Efficient Approaches for Solving Multi-Attribute Decision Making Problems with Imprecise Data

Abstract

IIn the last few years, several approaches have been proposed for solving multiattribute decision making (MADM) problems with imprecise data. In this thesis, limitations and/or flaws of some of these approaches are analysed. Also, to overcome these limitations and/or to resolve the flaws, the existing approaches are modified. Apart from these some new approaches have been proposed. The thesis comprises of eight chapters. A brief outline of the chapters is as follows: Chapter 1 is introductory in nature. In this chapter, the origin of MADM problems with imprecise data as well as the importance of fuzzy set theory in these problems is discussed. In Chapter 2, the flaws of the existing approaches for solving MADM problems with imprecise data, published in the following four papers, are analysed and the required modifications are suggested. [1] Wang, T. C., & Chen, Y. H. (2008). Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP. Information Sciences, 178(19), 3755-3765. [2] Chou, Y. C., Sun, C. C., & Yen, H. Y. (2012). Evaluating the criteria for human resource for science and technology (HRST) based on an integrated fuzzy AHP and fuzzy DEMATEL approach. Applied Soft Computing, 12(1), 64-71. v [3] Zheng, G., Zhu, N., Tian, Z., Chen, Y., & Sun, B. (2012). Application of a trapezoidal fuzzy AHP method for work safety evaluation and early warning rating of hot and humid environments. Safety Science, 50(2), 228-239. [4] Zhang, Z., & Zhang, S. (2013). A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets. Applied Mathematical Modelling, 37(7), 4948-4971. In Chapter 3, the work of Herrera and Martinez (International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 8, 539-562, 2000) has been analyzed. They have pointed out the limitations of existing fuzzy linguistic representation model and proposed 2-tuple fuzzy linguistic representation model. They proposed an approach for transforming numerical value into a linguistic 2-tuple and vice versa without any loss of information. Herrera and Martinez also proposed an approach for solving MADM problems having quantitative as well as qualitative attributes by considering the quantitative attributes as a linguistic variable whose linguistic terms are represented as triangular fuzzy numbers. Furthermore, Herrera and Martinez solved a MADM problem to illustrate their proposed approach. In this chapter, the drawbacks of this new approach, proposed by Herrera and Martinez, has been analyzed and in order to resolve them, the approach proposed by Herrera and Martinez has been modified. To illustrate the modified approach, the MADM problem, solved by Herrera and Martinez has been taken and solved. In Chapter 4, the work of Herrera and Martinez (International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 8, 539-562, 2000) has been extended by taking imprecise data as trapezoidal fuzzy numbers. They have shown that vi if the linguistic terms are represented as triangular fuzzy numbers then there will be no loss of information. However, if linguistic terms are represented as trapezoidal fuzzy numbers then there will be loss of information. On the basis of same, Herrera and Martinez (International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 8, 539-562, 2000) claimed that it is not genuine to represent the linguistic terms as trapezoidal fuzzy numbers. After a deep study of the published paper (International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 8, 539- 562, 2000), it is noticed that the reason for loss of information in case of trapezoidal fuzzy numbers is that the expression for evaluating the characteristic value of a trapezoidal fuzzy number, considered by Herrera and Martinez (International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 8, pp. 539-562, 2000), is not valid. Since, in the approach, modified in Chapter 3, there is no modification regarding the characteristic value of a fuzzy number so, if the characteristic value of trapezoidal fuzzy number, considered by Herrera and Martinez, will be used in the approach, modified in Chapter 3, then the same problem will occur. Keeping the same in mind, to overcome the limitations of the approach, modified in Chapter 3, in this chapter, an approach is proposed by using which the linguistic 2-tuple can be transformed into numerical value without using characteristic value of fuzzy numbers. Also, using this proposed approach, a new approach (named as Mehar-I approach) is proposed for solving such MADM problems in which there is a need to represent the linguistic terms as trapezoidal fuzzy numbers. The proposed Mehar-I approach can also be used if the linguistic terms are represented as triangular fuzzy numbers. To illustrate the proposed Mehar-I approach a MADM problem is solved. vii In Chapter 5, the work of Herrera and Martinez has been extended by taking imprecise data as vague sets. In fuzzy set, it is assumed that degree of non-membership (rejection) corresponding to an element will be 1−degree of membership corresponding to that element. However, in real life problems, there may be hesitation with acceptance and rejection in the mind of experts. Therefore, the assumption that degree of rejection will be 1−degree of acceptance is not always valid. Due to the same reason, Gau (IEEE Transactions on Systems, Man, and Cybernetics, 23, 610-614, 1993) introduced the concept of vague set, which is generalization of fuzzy set. In this chapter, a linguistic 3- tuple-representation model of a vague set is defined. Also, an approach for transforming the linguistic 3-tuple-representation model of a triangular/trapezoidal vague set into numerical value and vice versa without loss of information is proposed. Furthermore, an approach (named as Mehar-II approach) for solving MADM problems having qualitative as well as quantitative attributes is proposed by considering the quantitative attributes as triangular/trapezoidal vague sets. The proposed Mehar-II approach is illustrated with the help of a numerical example. In Chapter 6, a modified ratio ranking approach for solving MADM problems with imprecise parameters as traingular intuitionistic fuzzy numbers has been proposed. The intuitionistic fuzzy set, proposed by Atanassov (Fuzzy Sets and Systems, 20, 87-96, 1986), is also a generalization of fuzzy set (Information and Control, 8, 338-353, 1965). The difference between a vague set and an intuitionistic fuzzy set is that a vague set is characterized by a membership function and 1−non-membership function, while an intuitionistic fuzzy set is characterized by a membership function and a nonmembership function. viii Li (Computers & Mathematics with Applications, 60(6), 1557-1570, 2010) proposed the membership function and non-membership function of a special type of intuitionistic fuzzy set (named as triangular intuitionistic fuzzy number) as well as an approach (named as ratio ranking approach) for comparing triangular intuitionistic fuzzy numbers. Furthermore, Li used this approach for solving MADM problems with imprecise data as triangular intuitionistic fuzzy numbers. In this chapter, it is shown that the non-membership function of triangular intuitionistic fuzzy number, proposed by Li, is not valid and the exact non-membership function is proposed. Also, it is shown that of the expressions, proposed by Li for comparing triangular intuitionistic fuzzy numbers, are not valid and exact expressions are proposed. Furthermore, it is pointed out that Li has used the existing invalid expressions for finding the solution of a MADM problem. So, the solution of MADM problem, obtained by Li, is not exact solution and exact solution of MADM problem, considered by Li, is obtained. In Chapter 7, exact graphical representation of a triangular intuitionistic fuzzy number and a general intuitionistic fuzzy number is proposed. Recently, Li and Liu (IEEE Transactions on Fuzzy Systems, vol. 23, pp. 885-896, 2015) have proposed the graphical representation of a general intuitionistic fuzzy number with the help of the existing graphical representation of a triangular intuitionistic fuzzy number (International Journal of Computer, Electrical, Automation, Control and Information Engineering, 3, 350-357). In the last few years, several researchers have used triangular intuitionistic fuzzy number, proposed by Mahapatra and Roy, in their research work. In this chapter, it is shown that the graphical representation of a triangular intuitionistic ix fuzzy number, proposed by Mahapatra and Roy, as well as the graphical representation of a general intuitionistic fuzzy number, proposed by Li and Liu, are not valid. Furthermore, the exact graphical representation of a triangular intuitionistic fuzzy number and a general intuitionistic fuzzy number are proposed. In Chapter 8, based on the above study, future scope of the work is presented.