Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator

Single-valued neutrosophic (SVN) sets can successfully describe the uncertainty problems, and Schweizer-Sklar (SS) t-norm (TN) and t-conorm (TCN) can build the information aggregation process more ﬂexible by a parameter. To fully consider the advantages of SVNS and SS operations, in this article, we extend the SS TN and TCN to single-valued neutrosophic numbers (SVNN) and propose the SS operational laws for SVNNs. Then, we merge the prioritized aggregation (PRA) operator with SS operations, and develop the single-valued neutrosophic Schweizer-Sklar prioritized weighted averaging (SVNSSPRWA) operator, single-valued neutrosophic Schweizer-Sklar prioritized ordered weighted averaging (SVNSSPROWA) operator, single-valued neutrosophic Schweizer-Sklar prioritized weighted geometric (SVNSSPRWG) operator, and single-valued neutrosophic Schweizer-Sklar prioritized ordered weighted geometric (SVNSSPROWG) operator. Moreover, we study some useful characteristics of these proposed aggregation operators (AOs) and propose two decision making models to deal with multiple-attribute decision making (MADM) problems under SVN information based on the SVNSSPRWA and SVNSSPRWG operators. Lastly, an illustrative example about talent introduction is given to testify the eﬀectiveness of the developed methods. (cid:1) 2018 Elsevier B.V. All


Introduction
The main purpose of MADM problems is to select the best alternative from the limited alternatives according to the preference values given by decision makers (DMs) with respect to the criteria. However, because of the complexity of decision environment, it is difficult for DMs to express the preference values by a single real number in practical problems. To deal with such situation, intuitionistic fuzzy set (IFS) proposed by Atanassov (1986) is one of the flourishing generalizations of fuzzy set (FS) introduced by Zadeh (1965) to express uncertain and imprecise information more accurately (Liu, Mahmood, & Khan, 2017;Xu and Yager, 2006;Xu, 2007). However, in some situations, only truth-membership degree (TMD) and falsitymembership degree (FMD) cannot describe the inconsistent information accurately. To deal with such situation, Smarandache (1999) proposed neutrosophic set (NS) which describe the uncertain, imprecise and inconsistent information by TMD, indeterminacy-membership degree (IMD), and FMD. The three functions are independent and are standard or non-standard subsets 0 À ; 1 þ ½. As the NS has the IMD, therefore it can describe the uncertain information more accurately than FS and IFS, and it is also more consistent with human natural feelings and judgement. But NS is hard to be used in real problem due to the contained non-standard subsets of 0 À ; 1 þ ½. Therefore, in order to utilize NS easily in real problems, Wang, Smarandache, Zhang, and Sunderraman (2010) developed a SVNS, which is subclass of NS.
In real decision making, we need AOs to integrate the given information. In neutrosophic environment, many scholars have developed some AOs. For example, Ye (2014) firstly developed the operational rules for SVNNs and introduced the weighted averaging operator for SVNNs (SVNWA) and geometric average operator for SVNNs. Later on, Peng, Wang, Wang, Zhang, and Chen (2016) found out some limitations in the operational rules developed by Ye (2014), and introduced some improved operational laws for SVNNs, and proposed ordered weighted average for SVNNs (SVNOWA) and ordered weighted geometric operator for SVNNs (SVNOWG). Lu and Ye (2017) further developed some hybrid averaging and hybrid geometric operators for SVNNs and applied them to MADM. After these studies, several researchers developed different AOs, such as Liu, Chu, Li, and Chen (2014) proposed some generalized Hamacher AOs for NS and applied them to multiple-attribute group decision making (MAGDM). Wu, Wu, Zhou, Chen, & Guan, 2018) defined some AOs based on Hamacher TN and TCN and applied them to deal with group decision making under SVN 2-tuple linguistic environment. Garg (2016) developed some AOs based on Frank TN and TCN and applied them to solve MADM problems under SVN environment. Zhang, Liu, and Shi (2016) extended TODIM to neutrosophic environment and applied it to MAGDM problem. Mandal and Basu (2018) developed some vector AOs for solving MADM problems under neutrosophic environment. Recently, Karaaslan and Hayat (2018) proposed some new operational laws for SVN matrices and give their application in MAGDM. Garg (2017) developed some parametric distance measures for SVNSs and give their applications in pattern recognition and medical diagnosis. , Abdel-Basset, Zhou, Mohamed, and Chang (2018) developed AHP-SWOT, ANP-TOPSIS and VIKOR for NSs and applied them to solve strategic planning, supplier selection and e-government website evaluation problems. Peng and Jingguo (2018) developed MABAC, TOPSIS and new similarity measure for SVNSs and proposed three approaches for MADM. Abdel-Basset, Mohamed, Zhou, and Hezam (2017), Abdel-Basset, Mai, Smarandache, and Chang (2018) developed MAGDM method based on neutrosophic analytic hierarchy process and neutrosophic association rule mining algorithm for big data analysis. Abdel-Basset, Gunasekaran, Mohamed, and , Abdel-Basset, Manogaran, Gamal, and Smarandache (2018) developed a novel method for solving the full neutrosophic linear programming problems, and also developed a hybrid approach based on NSs and DEMATEL to solve supplier selection problems. Some other applications of NS were studied by researchers in Abdel-Basset, Gunasekaran, & Mai, 2018;Chang, Abdel-Basset, & Ramachandran, 2018).
All the above-stated operators are established based on the expectation that the aggregated input arguments are independent. But in some situations, it may be possible that there exists interaction between the decision making criteria under neutrosophic environment. To deal with such situation, Liu and Wang (2014) extended Bonferroni mean (BM) to neutrosophic environment and developed some normalized BM operators for SVNNs and applied them to MAGDM. Li, Liu, and Chen (2016) developed some Heronian mean (HM) operators for SVNNs and applied them to MAGDM problems under SVN environment. Yang and Li (2016) and Liu and Tang (2016) applied the power average operator to the neutrosophic environment and proposed a SVN power average operator and a generalized interval neutrosophic (IN) power averaging (GINPA) operator respectively, which have the property that they can remove the negative impact of the extreme evaluation values on the final ranking results. Wang, Yang, and Li (2016) introduced Maclaurin symmetric mean (MSM) operators to take the interrelationship among the aggregated arguments. Liu and You (2017) proposed Muirhead mean (MM) operator to deal with IN information. These existing AOs have not considered the situation in which the criteria have priority relationship among them. To solve this problem, Wu, Wang, Peng, and Chen (2016) extended prioritized aggregation (PA) operators (Yager, 2008) to SVN environment and proposed SVN prioritized weighted averaging (SVNPWA) and SVN prioritized weighted geometric aggregation (SVNPWG) operators, and applied them to MADM. Moreover, Liu and Wang (2016) developed some prioritized ordered weighted average/geometric operator to deal with neutrosophic information. Ji, Wang, and Zhang (2018) combined PA operators with BM operator and introduced some SVN prioritized BM operators by utilizing Frank operations. Recently, Wei and Wei (2018) proposed some PA operators based on Dombi TN and TCN and applied them to MADM.
From the above stated AOs, most of these AOs for NS or SVNS are based on algebraic, Hamacher, Frank and Dombi operational laws, which are special cases of Archimedean TN (ATN) and TCN (ATCN). Certainly, ATN and ATCN are the extensions of many TNs and TCNs, which have some special cases chosen to express the union and intersection of SVNS (Liu, 2016). Schweizer-Sklar (SS) operations (Deschrijver and Kerre, 2002) are the special cases from ATN and ATCN, they are with a variable parameter, so they are more flexible and superior than the other operations. However, the most researches about SS mainly concentrated on the fundamental theory and characteristics of Schweizer-Sklar TN (SSTN) and TCN (SSTCN) (Deschrijver, 2009;Zhang, He, & Xu, 2006). Recently, Liu and Wang (2018), Zhang (2018) combine SS operations with interval-valued IFS (IVIFS) and IFS, and proposed power average/geometric operators and weighted averaging operators for IVIFSs and IFSs respectively.
From the above discuss, we can know (1) SVNNs are better to describe uncertain information by defining TMD, IMD and FMD than FSs and IFSs in solving the MADM problems; (2) The SS operations are more flexible and superior than the other operations by a variable parameter; (3) there are many MADM problems in which the criteria have priority relationship, and some existing AOs can consider this situation only when the criteria take the form of real numbers. Now, there are no such AOs to deal with MADM problems under SVN information based on SSTN and SSTCN, so, in this paper, we combine the ordinary PA operator with SS operations to deal with the information of SVNNs.
Based on the above research motivation, the goals and contributions of this article are shown as follows.
(3) Proposing two MADM approaches based on the proposed AOs. (4) Verifying the effectiveness and practicality of the proposed approach.
To do so, the rest of this article is organized as follows. In Section 2, we initiated some basic ideas of SVNSs, PA operators, Shweizer-Sklar operations. In Section 3, we develop some Schweizer-Sklar operational laws for SVNNs. In Section 4, we propose SVNSSPRAWA and SVNSSPRAOWA operators, and discuss some properties and special cases of the proposed AOs. In Section 5, we propose SVNSSPRAWG and SVNSSPRAOWG operators, discuss some properties and special cases of the proposed AOs. In Section 6, we develop two MADM approaches based on these AOs. In Section 7, we solve a numerical example to show the validity and advantages of the proposed approach by comparing with other existing methods.

Some concepts of SVNSs
In this subpart, we review some basic concepts about SVNSs, SVNNs, score and accuracy functions, and their operational rules.
Definition 1 ( (Wang et al., 2010)). Let N be the domain set, with a general element expressed by w. A SVNS f SV in N is mathematically symbolized as be two SVNNs. Then the comparison rules of SVNNs are described as follow: (1) If f SO sv 1 ð Þ < f SO sv 2 ð Þ, then sv 2 is greater than sv 1 , and is denoted as sv 2 > sv 1 ; (2) If f SO sv 1 ð Þ ¼ f SO sv 2 ð Þ, and f AR sv 1 ð Þ < f AR sv 2 ð Þ then sv 2 is greater than sv 1 , and is denoted as sv 2 > sv 1 ; ð Þ then sv 1 is equal to sv 2 , and is denoted as sv 1 ¼ sv 2 :

PA operator
Definition 6 ( (Yager, 2008)). Let e C ¼ e C 1 ; e C 2 ; :::; e C g be set of attributes, and assure that there exist prioritization among the attributes expressed by a linear ordering e C 1 > e C 2 > ::: > e C gÀ1 > e C g , which indicates that the criterion e C a has a higher priority than e C b , if a < b: e C a s ð Þ is an evaluation value expressing the performance of the alternative s under the attribute e C a and satisfies e C a 2 0; 1 ½ .
If PA e C 1 s ð Þ; e C 1 s ð Þ; :: Obviously, the PA operator has been effectively applied to the situation where the attributes are real values.

Schweizer-Sklar (SS) operations
The SS operations consist of the SS product and SS sum, which are special cases of ATT, respectively.
where e T and e T Ã respectively, express T-norm (TN) and T-conorm (TCN).
The Schweizer-Sklar TN and TCN (Deschrijver and Kerre, 2002) are defined as follows: where q < 0; m; n 2 0; 1 ½ . Additionally, when q ¼ 0; we have e T q m; n ð Þ ¼ mn and e T Ã q m; n ð Þ ¼ m þ n À mn. That is, SS TN and TCN reduce to algebraic TN and TCN. Now, in the next section, based on TN e T q m; n ð Þ and TCN e T On the basis of Definition (7) and Eqs. (13), and (14), the SS operations of SVNNs are described as follows q < 0 ð Þ: be any two SVNNs, then Proof. We will prove Eq. (26) by utilizing mathematical induction (MI). The following steps of MI have been followed: SVNSSPRWA sv 1 ;sv 2 ;:::: Step 1. For g ¼ 2; we have From the operational laws for SVNNs, proposed in Definition 8, we have So, Eq. (27) becomes i.e. when g ¼ 2, Eq. (26) is true.

Single-valued neutrosophic Schweizer-Sklar prioritized weighted geometric operator
In this part, we develop single-valued neutrosophic Schweizer-Sklar prioritized weighted geometric (SVNSSPRWG) and single-valued neutrosophic Schweizer-Sklar prioritized ordered weighted geometric (SVNSSPROWG) operators. We also discuss some characteristics of the developed aggregation operators.
Theorem 6. For a group of SVNNs sv p ¼ TR p ; ID p ; FL p D E ; p ¼ 1; 2; :::; g ð Þ , the value aggregated by the developed SVNPRWG operator is still a SVNN and is specified by: Proof. We will prove Eq. (43) by utilizing MI. The following steps of MI have been followed: Step 1. For g ¼ 2; we have From the operational laws for SVNNs, proposed in Definition 8, we have SVNSSPRWG sv 1 ;sv 2 ;:::: and So, Eq. (44) becomes i.e., when g ¼ 2, Eq. (43) is true.

The method based on SVNSSPRWA operator
In the following, a process for ranking and selecting the most preferable alternative(s) is provided as follows.
Step 1. Standardize the decision matrix.
First, the decision making information m rs in the matrix M ¼ m rs À Á hÂg must be standardized. Consequently, the attribute can be grouped into the cost and benefit types. For benefit type attribute, the assessment information does not need to changed, but for cost type attribute, it must be modified with the complement set.
Step 6. End. Step 1 and step 2 are same.
Step 3. Use the SVNSSPRWG operator to get the overall SVNN m r r ¼ 1; Step 5. According to score values, ranking order of alternatives is G 3 > G 1 > G 2 > G 4 : So, the best provider is G 3 , while the worst one is G 4 :

Effect of the parameter q on decision result of this example
In order to see the effect of the parameter q on the decision-making result, we set the distinct values for the parameter q in step 3, to rank the alternatives. The score values and ranking order are described in Tables 3 and 4. As from Table 3, we can notice that the ranking orders by utilizing SVNSSPWA operator are slightly different when the parameter q takes the distinct values. When the value of the parameter q tends to zero, the best choice is G 3 and the worst choice is G 2 . When the value of the parameter q decreases from 0 then the best choice is G 3 while the worst one is G 4 . We can also see from Table 3, when the value of the parameter decreases the score values become bigger and bigger.
From Table 4, we can see that the ranking orders by utilizing SVNSSPWG operator do not change for different values of the parameter q, the best choice is G 3 , while the worst one is G 4 . We can also notice from Table 4, when the value of the parameter q decreases, the score values become smaller and smaller. Generally, different DMs can set different values of the parameter q according to their actual need.
Example 6 ( (Wei and Wei, 2018)). In order to reinforce the academic education, the school of management in a Chinese university wants to introduce excellent overseas teachers. This introduction caught much attention from the school, university president, dean of management school and human resource officer sets of a panel of decision makers who will take the whole responsibility for this introduction. The panel made strict assessment for five alternatives (candidates) G r r ¼ 1; 2; :::; 5 ð Þfrom four characteristics (attributes) namely, morality H 1 ; research potential H 2 ; skill of teaching H 3 , education background H 4 . The president of the university has absolute priority in decision making, and the dean of the school of management is next. In addition, this introduction will be in a strict

Comparison with the other methods
In order to further show the effectiveness of the proposed methods based on the proposed AOs, in this article, we solve Example 6 by seven existing methods based on different aggregation operators under SVN environment. SVN weighted averaging (SVNWA) operator proposed by Ye (2014), SVNWA operator proposed by Peng et al. (2016) based on improved operational laws for SVNNs, SVN-MABAC (Peng and Jingguo, 2018), SVN-TOPSIS (Peng and Jingguo, 2018), SVN prioritized weighted averaging (PRWA) operator developed by Wu et al. (2016), SVN Dombi prioritized weighted averaging (PRWA) operator developed by Wei and Wei (2018) and SVNN normalized BM (SVNNBM) operator developed by Liu and Wang (2014). The score values and ranking order are given in Table 8.
The weight vector of attributes for these methods is obtained using the PRA operator.
From Table 8, we can see that when value of the parameter q tends to zero, the ranking orders obtained by the proposed method based on the proposed aggregation oper-ators are same with the other five methods. This shows that our method is valid. Further, when we set the parameter value q ¼ À2, then, the ranking order is same as that obtained from the methods developed in (Peng and Jingguo, 2018) and (Wei and Wei, 2018) based on SVN-TOPSIS and SVN Dombi prioritized averaging operators.
Moreover, the comparison among our method with the existing seven methods can be pointed out as follows: (1) The methods developed by Ye (2014) and Peng et al. (2016) are based on SVNWA operators. These aggregation operators are based on algebraic operations, while the aggregation operators in this article are based on Schweizer-Sklar operations. Although the best alternative is same, however, when we change the value of the parameter q the best alternative changed. That's why our method is more flexible and effective than Ye (2014) and Peng et al. (2016).
(2) The method of Wu et al. (2016) is based on SVN prioritized weighted averaging operator. This is a special case of the developed aggregation operators, when the value of the parameter tends to Zero.  (3) The method developed by Liu and Wang (2014) is based on the SVNNNWBM operator, to solve the same example, we set p ¼ q ¼ 1, then the ranking order is same as the one obtained by the developed aggregation operators, when the value of the parameter tends to zero. This shows the effectiveness of the proposed approach based on the developed aggregation operator. But the advantage of the developed method in this article is that it can deal with the situation in which the attributes are with the prioritized relationship. (4) The methods developed by Peng and Jingguo (2018) is based on SVN-TOPSIS and SVN-MABAC method in which the weights of the attributes are obtained via gray system theory and cannot consider the prioritized relationship among the attributes. (5) The method developed by Wei and Wei (2018) is based on Dombi prioritized aggregation for SVNSs. The Dombi prioritized aggregation operator also consists of parameter, but the decision makers can considered the parameter greater than zero, while in the proposed aggregation operators in this article the decision makers can considered the parameter values less than zero.
Certainly, the developed methods in this article are more general and flexible by the parameter, and are more advanced to be used in practical decision-making problems.

Conclusion
Since SVNNs are a better tool to define uncertain information more accurately than the FS and IFS. In this article, we investigated some Schweizer-Sklar prioritized aggregation operator based on SVNNs and proposed two methods to deal with single-valued neutrosophic information. First, we have developed some new aggregation operators and studied their desirable properties such as idempotency, monotonicity and boundedness. Moreover, we have analyzed some special cases of the developed operators, and have presented two MADM methods based on the proposed aggregation operators to deal with SVN information. Lastly, some practical examples are given to show the verification of the developed methods and to demonstrate the effectiveness and practicality of the developed approaches and a comparison analysis is also given to verify the developed methods.
In future we shall combine SSTN and SSTCN with several generalizations of NSs such as interval neutrosophic sets, Double-valued neutrosophic sets, multi-valued neutrosophic sets and develop different aggregation operators such as Bonferroni mean operators, Heronian mean operators, Maclaurin symmetric mean operators for SVNNs. In addition, we will also apply the proposed method to solve some real decision problems Chang et al., 2018;Guan, Zhao, & Du, 2017).