A note on the estimation of the Pareto efficient set for multiobjective matrix permutation problems

There are a number of important problems in quantitative psychology that require the identification of a permutation of the n rows and columns of an n × n proximity matrix. These problems encompass applications such as unidimensional scaling, paired‐comparison ranking, and anti‐Robinson forms. The importance of simultaneously incorporating multiple objective criteria in matrix permutation applications is well recognized in the literature; however, to date, there has been a reliance on weighted‐sum approaches that transform the multiobjective problem into a single‐objective optimization problem. Although exact solutions to these single‐objective problems produce supported Pareto efficient solutions to the multiobjective problem, many interesting unsupported Pareto efficient solutions may be missed. We illustrate the limitation of the weighted‐sum approach with an example from the psychological literature and devise an effective heuristic algorithm for estimating both the supported and unsupported solutions of the Pareto efficient set.