Aggregation Operators for Various Extensions of Fuzzy Set and Its Applications in Transportation Problems

Abstract

There exist several methods in the literature to solve fuzzy transportation problems under its various extensions. But, as all these methods have been proposed by considering the assumption that the aggregated value of all the parameters are known. Therefore, if instead of the aggregated values of the parameters, the values of the various parameters, collected from experts, are provided. Then, the existing methods cannot be used to solve fuzzy transportation problems and its various extensions. To overcome this limitation, firstly, there is need to aggregate the provided values. But, as there exist several weighted averaging operators and weighted geometric operators for each extension of fuzzy set. Therefore, firstly, there is need to choose the appropriate weighted averaging operator and weighted geometric operator for each extension of fuzzy set. After aggregating the provided values by the selected aggregation operator, researchers may use the recently proposed methods for solving fuzzy transportation problems and its various extensions. However, after a deep study, some shortcomings have been observed in these methods. Therefore, it is scientifically incorrect to use these methods in their present form. Keeping all above in mind, the aim of this thesis is (i) To choose/propose an appropriate weighted averaging operator for various extensions of fuzzy set. Also, to show that the weighted geometric operator cannot be defined for fuzzy set and its extensions. (ii) To point out the limitations as well as the flaws of an existing method. Also, to propose a modified method to overcome the limitations as well as to resolve the flaws of the existing method. (iii) To propose a simplified approach for solving balanced fully intuitionistic fuzzy transportation problems as compared to the existing approach. Also, to generalize the proposed approach, with the help of the trapezoidal intuitionistic fuzzy weighted aggregation operator for solving such balanced fully intuitionistic fuzzy transportation problems in which the aggregated values of the parameters are not known. (iv) To point out the flaws of an existing method for transforming an unbalanced fully intuitionistic fuzzy transportation problems into a balanced fully intuitionistic fuzzy transportation problem as well as to propose a valid method for transforming an unbalanced fully intuitionistic fuzzy transportation problem into a balanced fully intuitionistic fuzzy transportation problem. (v) To point out the flaws of an existing method for transforming an unbalanced generalized interval valued trapezoidal fuzzy number transportation problem into a balanced generalized interval valued trapezoidal fuzzy number transportation problem as well as to propose a valid method for transforming an unbalanced generalized interval valued trapezoidal fuzzy number transportation problem into a balanced generalized interval valued trapezoidal fuzzy number transportation problem .