Abstract
Many aggregation operators in multiattribute decisions assume that attributes are independent of each other; this leads to an unreasonable situation in information aggregation and decision-making. Heronian mean is the aggregation operator that can embody the interaction between attributes. In this paper, we merge the linguistic neutrosophic cubic number (LNCN) and the Heronian mean operator together to develop a LNCN generalized weighted Heronian mean (LNCNGWHM) operator and a LNCN three-parameter weighted Heronian mean (LNCNTPWHM) operator and then discuss their properties. Further, two multiattribute decision methods based on the proposed LNCNGWHM or LNCNTPWHM operator are introduced under LNCN environment. Finally, an example is used to indicate the effectiveness of the developed methods.
1. Introduction
In multiple attribute decision-making problems, the general attribute values are usually difficult to be given as the form of real numbers and instead of the form of uncertain variables, such as fuzzy numbers, intuitionistic fuzzy numbers, and linguistic variables (LVs). Fuzzy set (FS) was proposed by Zadeh to measure things that cannot be described with accurate information or accurate probability distribution [1]. The intuitionistic fuzzy set (IFS) proposed by Atanassov consists of the membership and nonmembership information, which is an effective extension of fuzzy set theory [2]. LVs are used to evaluate attributes in multiattribute decision-making while attributes cannot be evaluated by numerical values. Linguistic set (LS) L = ( is an even number) was proposed by Zadeh to deal with the approximate reasoning problems [3, 4]. Since FS, IFS, and LS are effective tools to deal with the approximate reasoning problems, many researches have studied them [5–11]. However, in the literature [7–11], only incomplete information can be effectively expressed, and then indeterminate and inconsistent information cannot be described effectively. In order to make up for the deficiency of literature [7–11], Smarandache put forward the theory of neutrosophic set (NS) consisting of three parts: truth, falsity, and indeterminacy [12]. Wang and Smarandache also proposed a single value neutrosophic set (SVNS), which satisfies, and ++ [13]. IF the three components, , and in NS are interval numbers, Wang and Zhang introduced an interval neutrosophic set (INS) [14]. Next, some scholars put NS and LS together to develop their new set of concepts. First, Ye defined the interval neutrosophic uncertain linguistic set (INULS) and its multiple attribute decision-making (MADM) method [15]. Then, Ye proposed single valued neutrosophic linguistic numbers (SVNLNs) for multiple attribute group decision-making (MAGDM) [16]. Further, Fang and Ye proposed the linguistic neutrosophic number (LNN) consisting of the truth, falsity, and indeterminacy linguistic degrees [17]. Recently, Ye also proposed a new concept of a linguistic neutrosophic cubic number (LNCN) to extend neutrosophic cubic sets to linguistic neutrosophic arguments [18]. So far, there is little study on LNCN. However, many aggregation operators in multiattribute decisions assume that attributes are independent of each other; this leads to an unreasonable situation in information aggregation and decision-making. Then, Heronian mean is the aggregation operator that can embody the interaction between attributes. So, in this paper, we propose the LNCN generalized weighted Heronian mean (LNCNGWHM) operator and the LNCN three-parameter weighted Heronian mean (LNCNTPWHM) operator and investigate their properties.
The remaining organizations of this paper are listed as follows. Section 2 describes some concepts of LNN, LNCN, GWHM, and TPWHM. Section 3 proposes the LNCNGWHM and LNCNTPWHM operators and investigates their properties. Section 4 establishes MADM methods by using the LNCNGWHM and LNCNTPWHM operators. Section 5 provides an illustrative example to demonstrate the application and effectiveness of the proposed methods. Section 6 gives conclusions of this paper.
2. Some Basic Concepts
2.1. Linguistic Neutrosophic Numbers, Linguistic Neutrosophic Cubic Numbers, and Their Operational Laws
Definition 1 ([17]). Set as a language term set; is an even number. Then a LNN can be defined as follows:in which and ,, and represent the truth, indeterminacy, and falsity variables, respectively, in linguistic terms.
Definition 2 ([17]). Set , , and as three LNNs in and real number; then the operational laws of LNNs are as follows:
Definition 3 ([18]). Set as a language term set; is an even number. Then a LNCN can be defined as follows:where expresses an uncertain LNN and represent, respectively, the truth, the indeterminacy, and the falsity uncertain linguistic variables for and ,; expresses a LNN, in which and , and represent the truth, the indeterminacy, and the falsity linguistic variables, respectively, in linguistic terms.
Definition 4 ([18]). Set , as a LNCN in , then, one calls ①an internal LNCN if;②an external LNCN if, and
Definition 5 ([18]). Set =, , and as two LNCNs in and real number; then the operational laws of LNCNs are as follows:
Definition 6 ([18]). Set , as a LNCN in ; then the score , accuracy , and certain functions can be defined, respectively, as follows:
Definition 7 ([18]). Set , , , as two LNCNs in ; then if >, then; if = then if >, then; if and , then; if = and , then.
2.2. Generalized Weighted Heronian Mean and Three-Parameter Weighted Heronian Mean Operators
Definition 8 ([19]). Let be the relative weight of for , , and and . Thenwhere is called a generalized weighted Heronian mean (GWHM) operator.
Definition 9 ([19]). Let = be the relative weight of,, and and . Thenwhere is called a three-parameter weighted Heronian mean (TPWHM) operator.
3. Two Aggregation Operators of LNCNs
3.1. GWHM Operator of LNCNs
Definition 10. Set = , () as a collection of LNCNs in ; then the LNCNGWHM operator can be defined aswhere, and .
Then we can use Definitions 5 and 10 to get the following theorem.
Theorem 11. Set , () as a collection of LNCNs in L; then by LNCNGWHM operator, the aggregation result of is still a LNCN, which is given by the following form:where , and.
Proof. (1)(2)(3) (4) (5) (6) (7)The proof of Theorem 11 is completed.
Theorem 12 (idempotency). Set = as a collection of LNCNs in ; if = m, then
Proof. Since = m for , there is the following result: The proof of Theorem 12 is completed.
Theorem 13 (monotonicity). Set = and = () as two collections of LNCNs in ; if , , , , , , , , then
Proof. Because, , , , , and , , , we can easily obtainSimilarlyAndSo, . Therefore, the proof of Theorem 13 is completed.
Theorem 14 (boundedness). Set = as a collection of LNCNs in L; letting , , and , , , then
Proof. Based on Theorems 12 and 13, we can obtainThen .
The proof of Theorem 14 is completed.
3.2. Three-Parameter Weighted Heronian Mean Operator of LNCNs
Definition 15. Set = , as a collection of LNCNs in L; then the LNCNTPWHM operator can be defined as where , , and .
Then, we can use Definitions 5 and 15 to get the following theorem.
Theorem 16. Set , as a collection of LNCNs in ; then by LNCNTPWHM operator, the aggregation result of is still a LNCN, which is as follows: where , , and .
Theorem 17 (idempotency). Set = () as a collection of LNCNs in ; if , then
Theorem 18 (monotonicity). Set = and = () as two collections of LNCNs in ; if , , , , , , , , then
Theorem 19 (boundedness). Set = as a collection of LNCNs in L; letting , , and , , , then
The proofs of Theorems 16–19 are similar to those of Theorems 11–14, so we do not repeat them again.
4. MADM Methods Based on the LNCNGWHM or LNCNTPWHM Operator
This section uses the LNCNGWHM or LNCNTPWHM operator to deal with the MADM problems with LNCN information.
Let be a discrete set of alternatives with a set of attributes, and the weight vector of is , and . A linguistic assessment set is given. Some experts use LNCN to evaluate the alternatives under the attributes. The assessed values of the experts for with attribute are . Then, we can get the neutrosophic linguistic cubic decision evaluation matrix.
Then, the decision-making method based on the LNCNGWHM or LNCNTPWHM operator is described as follows.
Step 1. According to the weight vector and the LNCNGWHM or LNCNTPWHM operator, we can calculate or.
Step 2. Calculate the score value of S() (accuracy value of A() and certain value of C() if necessary) of the LNCN according to formula (11) (formula (12) and formula (13) if necessary).
Step 3. According to ranking method of Definition 7, we can rank the attributes corresponding to the values of S() (accuracy value A() and certain value C() if necessary).
Step 4. End.
5. Illustrative Example
This section considers a decision-making problem adapted from the literature [20]. A mechanical designer wants to design press machine; then he should consider the design of the reducing mechanism and the working mechanism. According to the press machine’s functional requirements, there are four design schemes/alternatives to be proposed by the designers, which are shown in Table 1. The four design schemes must satisfy the requirements of four attributes while being evaluated: the manufacturing cost (, the mechanical structure (, the transmission effectiveness (and the reliability (. The importance of four attributes is given as a weight vector . Then, the experts define the linguistic term set, where , , , , , , , , . Afterwards, they evaluate the four design schemes/alternatives under the four attributes by the form of LNCNs based on . Thus, the LNCN decision matrix can be established, which are shown in Table 2.
Next, the decision-making methods proposed in Section 4 are employed to deal with the decision problem; the description of decision procedures is shown as follows:
5.1. The Decision-Making Process Based on LNCNGWHM Operator or LNCNTPWHM Operator
Step 1. By using (17) (suppose p = q = 1) and the weight vector of attributes, we can obtain the comprehensive evaluation values of alternative as follows:
Step 2. Calculate the score values of according to (11) for :
According to the results of , we can rank the alternatives , and then the design scheme is the best among all the alternatives.
On the other hand, we can use the LNCNTPWHM operator (set p=1, q=1) to deal with this problem.
Step 1 ′. By using (34) (suppose p = q = 1) and the weight vector of attributes, we can obtain the comprehensive evaluation values of alternative as follows:
Step 2 ′. Calculate the score values of S() according to (11) for :
According to the results of S() , we can rank the alternatives , so the design scheme is the best among all the alternatives.
5.2. Analyzing the Effect of the Parameters p, q, and r
Different parameters , , and may have different effects on the decision results. Therefore, this section takes different values of , , and to sort the various alternatives, and then Tables 3-4 present the results.
From Tables 3 and 4, we can see that LNCNTPWHM operator can get more stable sorting than the LNCNGWHM operator, and, in addition, when the parameters , , and take different values, the best design scheme/alternative using either the LNCNGWHM or LNCNTPWHM operator is always . Therefore, the parameters , , and in the LNCNGWHM or LNCNTPWHM operator have little influence on decision-making.
5.3. Comparing with the Related Methods
Firstly, compared with the literature [20], this paper used the decision information under LNCN environment, while the literature [20] used the decision information under intuitionistic fuzzy environment. In the literature [20], only incomplete information can be effectively expressed, and the indeterminate and inconsistent information cannot be described effectively, while LNCN is composed of uncertain linguistic neutrosophic number and linguistic neutrosophic number, where the uncertain linguistic neutrosophic number is represented by the truth, the indeterminacy, and the falsity uncertain linguistic variables, respectively, and the linguistic neutrosophic number is represented by the truth, indeterminacy, and falsity linguistic variables, respectively. So LNCN contains more information than the intuitionistic fuzzy number in [20].
Second, compared with the existing related methods based on the LNCNWAA and LNCNWGA operators in literature [18], all the ranking results have been shown in Table 5.
The results given in Table 5 show that all the aggregated values of the LNCNGWHM and LNCNTPWHM operators are more or less close to moderate values between the aggregated values of the LNCNWAA and LNCNWGA operators. Then, all the ranking orders based on the LNCNWAA LNCNWGA, LNCNGWHM, and LNCNTPWHM operators are identical. However, the LNCNGWHM and LNCNTPWHM operators embody the interaction between attributes and consider the different , , and values to make the decision-making results more persuasive and comprehensive than the LNCNWAA and LNCNWGA operators in literature [18].
6. Conclusions
This paper proposed MADM methods based on the LNCNGWHM and LNCNTPWHM operators for LNCNs. First, a LNCN generalized weight Heronian mean (LNCNGWHM) operator and a LNCN three-parameter weighted Heronian mean (LNCNTPWHM) operator were proposed and the related properties of these two operators were discussed. Second, the two methods of MADM in a LNCN setting were put forward based on the LNCNGWHM operator and the LNCNTPWHM operator. Finally, these two methods are used to solve a practical problem. In order to make the decision-making result more convincing, the different values of the parameters p, q, and r were taken to observe the sorting results. From the sorting results, we found that the influence of three parameters on the decision results was very small. Furthermore, compared with the relative method, the proposed methods in this paper can get the same selection result as the existing method. Therefore, the proposed methods demonstrate potential applications in handling MADM problems under LNCN environment.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest.
Authors’ Contributions
Changxing Fan proposed the LNCNGWHM and LNCNTPWHM operators and investigated their properties, Changxing Fan presented the organization and decision-making method of this paper, and Jun Ye provided the calculation and analysis of the illustrative example; the authors wrote the paper together.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (under Grant no. 61703280, no. 61603258), Science and Technology Planning Project of Shaoxing City of China (no. 2017B70056), and Education Department of Zhejiang (no. Y 201635390).