Generalized nash fairness solutions for bi‐objective minimization problems

Abstract

In this article, we consider a particular case of bi‐objective optimization (BOO), called bi‐objective minimization (BOM), where the two objective functions to be minimized take only positive values. As well as for BOO, most of the methods proposed in the literature for solving BOM focus on computing the Pareto‐optimal solutions representing different trade‐offs between two objectives. However, it may be difficult for a central decision‐maker to determine the preferred solutions due to the huge number of solutions in the Pareto set. We propose a novel criterion for selecting the preferred Pareto‐optimal solutions by introducing the concept of ‐Nash Fairness (‐) solutions inspired by the definition of proportional fairness. The ‐ solutions are the feasible solutions achieving some proportional nash equilibrium between the two objectives. The positive parameter is introduced to reflect the relative importance of the first objective to the second one. For this work, we will discuss existential and algorithmic questions about the ‐ solutions by first showing their existence for BOM. Furthermore, the ‐ solution set can be a strict subset of the Pareto set. As there are possibly many ‐ solutions, we focus on extreme ‐ solutions achieving the smallest values for one of the objectives. Then, we propose two Newton‐based iterative algorithms for finding extreme ‐ solutions. Finally, we present computational results on some instances of the bi‐objective travelling salesman problem (BOTSP) and the bi‐objective shortest path problem.