Multiple Criteria Decision Making Approach with Multivalued Neutrosophic Linguistic Normalized Weighted Bonferroni Mean Hamacher Operator
The neutrosophic set and linguistic term set are widely applied in recent years. Motivated by the advantages of them, we combine the multivalued neutrosophic set and linguistic set and define the concept of the multivalued neutrosophic linguistic set (MVNLS). Furthermore, Hamacher operation is an extension of the algebraic and Einstein operation. Additionally, the normalized weighted Bonferroni mean (NWBM) operator can consider the weight of each argument and capture the interrelationship of different arguments. Therefore, the combination of NWBM operator and Hamacher operation is more valuable and agile. Firstly, MVNLS and multivalued neutrosophic linguistic number (MVNLN) are defined, then some new operational rules of MVNLNs on account of Hamacher operations are developed, and the comparison functions for MVNLNs are given. Secondly, multivalued neutrosophic linguistic normalized weighted Bonferroni mean Hamacher operator (MVNLNWBMH) is proposed, and a number of expected characteristics of new operator are investigated. Meanwhile, some special cases of different parameters , and are analyzed. Thirdly, the approach utilizing the MVNLNWBMH operator is introduced to manage multiple criteria decision making (MCDM) issue in multivalued neutrosophic linguistic environment. Ultimately, a practical example is presented and a comparative analysis is carried out, which validate the effectiveness and generalization of the novel approach.
In real world, due to the complexity of decision information, the fuzzy theory has attracted widespread attention and has been developed in various fields. Zadeh  firstly proposed the notion of fuzzy sets (FSs). Then, Atanassov  introduced the intuitionistic fuzzy sets (IFSs), which overcome the weakness of nonmembership degrees. Subsequently, in order to address the hesitation degree of decision makers, Torra  defined hesitant fuzzy sets (HFSs). Fuzzy set theory has been well promoted, but it still cannot manage the inconsistent and indeterminate information. Under this circumstance, Smarandache  proposed neutrosophic sets (NSs), whose indeterminacy degree is independent of both true and false membership. NS is an extension of IFS and makes decision makers express their preference more accurately, so some achievements on NSs and its extensions have been undertaken. Some various concepts of different NSs are defined. For example, Smarandache  and Wang et al.  introduced single-valued neutrosophic sets (SVNs) to facilitate its application. Ye  pointed out the concept of simplified neutrosophic sets (SNSs). Wang et al.  developed the concept of interval neutrosophic sets (INSs). However, under certain conditions, the decision makers likely give different evaluation numbers for expressing their hesitancy. Subsequently, the definition of single-valued neutrosophic hesitant fuzzy sets (SVNHFSs) was firstly proposed by Ye  in 2014, and then Wang and Li  also proposed multivalued neutrosophic sets (MVNSs) in 2015. Actually, the notions of SVNHFSs and MVNSs are equal. For simplicity, we adapt the term of MVNSs in this paper.
On the other hand, the aggregation operators, comparison method for neutrosophic numbers, have also been studied. For SVNSs, Liu and Wang  employed NWBM operator to solve multiple criteria problem in single-valued neutrosophic environment. Ye  gave the definitions of cross-entropy and correlation coefficient. For INSs, Zhang et al.  developed some aggregation operators. Liu and Shi  not only provided the definition of interval neutrosophic hesitant fuzzy sets (INHFSs), but also discussed the generalized hybrid weighted average operator. Broumi and Smarandache [15–17] studied the correlation coefficients, cosine similarity measure, and some new operations. Ye  proposed similarity measures between interval neutrosophic sets. For MVNSs, Ye  developed SVNHFWA and SVNHFWG operators for MCDM problem. Peng et al. [19, 20] extended power aggregation operators and defined some outranking relations under MVNS environment. Ji et al.  analyzed a novel TODIM method for MVNSs.
In real life, owing to the ambiguity of decision makers’ thinking, people prefer to utilize linguistic variables for describing their assessment value rather than the quantization value. Therefore, linguistic variable has attracted widespread attention in the field of MCDM. The linguistic variable was firstly proposed by Zadeh  and applied for the fuzzy reasoning. After that, a series of works on it have been made. Wang et al. [23–25] presented a new approach in view of hesitant fuzzy linguistic information. Meng et al.  developed linguistic hesitant fuzzy sets and studied hybrid weighted operator. Tian et al.  defined gray linguistic weighted Bonferroni mean operator for MCDM.
In order to indicate the true, indeterminate and false extents concerning a linguistic term, the NSs and linguistic set (LS) are combined. Several neutrosophic linguistic sets and their corresponding operators are defined, for example, single-valued or simplified neutrosophic linguistic sets and trapezoid linguistic sets [28–31], interval neutrosophic certain or uncertain linguistic sets [32–34]. However, due to the hesitancy of people’s thinking, the trueness of a linguistic term may be given several values, and the case is similar to the false and indeterminate extents. The existing literature does not consider this perspective. Therefore, the multivalued neutrosophic linguistic set (MVNLS) and multivalued neutrosophic linguistic number (MVNLN) in this article are proposed in order to better express the information.
Aggregation operator which can fuse multiple arguments into a single comprehensive value is an important tool for MCDM problem. Many researchers have developed some efficient operators [35–42], for instance, the weighted geometric average (WGA) or averaging (WA) operator, prioritized aggregation (PA) operator, Maclaurin symmetric mean operator, and Bonferroni mean (BM) operator. BM operator was originally defined by Bonferroni  and has attracted widespread attention because of its characteristics of capturing interrelationship among arguments. Some achievements have been made on it [11, 44–49]. In order to aggregate neutrosophic linguistic information, some researches on aggregation operators under neutrosophic linguistic and neutrosophic uncertain linguistic environments have also been applied [28–34, 50]. Until now, BM and NWBM fail to accommodate aggregation information for multivalued neutrosophic linguistic environment. Motivated by this limitation, we will extend the NWBM operator to MVNLS in this article.
T-norms and t-conorms are two functions that satisfy certain conditions, respectively. The Archimedean t-conorms and t-norms are well known, which include algebraic, Einstein, and Hamacher. Hamacher operation is an extension of algebraic and Einstein. Generally, the algebraic operators are common; there are also a few aggregation operations based on Einstein operations. Because Hamacher operator is more general, Liu et al. [51, 52] discussed the Hamacher operational rules. So far, there is no research for MVNLS based on Hamacher operations. Since it is better for MVNLS to depict the actual situation, NWBM operator can capture the interrelationship among arguments, and Hamacher operations are more general, it is of great meaning to study the NWBM Hamacher operators under multivalued neutrosophic linguistic environment for MCDM problems.
The main purposes of the paper are presented as follows:(1)To better express people’s hesitancy, combining the MVNS and LS, we give the notions of MVNLS and MVNLN; besides, the score, accuracy, and certainty functions are also investigated to compare MVNLNs.(2)Due to the generalization of Hamacher operational rules, we define new operations of MVNLNs based on Hamacher operational rules and discuss their operational relations.(3)The NWBM considering the interrelationship of different arguments has gained widespread concerns; we extend NWBM operator to MVNLN environment, the MVNLNWBMH operator is defined, and some desirable characteristics are also studied.(4)In order to verify the effectiveness, an example for MCDM problem utilizing MVNLNWBMH operator is illustrated and a comparative analysis is conducted. We also analyze the influences of different parameter values for the final outcomes, and the results demonstrate that the operator proposed is more general and flexible.
The article is arranged in this way. In Section 2, we review a number of notions and operations for MVNS, LS, NWBM operator and Hamacher. In Section 3, we propose the definitions of MVNLS and MVNLN and develop the operations of MVNLNs on the basis of Hamacher t-conorms and t-norms. Meanwhile, the algebraic as well as Einstein operations for MVNLNs are also presented, which are special cases of Hamacher operation. Moreover, the comparison method of MVNLNs is also defined. In Section 4, we propose the MVNLNWBMH operator and investigate its properties. Furthermore, when corresponding parameters are assigned different values, the special examples are also discussed. In Section 5, we establish the MCDM procedure on account of the proposed aggregation operators with MVNLS information. Section 6 presents a concrete example, and a comparison analysis is provided to show the practicability of utilizing our method. Finally, in Section 7, some results are presented.
Some notions and operation are introduced in this section, which will be useful in the latter analysis.
2.1. Linguistic Term Sets
Suppose that is an ordered and finite linguistic set, in which denotes a linguistic variable value and is an odd value. When is equal to seven, the corresponding linguistic sets are provided as follows:In order to avoid the linguistic information loss, the set above is expanded, that is, a contiguous set,
Definition 1 (see ). Let and be any two linguistic variables, the corresponding operations are presented:
2.2. Multivalued Neutrosophic Sets
Definition 2 (see [9, 10]). Suppose that is a collection of objects; MVNSs on is defined by where , , , and , , and are three collections of crisp numbers belonging to , representing the probable true-membership degree, indeterminacy-membership degree, and falsity-membership degree, where in belongs to , respectively, satisfying these conditions , and . If there is only one element in , is indicated by the three-tuple , that is, known as a multivalued neutrosophic number (MVNN). Generally, MVNSs are considered as the generalizations of the other sets, such as FSs, IFSs, HFSs, DHFs, and SVNSs.
2.3. Normalized Weighted Bonferroni Mean
Definition 3 (see ). Let as well as be a set of nonnegative values; then the BM is defined as
Definition 4 (see ). Let and be a set of nonnegative values, and the corresponding NWBM can be expressed as follows:where represents the corresponding weighted vector of , satisfying and . The weight vector can be given by decision makers in real problem.
Obviously, the NWBM operator possesses a few characteristics such as commutativity, reducibility, monotonicity, boundedness, and idempotency.
2.4. Hamacher Operations
We know aggregation operator is given in accordance with different t-norms and t-conorms; there are some exceptional circumstances listed as follows:(1)algebraic t-norm and t-conorm(2)Einstein t-norm and t-conorm(3)Hamacher t-norm and t-conorm
In particular, when , , the algebraic and Einstein operations are the simplifications of Hamacher t-norm and t-conorm.
3. Multivalued Neutrosophic Linguistic Set
3.1. MVNLS and Its Hamacher Operations
Definition 5. Let be a set of points; an MVNLS in is defined as follows:where , , , , and , , and are three sets of crisp values in , denoting three degrees of in belonging to , which are trueness, indeterminacy, and falsity, satisfying these conditions , and
Definition 6. Let be an MVNLS; supposing there is only one element in , then tuple is depicted as a multivalued neutrosophic linguistic number (MVNLN). For simplicity, the MVNLN can also be represented as
Definition 7. Let and be two MVNLNs, and ; then the operations of MVNLNs can be defined on the basis of Hamacher operations.If , then the operations based on Hamacher operational rules in Definition 7 will be simplified to the Algebraic operational rules as follows:Supposing , , , , , and contain only one value, then the operations defined above can be reduced to the operations of SVNLNs based on algebraic operations proposed by Ye .
If , then the operations based on Hamacher operational rules in Definition 7 will be simplified to the Einstein operations of MVNLNs presented below:Supposing , , , , , and contain only one value, then the operations defined above can be reduced to the operations of SVNLNs based on Einstein operations.
Theorem 8. Let , , and be any three MVNLNs, and ; then the properties below are correct:Then, (4) will be proved as follows.
Proof of (4). Since , Therefore, can be obtained.
Similarly, the other equations in Theorem 8 are easily certified in the light of Definition 7.
3.2. Comparison Method
The score, accuracy, and certainty functions are important indexes to rank MVNLNs, and their corresponding definition is given below.
Definition 9. Let be an MVNLN, and the score, accuracy, and certainty functions are achieved as below.where , , and are the numbers of the values in , , and , respectively.
The linguistic variable is important for an MVNLN. Therefore, the comparison functions defined above in Definition 9 are denoted as the linguistic variable. The bigger the truth degree concerning the variable is, the smaller the indeterminacy degree and the false degree concerning the linguistic variable are, and the higher the MVNLN is. Regarding the function of score, the greater the corresponding to is, the higher the affirmative statement is. Regarding the function of accuracy, the greater the minus is, the more certain the statement is. Regarding the function of certainty, the bigger the is, the more certain the statement is.
Based on Definition 9, the comparison method between MVNLNs is obtained.
Definition 10. Supposing and are two MVNLNs, the compared approach is achieved as follows:(1)Supposing that , then is greater than , represented as .(2)Supposing that , and , then is greater than , represented as .(3)Supposing that , , and , then is greater than , represented as .(4)Supposing that , , and , then equals , represented as .
4. The Multivalued Neutrosophic Linguistic Normalized Weighted Bonferroni Mean Hamacher Operator
The NWBM operator not only can take into account the advantages of BM and WBM, but also has the property of reducibility and idempotency. However, the NWBM operator has not been applied to the cases where the input arguments are MVNLNs.
Definition 11. Let be a space of MVNLNs, , and be the weighted vector for and . Then the operator of MVNLNWBMH is achieved as below, the aggregation result is still an MVNLN. According to the operational laws in Definition 7, the results are derived below:where
Proof. According to the operational rules for MVNLNs, the results below can be gainedFirstly, we need to testify the mathematical formula below.The mathematical induction on is adopted to prove (21).
Supposing , the equation below is obtained. where