Abstract
Pairwise comparison matrices play a prominent role in multiple-criteria decision-making, particularly in the analytic hierarchy process (AHP). Another form of preference modeling, called an incomplete pairwise comparison matrix, is considered when one or more elements are missing. In this paper, an algorithm is proposed for the optimal completion of an incomplete matrix. Our intention is to numerically minimize a maximum eigenvalue function, which is difficult to write explicitly in terms of variables, subject to interval constraints. Numerical simulations are carried out in order to examine the performance of the algorithm. The results of our simulations show that the proposed algorithm has the ability to solve the minimization of the constrained eigenvalue problem. We provided illustrative examples to show the simplex procedures obtained by the proposed algorithm, and how well it fills in the given incomplete matrices.
Highlights
In addition to having been employed in the analytic hierarchy process (AHP), the technique of pairwise comparisons and some of its variants are currently used in other multi-criteria decision-making techniques as, for instance, multi-attribute utility theory [4]; the best-worst method [5]; ELECTRE [6]; PROMETHEE [7]; MACBETH [8]; and PAPRIKA [9]
It is hard to underestimate the importance of pairwise comparisons in multi-criteria decision analysis
We studied an application of the Nelder-Mead algorithm to the constrained ‘λmax -optimal completion’ and provided numerical simulations to study its performance
Summary
Complex decisions are being made everyday and especially in positions of high responsibility, they must be “defensible” in front of stakeholders. We considered an optimal completion algorithm for incomplete PCMs. The first step in this direction consists of considering the missing entries as variables. Algorithms 2021, 14, 222 of the attributes giving a necessary flexibility to the preference assessment For this reason, we consider it important that the proposed algorithm, described, is able to solve a constrained optimization problem, where variables are subject to interval constraints. All entries of each matrix in the paper consist of numerical values restricted to the interval [1/9, 9] in order to meet Saaty’s proposal and comply with AHP formulation It must, be said that our choice is not binding and that this requirement can be relaxed by removing some constraints.
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