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.

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.[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:

Definition of neutrosophic set
Where TA(u), IA(u), FA(u) are the real standard and non-standard subset of]  0, 1 + [ such that  0 ≤ TA(u)+IA(u)+FA(u) ≤ 3 + Since, T A (u), I A (u), F A (u) assume the values from the subset of]  0, 1 + [, so we take [0, 1] instead of]  0, 1 + [ due to the application in real life situation because]  0, 1 + [will be complicated to apply the real life problem with neutrosophic nature.

Definition: Single valued neutrosophic set [24]
Let U be a space of points with generic element in U denoted by u i. e. uU.A single valued neutrosophic set G in U is characterized by a truth-membership function TG(u), an indeterminacy-membership function IG (u) and a falsity-membership function FG(u), for each point u in U, TG(u), IG (u), FG(u)[0, 1], when U is continuous then single-valued neutrosophic set G can be written as When U is discrete, single-valued neutrosophic set can be written as ∑ <   (  ),   (  ),   (  ) >/   =1 ,    Complement of neutrosophic set [19,20] The complement of a neutrosophic set A is denoted by Aand defined as A= {<u: T A (u), I A (u), F A (u)>, u ∈U} TA(u) ={1 + } -TA(u) IA(u) =1 + } -IA(u)  A neutrosophic set M is contained in another neutrosophic set L i.e.M  L if for all a ∈ U, TM(u) ≤ TL(u) IM(u) ≤ I(u) And FM(u) ≤ FL(u) Definition: 2.5 Generalized neutrosophic set [75] Let U be a space of point with generic element in U denoted by u.Let M be a neutrosophic set in U denoted by M = <u, TM(u), IM(u), FM(u), uU> is said to be generalized neutrosophic set if Where TM(u), IM(u), FM(u) represent degree of membership function, indeterminacy function and non-membership function respectively.Definition: 2.6 Soft set [3] Let U be an initial universe set P is the set of parameters.Let B be non-empty subset of P i.e.B⊂P.Let ℙ (U) be the power set of U. Then the order pair (S, B) is called soft set over U, where S is the mapping from B to ℙ(U) i. e. S: B⊂P → ℙ (U) Definition: 2.7 Neutrosophic soft set [15] Let U be the universe set and N(U) denote the set of all neutrosophic subset of U. Let P be the set of all parameter and B is the non-empty sub-set of P i.e.B⊂P, then the order pair (S, B) is said to be neutrosophic soft set if S: B → N(U).Definition: 2.8 Generalized neutrosophic soft set [17] Let U be an initial universe set and N(U) denote the set of all neutrosophic subset of U, Let P be the set of parameters and B be the non-empty subset of P i.e.B⊂P then the order pair (S, µ) is said to be generalized neutrosophic soft set over U if S: P → N(U) µ: P→ [0, 1] = I i.e. µ is the fuzzy set Combining this two mapping we represent generalized neutrosophic soft set as S µ : P → N (U) I S µ ={(S(p1), µ(p1)): p1  P, S(p1)N(U), µ(p1) [0,1] = I} For each p1 P  , S (p1) denotes the neutrosophic value of the parameter p1.S(p1) is presented as follows: S(p1) = <u, TS(p 1 ) (u) , IS(p 1 ) (u)   , FS(p 1 ) (u) , u U > where T, I, F: U → [0, 1] Where T, I, F are the truth, indeterminacy and falsity membership function of the element u U such that 0 ≤ TS(p)(u) + IS(p)(u) +FS(p)(u) ≤ 3. Here, S µ is the parameterized family of neutrosophic sets over U, which has the degree of possibility of the approximate value set, which is denoted by µ(p) for any parameter p Є P. Here, µ(p) also represents the importance of parameter p.The importance of the parameter p is provided by the decision maker.So S µ can be defined as follows:

Example:
Consider a generalized neutrosophic soft set S µ , where U is the set of location.We select a location for logistic center on the basis of the parameters (P), namely cost, distance to suppliers, distance to customers, conformance to governmental regulation and law, quality of service and environmental impacts i.e.U= {l1, l2, l3, l4} and P = {p1, p2, p3, p4, p5, p6} p1 (stand for) = Cost p2 (stand for) = Distance to suppliers p3 (stand for) = Distance to customers p4 (stand for) = Conformance to governmental regulation and law p5 (stand for) = Quality of service   We express the above generalized neutrosophic soft set in matrix form as follows: The above matrix has been constructed only for one generalized neutrosophic soft set i.e. for only one decision maker.If the problem consists of D decision makers and L locations/ objects and each location has p parameters, then we can obtain D no. of generalized neutrosophic soft set i.e.D number of matrix having p number of rows and L+1 number of columns.Last column of the matrix represents the degree of possibility of each parameter to the decision makers.Definition: 2.9 Null or empty generalized neutrosophic soft set [17] A generalized neutrosophic soft set S µ over U is said to be a null generalized neutrosophic soft set if µ (p) = 0 and S (p) =<u; T S(p) (u), I S(p) (u) , F S(p) (u), u U>=<u, 0, 0, 0; u U> i.e.T S(p) (u) = 0, I S(p) (u) = 0 , F S(p) (u) = 0 p P = parameter and  u U.Definition: 2.11Generalized neutrosophic soft subset [18] Let M µ and N η be two generalized neutrosophic soft set over U.
M µ is said to be generalized neutrosophic soft subset of N η if µ  η and M is the neutrosophic subset of N.
i.e.TM(u) ≤ TN(u), IM(u) ≤ IN(u) and FM(u) ≥ FN(u) for any u U and for any p P denoted by M µ  N η .
Again if N η ⊆ M µ then M µ = N η Definition: 2.12 Complement of generalized neutrosophic soft set [17] Complement of a generalized neutrosophic soft set S µ over U is denoted by S µ' and defined by [17] Suppose M  [17] Assume that M  1 and M  2 be two generalized neutrosophic soft set over the same universe U.The intersection of two sets denoted by M  A linguistic variable refers to a variable whose values are represented by words or sentences in natural or artificial languages.Importance of the decision makers in the decision making process may not be equal.It can be expressed using linguistic variables such as very important, important, medium important, unimportant, very unimportant, etc.We have presented a conversion method between linguistic variables and single valued neutrosophic number (see the Table-1).

GREY-RELATIONAL ANALYSIS [57]
We now present the process for finding the grey relational co-efficient to ranking the alternatives according the largest degree of grey relation coefficient.Let Y0 be the referential sequence and Yi be the comparative sequence at point t.Then grey relation co-efficient  (Y0(t), Yi(t)) satisfies the four conditions approaching larger then  reduces to smaller.The grey relational co-efficient [57] of the referential sequences and comparative sequence at point t, can be expressed as follows:  [0, 1] refers to the distinguishable co-efficient used to adjust the range of the comparison environmental and to control level of differences of the relation co-efficient.When  = 0 comparison environment disappears and when 1   , the compassion environment is unaltered.Generally,  =.5 is considered for decision making environment.
The weights of the decision makers are unknown but the weights of the parameters are known from definition of generalized neutrosophic soft set.The ratings of the alternatives and importance of the choice-parameters are provided by the decision makers in the form of generalized neutrosophic soft sets.The steps for solving MAGDM by proposed approach have been presented below.
Step  m).Therefore, the decision matrix of k-th decision maker can be represented as follows: Here a

Step: 2 Determination of the weight of the decision makers
In the group decision making process the weights of the decision makers are very crucial for decision making [58].Assume that the group decision making unit consists of n decision makers.The importance of the decision makers in the group decision making process may not be equal.The importance of the decision makers may be expressed as linguistic variables and the linguistic variables can be converted into single valued neutrosophic numbers (see table 1).Assume that q D = ( q  , q  , q  ) be a single valued neutrosophic number that represents the rating of the q-th decision maker.Then the weight of the q-th decision maker [76] can be presented as follows: q q q q q q q q q q (3) This expression is the extension of the work of Boran et al. [77] in intuitionistic fuzzy number.If we consider the importance of the all decision makers is same, then the weight of the decision makers will be (1/n).

Step: 3 Aggregation of the weights of the parameters
The importance of parameter depends on decision maker's choice.In this paper, we have defined generalized neutrosophic soft weighted aggregate operator for aggregation of the weights of the parameters as follows: Step:

Construction of the aggregated generalized neutrosophic soft decision matrix
In the group decision making situation, all the individual assessments require to be combined into a group opinion based on neutrosophic soft weighted average operator.Let A be the aggregate decision matrix, then A has been defined as follows: Step:

Determination of the reference sequence based on generalized neutrosophic soft set
Let a , where i = 1, 2, 3, ….., r; Reference sequence should be characterized by the optimal sequence of the criteria values.1, 0, 0 are the values of the aspired levels of the membership function, indeterminacy functions, falsity (non-membership) function, respectively.Therefore, the point consisting of highest membership value, minimum indeterminacy, minimum falsity (non-membership) value would represent the reference value or ideal point or utopia point.For generalized neutrosophic soft decision matrix the maximum value a  i = (1, 0, 0) can be used as the reference value, then the reference sequence can be represented as follows:

Step: 6 Calculation of the grey relational coefficient
The calculation of the grey relational coefficient for each alternative can be defined as follows: ij  is the grey relational coefficient and ρ ∈ [0, 1] is the distinguishing coefficient.

Step: 7 Calculation of degree of grey relational coefficient
We calculate the degree of the grey relation coefficient of each alternative using grey relational coefficient and aggregate parameter weights by the equation (9).
Step: 8 Ranking all the alternatives We arrange all alternatives according to their degree of grey relational coefficient and the best alternative corresponds to the greatest degree of grey relational coefficient.

ILLUSTRATIVE EXAMPLE
Suppose that a new modern logistic center is required in a town.There are four locations L1, L2, L3, L4.A committee of four decision makers or experts, namely, is formed to select the most appropriate location on the basis of six parameters are adapted from the study [78], namely, cost (P1), distance to suppliers (P2), distance to customers (P3), conformance to government regulation and law (P4), quality of service (P5) and environmental impact (P 6 ) are considered for selecting parameters.Since, there are four decision makers we obtained four generalized neutrosophic soft set i.e.M .Let U be the set of locations i.e.U= {L1, L2, L3, L4} and P is the set of parameters i.e.P = {P1, P2, P3, P4, P5, P6}.The four generalized neutrosophic soft sets in matrix form for four decision makers are given bellow respectively.Step: 1 Formation of generalized neutrosophic soft matrix The matrix form of above four generalized neutrosophic soft set in the form of (2) defined above have been presented as follows. A1=

Step: 2 Determination of the weight of the decision makers
The weights of the decision makers have presented in the Table 2. Step:4 Construction of the aggregated generalized neutrosophic soft matrix Using generalized neutrosophic soft weighted average operator given by the equation ( 6), (7), the aggregated matrix can be constructed as follows:

Step: 5 Determination of the reference sequence
The reference sequence based on generalized neutrosophic soft set can be constructed as follows:  The grey relational coefficient matrix has been obtained for 5 .0   using the table 2 as follows:

Step: 7 Calculation of the degree of grey relational coefficient
The calculation of the degree of grey relational coefficient using the equation ( 9) has been performed as follows: 4 , Step: 8 Ranking the alternatives Arrange the alternative according to the degree of grey relational coefficient ( ) j  (j=1, 2, 3, 4) in descending order.Greater value of j  implies the better alternative Lj.

Determination of ranking order when equal weights of the decision makers are considered
We present the ranking of logistics center location when weights of the decision makers are equal.The first two steps are same as above.Equal weights of the decision makers imply Step: 3Aggregated weights of the parameters Using the equation ( 4), aggregated weights of the parameters have been obtained as follows: ) p ( λ 1 = .Step: 4 Construction of the aggregated generalized neutrosophic soft matrix Using generalized neutrosophic soft weighted average operator given by the equation ( 6), (7), the aggregated matrix in the form of ( 5) can be formed as:   The grey relational coefficient matrix has been constructed for 5 .0   , using the table 2 as follows: Step: 7 Calculation of the degree of grey relational coefficient The calculation of the degree of grey relational coefficient using the above equation ( 9) has been performed as follows:  Here we have obtained  The ranking order for unequal weights of the decision makers is L3 > L2 > L1 > L4. ii.
The ranking order for the equal weight of the decision makers is L3 > L2 > L4 > L1.The ranks of the first two location centers i.e.L3 > L2 remain the same.But the ranks of and L4 change due to weights factor of decision makers.Therefore, the ranking order depends on the weights of the decision maker.

: 1
Formation of generalized neutrosophic soft decision matrixSelection of key parameters is regarded as one of the important issue in a MAGDM problem.The key parameters are generally provided by the evaluator.Assume that the rating of alternative Li (i = 1, 2, …, m)with respect to the parameter pi (i =1, 2, …, r) provided by the k-th(k = 1, 2, …, n) DM is represented by GNSSs (M k k  ) (k = 1,2, …, n) and they can be presented in the matrix form < a =1,2, …, n; i =1, 2, ..., r; j =1, 2, …, Global Journal of Engineering Science and Research Management http: // www.gjesrm.com© Global Journal of Engineering Science and Research Management [162]

Here
the four locations are as follows: L3 > L2 > L1 > L4 Therefore, L 3 is the best logistic center.

8
Ranking the alternativesArrange the alternative according to the degree of grey relational coefficient ( ) j  (j=1, 2, 3, 4) in descending order.Greater value of j  implies the better alternative Lj.

1 .
the locations are as follows: L3 > L2 > L4 > L1 Therefore, L3 is the best logistic center.Note Comparison of ranking order with weights factors of decision makers.
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S µ (p 1 ) =
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S µ (p 2 ) =
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Table 1 . Conversion between linguistic variables and single valued neutrosophic numbers
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Table 3 :
Calculation of min ijGlobal Journal of Engineering Science and Research Management http: // www.gjesrm.com© Global Journal of Engineering Science and Research Management [164] Global Journal of Engineering Science and Research Management http: // www.gjesrm.com© Global Journal of Engineering Science and Research Management [165]