Using tropical optimization techniques in bi-criteria decision problems

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.

Algebraic Solution to Constrained Bi-Criteria Decision Problem of Rating Alternatives through Pairwise Comparisons

“…Note that this problem can be reduced to an equivalent problem in the same variables, which does not involve logarithms (see, e.g., [20,21]). Indeed, representing the absolute value function as the maximum of two opposite values, using the monotonicity of the logarithm and the condition a ij = 1/a ji turn the objective function into…”

“…which has a complete Pareto-optimal solution given in the tropical algebra setting in [20], where the solution set is explicitly described in a parametric vector form.…”

“…In the framework of tropical algebra, the problem of rating alternatives by pairwise comparisons is examined in a range of papers, including [13][14][15][16][17]. A new approach is proposed in [18][19][20][21], which offers a direct complete solution in an explicit parametric form. Specifically, a tropical analogue of the AHP is developed in [19,21], and a constrained single-criterion problem is solved in [18].…”

“…Specifically, a tropical analogue of the AHP is developed in [19,21], and a constrained single-criterion problem is solved in [18]. For an unconstrained bi-criteria problem, a complete Pareto-optimal solution is obtained in [20].…”

“…We follow the general solution scheme proposed in [20] to solve the bi-criteria problem without constraints. This scheme is further extended below to develop a new general solution that accommodates constraints in an efficient way.…”