Fuzzy mathematical programming with application to game theory

Abstract

In this thesis, we aim to apply I-fuzzy sets to study I-fuzzy linear programming problems and investigate duality results. From application point of view, it is shown that solving a two person matrix game in I-fuzzy environment is equivalent to solving two I-fuzzy linear programming problems which are dual to each other in the sense of I-fuzzy duality developed herein. newlineThe thesis comprises of eight chapters. Chapter 1 is introductory in nature and presents literature survey and some basic definitions related to the study. Also it contains the summary of the research work presented in this thesis. newlineChapter 2 proposes a bi-objective I-fuzzy linear programming problem aiming to achieve a trade-off between the twin objectives of meeting the aspiration level and satisfaction of the given constraints . A new two phase approach is developed to find an efficient solution of this bi-objective programming problem. The solutions of the given I-fuzzy linear programming problem are interpreted using thus obtained efficient solutions. Further more, two additional approaches are also proposed to get efficient solution of the same on similar lines. newlineChapter 3 reviews an existing approach for formulating the crisp equivalent of an I-fuzzy linear programming problem wherein the crisp model involves the binary variables. Our contribution in this chapter is to apply a different technique to formulate the crisp equivalent of an I-fuzzy linear programming problem without any binary variables. This approach enables us to study the duality for an I-fuzzy linear programming problem; a weak duality theorem is established here in an appropriate sense. newlineChapter 4 is concerned with the study of two person zero-sum matrix games with I-fuzzy goals. The indeterminacy factor associated with each player s I-fuzzy goal has been resolved using Yager s indeterminacy resolution principle and Hurwicz optimism-pessimism criterion .