Interval neutrosophic hesitant fuzzy choquet integral in multicriteria decision making

. We deﬁne the Interval Neutrosophic Hesitant Fuzzy Choquet Integral (INHFCI) operator as a useful tool for multicriteria decision making (MCDM). The INHFCI operator generalizes both the interval neutrosophic hesitant fuzzy ordered weighted averaging operator and the interval neutrosophic hesitant fuzzy weighted averaging operator. A modiﬁed version of the score function to make comparison among Interval Neutrosophic Hesitant Fuzzy elements is proposed. We develop an approach for multicriteria decision making based on the interval neutrosophic hesitant fuzzy choquet integral operator that applies to our proposed score function. Finally the model is illustrated with the help of an example.


Introduction
In decision making process under multiple criteria, governing information are often incomplete, indeterminate and inconsistent. To deal with such imprecise information, fuzzy set [1] was introduced by Zadeh in 1965. A fuzzy set is characterized by a membership function which represents the degree of acceptance in a decision making problem. A fuzzy set, thus, converts the impreciseness or vagueness by attributing a degree to which a certain object belongs to a set. In real situation, however, there may be a hesitation or uncertainty about the membership degree of the object in that set. So, as its consequence, Atanassov [2,3] introduced the intuitionistic fuzzy sets (IFSs) in 1983 that is characterized by the degrees of membership and non-membership with the condition that sum of these two degrees * Corresponding author. Saifur Rahman, Department of Mathematics, Rajiv Gandhi University, Itanagar-791112, India. E-mail: saifur.rahman@rgu.ac.in.
should not exceed 1. In the case of IFS, the nonmembership grade expresses the degree of rejection in a decision making problem. Later, Atanassov and Gargov [4] introduced interval valued intuitionistic fuzzy set (IVIFS) as a further generalization of IFS in which intervals in [0,1] are used for membership and non-membership values rather than exact numerical values. Although, IFSs and IVIFSs have the ability to handle incomplete information like acceptance and non-acceptance, the issue of indeterminate and inconsistent information remains in paucity. To overcome this, Smarandache [5] introduced neutrosophic sets (NSs). A neutrosophic set generalizes the concept of a fuzzy set [1], intuitionistic fuzzy set [2], interval valued intuitionistic fuzzy set [4], paraconsistent set [5], dialetheist set [5], paradoxist set [5], tautological set [5] to name a few. In the neutrosophic set, indeterminacy is quantified explicitly, and truth, indeterminacy, and falsity memberships are expressed independently. Wang et al. [11,12] proposed the concepts of a single-valued neutrosophic set (SVNS) and an interval neutrosophic set (INS), which are the subclasses of a neutrosophic set. Ye [13] proposed a single-valued neutrosophic cross-entropy measure and used it to multicriteria decision making(MCDM) problems. It is observed that, in fuzzy multicriteria decision making (MCDM) problems, representation of membership degrees of objects to a certain set is not unique. To deal with such type of difficulty, Torra and Narukawa [14] and Torra [15] defined a hesitant fuzzy set (HFS) as an extension of the fuzzy set (FS). Chen et al. [16] extended this to include interval valued HFS (IVHFS) in which the membership degrees of an element to a given set are not exactly defined but denoted by several possible interval values. Ye [17] defined the single valued neutrosophic hesitant fuzzy sets (SVNHFS) by combining the single valued neutrosophic set with the hesitant fuzzy set and developed some weighted averaging and weighted geometric operators for SVNHFS. Further, Liu and Shi [22] proposed the concept of interval neutrosophic hesitant fuzzy sets (INHFSs) by combining the IVHFS and INS, and then developed interval neutrosophic hesitant fuzzy generalized weighted average (INHFGWA) operator, interval neutrosophic hesitant fuzzy generalized ordered weighted average (INHFGOWA) operator and an interval neutrosophic hesitant fuzzy generalized hybrid weighted average (INHFGHWA) operator. Most of these aggregation operators however specify to situations, where criteria and preferences of decision makers are independent of one another, therefore their combined effect is additive in nature. However, in real life decision-making problems, the criteria of the problems are often interdependent or interactive. Choquet integrals [23] have been used as an aggregation mechanism in various MCDM problems involving ordinary fuzzy sets in order to describe the relative importance of decision criteria and their interactions.
In the present study, we introduce the interval neutrosophic hesitant fuzzy choquet integral (INHFCI) operator as a tool for multicriteria decision making (MCDM), and discuss the relevant properties. It is shown that the interval neutrosophic hesitant fuzzy choquet integral (INHFCI) operator generalizes the interval neutrosophic hesitant fuzzy OWA operator, and the interval neutrosophic hesitant fuzzy weighted averaging operator. An approach for multicriteria decision making is also developed based on the interval neutrosophic hesitant fuzzy choquet integral operator.
The rest of the paper proceeds as follows. Section 2 deals with the preliminarily ideas of the model formulation, Section 3 describes an ordering approach for the INHFEs. Section 4 discusses the notion of interval neutrosophic hesitant fuzzy choquet operator. Section 5 discusses some of the properties of the interval neutrosophic hesitant fuzzy choquet operator. Section 6 proposes a multicriteria decision making approach based on the interval neutrosophic hesitant fuzzy choquet operator. Section 7 illustrates the model with an example followed by the concluding remarks in Section 8.

Preliminaries
We compile in this section the relevant notion required for the development of the present paper.
Definition 1. [8,10] , then some of their basic operations are as follows: Then the degree of possibility ofã b is denoted by p(ã b ) and is defined by Suppose that there are n interval numbersã i = [a L i , a U i ] (i = 1, 2, . . . , n), then each interval numberã i is compared to all interval numbersã j (i = 1, 2, . . . , n) by using Equation (2.1), as Then a complementary matrix can be constructed as follows: where p ij 0, p ij + p ji = 1, p ii = 0.5. A neutrosophic set [5] characterizes each logical statement in a three dimensional space, where each dimension represents respectively the truth (T ), the falsehood (F ), and the indeterminacy (I) of the statement under consideration with not necessarily any connection between T , F and I. Suppose x is a generic element of a set X, then x belongs to the set in the following way: it is t true in the set, i indeterminate in the set, and f false, where t, i, and f are real numbers taken from the subsets T , I, and F of R with no restriction on T , I, F , nor on their sum n = t + i + f . But it is difficult to apply neutrosophic set in practical problems without specifying T , I and F . In the following we give a formal definition of a neutrosophic set taking the subsets T , I and F as the unit intervals [0, 1].

Definition 3.
A neutrosophic set A in X can be characterized by a truth-membership function T A : X → [0, 1], an indeterminacy-membership function I A : X → [0, 1], and a falsity-membership function F A : X → [0, 1]. Note that 0 ≤ T A (x), I A (x), F A (x) ≤ 1, and so, in this case, we have 0 ≤ T A (x) + I A (x) + F A (x) ≤ 3. This is known as a single valued neutrosophic set (SVNS) [11].
The complement of a neutrosophic set A, An interval neutrosophic set (INS) [12] gives value that ranges for the truth, indeterminacy and the falsity rather than single values for each of these quantities. Formally we have, Definition 4. [12] Given a set X, an object of the Following operations on INNs due to [12] are important for the development of our model. Letã andb be two INNs, and let λ > 0 be a real number, then  [15] defined three basic operations as follows.
Also, Xia and Xu [28] defined four operations on the HFEs h, h 1 , h 2 with a positive scale n as follows.
An interval-valued hesitant fuzzy set (IVHFS) [16,29] on X is defined as: is a set of some distinct interval values in [0,1], that denote the possible membership degrees of the element x ∈ X to the set E. We callh E (x) an interval valued hesitant fuzzy element (IVHFE). Let us simplify our notation by takingh instead of h E so thath = {γ|γ ∈h}, whereγ = [γ L , γ U ] is an interval number.
For three IVHFEsh,h 1 ,h 2 and a positive scale n, Chen et al. [16] introduced the following operations.
In order to develop an MCDM using interval neutrosophic hesitant fuzzy elements, the score function for making their comparisons is defined in [22] as follows. Definition 9. [22] The score function of an interval neutrosophic hesitant fuzzy element (INHFE)ñ is given as: where l, p, q are the numbers of the interval values inγ,δ,η, respectively. It has been observed that S(ñ) is an interval value included in [0,1], and so for two INHFEsñ 1 ,ñ 2 , the score functions S(ñ 1 ) and S(ñ 2 ) are comparable using the degrees of possibility. If S(ñ 1 ) ≥ S(ñ 2 ) thenñ 1 ñ 2 , i.e.ñ 1 is superior than or equal toñ 2 .
Liu and Shi [22] further improved the score function using a simple average method as follows.
where l j , p j , q j are the numbers of the interval values inγ j ,δ j ,η j . Their scores using Equation (2.5) are found to be equal i.e., S(ñ 1 ) = S(ñ 2 ) = 0.6167. But clearlyñ 1 andñ 2 are not equal. Therefore it is necessary to further improve the ranking procedure of INHFEs. In Section 3 an improved ranking method for INHFEs is proposed to address this issue. This is applied in the further development of our proposed model.
Following weighted aggregation operators defined on interval neutrosophic hesitant fuzzy elements are deemed important to draw comparison with our proposed model.
A fuzzy measure is additive if for any two disjoints Further studies on fuzzy measures and their properties can be found in [32]. A fuzzy measure is symmetric if for any subsets denotes the subjective weight of criterion x i in the set of criteria X. Thus a fuzzy measure represents the weight of each criterion as well as combination of criteria in which all of the w i (i = 1, 2, . . . , n)'s are not necessarily equal to one. Therefore, in order to determine fuzzy measures on X = {x 1 , x 2 , . . . , x n }, we need to find 2 n − 2 values for n criteria, except the values m(∅) and m(X) which are always equal to 0 and 1, respectively. So the evaluation model obtained becomes quite difficult. To avoid this difficulty, Sugeno [24] proposed a special kind of fuzzy measure called λ-fuzzy measure which is defined as follows: Definition 13. [24,30] Let X = {x 1 , x 2 , . . . , x n } be a non-empty set, let P(X) be the power set of X. Given a real number λ > −1, a λ-fuzzy measure m on X is a function m : P(X) −→ [0, 1], satisfying the followings Note that if X represents a set of criteria, the parameter λ determines an interaction level between the criteria. If λ = 0, then the fuzzy measure reduces to an additive measure, and A and B have no interaction between them. For negative and positive λ, a λfuzzy measure reduces to the sub-additive To determine a normalized measure on X, Sugeno [24] provides the following expression.
(2.8) Also, for every subset A ⊆ X, we have (2.9) Using equation Equation (2.8) the parameter λ can be uniquely determined from the boundary condition m(X) = 1 which is equivalent to solving the following equation (1 + λm(x i )). (2.10) If there is no ambiguity in the parameter λ, we call a λ-fuzzy measure simply a fuzzy measure.

Definition 14.
[25] Let f be a positive real-valued function on X = {x 1 , x 2 , . . . , x n }, and let m be a fuzzy measure on X. The discrete choquet integral of f with respective to m, denoted by C m (f ) is defined as follows.
In what follows next we define a modified score function to compare INHFEs that applies to our proposed MCDM procedure using INHFCI.

An ordering between INHFEs
Following the ambiguities of comparing two INHFEs by using the score functions given by Equations (2.4) and (2.5) we propose here an alternative ordering for INHFEs. This ordering approach is based on the possibility degree ranking (PDR) for the interval numbers [9,10], the interval neutrosophic numbers (INNs) [18] and the interval valued hesitant fuzzy elements (IVHFEs) [19,20].

Definition 15. Let
be two INHFEs. Then the possibility degree ofñ i ñ j is defined as follows: where l i , p i , q i are the numbers of the interval values inγ i ,δ i ,η i , and l j , p j , q j are the numbers of the interval values inγ j ,δ j ,η j . Now, using Equation (2.2), we can observe that 0 ≤ P(ñ i ñ j ) ≤ 1, P(ñ i =ñ j ) = 0.5, and P(ñ i ñ j ) + P(ñ j ñ i ) = 1. Also, and, for i = 1, 2, . . . , n. Then the possibility degree P ij = P(ñ i ñ j ) of each pair of INHFEñ i , (i = 1, 2, . . . , n) is given by Equation (3.1); thus the matrix of possibility degrees P = (P ij ) n×n can be constructed, where P ij ≥ 0, P ij + P ji = 1, and P ii = 0.5. Also, the ranks r i of theñ i (i = 1, 2, . . . , n) is given in the line of [6,7] as follows  1, 2, . . . , n). We call the ranking : a Possibility Degree Ranking (PDR). It is evident from the above formulations that the PDR generalizes the ordering given by the score function over the class of INNs and IVHFEs. However it is worth mentioning here that the ordering among INHFEs in terms of the r i 's is only a partial ordering and therefore this ranking is also not unique. Now, the PDR values are given by, r 1 = 0.5417, r 2 = 0.4584. Since r 1 ≥ r 2 , therefore,ñ 1 ñ 2 . Thus it appears that the PDR is more efficient for ranking INHFEs.

Remark 2.
The main advantage of adopting this possibility degree ranking (PDR) approach is that the possibility degrees between any two adjacent INHFEs can be also be obtained from the possibility degree matrix in Definition 15. Also as explained in Example 1, in some typical situations it seems that ranking orders ( ) of INHFEs with PDR approach is more effective than the other ranking approach like score function. Moreover the possibility degree P(ñ 1 ñ 2 ) of any two INHFEsñ 1 ,ñ 2 can be interpreted from the probability point of view for ranking INHFEs. As for instance if P(ñ 1 ñ 2 ) ≈ 1 then there is the more possibility thatñ 1 ñ 2 , whereas no such interpretation can be seen in the ranking approach like score function.

The interval neutrosophic hesitant fuzzy choquet integral (INHFCI) operator
In this section, we propose the interval neutro- be a collection of INHFEs, where (·) is a permutation such that r (1) ≥ r (2) ≥ · · · ≥ r (n) , and Denote by card(A) the cardinality of set A. Then the INHFCI operator is defined as: Note that ⊕ denotes here the sum operation on INHFEs as defined in the rule (iii) of Definition 8, and that the Equation (4.1) does not depend on the considered permutaion (·). Moreover, the ranking ofñ j (j = 1, 2, . . . , n) is done with respect to the PDR of INHFEs. If there are no ties between values r (1) , .., r (n) , then Equation (4.1) can be rewritten as where A (j) = {(1), .., (j)}, with the convention A (0) = ∅. Following theorem is an immediate consequence.
Theorem 1. Let X = {x 1 , x 2 , . . . , x n } be a finite set of criteria and let m be a fuzzy measure on X. Let, be a collection of INHFEs. Then the value aggregated by the INHFCI is also an INHFE, moreover, where, Proof. We use mathematical induction to prove this theorem. For n = 1 from Equation (4.1), Therefore the result holds for n = 1. Next suppose that the result hold for n = k, i.e., Then for n = k + 1, we have Hence the theorem holds for n = k + 1, which completes the proof.
Note that in proving all the following results relating to the INHFCI operators, there is no loss of generality in considering the case when the ordering among INHFEs is unique i.e., using Equation (4.2) for the INHFCI as the related results using Equation  In this case the INHFCI operator reduces to an interval neutrosophic hesitant fuzzy ordered weighted average (IVIHFOWA) operator [22], i.e., ñ 1 ,ñ 2 , . . . ,ñ n ). (4.8) Corollary 2 follows directly from Proposition 1.

Properties of the INHFCI operator
Some of the properties of INHFCI operator are as follows.

Theorem 3. (Monotonicity). Let
and be two collections of INHFEs on X, and let m be a fuzzy measure on X such that for all j; then

Theorem 4. (Boundedness). Let
be a collection of INHFEs on X, let m a fuzzy measure on X, and let where, Proof of Theorem 4 can be easily obtained by using Theorem 2 and 3.

Theorem 5. (Shift invariant) Let
be a collection INHFEs, and let m be a fuzzy measure on X. If λ > 0, then  The proofs of the next two theorems directly follow from Theorems 5 and 6 and hence omitted.

An approach to multicriteria decision making with interval neutrosophic hesitant fuzzy choquet integral operator
Now we are in a position to propose multicriteria decision making (MCDM) based on interval neutrosophic hesitant fuzzy choquet integral (INHFCI). We assume that the evaluation information of the alternatives are given by INHFEs that allows interactions among the criteria.
Let X = {x 1 , x 2 , . . . , x m } and let C = {c 1 , c 2 , . . . , c n } be a set of criteria. Suppose that the evaluation information of the criteria c j ∈ C j (j = 1, 2, . . . , n) with respect to the alternative x i ∈ X is represented by an INHFEñ ij = {t ij ,ĩ ij ,f ij }, wheret ij , i ij andf ij are set of some interval values in [0,1], denoting the degrees of hesitant truth-membership, indeterminacy-membership and falsity-membership, respectively. The multicriteria decision making approach for obtaining the best alternative with respect to an interval neutrosophic hesitant fuzzy choquet integral involves the following steps.
Step 1 Construct the interval neutrosophic hesitant fuzzy decision matrix D = (ñ ij ) m×n , whereñ ij = {t ij ,ĩ ij ,f ij } is an INHFE, which represents the evaluation information of the criteria c j ∈ C j with respect to the alternative x i ∈ X.
Step 2 Identify the fuzzy measure m(A) of all the A ⊆ X for each of the criteria c j (j = 1, 2, . . . , n) and using Equation (2.9) with parameter λ is determined using Equation (2.10).
Step 3 Utilize the possibility degree ranking approach as mentioned in Definition 15 to reorder the partial evaluationñ ij of the alternatives x i (i = 1, 2, . . . , m). The PDR given by P(ñ i(j) ñ i(k) ) is defined as follows: where l j , p j , q j are the numbers of the interval values inγ i(j) ,δ i(j) ,η i(j) , and l k , p k , q k are the numbers of the interval values inγ i(k) ,δ i(k) ,η i(k) . Now, using Equation (2.2), it is observed that 0 ≤ p i jk ≤ 1, p i jj = P(ñ i(j) =ñ i(k) ) = 0.5, and p i jk + p i kj = 1. Also, and, The matrix of possibility degrees P i = (p i jk ) n×n , (i = 1, 2, . . . , m) is constructed for each of the alternative x i (i = 1, 2, . . . m). Then the ranking of the partial evalutionñ ij is obtained by using Equation (3.5) on the matrix of possibility degrees P i (i = 1, 2, . . . , m) The partial evalutionñ ij are reordered in a descending order n i(1) ñ i(2) · · · ñ i(n) in accordance with the values of r i j (j = 1, 2, . . . , n), where {(1), (2), . . . , (n)} is a permutation of {1, 2, . . . , n}.

Illustrative example
In this section, we present an example to illustrate the proposed decision making method under interval neutrosophic hesitant fuzzy environment. Suppose a manufacturing company wants to recruit a sales executive from a group of four candidates A i (i = 1, 2, 3, 4) on the basis of a set of following criteria: (1) C 1 : Management Knowledge (2) C 2 : Communication Skill (3) C 3 : Objection handling Skill (4) C 4 : Ability to Attain Targets, and the evaluation values are expressed by INHFEs. The evaluation steps of the four alternatives on the basis of above mentioned criteria are as follows: Step 1 Based on the experts' assesment the interval neutrosophic hesitant fuzzy decision matrix is constructed as shown in Table 1 in the Appendix.
Then the fuzzy measure m(A) of all the A ⊆ X for each of the criteria C j (j = 1, 2, 3, 4) corresponding to the alternative A 1 are given by Similarly the partial evaluationsñ ij (i = 2, 3, 4 j = 1, 2, 3, 4) corresponding to the alternative A i (i = 2, 3, 4) and the fuzzy measure m(A) of all the A ⊆ X for each of the criteria C j (j = 1, 2, 3, 4) corresponding to the alternative A i (i = 2, 3, 4) are shown in Table 2 in the Appendix.
Step 4 Using Equation (6.5) the overall valueñ 1 of the alternative A 1 is given as: Step 5 We calculate the possibility degree P ij = P(ñ i ñ j ) (i, j = 1, 2, 3, 4) by using Equation ( Again, from Definition 15, since P ii = 0.5, P ij + P ij = 1 (i, j = 1, 2, 3, 4), so we have Since r 2 > r 3 > r 4 > r 1 , therefore, the ranking order of the alternatives A i (i = 1, 2, 3, 4) is A 2 A 3 A 4 A 1 . Hence the alternative A 2 is the best alternative. That is, A 2 is the most suitable candidate for the job of scales executive.

Conclusion
In this paper, we have proposed the interval neutrosophic hesitant fuzzy choquet integral (INHFCI) operator for multi criteria decision making problem under interval neutrosophic hesitant fuzzy environment, and discussed their properties such as idempotency, monotonicity, homogeneity etc. It is shown that the INHFCI generalizes both the interval neutrosophic hesitant fuzzy weighted averaging (INHFWA) and interval neutrosophic hesitant fuzzy ordered weighted (INHFOW) operator. Also, it is shown that the interval valued intuitionistic fuzzy choquet integral (IVIHFCI) operator is a particular case of INHFCI operator. Further, an approach for multicriteria decision making is proposed. Finally, an illustrative example is presented to demonstrate the application of INHFCI in the multicriteria decision making process. It is to be expected that these investigations of generalized choquet integral may open the door for further study in this field.