Open AccessSolution of a bi-criteria problem of rating alternatives using tropical optimization
Author(s) -
Nikolai Krivulin,
Margarita A. Tsobenko
Publication year - 2019
Publication title -
kompʹûternye instrumenty v obrazovanii
Language(s) - English
Resource type - Journals
eISSN - 2071-2359
pISSN - 2071-2340
DOI - 10.32603/2071-2340-2019-4-15-32
Subject(s) - mathematics , pairwise comparison , parameterized complexity , mathematical optimization , metric (unit) , pareto principle , maxima and minima , optimization problem , algorithm , statistics , mathematical analysis , operations management , economics
A problem is considered to evaluate scores (priorities, weights) of alternatives through the results of pairwise comparisons according to two criteria. A formal derivation and computational procedures of the solution to the problem are described, using methods of tropical mathematics, which studies algebraic systems with specially defined operations of addition and multiplication. The problem is reduced to simultaneous approximation of two matrices of pairwise comparisons by a common consistent matrix, in the Chebyshev metric in logarithmic scale. First, auxiliary variables are introduced to represent the minima of the objective functions, and a parameterized inequality is derived, which determines the set of solutions to the original optimization problem. The necessary and sufficient conditions for the existence of solutions of the inequality are used to evaluate the values of parameters, which correspond to the Pareto front of the problem. All solutions of the inequality under the obtained values are taken as a Pareto-optimal solution for the problem. To illustrate the computational procedures used, numerical examples of evaluating scores of alternatives are given for problems with matrices of the third order.