Transportation Problems in Intuitionistic Fuzzy Environment

Abstract

In the last few years, several methods have been proposed for solving different types of intuitionistic fuzzy transportation problems. In this thesis, limitations and flaws of these existing methods are pointed out. Also, to resolve the flaws as well as to overcome the limitations of the existing method, new methods are proposed. The thesis comprises seven chapters. A brief outline of the chapters is as follows: Chapter 1 Introduction Chapter 1 is introductory in nature. In this chapter, a need of intuitionistic fuzzy transportation problem as well as different types of intuitionistic fuzzy transportation problems are discussed. Furthermore, the existing methods [131,132] for solving intuitionistic fuzzy transportation problems are presented in a detailed manner. Chapter 2 A simplified method for solving intuitionistic fuzzy transportation problems of type – I In this chapter, an alternative method for solving intuitionistic fuzzy transportation problems of type – I is proposed. Also, the advantages of proposed method over the existing methods [9,46,48,64-66,118,132] are discussed. Furthermore, to illustrate the proposed method, the numerical examples, considered by Singh and Yadav [132], are solved by the proposed method. Chapter 3 A simplified method for solving intuitionistic fuzzy transportation problems of type – II In this chapter, an alternative method for solving intuitionistic fuzzy transportation prob- lems of type – II is proposed. Also, the advantages of proposed method over the existing methods [1,3,119,131], are discussed. Furthermore, to illustrate the proposed method, the numerical examples, considered by Singh and Yadav [131], are solved by the proposed method. Chapter 4 Modified approach for solving intuitionistic fully fuzzy transportation problems Kumar and Hussain [84-86] proposed methods for solving intuitionistic fully fuzzy transportation problems. In this chapter, it is pointed out that for the ranking function, used by Kumar and Hussain [84-86], the linearity property is not satisfying. However, in the existing method [84-86], this property is used. Therefore, the existing methods [84-86] are not valid. Hence, the result of numerical problems, obtained by Kumar and Hussain [84-86] by their proposed method, is also not correct. Furthermore, it is shown that for the ranking function, used by Singh and Yadav [132], linearity property is satisfying. Hence, the existing methods [84-86], will be valid if the ranking function, used by Kumar and Hussain [84-86], is replaced with the ranking function, used by Singh and Yadav [132]. Also, the exact results of numerical problems, considered by Kumar and Hussain [86], are obtained. Chapter 5 A new method for solving intuitionistic fully fuzzy transportation problems In this chapter, flaws of the existing methods [18,29,45,49,112,118,123,133,134,138] for solving intuitionistic fully fuzzy transportation problems are pointed out. Also, a new method is proposed for solving intuitionistic fully fuzzy transportation problems. To illustrate the proposed method, the intuitionistic fully fuzzy transportation problem, considered by Roseline and Amirtharaj [123], is solved by proposed method. Chapter 6 A new method for solving generalized intuitionistic fully fuzzy transportation problems Chakraborty et al. [18] proposed the arithmetic operations on generalized trapezoidal intuitionistic fuzzy numbers and used these arithmetic operations to find the solution of such intuitionistic fully fuzzy transportation problems in which cost, availability and demand all are represented by generalized trapezoidal intuitionistic fuzzy numbers. In this chapter, it is shown that Chakraborty et al. [18] have used the property in their proposed method. While, for the ranking function R, considered by Chakraborty et al. [18], this property is not satisfying. Hence, it is not genuine to used the method, proposed by Chakraborty et al. [18], to find the solution of generalized intuitionistic fully fuzzy transportation problem. Furthermore, a new method is proposed to resolve the flaws of the existing method [18]. To illustrate the proposed method, the generalized intuitionistic fully fuzzy transportation problem, considered by Chakraborty et al. [18], is solved by proposed method. Chapter 7 Future scope It is noticed that ranking of generalized exponential trapezoidal fuzzy numbers, obtained by using the existing method [121], is independent from height of generalized exponential trapezoidal fuzzy numbers. While, the ranking of generalized exponential trapezoidal fuzzy numbers should be dependent on its height. Hence, it is not genuine to use the existing method [121] for comparing the generalized exponential trapezoidal fuzzy numbers. In this chapter, the flaws of the existing method [121] are pointed out and a modified method for ranking of generalized exponential trapezoidal fuzzy numbers is proposed. In future, the proposed ranking method may be extended for generalized exponential trapezoidal intuitionistic fuzzy numbers and a new method may be proposed to find the solution of generalized exponential trapezoidal intuitionistic fully fuzzy transportation problems (transportation problems in which cost, availability and demand are represented by generalized exponential trapezoidal intuitionistic fuzzy numbers).