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Anchored Regularized Direct Least Squares (ARDLS): Integrating Established Prioritization Operators for Priority Elicitation in the Analytic Hierarchy Process
Kevin Kam Fung Yuen
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Abstract
Abstract:Pairwise reciprocal matrices are fundamental to the Analytic Hierarchy Process (AHP), a decision-making model. While the Direct Least Squares (DLS) method provides an intuitive mechanism for deriving priority vectors without complex transformations, the DLS provides multiple solutions. Under high levels of inconsistency, such as cyclic contradictions, this non-convexity yields multiple distinct global minima, resulting in unstable priority rankings that critically depend on initial algorithmic guesses. To overcome this structural deficiency, this paper introduces the Anchored Regularized Direct Least Squares (ARDLS) optimization model. ARDLS integrates uniquely determined established prioritization operators, such as normalization techniques, the Eigenvector method, Singular Value Decomposition, Cosine Maximization, and the Pseudo-Inverse Gram Matrix (the closed-form solution of Weighted Least Squares), as theoretical anchors within a regularization penalty. This integration systematically breaks mathematical symmetries, tilting the optimization landscape to guarantee convergence upon a single, unique global minimum. Comprehensive numerical experiments and simulations validate that the ARDLS framework successfully reduces root mean square error among established priority operators, while guaranteeing strict mathematical uniqueness. The proposed ARDLS may be the ideal alternative for the AHP applied to many application domains.
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- Source: https://arxiv.org/abs/2608.21187v1
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Anchored Regularized Direct Least Squares (ARDLS): Integrating Established Prioritization Operators for Priority Elicitation in the Analytic Hierarchy Process Kevin Kam Fung YUEN 1* 1 School of Science, Monash University Malaysia, Sunway, Malaysia * E-mail: kevin.yuen@monash.edu; kevinkf.yuen@gmail.com; https://orcid.org/0000-0003-1497-2575 Abstract. Pairwise reciprocal matrices are fundamental to the Analytic Hierarchy Process (AHP), a decision-making model. While the Direct Least Squares (DLS) method provides an intuitive mechanism for deriving priority vectors without complex transformations, the DLS provides multiple solutions. Under high levels of inconsistency, such as cyclic contradictions, this non-convexity yields multiple distinct global minima, resulting in unstable priority rankings that critically depend on initial algorithmic guesses. To overcome this structural deficiency, this paper introduces the Anchored Regularized Direct Least Squares (ARDLS) optimization model. ARDLS integrates uniquely determined established prioritization operators, such as normalization techniques, the Eigenvector method, Singular Value Decomposition, Cosine Maximization, and the Pseudo-Inverse Gram Matrix (the closed-form solution of Weighted Least Squares), as theoretical anchors within a regularization penalty. This integration systematically breaks mathematical symmetries, tilting the optimization landscape to guarantee convergence upon a single, unique global minimum. Comprehensive numerical experiments and simulations validate that the ARDLS framework successfully reduces root mean square error among established priority operators, while guaranteeing strict mathematical uniqueness. The proposed ARDLS may be the ideal alternative for the AHP applied to many application domains. Keywords: Non-convex Optimisation, Regularization, Direct Least Squares, Priority Elicitation, Analytic Hierarchy Process. 1. Introduction Although the Analytic Hierarchy Process (AHP) [21] is widely used for complex prioritization and decision making, it remains subject to debates regarding arbitrary hierarchical composition [1; 9; 28], conflict of expected utility theory [11; 22], and rank reversals [2; 15; 24] . A central line of research focuses on prioritization operators (POs), where numerous algebraic and optimization-based methods have been developed [3; 7; 8; 10; 12; 16-18; 20; 21; 23; 25-27; 30; 33]. Despite this, broad evaluations [6; 13; 14; 19; 29; 31; 33] may conclude that no individual PO consistently outperforms the others in the presence of judgmental inconsistencies. The AHP relies on pairwise reciprocal matrices (PRMs) to derive priorities. While priority extraction from consistent PRMs is trivial, extracting stable vectors from inconsistent matrices remains challenging, as different methods often yield contradictory rankings. Although arithmetically appealing, the Direct Least Squares (DLS) [7] operator possesses severe structural deficiencies: it lacks a closed- form analytical solution and is fundamentally non-convex. This non-convexity frequently induces alternative local and global optimaβmeaning distinct priority vectors ν€ can yield the exact same minimized objective value [4; 5; 7; 31]. Consequently, resolving this optimization landscape via standard numerical solvers may be unstable and computationally challenging. To overcome the limitations of DSL while preserving its arithmetic advantages and integrating other prioritization operators (POs), this paper proposes the Anchored Regularized Direct Least Squares (ARDLS) optimization model. By incorporating robust, uniquely determined POs as theoretical anchors within a regularization penalty, ARDLS systematically addresses the DLS limitations that result in multiple minima. The core contributions of this article are organized as follows. Section 2 reviews established POs with explicit algebraic forms that can be integrated into the proposed ARDLS framework. Section 3 introduces a PO evaluation metric, Root Mean Squared Error (RMSE), to quantify native errors. It demonstrates that while DLS achieves the lowest RMSE among all POs, it suffers from non-uniqueness by inducing multiple solutions. Section 4 formulates the ARDLS model to preserve a unique solution and strategically reduce RMSE by leveraging the results of other established POs. Section 5 provides a comprehensive numerical analysis alongside graphical illustrations to enhance the mathematical interpretability of ARDLS, validating its usability, feasibility, and stability across various inconsistency thresholds. Section 6 summarizes the methodological contributions of the study and outlines prospective avenues for future research. 2 Prioritization Operators This section reviews the algebraic formulations of established prioritization operators (POs), including four foundational normalization techniques introduced by [21] . Because these elementary procedures were not assigned explicit nomenclature in the early seminal literature[21] , they are named here according to their operational calculation steps, following the conventions established in [29; 30] . 2.1 Normalization of the Row Sum (NRS) The NRS method computes the sum of the elements within each row and normalizes them by the grand total of all matrix elements, ensuring that the components of the derived priority vector sum to unity. The algebraic form of NRS is given by: νβ² ν =βν νν ν ν=1 βΉν€ ν = νβ² ν β νβ² ν ν ν=1 , ν=1,2,...,ν (1) 2.2 Normalization of Reciprocals of Column Sum (NRCS) The NRCS method sums the elements within each column, computes the reciprocal of each column sum, and subsequently normalizes these values so that the final elements sum to unity. It is defined as follows: νβ² ν = 1 β ν νν ν ν=1 βΉ ν€ ν = νβ² ν β νβ² ν ν ν=1 , j=1,2,...,ν (2) 2.3 Arithmetic Mean of Normalized Columns (AMNC) Under the AMNC framework, each element in the Pairwise Comparison Matrix (PCM) ν΄ is first divided by its respective column sum. The final priority weight ν€ ν is then derived by taking the arithmetic mean of these normalized elements across each row: νβ² νν = ν νν β ν νν ν ν=1 βΉ ν€ ν = 1 ν βνβ² νν ν ν=1 , ν,ν=1,2,...,ν (3) 2.4 Normalization of the Geometric Means of Rows (NGMR) / Logarithmic Least Squares (LLS) The NGMR operator calculates the geometric mean of the elements in each row by taking the ν-th root of their product, followed by a scaling normalization to achieve a sum of unity: ν€β² ν =βν νν 1/ν ν ν=1 βΉ ν€ ν = ν€β² ν β ν€β² ν ν ν=1 , ν=1,2,...,ν (4) The literature [8] demonstrates that the geometric mean vector serves as the exact, closed-form solution to the Logarithmic Least Squares (LLS) optimization problem, which is formulated as: ννν β(ννν νν β ( ννν€β² ν βννν€β² ν ) ) 2 ν ν>ν ν ν=1 S.T. βν€β² ν ν ν=1 =1, ν€β² ν >0,ν=1,2,...,ν (5) Because the LLS formulation yields an unnormalized intermediate solution vector ν€ ν β² , a subsequent scaling step [29; 30] is applied to obtain the final normalized priority vector ν€ ν . 2.5 Eigenvector (EV) An Eigenvector (EV) operator , introduced by Saaty [21], derives the non-normalized priority vector ν€ β² as the principal right eigenvector corresponding to the maximum eigenvalue ν max of the matrix ν΄ by solving the characteristic eigensystem: ν΄ν€β²=ν max ν€β², ν€β²= [ ν€β² 1 ,ν€β² 2 ,...,ν€β² ν ] ν (6) ν€β², which is normalized as ν€ ν , is given by ν€β²=ννν νββ ( ν΄ ν ν ν νν΄ ν ν ν ) βΉ ν€ ν = ν€β² ν β ν€β² ν ν ν=1 ,ν=1,2,...,ν. (7) The principal eigenvalue ν max can be computed via: ν ννν₯ =ν+ 1 ν β νΏ νν 2 1+νΏ νν 1β€ν<νβ€ν , νΏ νν =( ν νν ν€ ν /ν€ ν β1) (8) ν max is used to form the Consistency Ratio (νΆν ) to evaluate the consistency of a PCM. νΆν = νΆνΌ ν νΌ ,νΆνΌ= ν ννν₯ βν νβ1 (9) 2.6 Singular Value Decomposition (SVD) The SVD framework for priority elicitation in AHP was proposed by [12] . To summarize their idea, this approach factorizes PRM into three components: ν΄=νΞ£ν ν (10) The priority weights ν€ ν are obtained by extracting and combining elements from the principal left singular vector ν’ β ,1 and the reciprocal elements of the principal right singular vector ν£ β ,1 , followed by standard normalization: ν€β² ν =ν’ ν,1 + 1 ν£ ν,1 βΉ ν€ ν = ν€β² ν β ν€β² ν ν ν=1 ,ν=1,2,...,ν (11) 2.7 Cosine maximization (CosMax) The Cosine Maximization (CosMax) method [16] maximizes the directional alignment between the priority weight vector and the column profiles of the matrix: max νΆ=β β ν€ ν ν ν=1 ν νν β β ν€ ν 2 ν ν=1 β β ν νν 2 ν ν=1 ν ν=1 S.T. βν€ ν ν ν=1 =1,ν€ ν >0,ν=1,...,ν (12) Although the original work [16] did not explicitly frame this objective using geometric cosine notation, the objective function C can be rewritten as the sum of the cosines of the angles ν ν between the weight vector ν€ and the column vectors ν β ν : νΉ ( ν€ ) =βcos ( ν ν ) ν ν=1 =β ν€β ν β ν βν€β 2 βν β ν β 2 ν ν=1 =νΆ (13) where ν β ν is a ν -th column vector of ν΄ . While [16] outlined a five-step algorithmic procedure to optimize this model, this study further derives a direct, single-stage algebraic closed-form solution: ν€ ν = β ( ν νν β β ν νν 2 ν ν=1 ) ν ν=1 β ( ν ν ν β β ν νν 2 ν ν=1 ) ν ν=1 ν ν =1 ,ν=1....,ν (14) 2.8 Least Squares operators To achieve the least RMSV, the Direct Least Squares (DLS) operator proposed by [7] directly minimizes the sum of squared errors: ννν DSE=β ( ν νν β ν€ ν ν€ ν ) 2 ν ν=1 ν ν=1 s.t. βν€ ν ν ν=1 =1, ν€ ν >0,ν=1,2,...,ν . (15) However, the DLS optimization problem possesses severe structural deficiencies mentioned in the Section1. To address this issue, the Weighted Least Squares (WLS) optimization approach was introduced by [7] as follows. ννν WSE=β ( ν€ ν βν νν ν€ ν ) 2 ν ν=1 ν ν=1 s.t. βν€ ν ν ν=1 =1, ν€ ν >0,ν=1,2,...,ν . (16) For a long time, the WLS model lacked a closed-form analytic solution until [31] proposed the Inverse Gram Matrix family of methods. Within this family framework, the Pseudo Inverse Gram Matrix (PIGM) approach offers an explicit solution, defined as follows: ν€= νΊ β1 ν ν ν νΊ β1 ν ,ν νν = ( νβ1 ) +βν νν 2 ν ν=ν 1βν νν βν νν νβ ν ,βν νν βνΊ. (17) Here, νββ ν€νν‘β νβ 0, and ν=[1,1,...,1] ν represents a column vector of ones compatible with the dimensions of the matrix νΊ β1 . 3. Prioritization result variance measures Selecting the most appropriate Prioritization Operator (PO) requires a rigorous performance metric. Advancing beyond traditional metrics such as Total Deviation (TD) [13] and Euclidean Distance (ED) [19] , the authors in [29] proposed the Root Mean Square Error (RMSE) of the form below. ν νννΈ ( ν΄,ν ) = β 1 νΓν β ( ν νν β ν€ ν ν€ ν ) 2 ν ν=1 ν ν=1 (18) To facilitate efficient matrix-centric computations, the above form can be compactly expressed via element-wise matrix operations [32] as below: ν νννΈ ( ν΄,ν ) = 1 ν β β ( ν΄βν ( ν β1 ) ν ) β2 (19) where β2 denotes the entry-wise Hadamard power, and ν€ β1 =[1/ν€ 1 ,1/ν€ 2 ,...,1/ν€ ν ] ν . Although DLS yields the lowest RMSE, it often leads to multiple solutions. This paper proposes ARDLS to overcome this limitation by providing a unique solution while outperforming other POs in terms of RMSE. 4. Anchored Regularized Direct Least Squares 4.1 The Formulation of ARDSL The optimization model for Anchored Regularized Direct Least Squares (ARDLS) aims to minimize the Anchored Regularized Direct Least Squares Error (ARDLSE). This objective function is formulated as the combination of the standard Direct Squares Error (ν·ννΈ(ν€) ) and an Anchor Regularization Penalty (ν΄ν ν(ν€)): ννν ARDSE (w)= DSE(w)+ARP(w) =β ( ν νν β ν€ ν ν€ ν ) 2 ν ν=1 ν ν=1 +νβ ( ν€ ν βν€ ν νννβνν ) 2 ν ν=1 s.t βν€ ν ν ν=1 =1, ν€ ν >0,ν=1,2,...,ν . (20) The anchor weights vector, ν€ anchor =[ν€ 1 anchor ,ν€ 2 anchor ,...,ν€ ν anchor ] ν , represents a pre-established priority vector derived from other Prioritization Operators (POs), such as those discussed in Section 2. To integrate these prior weights into the objective function, the ν΄ν ν(ν€) is introduced to act as a penalty that prevents the estimated weights from deviating excessively from the anchor weights weights (ν€ anchor ). Consequently, ARDLS reduces the DSE or RMSE by refining the weights generated by the chosen PO. From a computational perspective, when utilizing an optimization solver to solve the model, setting ν€ anchor as the initial search value for ν€ can significantly accelerate convergence, offering a distinct advantage over the random initial values typically utilized by solvers. By adjusting the regularization parameter νβ₯0 , the ARDLS model serves as an elegant compromise between the purely data-driven weights of DLS and the prior weights of the selected PO. When the regularizer is small (i.e., νβ0 ), the regularization term vanishes, reducing the ARDLS model to standard DLS. Conversely, when the regularizer is large (i.e., νββ), the solution converges strictly to the anchor weight vector ν€ anchor . The optimal behavior of the ARDLS solution heavily depends on the appropriate setting of ν. 4.2 Convexity Analysis of DLS If the PRM is perfectly consistent, DLS yields a unique solution. If the PRM is only slightly inconsistent, DLS generally retains a unique solution. However, if the PRM is highly inconsistent, the optimization landscape may become non-convex, and DLS is highly likely to suffer from multiple local or global minima. To determine the convexity condition for a specific DLS solution, the following theorem holds. Theorem 1 (DLS Local Convexity Bound): For any optimal weight vector ν€ β evaluated at the minimum DSE, the DLS problem is locally strictly convex along its coordinate axes (guaranteeing that the solution is an isolated, unique minimum strictly within its immediate local basin) if the DLS Local Convexity Bound, Ξ ν·νΏν , is strictly positive: β³ ν·ννΏ =min νβ 1,β―,n [ β 2 ν·ννΈ ( ν€ ) βν€ ν 2 ] > 0, (21) Explicitly, this evaluates to: β³ ν·ννΏ =min νβ 1,β―,ν [ β( 2ν€ ν ν€ ν 4 ( 3ν€ ν β2ν νν ν€ ν ) ) νβ ν +β 2 ν€ ν 2 νβ ν ] > 0 (22) Proof: To ensure that the function curves strictly upwards along a specific weight axis ν€ ν , its second partial derivative with respect to ν€ ν must be strictly greater than zero. From Eq. (15), The DSE is: ν·ννΈ ( ν€ ) =β ( ν νν β ν€ ν ν€ ν ) 2 ν ν=1 ν ν=1 (23) To find the partial derivative with respect to a specific variable ν€ ν , we isolate the terms involving ν€ ν , i.e., where ν=ν or ν=ν . The term where ν=ν and ν=ν evaluates to ( ν ν βν€ ν /ν€ ν ) 2 =(1β 1) 2 =0 and is therefore omitted. ν·ννΈ ( ν€ ) = β (ν νν β ν€ ν ν€ ν ) 2 νβ ν +β(ν νν β ν€ ν ν€ ν ) 2 νβ ν +νΆ (24) where νΆ represents the remaining terms independent of ν€ ν . Taking the first derivative of ν·ννΈ(ν€) with respect to ν€ ν using the chain rule yields: νν·ννΈ ( ν€ ) νν€ ν =β2 νβ ν ( ν νν β ν€ ν ν€ ν )( β 1 ν€ ν ) +β2 νβ ν (ν νν β ν€ ν ν€ ν ) ( ν€ ν ν€ ν 2 ) = β( β 2ν νν ν€ ν + 2ν€ ν ν€ ν 2 ) νβ ν + β( 2ν νν ν€ ν ν€ ν 2 β 2ν€ ν 2 ν€ ν 3 ) νβ ν (25) Next, taking the second derivative with respect to ν€ ν gives: β 2 ν·ννΈ ( ν€ ) βν€ ν 2 = β( 0+ 2 ν€ ν 2 ) νβ ν + β( β 4ν νν ν€ ν ν€ ν 3 + 6ν€ ν 2 ν€ ν 4 ) νβ ν = β 2 ν€ ν 2 νβ ν + β( 6ν€ ν 2 β4ν νν ν€ ν ν€ ν ν€ ν 4 ) νβ ν = β( 2ν€ ν ν€ ν 4 (3ν€ ν β2ν νν ν€ ν ) ) νβ ν + β 2 ν€ ν 2 νβ ν ,βν (26) Thus, Eq. (22) holds. Q.E.D If Ξ ν·νΏν >0, the DLS landscape possesses a locally unique solution within that specific coordinate basin. However, this local property does not preclude the existence of multiple distinct global solutions elsewhere residing within its own locally convex basin. Conversely, if Ξ ν·νΏν β€0, the landscape is non- convex at that coordinate and lacks even local uniqueness. Thus, DLS may suffer from multiple local or global minima. 4.3 Convexity Analysis of ARDLS The anchor regularization penalty (ARP) term introduces a strongly convex quadratic component that reshapes the optimization landscape. When ν is sufficiently large, the total objective function (ν΄ν ν·ννΈ) is forced into strict convexity, guaranteeing a single, unique global minimum. The following theorems determine the necessary threshold for ν. Theorem 2 (νννννν νννν‘ννν ννννν£νν‘νν£ν νν ν΄ν ν ): The second partial derivative of ν΄ν ν(ν€) with respect to ν€ ν takes the following form: ν 2 ARP ( ν€ ) νν€ ν 2 =2ν (27) Proof: From Eq. (21), the anchor regularization penalty is defined as: ARP(ν€)=νβ( ν ν=1 ν€ ν βν€ ν anchor ) 2 (28) To evaluate the partial derivative with respect to a specific variable ν€ ν , we expand the summation by separating the ν-th term from all other terms (νβ ν): ARP(ν€)=ν(ν€ ν βν€ ν anchor ) 2 +νβ( νβ ν ν€ ν βν€ ν anchor ) 2 (29) Taking the first partial derivative with respect to ν€ ν gives: βARP(ν€) βν€ ν = β βν€ ν [ ν(ν€ ν βν€ ν anchor ) 2 ] + β βν€ ν [ νβ( νβ ν ν€ ν βν€ ν anchor ) 2 ] = β βν€ ν [ ν(ν€ ν βν€ ν anchor ) 2 ] + 0 =2ν ( ν€ ν βν€ ν anchor ) ( β βν€ ν (ν€ ν βν€ ν anchor )) =2ν(ν€ ν βν€ ν anchor ) (30) Subsequently, evaluating the second partial derivative with respect to ν€ ν yields: β 2 ARP(ν€) βν€ ν 2 = β βν€ ν [ 2ν(ν€ ν βν€ ν anchor ) ] = β βν€ ν ( 2νν€ ν β2νν€ ν anchor ) =2ν (31) Q.E.D. Theorem 3 (Minimum bound of Regulizer (ν Μ ) ): If Ξ ν·νΏν β€0, the ARDLS problem guarantees a unique solution if ν>ν Μ , where the minimum bound of Regulizer ν Μ is of the form below. ν>ν Μ =β 1 2 β³ ν·ννΏ =β 1 2 min νβ 1,β―,ν [ β( 2ν€ ν ν€ ν 4 ( 3ν€ ν β2ν νν ν€ ν ) ) νβ ν +β 2 ν€ ν 2 νβ ν ] (32) Proof: From Eq. (20), the total objective function is: ARDSE (w)= DSE(w)+ARP(w). (33) To identify the minimum bound of the regularizer (ν Μ ) required to ensure coordinate-wise strict convexity, we compute the second partial derivative with respect to any weight ν€ ν : β 2 ARDSE ( ν€ ) βν€ ν 2 = β 2 DSE ( ν€ ) βν€ ν 2 + β 2 ν΄ν ν ( ν€ ) βν€ ν 2 (34) Substituting the second term utilizing Theorem 2 yields: β 2 ARDSE ( ν€ ) βν€ ν 2 = β 2 DSE ( ν€ ) βν€ ν 2 +2ν (35) From theorem 1, the explicit form of the first term is: β 2 DSE βν€ ν 2 = β 2ν€ ν ν€ ν 4 νβ ν ( 3ν€ ν β2ν νν ν€ ν ) + β 2 ν€ ν 2 νβ ν (36) To guarantee strict local convexity, we require β 2 ν·ννΈ(ν€) βν€ ν 2 +2ν>0 for all coordinates ν . Isolating 2ν gives: 2ν>β ν 2 ν·ννΈ ( ν€ ) νν€ ν 2 (37) To ensure this inequality holds simultaneously across all ν , 2ν must be strictly greater than the maximum possible value of β β 2 ν·ννΈ(ν€) βν€ ν 2 . Mathematically, the maximum of a negative set is equivalent to the negative of its minimum. Therefore, we establish the threshold using the previously defined minimum coordinate-wise curvature, Ξ ν·νΏν : 2ν>ββ³ ν·ννΏ =βmin νβ 1,β―,ν [ β( 2ν€ ν ν€ ν 4 ( 3ν€ ν β2ν νν ν€ ν ) ) νβ ν +β 2 ν€ ν 2 νβ ν ] (38) Dividing by 2 proves the minimum bound of the regularizer (ν Μ ) as shown in Eq. (33). Q.E.D. Since ν€ is unknown, an optimization solver can first be run on the DLS problem to establish a baseline Ξ ν·νΏν . If the DLS optimization landscape is locally strictly convex at the found minimum (Ξ ν·νΏν >0) and this minimum is the globally unique solution, it can be directly utilized. However, if an inconsistent PRM leads to a DLS problem yielding a set of multiple global solutions ν β = ν€ 1 β ,ν€ 2 β ,...,ν€ ν β , the local curvature (Ξ ν·νΏν ) generally differs across these distinct minima. The set ν β can be identified by running the DLS solver for a sufficiently large number of times (e.g., ν=1000) using different random initial search values. To robustly configure the regularization parameter, Ξ ν·νΏν must be evaluated across all identified global solutions. We must distinguish between the minimum curvature (Ξ ν·νΏν min ), required to enforce convexity, and the maximum curvature (Ξ ν·νΏν max ), needed to break symmetry across multiple convex basins: Ξ ν·νΏν min =min ν€ β βν β [ Ξ ν·νΏν ( ν€ β ) ] (39) Ξ ν·νΏν max =max ν€ β βν β [ Ξ ν·νΏν ( ν€ β ) ] (40) To guarantee a sufficiently robust bound for the regularizer (ν Μ ) that enforces strict convexity globally, we define the baseline regularization threshold using the worst-case (minimum) curvature: ν Μ =β 1 2 Ξ ν·νΏν min =β 1 2 min ν€ β βν β [ Ξ ν·νΏν ( ν€ β ) ] (41) Let νΌ be a safety margin scalar strictly greater than 1 (νΌ>1). The safe regularizer (ν β ) for ARDLS takes the form: ν β = ν,Ξ ν·νΏν min >0 and |ν β |=1 νΌΞ ν·νΏν max ,Ξ ν·νΏν min >0 and |ν β |>1 νΌν Μ ,Ξ ν·νΏν min β€0 (42) Depending on the underlying landscape topology, the regularization mechanism operates across three distinct scenarios: for a unique convex minimum (Ξ ν·νΏν min >0,|ν β |=1), a simple scaling factor (νβ₯1) softly guides the anchor weights to the solution of less RMSE. For multiple convex minima (Ξ ν·νΏν min >0,|ν β |>1), scaling by maximum curvature (νΌΞ ν·νΏν max ) breaks inter-basin symmetry to isolate a single global minimum. For non-convex settings (Ξ ν·νΏν min β€0 ), setting ν β =β νΌ 2 Ξ ν·νΏν min directly counteracts negative curvature ensuring a strictly positive second derivative (2ν+Ξ ν·νΏν ( ν€ β ) >0) to forcibly reshape the entire optimization space into a strictly convex landscape. Alternatively, this boundary condition can be estimated prior to any optimization by substituting the anchor weights as an approximation for the target weights (i.e., assuming ν€ ν βν€ ν anchor ). Plugging these a priori weights directly into Eq. (22) computes an estimated Ξ ν·νΏν . This efficient heuristic allows researchers to preemptively determine whether the unregularized landscape is likely non-convex, thereby guiding the initial configuration of the ARDLS model. Furthermore, setting the initial search vector to the anchor weights ensures that, even in the presence of multiple local or global minima, the solver is consistently guided into the same local valley, producing a stable and reproducible solution. 5. Graphical and Numerical analysis To demonstrate the practical behavior and landscape topology of the proposed model, this section investigates four numerical scenarios. The first three cases utilize 3Γ3 PRMs, allowing for direct graphical analysis and visual intuition of the optimization landscapes. The final case examines a higher- dimensional 6Γ6 PRM to evaluate the algorithmic performance and robustness on a more complex problem. 5.1. Graphical construction While prioritizing two criteria yields a trivially consistent matrix, three criteria represent the maximum dimensionality for direct human visualization. For problems involving more than three criteria, geometric representation becomes impossible. Therefore, 2D and 3D representations are employed for 3Γ3 prioritization problems to provide intuitive, graphical insights into the optimization mechanics. Consider a 3Γ3 PRM with the corresponding priority weight set ν= ν€ 1 ,ν€ 2 ,ν€ 3 : ν΄= ( 1ν 12 ν 13 1 ν 12 1ν 23 1 ν 13 1 ν 23 1 ) (43) Based on the core assumption ν νν β ν€ ν ν€ ν , the following system of linear equations is formed: ν€ 1 βν 12 ν€ 2 =0 ν€ 1 βν 13 ν€ 3 =0 ν€ 2 βν 23 ν€ 3 =0 (44) By applying the normalization constraint ν€ 1 +ν€ 2 +ν€ 3 =1, we can eliminate the variable ν€ 3 to project the solution points onto the 2D ( ν€ 1 ,ν€ 2 ) plane. For a two-dimensional diagram, the system of linear relationships can be rewritten as: ν€ 2 = ν€ 1 ν 12 ν€ 2 =1β ν€ 1 ( ν 13 +1 ) ν 13 ν€ 2 = ν 23 ( 1βν€ 1 ) 1+ν 23 (45) For a 3D diagram, the objective function's evaluated error is plotted along the z-axis against the feasible ( ν€ 1 ,ν€ 2 ) plane. We will now examine three distinct topological cases that arise from differently structured 3Γ3 PRMs. 5.2 Case 1: Perfect consistent PRM Let ν 12 =2 , ν 13 =6 , and ν 23 =3 . Because ν 13 =ν 12 ν 23 , this PRM is perfectly consistent. The graphical solutions derived from various Prioritization Operators (POs) are illustrated in Figure 1. In the 2D projection, the three linear boundary equations intersect at exactly one precise coordinate. This singular convergence indicates that all evaluated POs, including the proposed ARDLS, unanimously deduce the exact same unique priority vector: ν€= ( 0.6,0.3,0.1 ) . Figure 1. Unique solution point of perfectly consistent PRM 5.3 Case 2: Unique solution from DLS Let ν 12 =1/4 , ν 13 =1/2 , and ν 23 =3 . Unlike the first case, this PRM exhibits structural inconsistency (ν 13 β ν 12 ν 23 ). However, with a Consistency Ratio of CR=0.0156 , the deviation remains well within the standard recommended acceptable threshold (CRβ€0.1). The DLS method achieves the absolute lowest objective value (0.3491) with a globally unique solution, ν€= ( 0.148,0.622,0.229 ) . The 2D and 3D graphical representations comparing the behaviors of different Prioritization Operators (POs) under this slight inconsistency are shown in Figure 2. Figure 2. Solution region of inconsistent 3x3 PRM As illustrated in Figure 2, PRM inconsistency prevents boundary lines from intersecting at a single point, giving rise to a central "region of compromise" around which baseline prioritization operators (POs) disperse. However, 3D topology analysis reveals that the underlying optimization landscape retains strict unimodal convexity; matrix inconsistency merely elevates the global error floor without inducing non-convex ridges or competing local minima. Consequently, DLS reliably converges to a unique global optimal minimum, demonstrating structural stability and variance-minimizing efficacy under mild inconsistency. Table 1. Numerical Comparison of POs and ARDLS (ν=1) for an Inconsistent 3Γ3 PRM Metric NRS NRCS AMNC NGMR EV SVD CosMax PIGM ν€ 1 anchor 0.1338 0.1433 0.1373 0.1365 0.1365 0.1473 0.1377 0.1476 ν€ 2 anchor 0.6115 0.6337 0.6232 0.625 0.625 0.6302 0.6219 0.6316 ν€ 3 anchor 0.2548 0.223 0.2395 0.2385 0.2385 0.2225 0.2404 0.2208 ν€ 1 β 0.1484 0.1485 0.1485 0.1485 0.1485 0.1485 0.1485 0.1485 ν€ 2 β 0.6219 0.6221 0.622 0.622 0.622 0.6221 0.622 0.6221 ν€ 3 β 0.2296 0.2294 0.2295 0.2295 0.2295 0.2294 0.2295 0.2294 ν΄ν ν·ννΈ(ν€ β ) 0.3501 0.3494 0.3494 0.3494 0.3494 0.3493 0.3494 0.3493 ν·ννΈ(ν€ β ) 0.3492 0.3491 0.3491 0.3491 0.3491 0.3491 0.3491 0.3491 ν·ννΈ(ν€ anchor ) 0.7041 0.4211 0.5234 0.5514 0.5514 0.3719 0.5106 0.3806 ν΄ν ν(ν€ β ) 0.001 0.0002 0.0002 0.0002 0.0002 0.0001 0.0002 0.0002 ν νννΈ(ν€ anchor ) 0.2797 0.2163 0.2412 0.2475 0.2475 0.2033 0.2382 0.2056 ν νννΈ(ν€ β ) 0.197 0.197 0.197 0.197 0.197 0.197 0.197 0.197 The numerical results in Table 1 demonstrate that ARDLS exhibits robust convergence, consistently yielding an almost identical optimal weight vector (ν€ β β ( 0.1485,0.6221,0.2295 ) ) regardless of the starting base operator. The framework uniformly minimizes wide-ranging baseline Direct Squared Errors (0.3719β0.7041) down to 0.3491 while incurring a negligible Anchor Regularization Penalty (0.0001β€ν΄ν ν ( ν€ β ) β€0.0010), confirming that the regularizer gently guides the optimization without distorting the underlying error landscape. Ultimately, ARDLS achieves a uniform ν νννΈ ( ν€ β ) =0.1970 that strictly outperforms every individual base operator, successfully harmonizing conflicting baseline metrics into a single, superior priority vector. 5.4 Case 3: DLS multiple solution Let ν 12 =1/4, ν 13 =4, and ν 23 =1/4. This matrix exhibits severe cyclic inconsistency (CR=1.94). Under such extreme cyclicality, standard Direct Least Squares (DLS) optimization produces a non- convex, multi-modal objective landscape with three distinct global solutions achieving the minimum objective value of 28.445. Conversely, all selected POs yield symmetric weights ν€= ( 0.333,0.333,0.333 ) due to the cyclic structure of the matrix. Table 2. DLS Optimization Results for Inconsistent 3Γ3 PRM (CR=1.94) Initial value Final W solution DS objective value solution Ξ ν·νΏν DLS1 (0.33, 0.33, 0.33) (0.333,0.333,0.333) 28.688 local -9 DLS2 (0.34, 0.33, 0.33) (0.468,0.317, 0.215) 28.445 global 29.941 DLS3 (0.33, 0.34, 0.33) (0.215,0.468,0.317) 28.445 global 29.941 DLS4 (0.33, 0.33, 0.34) (0.317, 0.215,0.468) 28.445 global 29.941 The unregularized DLS method exhibits high sensitivity to initial search vectors under extreme cyclic inconsistency. As shown in Table 2 and Figure 3, initializing at the central symmetric vector ν€ initial = ( 0.33,0.33,0.33 ) traps the optimization at a local saddle region with an objective value of 28.688 and negative curvature (Ξ ν·νΏν =β9). While standard POs collapse onto this central ridge, small initial perturbations allow DLS to descend into one of three isolated, asymmetric global basins, achieving a lower objective value (28.445) and positive curvature (Ξ ν·νΏν =29.941). Figure 3. Unregularized DLS Multi-Modal Landscape To resolve this multi-modal instability, applying a safe regularization parameter (ν β =36, via Eq. (43)) transforms the non-convex landscape into a strictly unimodal basin. As illustrated in Figure 4, this regularization term eliminates competing global basins and enforces strict convexity. Consequently, the ARDLS framework guarantees stable convergence to a unique, robust global minimum, independent of initial search values, even when utilizing standard anchor vectors. Figure 4. Convexification via Safe Regularization (ν=36) 5.4 Case 4: Evaluating a Higher-Dimensional PRM The 6Γ6 PRM shown in Eq. (46), adapted from Saaty (2000), evaluates a higher-dimensional scenario characterized by a severe lack of consistency (CR=0.229). Despite the high inconsistency ratio, conducting a DLS optimization from 1,000 random starting points yielded a strictly unique global minimum. The optimization converged to the optimal weight vector w β = (0.184,0.220,0.037,0.150,0.210,0.197) , achieving DSE(w β )=60.015 . Because the landscape presents a single and strictly convex global minimum (β£ν β β£=1 and Ξ ν·νΏν min >0 ), this scenario is classified as Case 1. Consequently, the default structural scaling parameter ν=1 is sufficient, setting the safe regularizer to ν β =1. ν΄= [ 143134 1/41731/51 1/31/711/51/51/6 11/35111/3 1/355113 1/41631/31 ] (46) Table 3 presents a numerical comparison between standard POs and the ARDLS using a safe regularizer (ν=1 ). Under high matrix inconsistency, unregularized POs yield widely diverging baseline priority vectors (ν€ 0 ). This volatility is evident in their large objective error variances, with ν·ννΈ ( ν€ 0 ) ranging from 73.16 (CosMax) to 138.22 (PIGM), and baseline volatility (ν ννν ( ν€ 0 ) ) fluctuating between 1.42 and 1.95. By applying the ARDLS framework, the optimization reliably converges to almost the same regularized priority vector (ν€ β ), regardless of which highly variable PO serves as the initial anchor. The ARDLS approach successfully minimizes and locks the objective error at a uniform ν·ννΈ ( ν€ β ) = 60.015 across all methods. Additionally, it stabilizes vector variance to a consistent ν ννν ( ν€ β ) = 1.2912 while incurring negligible Anchor-Regularized Penalties (ARP). Ultimately, this demonstrates the regularizer's capability to robustly correct PO discrepancies in highly inconsistent, higher- dimensional matrices. Table 3. Numerical Comparison of POs and ARDLS (ν=1) for a highly Inconsistent 6Γ6 PRM NRS NRCS AMNC NGMR EV SVD CosMax PIGM ν€ 1 anchor 0.2421 0.3812 0.3047 0.316 0.3208 0.401 0.2926 0.415 ν€ 2 anchor 0.1884 0.1052 0.1486 0.1391 0.1395 0.1032 0.1554 0.0936 ν€ 3 anchor 0.0309 0.0447 0.0382 0.036 0.0348 0.0412 0.0395 0.0348 ν€ 4 anchor 0.1312 0.1312 0.1414 0.1251 0.1285 0.1212 0.1465 0.1123 ν€ 5 anchor 0.2321 0.2106 0.2208 0.236 0.2374 0.2112 0.2144 0.219 ν€ 6 anchor 0.1753 0.1271 0.1463 0.1477 0.1391 0.1221 0.1517 0.1253 ν€ 1 β 0.1845 0.1847 0.1846 0.1846 0.1846 0.1847 0.1846 0.1847 ν€ 2 β 0.2204 0.2203 0.2203 0.2203 0.2203 0.2203 0.2203 0.2203 ν€ 3 β 0.0371 0.0371 0.0371 0.0371 0.0371 0.0371 0.0371 0.0371 ν€ 4 β 0.1504 0.1504 0.1504 0.1504 0.1504 0.1504 0.1504 0.1504 ν€ 5 β 0.2104 0.2104 0.2104 0.2104 0.2104 0.2104 0.2104 0.2104 ν€ 6 β 0.1973 0.1972 0.1972 0.1972 0.1972 0.1972 0.1973 0.1972 ARDSE(w β ) 60.0212 60.0728 60.0379 60.0431 60.0452 60.0826 60.0335 60.0914 DSE(w β ) 60.0155 60.0155 60.0155 60.0155 60.0155 60.0156 60.0155 60.0156 DSE(w anchor ) 74.9265 95.1335 77.3477 85.2792 89.8472 107.7832 73.1656 138.2288 ARP(w β ) 0.0057 0.0572 0.0224 0.0276 0.0297 0.067 0.018 0.0758 RMSV(w anchor ) 1.4427 1.6256 1.4658 1.5391 1.5798 1.7303 1.4256 1.9595 RMSV(w β ) 1.2912 1.2912 1.2912 1.2912 1.2912 1.2912 1.2912 1.2912 6. Conclusion and future study This study addresses the challenges of estimating priority vectors from highly inconsistent Pairwise Reciprocal Matrices. Severe matrix inconsistency fractures the Direct Least Squares optimization landscape into a non-convex, multi-modal surface, causing most established Prioritization Operators to produce volatile baseline vectors or become trapped in local saddle points and competing global minima. To resolve these limitations, we introduced the Anchored Regularized Direct Least Squares (ARDLS) method. By dynamically evaluating the local curvature bounds (Ξ ν·νΏν ) of the objective function, the proposed algorithm determines the safe regularization penalty (ν β ) required to forcibly restore strict global convexity to the optimization space. Numerical evaluations demonstrate that ARDLS reliably eliminates initial-value sensitivity across both low-dimensional cyclic matrices (3Γ3 ) and highly inconsistent higher-dimensional settings (6Γ6). By enforcing strict convexity, the framework guarantees convergence to a unique and robust global priority vector (ν€ β ), drastically reduces Root Mean Square Error (normalized Direct Squared Error) across diverse POs. Ultimately, ARDLS bridges the gap between competing prioritization operators, offering a unified and mathematically sound approach to deriving stable priority vectors from inconsistent human judgments. To further extend the theoretical and practical applicability of ARDLS, future work should adapt the regularizer algorithm for incomplete pairwise comparisons with missing entries (ν νν ) and explore its extension to alternative non-linear objectives like Weighted Least Squares. Finally, integrating ARDLS into practical software applications will allow for empirical testing against traditional methods in real- world scenarios, such as supply chain management, resource allocation, and public policy evaluation. Measuring user satisfaction and decision stability in these environments will further validate the practical utility of the algorithm. Funding: The author received no financial support for the research. Author Contribution: The sole author, K.K.F. Yuen, was responsible for the entirety of this work, including conceptualization, methodology, analysis, and manuscript preparation. Conflicts of interest/Competing interests: The author declares that there are no conflicts of interest. Ethical Approval and Consent: This research did not involve human or animal subjects; therefore, ethics approval and consent for publication are not applicable. References [1] J. Barzilai, On the decomposition of value functions11Research supported in part by NSERC, Operations Research Letters 22 (1998), 159-170. [2] V. Belton and T. Gear, On a short-coming of Saaty's method of analytic hierarchies, Omega 11 (1983), 228-230. [3] J. BenΓtez and C. Serra-JimΓ©nez, A new and simple method to get the priority vector for reciprocal matrices, Operations Research and Decisions 35 (2025), 1-10. [4] S. BozΓ³ki, Solution of the least squares method problem of pairwise comparison matrices, Central European Journal of Operations Research 16 (2008), 345-358. [5] E. Carrizosa and F. Messine, An exact global optimization method for deriving weights from pairwise comparison matrices, Journal of Global Optimization 38 (2007), 237-247. [6] E.U. Choo and W.C. Wedley, A common framework for deriving preference values from pairwise comparison matrices, Computers & Operations Research 31 (2004), 893-908. [7] A.T.W. Chu, R.E. Kalaba, and K. Spingarn, A comparison of two methods for determining the weights of belonging to fuzzy sets, Journal of Optimization Theory and Applications 27 (1979), 531-538. [8] G. Crawford and C. Williams, A note on the analysis of subjective judgment matrices, Journal of Mathematical Psychology 29 (1985), 387-405. [9] J.S. Dyer, Remarks on the analytic hierarchy process, Management Science 36 (1990), 249-258. [10] S. Furtado and C.R. Johnson, Efficient vectors in priority setting methodology, Annals of Operations Research 332 (2024), 743-764. [11] S.I. Gass, Model world: The great debate - MAUT versus AHP, Interfaces 35 (2005), 308-312. [12] S.I. Gass and T. RapcsΓ‘k, Singular value decomposition in AHP, European Journal of Operational Research 154 (2004), 573-584. [13] B. Golany and M. Kress, A multicriteria evaluation of methods for obtaining weights from ratio-scale matrices, European Journal of Operational Research 69 (1993), 210-220. [14] J. Gyani, A. Ahmed, and M.A. Haq, MCDM and Various Prioritization Methods in AHP for CSS: A Comprehensive Review, IEEE Access 10 (2022), 33492-33511. [15] P.T. Harker and L.G. Vargas, The Theory of Ratio Scale Estimation: Saaty's Analytic Hierarchy Process, Management Science 33 (1987), 1383-1403. [16] G. Kou and C. Lin, A cosine maximization method for the priority vector derivation in AHP, European Journal of Operational Research 235 (2014), 225-232. [17] C.C. Lin, An enhanced goal programming method for generating priority vectors, Journal of the Operational Research Society 57 (2006), 1491-1496. [18] L. Mikhailov, A fuzzy programming method for deriving priorities in the analytic hierarchy process, Journal of the Operational Research Society 51 (2000), 341-349. [19] L. Mikhailov and M.G. Singh, Comparison analysis of methods for deriving priorities in the analytic hierarchy process, in: IEEE SMC'99 Conference Proceedings. 1999 IEEE International Conference on Systems, Man, and Cybernetics (Cat. No.99CH37028), 1999, p. 1037-1042 vol.1031. [20] T.L. Saaty, A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology 15 (1977), 234-281. [21] T.L. Saaty, Analytic Hierarchy Process: Planning, Priority, Setting, Resource Allocation, McGraw-Hill, New York (1980). [22] J.E. Smith and D.v. Winterfeldt, Anniversary Article: Decision Analysis in Management Science, Management Science 50 (2004), 561-574. [23] J. Szybowski, K. KuΕakowski, and S. Ernst, Almost optimal manipulation of pairwise comparisons of alternatives, Journal of Global Optimization 90 (2024), 243-259. [24] J. Tu and Z. Wu, Analytic hierarchy process rank reversals: causes and solutions, Annals of Operations Research 346 (2025), 1785-1809. [25] H. Wang, Y. Peng, and G. Kou, A two-stage ranking method to minimize ordinal violation for pairwise comparisons, Applied Soft Computing 106 (2021), 107287. [26] Y.-M. Wang, C. Parkan, and Y. Luo, Priority estimation in the AHP through maximization of correlation coefficient, Applied Mathematical Modelling 31 (2007), 2711-2718. [27] Y.-M. Wang, C. Parkan, and Y. Luo, A linear programming method for generating the most favorable weights from a pairwise comparison matrix, Computers & Operations Research 35 (2008), 3918-3930. [28] R. Whitaker, Criticisms of the Analytic Hierarchy Process: Why they often make no sense, Mathematical and Computer Modelling 46 (2007), 948-961. [29] K.K.F. Yuen, Analytic hierarchy prioritization process in the AHP application development: A prioritization operator selection approach, Applied Soft Computing Journal 10 (2010), 975-989. [30] K.K.F. Yuen, The Least Penalty Optimization Prioritization Operators for the Analytic Hierarchy Process: A Revised Case of Medical Decision Problem of Organ Transplantation, Systems Engineering 17 (2014), 442-461. [31] K.K.F. Yuen, Inverse gram matrix methods for prioritization in analytic hierarchy process: Explainability of weighted least squares optimization method, arXiv preprint arXiv:2401.01190 (2024). [32] K.K.F. Yuen, POO-LPSP: Parallel Osprey Optimized Least Penalty-Squared Prioritization Methods for Priority Derivation in the Analytic Hierarchy Process, arXiv preprint arXiv:2607.07313 (2026). [33] J. Zhang, G. Kou, Y. Peng, and Y. Zhang, Estimating priorities from relative deviations in pairwise comparison matrices, Information Sciences 552 (2021), 310-327.