Using Shapley Values and Genetic Algorithms to Solve Multiobjective Optimization Problems

Abstract

This paper proposes a new methodology to solve multiobjective optimization problems by invoking genetic algorithms and the concept of the Shapley values of cooperative games. It is well known that the Pareto-optimal solutions of multiobjective optimization problems can be obtained by solving the corresponding weighting problems that are formulated by assigning some suitable weights to the objective functions. In this paper, we formulated a cooperative game from the original multiobjective optimization problem by regarding the objective functions as the corresponding players. The payoff function of this formulated cooperative game involves the symmetric concept, which means that the payoff function only depends on the number of players in a coalition and is independent of the role of players in this coalition. In this case, we can reasonably set up the weights as the corresponding Shapley values of this formulated cooperative game. Under these settings, we can obtain the so-called Shapley–Pareto-optimal solution. In order to choose the best Shapley–Pareto-optimal solution, we used genetic algorithms by setting a reasonable fitness function.

Keywords

• cooperative games
• genetic algorithms
• Pareto-optimal solutions
• Shapley values
• weighting problems