# Low-Rank Multi-Objective Linear Programming

## TL;DR

This paper presents a method to reduce the number of objectives in multi-objective linear programs by transforming them into equivalent vector linear programs with fewer objectives, leading to faster computation and better visualization.

## Contribution

It introduces a novel transformation technique for MOLPs with non-full rank objective matrices, improving computational efficiency and interpretability.

## Key findings

- Significant speedup in solving MOLPs.
- Enhanced visualization and understanding of objectives.
- Effective reduction of objectives without losing information.

## Abstract

When solving multi-objective programs, the number of objectives essentially determines the computing time. This can even lead to practically unsolvable problems. Consequently, it is worthwhile to reduce the number of objectives without losing information. In this article, we discuss multi-objective linear programs (MOLP) with objective matrices that do not have a full row rank. We introduce a method to transform an MOLP into an equivalent vector linear program (VLP) with merely as many objectives as the rank of the original objective matrix. To this end, only a factorization of this matrix is needed to be calculated. One of the factors then forms a new ordering cone while the other factor remains as the objective matrix. Through a series of numerical experiments, we will show that this approach indeed leads to a significant speedup of computing time, and therefore provides a powerful technique for solving 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 of the MOLP and the corresponding VLP can be used to derive statements about the concept of nonessential objectives.