Interactive Fuzzy Approaches for Solving Multiobjective Two-Person Zero-Sum Games

Hitoshi Yano,Ichiro Nishizaki

doi:10.4236/am.2016.75036

Hitoshi Yano, Ichiro Nishizaki

Open Access

https://doi.org/10.4236/am.2016.75036

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Journal: Applied mathematics	Publication Date: Jan 1, 2016
Citations: 15	License type: CC BY 4.0

Affiliation: Nagoya City University, Hiroshima University

Abstract

In this paper, we consider multiobjective two-person zero-sum games with vector payoffs and vector fuzzy payoffs. We translate such games into the corresponding multiobjective programming problems and introduce the pessimistic Pareto optimal solution concept by assuming that a player supposes the opponent adopts the most disadvantage strategy for the self. It is shown that any pessimistic Pareto optimal solution can be obtained on the basis of linear programming techniques even if the membership functions for the objective functions are nonlinear. Moreover, we propose interactive algorithms based on the bisection method to obtain a pessimistic compromise solution from among the set of all pessimistic Pareto optimal solutions. In order to show the efficiency of the proposed method, we illustrate interactive processes of an application to a vegetable shipment problem.

Highlights

We propose interactive algorithms for multiobjectve two-person zero-sum games with vector payoffs and vector fuzzy payoffs under the assumption that each player has fuzzy goals for his/her multiple expected payoffs
The concept of security levels is inherent in the definition of maximin solutions in two-person zero-sum games
In this paper, we focus on two-person zero-sum games with vector fuzzy payoffs under the assumption that a player has fuzzy goals for the expected payoffs which are defined as nonlinear membership functions

Summary

Introduction

We propose interactive algorithms for multiobjectve two-person zero-sum games with vector payoffs and vector fuzzy payoffs under the assumption that each player has fuzzy goals for his/her multiple expected payoffs. Sakawa and Nishizaki [5] proposed a fuzzy approach for two-person zero-sum games with vector payoffs to obtain maximin solutions which are defined from the viewpoint of maximization of the degree of minimal goal attainment [6] [7]. They showed that such a problem is reduced to a linear programming problem. By introducing the fuzzy goals, they formulated two-person zero-sum games with vector payoffs as a linear programming problem to obtain maximin solutions.

Two-Person Zero-Sum Games with Vector Payoffs

Two-Person Zero-Sum Games with Vector Fuzzy Payoffs

An Application to Multi-Variety Vegetable Shipment Planning

Conclusion