Abstract
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.
Highlights
The problem of evaluating the ratings of alternatives based on the results of pairwise comparisons in accordance with two criteria is considered
The problem is reduced to the simultaneous approximation of two matrices of pairwise comparisons by a common consistent matrix, in the Chebyshev metric in logarithmic scale
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
Summary
Многокритериальная оптимизация представляет собой процесс одновременной максимизации (минимизации) двух или более целевых функций, которые могут отвечать взаимно противоречивым критериям. Одно из применений многокритериальной оптимизации связано с оценкой рейтингов (приоритетов, весов) альтернатив принятия решений на основе их парных сравнений [5, 6]. Результатом являются матрицы парных сравнений альтернатив по всем критериям, а также самих критериев, если они неравнозначны. Для решения многокритериальных задач определения рейтингов альтернатив с помощью матриц парных сравнений чаще всего используется метод анализа иерархий [2, 6]. Представление и решение задач оценки рейтингов альтернатив на основе парных сравнений с помощью методов тропической математики изучались в работах [13,14,15,16,17,18,19]. В настоящей работе описывается формальное построение и вычислительные процедуры решения двухкритериальной задачи оценки рейтингов альтернатив на основе предложенного в [18] подхода с использованием методов тропической оптимизации.
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