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Using tropical optimization techniques in bi-criteria decision problems

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  • Nikolai Krivulin

    (St. Petersburg State University)

Abstract

We consider decision problems of rating alternatives based on their pairwise comparisons according to two criteria. Given pairwise comparison matrices for each criterion, the problem is to find the overall scores of the alternatives. We offer a solution that involves the minimax approximation of the comparison matrices by a common consistent matrix of unit rank in terms of the Chebyshev metric in logarithmic scale. The approximation problem reduces to a bi-objective optimization problem to minimize the approximation errors simultaneously for both comparison matrices. We formulate the problem in terms of tropical (idempotent) mathematics, which focuses on the theory and applications of algebraic systems with idempotent addition. To solve the optimization problem obtained, we apply methods and results of tropical optimization to derive a complete Pareto-optimal solution in a direct explicit form ready for further analysis and straightforward computation. We then exploit this result to solve the bi-criteria decision problem of interest. As illustrations, we present examples of the solution of two-dimensional optimization problems in general form, and of a decision problem with four alternatives in numerical form.

Suggested Citation

  • Nikolai Krivulin, 2020. "Using tropical optimization techniques in bi-criteria decision problems," Computational Management Science, Springer, vol. 17(1), pages 79-104, January.
    • Handle: RePEc:spr:comgts:v:17:y:2020:i:1:d:10.1007_s10287-018-0341-x
    DOI: 10.1007/s10287-018-0341-x
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    References listed on IDEAS

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    1. Massimo Pappalardo, 2008. "Multiobjective Optimization: A Brief Overview," Springer Optimization and Its Applications, in: Altannar Chinchuluun & Panos M. Pardalos & Athanasios Migdalas & Leonidas Pitsoulis (ed.), Pareto Optimality, Game Theory And Equilibria, pages 517-528, Springer.
    2. Nikolai Krivulin, 2018. "Methods of Tropical Optimization in Rating Alternatives Based on Pairwise Comparisons," Operations Research Proceedings, in: Andreas Fink & Armin Fügenschuh & Martin Josef Geiger (ed.), Operations Research Proceedings 2016, pages 85-91, Springer.
    3. Ahn, Byeong Seok, 2017. "The analytic hierarchy process with interval preference statements," Omega, Elsevier, vol. 67(C), pages 177-185.
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    Cited by:

    1. Nikolai Krivulin, 2020. "Tropical optimization technique in bi-objective project scheduling under temporal constraints," Computational Management Science, Springer, vol. 17(3), pages 437-464, October.
    2. Nikolai Krivulin, 2021. "Algebraic Solution of Tropical Polynomial Optimization Problems," Mathematics, MDPI, vol. 9(19), pages 1-18, October.
    3. Nikolai Krivulin & Alexey Prinkov & Igor Gladkikh, 2022. "Using Pairwise Comparisons to Determine Consumer Preferences in Hotel Selection," Mathematics, MDPI, vol. 10(5), pages 1-25, February.
    4. Toly Chen, 2021. "A diversified AHP-tree approach for multiple-criteria supplier selection," Computational Management Science, Springer, vol. 18(4), pages 431-453, October.
    5. Nikolai Krivulin, 2021. "Algebraic Solution to Constrained Bi-Criteria Decision Problem of Rating Alternatives through Pairwise Comparisons," Mathematics, MDPI, vol. 9(4), pages 1-22, February.

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