Application of tropical optimization for solving multicriteria problems of pairwise comparisons using log-Chebyshev approximation
Int. J. Approx. Reason.(2022)
Abstract
We consider a decision-making problem to find absolute ratings of
alternatives that are compared in pairs under multiple criteria, subject to
constraints in the form of two-sided bounds on ratios between the ratings.
Given matrices of pairwise comparisons made according to the criteria, the
problem is formulated as the log-Chebyshev approximation of these matrices by a
common consistent matrix (a symmetrically reciprocal matrix of unit rank) to
minimize the approximation errors for all matrices simultaneously. We rearrange
the approximation problem as a constrained multiobjective optimization problem
of finding a vector that determines the approximating consistent matrix. The
problem is then represented in the framework of tropical algebra, which deals
with the theory and applications of idempotent semirings and provides a formal
basis for fuzzy and interval arithmetic. We apply methods and results of
tropical optimization to develop a new approach for handling the multiobjective
optimization problem according to various principles of optimality. New
complete solutions in the sense of the max-ordering, lexicographic ordering and
lexicographic max-ordering optimality are obtained, which are given in a
compact vector form ready for formal analysis and efficient computation. We
present numerical examples of solving multicriteria problems of rating four
alternatives from pairwise comparisons to illustrate the technique and compare
it with others.
MoreTranslated text
Key words
Idempotent semifield,Tropical optimization,Log-Chebyshev approximation,Constrained multiobjective optimization,Multiple criteria evaluation
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