NH-MADM Strategy in Neutrosophic Hesitant Fuzzy Set Environment Based on Extended GRA

Pranab BISWAS, Surapati PRAMANIK, Bibhas C. GIRI 1Department of Mathematics, Jadavpur University, Kolkata–700032, West Bengal, India 2Department of Mathematics, Nandalal Ghosh B.T College, Panpur, Narayanpur 743126, West Bengal, India 3Department of Mathematics, Jadavpur University, Kolkata–700032, West Bengal, India e-mail: prabiswas.jdvu@gmail.com, paldam2010@gmail.com, surapati.math@gmail.com, bcgiri.jumath@gmail.com

However, when decision makers are in doubt, they often hesitate to assign single value for rating alternatives, instead they prefer to assign a set of possible values.To deal with the issue, Torra (2010) introduced hesitant fuzzy set, which permits the membership degree of an element to a given set to be represented by the set of possible numerical values in [0,1].This set, an extension of fuzzy set, is useful for handling uncertain information in MADM process.Xia and Xu (2011) proposed some aggregation operators for hesitant fuzzy information and applied these operators to solve MADM.Wei (2012) studied hesitant fuzzy MADM by developing some prioritized aggregation operators for hesitant fuzzy information.Xu and Zhang (2013) developed TOPSIS method for hesitant fuzzy MADM with incomplete weight information.Li (2014) extended the MULTIMOORA method for multiple criteria group decision making with hesitant fuzzy sets.Mu et al. (2015) investigated a novel aggregation principle for hesitant fuzzy elements.
In a hesitant fuzzy MADM, decision maker does not consider non-membership degree for rating alternatives.However, this degree is equally important to express imprecise information.Zhu et al. (2012) presented the idea of dual hesitant fuzzy set, in which membership degrees and non-membership degrees are in the form of sets of values in [0, 1].Ye (2014c) and Chen et al. (2014) proposed correlation method of dual hesitant fuzzy sets to solve MADM with hesitant fuzzy information.Singh (2017) defined some distance and similarity measures for dual hesitant fuzzy set and utilized these measures in MADM.
Dual hesitant fuzzy set cannot properly capture indeterminate information in a decision making situation.Because of the inherent neutrosophic nature of human preferences, the rating values of alternatives and/or weights of attributes involved in the MADM problems are generally uncertain, imprecise, incomplete and inconsistent.Ye (2015b) introduced single-valued neutrosophic hesitant fuzzy set (SVNHFS) by coordinating hesitant fuzzy set and single-valued neutrosophic set.SVNHFS is characterized by truth hesitancy, indeterminacy hesitancy and falsity-hesitancy membership functions which are independent in nature.Therefore, SVNHFS can express the three kinds of hesitancy information existing in MADM problems.Ye (2015b) developed single valued neutrosophic hesitant fuzzy weighted averaging and single valued neutrosophic hesitant fuzzy weighted geometric operators for SVNHFS information and applied these two operators in MADM problems.Şahin and Liu (2017) defined correlation co-efficient between SVNHFSs and used to MADM.Biswas et al. (2016a) proposed GRA strategy for solving MADM in SVNHFS environment.Wang and Li (2018) proposed generalized SVNHF prioritized aggregation operators to solve MADM problem.Li and Zhang (2018) developed SVNHF based choquet aggregation operators for solving MADM problems.Liu and Shi (2015) introduced interval neutrosophic hesitant fuzzy set (INHFS) which consists of three membership hesitancy functions: the truth, the indeterminacy and the falsity.The three membership functions of an element to a given set are individually expressed by a set of interval values contained in [0, 1].Liu and Shi (2015) proposed hybrid weighted average operator for interval neutrosophic hesitant fuzzy set and utilized the operators in MADM.Ye (2016) put forward correlation coefficients of INHFSs and applied it to solve MADM problems.Biswas (2018) proposed an extended GRA strategy for solving MADM in neutrosophic hesitant fuzzy environment with incomplete weight information.We observe that the SVNHFS as well as INHFS can be considered in many practical MADM problems in which the decision maker has to make a decision in neutrosophic hesitant fuzzy environment.
Up until now very few number of studies exist in the literature about GRA strategy of MADM under SVNHFS and INHFS environment.Therefore, we have an opportunity to extend the traditional methods or to propose new methods regarding GRA to deal with MADM in neutrosophic hesitant fuzzy environment.
In this study, our objectives are as follows: 1. show the feasibility and effectiveness of the proposed strategies.
To do so, the paper is organized as follows: Section 2 presents some basic concepts related to single valued neutrosophic set, interval neutrosophic set, hesitant fuzzy set, SVNHFS and INHFS.In Section 3, we define score function, accuracy function, Hamming distance measure for SVNHFS and INHFS.We propose NH-MADM strategy in SVNHFS environment in Section 4 and INH-MADM strategy in INHFS environment in Section 5.In Section 6, we illustrate the proposed NH-MADM and INH-MADM strategies with two examples.Finally, in Section 7 we present some concluding remarks and future scope of research of the study.

Preliminaries
In this section, we review some basic definitions regarding single valued neutrosophic set, interval neutrosophic set, hesitant fuzzy set, SVNHFS and INHFS.Wang et al., 2010.)A single valued neutrosophic set A in a universe of discourse X = (x 1 , x 2 , . . ., x n ) is defined by

Neutrosophic Set
where the functions T A (x), I A (x) and F A (x), respectively, denote the truth, indeterminacy and falsity membership functions of x ∈ X to the set A, with the conditions Wang et al., 2005.)Let X be a non-empty finite set.Let D[0, 1] be the set of all closed sub intervals of the unit interval [0, 1].An interval neutrosophic set Ã in X is an object having the form: where T Ã : X → D[0, 1], I Ã : X → D[0, 1], F Ã : X → D[0, 1] with the condition 0 T Ã(x) + I Ã(x) + F Ã(x) 3 for any x ∈ X.The intervals T Ã(x), T Ã(x) and T Ã(x) denote, respectively, the degree of truth, the indeterminacy and the falsity membership degree of x to Ã. Then for each x ∈ X the lower and the upper limit points of closed intervals T Ã(x), I Ã(x) and F Ã(x) are denoted by T L Ã (x), T U Ã (x), I L Ã (x), I U Ã (x), T L Ã (x), T U Ã (x), respectively.Thus Ã can also be presented in the following form: 2.2.Hesitant Fuzzy Set D 3. (See Torra and Narukawa, 2009;Torra, 2010.)Let X be a universe of discourse.A hesitant fuzzy set on X is defined by where h F (x), referred to as the hesitant fuzzy element, is a set of some values in [0, 1] denoting the possible membership degree of the element x ∈ X to the set F .

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4. (See Chen et al., 2013.)Let X be a non-empty finite set.An interval hesitant fuzzy set on X is represented by where hE (x) is a set of some different interval values in [0, 1], which denotes the possible membership degrees of the element x ∈ X to the set E, hE (x) can be represented by an interval hesitant fuzzy element h that is denoted by { γ | γ ∈ h}, where γ = [γ L , γ U ] is an interval number.

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5. (See Ye, 2015b.)Let X be fixed set.Then a single valued neutrosophic hesitant fuzzy set n on X is defined as in which t (x), i(x) and f (x) represent three sets of some values in [0, 1], denoting, respectively, the possible truth, indeterminacy and falsity membership degrees of the element x ∈ X to the set N .The membership degrees t (x), i(x) and f (x) satisfy the following conditions: For convenience, the triplet n(x) = t (x), i(x), f (x) is denoted by n = t, i, f which we call single-valued neutrosophic hesitant fuzzy element (SVNHFE).

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6. (See Liu and Shi, 2015.)Let X be a non-empty finite set, an interval neutrosophic hesitant fuzzy set on X is represented by where } are three sets of some interval values in real unit interval [0, 1], which denotes the possible truth, indeterminacy and falsity membership hesitant degrees of the element x ∈ X to the set ñ.

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7. (See Biswas et al., 2016b.)Let n i = t i , i i , f i be an SVNHFE and l t , l i and l f are the number of elements in t i , i i , f i , respectively.Then the score function S(n i ), the accuracy function A(n i ) and certainty function C(n i ) of n i are defined as Then the following rules can be defined for comparison purposes: Then the score function S( ñi ), the accuracy function A( ñi ) and certainty function C( ñi ) of ñi are defined as follows: Biswas et al., 2016b.)Let n 1 = t 1 , i 1 , f 1 and n 2 = t 2 , i 2 , f 2 be any two SVNHFEs.Then the normalized Hamming distance between n 1 and n 2 is defined as where The distance function d(n 1 , n 2 ) of n 1 and n 2 satisfies the following properties: D 11.Let ñ1 = t1 , ĩ1 , f1 and ñ2 = t2 , ĩ2 , f2 be any two INHFEs.Then the normalized Hamming distance between ñ1 and ñ2 is defined as follows: where l tk , l ĩk and l fk are the number of possible membership values tk , ĩk , fk for k = 1, 2, respectively.
The distance function d( ñ1 , ñ2 ) of ñ1 and ñ2 satisfies the following properties:

GRA Strategy for MADM with SVNHFS
In this section, we propose GRA based strategy to find out the best alternative in MADM under SVNHFS environment.Assume that A = {A 1 , A 2 , . . ., A m } be the discrete set of m alternatives and C = {C 1 , C 2 , . . ., C n } be the set of n attributes for an SVNHFSs based MADM problem.Suppose that the rating value of the i-th alternative A i (i = 1, 2, . . ., m) over the attribute indicate the possible truth, indeterminacy and falsity membership degrees of the rating x ij for i = 1, 2, . . ., m and j = 1, 2, . . ., n.With these rating values, we construct a decision matrix X = (x ij ) m×n , where the entries assume the form of SVN-HFSs.The decision matrix is constructed as follows: ( We now propose MADM based on GRA to determine the most desirable alternative under the following cases: Case 1a.Completely known attribute weights. The weight vector of attributes prescribed by the decision maker is w = (w 1 , w 2 , . . ., w n ), where w j ∈ [0, 1] and n j =1 w j = 1.
Step 1. Determine the single valued neutrosophic hesitant fuzzy positive ideal solution (SVNHFPIS) A + and single valued neutrosophic hesitant fuzzy negative ideal solution (SVNHFNIS) A − of alternatives in the matrix X = (x ij ) m×n by the following equations, respectively.
• For benefit type attributes: • For cost type attributes: Using Definition 8, we compare attribute values x ij by using score, accuracy and certainty values of SVNHFEs.
Step 2. Determine the grey relational coefficient of each alternative from A + and A − by using the following equations: where the identification coefficient is generally set as ρ = 0.5.Step 3. Calculate the degree of grey relational coefficient of each alternative A i (i = 1, 2, . . ., m) from A + and A − by using Eq. ( 10) and Eq. ( 11), respectively, for i = 1, 2, . . ., m: Step 4. Calculate the relative closeness coefficient ξ i for each alternative A i (i = 1, 2, . . ., m) with respect to A + by the following equation: Step 5. Rank the alternatives according to the descending order of relative closeness coefficient values of alternatives to determine the best one.
However, in a real decision making, the information about the attribute weights is often incompletely known or completely unknown due to decision makers' limited expertise about the public domain.In this case, we propose some models for determining the weight vector of the attributes under the following cases:

Case 2a. Incompletely known attribute weights.
In this case, we have to determine the attribute weights to find out the best alternative.
We can take the weights of attributes as a subset of the above relationships for a particular decision making problem.We denote the considered weight information set by H .The similarity measure of x ij to A + j is defined as follows: where D(x ij , A + j ) (i = 1, 2, . . ., m) denotes the Hamming distance between x ij to A + j .Similarly, the similarity measure of x ij to A − j is determined as follows: where D(x ij , A − j ) (i = 1, 2, . . ., m) denotes the Hamming distance between x ij to A − j .Then the weighted similarity measures of the alternative A i (i = 1, 2, . . ., m) to are respectively determined as follows: An acceptable weight vector w = (w 1 , w 2 , . . ., w n ) should make all the similarities (S + 1 , S + 2 , . . ., S + m ) to ideal solution (A + i1 , A + i2 , . . ., A + in ) as large as possible and the similarities as small as possible under the condition w ∈ H . Therefore, we can set the following multiple objective non-linear optimization model to determine the weight vector: Since each alternative is non-inferior, there exists no preference relation among the alternatives.Then we can aggregate the above multiple objective optimization models with equal weights into the following single objective optimization model: , By solving the Model-2, we obtain the optimal solution w = (w 1 , w 2 , . . ., w n ) that can be used as the weight vector of the attributes.Using this weight vector and following Step 3 to Step 5, we can easily determine the relative closeness coefficient S i (i = 1, 2, . . ., m) of each alternative to find out an optimal alternative.

Case 3a. Completely unknown attribute weights.
Here we construct the following non-linear programming model to determine the weights of attributes. Model-3 We now aggregate the multiple objective optimization models (see Eq. ( 21)) to obtain the following single-objective optimization model: To solve the Model-4, we consider the following Lagrange function: where the real number φ is called the Lagrange multiplier.Differentiating partially L with respect to w j and φ, we have the following equations: It follows from Eq. ( 24) that Putting this value of w j in Eq. ( 25), we have where φ < 0 and 2 implies the sum of similarities with respect to the j -th attribute.Combining Eq. ( 26) and Eq. ( 27), we obtain Then, by normalizing w j (j = 1, 2, . . ., n), we make their sum into a unit and obtain the normalized weight of the j -th attribute: and consequently, we obtain the weight vector of the attribute as w = ( w1 , w2 , . . ., wn ).

Following Step 3 to
Step 5, presented in the current section, we can easily determine the best alternative by using this weight vector.

GRA Strategy for MADM with INHFS
We consider a MADM problem in which A = {A 1 , A 2 , . . ., A m } is the set of alternatives and C = {C 1 , C 2 , . . ., C n } is the set of attributes.We assume that the rating value of the alternative A i (i = 1, 2, . . ., m) over the attribute where γ + = γ ∈t(x) max{γ U ij }, δ+ = δ∈ ĩ(x) max{δ U ij }, and η+ = η∈ f (x) max{η U ij }.Then we construct an interval neutrosophic hesitant fuzzy decision matrix N as We solve the interval neutrosophic hesitant fuzzy MADM considering the following two cases:

Case 1b. Completely known attribute.
Assume that = (λ 1 , λ 2 , . . ., λ n ) be the weight vector such that λ j ∈ [0, 1] and n j =1 λ j = 1.We now consider the following steps required for the proposed strategy: Step 1. Determine the interval neutrosophic hesitant fuzzy positive ideal solution (INHFPIS) Ã+ and interval neutrosophic hesitant fuzzy negative ideal solution (INHFNIS) Ã− of alternatives from the decision matrix N = ( ñij ) m×n with the equations: • For benefit type attributes: • For cost type attributes: We compare attribute values ñij by using score, accuracy and certainty values of INHFSs defined in Definition 9.
Step 2. Calculate the grey relational coefficient of each alternative from Ã+ and Ã− with the equations: where the identification coefficient β = 0.5.
Step 3. Calculate the degree of grey relational coefficient of each alternative A i (i = 1, 2, . . ., m) from Ã+ and Ã− by using Eq. ( 10) and Eq. ( 11), respectively: Step 4. Determine the relative closeness coefficient χ i for each alternative A i with respect to the positive ideal solution Ã+ : Step 5. Rank the alternatives according to the descending order of relative closeness coefficient values of alternatives and choose the best alternative.

Case 2b. Incompletely known attribute weights.
If the information about the attribute weights is incomplete, then we can follow the same procedures discussed in Case 2a of Section 4. Then we determine the similarity measure between ñij and Ã+ j as where d( ñij , Ã+ j ) (i = 1, 2, . . ., m) denotes the Hamming distance between ñij and Ã+ j .Similarly, we obtain the similarity measure between ñij and Ã− j as where d( ñij , Ã− j ) (i = 1, 2, . . ., m) denotes the Hamming distance between ñij to Ã− j .The weighted similarity measures of the alternative are determined as follows: We find a suitable weight vector = (λ 1 , λ 2 , . . ., λ n ) that makes all the similarities (s + 1 , s + 2 , . . ., s + m ) to ideal solution ( Ã+ 1 , Ã+ 2 , . . ., Ã+ n ) as large as possible and the sim- as small as possible with the condition ∈ H . Therefore, we can set the following multiple objective non-linear optimization model to determine the weight vector: Since each alternative is non-inferior, then we construct the following single objective optimization model by aggregating the above multiple objective optimization models with equal weights: , Solving the Model-6, we have the optimal solution = (λ 1 , λ 2 , . . ., λ n ) that can be used as the weight vector of the attributes.Then we follow Step 3 to Step 5 to determine an optimal alternative.

Case 3b. Completely unknown attribute weights.
In this case we develop the following non-linear programming model to determine the attribute weights.
Similarly, we can aggregate all above multiple objective optimization model: To solve the Model-4, we consider the following Lagrange function: where the real number ψ is the Lagrange multiplier.Then differentiating partially L with respect to λ j and ψ, we have It follows from equation ( 49) that Putting this value of λ j in equation ( 50), we have where, ψ < 0 and 2 implies the sum of similarities with respect to the j -th attribute.Combining Eqs. ( 26) and ( 27), we obtain To normalize λ j (j = 1, 2, . . ., n), we make their sum into a unit and obtain the normalized weight of the j -th attribute: Consequently, we obtain the weight vector of the attribute as ¯ = ( λ1 , λ2 , . . ., λn ).With this weight vector, we follow Step 3 to Step 5 to find out an optimal alternative.We briefly present the steps of the proposed strategies in Fig. 1.

Illustrative Examples
In this section, we consider two examples: one for MADM with single valued neutrosophic hesitant fuzzy sets and other for MADM with interval neutrosophic hesitant fuzzy sets.

Example 1
We consider an example, adapted from Ye (2015b), to illustrate the applicability of the extended GRA method for MADM with incomplete information.Assume that an investment company wants to invest a sum of money in the following four possible alternatives (companies): • the car company (A 1 ), • the food company (A 2 ), • the computer company (A 3 ), • the arms company (A 4 ).
While making a decision, the company considers the following attributes: • the risk analysis (C 1 ), • the growth analysis (C 2 ), • the environmental impact analysis (C 3 ).
We assume that the rating values of the alternatives A i (i = 1, 2, 3, 4) are represented by neutrosophic hesitant fuzzy decision matrix R ij = (r ij ) 4×3 (see Table 1).The information of the attribute weights is incompletely known and the known weight information is given as follows: Step 1.Using Eqs. ( 6) and ( 9), we determine A + and A − by the equations:  Step 2. We assume d i (j ) as the deviation sequence of A + and then the sequence of the rating values of the alternative A i is considered as follows: We calculate the deviation sequences of alternatives A i (i = 1, 2, 3, 4) from A + in Table 2.
Similarly, we obtain the deviation sequences of alternatives A i (i = 1, 2, 3, 4) from A − in Table 3.
Pairwise comparison of relative closeness coefficient of each alternative is presented in Fig. 2.

Example 2
In this section, we consider the same problem presented in Example 1, but we assume that the rating values of the alternatives A i (i = 1, 2, 3, 4) are expressed by interval neutrosophic hesitant fuzzy sets.The decision matrix U ij = (u ij ) 4×3 are presented in Table 4.
The information of the attribute weights is partially known and the known weight information is given as follows: = 0.30 w 1 0.40, 0.20 w 2 0.30, 0.35 w 3 0.45, We now consider the following steps to the problem.
Step 1. Similar to Example 1, we can determine positive ideal solution Ã+ and the negative ideal solution Ã− by Eq. ( 31) and Eq. ( 32), respectively: Step 2. The deviation sequences of alternatives from Ã+ and Ã− are presented Tables 5  and 6, respectively.

Advantages of the Proposed Strategy
We now point out the advantages of our proposed strategy compared with the other existing strategies.First, we compare our proposed strategy with the Ye's strategy (Ye, 2015b).We have considered the same MADM problem under SVNHF environment and obtained similar ranking result.The strategy (Ye, 2015b) considers two aggregation operators to solve MADM problem with known weight information of the attributes.Similarly, the strategies (Liu et al., 2016;Peng et al., 2015;Wang and Li, 2018;Li and Zhang, 2018) offer aggregation operator to solve MADM under SVNFS environment.All these strategies are suitable when the weight information of attributes is known in advance.Then we consider Liu and Shi's strategy (Liu and Shi, 2015) which deals with two aggregation operator for solving MAGDM under INHFS environment, where the weights of attributes are completely known.On the other hand, our proposed strategies are comprehensive: they cover all the three cases, namely completely known, incompletely known, and completely unknown weight information of attributes.Therefore, our proposed approaches have a clear advantage over the existing strategy.
The main advantages of the paper are presented as follows: 1.The proposed strategy is more convenient as the preference values of the alternatives are considered by either SVNHFSs or INHFSs.2. The proposed strategy is flexible because it can give the decision makers more choices for choosing the importance of attribute weights.3. The proposed strategy is presented in a logical way to make it simple and understandable.4. The proposed strategy does not consider any complex aggregation operators, it only considers relative closeness coefficient obtained from GRA to rank the alternatives.5.There is no information loss occurred in the proposed strategy due to transformation of neutrosophic hesitant fuzzy set based attribute values into crisp value or interval.

Concluding Remarks
Neutrosophic hesitant fuzzy set is a powerful mathematical tool for dealing with imprecise, indeterminate, and incomplete information existing in real MADM process.In this paper, we have extended grey relational analysis strategy for solving MADM in singlevalued neutrosophic hesitant fuzzy environment.We have developed an optimization model to determine the weights of attributes, in which the weight information is incompletely known.We have also developed another model for determining attributes' weights where the information is completely unknown.Then we have ranked the alternative, based on the proposed NH-MADM strategy.Further, we have extended the NH-MADM strategies to interval neutrosophic hesitant fuzzy environment.Finally, we have provided two illustrative examples, one for the case of SVNHFSs and other, for the case of IVNHFSs to show the validity and effectiveness of the proposed strategies.We hope that the proposed strategies can be applied in many real applications, where the information is neutrosophic hesitant in nature.
are three sets of some interval values in the real unit interval [0, 1].These values denote the possible truth, indeterminacy and falsity membership hesitant degrees with the following limits:

Fig. 1 .
Fig. 1.The schematic diagram of the proposed strategy.

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Fig. 3. Pairwise comparison of each alternative.
To propose GRA based strategy for handling MADM problem under SVNHFS and INHFS environments.4. Finally, to present illustrative examples, one for SVNHFS and other for IVNHFs to To present the idea of MADM problem in SVNHFS and INHFS environments, where the preference values of alternatives are considered with either SVNHFSs or INHFSs and the weight information of attributes are assumed to be completely known, incompletely known, and completely unknown.2. To develop some optimization models for determining weight information of attributes, when attributes' weights are incompletely known, or completely unknown.3.

Table 1
Single valued neutrosophic hesitant fuzzy decision matrix.

Table 2
Distance of alternatives from SVNHFPIS.

Table 5
Distance of alternatives from INHFPIS.