Low-Rank Multi-Objective Linear Programming

Abstract

When solving multi-objective programs (MOLPs), the number of objectives essentially determines the computing time. This can even lead to practically unsolvable problems. Consequently, it is of huge interest to reduce the number of objectives without losing information. There exist approaches for transforming vector linear programs (VLPs), i.e., the ordering cone is non-standard, into MOLPs in the literature. We here propose a method for solving MOLPs more efficiently by applying the transformation in reverse. In particular, we discuss MOLPs with linear dependent objective functions. The resulting VLPs have merely as many objectives as the rank of the objective matrices of the MOLPs. To achieve this, only a factorization of this matrix needs to be calculated. One factor then forms a new ordering cone, while the other remains as the objective matrix. Through multiple series of numerical experiments, we show that this approach indeed significantly reduces computing time, and therefore provides a technique for solving low-rank MOLPs in practice. As there are fewer objectives to consider, the approach additionally helps decision makers to get a better visualization as well as understanding of the actual problem. Moreover, we will point out that the equivalence between MOLPs and corresponding VLPs can be used to derive statements about the well-known concept of nonessential objectives.