Weakly admissible transformations for solving algebraic assignment and transportation problems

Abstract

Weakly admissible transformations are introduced for solving algebraic assignment and transportation problems, which cover so important classes as problems with sum objectives, bottleneck objectives, lexicographical objectives and others. A transformation of the cost matrix is called weakly admissible, if there are two constants α and β in the underlying semigroup that for all feasible solutions the composition of α and the objective value with respect to the original cost coefficients is equal to the composition of β and the objective value with respect to the transformed cost coefficients. The elements α and β can be determined by shortest path algorithms. An optimal solution for the algebraic assignment problem can be found after at most n weakly admissible transformations, therefore the proposed method yields an O(n ³) algorithm for algebraic assignment problems.