Fuzzy Linear Programming and its Applications

Abstract

In many real life situations, the decision maker may not be in a position to specify the objective and/or constraint functions precisely but rather can specify them in a “fuzzy sense”. In such situations, it is desirable to use some fuzzy linear programming type of modeling. The technique of fuzzy linear programming enlarges the range of applications of the linear programming method. It enables us to consider tolerances for values of decision model parameters in a more natural and direct way. It is of special importance in a situation when the necessity of taking into account tolerances for parameters is due to the impossibility of determining them precisely as well as the situation when some tolerances for parameters are consciously assumed by the decision maker. This thesis is devoted to fuzzy linear programming and its applications. The main topics are various models of fuzzy linear programming problem, fuzzy transportation problem, fuzzy assignment problem, fuzzy travelling salesman problem and multiobjective transportation problem. The chapter-wise summary of the thesis is as follows: Chapter 1 is introductory in nature. This chapter includes basic definitions, operations and concepts used throughout the work. Chapter 2 presents brief review of the work done in the area of fuzzy linear programming problem, fuzzy transportation problem, fuzzy assignment problem, fuzzy travelling salesman problem and multiobjective transportation problem. In Chapter 3, decision making under fuzzy environment and various models of fuzzy linear programming problem have been studied. In Chapter 4, fuzzy transportation problem has been studied. Three new methods have been proposed to obtain the initial fuzzy basic feasible solution of a particular type of fuzzy transportation problem. Also a new method has been proposed to find the fuzzy optimal solution from initial basic feasible solution. To illustrate the proposed methods, a real life problem has been solved. In Chapter 5, fuzzy assignment problem which is a special type of fuzzy linear programming problem has been studied. A new algorithm has been proposed to find the fuzzy optimal solution for the fuzzy assignment problem and is illustrated by a numerical example. In Chapter 6, fuzzy travelling salesman problem is studied. A new algorithm has been proposed using ranking to characterize the fuzzy optimality of travelling salesman problem. To illustrate the proposed algorithm a numerical has been solved. In Chapter 7, the importance of the fuzzy programming approach, goal programming approach and their combination called the fuzzy goal programming approach to solve the multiobjective transportation problem has been discussed. A new approach has been proposed for the problem solved by Zangiabadi and Maleki [58]. Also the results of existing and proposed approaches are compared from different point of views.