Interval Neutrosophic Numbers Choquet Integral Operator for Multi-criteria Decision Making

In this paper, the Choquet integral and the interval neutrosophic set theory are combined to make multi-criteria decision for problems under neutrosophic fuzzy environment. Firstly, a ranking index is proposed according to its geometrical structure, and an approach for comparing two interval neutrosophic numbers is given. Then, a ≤ L implied operation-invariant total order which satisfies order-preserving condition is proposed. Secondly, an interval neutrosophic number Choquet integral (INNCI) operator is established and a detailed discussion on its aggregation properties is presented. In addition, the procedure of multi-criteria decision making based on INNCI operator is given. Finally, a practical example for selecting the third party logistics providers is provided to illustrate the feasibility of the developed approach.


Introduction
The concept of neutrosophic set (NS) is introduced by Smarandache [22], which generalizes the classic set, fuzzy set (FS), interval valued fuzzy set (IVFS), intuitionistic fuzzy set (IFS), as well as interval valued intuitionistic fuzzy set (IVIFS).A NS is characterized independently by a truth-membership, an indeterminacy-membership and a falsity-membership.It is a powerful tool to deal with incomplete, indeterminate and inconsistent information.Comparing with NS, IFSs and IVIFSs can only handle incomplete information but not the indeterminate information and inconsistent information which exist commonly in real Technology and Business University, No.33 Fucheng Road, Haidian District, Beijing, China.Tel.: +86 10 68985991; Fax: +86 10 68984948; E-mail: sunhongxia@btbu.edu.cn.situations.For example, in a decision making process, a manager decides whether he should select the third party logistics provider A or not.When we ask about the opinion of an expert about a certain statement, he may say that the possibility that he select A is between 0.5 and 0.6, that he does not select A is between 0.1 and 0.2, and the degree that he is not sure is between 0.2 and 0.3.For a neutrosophic notation, it can be expressed as x([0.5, 0.6], [0.1, 0.2], [0.2, 0.3]).The sum of the degree of truth, indeterminacy, and falsity may be greater or less than 1.For example, if x = (0.6, 0.2, 0.3), x is a NS but is not an IFS.If x = (0.5, 0.1, 0.2), x is a NS but is not an IFS.If x = (0.5, 0.2, 0.3), x is not only a NS, but also an IFS.Another example, assuming there are 10 voters during a voting process, in time t 1 , four vote 'yes', three vote 'no' and three are undecided.For neutrosophic notation, it can be expressed as x(0.4,0.3, 0.3); in time t 2 , two vote 'yes', three vote 'no', two give up, and three are undecided, then it can be expressed as x(0.2, 0.3, 0.3).That is beyond the scope of the IFS.So the notion of neutrosophic set is more general [22,33].
In a NS, the degree of truth, indeterminacy, and falsity belong to ]0 − , 1 + [, where ]0 − , 1 + [ is the non-standard unit interval [21].Obviously, it is difficult to apply in real applications.Therefore, Wang et al. [27] proposed the concept of a single valued neutrosophic set (SVNS), which is the subclass of a NS.Sometimes the degree of truth, falsity, and indeterminacy of a certain statement cannot be defined exactly in the real situations but denoted by several possible interval values.In order to research this problem, Wang et al. [26] proposed the concept of interval neutrosophic set (INS) and gave the set-theoretic operators of INS.Recently, many researchers have shown great interest in multi-criteria decision making (MCDM) problems with neutrosophic information.Ye [30,31] proposed correlation coefficients between SVNSs and applied them to MCDM problems with single valued neutrosophic information.Ye [32] proposed single valued neutrosophic cross entropy and applied it to MCDM.Ye [29] introduced the concept of simplified neutrosophic sets (SNSs), which can be described by singleton subintervals/subsets in the real unit interval [0, 1], and proposed a MCDM method using aggregation operators for SNSs.Chi and Liu [2] extended a TOPSIS method to interval neutrosophic multiple attribute decision-making problems.Zhang et al. [34] defined the operations of INSs and gave the aggregation operators of interval neutrosophic number weighted averaging(INNWA) and interval neutrosophic number weighted geometric(INNWG), then a MADM method is established based on the proposed operators.Broumi and Smarandache [1] introduced the concept of correlation coefficients of interval valued neutrosophic set.
Aggregation function plays an important role in MCDM problems.All above aggregation operators only consider situations where criteria (attribute) and preferences of decision makers are independent of one another, which means that their effects are viewed as additive.However, in many real decision making problems, it is common to find that there is interaction among preference of decision makers.As an aggregation function, the Choquet integral [3] with respect to fuzzy measures [23] is able to flexibly describe the relative importance of decision criteria as well as their interactions [11,19].Therefore, it is interesting to combine the Choquet integral and the INS theory for MCDM under neutrosophic fuzzy environment.On one hand, we can deal with the imprecise and uncertain decision information; on the other hand, we can efficiently take into account of the various interactions among the decision criteria.Based on the above discussion, there are three aims in the paper.First, it will propose a new ranking method for interval neutrosophic numbers (INNs).Second, it will propose an interval neutrosophic number Choquet integral (INNCI) operator and discuss its properties.Third, it will establish a decision making method based on the proposed ranking method and the INNCI operator to handle decision making problems with interval neutrosophic information.
The paper is organized as follows.In Section 2, the concepts of NS, SNS, INS and operations of INS are reviewed.In Section 3, a ranking index is proposed according to geometrical quantities which reflect the inferiors and superiors of INNs, and an approach to compare two INNs is proposed.Furthermore, a ≤ L implied operation-invariant total order which satisfies order-preserving condition is proposed.In Section 4, INNCI operator is proposed, and some of its properties are researched.In Section 5, the MCDM procedure based on INNCI operator is presented under neutrosophic environment.In Section 6, an example is given to illustrate the concrete application of the method and to demonstrate its feasibility and applicability.Conclusions are made in Section 7.

Interval neutrosophic Set
Definition 2.1.[22] Let X be a space of points (objects), with a generic element in X denoted by x.A NS A in X is characterized by a truth-membership function T A (x), an indeterminacy-membership function I A (x), and a falsity-membership function Since it is difficult to apply NSs to practical problems, Wang et al. [27] introduced the concept of a SVNS, which is an instance of a NS and can be used in real scientific and engineering applications.Definition 2.2.Let X be a space of points (objects) with generic elements in X denoted by x.A SVNS A in X is characterized by truth-membership function T A (x),

indeterminacy-membership function I A (x), and falsitymembership function F
Similar to interval-valued intuitionistic fuzzy set, Wang et al. [26] proposed the concept of INS.Definition 2.3.[26] Let X be a space of points (objects) with generic elements in X denoted by x.An INS A in X is characterized by a truth-membership function T A (x), an indeterminacy-membership function I A (x), and a falsity-membership function F A (x).For each point x in X, we have that

Remark 2.1. From Definition 2, an INS A can be expressed as
Then, the sum of T A (x), I A (x) and Definition 2.4.[26] denotes an INN and L be the set of all INNs in X. Definition 2.5.[2] Let ã and b be two INNs, and λ be a real number.Then, the operational rules are defined as follows: (1) The operations of two INNs has been defined by Zhang et al. [34] by using of a strict Archimedean tnorm T (x, y) = k − (k(x) + k(y)) and its dual t-conorm S(x, y) = l − (l(x) + l(y)) with l(x) = k(1 − x).When k(x) = − log(x), the operational rules defined in [2] and [34] are identifiable.The results of Theorem 2.1 are obvious.Furthermore, the operations of two INNs have the following properties [34].

Comparison method of INNs
In this section, we will propose two approaches to compare two INNs.Firstly, a ranking index is developed according to some geometrical quantities which reflect the inferiors and superiors of INNs, and then order relation ≤ H is proposed.Furthermore, due to limitations of ≤ H , a ≤ L implied operation-invariant order is proposed.

Ranking method for INNs based on geometrical structure
Let us consider a three-dimensional coordinate of the technical neutrosophic cube that shown in Fig. 1, where x is the truth axis with value range in [0, 1], y is the false axis with value range in [0, 1], and similarly z z x y A(0,0,0)  is the indeterminate axis with value range in [0, 1].The neutrosophic cube can be divided into three disjoint regions [5]: (1) Triangle BDE , whose sides are equal to √ 2, represents the geometrical locus of the points whose sum of the coordinates is 1.
(2) The pyramid EADB is the locus of the points whose sum of coordinates is less than 1.
(3) the solid EHDCGFB ( excluding BDE ) is the locus of points whose sum of their coordinates is greater than 1.
Considering a point a in the technical neutrosophic cube.The superiors point is B(1, 0, 0) and the inferiors point is H(0, 1, 1), therefore, the shorter distance between a and B(1, 0, 0) is, and the longer distance between a and H(0, 1, 1) is, the bigger a is.
Given a INN ã, the two-dimensional projection of ã in plan (x, y) is shown in Fig. 2. The smaller area of S 1 is, and the bigger area of S 2 is, the bigger ã is.The shorter distance between ) is, the bigger ã is.According to the geometrical description of INNs, the ranking index of INNs is proposed.

Definition 3.1. Let ã be an INN. The ranking index of ã is defined as follow
where From the Definition 3.1, it is easy to get the following proposition.

Proposition 3.1. For any INNs
The proof of Theorem is completed.
Based on the ranking index of INNs, an approach to compare two INNs is proposed as follow.The order relation ≤ H is reflexive, antisymmetric, transitive and total, and hence defines a total order on INNs.

≤ L implied operation-invariant order
The order relation ≤ L is reflexive, antisymmetric and transitive, and so a partial order.As compared with the order relation ≤ H , the order relation ≤ L has some important properties although it is not a total order.According to Theorem 3.1 and Definition 3.3, it is easy to obtain the following proposition.( Monotonicity is one of the most important properties of an aggregation function.Therefore it is necessary to define an operation-invariant total order.Inspired by the concept of the operation-invariant total order on IFVs proposed by Liu [15][16][17] and an order implied operation-invariant total order proposed by Wu et al. [25], the following definition is given.In the following, we will propose a ≤ L implied operation-invariant total order.
operation-invariant total order, denoted by ≤ S , can be defined as follows: It is easy to verify that the order relation ≤ S satisfies the implication conditions and the order-preserving conditions given in the Definition 3.4.

Comparative analysis with score, accuracy and certainty functions for INNs
Zhang et al. [34] defined the score function, accuracy function and certainty function for an where s(ã), a(ã) and c(ã) represent the score function, accuracy function, and certainty function of the INN ã, respectively.Zhang et al. [34] gave the ranking method as follows:  ( One can see that a fuzzy measure is a normal monotone set function which vanishes at the empty set.Furthermore, A fuzzy measure on X is said to be • cardinality-based if, for any A ⊆ X, µ(A) depends only on the cardinality of A.
In the framework of the MCDM, X can be interpreted as a finite decision criteria set.µ(A) can be viewed as the grade of subjective importance of subset A ⊆ X.The monotonicity of the fuzzy measure means that the importance of a subset of criteria cannot decrease when new criteria are added to it [12].Definition 4.2.Let f be a real-valued function on X, the Choquet integral of f with respect to a fuzzy measure µ on X is defined as where (•) indicates a permutation on X such that When using a fuzzy measure to model the importance of the subsets of criteria, the Choquet integral can be viewed as a aggregation function [4,24,25].It has been proposed by many authors as an adequate substitute to weighted arithmetic mean (WAM) or ordered weighted averaging (OWA) operator to aggregate interacting criteria [10,14,19].The Choquet integral identifies with a WAM (resp.OWA) as soon as the fuzzy measure is additive (resp.cardinality-based).

Interval neutrosophic numbers Choquet integral operator
Definition 4.3.Let f : X → L be an interval neutrosophic number function on X, and µ be a fuzzy measure on X.The interval neutrosophic number Choquet integral (INNCI) of f with respective to µ is defined as where (•) indicates a permutation on X according to a given total order ≤ such that f Assume that fuzzy measure µ : X → [0, 1] is given by µ(∅) = 0, µ({x 1 }) = 0.5, µ({x 2 }) = 0.3, µ({x 1 , x 2 }) = 1.By Eq. ( 5), we have = [0.6090,0.6861], [0.0810, 0.1934], [0.1129, 0.2098] Theorem 4.1.Let f : X → L be an interval neutrosophic number function on X, and µ be a fuzzy measure on X.Then their aggregated value by using the INNCI operator is also an interval neutrosophic number, and where (•) indicates a permutation on X according to a given total order ≤ such that f (x ( 1) Proof.The first result can be directly obtained from Definition 4.3 and Theorem 2.1.Eq. ( 6) is easily proved by using mathematical induction on n.Definition 4.4.Let f and g be two interval neutrosophic number functions on X. f and g are said to be comonotonic about a given order relation ≤ if The following propositions show some properties of the INNCI operator.Proposition 4.1.(Idempotency).Let f , g be interval neutrosophic number functions on X, and ã be an Proposition 4.2.Let f , g be interval neutrosophic number functions on X and µ, ν be fuzzy measures on X, then However, we can also calculate the two expressions, (C) f dµ ⊕ (C) f dν and (C) f d(αµ), by Eqs.(5) and (6).Proposition 4.4.(monotonicity).Let f , g be interval neutrosophic number functions on X.For a ≤ L implied operation-invariant total order ≤, if f (x i ) ≤g(x i ) for all x i ∈ X, then If f and g are comonotonic about the permutations (π(1), π( 2), • • • , π(n)), then the conclusion is completed.
We will consider two cases.
By the definition of f ( 2) , If there exists at least one point x k 0 such that g(x π (2) (k 0 +1) ) ≥g(x π (2) (k 0 +1) ), repeat the above steps until k 0 = n.Then we will define a series interval neutrosophic numbers functions f (k) (k ≤ n) and permutations π (k) , then the function f (k) and g are comonotonic about the permutation By the definition of f (k) , Proposition 4.5.Let f , g be interval neutrosophic number functions on X.For a ≤ L implied operationinvariant total order ≤, if f and g are comonotonic about order relation ≤, then Proof.According to Definition 4.4 and Definition 3.4, it is easy to get the conclusion.Proposition 4.6.Let f be interval neutrosophic number function on X.For a ≤ L implied operation-invariant total order ≤ and λ ≥ 0, Proof.According to Definition 3.4, it is easy to get the conclusion.
From Proposition 4.5 and 4.6, we can obtain the following corollary.Corollary 4.1.Let f , g be interval neutrosophic number functions on X, ã be an INNs.For a ≤ L implied operation-invariant total order ≤ and λ ≥ 0, It is not hard to see that the aggregation properties of the INNCI greatly depend on the given order relation.Based on the order relations ≤ H and ≤ P , the INNCI only has the Properties 4.1 − 4.2.However, based on a ≤ L implied operation-invariant total order, the INNCI has all the properties presented above.
Zhang et al. [34] have developed interval neutrosophic number weighted averaging (INNWA) operator and interval neutrosophic number weighted geometric (INNWG) operator, which are respectively defined as follows.
The following theorems show that INNCI operator is a generalization of INNWA and INNWG.Theorem 4.2.Let f be an interval neutrosophic number function on X, µ be a fuzzy measure on X. ( Proof.(1) If µ is additive, let ω i = µ({i}), then the conclusion is obvious.
(2) If µ is cardinality-based, ω is defined as follows: Remark 4.2.The relationship between fuzzy measure and weight vector can be given as follows: (1) The additive fuzzy measure µ associated to an INNWA is given by µ(A) = x i ∈A ω i for A ⊆ X.
(2) The cardinality-based fuzzy measure µ associated to an INNWG is given by µ(A) = n i=n−|A|+1 ω i for any non-empty subset A ⊆ X.

Multicriteria decision-making method based on interval neutrosophic numbers Choquet integral operator
This section presents a new method for MCDM, in which the partial evaluations of the alternatives are given by INNs and the interaction among the criteria are allowed.
For a MCDM problem, let Y = {y 1 , y 2 , • • • , y m } be a set of alternatives, X = {x 1 , x 2 , • • • , x n } be a set of criteria.To get the best alternative, the MCDM procedure based on INNCI operator is proposed as follows.
Step 1. Construct the INNs decision matrix.Assume that the partial evaluation of the alternative where T ij indicates the degree to which the alternative y i satisfies the criterion x j , I ij indicates the indeterminacy degree to which the alternative y i satisfies or does not satisfy the criterion x j , F ij indicates the degree to which the alternative y i does not satisfy the criterion x j , and Then we can obtain a decision making matrix as follow: Step 2. Reorder the partial evaluation dij of the alternative y j such that di(j) ≤ di(j+1) for a given order relation.For example, by order relation ≤ H of Definition 3.2, we calculate H( dij ) of the partial evaluation dij of the alternative y j (j = 1, 2, • • • , m) on the criteria x i (i = 1, 2, • • • , n), and then rank the partial evaluation dij such that di(j) ≤ H di(j+1) .
Step 4. Choose the INNCI operator to aggregate the partial evaluations of each alternative and get the overall evaluations.
Step 5. Rank those overall evaluations according to the given total order relation on INNs, and select the best one(s).

An illustrative example
This section presents an application of the proposed method to select the third party logistics providers.Suppose that there are four providers (y 1 , y 2 , y 3 , y 4 ) whose core competencies are evaluated by means of the following four criteria (x 1 , x 2 , x 3 , x 4 ): (1) the cost of service (x 1 ); (2) the operational experience in the industry (x 2 ); (3) customer satisfaction (x 3 ); (4) market reputation (x 4 ).

Procedures of decision making based on three order relation ≤ H , ≤ P and ≤ S
Step 1.The INNs decision matrix of the third party logistics providers is made up according to the four evaluating criteria.The evaluation of a provider y i (i = 1, 2, 3, 4) with respect to a criteria x j (j = 1, 2, 3, 4) is obtained from the experts.Suppose that the INNs decision matrix is constructed as shown in Table 1.
Step 2. According to Table 1, by Definition 3.2, Definition 3.5 and Definition 3.6, the partial evaluation dij of the candidate y j is reordered such that di(j) ≤ * di(j+1) (i = 1, 2, 3, 4), where ≤ * denotes the order relations ≤ H , ≤ P and ≤ S , respectively.The results are shown in Table 2. Therefore Step 3. Suppose that the fuzzy measures of criteria of X and subsets of X are shown in Table 3. Step 4. By Eq. ( 6), utilizing the INNCI operator to aggregate the partial evaluations of each alternative y i (i = 1, 2, 3, 4).The results are shown in Table 4.
Step 5.According to the overall evaluations of the third party logistic providers, by Definition 3.2,  5.

Comparison analysis of results
In Section 6.1, we have given the ranking order based on three order relations ≤ H , ≤ P and ≤ S , but from the results shown in Table 5, the raking order calculated based on three order relations is different.Therefore, it is very difficult to decide which alternative is the best choice and whether an alternative is definitely better than another one.
Which mean that, on each criterion, the partial evaluation of y 1 is larger or at least equal to that of y 4 (denoted as y 1 y 4 ).Hence, the overall evaluation of y 1 should be larger or at least equal to that of y It is obvious that the above result is unreasonable.Table 6 shows the unreasonable results in this illustrative example based on the order relation ≤ H and ≤ P .The main reason for such unreasonable results is that, as mentioned in Section 3.3, the order relation ≤ H and ≤ P are not order-preserving for operations, that is , these two order relations are not an operationinvariant total order on INNs.The increasingness property of an aggregation function strongly depends on the order-preserving property of the given order relation.Therefore, the interval neutrosophic numbers Choquet integral operator, INNCI, is not increasing with respect to these two order relation.
The key to avoiding such unreasonable results is to adopt a ≤ L implied operation-invariant total order (see Definition 3.5) in the aggregation process.Based on the ≤ L implied operation-invariant total order ≤ S , we can generate the overall evaluations of the alternatives by using the INNCI, as shown in Table 5, and then we will select provider y 1 .

Conclusions
Interval neutrosophic set (INS) is a subclass of a neutrosophic set, which can be applied in the problems with uncertain, imprecise, incomplete, and inconsistent information existing in real applications.As an aggregation function, the Choquet integral with respect to fuzzy measures is able to flexibly describe the relative importance of decision criteria as well as their interactions.In this paper, we combined the Choquet Increasingness is a natural requirement for an aggregation function in MCDM.The increasingness property of INNCI aggregation function strongly depends on the order-preserving property of the given order relation.In this paper, we proposed two approaches to compare two INNs.According to its geometrical structure, the ranking index is developed according to some geometrical quantities which reflect the inferiors and superiors of INNs.Based on the ranking index, an order relation, denoted by ≤ H , is proposed.Examples shown that order relations ≤ H and ≤ P have some limitations, but they can be viewed as mutually complementary if the comparing results is obviously irrational.Because order relation ≤ H and ≤ P are not order-preserving for operations, we furthermore proposed a ≤ L implied operation-invariant total order to ensure the increasingness of INNCI operator, and it is more superior than the order relations ≤ H and ≤ P .
We only have proposed a kind of a ≤ L implied operation-invariant total order (Definition 3.5), which does not adequately reflect the characteristics of INNs.Therefore, it is of great interest to find a ≤ L implied operation-invariant total order which can better reflect the characteristics of INNs.Furthermore, how to find a good aggregation operation is also an important key issue in netrosophic MCDM problems.

Theorem 2 . 1 .
Let ã and b be two INNs, then ã ⊕ b and λ ⊗ ã are also INNs.

Proposition 2 . 1 .
Let ã , b and c be INNs and α

Definition 3 . 2 .
Let ã and b be two INNs, H(ã) and H( b) be ranking index of ã and b, respectively, then (1) If H(ã) < H( b), then ã is smaller than b, denoted by ã < H b.
integral and the INS theory to propose INNCI operator for MCDM problem with netrosophic information and investigated their aggregation properties, such as idempotency and monotonicity.INNCI operator can represent INNWA and INNWG.Therefore, INNCI operator is superior to existing operators.
Let λ = 0.2, then ãλ > H bλ , then ãλ > P bλ .From Example 3.1 and Example 3.2, in some situations, if we use ranking method as given in Definition 3.2 and Definition 3.6 to compare two INNs, the results might be irrational.We can use the order relation ≤ H to compare two INNs if the comparing results is obviously irrational by using the order relation ≤ P and vice versa.Therefore, the order relation ≤ H and ≤ P can be viewed as mutually complementary.

Table 1 The
INNs decision matrix of the third party logistics providers

Table 2
Ranking order of dij based on the order relation ≤ * .

Table 5
Ranking order of the third party logistics providers.

Table 6
The unreasonable results based on the order relation≤ H and ≤ P .Corresponding unreasonable ranking ordery 1 > Z y 4 y 1 < Z y 4 y 2 < Z y 3 y 2 > Z y 3 y 2 < H y 3 y 2 > H y 3