Tropical optimization techniques in multi-criteria decision making with Analytical Hierarchy Process
N. Krivulin (1), S. Sergeev (2) ((1) Saint Petersburg State, University, Russia, (2) University of Birmingham, UK)

TL;DR
This paper introduces a novel tropical optimization framework for implementing the Analytic Hierarchy Process in multi-criteria decision making, providing closed-form solutions and techniques for handling non-unique solutions.
Contribution
It develops a new theoretical and computational approach using tropical mathematics for the AHP, including solutions for non-unique cases and differentiation of alternatives.
Findings
Closed-form solutions for rating problems
Tropical optimization techniques for non-unique solutions
Methods to identify most and least differentiating vectors
Abstract
We apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the Analytic Hierarchy Process in multi-criteria problems of rating alternatives from pairwise comparison data. The framework involves the Chebyshev approximation of pairwise comparison matrices by consistent matrices in the logarithmic scale. We reduce the log-Chebyshev approximation to multidimensional tropical optimization problems, and offer complete direct solutions to the problems in the framework of tropical mathematics. The results obtained provide a closed-form solution to the rating problem of interest as either a unique score vector (up to a positive factor) or as a set of different score vectors. To handle the problem when the solution is not unique, we develop tropical optimization techniques to find those vectors from the solution set…
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Taxonomy
TopicsMulti-Criteria Decision Making · Optimization and Mathematical Programming · Data Management and Algorithms
