Parametric Biobjective Linear Programming

TL;DR

This paper explores the behavior and solutions of biobjective linear programming problems with a single varying parameter, connecting them to multi-objective problems and providing structural insights and solution strategies.

Contribution

It introduces a novel analysis of parametric biobjective linear programs, linking them to triobjective problems and offering new solution methods.

Findings

Established a one-to-one correspondence between parametric biobjective and triobjective solutions.

Analyzed the behavior of solutions with respect to extreme weights in the weight set.

Developed solution strategies for specific parametric biobjective linear programming cases.

Abstract

We consider parametric linear programming problems with multiple objective functions depending linearly on some parameter. Both parametric (single-objective) linear programming and (non-parametric) multi-objective linear programming are well-researched topics. However, literature on the combination of both, parametric linear programming with multiple objectives, is scarce. This research gap encourages our work in this field. Our main focus is on biobjective linear programs with a single parameter. We establish a connection of this problem to non-parametric multi-objective problems. Using the so-called weight set decomposition, we are able to explain the behavior of parametric biobjective linear programs when the parameter value is variated. We investigate two special cases of parametric biobjective linear programs: In the first, there is only one parametric objective and, in the second, the parametric dependency is the same for both objectives. We prove that there is a one-to-one correspondence between the solution of the parametric program and the solution of the triobjective program using the weighted sum scalarization. We provide structural insights to the solution of the parametric biobjective linear program with respect to extreme weights of the weight set of the triobjective linear program and develop solution strategies for the parametric program.