International Journal of Latest Engineering Research and Applications (IJLERA) ISSN: 2455-7137 Volume – 02, Issue – 07, July – 2017, PP – 41-45
Vikor Method for Decision Making Problem Using Octagonal Neutrosophic Soft Matrix K. Selvakumari1, M. Ajitha Priyadharshini2 1
Assistant Professor, Department of Mathematics, Vels University, Chennai, Tamil Nadu 2 Research Scholar, Department of Mathematics, Vels University, Chennai, Tamil Nadu
Abstract: The scope of this paper is to introduce a new concept of buying a car on the basis of choice parameter by using vikor method. Here octagonal fuzzy number plays a vital role for solving many decision making problem involving uncertainty. In this paper, the defuzzification is done using the area of region, the fuzzified value can be converted into neutrosophic soft matrix and vikor method is applied to the defuzzified data to find the best solution. Finally, an illustrative example gives the effectiveness and feasibility of the proposed approach. Key words: Fuzzy soft set, Neutrosophic soft set, Octagonal fuzzy number, Vikor method.
1. Introduction The abstract idea of fuzzy set was given by L.A. Zadeh[21] in 1965. On fuzzy soft sets, Advances in fuzzy system was given by B. Ahmed and A. Kharal[1] (2009). The neutrosophic set was developed by F. Smarandache[19] it applied in the field of decision making problems in mass scale of vague, contingency and incompative data. Neutrosophic is related to actual membership function, unstable membership function and not actual membership function. Molodtsov[13] put many research in the field of soft set theory in the application of computer and mathematics in 1999. The octagonal fuzzy numbers are proposed by Malini.S.U and Kennedy Felbin.C[12] in 2013. The octagonal fuzzy number is defuzzified into singleton (crisps) value by area of region. The vikor method is proposed by Opricovis.S and Tzeng[16], in 2007 the extended vikor method in comparison with outranking methods. Compromise solution by multi criteria decision making methods and a comparative analysis of vikor method is developed by Opricovis.S and Tzeng[17], in 2004. The Vikor algorithm is remarkably successful method for solving the uncertainty. The importance feature of the proposed pattern is its level of commiserate are measured by a fixed remark and other possibilities. This gives a successful in producing a desired or intended result.
2. Preliminaries 2.1 Definition: (Fuzzy Soft Set)[1,2] Let U be an initial universal set and E be a set of parameters. Let A ⊆ E. A pair (đ??šđ??´ , E) is called a fuzzy soft set over U. Where đ??šđ??´ is a mapping given by đ??šđ??´ : E → đ??ź đ?‘ˆ . Where đ??ź đ?‘ˆ denotes the collection of fuzzy subsets U. 2.2 Definition: (Fuzzy Soft Matrices)[4,5] Let (đ??šđ??´ , E) be fuzzy soft set over U. Then a subset of UxE is uniquely defined by đ?‘…đ??´ = {(u,e): e∈ đ??´, u∈ đ??šđ??´ (đ?‘’)} which is called relation form of (đ??šđ??´ , E). The characteristic function of đ?‘…đ??´ is written by, đ?œ‡đ?‘…đ??´ : UxE→[0,1]. Where đ?œ‡đ?‘…đ??´ is the membership value of u∈U. đ?œ‡11 â‹Ż đ?œ‡1đ?‘› â‹ą â‹Ž If đ?œ‡đ?‘–đ?‘— = đ?œ‡đ?‘…đ??´ (đ?‘˘đ?‘– , đ?‘’đ?‘— ). [đ?œ‡đ?‘–đ?‘— ]đ?‘š Ă—đ?‘› = â‹Ž đ?œ‡đ?‘š 1 â‹Ż đ?œ‡đ?‘šđ?‘› mĂ—n is soft matrix of the soft set (đ??šđ??´ , E) over U. 2.3 Definition: (Neutrosophic Soft Set)[3] Let U = đ?‘“1 , đ?‘“2 , . . , đ?‘“đ?‘š be the universal set and E be the set of parameters given by E= đ?‘’1 , đ?‘’2 , . . , đ?‘’đ?‘š . Let A ⊆ E a pair (F,A) be a fuzzy neutrosophic soft set. Then the fuzzy neutrosophic soft set (F,A) in a matrix form as A = đ?‘Žđ?‘–đ?‘— where i=1,2,..,m and j=1,2,..,n. Let đ?‘‡đ?‘–đ?‘— , đ??źđ?‘–đ?‘— , đ??šđ?‘–đ?‘— be the neutrosophic soft matrix. We can define a matrix in the form
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