Gra Based Multi Criteria Decision Making in Generalized Neutrosophic Soft Set Environment

The main objective of the paper is to propose generalized neutrosophic soft multi criteria decision making based on grey relational analysis. The concept of generalized neutrosophic soft sets has been derived from the hybridization of the concepts of neutrosophic set and soft set. In this paper we have defined neutrosophic soft weighted average operator in order to aggregate the individual decision maker’s opinion into a common opinion based on choice parameters of the evaluators. In the decision making process, the decision makers provide the rating of alternatives with respect to the parameters in terms of generalized neutrosophic soft set. We determine the order of the alternatives and identify the most suitable alternative based on grey relational coefficient. Finally, in order to demonstrate the effectiveness and applicability of the proposed approach, a numerical example of logistics center location selection problem has been solved. INTRODUCTION Evolution of human society evokes complexity in their life and human beings have to deal with uncertainty, imprecise data to solve their real life problems. To deal uncertainty, mathematicians proposed a number of theories such as probability [1], fuzzy sets [2], interval mathematics, etc. Molodtsov [3] described the limitations of these theories in his study and grounded the concept of soft set theory to overcome the difficulties in 1999. Soft set theory has been successfully applied in data analysis [4], optimization [5], etc. The researchers have showed great interest in the theory and they proposed different hybrid soft sets and their applications such as fuzzy soft set [6,7,8], generalized fuzzy soft set [9,10], intuitionistic fuzzy soft set [11,12], possibility intuitionistic fuzzy soft set [13], vague soft set [14], neutrosophic soft set [15], weighted neutrosophic soft set [16],generalized neutrosophic soft set [17, 18]. However, neutrosophic set [19, 20, 21, 22, 23], is the generalization of fuzzy set, and intuitionistic fuzzy set. In 2010, Wang et al. [24] defined single valued neutrosophic set, which is an instance of neutrosophic set. Neutrosophic set and single valued neutrosophic sets have been successfully applied in different research areas such as social sciences [25, 26, 27], conflict resolution [28], artificial intelligence and control systems [29], medical diagnosis [30, 31, 32, 33], decision making [34, 35, 36, 37, 38, 39, 40, 41, 42, 43], image processing [44, 45], decision making in neutrosophic hybrid environment [ 46, 47,48, 49, 50, 51, 52, 53, 54]. Neutrosophic sets and soft set sets are two different concepts. Literature review suggests that both are capable of handling uncertainty and incomplete information. It seems that the hybrid system called ‘generalized neutrosophic soft set’ is capable of dealing with uncertainty, indeterminacy and incomplete information. It seems that generalize neutrosophic soft set is very interesting and applicable in realistic problems. Literature review reveals that only few studies on generalized neutrosophic soft sets [17, 18, 55, 56] have been done. Deng [57] studied grey relational analysis (GRA). GRA has been applied widely different areas of research such as teacher selection [58], weaver selection [59], brick optimal welding parameter selection [60], failure mode and effects analysis [61], multi attribute decision making (MADM) [62], multi criteria decision making [63, 64, 65], medical diagnosis [66], etc. Biswas et al. [67, 68] at first used the concept of GRA in neutrosophic environment for MADM problems. Pramanik and Mondal [69] studied interval neutrosophic multi-attribute decision-making based on grey relational analysis. Mondal and Pramanik [70] presented a neutrosophic school choice model based [Pramanik et al., 3(5): May, 2016] ISSN 2349-4506 Impact Factor: 2.265 Global Journal of Engineering Science and Research Management http: // www.gjesrm.com © Global Journal of Engineering Science and Research Management [154] on modified GRA method. Mondal and Pramanik [71] also introduced rough neutrosophic MADM based on modified GRA. Dey et al. [72] studied neutrosophic soft multi-attribute group decision making based on grey relational analysis. Dey et al. [73] presented neutrosophic soft multi-attribute decision making based on grey relational projection method. Dey et al. [74] studied extended grey relational analysis method for multiple attribute decision making problems under interval neutrosophic uncertain linguistic environment. GRA based MCDM problem in generalized neutrosophic soft set environment has not yet been addressed in the literature. In this paper, we have presented a generalized neutrosophic soft multi-attribute group decision making model with grey relational analysis. Rest of the paper is organized in the following way. Section 2 presents some basic definitions of neutrosophic set, generalized neutrosophic set, soft set, generalized neutrosophic soft set and real life example on generalizedneutrosophic soft set. Section 3 presents the grey relational analysis. Section 4 is devoted to present a compact model for a generalized neutrosophic-soft MAGDM based on grey-relational analysis. Section 5 presents an illustrative example to show the applicability of the proposed model. Finally, section 6 presents the conclusion and future direction of research work. PRELIMINARIES In this section, we will give the basic concept of neutrosophic set, generalized neutrosophic set, soft set and neutrosophic soft set, generalized neutrosophic soft set. Definition of neutrosophic set [19, 20] Let U be a space of points (objects) with generic element in U denoted by u i.e. u ∈U. A neutrosophic set A in U is denoted by A= {< u: TA(u), IA(u), FA(u)> u U } where A T , A I , A F represent membership, indeterminacy and non-membership function respectively. A T , A I , A F are defined as follows: