Generalized Nash Fairness Solutions for Bi-Objective Discrete Optimization: Theory and Algorithms

Abstract

This paper deals with a particular case of Bi-Objective Optimization called Bi-Objective Discrete Optimization (BODO), where the feasible set is discrete, and the two objectives take only positive values. Since the feasible set of a BODO problem is discrete and usually finite, it can theoretically be enumerated to identify the Pareto set, which consists of all Pareto-optimal solutions representing different trade-offs between the two objectives. However, in general, this problem is challenging due to two main issues: time complexity, as the number of Pareto-optimal solutions can be exponentially large, and lack of decisiveness. From a practical point of view, the Central Decision Maker (CDM) may be interested in a reduced Pareto set that reflects the CDM’s preferences, which can be obtained by a computationally efficient algorithm.

In this paper, we propose a new criterion for selecting solutions within the Pareto set of BODO. For this purpose, we focus on solutions achieving proportional fairness between two objectives, called generalized Nash Fairness solutions ($\rho$-NF solutions). The positive parameter $\rho$, provided by the CDM, reflects the relative importance of the first objective compared to the second one.

We first introduce the $\rho$-NF solution concept for BODO. We then show that the $\rho$-NF solution set is a subset of the Pareto set, and this inclusion can be strict. We also propose a recursive Newton-like algorithm for determining the $\rho$-NF solution set. Finally, an illustrative example of BODO is given.