4.1 A concise comparison of fuzzy weight-elicitation methods. . . . . . . . . . . . . . . . . . . . 93 4.2 Related uncertainty-model variants of AHP (classified by the degree-domain dimensionk). . 98 4.3 Related concepts of LOPCOW under uncertainty-aware models. . . . . . . . . . . . . . . . . 102 4.4 Related concepts of SIWEC under uncertainty-aware models. . . . . . . . . . . . . . . . . . 107 4.5 Related concepts of judgment matrices under uncertainty-aware models. . . . . . . . . . . . 109 4.6 Related concepts of ANP under uncertainty-aware models. . . . . . . . . . . . . . . . . . . . 114 4.7 Related concepts of Ordinal Priority Approach (OPA) under uncertainty-aware models. . . 118 4.8 Related concepts of PIPRECIA under uncertainty-aware models. . . . . . . . . . . . . . . . 122 4.9 Related uncertainty-model variants of SWARA (classified by the degree-domain dimensionk).125 4.10 Related concepts of CILOS under uncertainty-aware models. . . . . . . . . . . . . . . . . . 131 4.11 Related concepts of BWM under uncertainty-aware models. . . . . . . . . . . . . . . . . . . 141 4.12 Related concepts of CRITIC under uncertainty-aware models. . . . . . . . . . . . . . . . . . 146 4.13 Related concepts of MEREC under uncertainty-aware models. . . . . . . . . . . . . . . . . . 149 380 Appendix (List of Tables) 5.1 A concise comparison of four structure/causality decision-modeling frameworks. . . . . . . . 155 5.2 Related concepts of DEMATEL under uncertainty-aware models. . . . . . . . . . . . . . . . 159 5.3 Related concepts of Interpretive Structural Modeling under uncertainty-aware models. . . . 165 5.4 Related concepts of MICMAC under uncertainty-aware models. . . . . . . . . . . . . . . . . 171 5.5 A compact catalogue of Cognitive Map variants by the uncertainty encoding (indexed by a convenient degree-domain dimensionk). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.1 A concise comparison of representative decision-matrix methods discussed in this chapter. . 175 6.2 COPRAS and representative uncertainty-aware variants (organized by the degree-domain dimensionk). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.3 Related concepts of MACBETH under uncertainty-aware models. . . . . . . . . . . . . . . . 184 6.4 CoCoSo and representative uncertainty-aware variants (organized by the degree-domain di- mensionk). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.5 Related concepts of SAW under uncertainty-aware models. . . . . . . . . . . . . . . . . . . . 193 6.6 Related concepts of RAFSI under uncertainty-aware models. . . . . . . . . . . . . . . . . . 197 6.7 Related concepts of RATMI under uncertainty-aware models. . . . . . . . . . . . . . . . . . 202 6.8 Related concepts of RANCOM under uncertainty-aware models. . . . . . . . . . . . . . . . 205 6.9 Related concepts of AROMAN under uncertainty-aware models. . . . . . . . . . . . . . . . 212 6.10 Related concepts of MAUT under uncertainty-aware models. . . . . . . . . . . . . . . . . . 216 6.11 Related concepts of SMART under uncertainty-aware models. . . . . . . . . . . . . . . . . . 220 6.12 Related concepts of TODIM under uncertainty-aware models. . . . . . . . . . . . . . . . . . 227 6.13 Related concepts of GRA under uncertainty-aware models. . . . . . . . . . . . . . . . . . . 231 6.14 Related concepts of ARAS under uncertainty-aware models. . . . . . . . . . . . . . . . . . . 235 6.15 Related concepts of WASPAS under uncertainty-aware models. . . . . . . . . . . . . . . . . 238 6.16 Related concepts of MOORA under uncertainty-aware models. . . . . . . . . . . . . . . . . 241 6.17 Related concepts of Preference Selection Index under uncertainty-aware models. . . . . . . . 245 6.18 Related concepts of Range of Value under uncertainty-aware models. . . . . . . . . . . . . . 248 6.19 Related concepts of MOOSRA under uncertainty-aware models. . . . . . . . . . . . . . . . . 250 7.1 A concise comparison of representative distance-to-reference, border-based, and compromise- index methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 7.2 Related concepts of TOPSIS under uncertainty-aware models. . . . . . . . . . . . . . . . . . 255 7.3 Related concepts of MARCOS under uncertainty-aware models. . . . . . . . . . . . . . . . . 259 7.4 Related concepts of CODAS under uncertainty-aware models. . . . . . . . . . . . . . . . . . 263 7.5 Related concepts of EDAS under uncertainty-aware models. . . . . . . . . . . . . . . . . . . 266 7.6 Related concepts of VIKOR under uncertainty-aware models. . . . . . . . . . . . . . . . . . 270 7.7 Related concepts of MABAC under uncertainty-aware models. . . . . . . . . . . . . . . . . 273 7.8 Related concepts of MAIRCA under uncertainty-aware models. . . . . . . . . . . . . . . . . 276 8.1 A concise comparison of representative outranking decision methods. . . . . . . . . . . . . . 277 8.2 Related concepts of ELECTRE under uncertainty-aware models. . . . . . . . . . . . . . . . 282 8.3 Related concepts of FlowSort under uncertainty-aware models. . . . . . . . . . . . . . . . . 286 8.4 Related concepts of PROMETHEE under uncertainty-aware models. . . . . . . . . . . . . . 289 8.5 Related concepts of QUALIFLEX under uncertainty-aware models. . . . . . . . . . . . . . . 292 8.6 Related concepts of ORESTE under uncertainty-aware models. . . . . . . . . . . . . . . . . 296 9.1 Concise comparison of representative rule-induction, learning, evidence-aggregation, and sequential-decision methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 9.2 Related concepts of decision trees under uncertainty-aware models. . . . . . . . . . . . . . . 301 9.3 Related concepts of Markov decision processes under uncertainty-aware models. . . . . . . . 308 9.4 Related concepts of evidential reasoning under uncertainty-aware models. . . . . . . . . . . 312 10.1 Concise comparison of representative other related decision methods. . . . . . . . . . . . . . 313 Appendix (List of Tables) 10.2 Related concepts of goal programming under uncertainty-aware models. . . . . . . . . . . . 318 10.3 Related concepts of data envelopment analysis under uncertainty-aware models. . . . . . . . 320 10.4 Related concepts of decision tables under uncertainty-aware models. . . . . . . . . . . . . . 326 10.5 Related concepts of ultrafilters under uncertainty-aware models. . . . . . . . . . . . . . . . . 331 10.6 Related concepts of SWOT under uncertainty-aware models. . . . . . . . . . . . . . . . . . 334 10.7 Related concepts of cost-benefit analysis under uncertainty-aware models. . . . . . . . . . . 338 12.1 Concise comparison of the new decision-making methods introduced in this chapter. . . . . 353 * Appendix (List of Figures) 1.1 Conceptual diagram: combining an uncertain-set paradigm with a decision-making technique to obtain novel and practically meaningful outcomes. . . . . . . . . . . . . . . . . . . . . . .9 2.1 Conceptual illustration of a neutrosophic set. 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