Algebraic solution to box-constrained bi-criteria problem of rating alternatives through pairwise comparisons

Title:

Algebraic solution to box-constrained bi-criteria problem of rating alternatives through pairwise comparisons

Creator:

Krivulin, Nikolai

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:9c7ff31a-4680-4fec-8d6e-dc20986bc521
uuid:9c7ff31a-4680-4fec-8d6e-dc20986bc521
doi:10.14736/kyb-2022-5-0665

Subject:

idempotent semifield, tropical optimization, constrained bi-criteria decision problem, Pareto-optimal solution, box constraints, and pairwise comparisons

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

We consider a decision-making problem to evaluate absolute ratings of alternatives that are compared in pairs according to two criteria, subject to box constraints on the ratings. The problem is formulated as the log-Chebyshev approximation of two pairwise comparison matrices by a common consistent matrix (a symmetrically reciprocal matrix of unit rank), to minimize the approximation errors for both matrices simultaneously. We rearrange the approximation problem as a constrained bi-objective optimization problem of finding a vector that determines the approximating consistent matrix, and then represent the problem in terms of tropical algebra. We apply methods and results of tropical optimization to derive an analytical solution of the constrained problem. The solution consists in introducing two new variables that describe the values of the objective functions and allow reducing the problem to the solution of a system of parameterized inequalities constructed for the unknown vector, where the new variables play the role of parameters. We exploit the existence condition for solutions of the system to derive those values of the parameters that belong to the Pareto front inherent to the problem. Then, we solve the system for the unknown vector and take all solutions that correspond to the Pareto front, as a complete solution of the bi-objective problem. We apply the result obtained to the bi-criteria decision problem under consideration and present illustrative examples.

Language:

English

Rights:

http://creativecommons.org/licenses/by-nc-sa/4.0/
policy:public

Coverage:

665-690

Source:

Kybernetika | 2022 Volume:58 | Number:5

Harvested from:

CDK

Metadata only:

false