Mathematical Problems in Engineering

Volume 2019, Article ID 9143624, 14 pages

https://doi.org/10.1155/2019/9143624

## Induced Choquet Integral Aggregation Operators with Single-Valued Neutrosophic Uncertain Linguistic Numbers and Their Application in Multiple Attribute Group Decision-Making

^{1}School of Management, Hefei University of Technology, Hefei 230009, China^{2}Ministry of Education Engineering Research Center for Intelligent Decision-Making and Information Systems Technologies, Hefei 230009, China

Correspondence should be addressed to Dong Wang; moc.361@1088dgnaw

Received 19 September 2018; Revised 15 December 2018; Accepted 23 December 2018; Published 11 February 2019

Academic Editor: Peide Liu

Copyright © 2019 Shuping Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For real decision-making problems, aggregating the attributes which have interactive or correlated characteristics by traditional aggregation operators is unsuitable. Thus, applying Choquet integral operator to approximate and simulate human subjective decision-making process, in which independence among the input arguments is not necessarily assumed, would be suitable. Moreover, using single-valued neutrosophic uncertain linguistic sets (SVNULSs) can express the indeterminate, inconsistent, and incomplete information better than FSs and IFSs. In this paper, we studied the MAGDM problems with SVNULSs and proposed two single-valued neutrosophic uncertain linguistic Choquet integrate aggregation operators where the interactions phenomena among the attributes or the experts are considered. First, the definition, operational rules, and comparison method of single-valued neutrosophic uncertain linguistic numbers (SVNULNs) are introduced briefly. Second, induced single-valued neutrosophic uncertain linguistic Choquet ordered averaging (I-SVNULCA) operator and induced single-valued neutrosophic uncertain linguistic Choquet geometric (I-SVNULCG) operator are presented. Moreover, a few of its properties are discussed. Further, the procedure and algorithm of MAGDM based on the above single-valued neutrosophic uncertain linguistic Choquet integral operator are proposed. Finally, in the illustrative example, the practicality and effectiveness of the proposed method would be demonstrated.

#### 1. Introduction

In the real-life, due to the complexity of environment and the limitation of human knowledge, human preference judgments may be difficult to express by crisp numbers. The decision information in real-world situations is uncertain, incomplete, and inconsistent. Therefore, numerous decision methods based on fuzzy information which can be considerably appropriate to express decision makers’ preference were developed. Zadeh [1] first proposed fuzzy sets (FSs), which are regarded as important tools to solve decision-making problems. Given that FSs only consider a membership function, Atanassov [2] introduced the intuitionistic fuzzy sets (IFSs), which are an extension of FSs and considered membership and nonmembership and have been extensively applied in solving MAGDM problems [3–11]. However, FSs and IFSs can only express incomplete information rather than indeterminate and inconsistent information. Accordingly, Smarandache [12] introduced neutrosophic sets (NSs), which can substantially express indeterminate information, since it is difficult to apply NSs in the real decision-making process without specific description. Therefore, Wang [13] defined the concept of single-valued neutrosophic sets (SVNS), which is an instance of NSs. Moreover, interval neutrosophic sets (INSs) [14], as a particular extension of an NS, have also been proposed. Subsequently, various aspects of SVNSs and INSs have been studied by more and more scholars and experts [15–23]. Ye [24] presented correlation and weighted correlation coefficients of SVNSs. Liu [21] proposed some aggregation operators by combined the PA operator and GWA operator to INS and discussed these operators’ properties.

NSs are progressing rapidly in theory and application. Shahzadi [25] proposed distance measures and similarity measures of SVNSs which are applied to medical diagnosis. Then, the tangent similarity measure and cosine similarity measure of SVNS were proposed and applied in medical diagnosis [26, 27]. Tan [28] proposed exponential aggregation operator of interval neutrosophic numbers for Typhoon Disaster Evaluation. Abdel-Basset [29] developed a hybrid method with neutrosophic sets and applied it in developing supplier selection criteria. Abdel-Basset [30] developed three-way decisions based on NSs and applied it in supplier selection problem. Ye [31] presented some exponential aggregation operator in INSs and applied it in the selection problem of global suppliers. Yang [32] developed a generalized interval neutrosophic fuzzy correlated averaging operator and a linear assignment method to accommodate the interval neutrosophic sets based on Choquet integral to solve the selection problem of investment companies.

The people’s thinking is ambiguous and the objective things are complex, which have occasionally prevented us from using a few real numbers to express several pieces of qualitative information in real MAGDM problems. However, this qualitative information can be easily expressed by linguistic terms. However, there is a downside to using linguistic variables; it cannot handle uncertain and inconsistent information. Therefore, Ye [33] proposed the concepts of single-valued neutrosophic linguistic set (SVNLS) and single-valued neutrosophic linguistic number (SVNLN) and introduced their operational rules. Thereafter, motivated by SVNS and intuitionistic uncertain linguistic set [34], Liu [35] proposed the concepts of single-valued neutrosophic uncertain linguistic sets (SVNULSs) and single-valued neutrosophic uncertain linguistic numbers (SVNULNs). SVNULSs can deal with fuzzy, uncertain, inconsistent, or indeterminate information and are the generalization of SVNLSs. At present, research on SVNULSs is minimal [35, 36].

Information aggregation operators generally play an important part in the process of MAGDM problems and thereby attract the attention of an increasing number of researchers [37–44]. However, these aggregation operators are assembled based on the DM’s preference and attributes are independent of each other [45]. For real decision-making problems, mutual influence and interaction among attributes or experts are constantly present and ignored in the decision-making. Therefore, the concept of fuzzy measure introduced by Sugeno [46] is an effective tool for addressing the interaction phenomena among input arguments, has attracted increasing attention from researchers, and has been applied in numerous application domains [9, 45, 47–52]. To date, no research has been conducted on neutrosophic uncertain linguistic decision-making that considers the various interactions between input single-valued neutrosophic uncertain linguistic information and various decision makers. The current study is motivated by the Choquet integral [53] and (1) extends the induced Choquet integral to aggregate the decision variables with SVNULNs, (2) develops two single-valued neutrosophic uncertain linguistic Choquet integral operators, (3) investigates their various properties, and (4) discusses a few of their special cases. Thereafter, we propose one procedure for MAGDM under the environments of SVNULNs based on the proposed operators in this paper.

The rest of this article is organized as follows. First, we simply introduce some basic concepts of SVNLSs and LVs and a few operational laws of SVNULNs in Section 2. Then, Section 3 reviews fuzzy measure, Choquet integral, and I-COA operator. Section 4 proposes some aggregation operators with SVNULNs. This section also discusses their properties of their operators. Section 5 develops a MAGDM method. We propose an illustrative example to demonstrate the application of the proposed aggregation operator and method and analyze the differences compared with other methods in Section 6. The conclusions and further future research are provided in the end of this paper.

#### 2. Preliminaries

In this section, some concepts and definitions of linguistic term sets (LTSs) provided by Zadeh, NSs, simplified neutrosophic sets (SNSs), and SVNULS are introduced, and these concepts and definitions will be used in the remainder of this paper.

##### 2.1. Uncertain Linguistic Term Set

As an effective tool to express qualitative information, linguistic variables are proposed by Zadeh [54]. Suppose that is a finite and totally ordered discrete linguistic term set, in which is LVs. Accordingly, and satisfy the following properties [55, 56]:(1)The set is ordered: if and only if .(2)If , .(3)Negation operator: .

For any LVs and , the operations are defined as follows [55, 56]:(1).(2), .(3).(4), .

For better completeness of the discrete linguistic term set, we extend the discrete term set into a continuous linguistic term set , where is a sufficient large number; would be called the original term when ; otherwise, is the virtual term [57]. Generally, the original linguistic terms would be utilized to evaluate alternatives by the DM, while only in the operation process the virtual linguistic terms appear.

*Definition 1 (see [58]). *Let , where , and . is the lower bound of and is the upper bound of . Hence is called an uncertain linguistic variable (ULV).

*Definition 2 (see [55]). *For two ULVs and , the operations are defined as follows:

##### 2.2. NSs and SNSs

*Definition 3 (see [12]). *Let be a space of points, and let denote a generic element in . A neutrosophic set in is characterized by the truth-membership function , an indeterminacy-membership function , and a falsity-membership function . The functions , , and are real standard or nonstandard subsets of . Thus, all three neutrosophic components , , and meet the condition .

However, NSs were difficult to be applied to practical problems. Then, by changed nonstandard interval numbers of NSs, Ye [59] introduced the concept and operations of SNSs, which is a subclass of NS and can be defined as follows.

*Definition 4 (see [59]). *Let X be a universal set. A NS in is characterized by , , and , which are single subintervals/subsets in the real standard 0, 1]. That is, , , and satisfy the condition . Thereafter, can be simply denoted by , which is called SNSs.

##### 2.3. SVNULNs and Their Operations

Liu et al. [35] used uncertain linguistic term sets and SNSs as bases to introduce the concept and operations of SVNULNs and define a method with the proposed score and accuracy functions for comparing two SVNULNs.

*Definition 5 (see [35]). *Let , X, and be the given discourse domains. Accordingly, is called SVNULSs, where , , and , , and are three sets of certain single value in real unit interval and express the truth-membership, indeterminacy-membership, and falsity-membership function, respectively, of the element to . For convenience, is defined SVNULNs. Furthermore, degenerates to an ULV when , and .

SVNULSs are extension of an uncertain linguistic term and SVNS. Compared with ULVs, SVNULSs can more accurately reflect uncertainty and fuzziness. Compared with SVNSs, SVNULSs integrate ULVs and SVNSs, and assign truth-membership, indeterminacy-membership, and falsity-membership functions to a specific ULV. Thus, SVNULSs are effective tools to address the problems which are defined by qualitative expression that involve incomplete, indeterminate, and inconsistent information.

The following section introduces the operations of SVNULNs [35].

*Definition 6 (see [35]). *Suppose and be two SVNULNs, . The algebraic operations between and can be defined as follows:These operational results remain to be SVNULNs.

Theorem 7. *Suppose that and be two SVNULNs, . The operational laws have the following characteristics:*(1)*.*(2)*.*(3)*.*(4)*.*

*Definition 8 (see [35]). *Let be a SVNULN. The score function and accuracy function of can be defined, as follows:

*Definition 9 (see [35]). *Let and be any two SVNULNs. The comparison method between and can be defined as follows:(1)If , then .(2)If and , then .(3)If and , then .(4)If and , then .

*3. Fuzzy Measure and Choquet Integral*

*In real decision problems, a certain degree of interdependent or interactive characteristics often exist among the attributes or experts [60]. However, measuring the importance of attributes by using additive measures is not suitable because the independence of these attributes is often violated [61]. The concept of fuzzy measure introduced by Sugeno [46] is an effective tool for addressing the interaction phenomena among input arguments [45, 60, 62–65].*

*Definition 10 (see [66]). *Let a universal set , while is the power set of . A fuzzy measure on is a set function that meets the following conditions:

(1) and .

(2) If and , then .

(3) , for all and , where .

In particular, if , then condition (3) in Definition 10 is reduced to the axiom of the additive measure , thereby indicating a lack of interaction between B and C; if , then ; that is to say, sets A and B have a multiplicative effect. If , then , thereby expressing that sets A and B have a substitutive effect. The use of parameter can adequately represent the interaction between sets in MAGDM.

Let be a finite set, in which . Sugeno [46] provided the following equation to avoid the computational complexity of fuzzy measure on X:where for all and . Note that for a subset with a single is called a fuzzy density and can be denoted as .

*In particular, we have the following equation for every subset :The value of can be uniquely determined by based on (2) and can be expressed as follows:*

*Definition 11 (see [53, 60]). *Let be a positive real-valued function on and be a fuzzy measure on . The discrete Choquet integral of with respective to is defined as follows:where the subscript indicates a permutation on such that , is the largest value in the set , while when , .

*The Choquet integral can aggregate the attributes even when the mutual preferential independence assumption is violated [67]. Inspired by the Induced ordered weight averaging (IOWA) operator [68], Yager [69] considered a considerably general policy toward ordering the arguments and formulating the ordered argument vector and defined a considerably general type of Choquet integral operator (i.e., I-COA operator), as follows.*

*Definition 12 (see [69]). *Let be a positive real-valued function on and be a fuzzy measure on . An induced Choquet ordered averaging operator of dimension is a function I-COA: , which is defined to aggregate the set of second arguments of a list of 2-tuples based on the following expression:where is a permutation, such that , indicates the 2-tuple with as the largest value in the set , when , and .

*4. Induced Simplified Neutrosophic Linguistic Choquet Integral Operators*

*The I-COA operator [69] can only aggregate crisp numbers and has not been used in conditions where the input arguments are SVNULNs. We use Definitions 6, 10, and 12, as base to (1) extend the I-COA operator to accommodate the conditions of the input arguments are SVNULNs, (2) define the I-SVNULCA and I-SVNULCG operators, and (3) analyze a few necessary properties under the SVNUL environments.*

*Definition 13. *Let be a collection of SVNULNs on and be a fuzzy measure on . An I-SVNULCA operator of dimension is a function I-SVNULCA: , which is defined to aggregate the set of second arguments of a collection of 2-tuples based on the following expression:where the subscript is a permutation, such that . That is, is 2-tuple with being the largest value in the set , , , and .

*Theorem 14. Let be a collection of SVNULNs on and be a fuzzy measure on . Their aggregated value by using the I-SVNULCA operator is also an SVNULN,*

*Proof. *(1) For , according to the operational laws of Definition 2, we haveSince , then(2) When , we obtain (21) by using (18). (3) When , by utilizing (18) and Definition 8, we obtainTherefore, we obtain (18) for any based on the previous results. This condition completes the proof.

*Theorem 15 (commutativity). Let be a collection of SVNULNs, while is any permutation of . Thus, we have*

*Proof. *LetSince is any permutation of , we have , and then

*Theorem 16 (idempotency). Let be a collection of SVNULNs. If for all , then*

*Proof. *Since for all , then

*Theorem 17 (monotonicity). Let and be a collection of SVNULNs. If and are two collections of 2-tuples, such that , , then*

*Proof. *LetSince , it follows that , then

*Theorem 18 (boundedness). Let and , thenwhere is a permutation such that is the 2-tuple with being the th largest value in the set , when and .*

*Proof. *Let be a permutation such that is the 2-tuple with the th largest value in the set , then .

So, we haveSince ,

thus ≤ , , ≤ .

If the order-inducing variable is the argument variable, then the I-SVNULCA operator is reduced to the SVNULCA operator. That is, if , for all , then the I-SVNULCA operator (see (18)) is reduced to the single-valued neutrosophic uncertain linguistic Choquet ordered averaging (SVNULCA) operator.where the subscript is a permutation, such that , , , and .

The SVNULCA operator has the same properties as those of the I-SVNULCA operator, such as commutativity, idempotency, and monotonicity.

*In the following, we propose an I-SVNULCG operator based on Definition 13 and the OWG operator.*

*Definition 19. *Let be a collection of SVNULNs on and be a fuzzy measure on . An I-SVNULCG operator of dimension is a function I-SVNULCG: , which is defined to aggregate the set of second arguments of a collection of 2-tuples based on the following expression:where the subscript is a permutation such that . That is, is 2-tuple with as the largest value in the set , , , and .

*Theorem 20. Let be a collection of SVNULNs on and be a fuzzy measure on . Their aggregated value by using the I-SVNULCG operator is also an SVNULN.The I-SVNULCG operator has the following properties that are similar to those of the I-SVNULCA operator.*

*Theorem 21 (commutativity). Let be a collection of SVNULNs; is any permutation of . Thus, we have the following equation:*

*Theorem 22 (idempotency). Let be a collection of SVNULNs. If for all , then*

*Theorem 23 (monotonicity). Let and be a collection of SVNULNs. If and are two collections of 2-tuples, such that , , then*

*Theorem 24 (boundedness). Let and , thenwhere is a permutation, such that is the 2-tuple with as the th largest value in the set , when and .If the order-inducing variable is the argument variable, then the I-SVNULCG operator is reduced to the SVNULCG operator. That is, if , for all , then the I-SVNULCG operator (see (35)) is reduced to the single-valued neutrosophic uncertain linguistic Choquet ordered geometric (SVNULCG) operator.*