Rating alternatives from pairwise comparisons by solving tropical optimization problems
For researchers in decision-making and optimization, this provides a more efficient solution to a known problem, but the improvement is incremental.
This paper addresses the problem of rating alternatives from pairwise comparisons under constraints, using tropical optimization to find direct solutions that reduce computational effort compared to existing methods.
We consider problems of rating alternatives based on their pairwise comparison under various assumptions, including constraints on the final scores of alternatives. The problems are formulated in the framework of tropical mathematics to approximate pairwise comparison matrices by reciprocal matrices of unit rank, and written in a common form for both multiplicative and additive comparison scales. To solve the unconstrained and constrained approximation problems, we apply recent results in tropical optimization, which provide new complete direct solutions given in a compact vector form. These solutions extend known results and involve less computational effort. As an illustration, numerical examples of rating alternatives are presented.