TY - JOUR

T1 - Using tropical optimization techniques in bi-criteria decision problems

AU - Krivulin, Nikolai

N1 - Funding Information: This work was supported in part by the Russian Foundation for Basic Research (Grant No. 18-010-00723).

PY - 2020/1/1

Y1 - 2020/1/1

N2 - 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.

AB - 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.

KW - ANALYTIC HIERARCHY PROCESS

KW - MAX-ALGEBRA

KW - PAIRWISE

UR - https://arxiv.org/abs/1810.08662

UR - http://www.scopus.com/inward/record.url?scp=85057571878&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/4627caa7-ac1c-3287-aae5-b41e00a9efaa/

U2 - 10.1007/s10287-018-0341-x

DO - 10.1007/s10287-018-0341-x

M3 - Article

VL - 17

SP - 79

EP - 104

JO - Computational Management Science

JF - Computational Management Science

SN - 1619-697X

IS - 1

T2 - XV International Conference on Computational Management Science

Y2 - 29 May 2018 through 31 May 2018

ER -