POO-LPSP: Parallel Osprey Optimized Least Penalty-Squared Prioritization Methods for Priority Derivation in the Analytic Hierarchy Process Kevin Kam Fung YUEN1* 1 School of Science, Monash University Malaysia, Sunway, Malaysia *E-mail: [email protected]; [email protected];
Abstract. Pairwise comparison (PC) via pairwise reciprocal matrices (PRMs) is central to the Analytic Hierarchy Process (AHP). Although the traditional eigenvector method is widely applied to derive priorities, its theoretical robustness in reflecting true priority vectors remains debated. Building upon a previous iteration of this study, this research develops the revised Least PenaltySquared Prioritization (LPSP) optimization models, including the revised Least Product of Penalty and Direct Squares (LPPDS) and revised Weighted Squares (LPPWS), to minimize the revised Root Mean Penalty-Squared Variance (RMPSV) and the revised Root Mean Penalty-Weighted Square Variance (RMPSWV). However, solving these non-linear formulations is computationally complex for decision-makers. To overcome these limitations, this study proposes the Parallel Osprey Optimized Least Penalty-Squared Prioritization (POO-LPSP) method. By integrating an improved bio-inspired metaheuristic Parallel Osprey Optimization Algorithm (POOA), this framework efficiently solves complex LPSP models to minimize RMPSV and RMPSWV, thereby enhancing prioritization reliability. The practical utility and computational efficiency of the POO-LPSP method are validated through a numerical application focusing on a Generative AI (GAI) vendor selection problem. To extend, POO-LPSP can serve as a robust alternative to Saaty’s Eigen system method for AHP applications. Keywords: Osprey Optimization Algorithm, Pairwise Comparison, Parallel computing, Analytic Hierarchy Process, Multiple Criteria Decision Making.
1. Introduction The origins of the Pairwise Comparison (PC) method may be traced back to the pioneering 13thcentury social choice and voting theories of the philosopher R. Llull [7]. In 1927, L. L. Thurstone introduced "The Law of Comparative Judgment" [23], establishing a formal psychometric continuum that mathematically converted subjective binary choices into scalable interval measurements. Thomas L. Saaty introduced a structured ratio scale framework in pairwise reciprocal matrices [21] leading to the development of the Analytic Hierarchy Process (AHP) [22], providing a mathematically rigorous foundation for multi-criteria decision-making that uses subjective human perception for priority vector derivation. However, given the existence of diverse PC methodologies, the AHP should not be conflated with the broader concept of PCs [13].
1
The AHP systematically decomposes a complex decision problem through four core stages: structural definition, assessment, local prioritization, and global synthesis [29]. During the definition stage, decision-makers define the objective, a set of alternatives 𝑇 = {𝑡1 , … , 𝑡𝑗 , … , 𝑡𝑚 }, and a set of evaluation criteria 𝐶 = {𝑐1 , … , 𝑐𝑖 , … , 𝑐𝑛 }. In the assessment stage, pairwise comparisons are executed to construct Pairwise Reciprocal Matrices (PRMs). Each PRM is denoted by 𝐴 = [𝑎𝑖𝑗 ] such that 0 < 𝑎𝑖𝑗 = 𝑎𝑗𝑖−1 , ∀𝑖, 𝑗 ∈ 1,2, … , 𝑛 (1) The entries of 𝐴 are assigned values via a 1-to-9 assessment system, measuring the dominance of item 𝑖 against item 𝑗. To evaluate PRM consistency, the Consistency Ratio (𝐶𝑅) is determined by quotient of the Consistency Index (CI) and Random Index (RI). 𝐶𝐼 (2) 𝐶𝑅 = 𝑅𝐼 The 𝐶𝐼 is a function of the matrix dimension 𝑛 and its principal eigenvalue 𝜆𝑚𝑎𝑥 , defined as: 𝜆𝑚𝑎𝑥 − 𝑛 (3) 𝐶𝐼 = 𝑛−1 Furthermore, 𝜆𝑚𝑎𝑥 can be calculated by: 𝛿𝑖𝑗2 1 𝑎𝑖𝑗 𝜆𝑚𝑎𝑥 = ∑ , 𝛿𝑖𝑗 = ( −1) (4) 𝑤 𝑛 1 + 𝛿𝑖𝑗 𝑖 /𝑤𝑗 1≤𝑖<𝑗≤𝑛
A perfectly consistent PRM always has 𝜆𝑚𝑎𝑥 = 𝑛. The condition 𝜆𝑚𝑎𝑥 < 𝑛 is mathematically impossible, meaning that any deviation from perfect consistency will strictly result in 𝜆𝑚𝑎𝑥 > 𝑛. In the prioritization stage, extracting a normalized priority vector 𝑊 = [𝑤1 , … , 𝑤𝑛 ] such that ∑𝑛𝑖=1 𝑤𝑖 = 1 from a PRM requires applying a Prioritization Operator (PO). The conventional Analytic Hierarchy Process (AHP) utilizes the Eigen System to find 𝑊 by solving the principal eigenvalue limit problem: 𝐴𝑘 𝑒 𝑇 𝑤′𝑖 (5) 𝑤′ = 𝑙𝑖𝑚 ( 𝑘 𝑇 ) ⟹ 𝑤𝑖 = 𝑛 , 𝑖 = 1,2, … , 𝑛. 𝑘→∞ 𝑒𝐴 𝑒 ∑𝑖=1 𝑤′𝑖 During the synthesis phase, a weighted average operator aggregates these weighted local priories into a comprehensive global priority vector, 𝑇 = [𝑡1 , 𝑡2 , … , 𝑡𝑗 , … , 𝑡𝑚 ], which ultimately determines the final ranks for the alternatives. Despite the wide adoption of the AHP, these four stages frequently introduce operational anomalies, the phenomenon of rank reversal, where the relative preference ordering of alternatives shifts unexpectedly. [29] indicated that these vulnerabilities fundamentally stem from four problem domains: i) the structural selection of criteria in the definition stage; ii) the selection of numerical scales during assessment; iii) the selection of prioritization operators (POs) during prioritization; iv) selection of aggregation operators during final synthesis. This research specifically investigates Problem iii). The choice of PO is a critical pivot point in multi-criteria decision-making, as different mathematical formulations can yield completely contradictory priority rankings from the exact same underlying PRMs. Although there are various POs ranging from the algebraic forms to optimization forms such as [21; 22],[10], [25], [14], [6], [8], [16], [18], [26], [30] [32], [24], extensive comparative analyses [11] , [19], [5], [29], [12], [31] have established that no single prioritization technique achieves universal superiority when handling inconsistent judgments. Ultimately, if a unique or novel operator is proposed, evidence must rigorously demonstrate its superior performance against existing methods based on convincing and reasonable measurement criteria. To achieve this, the present research addresses problem 3 by improving a 2
least penalty square prioritization method originally proposed by [28; 30]. As a closed-form solution cannot be analytically derived for this method, numerical optimization is required. The Osprey Optimization Algorithm (OOA) is revised and tailor-made to solve this optimization problem. OOA represents a metaheuristic approach inspired by osprey predatory behaviors [9]. The algorithm is claimed to mathematically simulate the osprey's dual-phase foraging strategy, which consists of detecting and capturing prey, followed by relocating to consume it. By effectively simulating this behavior to balance global exploration and local exploitation, OOA demonstrates superior optimization performance across various benchmark functions compared to classical well-known algorithms [9]. Recent literature highlights the expanding utility and algorithmic evolution of OOA across diverse engineering and optimization domains. The baseline algorithm has been successfully deployed for lithium-ion battery parameter estimation [1], complex distributed generation placement [20], random forest regression tuning [15] , and joint antenna array vector configurations [27]. To further enhance its mathematical framework, researchers have introduced advanced structural variants, including a Pareto-based OOA (POOA) for multioutput support vector regression[2] and an exploitation-enhanced levy osprey optimization algorithm (LOOA) [17] . The contribution of the article is presented in several key areas. Section 2 introduces a new PO measurement metric, the Root Mean Penalty-Squared Variance (RMPSV), which integrates violation with squared variance. Section 3 formulates the Least Penalty-Squared Prioritization (LPSP) optimization model to derive an optimized priority vector by minimizing the RMPSV. Section 4 designs a customized Parallel Osprey Optimization Algorithm (POOA) to solve the LPSO models and classical optimization models such as Direct Least Squares (DLS) [6] and Weighted Least Squares (WLS) [6] to calculate priorities (or weights) from pairwise reciprocal matrices. Section 5 validates the practical applicability of the proposed framework via a Generative AI (GAI) service vendor selection case study, benchmarking its performance against established approaches including the Eigenvector, Geometric Mean (GM) [8], DLS, and the Pseudo Inverse Gram Matrix (PIGM) method [31] . Section 6 offers a comprehensive summary of methodological contributions alongside prospective future work. 2. Root Mean Penalty-Squared Variance 2.1 Foundational Deviation and Variance Metrics Selecting the most appropriate Prioritization Operator (PO) requires a robust measurement model supported by empirical validation. PO measurement models evaluate operator fitness, allowing decision-makers to systematically benchmark and select the optimal PO. This section reviews several foundational measurement models and, by synthesizing their advantages while mitigating their limitations, introduces a novel variance framework. [11] introduced the Total Deviation (TD) to quantify the sum of squared deviations between the derived weight ratios and their corresponding matrix entries. Building upon this, [19] defined Euclidean Distance (ED) as the square root of TD: 𝑛
𝑛
𝑤𝑖 𝐸𝐷(𝐴, 𝑊) = √∑ ∑ (𝑎𝑖𝑗 − ) 𝑤𝑗
2
(6)
𝑖=1 𝑗=1
To enhance computational efficiency via matrix operations, this study reformulates ED using the squared Frobenius norm: 3
(7)
𝐸𝐷(𝐴, 𝑊) =∥ 𝐴 − w(w −1 )𝑇 ∥𝐹 = √∑(𝐴 − 𝑊(𝑊 −1 )𝑇 )∘2
As ED scales with the 𝑛 × 𝑛 dimension of the reciprocal matrix 𝐴, averaging the values yields a more interpretable metric. [29] proposed the Root Mean Square Variance which calculates the root of the averaged sum of squared deviations: 𝑛
𝑛
1 𝑤𝑖 𝑅𝑀𝑆𝑉(𝐴, 𝑊) = √ ∑ ∑ (𝑎𝑖𝑗 − ) 𝑛×𝑛 𝑤𝑗
2
(8)
𝑖=1 𝑗=1
To facilitate efficient matrix computation, this paper expresses RMSV as: 1 (9) 𝑅𝑀𝑆𝑉(𝐴, 𝑊) = √∑(𝐴 − 𝑊(𝑊 −1 )𝑇 )∘2 𝑛 A variation of this metric is the Root Mean Square-Weighted Variance (RMSWV), defined as: 𝑛
𝑛
1 2 𝑅𝑀𝑆𝑊𝑉(𝐴, 𝑊) = √ ∑ ∑(𝑤𝑖 − 𝑎𝑖𝑗 𝑤𝑗 ) 𝑛×𝑛
(10)
𝑖=1 𝑗=1
In matrix form, RMSWV is calculated as: 1 𝑅𝑀𝑆𝑊𝑉(𝐴, 𝑊) = ∥ 𝑊𝑒𝑛𝑇 − 𝐴 × diag(𝑊) ∥𝐹 𝑛 1 = √∑(𝑊𝑒𝑛𝑇 − 𝐴 × diag(𝑊))∘2 𝑛
(11)
2.2 Penalty Weights A critical limitation of TD, ED, RMSV, and RMSWV is their failure to account for consistency 𝑤
violation penalties. For example, the penalty condition(𝑤𝑖 > 𝑤𝑗 ∧ 𝑎𝑖𝑗 < 1 ∧ 𝑎𝑖𝑗 ≠ 𝑤 𝑖 ) is 𝑗
fundamentally different from the
𝑤 condition (𝑤𝑖 > 𝑤𝑗 ∧ 𝑎𝑖𝑗 > 1 ∧ 𝑎𝑖𝑗 ≠ 𝑖 ). 𝑤𝑗
These distinct
violations should not be weighed equally. To quantify the variance linked to specific penalty weights, [11] introduced Minimum Violation (MV) index. 𝑀𝑉(𝐴, 𝑊) = ∑ ∑ 𝐼𝑖𝑗 , 𝑖
𝑗
1 , 𝑤𝑖 > 𝑤𝑗 & 𝑎𝑖𝑗 < 1 0.5 , 𝑤𝑖 = 𝑤𝑗 & 𝑎𝑖𝑗 ≠ 1 𝑤ℎ𝑒𝑟𝑒 𝐼𝑖𝑗 = { 0.5 , 𝑤𝑖 ≠ 𝑤𝑗 & 𝑎𝑖𝑗 = 1 0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(12)
[29; 30] identified a logical flaw in the original MV piecewise function above, correcting it to ensure 𝐼𝑖𝑗 equals 1 if (𝑤𝑖 < 𝑤𝑗 ∧ 𝑎𝑖𝑗 > 1). Furthermore, because the raw MV value is dependent on the matrix size (𝑛2 ), adopting a mean value provides a more stable metric for comparing POs. Based on these improvements, Mean MV (MMV) was formulated. Leveraging reciprocal properties, this study refines the flipped inequality signs of the MMV to properly scale severity: 𝑀𝑀𝑉(𝐴, 𝑊) =
1 (∑ ∑ 𝐼𝑖𝑗 ) 𝑛2 𝑖
(13)
𝑗
4
1,
(𝑤𝑖 > 𝑤𝑗 ∧ 𝑎𝑖𝑗 < 1)
1,
(𝑤𝑖 < 𝑤𝑗 ∧ 𝑎𝑖𝑗 > 1)
Where 𝐼𝑖𝑗 = 0.5, (𝑤𝑖 = 𝑤𝑗 ∧ 𝑎𝑖𝑗 ≠ 1) 0.5, (𝑤𝑖 ≠ 𝑤𝑗 ∧ 𝑎𝑖𝑗 = 1) { 0, otherwise 2.3 The revised Framework: Root Mean Penalty-Squared Variance A structural pitfall of MMV is that it relies strictly on discrete penalty scores, completely ignoring the magnitude of the variance values. To integrate RMSV/RMSWV variance tracking with MMV structural penalty logic, this study improves upon the hybrid forms from [28; 30] by proposing improved Root Mean Penalty-Squared Variance (RMPSV). This metric is formalized in two variations. The standard RMPSV is proposed as follows. 𝑅𝑀𝑃𝑆𝑉(𝐴, 𝑊) =
1 1 𝑤𝑖 √∑ 𝑃(𝐴 − 𝑊(𝑊 −1 )𝑇 )∘2 = √∑ ∑ 𝑃𝑖𝑗 (𝑎𝑖𝑗 − ) 𝑛 𝑛 𝑤𝑗 𝑖
2
(14)
𝑗
where the penalty matrix, 𝑃 = [𝑃𝑖𝑗 ], is determined by: 𝛽1 , (𝑤𝑖 > 𝑤𝑗 ∧ 𝑎𝑖𝑗 > 1) 𝛽1 , (𝑤𝑖 < 𝑤𝑗 ∧ 𝑎𝑖𝑗 < 1) 𝑃𝑖𝑗 =
𝛽1 ,
(𝑤𝑖 = 𝑤𝑗 ∧ 𝑎𝑖𝑗 = 1)
(15)
𝛽2 , (𝑤𝑖 = 𝑤𝑗 ∧ 𝑎𝑖𝑗 ≠ 1)
𝛽2 , (𝑤𝑖 ≠ 𝑤𝑗 ∧ 𝑎𝑖𝑗 = 1) otherwise {𝛽3 , The weighted variant, Root Mean Penalty-Weighted Square Variance (RMPSWV), is formulated as: 1 𝑅𝑀𝑃𝑆𝑊𝑉(𝐴, 𝑊) = √∑ 𝑃 × (𝑊𝑒𝑛𝑇 − 𝐴 × diag(𝑊))∘2 𝑛 (16) 1 2 = √∑ ∑ 𝑃𝑖𝑗 (𝑤𝑖 − 𝑎𝑖𝑗 𝑤𝑗 ) 𝑛 𝑖
𝑗
Crucially, the baseline consistency condition (𝑤𝑖 = 𝑤𝑗 ∧ 𝑎𝑖𝑗 = 1) for 𝑃𝑖𝑗 , which was omitted in my previous versions [28; 30], is formally integrated here. The penalty weight vector is defined as 𝛽 = {𝛽1 , 𝛽2 , 𝛽3 }, subject to the constraint 1 ≤ 𝛽1 ≤ 𝛽2 ≤ 𝛽3 . Under this generalized framework, RMSV and RMSWV are simply special cases of RMPSV and RMPSWV where 𝛽1 = 𝛽2 = 𝛽3 = 1. In the historical MMV framework, the implied weights were 𝛽1 = 0, 𝛽2 = 0.5, 𝛽3 = 1. However, assigning a zero value to 𝛽1 inadvertently cancels out valid variance data. To resolve this and ensure mathematical stability, the default penalty settings proposed for this updated model are defined as 𝛽1 = 1, 𝛽2 = 3, and 𝛽3 = 10. 3. Least Penalty-Squared Optimization Prioritization Operators [6] introduced the Direct Least Squares (DLS) method, which derives priority weights by 𝑤 minimizing the total variance between the reciprocal matrix entry 𝑎𝑖𝑗 and the weight ratio 𝑤 𝑖 : 𝑗
5
𝑛
𝑛
DS= ∑ ∑ (𝑎𝑖𝑗 −
𝑀𝑖𝑛
𝑛
S.T
𝑖=1 𝑗=1
𝑤𝑖 ) 𝑤𝑗
2
(17)
∑ 𝑤𝑖 = 1, 𝑤𝑖 > 0, 𝑖 = 1,2, … , 𝑛 . 𝑖=1
However, DLS method lacks a closed-form analytical solution, produces alternative optima, which means that different W values leading to the exact same minimized DS value, and is very challenging to resolve via numerical optimization(some discussion can be found in [6][3][4][31]). Thus, [6] transformed the DLS objective function to the Weighted Least Squares (WLS): 𝑛
𝑀𝑖𝑛
𝑛 2
WS= ∑ ∑(𝑤𝑖 − 𝑎𝑖𝑗 𝑤𝑗 ) 𝑖=1 𝑗=1
𝑛
S.T.
(18)
∑ 𝑤𝑖 = 1, 𝑤𝑖 > 0, 𝑖 = 1,2, … , 𝑛 . 𝑖=1
A closed-form solution for WLS remained absent in the literature until [31] introduced four variants of the Inverse Gram Matrix methods. Among these, the Pseudo Inverse Gram Matrix (PIGM) method [31] is formulated as follows: 𝐺 −1 𝑒 , 𝑒 𝑇 𝐺 −1 𝑒
2 (𝑛 − 1) + ∑ 𝑎𝑘𝑗
𝑖=𝑗
, ∀𝑔𝑖𝑗 ∈ 𝐺. (19) 1 − 𝑎𝑖𝑗 − 𝑎𝑗𝑖 𝑖≠𝑗 where 𝑒 = (1, ⋯ ,1)𝑇 is a column vector of ones with a size equal to the row dimension of 𝐺 −1 . [8] proposed Logarithm Least Square (LLS) PO has the following form: 𝑤=
𝑛
𝑔𝑖𝑗 = {
𝑘
𝑛
2
∑ ∑ (𝑙𝑛 𝑎𝑖𝑗 − (𝑙𝑛 𝑤 ′𝑖 − 𝑙𝑛 𝑤 ′𝑗 ))
𝑀𝑖𝑛
𝑖=1 𝑗>𝑖 𝑛
S.T.
(20)
∏ 𝑤′𝑖 = 1, 𝑤′𝑖 > 0, 𝑖 = 1,2, … , 𝑛 𝑖=1
Geometric Mean (GM) is the LLS closed form solution, which produces unnormalized solution {𝑤′𝑖 }, and the normalization [29; 30] is needed. 𝑛
1/𝑛
𝑤′𝑖 = ∏ 𝑎𝑖𝑗
⟹ 𝑤𝑖 =
𝑗=1
𝑤′𝑖 , 𝑛 ∑𝑖=1 𝑤′𝑖
𝑖 = 1,2, … , 𝑛
(21)
Least Penalty-Squared Optimization (LPSO) POs mathematically integrate penalty parameters directly into the optimization model to enforce ordinal consistency. Incorporating the updated penalty structures 𝑃𝑖𝑗 , this study refines two primary LPSO operators established in prior iteration of the work [28; 30], namely the Least Product of Penalty and Direct Squares (LPPDS) and the Least Product of Penalty and Weighted Squares (LPPWS). The LPPDS maps the dynamic penalty weights (𝑃𝑖𝑗 ) onto the DLS objective function, yielding the following formulation: 𝑛
𝑀𝑖𝑛
PPDS = ∑ ∑ 𝑃𝑖𝑗 ⋅ (𝑎𝑖𝑗 − 𝑛
S.T.
𝑛
𝑖=1 𝑗=1
𝑤𝑖 ) 𝑤𝑗
2
(22)
∑ 𝑤𝑖 = 1, 𝑤𝑖 > 0, 𝑖 = 1,2, … , 𝑛 𝑖=1
6
𝑃𝑖𝑗 is shown in Eq. (15). The penalty weights 𝛽1 , 𝛽2 , and 𝛽3 , by default, are defined as 1,3, and 10 respectively to penalize consistency violations. Similarly, the LPPWS applies the penalty matrix 𝑃𝑖𝑗 to the WLS formulation, defined as follows: 𝑛
𝑀𝑖𝑛
2
PPWS= ∑ ∑ 𝑃𝑖𝑗 ⋅ (𝑤𝑖 − 𝑎𝑖𝑗 𝑤𝑗 ) 𝑛
S.T
𝑛
𝑖=1 𝑗=1
(23)
∑ 𝑤𝑖 = 1, 𝑤𝑖 > 0, 𝑖 = 1,2, … , 𝑛 𝑖=1
Unlike WLS, neither LPPDS nor LPPWS possesses a tractable closed-form solution. Because both models are non-convex, non-differentiable, and governed by piecewise step-functions based on the decision variables, traditional gradient-based solvers, such as Gradient descent, NewtonRaphson or interior-point methods, are fundamentally ineffective. These classical approaches depend on smooth derivatives and will inevitably become trapped in local minima. Consequently, employing a metaheuristic approach is an effective way to reliably explore the global search space. However, calculating the dynamic penalty conditions of 𝑃𝑖𝑗 for every candidate solution within a metaheuristic population across continuous iterations imposes a severe computational bottleneck. To overcome these challenges, this study proposes a Parallel Osprey Optimization Algorithm (POOA), inheriting the OOA metaheuristic search mechanism, enable it to successfully escape local minima and approximate the global optimum within a non-convex space. Furthermore, integrating parallel computing architecture allows the algorithm to simultaneously evaluate the complex penalty matrices across the entire candidate population. This parallelization drastically reduces execution time, rendering the LPPDS and LPPWS models highly viable for integration into real-time decision-making frameworks. 4. Parallel Osprey Optimization Algorithm To address the computational limitations and strict structural constraints inherent in decision matrix prioritization, this study proposes Parallel OOA (POOA), an enhanced variant of the Osprey Optimization Algorithm (OOA) [9]. Although the baseline OOA features strong global exploration capabilities, its original design is ill-equipped for this domain due to two primary limitations: • Unconstrained Mechanics: The original algorithm lacks explicit constraint-handling, making it unsuited for the strict mathematical boundaries (∑wi = 1, wi > 0) governing priority weight vectors. • Sequential Framework: Its single-threaded, linear execution becomes computationally inefficient when navigating highly non-convex and non-smooth optimization problem. To achieve real-time operational capability, the proposed POOA significantly advances the standard metaheuristic foundation through four core architectural modifications. The complete POOA execution flow, implemented in R, is detailed in Algorithm 1. The first major enhancement introduces a dynamic parallel evaluation mechanism to mitigate the severe computational bottlenecks caused by search space to solve the complex objective functions. This multi-core processing layer evaluates population fitness concurrently, governed by an intelligent auto-gatekeeper (Step 2) that dynamically reverts to sequential processing when search dimensions are small, specifically (𝑁 × 𝑚 × 𝑆) < 1000, to prevent operating system thread-spawning overhead. Second, to address the tendency of traditional Rbased parallel clusters to inadvertently clone heavy data structures and risk memory leakage, the 7
POOA incorporates a memory-safe environment export filter. Operating within Step 2, this parsing routine utilizes custom scoping functions to strip raw datasets from the active workspace, successfully exporting only core functional components prior to mapping. Third, the POOA abandons the static maximum iteration limits of classical OOA, which often result in thousands of wasted computational cycles. Instead, it integrates an adaptive earlystopping criterion (Step 12) featuring a stagnation monitor. An active loop-break triggers immediately if a non-improving metric, 𝑛𝑜_𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑚𝑒𝑛𝑡_𝑐𝑜𝑢𝑛𝑡𝑒𝑟, breaches a user-defined threshold 𝑆. Finally, because priorities satisfy normalization axiom (e.g., summing to 1), the POOA uses an explicit constraint repair mechanism (𝑓𝑟𝑒𝑝𝑎𝑖𝑟 ). Implemented during vector initialization (Step 3) and within the core exploration and exploitation loops (Steps 6 and 9), this operator guarantees all candidate positions remain within a mathematically valid. Algorithm 1: Parallel Osprey Optimization Algorithm (POOA) Step
Algorithmic Logic
Input:
{Objective function 𝑜𝑏𝑗_𝑓𝑛; search space boundary vectors 𝑙𝑏, 𝑢𝑏 ∈ ℝ𝑚 ; maximum iterations 𝑚𝑎𝑥_𝑖𝑡𝑒𝑟; population size 𝑁; early-stopping stagnation threshold 𝑆; hyperplane projection rule 𝑟𝑒𝑝𝑎𝑖𝑟_𝑓𝑛; requested computing cores 𝑛_𝑐𝑜𝑟𝑒𝑠}
Output:
{Normalized optimal priority vector 𝑤 ∗ , global minimum fitness score 𝐹𝑏𝑒𝑠𝑡 , convergence track.}.
Initialization Step 1:
Define problem dimension: 𝑚 ← 𝑙𝑒𝑛𝑔𝑡ℎ(𝑙𝑏).
Step 2:
Parallel Infrastructure Configuration Gatekeeper: if 𝑛_𝑐𝑜𝑟𝑒𝑠 > 1, then if (𝑁 × 𝑚 × 𝑆) < 1000, then 𝑛_𝑐𝑜𝑟𝑒𝑠 ← 1 else •
Initialize parallel compute cluster: 𝑐𝑙 ← 𝑚𝑎𝑘𝑒𝐶𝑙𝑢𝑠𝑡𝑒𝑟(𝑛_𝑐𝑜𝑟𝑒𝑠).
•
Register an execution panic-hook to prevent cluster hangs: 𝑜𝑛. 𝑒𝑥𝑖𝑡(𝑠𝑡𝑜𝑝𝐶𝑙𝑢𝑠𝑡𝑒𝑟(𝑐𝑙)).
•
Memory-Safe Environment Export Filter: Parse environments (.GlobalEnv and parent.frame() in R); Identify and strip away heavy data structures or lists; Export only isolated function definitions and scalar rules to worker nodes via clusterExport to mitigate system memory leakage.
end if end if Step 3:
Population Initialization and Normalization: for each osprey 𝑖 = 1 to 𝑁 do 𝑋[𝑖,⋅] ← 𝑙𝑏 + 𝑟𝑎𝑛𝑑(1, 𝑚) ⊙ (𝑢𝑏 − 𝑙𝑏). if 𝑟𝑒𝑝𝑎𝑖𝑟_𝑓𝑛 is defined, then Normalize the values in repair function: 𝑋[𝑖,⋅] ← 𝑟𝑒𝑝𝑎𝑖𝑟_𝑓𝑛(𝑋[𝑖,⋅]). end for
8
Step 4:
Initial Performance Evaluation: if 𝑛_𝑐𝑜𝑟𝑒𝑠 > 1, then evaluate initial population fitness vector concurrently: 𝐹_𝑣𝑒𝑐 ← parApply(𝑐𝑙, 𝑋, 𝑜𝑏𝑗_𝑓𝑛).
Step 5:
State Tracking Initialization: • Identify absolute best position index: 𝑏𝑒𝑠𝑡_𝑖𝑑𝑥 ← arg min(𝐹_𝑣𝑒𝑐). • Extract initial global target: 𝑋𝑏𝑒𝑠𝑡 ← 𝑋[𝑏𝑒𝑠𝑡_𝑖𝑑𝑥,⋅], • Extract baseline score: 𝐹𝑏𝑒𝑠𝑡 ← 𝐹_𝑣𝑒𝑐[𝑏𝑒𝑠𝑡_𝑖𝑑𝑥]. • Initialize tracking registers: 𝑛𝑜_𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑚𝑒𝑛𝑡_𝑐𝑜𝑢𝑛𝑡𝑒𝑟 ← 0.
Main Loop Metaheuristic Two-Phase Optimization Loop: for 𝑡 = 1 to 𝑚𝑎𝑥_𝑖𝑡𝑒𝑟 do Step 6:
PHASE 1: Position Identification and Hunting (Exploration) for each osprey 𝑖 = 1 to 𝑁 do Isolate the set of strictly superior candidate solutions: 𝐹𝑃𝑖 ← {𝑋[𝑘,⋅] ∣ 𝐹_𝑣𝑒𝑐[𝑘] < 𝐹_𝑣𝑒𝑐[𝑖]}. if 𝐹𝑃𝑖 is empty then Default target set to global leader: 𝐹𝑃𝑖 ← {𝑋𝑏𝑒𝑠𝑡 }. else Append global leader to the candidate set: 𝐹𝑃𝑖 ← 𝐹𝑃𝑖 ∪ {𝑋𝑏𝑒𝑠𝑡 }. end if Randomly select a target fish position from the set: 𝑆𝐹𝑖 ← sample(𝐹𝑃𝑖 , 1). Generate determination steps 𝐼𝑖𝑗 ∈ {1,2}𝑚 and stochastic parameters 𝑟𝑖𝑗 ∈ [0,1]𝑚 . Compute exploration candidate position: 𝑋𝑃1 [𝑖,⋅] ← 𝑋[𝑖,⋅] + 𝑟𝑖𝑗 ⊙ (𝑆𝐹𝑖 − 𝐼𝑖𝑗 ⊙ 𝑋[𝑖,⋅]). Enforce boundary constraints: 𝑋𝑃1 [𝑖,⋅] ← max(min(𝑋𝑃1 [𝑖,⋅], 𝑢𝑏), 𝑙𝑏). if 𝑟𝑒𝑝𝑎𝑖𝑟_𝑓𝑛 is defined, then Map coordinates to priority domain: 𝑋𝑃1 [𝑖,⋅] ← 𝑟𝑒𝑝𝑎𝑖𝑟_𝑓𝑛(𝑋𝑃1 [𝑖,⋅]). end for
Step 7:
Phase 1 Evaluation: Concurrently evaluate fitness vector: 𝐹_𝑃1 ← evaluate_population(𝑋𝑃1 ).
Step 8:
Phase 1 Greedy Selection Strategy: for each individual osprey 𝑖 = 1 to 𝑁 do if 𝐹_𝑃1[𝑖] < 𝐹_𝑣𝑒𝑐[𝑖] then update position and fitness: 𝑋[𝑖,⋅] ← 𝑋𝑃1 [𝑖,⋅]; 𝐹_𝑣𝑒𝑐[𝑖] ← 𝐹_𝑃1[𝑖].
Step 9:
PHASE 2 (Exploitation): Carrying Fish to Suitable Positions for each individual osprey 𝑖 = 1 to 𝑁 do
9
•
Generate localized stochastic sampling array: 𝑟 ∈ [0,1]𝑚 .
•
Compute exploitation position using a time-decaying contraction factor: 𝑋𝑃2 [𝑖,⋅] ← 𝑋[𝑖,⋅] +
•
𝑙𝑏+𝑟⊙(𝑢𝑏−𝑙𝑏) 𝑡
.
Enforce boundary constraints: 𝑋𝑃2 [𝑖,⋅] ← max(min(𝑋𝑃2 [𝑖,⋅], 𝑢𝑏), 𝑙𝑏).
•
if 𝑟𝑒𝑝𝑎𝑖𝑟_𝑓𝑛 is defined, then 𝑋𝑃2 [𝑖,⋅] ← 𝑟𝑒𝑝𝑎𝑖𝑟_𝑓𝑛(𝑋𝑃2 [𝑖,⋅]).
end for Step 10:
Phase 2 Greedy Evaluation Concurrently Phase 2 candidate fitness array: 𝐹_𝑃2 ← evaluate_population(𝑋𝑃2 ).
Step 11:
Phase 2 Greedy Selection Strategy: for each individual osprey 𝑖 = 1 to 𝑁 do if 𝐹_𝑃2[𝑖] < 𝐹_𝑣𝑒𝑐[𝑖], then update position and fitness: 𝑋[𝑖,⋅] ← 𝑋𝑃2 [𝑖,⋅]; 𝐹_𝑣𝑒𝑐[𝑖] ← 𝐹_𝑃2[𝑖]. end for
Step 12:
Global Memory Management & Adaptive Stopping Check: current_best_idx←argmin(F_vec); 𝑐𝑢𝑟𝑟𝑒𝑛𝑡_𝑏𝑒𝑠𝑡_𝐹 ← 𝐹_𝑣𝑒𝑐[𝑐𝑢𝑟𝑟𝑒𝑛𝑡_𝑏𝑒𝑠𝑡_𝑖𝑑𝑥]; if 𝑐𝑢𝑟𝑟𝑒𝑛𝑡_𝑏𝑒𝑠𝑡_𝐹 < 𝐹𝑏𝑒𝑠𝑡 then Record new global optimum: •
𝐹𝑏𝑒𝑠𝑡 ← 𝑐𝑢𝑟𝑟𝑒𝑛𝑡_𝑏𝑒𝑠𝑡_𝐹;
•
𝑋𝑏𝑒𝑠𝑡 ← 𝑋[𝑐𝑢𝑟𝑟𝑒𝑛𝑡_𝑏𝑒𝑠𝑡_𝑖𝑑𝑥,⋅];
Reset stagnation tracking register: 𝑛𝑜_𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑚𝑒𝑛𝑡_𝑐𝑜𝑢𝑛𝑡𝑒𝑟 ← 0. else Increment stagnation tracking register: 𝑛𝑜_𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑚𝑒𝑛𝑡_𝑐𝑜𝑢𝑛𝑡𝑒𝑟 ← 𝑛𝑜_𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑚𝑒𝑛𝑡_𝑐𝑜𝑢𝑛𝑡𝑒𝑟 + 1. end if if 𝑛𝑜_𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑚𝑒𝑛𝑡_𝑐𝑜𝑢𝑛𝑡𝑒𝑟 ≥ 𝑆, then Terminate optimization loop prematurely due to Stagnation Limit Reached. end if end for Step 13:
Normalization & Cleanup: If Cluster instance 𝑐𝑙 is active, then execute 𝑠𝑡𝑜𝑝𝐶𝑙𝑢𝑠𝑡𝑒𝑟(𝑐𝑙).
10
Map the unbounded optimal array into an exactly normalized priority weight vector: 𝑤∗ ← return {𝑤 ∗ , 𝐹𝑏𝑒𝑠𝑡 , convergence track}
𝑋𝑏𝑒𝑠𝑡 𝑚
∑𝑘=1 𝑋𝑏𝑒𝑠𝑡,𝑘
5. Application of Generative AI Service Provider Selection A prominent healthcare enterprise leverages Generative AI (GAI) to automate administrative workflows, clinical documentation, and unstructured literature synthesis. This minimizes clinicians' cognitive burdens, accelerating research and diagnostics while maintaining data security. Choosing an optimal vendor is vital to balance clinical efficacy, technological agility, and operational efficiency. To address this vendor selection challenge, this study utilizes the enhanced AHP with seven POs, Eigenvector, Geometric Mean (a closed form of LLM), DLS, PIGM (a closed form of WLS), LPPWS and LPPDS, proposed in Section 3, with respect to the five PO metrics, MMV, RMSV, RMSWV, RMPSV and RMPSWV, proposed in Section 4. 5.1 Decision Problem Structure The hierarchical decision model, presented in Figure 1, is designed to identify an optimal GAI service provider for sustainable business operations. The evaluation framework comprises six core criteria (𝐶1 –𝐶6 ) applied to a shortlist of five anonymized alternatives (𝑡1 –𝑡5 ). For the purposes of vendor neutrality and data masking, these alternatives are representative of major platforms (e.g., Google’s Gemini 2.0, OpenAI’s GPT-5.5, Claude 5, Meta’s Llama 4, Cohere’s Command A+, DeepSeek’s DeepSeek-V4, xAI’s Grok 4.5, Mistral Large 3, etc.). The operational definitions of these criteria are provided below.
Figure 1. Structure for selecting the best GAI service provider
• •
•
Performance (𝐶1 ): evaluates cognitive reasoning in complex biomedical tasks, measuring hallucination rates, semantic coherence, and diagnostic accuracy to ensure clinical fidelity. Pricing attractiveness (𝐶2 ): assesses total cost of ownership across token pricing schedules, subscription models, enterprise volume discounts, and batch-processing cost-mitigation structures. Data protection (𝐶3 ): verifies privacy and security compliance for protected health information (PHI), encompassing enterprise encryption, data residency, zero-retention training policies, and cybersecurity resilience. 11
• • •
Speed (𝐶4 ): quantifies real-time responsiveness via time-to-first-token (TTFT), throughput velocity, generation speed, and SLA-backed API uptime guarantees. API capability (𝐶5 ): gauges interoperability via developer documentation, SDK robustness, and integration compatibility with legacy enterprise infrastructure and clinical software tools. Customization ( 𝐶6 ): evaluates a system's adaptability to clinical terminology through retrieval-augmented generation (RAG), hyperparameter tuning, and parameter-efficient finetuning (PEFT).
Table 1. PRM for Six Criteria (A0 , CR= 0.094) A0
C1
C2
C3
C4
C5
C6
C1
1
2
1/4
6
5
2
C2
1/2
1
1/6
3
1/3
1/4
C3
4
6
1
8
7
5
C4
1/6
1/3
1/8
1
1/4
1/5
C5
1/5
3
1/7
4
1
1/3
C6
1/2
4
1/5
5
3
1
Table 2. PRMs for priorities among five alternatives with respect to each criterion.
T1
A1 (CR= 0.019) T2 T3 T4
T1
A2 (CR = 0.020) T2 T3 T4
T5
T5
T1
1
2
1
4
5
1
1/3
1/3
1/5
1/6
T2
1/2
1
1/2
2
3
3
1
2
1/2
1/3
T3
1
2
1
2
3
3
1/2
1
1/3
1/4
T4
1/4
1/2
1/2
1
2
5
2
3
1
1/2
T5
1/5
1/3
1/3
1/2
1
6
3
4
2
1
T1
1
1
A3 (CR=0.017) 1
4
5
1
1
A4 (CR=0.024) 2 1/4
1/5
T2
1
1
1
2
5
1
1
3
1/2
1/3
T3
1
1
1
3
5
1/2
1/3
1
1/4
1/5
T4
1/4
1/2
1/3
1
3
4
2
4
1
1
T5
1/5
1/5
1/5
1/3
1
5
3
5
1
1
T1
1
3
A5 (CR=0.036) 1/2
5
5
1
1/2
1/6
1/5
1/4
T2
1/3
1
1/2
3
3
2
1
1/2
1/4
1/3
T3
2
2
1
4
4
6
2
1
1/3
1/2
T4
1/5
1/3
1/4
1
1
5
4
3
1
3
T5
1/5
1/3
1/4
1
1
4
3
2
1/3
1
A6 (CR = 0.051)
Table 1 delineates the initial PRM formulated to assess the comparative significance of the six criteria. Building upon this, Table 2 provides the specific matrices constructed to rank the five
12
alternatives under each criterion. Given that the Consistency Ratio (CR) across all computed matrices falls safely beneath the guided 0.1 limit, within the acceptance range. Table 3. Weighted Decision matrices and final results with respect to each PO. Eigen
GM/LLS
C1
C2
C3
C4
C5
C6
C1
C2
C3
C4
C5
C6
W
0.2
0.06
0.483
0.029
0.081
0.148
Final
0.197
0.059
0.486
0.03
0.078
0.151
Final
T1
0.353
0.051
0.302
0.102
0.328
0.053
0.257
0.355
0.051
0.299
0.101
0.326
0.052
0.255
T2
0.183
0.156
0.258
0.139
0.168
0.091
0.202
0.184
0.156
0.26
0.139
0.171
0.095
0.204
T3
0.282
0.104
0.28
0.062
0.369
0.18
0.256
0.279
0.103
0.282
0.061
0.363
0.179
0.255
T4
0.112
0.267
0.11
0.319
0.068
0.442
0.172
0.112
0.268
0.108
0.32
0.07
0.439
0.172
T5
0.069
0.422
0.05
0.378
0.068
0.234
0.114
0.069
0.422
0.05
0.379
0.07
0.236
0.115
W
0.235
0.07
0.41
0.044
0.061
0.181
Final
0.165
0.069
0.537
0.043
0.065
0.121
Final
T1
0.378
0.059
0.301
0.084
0.365
0.057
0.253
0.352
0.059
0.297
0.083
0.295
0.056
0.251
T2
0.195
0.158
0.262
0.145
0.181
0.096
0.199
0.175
0.148
0.271
0.139
0.152
0.102
0.213
T3
0.249
0.104
0.281
0.072
0.31
0.264
0.251
0.299
0.1
0.285
0.07
0.406
0.163
0.258
T4
0.106
0.292
0.102
0.31
0.073
0.346
0.167
0.103
0.255
0.093
0.332
0.074
0.478
0.162
T5
0.074
0.387
0.055
0.389
0.073
0.236
0.131
0.073
0.438
0.054
0.376
0.074
0.202
0.117
DLS
PIGM/WLS
LPPDS
LPPWS
W
0.234
0.066
0.409
0.044
0.066
0.183
Final
0.166
0.067
0.537
0.043
0.067
0.121
Final
T1
0.365
0.059
0.296
0.088
0.337
0.058
0.247
0.338
0.059
0.29
0.087
0.295
0.056
0.246
T2
0.194
0.158
0.266
0.133
0.182
0.095
0.2
0.175
0.148
0.279
0.131
0.152
0.102
0.217
T3
0.262
0.104
0.281
0.073
0.337
0.25
0.254
0.313
0.1
0.285
0.07
0.406
0.163
0.261
T4
0.106
0.292
0.102
0.319
0.073
0.348
0.168
0.102
0.255
0.093
0.346
0.074
0.478
0.162
T5
0.073
0.387
0.055
0.387
0.073
0.25
0.132
0.072
0.438
0.054
0.367
0.074
0.202
0.115
5.2 Comparisons and discussion To resolve the complex, non-linear objective functions underlying DLS, WLS, LPPDS, and LPPWS, the POOA is implemented as the primary metaheuristic solver. POOA efficiently explores the continuous priority weight space to avoid local minima and reliably discover global optimal solutions under varying penalty constraints. Crucially, the POOA framework configured for the WLS objective function yields priority vectors nearly identical to the closed-form PIGM solution, validating the algorithm’s high convergence precision and mathematical consistency. Table 3 details the weighted decision matrices and final rankings derived across different POs. For selection outcome of the best provider, GM, PIGM, LPPDS, and LPPWS yield 𝑇3 , whereas Eigen and DLS yield 𝑇1 . That means you may make a wrong best choice if you use classical Eigen method for AHP. Conversely, all evaluated methods consistently rank 𝑇5 as the least optimal alternative. Table 4 reports error measurements, bolding the minimum values for each metric. All PRMs exhibiting a CR less than 0.1 are structurally valid. However, non-zero MMV values indicate that not all POs consistently generate feasible solutions.
13
Table 4. Error measurement metrics. A0 (CR= 0.094)
A1 (CR= 0.019)
MMV
RMSV
RMSWV
RMPSV
RMPSWV
MMV
RMSV
RMSWV
RMPSV
RMPSWV
Eigen
0
1.685
0.100
1.685
0.100
0.04
0.345
0.042
0.357
0.050
GM
0
1.665
0.098
1.665
0.098
0.04
0.338
0.042
0.352
0.052
PIGM
0.056
1.304
0.075
1.699
0.106
0.04
0.371
0.038
0.377
0.044
DLS
0.056
0.998
0.134
1.510
0.155
0.04
0.281
0.051
0.331
0.073
LPPWS
0
1.306
0.075
1.306
0.075
0.04
0.419
0.040
0.420
0.041
LPPDS
0
1.005
0.135
1.005
0.135
0.04
0.290
0.046
0.321
0.062
A2 (CR = 0.020)
A3 (CR=0.017)
Eigen
0
0.519
0.041
0.519
0.041
0.12
0.402
0.038
0.409
0.044
GM
0
0.531
0.042
0.531
0.042
0.12
0.400
0.037
0.406
0.041
PIGM
0
0.463
0.036
0.463
0.036
0.12
0.375
0.031
0.378
0.033
DLS
0
0.359
0.053
0.359
0.053
0.12
0.358
0.033
0.365
0.038
LPPWS
0
0.463
0.036
0.463
0.036
0.12
0.386
0.031
0.387
0.032
LPPDS
0
0.359
0.053
0.359
0.053
0.12
0.359
0.032
0.363
0.036
A4 (CR=0.024)
A5 (CR=0.036)
Eigen
0.080
0.500
0.046
0.521
0.054
0
0.525
0.081
0.525
0.081
GM
0.080
0.499
0.045
0.521
0.054
0
0.487
0.081
0.487
0.081
PIGM
0.080
0.381
0.035
0.443
0.045
0
0.644
0.073
0.644
0.073
DLS
0.080
0.360
0.038
0.442
0.055
0.120
0.376
0.102
0.885
0.300
LPPWS
0.080
0.418
0.037
0.452
0.042
0.040
0.644
0.073
0.644
0.073
LPPDS
0.080
0.376
0.037
0.421
0.050
0.040
0.411
0.089
0.411
0.089
A6 (CR = 0.051)
Sum of variance
Eigen
0
0.930
0.082
0.930
0.082
0.240
4.906
0.429
4.946
0.451
GM
0
0.932
0.082
0.932
0.082
0.240
4.853
0.426
4.894
0.449
PIGM
0
1.005
0.072
1.005
0.072
0.296
4.545
0.359
5.010
0.408
DLS
0.080
0.672
0.141
1.016
0.242
0.496
3.406
0.552
4.908
0.915
LPPWS
0
1.005
0.072
1.005
0.072
0.280
4.644
0.364
4.679
0.370
LPPDS
0
0.678
0.135
0.678
0.135
0.280
3.479
0.529
3.558
0.560
Regarding the specific error metrics shown in Table 4, DLS consistently minimizes RMSV, while PIGM (WLS) minimizes RMSWV. The penalty-based models, LPPDS and LPPWS, consistently 14
produce the lowest RMPSV and RMPSWV, respectively. When MMV equals zero, DLS converges with LPPDS, and PIGM (WLS) converges with LPPWS; they produce identical corresponding error values due to equivalent priority outputs. The overall variance sums confirm this behavior: DLS (3.406) and PIGM (0.359) minimize RMSV and RMSWV respectively, whereas LPPDS (3.558) and LPPWS (0.370) effectively minimize RMPSV and RMPSWV respectively. In summary, while vendor rankings fluctuate based on the applied PO, 𝑇3 emerges as the most robust choice across most mathematical methods. Furthermore, the error analysis validates the efficacy of the proposed LPPDS and LPPWS optimization models. By successfully minimizing penalty-weighted variances while maintaining structural consistency, these models provide a highly reliable prioritization framework for decision-making scenarios where feasibility violations must be strictly penalized. 6. Conclusion While the mathematical robustness of Saaty’s eigenvector method for PRMs remains a subject of ongoing academic debate, this study introduces the Parallel Osprey Optimized Least PenaltySquared Prioritization method to resolve the computational complexity and opacity inherent in optimization-based prioritization within the Analytic Hierarchy Process. By leveraging the bioinspired POOA, the proposed framework efficiently solves intricate LPSP formulations to derive accurate priority vectors while explicitly minimizing RMPSV and RMPSWV. A practical enterprise application focusing on Generative AI vendor selection validates the operational feasibility, clarity, and computational efficiency of the POO-LPSP method, establishing it as a highly rigorous and scalable decision-support tool for complex expert evaluations. The experimental results based on application demonstrate that the POOA framework serves as a powerful, versatile metaheuristic solver capable of discovering global optimal solutions across LPSP, WLS, and DLS optimization landscapes. Furthermore, the comparative error analysis indicates that eigenvector method cannot select the best one choice when evaluated against the demonstrated metrics, emphasizing the measurement of penalty and variance. Future research may focus on deploying the POO-LPSP framework across broader industrial applications, actively replacing classical AHP in high-stakes domains where traditional eigenvector methods yield suboptimal variance resolution. Furthermore, extending the algorithm’s highly scalable parallel architecture to support large-scale group decision-making may be explored to efficiently aggregate massive, highly diverse expert evaluation matrices. Finally, integrating POO-LPSP with other MCDM models such as TOPSIS or PROMETHEE may be investigated to further enhance its versatility and cross-disciplinary applicability. References [1] A.H. Alqahtani, H.M. Fahmy, H.M. Hasanien, M. Tostado-Veliz, A. Alkuhayli, and F. Jurado, Parameters estimation and sensitivity analysis of lithium-ion battery model uncertainty based on osprey optimization algorithm, Energy 304 (2024), 132204. [2] M. Alrbai, S. Al-Dahidi, H. Alahmer, L. Al-Ghussain, R. Al-Rbaihat, H. Hayajneh, and A. Alahmer, Integration and Optimization of a Waste Heat Driven Organic Rankine Cycle for Power Generation in Wastewater Treatment Plants, Energy 308 (2024), 132829. [3] S. Bozoki, Solution of the least squares method problem of pairwise comparison matrices, Central European Journal of Operations Research 16 (2008), 345-358. [4] 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.
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