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

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Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator Peide Liu a,⇑, Qaisar Khan b,⇑, Tahir Mahmood b a

School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan, Shandong 250014, China b Department of Mathematics and Statistic, International Islamic University, Islamabad, Pakistan Received 3 September 2018; received in revised form 11 October 2018; accepted 11 October 2018

Abstract 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 singlevalued neutrosophic Schweizer-Sklar prioritized weighted averaging (SVNSSPRWA) operator, single-valued neutrosophic SchweizerSklar 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. Ó 2018 Elsevier B.V. All rights reserved.

Keywords: Single-valued neutrosophic sets; Schweizer-Sklar operations; Prioritized aggregation operator; Multiple-attribute decision making

1. 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 ⇑ Corresponding authors.

E-mail addresses: [email protected] (P. Liu), qaisar.phdma46@iiu. edu.pk (Q. Khan).

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

https://doi.org/10.1016/j.cogsys.2018.10.005 1389-0417/Ó 2018 Elsevier B.V. All rights reserved.

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

2

P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

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
; 1þ 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
; 1þ 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, Mai, and Smarandache (2018), 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 Smarandache (2018), 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 and Mai, 2018; AbdelBasset, Mai, & Chang, 2018; 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

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

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. (1) Developing single-valued neutrosophic SchweizerSklar prioritized weighted averaging (SVNSSPRWA) operator and single-valued neutrosophic SchweizerSklar prioritized ordered weighted averaging (SVNSSPROWA) operator. (2) Discussing properties and specific cases of these proposed AOs. (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

3

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. 2. Preliminaries 2.1. 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, f in N is with a general element expressed by w. A SVNS SV mathematically symbolized as nD E o f ¼ SV w; TR e ðwÞ; IDSV ðwÞ; FL e ðwÞ jw 2 N ð1Þ SV SV where TR e ðwÞ; ID e ðwÞ and FL e ðwÞ represents truthSV SV SV membership (TM) function, indeterminacy-membership (IM) function and falsity-membership (FM) function, respectively, and they are single subsets of the standard unit

½0; 1.

interval

That

TR e ðwÞ : N ! ½0; 1,

is,

SV

ID e ðwÞ : N ! ½0; 1 and FL e ðwÞ : N ! ½0; 1, and the SV SV sum of these three functions must satisfy the condition 0 6 TR e ðwÞ þ IDSV ðwÞ þ FL e ðwÞ 6 3: The triplet SV D ESV TR e ðwÞ; ID e ðwÞ; FL e ðwÞ is said to be a SVN number SV

SV

SV

(SVNN). For computational D E simplicity, we shall denote a SVNN by sv ¼ TR; ID; FL and the set of all SVNNs is designated by H. Definition 2 ((Peng et al., 2016)). Let sv; sv1 and sv3 be any three SVNNs and n > 0: Then some operational laws for SVNNs are defined as follows: D E ð1Þ sv1  sv2 ¼ TR1 þ TR2
TR1 TR2 ; ID1 ID2 ; FL1 FL2 ; ð2Þ

D E ð2Þ sv1  sv2 ¼ TR1 TR2 ;ID1 þ ID2
ID1 ID2 ;FL1 þ FL2
FL1 FL2 ;

 ð3Þ nsv ¼ ð4Þ sv ¼

n

n

1
1
TR ; ID ; FL 

n



n



n

n

ð3Þ

 ; 

TR ; 1
1
ID ; 1
1
ID

ð4Þ n 

:

ð5Þ

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

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P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

D E Definition 3 ((Peng et al., 2016)). Let sv ¼ TR; ID; FL be f can be expressed as a SVNN, a score function SO follows:   f ðsvÞ ¼ 1 TRsv þ 2
IDsv
FLsv ; SO f ðsvÞ 2 ½0; 1 SO 3

ð6Þ

D E Definition 4 ((Peng et al., 2016)). Let sv ¼ TR; ID; FL be

f1 ¼ Definition 7 ((Wang et al.,D 2010)). Let SV D E E f 2 ¼ TR ; ID ; FL TR e ; ID e ; FL e and SV be e e e SV 1 SV 1 SV 1 SV 2 SV 2 SV 2 two SVNNs. Then the generalized union and intersection are defined as follows: f1 SV

SV 1

f can be expressed as a SVNN, an accuracy function AR follows:   f ðsvÞ ¼ TRsv
FLsv ; AC f ðsvÞ 2 ½
1; 1 AR

ð7Þ

D E Definition 5 ((Peng et al., 2016)). Let sv1 ¼ TR1 ; ID1 ; FL1 D E and sv2 ¼ TR2 ; ID2 ; FL2 be two SVNNs. Then the comparison rules of SVNNs are described as follow: f ðsv2 Þ, then sv2 is greater than sv1 , and f ðsv1 Þ < SO (1) If SO is denoted as sv2 > sv1 ; f ðsv1 Þ ¼ SO f ðsv2 Þ, and AR f ðsv1 Þ < AR f ðsv2 Þ then sv2 (2) If SO is greater than sv1 , and is denoted as sv2 > sv1 ; f ðsv2 Þ, and AR f ðsv1 Þ ¼ AR f ðsv2 Þ then sv1 f ðsv1 Þ ¼ SO (3) If SO is equal to sv2 , and is denoted as sv1 ¼ sv2 : 2.2. PA operator   e¼ C e 1; C e 2 ; :::; C e g be Definition 6 ((Yager, 2008)). Let C set of attributes, and assure that there exist prioritization among the attributes expressed by a linear ordering e 2 > ::: > C e g
1 > C e g , which indicates that the crie1 > C C e b , if a < b: C e a ðsÞ is e a has a higher priority than C terion C an evaluation value expressing the performance of the e a and satisfies alternative s under the attribute C e C a 2 ½0; 1. If   g e 1 ðsÞ; :::; C e g ðsÞ ¼  -a C e a ðsÞ e 1 ðsÞ; C PA C a¼1

ð8Þ

Obviously, the PA operator has been effectively applied to the situation where the attributes are real values. 2.3. Schweizer-Sklar (SS) operations The SS operations consist of the SS product and SS sum, which are special cases of ATT, respectively.

nD   [ f 2 ¼ w; Te  TR ;TR  SV e1 e2 ; eT ;eT SV SV    E o Te ID e ;ID e ; Te FL e ;FL e jw 2 N

f1 SV

\  f eT ;eT SV 2 ¼

nD

SV 2



SV 1

SV 2



 

w; Te TR e ; TR e ; Te SV 1 SV 2  E o  Te FL e ; FL e jw 2 N SV 1 SV 2

ð9Þ  ID e ; ID e ; SV 1 SV 2 ð10Þ

 where Te and Te respectively, express T-norm (TN) and T-conorm (TCN).

The Schweizer-Sklar TN and TCN (Deschrijver and Kerre, 2002) are defined as follows: Te SS ðm; nÞ ¼ ðmq þ nq
1Þq 1

ð11Þ

 q q Te SS ðm; nÞ ¼ 1
ðð1
mÞ þ ð1
nÞ
1Þq 1

ð12Þ

where q < 0; m; n 2 ½0; 1. Additionally, when q ¼ 0; we have Te q ðm; nÞ ¼ mn and  e T q ðm; nÞ ¼ m þ n
mn. That is, SS TN and TCN reduce to algebraic TN and TCN. Now, in the next section, based on TN Te q ðm; nÞ and  TCN Te q ðm; nÞ of SS operations, we can give the following definition about SS operations of SVNNs. 3. Schweizer-Sklar operations for SVNNs D E Definition 8. Assume sv1 ¼ TR1 ; ID1 ; FL1 and D E sv2 ¼ TR2 ; ID2 ; FL2 are any two SVNNs. Then based on SS operations, the generalized union and intersection are introduced as follows: D      E sv1 e e sv2 ¼ Te TR1 ; TR2 ; Te ID1 ; ID2 ; Te FL1 ; FL2 ; T ;T

ð13Þ D      E sv1 e e sv2 ¼ Te TR1 ; TR2 ; Te ID1 ; ID2 ; Te FL1 ; FL2 : T ;T

ð14Þ

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

On the basis of Definition (7) and Eqs. (13), and (14), the SS operations of SVNNs are described as follows ðq < 0Þ:

ð1Þ sv1 SS sv2 ¼



* ð 3Þ

sv´1

¼

1


 q1  q q1  q q1  q  q q q 1
TR1 þ 1
TR2
1 ; ID1 þ ID2
1 ; FL1 þ FL2
1

and

sv2 ¼

ð1Þ sv1 SS sv2 ¼ sv2 SS sv1 ;

ð19Þ

ð2Þ sv1 SS sv2 ¼ sv2 SS sv1 ;

ð20Þ

ð3Þ ´ðsv1 SS sv2 Þ ¼ ´sv1 SS ´sv2 ; ´ P 0;

ð21Þ

ð4Þ ´1 sv1 SS ´2 sv1 ¼ ð´1 þ ´2 Þsv1 ; ´1 ; ´2 P 0;

ð22Þ

ð 5Þ

ð17Þ

+   q1  q1  q1 ´ q q 1
´ 1
TR1
ð´
1Þ ; ´ID1
ð´
1Þ ; ´FL1
ð´
1Þ

D E Theorem 1. Let sv1 ¼ TR1 ; ID1 ; FL1 D E TR2 ; ID2 ; FL2 be any two SVNNs, then

sv´1 SS sv´2

´

¼ ðsv1 SS sv2 Þ ;

ð6Þ sv´1 1 SS sv´1 2 ¼ ðsv1 Þ

ð´1 þ´2 Þ

´ P 0;

ð18Þ

4.1. Single valued neutrosophic Schweizer-Sklar prioritized weighted averaging (SVNSSPRAWA) operator Definition 9. A single valued neutrosophic SchweizerSklar prioritized weighted averaging (SVNSSPRWA) operator is a function SVNSSPRWA : H g ! H, which is described as g   SVNSSPRWA sv1 ; sv2 ; :::; svg ¼ 

p¼1

ð23Þ

; ´1 ; ´2 P 0:

ð24Þ

ð15Þ ð16Þ

  q1   q1 +  q1 ´ ´ q ´TR1
ð´
1Þ ; 1
´ 1
ID1
ð´
1Þ ; 1
´ 1
FL1
ð´
1Þ

* ð4Þ ´sv1 ¼

weighted averaging (SVNSSPRAOWA) operator, and discuss some characteristics of these developed aggregation operators.

q1  q1  q1  q  q q  q q q TR1 þ TR2
1 ; 1
1
ID1 þ 1
ID2
1 ; 1
1
FL1 þ 1
FL2
1

 ð2Þ sv1 SS sv2 ¼

5

Tp g

 Tp

svp

ð25Þ

p¼1 p
1

f ðsvh Þ; ðp ¼ 2; 3; ::::; gÞ. Here, where T 1 ¼ 1; and T p ¼  SO h¼1

4. Single-valued neutrosophic Scheweizer-Sklar prioritized weighted operator In D this part,E for a group of SVNNs svp ¼ TRp ; IDp ; FLp ; ðp ¼ 1; 2; :::; gÞ symbolized by H, we introduce a few new PA operators for SVNNs, namely single valued neutrosophic Schweizer-Sklar prioritized

f ðsvh Þ expresses the score value of SVNN svh : SO Theorem 2. For a group of SVNNs svp ¼ D E TRp ; IDp ; FLp ; ðp ¼ 1; 2; :::; gÞ, the value aggregated by the developed SVNPRWA operator is still a SVNN and is specified by:

1q1 0 1q1 0 1q1 +   T T T T T T q C B g C B g C B g g g g   q q p p p p p p C C C B B SVNSSPRWA sv1 ;sv2 ;::::;svg ¼ 1
B 1
TR
 þ1 ;  ID
 þ1 ;  FL
 þ1  p g g g p p A @p¼1 g A @p¼1 g A : @p¼1 g p¼1 p¼1 p¼1  Tp  Tp  Tp  Tp  Tp  Tp *

0

p¼1

p¼1

p¼1

p¼1

p¼1

p¼1

ð26Þ

weighted averaging (SVNSSPRAWA) operator, single valued neutrosophic Schweizer-Sklar prioritized ordered

Proof. We will prove Eq. (26) by utilizing mathematical induction (MI). The following steps of MI have been followed:

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

6

P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

Then, for g ¼ b þ 1, according to the operational rules developed for SVNNs in Definition 8, we have

Step 1. For g ¼ 2; we have T

2

SVNSSPRWAðsv1 ; sv2 Þ ¼  T

¼

1

2

 T

p¼1

svp ;

p

2

p¼1  T p¼1

p

sv1 

2

T

 T

p

ð27Þ

sv2

2 p

p¼1

From the operational laws for SVNNs, proposed in Definition 8, we have

*

T1

sv1 ¼

2

 Tp

p¼1

0

0

11q1 0

0

11q1 0

0

11q1

q1 B T B T1  CC B T 1 B T1 CC B T 1 B T1 CC q q 1 C B C B C 1
B 1
TR1
B
1C ID1
B
1C FL1
B
1C @ 2 @ 2 AA ; @ 2 @ 2 AA ; @ 2 @ 2 AA  Tp  Tp  Tp  Tp  Tp  Tp p¼1

p¼1

p¼1

p¼1

p¼1

+ ;

p¼1

and *

T2

sv2 ¼

2

 Tp

p¼1

0

0

11q1 0

0

11q1 0

0

11q1

q1 B T B T2  CC B T 2 B T2 CC B T 2 B T2 CC q q 2 C B C B C 1
B 1
TR2
B
1C ID2
B
1C FL2
B
1C @ 2 @ 2 AA ; @ 2 @ 2 AA ; @ 2 @ 2 AA  Tp  Tp  Tp  Tp  Tp  Tp p¼1

p¼1

p¼1

p¼1

p¼1

+ :

p¼1

So, Eq. (27) becomes

00

*

T

sv1 

1

2

T

p¼1 p

T

p¼1 p

000 BB@ @@

00

BB B sv2 ¼ 1
@@1
@1
@@

2

2

T

0

T

1

2

q

ID1
@

p¼1 p

1 0  q 1
TR1 A
@

1

2

T

p¼1 p

0 T

T

T

1

2

T

11q1 1q 00 C B
1AA A þ @@

p¼1 p

T

q

ID2
@

2

2

p¼1 p

*

0 2

¼ 1
@ 

T

p

2

p¼1  T

1

2

T

11q1 11q 0 0 00 CC B B
1AA AA þ @1
@1
@@

p¼1 p

0 T

T

T

2

2

T

1q1 000 11q1 1q C BB C
1AA A
1A ; @@@ 1q1 0 p

2

p¼1  T

p¼1 p

0 T

q

1

2

T

FL1
@

p¼1 p

T

T

1

2

T

þ 1A ; @ 

T

p

2

p¼1  T

q

11q1 1q 00 C B
1AA A þ @@

p¼1 p

T

2

TRp


p

2

p¼1  T

p¼1 p

p¼1 p

T

2

2

T

0 T

q

T

2

q

2

p¼1  T

2

FLp


p

T

2

2

T

1q1 11q1 1q + C C
1AA A
1A

p¼1 p

1q1 + p

2

p¼1  T

p¼1 p

T

FL2
@

2

2

T

p¼1 p

þ 1A ; @ 

1q1 11q1 11q C CC
1AA AA
1A ;

p¼1 p

1q1 0 2

p¼1 p

2

2

T

1 0  q 1
TR2 A
@

p¼1 p

p¼1 p

 q 2 1
TRp


T

þ 1A

:

p¼1 p

i.e. when g ¼ 2, Eq. (26) is true. Step 2. Assume that for g ¼ b, Eq. (26) is true. i.e.,

SVNSSPRWAðsv1 ; sv2 ; ::::; svb Þ ¼ 0 *  q T b b 1
@  b p 1
TRp
 p¼1  T p¼1

p

T g

1q1 0

p¼1

þ 1A ; @ 

p

p¼1  T

b

p

T

q

IDp


p

b

p¼1  T p¼1

b

p

T

1q1 0

þ 1A ; @ 

p

b

p¼1  T p¼1

b

p

T

q

FLp


p

b

p¼1  T p¼1

b

p

T

1q1 + þ 1A

p

b

p¼1  T p¼1

:

ð28Þ

p

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

0

*

0

11q1 0

q1 B T B T bþ1  CC bþ1 B CC sv ¼ 1
1
TR
B bþ1 bþ1 @bþ1 @bþ1
1AA bþ1  Tp  Tp  Tp T bþ1

p¼1

p¼1

0

7

11q1 0

B T bþ1 q B T bþ1 CC B CC ;B @bþ1 IDbþ1
@bþ1
1AA  Tp  Tp

p¼1

p¼1

p¼1

0

11q1

B T bþ1 q B T bþ1 CC B CC ;B @bþ1 FLbþ1
@bþ1
1AA  Tp  Tp p¼1

+ ;

p¼1

  SVNSSPRWA sv1 ; sv2 ; ::::; svg ¼ 0 1q1 0 1q1 *   Tp qC B g B g Tp qC B 1
TRp C IDp C 1
B  g g @p¼1 A ; @p¼1 A;  Tp  Tp

and SVNSSPRWAðsv1 ; sv2 ; ::::; svb ; svbþ1 Þ T ¼ SVNSSPRWAðsv1 ; sv2 ; ::::; svb ÞSS bþ1bþ1 svbþ1  Tp

p¼1

0

1q1

B g Tp qC B FLp C @p¼1 g A  Tp

p¼1

p¼1

+ :

ð29Þ

p¼1

*

00

0

0

BB B ¼ 1
@@ 1
@1
@  b

T b

b



p¼1  T

1
TRb

q

b

T




b

b

p¼1  T

p¼1 p

1q1 11q 0 0 0 0  q CC B B @ T bþ1 A þ 1 AA þ @1
@1
bþ1 1
TRbþ1
@ T

p¼1 p

00

p¼1  T

T

p¼1 p

bþ1 T



p

bþ1

p¼1  T

1
TRp

q

bþ1 T




1q1 0 p

bþ1

p¼1  T

p¼1 p

T g

 T

p¼1

T

g

P 0; such that 

p

g

p

p¼1

bþ1 T bþ1

p¼1 p

p

SVNSSPRAWAðsv1 ; sv2 ; sv3 Þ ¼

0 3

1
@

T



p

3

p¼1  T p¼1

1
TRp

p

q

p

p¼1  T

degenerates into the following form:

*

3




T

q

p

q

b

T

IDb


b

p¼1 p

b

b

p¼1  T

1q1 1q C þ 1A A

p¼1 p

p¼1  T

T

bþ1 T

IDp


1q1 0

bþ1 T

þ 1A ; @ 

p

bþ1

p¼1  T

p¼1 p

q

bþ1 T

FLp


p

bþ1

p¼1  T

p¼1 p

1q1 + p

bþ1

p¼1  T

p¼1 p

þ 1A

:

p¼1 p

Example 1. Let sv1 ¼ h0:3;0:4;0:5i, sv2 ¼ h0:4;0:2;0:1i and sv3 ¼ h0:6;0:1;0:2i be three SVNNs. Based on the score f ðsv1 Þ ¼ 0:4667; SO f ðsv2 Þ ¼ function of SVNNs, we get SO f 0:7 and SO ðsv3 Þ ¼ 0:7667, and hence T 1 ¼ 1; T 2 ¼ 0:4667 and T 3 ¼ 0:3267: By using this information ðq ¼
2Þ, we can get

1q1 0

þ 1A ; @ 

p

b

p¼1  T

p¼1 p

3

3

T

1q1 1q1 1q 00 0 11q1 1q + q bþ1 T T C B C C bþ1 bþ1 þ 1A A þ @@bþ1 FLbþ1
@  bþ1
1AA A
1A

1q1 0

p¼1  T p¼1

b

T

p¼1 p

¼ 1; then, Eq. (26)

p

p¼1  T

b

b

p¼1  T

þ 1A ; @ 

p¼1 p

T

FLb


So, when g ¼ b þ 1, Eq. (26) is true. Therefore, Eq. (26) is true for all g: When

b

p¼1 p

0 1
@ 

b

q

b

p¼1  T

p¼1 p

*

T

b

1q1 000 11q1 11q CC C BB b
1 AA AA
1 A ; @@@ 

p¼1 p

p¼1 p

1q1 000 0 11q1 1q bþ1 T B@ T bþ1 q C C BB b bþ1 þ@ bþ1 IDbþ1
@  bþ1
1AA A
1A ; @@@ 

T

T

q

TRp


p

3

p¼1  T p¼1

3

p

T

3

þ 1A ; @ 

p

3

p¼1  T p¼1

p

T

q

FLp


p

3

p¼1  T p¼1

3

p

T

1q1 + þ 1A

p

3

p¼1  T p¼1

p

¼ h0:4226;0:1883;0:1746i:

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

:

8

P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

Theorem 3. For a group of SVNNs svp ¼ D E TRp ; IDp ; FLp ; ðp ¼ 1; 2; :::; gÞ, the SVNPRWA operator

and

0q

satisfies the following properties:

T0

g



g

p¼1

(1) (Idempotency) If all svp ðp ¼ 1; 2; :::; gÞ are equal, i.e., D E svp ¼ sv ¼ TR; ID; FL ; then

p¼1

p¼1 p

T0

g

0q

p

T0 g

T0

p¼1

p¼1  T 0

T0

IDp


p

g

p¼1

þ1

6

p

T

g

q

IDp


p

g

þ 1;

p

p¼1  T

p¼1 p

!q1

p

p¼1  T 0

p

g

p¼1  T

T0

g

0q

p

T

g

þ16 

p

g

p¼1 

p¼1

p¼1 p

T

g



g

p

p¼1  T

q

g

IDp


p¼1 p

T g

!q1 p

p¼1  T

þ1

p¼1 p

ð33Þ

E Proof. Since svp ¼ sv ¼ TR; ID; FL ; for all p: so,

Similarly, we have

1q1 0

0

*



q

IDp ;

p

IDp


p

g

)

D

  SVNSSPRWA sv1 ; sv2 ; ::::; svg ¼

T0 g

p¼1 

ð30Þ

g

p¼1  T

p

g

)

T

g

0q

IDp 6 

p

p¼1  T 0

  SVNSSPRWA sv1 ; sv2 ; ::::; svg ¼ sv:

q

Since IDp 6 IDp

1q1 0

1q1

q Tp Tp Tp C B g Tp C B g Tp C B g Tp  g g g q q B B 1
B  g 1
TRp
 g þ 1C IDp
 g þ 1C FLp
 g þ 1C g g A ; @p¼1 A ; @p¼1 A @p¼1 p¼1 p¼1 p¼1  Tp  Tp  Tp  Tp  Tp  Tp 0

* ¼ 

p¼1

p¼1

1q1 0

1q1 0

p¼1

1q1

q C B g T Tp B g Tp  qC B g qC p B B 1
TR C ID C FL C 1
B  g g g A ; @p¼1 A ; @p¼1 A @p¼1  Tp  Tp  Tp p¼1

p¼1

p¼1

p¼1

+ ;

p¼1

+

p¼1

q q1  q q1  q q1  ¼ 1
1
TR ; ID ; FL ; D E ¼ TR; ID; FL : 

D 0 0 0E (2) (Monotonicity) If sv0p ¼ TRp ; IDp ; FLp and D E svp ¼ TRp ; IDp ; FLp are two groups of SVNNs, such 0

0

0 0

1q1

T 0p g

B g Tp C g 0q B  þ 1C @p¼1 g 0 FLp
p¼1 A Tp  T 0p

0

p¼1

that sv0p P svp : i.e., TRp P TRp ; IDp 6 IDp ; FLp 6 FLp and for all p; then

p¼1

0

1q1 Tp

B g 6B @

    SVNSSPRWA sv01 ; sv02 ; ::::; sv0g P SVNSSPRWA sv1 ; sv2 ; ::::; svg :

p¼1

g

q

FLp


g

p¼1

 Tp

p¼1

ð31Þ

Tp g

 Tp

C þ 1C A

ð34Þ

p¼1

From Eqs. (32)–(34), we get     SVNSSPRWA sv01 ;sv02 ;::::;sv0g P SVNSSPRWA sv1 ;sv2 ;::::;svg :

0

Proof. Since sv0p P svp , which implies TRp P TRp and so

  q 0 0 q 1
TRp 6 1
TRp ; and 0 6 1
TRp 6 1
TRp 6 1;  q T0  T g g 0 q )  g p 0 1
TRp 6  g p 1
TRp ; p¼1  T p¼1

T0

g

) 

g

 T0 p¼1 p

p¼1

)

)1


g



T0 p g

p¼1

 T0 p¼1 p

g

T0



g

p

p¼1  T 0 p¼1

p

p

p¼1  T

p

p¼1

 g 0 q 1
TRp


T0 g

 T0 p¼1 p

p¼1

 g 0 q 1
TRp


p¼1

 g 0 q 1
TRp


T0 p g

 T0 p¼1 p T0 g

p

p¼1  T 0 p¼1

p

p

T

g

þ16 

g

þ1

p¼1

6

g



 q g 1
TRp


p

T g



p

g



1
TRp

p

q

g




þ 1;

p

p¼1

p¼1

P1


T g

p¼1  T

p¼1  T

!q1 þ1

p

p¼1  T

!q1

p

p

T g

g

p¼1  T p¼1

 q g 1
TRp


p p

þ1

p

p¼1  T p¼1

T

ð32Þ

!q1 ;

p

T g

!q1

p¼1  T p¼1

þ1

p

:

p

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

D E (3) (Boundedness) Let svp ¼ TRp ; IDp ; FLp be a group   g g g ¼ max TR ; min ID ; min FL of SVNNs, and svþ p p p ; p p¼1 p¼1 p¼1  g  g g min TRp ; max IDp ; max FLp , then sv
p ¼ p¼1

p¼1

9

Definition 10. A SVNPROWA operator is a function SVNPROWA : H g ! H; described as follows: g   SVNSSPRWA sv1 ; sv2 ; ::::; svg ¼ 

p¼1

Tp g

 Tp

svfð pÞ

ð37Þ

p¼1

p¼1

p
1



f ðsvh Þ; ðp ¼ 2; 3; ::::; gÞ, f is a where T 1 ¼ 1; and T p ¼  SO h¼1



ð35Þ

permutation of ð1; 2; :::; gÞ such that fð pÞ P fð p
1Þ for p ¼ 2; 3; :::; g:

min TRp 6 TRp 6 max TRp ; min IDp 6 IDp 6

D E Theorem 4. For a group of SVNNs svp ¼ TRp ; IDp ; FLp ;




þ

sv 6 SVNSSPRWA sv1 ; sv2 ; ::::; svg 6 sv : Proof. Since g

g

g

g

p¼1 g

p¼1

p¼1

g

ðp ¼ 1; 2; :::; gÞ, the value aggregated by the developed SVNPROWA operator is still a SVNN and is specified by:

max IDp , and min FLp 6 FLp 6 max FLp ; p¼1

p¼1

p¼1

  SVNSSPROWA sv1 ; sv2 ; ::::; svg ¼

*

1q1 0

0

q Tp C B g Tp  g 1
B  g 1
TRfðpÞ
 g þ 1C A @p¼1 p¼1  Tp  Tp p¼1

p¼1

1q1 0

Tp C B g Tp g q ;B  g IDfðpÞ
 g þ 1C A @p¼1 p¼1  Tp  Tp p¼1

p¼1

1q1

Tp C B g Tp g q ;B  g FLfðpÞ
 g þ 1C A @p¼1 p¼1  Tp  Tp p¼1

+

p¼1

ð38Þ

Hence, from property (2), we have     SVNSSPRWA sv01 ; sv02 ; ::::; sv0g P SVNSSPRWA sv1 ; sv2 ; ::::; svg : When q ¼ 0; the SVNPRASSWA operator reduces to the PA operator based on the algebraic operational laws for SVNNs. That is,

* SVNSSPROWAðsv1 ; sv2 ; sv3 Þ ¼

0 T

3

1
@



p

3

p¼1  T p¼1

1
TRfðpÞ

q

3




T

3

þ 1A ; @ 

p

3

p¼1

Example 2. Consider the SVNNs given in Example 1, we have T 1 ¼ 1; T 2 ¼ 0:4667 and T 3 ¼ 0:3267: the score values f ðsv2 Þ ¼ 0:7 and SO f ðsv3 Þ ¼ 0:7667. f ðsv1 Þ ¼ 0:4667; SO are SO f ðsv3 Þ > SO f ðsv2 Þ > SO f ðsv1 Þ and hence, So, we have SO svfð1Þ ¼ sv3 ; svfð2Þ ¼ sv2 ; svfð3Þ ¼ sv1 : By using this information ðq ¼
2Þ, we can get 1q1 0

p¼1  T

p

Proof. Same as Theorem 2, it is omitted.

p

T

q

IDfðpÞ


p

3

p¼1  T p¼1

3

p

T

1q1 0 3

þ 1A ; @ 

p

3

p¼1  T p¼1

p

T

q

FLfðpÞ


p

3

p¼1  T p¼1

3

p

T

1q1 + þ 1A

p

3

p¼1  T p¼1

p

¼ h0:5327;0:1256;0:1568i:

  SVNSSPRWAq¼0 sv1 ;sv2 ;::::;svg *

g



¼ 1
 1
TRp p¼1



T p g  T p¼1 p

g



;  IDp p¼1



T p g  T p¼1 p

g



;  FLp p¼1



T p g  T p¼1 p

+ ð36Þ

4.2. Single-valued neutrosophic Schweizer-Sklar prioritized ordered weighted averaging operator In this subpart, we consider ordered weighted average (OWA), and propose the SVNSSOWA operator.

D E Theorem 5. For a group of SVNNs svp ¼ TRp ; IDp ; FLp ; ðp ¼ 1; 2; :::; gÞss (1) (Idempotency). If all svp ðp ¼ 1; 2; :::; gÞ are equal, i.e., D E svp ¼ sv ¼ TR; ID; FL ; then   SVNSSPROWA sv1 ; sv2 ; ::::; svg ¼ sv:

ð39Þ

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

:

10

P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

D 0 0 0E (2) (Monotonicity). If sv0p ¼ TRp ; IDp ; FLp and D E svp ¼ TRp ; IDp ; FLp are two groups of SVNNs, such 0

sv0p

0

0

P svp : i.e., TRp P TRp ; IDp 6 IDp ; FLp 6 FLp that for all p; then SVNSSPROWA



sv01 ; sv02 ; ::::; sv0g





 P SVNSSPROWA sv1 ; sv2 ; ::::; svg : ð40Þ

D

p¼1

*

Definition 11. A single valued neutrosophic SchweizerSklar prioritized weighted geometric (SVNSSPRWG) operator is a function SVNSSPRWA : H g ! H, which is described as T p g  T p¼1 p

g   SVNSSPRWG sv1 ; sv2 ; :::; svg ¼  svp p
1

p¼1

0

ð42Þ

p¼1

f ðsvh Þ; ðp ¼ 2; 3; ::::; gÞ. Where T 1 ¼ 1; and T p ¼  SO

E

(3) (Boundedness). Let svp ¼ TRp ; IDp ; FLp be a group of   g g g ¼ max TR ; min ID ; min FL SVNNs, and svþ ; p p p p p¼1 p¼1 p¼1  g  g g min TRp ; max IDp ; max FLp , then sv
p ¼ p¼1

5.1. Single-valued neutrosophic Schweizer-Sklar prioritized weighted geometric (SVNSSPRWG) operator

1q1

h¼1

D E Theorem 6. For a group of SVNNs svp ¼ TRp ; IDp ; FLp ; ðp ¼ 1; 2; :::; gÞ, the value aggregated by the developed SVNPRWG operator is still a SVNN and is specified by:

0

1q1

0

1q1

q g T q g T Tp B g Tp C B g Tp  C B g Tp  C g q p p B B SVNSSPRWG sv1 ;sv2 ;::::;svg ¼ B TRp
 g þ 1C 1
IDp
 g þ 1C 1
FLp
 g þ 1C  g g g @p¼1 A ;1
@p¼1 A ;1
@p¼1 A p¼1 p¼1 p¼1  Tp  Tp  Tp  Tp  Tp  Tp 



p¼1

p¼1

p¼1

p¼1

p¼1

+ :

p¼1

ð43Þ

  sv 6 SVNSSPROWA sv1 ; sv2 ; ::::; svg 6 svþ :


ð41Þ

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

5. 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.

T 1 2  T p¼1 p

sv1

* ¼

0

0

11q1

2

SVNSSPRWAðsv1 ; sv2 Þ ¼  svp p¼1

T

¼ svp

T

1

2  T p¼1 p

;

ð44Þ

2

2  T p¼1 p

 sv2

From the operational laws for SVNNs, proposed in Definition 8, we have

0

0

T p 2  T p¼1 p

11q1

0

0

11q1

q1 B T q1 B T B T1 CC CC CC B T1 B T1  B T1  q 1 1 B B B C C C TR1
B
1C 1
ID1
B
1C 1
FL1
B
1C @ 2 @ 2 @ 2 AA ; 1
@ 2 AA ; 1
@ 2 AA @ 2  Tp  Tp  Tp  Tp  Tp  Tp p¼1

p¼1

p¼1

p¼1

p¼1

+ ;

p¼1

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

11

and

T 2 2  T p¼1 p

*

sv2

¼

0

11q1

0

0

B T2 B T2 CC B T2 q C B B TR2
B
1C 1
ID2 @ 2 AA ; 1
@ 2 @ 2  Tp  Tp  Tp p¼1

p¼1

11q1

0 

q1

p¼1

0

B T2 CC B T2 C B
B
1C 1
FL2 AA ; 1
@ 2 @ 2  Tp  Tp p¼1

11q1

0 

q1

p¼1

CC B T2 C
B
1C AA @ 2  Tp

+ ;

p¼1

So, Eq. (44) becomes

T 1 2  T p¼1 p

sv1

000 * BB ¼ @@@

T 2 2  T p¼1 p

 sv2

T

0

00

B BB 1
@@1
@1
@@

T

 1

2

 T

p¼1

00

0

B BB 1
@@1
@1
@@ *0 2 ¼ @

T

q

p¼1

p

p¼1

1
ID1

1

2

q

1

p

1
FL1

q

1

T

1

2

T

A
@

T

1q1

1

p¼1

p

T

 2

2

 T

T

T

 q 2 1
IDp


p

2

p¼1

T

2

i.e., when g ¼ 2, Eq. (43) is true.

1
FL2

q

1

T

2

2

 T

T

A
@

p

2

p¼1  T

p



p¼1

p

1q1 11q1 11q + CC C A A
1 AA
1A

0 T

2

2

 T

p¼1

0

1q1 11q1 11q CC C
1AA AA
1A ;

0

A
@

p

þ 1A ; 1
@ 

p

1

p¼1

2

2

p¼1

q

p

2

 T

1q1

p¼1  T

p

1
ID2



p¼1

p¼1  T

p

2

p¼1

0

þ 1A 1
@ 

p

p¼1

p

2

2

p¼1  T

p

T 2

 T

0 11q1 11q 0 00 CC B B
1AA AA þ @1
@1
@@

2

 T

q

TR2
@

2

p

0 p¼1

2

T 2

 T

0 11q1 11q 0 00 C C B B
1AA AA þ @1
@1
@@

0  T

1q1 11q1 1q C C
1AA A
1A ;

0

p¼1

A
@

p

TRp


p

2

p¼1  T

p

1

p¼1



 T

p¼1

T 2

 T

p

00 T

q

TR1
@

1

2

 T

p¼1

00

11q1 1q 00 C B
1AA A þ @@

0

1
FLp

q

p

2




T

1q1 + þ 1A

p

2

p¼1  T

p

p¼1

:

p

Then, for g ¼ b þ 1, according to the operational rules developed for SVNNs in Definition 8, we have

Step 2. Assume that for g ¼ b, Eq. (43) is true, i.e.,

*

0

1q1

0

1q1

0

1q1

q b T q b T Tp B b Tp C B b Tp  C B b Tp  C q b p p B B SVNSSPRWGðsv1 ;sv2 ;::::;svb Þ ¼ B  b TRp
 b þ 1C 1
IDp
 g þ 1C 1
FLp
 g þ 1C @p¼1 A ;1
@p¼1 A ;1
@p¼1 A b b p¼1 p¼1 p¼1  Tp  Tp  Tp  Tp  Tp  Tp p¼1

p¼1

p¼1

p¼1

+ :

p¼1

p¼1

ð45Þ

T bþ1 bþ1  T p¼1 p

svbþ1

*

0

11q1

0

0

0 

B T bþ1 q B T bþ1 CC B T bþ1 B CC B ¼ B 1
IDbþ1 @bþ1 TRbþ1
@bþ1
1AA ;1
@bþ1  Tp  Tp  Tp p¼1

p¼1

p¼1

q1

11q1

0

B T bþ1 CC B T bþ1 B CC
B 1
FLbþ1 @bþ1
1AA ;1
@bþ1  Tp  Tp p¼1

p¼1

11q1

0 

q1

CC B T bþ1 CC
B @bþ1
1AA  Tp

+ ;

p¼1

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

12

P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

Example 3. Let sv1 ¼ h0:5;0:2;0:4i, sv2 ¼ h0:7; 0:3; 0:4i and sv3 ¼ h0:3; 0:4; 0:6i be three SVNNs. Based on the score f ðsv1 Þ ¼ 0:6333; SO f ðsv2 Þ ¼ function of SVNNs, we get SO

and SVNSSPRWGðsv1 ; sv2 ; ::::; svb ; svbþ1 Þ T bþ1 bþ1  T p¼1 p

¼ SVNSSPRWGðsv1 ; sv2 ; ::::; svb ÞSS svbþ1

1q1 000 1q1 1q 00 11q1 1q 0 * C BBB b T B T CC CC C Bbþ1 T bþ1 T q q b C BB C BB bþ1 CC C B ¼ BBB  b b TRb
 b b þ 1C TR

1
1 C þBB C C; bþ1 A @ A A @ @ A @ bþ1 A A @@ p¼1 p¼1 p¼1 bþ1  Tp  Tp  Tp  Tp p¼1

00

0

p¼1

0

B BB 1
@@1
@1
@  b

T b

b

p¼1  T p¼1

00

0

0

b B BB 1
@@1
@1
@ 

T b

1
IDb

q

T




b

b

T




b

p

p¼1

0



T

 T

p

p¼1

q

 T

p

p¼1

1q1 11q1 11q CC C
1AA AA
1A ;

0
@

p

T b

b

 T

p¼1

0 1q1 11q 0 0 0  q T C B C B þ 1A AA þ @1
@1
@bþ1bþ1 1
FLbþ1
@

b

p¼1  T

0

B CC B þ 1A AA þ @1
@1
@bþ1bþ1 1
IDbþ1

b

p¼1

1
FLb

0

q

b

p¼1  T

q

p¼1

1 1 11q

p



b

p¼1  T p¼1



p¼1

p

T b

1q1 11q1 11q + CC C
1AA AA
1A

b

 T

p¼1

p

p

1q1 1q1 1q1 0 0 0 * + q bþ1 T q bþ1 T Tp C C C Bbþ1 T p Bbþ1 T p  Bbþ1 T p  q bþ1 p p C C C B B B ¼ @  bþ1 TRp
 bþ1 þ 1A ; 1
@  bþ1 1
IDp
 bþ1 þ 1A ; 1
@  bþ1 1
FLp
 bþ1 þ 1A : p¼1 p¼1 p¼1 p¼1 p¼1 p¼1  Tp  Tp  Tp  Tp  Tp  Tp p¼1

p¼1

p¼1

p¼1

T g

 T

p¼1

g

P 0; such that 

p

T g

p

¼ 1; the Eq. (43)

p

p¼1  T p¼1

p¼1

f ðsv3 Þ ¼ 0:4333, and hence T ¼ 1; T ¼ 0:6667 and SO 1 2 0:6333 and T 3 ¼ 0:42222: By using this information ðq ¼
2Þ, we can obtain

So, when g ¼ b þ 1, Eq. (43) is true. Therefore, Eq. (43) is true for all g: When

p¼1

p

degenerates into the following form:

  SVNSSPRWG sv1 ; sv2 ; ::::; svg ¼

*

0

1q1

0

1q1

T

q

3

p¼1  T p¼1

3

TRp


p p

1q1

q C q C B g Tp B g Tp  B g Tp  qC B C ;1
B C ;1
B C TR 1
ID 1
FL p p pA @p¼1 g @p¼1 g @p¼1 g A A  Tp  Tp  Tp p¼1

*0 3 SVNSSPRWGðsv1 ; sv2 ; sv3 Þ ¼ @ 

0

T

p¼1

1q1

þ 1A ; 1
@ 

3

p¼1  T p¼1

0 3

p p

T 3

p¼1  T p¼1

p

:

ð46Þ

p¼1

 q 3 1
IDp


p

+

T

1q1

þ 1A ; 1
@ 

3

p¼1  T p¼1

0 3

p p

T

 q 3 1
FLp


p

3

p¼1  T p¼1

p

T

1q1 + þ 1A

p

3

p¼1  T p¼1

:

p

¼ h0:4537;0:2856;0:4648i:

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

D E Theorem 7. For a group of SVNNs svp ¼ TRp ; IDp ; FLp ;

13

  SVNSSPRWGq¼0 sv1 ;sv2 ;::::;svg

ðp ¼ 1; 2; :::; gÞ, the SVNPRWG operator satisfies the following properties:

* ¼

g



 TRp



T p g  T p¼1 p

g



;1
 1
IDp

p¼1



T p g  T p¼1 p

p¼1

  ;1
 1
FLp g

T p g  T p¼1 p

p¼1

(1) (Idempotency). If all svp ðp ¼ 1; 2; :::; gÞ are equal, i.e., D E svp ¼ sv ¼ TR; ID; FL ; then   SVNSSPRWG sv1 ; sv2 ; ::::; svg ¼ sv:

ð50Þ

5.2. Single-valued neutrosophic Schweizer-Sklar prioritized ordered weighted geometric operator

ð47Þ

D 0 0 0E (2) (Monotonicity). If sv0p ¼ TRp ; IDp ; FLp and svp ¼ D E TRp ; IDp ; FLp are two groups of SVNNs, such that 0

0

Definition 12. A SVNPROWG operator is a function SVNPROWG : H g ! H; described as follows: T p g  T p¼1 p

g   SVNSSPROWG sv1 ; sv2 ; ::::; svg ¼  svfð pÞ

0

sv0p P svp : i.e., TRp P TRp ; IDp 6 IDp ; FLp 6 FLp for all p; then

p
1 h¼1

permutation of ð1; 2; :::; gÞ such that fð pÞ P fð p
1Þ for p ¼ 2; 3; :::; g:

ð48Þ

D E Theorem 8. For a group of SVNNs svp ¼ TRp ; IDp ; FLp ;

D E (3) (Boundedness). Let svp ¼ TRp ; IDp ; FLp be a group of   g g g ¼ max TR ; min ID ; min FL SVNNs, and svþ p p p ; p p¼1 p¼1 p¼1  g  g g min TRp ; max IDp ; max FLp , then sv
p ¼ p¼1

ð51Þ

p¼1

f ðsvh Þ; ðp ¼ 2; 3; ::::; gÞ, f is a where T 1 ¼ 1; and T p ¼  SO

  SVNSSPRWG sv01 ; sv02 ; ::::; sv0g   P SVNSSPRWG sv1 ; sv2 ; ::::; svg :

p¼1

+

ðp ¼ 1; 2; :::; gÞ, the value aggregated by the developed SVNPROWG operator is still a SVNN and is specified by:

p¼1

1q1 1q1 0 +     Tp Tp Tp q q C C C B g Tp B g Tp B g Tp g g g   q C ;1
B  C C B SVNSSPROWG sv1 ;sv2 ;::::;svg ¼ B TR
 þ 1 1
ID
 þ 1 ;1
 1
FL
 þ 1  fðpÞ f ðp Þ fðpÞ g g g A A A : @p¼1 g @p¼1 g @p¼1 g p¼1 p¼1 p¼1  Tp  Tp  Tp  Tp  Tp  Tp *

1q1

0

p¼1

0

p¼1

p¼1

p¼1

p¼1

p¼1

ð52Þ

  sv
6 SVNSSPRWG sv1 ; sv2 ; ::::; svg 6 svþ :

ð49Þ

Proof. Same as Theorem 6, it is omitted. Example 4. Consider the SVNNs given in Example 3, we have T 1 ¼ 1; T 2 ¼ 0:6333 and T 3 ¼ 0:42222: the score values f ðsv2 Þ ¼ 0:6667 and SO f ðsv3 Þ ¼ f ðsv1 Þ ¼ 0:6333; SO are SO f f f 0:4333. So, we have SO ðsv2 Þ > SO ðsv1 Þ > SO ðsv3 Þ and hence, svfð1Þ ¼ sv2 ; svfð2Þ ¼ sv1 ; svfð3Þ ¼ sv3 : By using this information ðq ¼
2Þ, we can obtain

When q ¼ 0; the SVNPRASSWG operator reduces to the PG operator based on the algebraic operational laws for SVNNs. That is,

*0 3 SVNSSPROWGðsv1 ;sv2 ;sv3 Þ ¼ @ 

T

p

3

p¼1  T

p¼1 p

q

3

TRfðpÞ


T

1q1 p

3

p¼1  T

p¼1 p

0 3

þ 1A ;1
@ 

T

p

3

p¼1  T

p¼1 p

 q 3 1
IDfðpÞ


T

1q1 p

3

p¼1  T

p¼1 p

0 3

þ 1A ;1
@ 

T

p

3

p¼1  T

p¼1 p



1
FLfðpÞ

q

3




T g

1q1 + p

p¼1  T

þ 1A

p¼1 p

¼ h0:4710;0:3007;0:4648i:

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

14

P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

D E Theorem 9. For a group of SVNNs svp ¼ TRp ; IDp ; FLp ; ðp ¼ 1; 2; :::; gÞ, the SVNPROWG operator satisfies the following properties: (1) (Idempotency). If all svp ðp ¼ 1; 2; :::; gÞ are equal, i.e., D E svp ¼ sv ¼ TR; ID; FL , then   SVNSSPROWG sv1 ; sv2 ; ::::; svg ¼ sv:

ð53Þ D

0

0

0E

and (2) (Monotonicity). If sv0p ¼ TRp ; IDp ; FLp D E svp ¼ TRp ; IDp ; FLp are two groups of SVNNs, such 0

0

0

that sv0p P svp : i.e., TRp P TRp ; IDp 6 IDp ; FLp 6 FLp for all p; then   SVNSSPROWG sv01 ; sv02 ; ::::; sv0g   P SVNSSPROWG sv1 ; sv2 ; ::::; svg : D

ð54Þ

p¼1

6.1. 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 mrs in the matrix   M ¼ mrs hg 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. The decision matrix can be standardized by the following formula:

mrs ¼

8 > <

D

TRrs ; IDrs ; FLrs

E

D E > : FL ; 1
ID ; TR rs rs rs

for benefit type attribute H rs for cost type attribute H rs ð56Þ

E

(3) (Boundedness). Let svp ¼ TRp ; IDp ; FLp be a group of  g g g ¼ max TRp ; min IDp ; min FLp i; SVNNs, and svþ p p¼1 p¼1 p¼1  g  g g sv
min TRp ; max IDp ; max FLp , then p ¼ p¼1

TRrs 2 ½0;1;IDrs 2 ½0;1;FLrs 2 ½0;1;0 6 TRrs þ IDrs þ FLrs 6 3; ðr ¼ 1; 2; :::; g;s ¼ 1; 2; :::; hÞ. The goal of this problem is to rank the alternatives.

Step 2. Determine the values of T rs ðr ¼ 1; 2; :::; h; s ¼ 1; 2; :::; gÞ by using the following formula:

p¼1

  sv
6 SVNSSPROWG sv1 ; sv2 ; ::::; svg 6 svþ :

ð55Þ

6. The MADM methods based on the proposed aggregation operators In this part, we shall use the SVNPRWA and SVNPRWG operators with SVNNs to solve the MADM problem. The following presumptions or notations are utilized to express the MADM problems. n Let the discrete o set of alternatives be expressed by G ¼ G1 ; G2 ; :::; Gh , and

the n set of oattributes be expressed by H ¼ H 1 ; H 2 ; :::; H g , and that there is a prioritization among the attributes represented by the linear-ordering H 1 > H 2 > :::: > H g
1 > H g , the specified attribute H n has a higher priority than Hm if n < m: Assume that    M ¼ mrs hg ¼ TRrs ; IDrs ; FLrs is the SVNN decision hg

matrix, where TRrs ; IDrs and FLrs express the TM function, IM function and FM function respectively, such that

T rs ¼

s
1 Y

  f mrl ðr ¼ 1; 2; :::; h; s ¼ 2; 3; ::::; gÞ SO

ð57Þ

l¼1

where T r1 ¼ 1 for r ¼ 1; 2; :::; h: Step 3. Use the decision information from decision   matrix M ¼ mrs hg and the SVNSSPRWA operator given in Eq. (26), D E   mr ¼ TRr ; IDr ; FLr ¼ SVNSSPRWA mr1 ; mr2 ; :::; mrg

ð58Þ

to get the overall SVNN mr ðr ¼ 1; 2; :::; hÞ. Step 4. Determine the score values   f mr ðr ¼ 1; 2; :::; hÞ SO of the overall SVNNs mr ðr ¼ 1; 2; ::::; hÞ by Definition 3 to rank all the alternatives Gr ðr ¼ 1; 2; :::; hÞ. Step 5. Rank all the alternatives Gr ðr ¼ 1; 2; :::; hÞ and select best one utilizing Definition 5. Step 6. End.

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

6.2. The method based on SVNSSPRWA operator

Table 1 Decision matrix M of Example 5.

Steps 1 and 2 are same. Step 3. Use the decision information permitted decision   matrix M ¼ mrs hg and the SVNSSPRWG operator given in Eq. (42) D E   mr ¼ TRr ; IDr ; FLr ¼ SVNSSPRWG mr1 ; mr2 ; :::; mrg

15

ð59Þ

To get the overall SVNN mr ðr ¼ 1; 2; :::; hÞ. Step 4. Determine the score values   f mr ðr ¼ 1; 2; :::; hÞ SO of the overall SVNNs mr ðr ¼ 1; 2; ::::; hÞ by Definition 3 to rank all the alternatives Gr ðr ¼ 1; 2; :::; hÞ. Step 5. Rank all the alternatives Gr ðr ¼ 1; 2; :::; hÞ and select best one utilizing Definition 5. Step 6. End.

7. An illustrative example In this part, we use a numerical example of selecting third-party logistics (TPL) providers with SVNNs (Ji et al., 2018) to show the effectiveness and advantages of the developed approach. Example 5. An electronic commerce distributer expects to select a suitable TPL provider. Initially, four providers (alternatives) Gr ðr ¼ 1; 2; :::; 4Þ are available for selection and are evaluated by experts with respect to the following four attributes (1) customer satisfaction H 1 ; (2) service cost H 2 ; (3) market reputation H 3 ; and (4) operational experience in the industry H 4 . The following priority relationship H 1 > H 2 > H 3 > H 4 among the four attributes is considered by the electronic commerce distributer. The assessment values of the four providers with respect to the four attributes are provided by expert in the form of SVNNs and listed in Table 1. Step 1. Normalize the decision matrix. Since H 1 , H 3 and H 4 are of benefit type, and H 2 is of cost type attribute. Hence, by using Eq. (56), the normalized decision matrix is given in Table 2. Step 2. Determine the values of T rs ðr ¼ 1; 2; :::; 4; s ¼ 1; 2; :::; 4Þ by using the formula (57), and get

G1 G2 G3 G4

H1

H2

H3

H4

h0:7; 0:1; 0:2i h0:9; 0:1; 0:1i h0:5; 0:1; 0:4i h0:4; 0:3; 0:2i

h0:3; 0:9; 0:5i h0:3; 0:8; 0:4i h0:1; 0:8; 0:7i h0:2; 0:9; 0:6i

h0:3;0:2; 0:1i h0:5; 0:3; 0:2i h0:6; 0:2; 0:2i h0:7; 0:2;0:1i

h0:5;0:1; 0:4i h0:3; 0:2;0:4i h0:8; 0:1; 0:3i h0:2; 0:2;0:5i

Table 2 Normalize decision matrix M. G1 G2 G3 G4

H1

H2

H3

H4

h0:7; 0:1; 0:2i h0:9; 0:1; 0:1i h0:5; 0:1; 0:4i h0:4; 0:3; 0:2i

h0:5; 0:1; 0:3i h0:4; 0:2; 0:3i h0:7; 0:2; 0:1i h0:6; 0:1; 0:2i

h0:3;0:2; 0:1i h0:5; 0:3; 0:2i h0:6; 0:2; 0:2i h0:7; 0:2;0:1i

h0:5;0:1; 0:4i h0:3; 0:2;0:4i h0:8; 0:1; 0:3i h0:2; 0:2;0:5i

2

1

0:800

0:700

0:6667

3

6 61 6 T rs ¼ 6 61 4

0:6667

7 0:6333 0:6667 7 7 7 0:800 0:7333 7 5

1

0:6333

0:7667 0:8000

0:900

Step 3. Use the SVNSSPRWA given in Eq. (58) to get the overall SVNN mr ðr ¼ 1; 2; :::; 4Þ (assume q ¼
2), and obtain m1 ¼ h0:5852;0:1095;0:1712i; m2 ¼ h0:8266;0:1479;0:3213i; m3 ¼ h0:6965;0:1234;0:1779i; m4 ¼ h0:5597;0:1678;0:1628i: Step 4. Determine the score   f of the overall SO mr ðr ¼ 1; 2; :::; 4Þ mr ðr ¼ 1; 2; ::::; 4Þ by Definition 3, and have

values SVNNs

      f m1 ¼ 0:7682; SO f m2 ¼ 0:7858; SO f m3 SO   f m4 ¼ 0:7430: ¼ 0:7984; SO So, we get m3 > m2 > m1 > m4 : Step 5. According to score values, ranking order of alternatives is G3 > G2 > G1 > G4 : So the best provider is G3 , while the worst one is G4 : Similarly, we solve the above Example 5 by utilizing SVNSSPWG operator:

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

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P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

Step 1 and step 2 are same. Step 3. Use the SVNSSPRWG operator to get the overall SVNN mr ðr ¼ 1; 2; :::; 4Þ. (assume q ¼
2), and have

m1 ¼ h0:4493;0:1253;0:2681i;

G3 and the worst choice is G2 . When the value of the

m2 ¼ h0:4364;0:1980; 0:2643i;

parameter q decreases from 0 then the best choice is G3

m3 ¼ h0:6055;0:1502;0:2900i; m4 ¼ h0:3289;0:2208;0:3088i: Step 4. Determine   f mr ðr ¼ 1; 2; :::; 4Þ of SO mr ðr ¼ 1; 2; ::::; 4Þ, and have

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

the the

score overall

values SVNNs

while the worst one is G4 . 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 G3 , while the worst one is G4 . 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.

  f m1 ¼ 0:6853; SO   f m2 ¼ 0:6580; SO   f m3 ¼ 0:7218; SO   f m4 ¼ 0:5998: SO So, m3 > m1 > m2 > m4 : Step 5. According to score values, ranking order of alternatives is G3 > G1 > G2 > G4 : So, the best provider is G3 , while the worst one is G4 : 7.1. 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

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) Gr ðr ¼ 1; 2; :::; 5Þ from 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

Table 3 Score values and ranking order for different values of q.utilizing SVNSSPWA operator for example 5. q q!0 q ¼
1 q ¼
2 q ¼
7 q ¼
20 q ¼
100 q ¼
200

Score values     f m1 ¼ 0:7350; SO f m2 ¼ 0:6853; SO     f m3 ¼ 0:7607; SO f m4 ¼ 0:6950: SO     f m1 ¼ 0:7526; SO f m2 ¼ 0:7480; SO     f m4 ¼ 0:7211: f m3 ¼ 0:7806; SO SO     f m2 ¼ 0:7858; f m1 ¼ 0:7682; SO SO     f m3 ¼ 0:7984; SO f m4 ¼ 0:7430: SO     f f SO m1 ¼ 0:8070; SO m2 ¼ 0:8384;     f m3 ¼ 0:8404; SO f m4 ¼ 0:7967: SO     f m1 ¼ 0:8244; SO f m2 ¼ 0:8571; SO     f m4 ¼ 0:8207: f m3 ¼ 0:8578; SO SO     f m1 ¼ 0:8316; SO f m2 ¼ 0:8648; SO     f m4 ¼ 0:8309: f m3 ¼ 0:8649; SO SO     f m2 ¼ 0:8657; f m1 ¼ 0:8325; SO SO     f m3 ¼ 0:8658; SO f m4 ¼ 0:8321: SO

Ranking order G3 > G1 > G4 > G2 : G3 > G1 > G2 > G4 : G3 > G2 > G1 > G4 : G3 > G2 > G1 > G4 : G3 > G2 > G1 > G4 : G3 > G2 > G1 > G4 : G3 > G2 > G1 > G4 :

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

P. Liu et al. / Cognitive Systems Research xxx (xxxx) xxx

17

Table 4 Score values and ranking order for different values of q utilizing SVNSSPWG operator. q

Score values     f m2 ¼ 0:6895; f m1 ¼ 0:7069; SO SO     f m4 ¼ 0:6446: f m3 ¼ 0:7349; SO SO     f m2 ¼ 0:6722; f m1 ¼ 0:6961; SO SO     f m4 ¼ 0:6214: f m3 ¼ 0:7282; SO SO     f m2 ¼ 0:6580; f m1 ¼ 0:6853; SO SO     f m3 ¼ 0:7218; SO f m4 ¼ 0:5998: SO     f m2 ¼ 0:6163; f m1 ¼ 0:6447; SO SO     f m4 ¼ 0:5367: f m3 ¼ 0:6953; SO SO     f m1 ¼ 0:6069; SO f m2 ¼ 0:5747; SO     f m4 ¼ 0:4961: f m3 ¼ 0:6639; SO SO     f m2 ¼ 0:5419; f m1 ¼ 0:5754; SO SO     f m3 ¼ 0:6397; SO f m4 ¼ 0:4727: SO     f m1 ¼ 0:5710; SO f m2 ¼ 0:5376; SO     f m3 ¼ 0:6365; SO f m4 ¼ 0:4697: SO

q!0 q ¼
1 q ¼
2 q ¼
7 q ¼
20 q ¼
100 q ¼
200

Ranking order G3 > G1 > G2 > G4 : G3 > G1 > G2 > G4 : G3 > G1 > G2 > G4 : G3 > G1 > G2 > G4 : G3 > G1 > G2 > G4 : G3 > G1 > G2 > G4 : G3 > G1 > G2 > G4 :

accordance with the principle of combining ability with political integrity. The prioritization among the attributes

m1 ¼ h0:5151; 0:4787; 0:1176i;

is as follow, H 1 > H 2 > H 3 > H 4 : The decision makers

m3 ¼ h0:5626; 0:4490; 0:1764i; m4 ¼ h0:7246; 0:1434; 0:1681i; m5 ¼ h0:5366; 0:5754; 0:1462i:

assess possible 5 alternatives Gr ðr ¼ 1; 2; :::; 5Þ with respect to the 4 attributes H s ðs ¼ 1; 2; :::; 4Þ and construct the following SVN decision matrix given in Table 5. Step 1. Normalize the decision matrices by using Eq. (56). Since all the attributes are of benefit type so there is no need to normalize it. Step 2. Determine the values of T rs ðr ¼ 1; 2; :::; 5; s ¼ 1; 2; :::; 4Þ by using the formula (57), and get 2

1

6 61 6 6 T rs ¼ 6 61 6 61 4 1

0:5333

0:3556

0:8000

0:6133

0:6333

0:3167

0:8000

0:5600

0:5333

0:2667

0:1011

3

  f mr Step 4. Determine the score values SO ðr ¼ 1; 2; :::; 5Þ of the overall SVNNs mr ðr ¼ 1; 2; ::::; 5Þ by using Definition (3), and have       f m1 ¼ 0:6396; SO f m2 ¼ 0:7963; SO f m3 SO     f m4 ¼ 0:8044; SO f m5 ¼ 0:6050: ¼ 0:6458; SO So, m4 > m2 > m3 > m1 > m5 :

7 0:3435 7 7 7 0:1404 7 7 7 0:3435 7 5 0:0948

Step 5. According to score values, ranking order of alternatives is G4 > G2 > G3 > G1 > G5 : So, the best candidate is G4 , while the worst one is G5 : Similarly, we solve the above Example 6 by the SVNSSPWG operator:

Step 3. Use the SVNSSPRWA given in Eq. (58) to get the overall SVNN mr ðr ¼ 1; 2; :::; 5Þ (assume q ¼
2), and obtain Table 5 Decision matrix M of Example 6. G1 G2 G3 G4 G5

m2 ¼ h0:7209; 0:2000; 0:1320i;

H1

H2

H3

H4

h0:5;0:8;0:1i h0:7;0:2;0:1i h0:6;0:5;0:2i h0:8;0:1;0:3i h0:5;0:5;0:4i

h0:6;0:3;0:3i h0:7;0:2;0:2i h0:5;0:7;0:3i h0:6;0:3;0:2i h0:4;0:8;0:1i

h0:3;0:6;0:1i h0:7;0:2;0:4i h0:5;0:3;0:1i h0:6;0:2;0:1i h0:7;0:6;0:1i

h0:5;0:7;0:2i h0:8;0:2;0:1i h0:6;0:3;0:2i h0:6;0:2;0:2i h0:5;0:8;0:2i

Step 1 and step 2 are same. Step 3. Use the SVNSSPRWG operator given Equation (Step 3) to get the overall SVNN mr ðr ¼ 1; 2; :::; 5Þ (assume q ¼
2), and have m1 ¼ h0:4498; 0:7400; 0:1745i; m2 ¼ h0:7104; 0:2000; 0:2269i; m3 ¼ h0:5477; 0:5821; 0:2233i; m4 ¼ h0:6554; 0:2051; 0:2265i; m5 ¼ h0:4790; 0:7022; 0:3042i:

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  f mr Step 4. Determine the score values SO ðr ¼ 1; 2; :::; 5Þ of the overall SVNNs mr ðr ¼ 1; 2; ::::; 5Þ by using Definition 3, and get       f m1 ¼ 0:5118; SO f m2 ¼ 0:7612; SO f m3 SO     f m5 ¼ 0:4909: f m4 ¼ 0:7412; SO ¼ 0:5808; SO So, m2 > m4 > m3 > m1 > m5 : Step 5. According to score values ranking order of alternatives is G2 > G4 > G3 > G1 > G5 : So, the best candidate is G2 , while the worst one is G5 : 7.2. Effect of the parameter q on decision result of this example 6 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 6 and 7, and Figs. 1 and 2. From Table 6, 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 is -1 and tends to zero, the best choice is G2 . When the value of the parameter q decreases from -1 then the best choice is G4 . We can also see from Table 6, when the value of the parameter decreases the score values become bigger and bigger. From Table 7, 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 G2 . We can also notice from Table 7, 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.

Table 6 Score values and ranking order for different values of q utilizing SVNSSPWA operator for example 6. q q!0 q ¼
1 q ¼
2 q ¼
7 q ¼
20 q ¼
100 q ¼
200

Score values       f m1 ¼ 0:5935; SO f m2 ¼ 0:7828; SO f m3 ¼ 0:6195 SO     f m4 ¼ 0:7716; SO f m5 ¼ 0:5652: ; SO       f m1 ¼ 0:6179; SO f m2 ¼ 0:7908; SO f m3 ¼ 0:6325 SO     f m4 ¼ 0:7887; SO f m5 ¼ 0:5877: ; SO       f m2 ¼ 0:7963; SO f m3 ¼ 0:6458 f m1 ¼ 0:6396; SO SO     f m4 ¼ 0:8044; SO f m5 ¼ 0:6050: ; SO       f f m3 ¼ 0:6913 f SO m1 ¼ 0:6916; SO m2 ¼ 0:8110; SO     f m5 ¼ 0:6521: f m4 ¼ 0:8433; SO ; SO       f f f m3 ¼ 0:7181 SO m1 ¼ 0:7170; SO m2 ¼ 0:8248; SO     f m5 ¼ 0:6829: f m4 ¼ 0:8588; SO ; SO       f f m3 ¼ 0:7304 f SO m1 ¼ 0:7301; SO m2 ¼ 0:8317; SO     f m4 ¼ 0:8651; SO f m5 ¼ 0:6967: ; SO       f f f m3 ¼ 0:7318 SO m1 ¼ 0:7317; SO m2 ¼ 0:8325; SO     f m5 ¼ 0:6983: f m4 ¼ 0:8659; SO ; SO

Ranking order G2 > G4 > G3 > G1 > G5 : G2 > G4 > G3 > G1 > G5 : G4 > G2 > G3 > G1 > G5 : G4 > G2 > G1 > G3 > G5 : G4 > G2 > G3 > G1 > G5 : G4 > G2 > G3 > G1 > G5 : G4 > G2 > G3 > G1 > G5 :

Table 7 Score values and ranking order for different values of q utilizing SVNSSPWG operator for example 6. q q!0 q ¼
1 q ¼
2 q ¼
7 q ¼
20 q ¼
100 q ¼
200

Score values       f m1 ¼ 0:5463; SO f m2 ¼ 0:7688; SO f m3 ¼ 0:5985 SO     f f   ; SO m4 ¼ 0:7503;   SO m5 ¼ 0:5241:   f m1 ¼ 0:5271; SO f m2 ¼ 0:7651; SO f m3 ¼ 0:5894 SO     f m5 ¼ 0:5067: f m4 ¼ 0:7457; SO ; SO       f m1 ¼ 0:5118; SO f m2 ¼ 0:7612; SO f m3 ¼ 0:5808 SO     f m4 ¼ 0:7412; SO f m5 ¼ 0:4909: ; SO       f m2 ¼ 0:7414; SO f m3 ¼ 0:5514 f m1 ¼ 0:4651; SO SO     f m4 ¼ 0:7224; SO f m5 ¼ 0:4470: ; SO       f m2 ¼ 0:7170; SO f m3 ¼ 0:5255 f m1 ¼ 0:4268; SO SO     f m5 ¼ 0:4188: f m4 ¼ 0:6959; SO ; SO       f f m3 ¼ 0:5053 f SO m1 ¼ 0:4053; SO m2 ¼ 0:7033; SO     f m5 ¼ 0:4037: f m4 ¼ 0:6728; SO ; SO       f f m3 ¼ 0:5027 f SO m1 ¼ 0:4026; SO m2 ¼ 0:7017; SO     f m4 ¼ 0:6697; SO f m5 ¼ 0:4019: ; SO

Ranking order G2 > G4 > G3 > G1 > G5 : G2 > G4 > G3 > G1 > G5 : G2 > G4 > G3 > G1 > G5 : G2 > G4 > G3 > G1 > G5 : G2 > G4 > G3 > G1 > G5 : G2 > G4 > G3 > G1 > G5 : G2 > G4 > G3 > G1 > G5 :

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

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Fig. 1. Chart for different values of parameter q utilizing SVNSSPWA operator in Example 6.

Fig. 2. Chart for different values of parameter q utilizing SVNSSPWG operator in Example 6.

7.3. 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
2, 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 SVNTOPSIS 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.

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

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Table 8 Score values and ranking orders with different methods. Methods SVNWA operator (Ye, 2014)

SVNWA operator (Peng et al., 2016) SVNDPAW (Wei and Wei, 2018) q ¼ 2 SVN-TOPSIS (Peng and Jingguo, 2018) SVN-MABAC (Peng and Jingguo, 2018) SVNPWA operator (Wu et al., 2016) SVNNBM operator (p=q=1) (Liu and Wang, 2014) SVNSSPRWA operator (in this article) (q ! 0) SVNSSPRWA operator (in this article) (q ¼
2)

Score values       f m1 ¼ 0:5532; SO f m2 ¼ 0:7698; SO f m3 ¼ 0:6001 SO     f m4 ¼ 0:7580; SO f m5 ¼ 0:5301: ; SO       f f m3 ¼ 0:6195 f SO m1 ¼ 0:5934; SO m2 ¼ 0:7828; SO     f m4 ¼ 0:7716; SO f m5 ¼ 0:5652: ; SO       f f m3 ¼ 0:6529 f SO m1 ¼ 0:6540; SO m2 ¼ 0:7973; SO     f m5 ¼ 0:6136: f m4 ¼ 0:8080; SO ; SO       C m1 ¼
3:3557;C m2 ¼
0:8123;C m3 ¼
2:7509;     C m4 ¼
0:4144;C m5 ¼
3:8097:       Q m1 ¼ 0:2637;Q m2 ¼ 0:6122;Q m3 ¼ 0:2176;     Q m4 ¼ 0:5891;Q m5 ¼ 0:1903:       f m2 ¼ 0:7828; SO f m3 ¼ 0:6195 f m1 ¼ 0:5934; SO SO     f m4 ¼ 0:7716; SO f m5 ¼ 0:5652: ; SO       f m2 ¼ 0:774729; SO f m3 ¼ 0:60837 f m1 ¼ 0:565597; SO SO     f m5 ¼ 0:5391: f m4 ¼ 0:754169; SO ; SO       f m1 ¼ 0:5935; SO f m2 ¼ 0:7828; SO f m3 ¼ 0:6195 SO     f m4 ¼ 0:7716; SO f m5 ¼ 0:5652: ; SO       f m2 ¼ 0:7963; SO f m3 ¼ 0:6458 f m1 ¼ 0:6396; SO SO     f m4 ¼ 0:8044; SO f m5 ¼ 0:6050: ; SO

(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. 8. 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

Ranking order G2 > G4 > G3 > G1 > G5 : G2 > G4 > G3 > G1 > G5 : G4 > G2 > G1 > G3 > G5 : G4 > G2 > G3 > G1 > G5 : G2 > G4 > G1 > G3 > G5 : G2 > G4 > G3 > G1 > G5 : G2 > G4 > G3 > G1 > G5 : G2 > G4 > G3 > G1 > G5 : G4 > G2 > G3 > G1 > G5 :

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 (Abdel-Basset and Mai, 2018; Abdel-Basset, Mai, et al., 2018; Abdel-Basset, Gunasekaran, et al., 2018; Chang et al., 2018; Guan, Zhao, & Du, 2017). Acknowledgement This paper is supported by National Natural Science Foundation of China (Nos.71771140, 71471172), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045), Shandong Provincial Social Science Planning Project (Nos. 17BGLJ04, 16CGLJ31 and 16CKJJ27), and Key research and development program of Shandong Province (No. 2016GNC110016).

Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005

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Please cite this article as: P. Liu, Q. Khan and T. Mahmood, Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.005