Linear Programming Problems with Generalized Fuzzy Sets

Abstract

In this thesis, the limitations and shortcomings of existing methods for solving linear programming problems with fuzzy sets are pointed out. To overcome the limitations and shortcomings of existing methods, some new ranking approaches are proposed for comparing generalized fuzzy sets and vague sets. On the basis of proposed ranking approaches, some new methods are proposed to nd the appropriate solution of such linear programming problems with generalized fuzzy sets and vague sets in which only the parameters cost (or pro t) are represented by fuzzy sets or vague sets. The chapter wise summary of the thesis is as follows: In Chapter 1, a brief review of the work done in the area of linear programming problems with fuzzy sets is presented. In Chapter 2, it is shown that the existing method for solving linear programming problems with fuzzy sets can be used only for solving such linear programming problems in which either the parameters are represented by normal fuzzy sets or generalized fuzzy sets having equal height. Although, the limitations of the existing method can be removed by replacing an appropriate ranking approach instead of already used existing ranking approach for comparing fuzzy sets but there are some shortcomings in all the existing ranking approaches so none of the existing ranking iii iv approach can be used to overcome the limitations of existing method. To overcome the limitations of existing method, a new ranking approach is proposed for comparing generalized trapezoidal fuzzy sets and on the basis of proposed ranking approach a new method is proposed for solving linear programming problems with generalized fuzzy sets. In Chapter 3, the limitations of some existing results for comparing generalized p-norm fuzzy sets are pointed out and with the help of the ranking approach, proposed in Chapter 2, some new results are proposed by modifying the existing results to overcome the limitations of existing results. It is shown that the existing results are the particular cases of the proposed results. Also, the method for solving linear programming problems with generalized fuzzy sets, proposed in Chapter 2, is used to solve a linear programming problem with generalized p-norm trapezoidal fuzzy sets. In Chapter 4, the shortcomings of the ranking approach, proposed in Chapter 2, are pointed out and to overcome these shortcomings a new ranking approach, named as RM ranking approach, is proposed for comparing generalized trapezoidal fuzzy sets. Also, with the help of RM ranking approach, a new method is proposed for solving linear programming problems with generalized trapezoidal fuzzy sets. In Chapter 5, the limitations of some existing results related to comparison of intuitionistic fuzzy sets and shortcomings of an existing ranking approach for comparing triangular intuitionistic fuzzy sets are pointed out. To overcome the limitations of existing results, some new results are proposed by modifying the existing results and to overcome the shortcomings of existing ranking approach, a new ranking approach is proposed for comparing trapezoidal vague sets. Also, the method v for solving linear programming problems with generalized fuzzy sets, proposed in Chapter 4, with proposed ranking approach is used to solve a linear programming problem with trapezoidal vague sets. In Chapter 6, it is shown that the results of the linear programming problems with fuzzy sets obtained by using the existing and proposed methods are not appropriate. It is pointed out that all the shortcomings in the results are occurring due to used ranking approaches. To overcome the shortcomings of existing and proposed ranking approaches, a new ranking approach, named as RMDS ranking approach, is proposed for comparing trapezoidal vague sets. On the basis of proposed RMDS ranking approach, a new method is proposed for solving linear programming problems with trapezoidal vague sets. To show the advantage of the proposed method the linear programming problems with fuzzy and vague sets, for which the results obtained by using the existing and other proposed methods are not appropriate, are solved by using the proposed method and it is shown that the obtained results are appropriate. Finally, in Chapter 7, based on the present study, conclusions are drawn and future work have been suggested.