POO-LPSP Parallel Osprey Optimized Least Penalty-Squared Prioritization Methods for Priority Derivation in the Analytic Hierarchy Process

Q: What problem does POO-LPSP address in AHP priority derivation?

It targets the computational complexity of nonlinear revised LPSP optimization models used to derive priorities from PRMs, while aiming to improve prioritization reliability by minimizing revised variance measures.

Q: Which LPSP formulations are revised in this study?

The revised models include Least Product of Penalty and Direct Squares (LPPDS) and revised Weighted Squares (LPPWS), both designed to minimize revised Root Mean Penalty-Squared Variance (RMPSV) and revised Root Mean Penalty-Weighted Square Variance (RMPSWV).

Q: How is the optimization for LPSP carried out efficiently?

POO-LPSP integrates an improved bio-inspired metaheuristic Parallel Osprey Optimization Algorithm (POOA) to solve the complex LPSP formulations in a parallel and computationally efficient manner.

Jul 16, 2026

Upload by Quinn

Pairwise comparison (PC) using pairwise reciprocal matrices (PRMs) is foundational to the Analytic Hierarchy Process (AHP), yet the eigenvector-based priority derivation remains under debate for accurately reflecting true priority vectors. Building on revised Least Penalty-Squared Prioritization (LPSP) optimization models (including revised LPPDS and LPPWS) that minimize revised RMPSV and RMPSWV, this work addresses their nonlinear computational burden. It proposes POO-LPSP, integrating an improved Parallel Osprey Optimization Algorithm (POOA) to solve LPSP efficiently. A numerical application for generative AI vendor selection validates reliability and computational efficiency, positioning POO-LPSP as an alternative to Saaty’s eigen system.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages16

Reading Time40 min

Introduction

Pairwise comparison and AHP background

AHP four stages and consistency evaluation

Prioritization operator role and rank reversal

Focus of the study: revising priority operators

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: kevin.yuen@monash.edu](E-mail: kevin.yuen@monash.edu); [kevinkf.yuen@gmail.com](kevinkf.yuen@gmail.com); 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 ofthe 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] . 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 􀜶 = {􀝐1,…, 􀝐􀯝 ,…, 􀝐􀯠 }, and a set of evaluation criteria 􀜥 = {􀜿1,…, 􀜿􀯜 ,…, 􀜿􀯡 } . In the assessment stage, pairwise comparisons are executed to construct Pairwise Reciprocal Matrices (PRMs) . Each PRM is denoted by 􀜣 = [􀜽􀯜􀯝 ] such that 0 < 􀜽 􀯜􀯝 = 􀜽1, ∀􀝅, 􀝆 ∈ 1,2,…, 􀝊 (1) The entries of 􀜣 are assigned values via a 1-to-9 assessment system, measuring the dominance of item 􀝅 against item 􀝆 . To evaluate PRM consistency, the Consistency Ratio (􀜥􀜴) is determined by quotient of the Consistency Index (CI) and Random Index (RI) . 􀜥􀜴 = 􀜥􀜴􀜫􀜫~~ ~~ (2) The 􀜥􀜫 is a function of the matrix dimension 􀝊 and its principal eigenvalue 􀟣􀯠􀯔􀯫 , defined as: 􀟣􀯠􀯔􀯫 − 􀝊 􀜥􀜫 = (3) 􀝊 − 1 Furthermore, 􀟣􀯠􀯔􀯫 can be calculated by: 􀟣􀯠􀯔􀯫 = 1􀝊 ∑ 1~~ ~~2􀯜􀯝􀟜􀯜􀯝~~ ~~ , 􀟜􀯜􀯝 = (~~􀝓~~􀜽􀯜~~/~~􀯜􀯝~~􀝓~~􀯝 −1) (4) 1≤􀯜<􀯝≤􀯡 A perfectly consistent PRM always has 􀟣􀯠􀯔􀯫 = 􀝊 . The condition 􀟣􀯠􀯔􀯫 < 􀝊 is mathematically impossible, meaning that any deviation from perfect con

FAQs

What problem does POO-LPSP address in AHP priority derivation?

It targets the computational complexity of nonlinear revised LPSP optimization models used to derive priorities from PRMs, while aiming to improve prioritization reliability by minimizing revised variance measures.

Which LPSP formulations are revised in this study?

The revised models include Least Product of Penalty and Direct Squares (LPPDS) and revised Weighted Squares (LPPWS), both designed to minimize revised Root Mean Penalty-Squared Variance (RMPSV) and revised Root Mean Penalty-Weighted Square Variance (RMPSWV).

How is the optimization for LPSP carried out efficiently?

POO-LPSP integrates an improved bio-inspired metaheuristic Parallel Osprey Optimization Algorithm (POOA) to solve the complex LPSP formulations in a parallel and computationally efficient manner.

Jul 16, 2026

Upload by Quinn