greensupplierselectionbasedondombiprioritized … · 1092...

International Journal of Computational Intelligence SystemsVol. 12(2), 2019, pp. 1091–1101

DOI: https://doi.org/10.2991/ijcis.d.190923.001; ISSN: 1875-6891; eISSN: 1875-6883https://www.atlantis-press.com/journals/ijcis/

Green Supplier Selection Based on Dombi PrioritizedBonferroni Mean Operator with Single-Valued TriangularNeutrosophic Sets

Jianping Fan, Xuefei Jia, Meiqin Wu*

School of Economics and Management, Shanxi University, Taiyuan 030006, China

ART I C L E I N FOArticle History

Received 02 Jul 2019Accepted 15 Sep 2019

Keywords

Single-valued triangularNeutrosophic sets

Dombi operationsPrioritized operatorBonferroni mean operatorDombi prioritized normalized

Bonferroni mean operator

ABSTRACTThe choice of green suppliers involves a large amount of inaccurate, incomplete, and inconsistent information, and the single-valued triangular Neutrosophic number that is an extension of the single-valued Neutrosophic number can effectively handlesuch problems. Considering the advantages of the single-valued triangular Neutrosophic number, this paper proposes a newaggregate operator to solve the problem of multi-criteria decision making. The new aggregate operator takes into account thepriority relationship and the interrelationship between the criteria. To make the new aggregate operator more flexible, this paperintroduces the Dombi operations. This paper combines the Dombi operations with the prioritized average operator and theBonferronimean operator to propose the single-valued triangularNeutrosophicDombi prioritized normalized Bonferronimean(SVTNDPNBM) operator. Finally, the SVTNDPNBM operator is applied to the problem of the green supplier selection, whichproves its feasibility and stability.

© 2019 The Authors. Published by Atlantis Press SARL.This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

In recent years, consumers, business operators, and governmentsare paying more and more attention to green development andenvironmental performance issues, with the rapid consumptionof resources and serious environmental pollution. In this context,companies that want to have competitive advantage in the mar-ket cannot ignore the environmental factors in supplier selection.Choosing the appropriate green supplier can improve market com-petitive advantage and environmental performance. But choosingthe right green supplier requires consideration of criteria such asproduct quality, cost, service capability, green image, and innova-tion capability. In essence, the process of choosing the best greensupplier is a multi-criteria decision-making (MCDM) problem.

After Zadeh [1] proposed the concept of fuzzy sets (FS), FSwere widely used. But the FS represent the uncertainty of deci-sion information only by using the membership degree. Atanassov[2] proposed intuitionistic fuzzy sets (IFS) by introducing non-membership degree, which can effectively deal with the problemsthat FS cannot handle. Gargov et al. [3] and Atanassov [4] extendedthe IFS to interval numbers, and proposed interval IFS (IIFS). Torra[5] defined hesitant fuzzy sets (HFS), which can deal with theuncertainty caused by the decision maker’s hesitation well. Qianet al. [6] and Zhu et al. [7] defined generalized HFS and doubleHFS, respectively. However, sometimes the sum of membership

*Corresponding author. Email: [email protected]

degree and non-membership degree is greater than one. Yager [8]proposed Pythagorean fuzzy sets (PFS) to solve this problem, allow-ing the sum of membership degree and non-membership degree tobe greater than one, while satisfying the sum of the squares of themembership degree and the non-membership degree is less thanor equal to one. Although FSs theory has been extensively studiedand expanded, FS and their extension sets cannot handle discontin-uous and inconsistent information. The emergence of the Neutro-sophic sets (NS) just makes up for this deficiency. Smarandache [9]proposed the concept of the NS, which uses the truth-membershipfunction, the indeterminacy-membership function and the falsity-membership function to depict the fuzzy information, and these areindependent of each other.

Although the NS expand the expression of uncertain information,it is very inconvenient in practical applications. To simplify theNS, Ye [10] proposed the concept of simplified Neutrosophic sets(SNS), and pointed out SNS contain single-valued Neutrosophicsets (SVNS) and interval Neutrosophic sets (INS). Ye [11] proposedaggregation operators and cosine similarity measurement of SNS.Peng, Wang et al. [12] improved the operations of SNS in the lit-erature [11]. Biswas et al. [13] combined triangular fuzzy numberswith SVNS, and proposed single-valued triangular Neutrosophicsets (SVTNS). Wang et al. [14] considered the conflicting criteria,extended the original VIKORmodel to SVTNS, and introduced thespecific steps to apply the method.

Pdf_Folio:1091

https://doi.org/10.2991/ijcis.d.190923.001https://www.atlantis-press.com/journals/ijcis/http://creativecommons.org/licenses/by-nc/4.0/

1092 J. Fan et al. / International Journal of Computational Intelligence Systems 12(2) 1091–1101

The aggregation operators present the powerful tool for handlingMCDM problem. Dombi [15] proposed Dombi operations, includ-ing T-norm and T-conorm, which show the advantage of good flex-ibility with the operation of parameters. Recently, several authorsdefined Dombi operations in IFS [16], SVNS [17], HFS [18]. Yager[19] proposed prioritized average (PA) operator by considering thepriority relationship between the criteria. The PA operator has beenwidely used in a variety of fuzzy numbers, such as HFS [20], uncer-tain linguistic sets [21], interval-valued hesitant fuzzy [22], INS[23]. In order to solve the relationship between the criteria, Bonfer-roni [24] proposed the Bonferroni mean (BM) aggregation oper-ator. Yager et al. [25] defined weighted Bonferroni mean (WBM)operator based on the BM operator, but WBM has the disadvan-tage of non-reducibility. In order to overcome this deficiency, Zhouand He [26] defined the normalized weighted Bonferroni mean(NWBM) operator based on the WBM operator. Tian et al. [27]proposed the gray linguistic BM operator to address the situationswhere the criterion values take the form of gray linguistic num-bers and the criterion weights are known. Each operator has itsown unique advantages. Liu et al. [16] proposed some Dombi BMoperators in the IFS environment. Khan et al. [28] proposed Dombipower BM operators in the INS environment. Nie et al. [29] pro-posed partitioned NWBM operator based on Shapley fuzzy mea-sures, in the PFS environment. Considering the advantages of theHamy mean operator, Li et al. [30] proposed Dombi Hamy meanoperators in the IFS environment, and Wu et al. [31] proposedDombi Hamy mean operators in the IIFS environment. Yager et al.[32] applied the Dombi operators to Picture fuzzy, and Zhang et al.[33] proposed Picture fuzzy Dombi Heronian mean operators. Weiand Zhang [34] introduced Bonferroni power operators into SVNSenvironment. Wang et al. [35] combined Frank operational laws toproposed Frank prioritized BM operator.

Choosing the right green supplier can eliminate some environmen-tal impact and improve environmental performance. So far, theresearch of green supplier selection has achieved certain results.Noci [36] pointed out for the first time that measuring the environ-mental performance of green suppliers includes quantitative andqualitative indicators, and proposed a method of selecting greensuppliers from the environmental perspective. Lee et al. [37] usedDelphi method to distinguish traditional suppliers from green sup-pliers and proposed a fuzzy extended analytic hierarchy processmodel to evaluate green suppliers. Gao et al. [38] used intuitionis-tic fuzzy numbers for green supplier selection when criteria weightsare unknown. Due to the complexity of the environment, the infor-mation available for evaluation selection is increasingly uncertain.Liang et al. [39] proposed the single-valued trapezoidal Neutro-sophic preference relations as a strategy for tackling green supplierselection problems. Qin et al. [40] considered that decision mak-ers are not entirely reasonable in making decisions, and extend theTODIM technique to solveMCDMproblems, then proposed a newmethod for select the optimal green supplier in the interval type-2 FS environment. Yazdani et al. [41] comprehensively consideredthe evaluation criteria of traditional suppliers and green suppliers,and sorted green suppliers based on the quality function deploy-ment model. Li et al. [42] demonstrated the advantages of proba-bility HFS in decision making process, and combined probabilityHFS and the extended qualitative flexible multiple method to solvethe problem of green supplier selection. Ji et al. [43] conducted astudy on green supplier selection in the context of the single-valuedNeutrosophic linguistic sets.

According to the existing literature, nobody proposed the Dombioperations of SVTNS, and nobody combined the PA operator withthe BM operator for the SVTNS environment. Therefore, it is nec-essary to propose SVTNDPNDM operator. The SVTNDPNBMoperator has some flexibility, and simultaneously considers thepriority relationship and interaction between the criteria by inte-grating the Dombi operations, the PA operator and NWBMoperator. In this paper, the SVTNS are used to represent the eval-uation value corresponding to different green suppliers, which canrepresent more uncertain information. The selection of the PAoperator can take into account the priority relationship between thecriteria. The selection of the BM operator can take into accountthe interrelationship between the criteria, and introduce the Dombioperations for the flexibility of the operation.

This paper firstly defines the Dombi operations in the SVTNS envi-ronment. Then, based on theDombi operations, we combine the PAoperator and BM operator, and propose the single-valued triangu-lar Neutrosophic Dombi Bonferroni mean (SVTNDBM) operatorand the single-valued triangular Neutrosophic Dombi prioritizednormalized Bonferroni mean (SVTNDPNBM) operator. Finally,based on the proposed new operator, a model is established to solveMCDM problem.

The rest of the paper is structured as follows: In the second section,the related concepts of the SVTNS, Dombi operations, PA oper-ator and BM operator are introduced in detail. The third sectionproposes the new operator of SVTNS. The forth section builds amodel for selecting suitable green suppliers. The fifth section givesa numerical example to prove the feasibility and adaptability of theproposed method. The last part is the conclusion.

2. PRELIMINARIES

This section introduces some concepts about the SVTNS, Dombioperations, PA operator, and BM operator. These will be used inlater papers.

2.1. Single-Valued Triangular NeutrosophicSets

The SVNS is a good representation of uncertain, incomplete, andinconsistent information in the real world, but decision makersoften use fuzzy numbers rather than precise numbers to representmembership function. Biswas et al. [13] defined the SVTNS bycombining the triangular fuzzy number and the SVNS.

Definition 1. [13] Let X be a finite set of points (objects), let xdenote a generic element in X, and E [0,1] be the set of all triangu-lar fuzzy numbers on [0,1]. The SVTNSA in X is characterized by atruth-membership functionTA (x), an indeterminacy-membershipfunction IA (x) and a falsity-membership function FA (x). Then, theSVTNS A can be depicted as:

A = {⟨x,TA (x) , IA (x) , FA (x)⟩ |x ∈ X}

where, TA (x) ∶ X → E [0, 1], IA (x) ∶ X → E [0, 1] andFA(x) ∶ X → E[0, 1]. TA (x) , IA (x) , FA (x) can be expressed as fol-lows: TA (x)=

(T1A (x) ,T2A (x) ,T3A (x)

), IA(x)=

(I1A(x), I2A(x), I3A(x)

)and FA (x) =

(F1A (x) , F2A (x) , F3A (x)

), for every x ∈ X, satisfy

0 ≤ T3A (x) + I3A (x) + F3A (x) ≤ 3.Pdf_Folio:1092

J. Fan et al. / International Journal of Computational Intelligence Systems 12(2) 1091–1101 1093

For convenience, we consider A = ⟨(a, b, c) ,(e, f, g

), (r, s, t)⟩

as SVTN number, where(T1A (x) ,T2A (x) ,T3A (x)

)= (a, b, c),(

I1A (x) , I2A (x) , I3A (x))=

(e, f, g

)and

(F1A (x) , F2A (x) , F3A (x)

)=

(r, s, t).Definition 2. [13] Let

A1 = ⟨(a1, b1, c1) ,(e1, f1, g1

), (r1, s1, t1)⟩, A2 =

⟨(a2, b2, c2) ,(e2, f2, g2

), (r2, s2, t2)⟩ be two SVTN numbers, the

rules of operations can be defined as follows:

1. A1⊕A2 = ⟨(a1 + a2 – a1a2, b1 + b2 – b1b2, c1 + c2 – c1c2) ,(e1e2, f1f2, g1g2

), (r1r2, s1s2, t1t2)

⟩

2. A1 ⊗ A2 = ⟨(a1a2, b1b2, c1c2) ,(e1 + e2 – e1e2, f1 + f2 – f1f2, g1 + g2 – g1g2

),

(r1 + r2 – r1r2, s1 + s2 – s1s2, t1 + t2 – t1t2)⟩

3. 𝜆A1 = ⟨(1 – (1 – a1)𝜆 , 1 – (1 – b1)𝜆 , 1 – (1 – c1)𝜆

),(

e𝜆1 , f𝜆1 , g

𝜆1),(r𝜆1 , s

𝜆1 , t

𝜆1) ⟩ ,

𝜆 > 0

4. A𝜆1 = ⟨

(a𝜆1 , b

𝜆1 , c

𝜆1),(

1 – (1 – e1)𝜆 , 1 –(1 – f1

)𝜆 , 1 – (1 – g1)𝜆) ,(1 – (1 – r1)𝜆 , 1 – (1 – s1)𝜆 , 1 – (1 – t1)𝜆

) ⟩ ,𝜆 > 0

The above operations satisfy the following properties:

1. A1 ⊕ A2 = A2 ⊕ A1;A1 ⊗ A2 = A2 ⊗ A12. 𝜆 (A1 ⊕ A2) = 𝜆A1 ⊕ 𝜆A2;

(A1 ⊗ A2)𝜆 = A𝜆1 ⊗ A𝜆2 𝜆 > 0

3. 𝜆1A1 ⊕ 𝜆2A1 = (𝜆1 + 𝜆2)A1;A𝜆11 ⊗ A

𝜆21 = A

𝜆1+𝜆21 𝜆1, 𝜆2 > 0

Definition 3. [13] Let A = ⟨(a, b, c) ,(e, f, g

), ((r, s, t))⟩ be a SVTN

number, the score function S and accuracy function H can beexpressed as follows:

S (A) = 112 [8 + (a + 2b + c) –(e + 2f + g

)– (r + 2s + t)]

S (A) ∈ [0, 1] (1)

H (A) = 14 [(a + 2b + c) – (r + 2s + t)] ,H (A) ∈ [–1, 1] (2)

Definition 4. [13] Let A1 , A2 be two SVTN numbers, accordingDefinition 3 the order relations are defined as follows:

1. if S (A1) < S (A2) , thenA1 ≺ A2 ;2. if S (A1) > S (A2) , thenA1 ≻ A2 ;3. if S (A1) = S (A2) ,H (A1) < H (A2) , thenA1 ≺ A2 ;4. if S (A1) = S (A2) ,H (A1) > H (A2) , thenA1 ≻ A2 ;5. if S (A1) = S (A2) ,H (A1) = H (A2) , thenA1 ∼ A2 ;

2.2. Dombi Operations

Information aggregation in multi-criteria decision making is acrucial step. However, the existing aggregation operators are flex-ibility lack. To overcome this deficiency, Dombi [15] proposedDombi operations, including T-norm and T-conorm.

Definition 5. [15] Let s and t be any two real numbers. Then, theDombi T-norm and Dombi T-conorm among s and t are depictedas follows:

OD (s, t) =1

1 +(( 1–s

s

)𝛾+( 1–t

t

)𝛾)1/𝛾 (3)

OCD (s, t) = 1 –1

1 +((

s1–s

)𝛾+(

t1–t

)𝛾)1/𝛾 (4)where, 𝛾 ≥ 1 and (s, t) ∈ [0,1] × [0,1].

2.3. PA Operator

When a decision maker makes a multi-criteria decision, all the cri-teria are not equally important, and there is a priority relationshipbetween the criteria. Therefore, Yager [19] considering the priorityrelationship between the criteria proposed a PA operator.

Definition 6. [19] Let C = {C1,C2,⋯ ,Cn} be a set of criteria, andexists C1 ≻ C2 ≻ ⋯ ≻ Cn priority relationship between the crite-ria.Where the value y under the criteriaCi isCi

(y)(i = 1, 2,⋯ , n),

and satisfies Ci(y)∈ [0, 1]. Then the PA operator is depicted as

follows:

PA(C1

(y),C2

(y),⋯ ,Cn

(y))

=n

∑i=1

wiCi(y)

(5)

where wi =Hi

∑ni=1 Hi, Hi =∏i–1k=1 Ck

(y)(i ≥ 2), and H1 = 1.

2.4. BM Operator

In some special cases, the criteria are dependent criteria. In orderto solve the relationship between the criteria, Bonferroni [24] pro-posed the BM aggregation operator.

Definition 7. [24] Let bi (i = 1, 2,⋯ , n) be a collection of non-negative real numbers and p, q > 0. The BM operator is depicted asfollows:

BMp,q (b1, b2,⋯ , bn) =

⎛⎜⎜⎜⎜⎜⎝1

n (n – 1)n

∑i, j = 1,i ≠ j

bpi bqj

⎞⎟⎟⎟⎟⎟⎠

1/p+q

(6)

However, in practical problems, multi-criteria decision usuallyneeds to consider the importance of the criteria, and assign differentweights to different criteria. Yager et al. [25] defined WBM opera-tor based on the BM operator.Pdf_Folio:1093


Definition 8. [25] Let bi (i = 1, 2,⋯ , n) be a collection of non-negative real numbers and p, q > 0. w = (w1,w2,⋯wn)T is theweight vector of bi,∑ni=1 wi = 1, and wi ∈ [0, 1]. The WBM oper-ator is depicted as follows:

WBMp,q (b1, b2,⋯ , bn) =

⎛⎜⎜⎜⎜⎜⎝1

n (n – 1)n

∑i, j = 1,i ≠ j

(wibi)p (wjbj)q

⎞⎟⎟⎟⎟⎟⎠

1/p+q

(7)

There is a defect in this WBM operator that is non-reducibility.Zhou and He [26] further defined the NWBM operator based onthe WBM operator.

Definition 9. [26] Let bi (i = 1, 2,⋯ , n) be a collection of non-negative real numbers and p, q > 0. w = (w1,w2,⋯wn)T is theweight vector of bi, ∑ni=1 wi = 1, and wi ∈ [0, 1]. The NWBMoperator is depicted as follows:

NWBMp,q (b1, b2,⋯ , bn) =

⎛⎜⎜⎜⎜⎜⎝n

∑i, j = 1,i ≠ j

wiwj1 – wi

bpi bqj

⎞⎟⎟⎟⎟⎟⎠

1/p+q

(8)

1. A1 ⊕D A2 = ⟨

⎛⎜⎜⎜⎜⎝1 – 1

1+((

a11–a1

)𝛾+(

a21–a2

)𝛾) 1𝛾 , 1 –1

1+((

b11–b1

)𝛾+(

b21–b2

)𝛾) 1𝛾 , 1 –1

1+((

c11–c1

)𝛾+(

c21–c2

)𝛾) 1𝛾⎞⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎝1

1+((

1–e1e1

)𝛾+(1–e2e2

)𝛾) 1𝛾 ,1

1+((

1–f1f1

)𝛾+(1–f2f2

)𝛾) 1𝛾 ,1

1+((

1–g1g1

)𝛾+(1–g2g2

)𝛾) 1𝛾⎞⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎝1

1+((

1–r1r1

)𝛾+(1–r2r2

)𝛾) 1𝛾 ,1

1+((

1–s1s1

)𝛾+(1–s2s2

)𝛾) 1𝛾 ,1

1+((

1–t1t1

)𝛾+(1–t2t2

)𝛾) 1𝛾⎞⎟⎟⎟⎟⎠⟩

2. A1 ⊗D A2 = ⟨

⎛⎜⎜⎜⎜⎝1

1+((

1–a1a1

)𝛾+(1–a2a2

)𝛾) 1𝛾 ,1

1+((

1–b1b1

)𝛾+(1–b2b2

)𝛾) 1𝛾 ,1

1+((

1–c1c1

)𝛾+(1–c2c2

)𝛾) 1𝛾⎞⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎝1 – 1

1+((

e11–e1

)𝛾+(

e21–e2

)𝛾) 1𝛾 , 1 –1

1+((

f11–f1

)𝛾+(

f21–f2

)𝛾) 1𝛾 , 1 –1

1+((

g11–g1

)𝛾+(

g21–g2

)𝛾) 1𝛾⎞⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎝1 – 1

1+((

r11–r1

)𝛾+(

r21–r2

)𝛾) 1𝛾 , 1 –1

1+((

s11–s1

)𝛾+(

s21–s2

)𝛾) 1𝛾 , 1 –1

1+((

t11–t1

)𝛾+(

t21–t2

)𝛾) 1𝛾⎞⎟⎟⎟⎟⎠⟩

3. DOMBI PRIORITIZED NORMALIZED BMOPERATOR OF SVTNS

This section proposes SVTNDBM operator. Afterwards, we definethe SVTNDPNBM operator on the basis of the Dombi operations,the PA operator and NWBM operator. The SVTNDPNBM opera-tor has some flexibility, and simultaneously considers the priorityrelationship and interaction between the criteria by integrating theDombi operations, the PA operator andNWBMoperator. Then, weprove several properties of the SVTNDPNBM operator.

3.1. Dombi Operations of SVTNS

This part introducesDombi operations of SVTNSbased on theDef-inition 2 and the Definition 5.

Definition 10. Let

A1 = ⟨(a1, b1, c1) ,(e1, f1, g1

), (r1, s1, t1)⟩, A2 =

⟨(a2, b2, c2), (e2, f2, g2), (r2, s2, t2)⟩ be two SVTN numbers, 𝛾 ≥ 1and 𝜆 > 0. Then, the Dombi T-norm and Dombi T-conorm ofSVTNS can be depicted as follows:

Pdf_Folio:1094


3. 𝜆 ⋅D A1 = ⟨

⎛⎜⎜⎜⎜⎝1 – 1

1+(𝜆(

a11–a1

)𝛾) 1𝛾 , 1 –1

1+(𝜆(

b11–b1

)𝛾) 1𝛾 , 1 –1

1+(𝜆(

c11–c1

)𝛾) 1𝛾⎞⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎝1

1+(𝜆(1–e1e1

)𝛾) 1𝛾 ,1

1+(𝜆(1–f1f1

)𝛾) 1𝛾 ,1

1+(𝜆(1–g1g1

)𝛾) 1𝛾⎞⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎝1

1+(𝜆(1–r1r1

)𝛾) 1𝛾 ,1

1+(𝜆(1–s1s1

)𝛾) 1𝛾 ,1

1+(𝜆(1–t1t1

)𝛾) 1𝛾⎞⎟⎟⎟⎟⎠⟩

4. (A1)∧D𝜆 = ⟨

⎛⎜⎜⎜⎜⎝1

1+(𝜆(1–a1a1

)𝛾) 1𝛾 ,1

1+(𝜆(1–b1b1

)𝛾) 1𝛾 ,1

1+(𝜆(1–c1c1

)𝛾) 1𝛾⎞⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎝1 – 1

1+(𝜆(

e11–e1

)𝛾) 1𝛾 , 1 –1

1+(𝜆(

f11–f1

)𝛾) 1𝛾 , 1 –1

1+(𝜆(

g11–g1

)𝛾) 1𝛾⎞⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎝1 – 1

1+(𝜆(

r11–r1

)𝛾) 1𝛾 , 1 –1

1+(𝜆(

s11–s1

)𝛾) 1𝛾 , 1 –1

1+(𝜆(

t11–t1

)𝛾) 1𝛾⎞⎟⎟⎟⎟⎠⟩

3.2. The SVTNDPNBM Operator

Now, based on these new Dombi T-norm and Dombi T-conorm ofSVTNS, we define the SVTNDBMoperator and the SVTNDPNBMoperator.

Definition 11. Let

xi = ⟨(ai, bi, ci) ,(ei, fi, gi

), (ri, si, ti)⟩ (i = 1, 2,⋯ , n) be a set of

SVTN numbers, and p, q > 0. Then, the SVTNDBM operator canbe depicted as follows:

SVTNDBMp,q (x1, x2,⋯ , xn) =

⎛⎜⎜⎜⎜⎝1

n (n – 1) ⋅Dn⊕Di, j = 1,i ≠ j

((xi)∧Dp ⊗D

(xj)∧Dq)⎞⎟⎟⎟⎟⎠

∧D1

p+q

(9)

This part proposes the SVTNDPNBM operator based on the PAoperator and NWBM operator as Definitions 6 and 9. The SVT-NDPNBM operator is defined as follows:

Definition 12. Let C = {C1,C2,⋯ ,Cn} be a set of crite-ria, and exists C1 ≻ C2 ≻ ⋯ ≻ Cn priority rela-tionship between the criteria. The performance value of objectx under criterion Ci is denoted by SVTN numbers xi =⟨(ai, bi, ci) ,

(ei, fi, gi

), (ri, si, ti)⟩(i = 1, 2,⋯ , n). Then the SVT-

NDPNBM operator is depicted as follows:

SVTNDPNBMp,q (x1, x2,⋯ , xn) =

⎛⎜⎜⎜⎜⎝n⊕Di, j = 1,i ≠ j

( wiwj1 – wi

⋅D((xi)∧Dp ⊗D

(xj)∧Dq))⎞⎟⎟⎟⎟⎠

∧D1

p+q

(10)

where wi =Hi

∑ni=1 Hi, Hi = ∐i–1k=1 S (xk) (i ≥ 2), H1 = 1, and S (xk)

is the score function of SVTN number xk obtained by Definition 4.

Theorem 1. The SVTNDPNBM operator in Definition 12 is still anSVTN number, anPdf_Folio:1095


SVTNDPNBMp,q (x1, x2,⋯ , xn)

= ⟨

⎛⎜⎜⎜⎜⎜⎜⎝

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝1/

n

∑i, j = 1,i ≠ j

(p+q)WijpA𝛾i +qA

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

,

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝1/

n

∑i, j = 1,i ≠ j

(p+q)WijpB𝛾i +qB

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

, 1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝1/

n

∑i, j = 1,i ≠ j

(p+q)WijpC𝛾i +qC

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎜⎜⎝1 – 1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝1/

n

∑i, j = 1,i ≠ j

(p+q)WijpE𝛾j +qE

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

, 1 – 1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝1/

n

∑i, j = 1,i ≠ j

(p+q)WijpF𝛾i +qF

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

, 1 – 1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝1/

n

∑i, j = 1,i ≠ j

(p+q)WijpG𝛾i +qG

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎜⎜⎝1 – 1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝1/

n

∑i, j = 1,i ≠ j

(p+q)WijpE𝛾i +qE

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

, 1 – 1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝1/

n

∑i, j = 1,i ≠ j

(p+q)WijpF𝛾i +qF

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

, 1 – 1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝1/

n

∑i, j = 1,i ≠ j

(p+q)WijpG𝛾i +qG

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎠⟩

Theorem 1 can be proved by mathematical operations as follows:

Proof. Let1 – aiai

= Ai,1 – ajaj

= Aj,1 – bibi

= Bi,1 – bjbj

= Bj,

1 – cici

= Ci,1 – cjcj

= Cj,ei

1 – ei= Ei,

ej1 – ej

= Ej,fi

1 – fi= Fi,

fj1 – fj

= Fj,gi

1 – gi= Gi,

gj1 – gj

= Gj,ri

1 – ri= Ri,

rj1 – rj

= Rj,

si1 – si

= Si,sj

1 – sj= Sj,

ti1 – ti

= Ti,tj

1 – tj= Tj,

wiwj1 – wi

= Wij.According to Definition 10, we have,

(xi)∧Dp ⊗D(xj)∧Dq =

⟨

⎛⎜⎜⎜⎜⎝1

1 +(pA𝛾i + qA

𝛾j

) 1𝛾, 1

1 +(pB𝛾i + qB

𝛾j

) 1𝛾, 1

1 +(pC𝛾i + qC

𝛾j

) 1𝛾

⎞⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎝1 – 1

1 +(pE𝛾i + qE

𝛾j

) 1𝛾, 1 – 1

1 +(pF𝛾i + qF

𝛾j

) 1𝛾,

1 – 1

1 +(pG𝛾i + qG

𝛾j

) 1𝛾

⎞⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎝1 – 1

1 +(pR𝛾i + qR

𝛾j

) 1𝛾,

1 – 1

1 +(pS𝛾i + qS

𝛾j

) 1𝛾, 1 – 1

1 +(pT𝛾i + qT

𝛾j

) 1𝛾

⎞⎟⎟⎟⎟⎠⟩

Then,

wiwj1 – wi

⋅D (xi)∧Dp ⊗D(xj)∧Dp = ⟨

⎛⎜⎜⎜⎜⎝1 – 1/

⎛⎜⎜⎜⎜⎝1 +

W1𝛾i j(

pA𝛾i + qA𝛾j

) 1𝛾

⎞⎟⎟⎟⎟⎠,

1 – 1/

⎛⎜⎜⎜⎜⎝1 +

W1𝛾i j(

pB𝛾i + qB𝛾j

) 1𝛾

⎞⎟⎟⎟⎟⎠, 1 – 1/

⎛⎜⎜⎜⎜⎝1 +

W1𝛾i j(

pC𝛾i + qC𝛾j

) 1𝛾

⎞⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎝1/

⎛⎜⎜⎜⎜⎝1 +

W1𝛾i j(

pE𝛾i + qE𝛾j

) 1𝛾

⎞⎟⎟⎟⎟⎠, 1/

⎛⎜⎜⎜⎜⎝1 +

W1𝛾i j(

pF𝛾i + qF𝛾j

) 1𝛾

⎞⎟⎟⎟⎟⎠,

1/

⎛⎜⎜⎜⎜⎝1 +

W1𝛾i j(

pG𝛾i + pG𝛾j

) 1𝛾

⎞⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎝1/

⎛⎜⎜⎜⎜⎝1 +

W1𝛾i j(

pR𝛾i + qR𝛾j

) 1𝛾

⎞⎟⎟⎟⎟⎠,

1/

⎛⎜⎜⎜⎜⎝1 +

W1𝛾i j(

pS𝛾i + qS𝛾j

) 1𝛾

⎞⎟⎟⎟⎟⎠, 1/

⎛⎜⎜⎜⎜⎝1 +

W1𝛾i j(

pT𝛾i + qT𝛾j

) 1𝛾

⎞⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎠⟩

Pdf_Folio:1096


And,

n⊕D

i, j = 1,i ≠ j

( wiwj1 – wi

⋅D((xi)∧Dp ⊗D

(xj)∧Dq))

= ⟨

⎛⎜⎜⎜⎜⎜⎜⎝1 – 1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝n

∑i, j = 1,i ≠ j

WijpA𝛾i +qA

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

, 1 – 1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝n

∑i, j = 1,i ≠ j

WijpB𝛾i +qB

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

, 1 – 1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝n

∑i, j = 1,i ≠ j

WijpC𝛾i +qC

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎜⎜⎝1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝n

∑i, j = 1,i ≠ j

WijpE𝛾i +pE

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

, 1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝n

∑i, j = 1,i ≠ j

WijpF𝛾i +qF

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

, 1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝n

∑i, j = 1,i ≠ j

WijpG𝛾i +qG

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎠,

⎛⎜⎜⎜⎜⎜⎜⎝1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝n

∑i, j = 1,i ≠ j

WijpR𝛾i +qR

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

, 1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝n

∑i, j = 1,i ≠ j

WijpS𝛾i +qS

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

, 1/

⎛⎜⎜⎜⎜⎜⎜⎝1 +

⎛⎜⎜⎜⎜⎜⎝n

∑i, j = 1,i ≠ j

WijpT𝛾i +qT

𝛾j

⎞⎟⎟⎟⎟⎟⎠

1𝛾 ⎞⎟⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎠⟩

Furthermore, according to Definition 10 we can prove thatTheorem 1 is established.

Theorem 2. (Reducibility)

Let C = {C1,C2,⋯ ,Cn} be a set of criteria, and Ci (i = 1, 2,⋯ , n)have the priority relationship. When wi =

1n (i = 1, 2,⋯ , n) is sat-

isfied, the SVTNDPNBM operator is equivalent to the SVTNDBMoperator.

Proof. When wi =1n (i = 1, 2,⋯ , n), then

wiwj1 – wi

= 1n (n – 1) ,according to Definition 10, we can obtain that

SVTNDPNBMp,q (x1, x2,⋯ , xn) = SVTNDBMp,q (x1, x2,⋯ , xn)

Then, we can prove that Theorem 2 is established.

Theorem 3. (Idempotency)

Let xi = ⟨(ai, bi, ci) ,(ei, fi, gi

), (ri, si, ti)⟩(i = 1, 2,⋯ , n) be a set of

SVTN numbers, if all SVTN number are equal, i.e., xi = x. Then,

SVTNDPNBMp,q (x1, x2,⋯ , xn) = x

Proof. When xi = x, according to Definition 10, we can obtain that

SVTNDPNBMp,q (x1, x2,⋯ , xn)

=

⎛⎜⎜⎜⎜⎝n⊕Di, j = 1,i ≠ j

( wiwj1 – wi

⋅D((xi)∧Dp ⊗D

(xj)∧Dq))⎞⎟⎟⎟⎟⎠

∧D1

p+q

=

⎛⎜⎜⎜⎜⎝n⊕Di, j = 1,i ≠ j

( wiwj1 – wi

⋅D (xi)∧Dp+q)⎞⎟⎟⎟⎟⎠

∧D1

p+q

=

⎛⎜⎜⎜⎜⎜⎝n

∑i, j = 1,i ≠ j

wiwj1 – wi

⋅D (xi)∧Dp+q

⎞⎟⎟⎟⎟⎟⎠

∧D1

p+q

=((xi)∧Dp+q

)∧D 1p+q= x

Then, we can prove that Theorem 3 is established.

Pdf_Folio:1097


4. METHOD FOR SELECTING GREENSUPPLIER

Suppose that there are m green providers X = {x1, x2,⋯ , xm} andn criteria C = {c1, c2,⋯ , cn}. There is a correlation between thecriteria, and different criteria have a strict priority relationship. LetU =

(aij)m×n

be a single-valued triangular Neutrosophic decisionmatrix, where aij = ⟨Tij, Iij, Fij⟩ is the evaluation value of each greensupplier under different criteria.

The following section is a sorting process for green suppliers basedon the SVTNDPNBM operator.

• Step 1: Calculate the score value sij of each aij according toDefinition 3.

• Step 2: Determine the relevant weight according to each sij.

• Step 3: Assume 𝛾 = 1, p = q = 1 and aggregate the evaluationvalues of each green suppliers based on Definition 12 and theresults of Step 2.

• Step 4: Calculate the value S (xi) , H (xi) (i = 1, 2,⋯ ,m)of thegreen suppliers after aggregation according to Definition 3.

• Step 5: Sort by the results of step 4.

• Step 6: Replace 𝛾, p, q with different values and compareanalysis.

• Step 7: Comparative analysis.

5. NUMERICAL EXAMPLE

Consider the green supplier selection problem in which a criterionhas a priority relationship and amutual relationship and cannot givecorresponding weights to the example of the section. There are fivegreen suppliers to choose X = {x1, x2,⋯ , x5} and five evaluationcriteria C = {c1, c2,⋯ , c5}. The five criteria are product quality,cost, service capability, green image, and innovation capability. Thepriority relationship between the criteria is c1 ≻ c2 ≻ c3 ≻ c4 ≻ c5.Table 1 is a single-valued triangular Neutrosophic decision matrix.The elements in the Table 1 represent the corresponding evaluationvalues under the five criteria.

Step 1: Calculate the score value sij of each aij according to Defini-tion 3. The calculation results are shown in Table 2.

Step 2: Determine the relevant weight according to each sij. The cal-culation results are shown in Table 3.

Step 3: Assume 𝛾 = 1, p = q = 1 and aggregate the evaluationvalues of each green suppliers based onDefinition 12 and the resultsof Step 2. According to this, the comprehensive evaluation value ofthe green suppliers can be obtained as follows:

x1 = ⟨(0.5319, 0.6211, 0.7872) , (0.3547, 0.4150, 0.5237) ,(0.1884, 0.3404, 0.4666) ⟩

x2 = ⟨(0.5720, 0.6539, 0.7654) ,

(0.3966, 0.4953, 0.6066

),

(0.2050, 0.3372, 0.4159) ⟩

x3 = ⟨(0.6490, 0.7340, 0.8307) , (0.4666, 0.5469, 0.6115) ,(0.3559, 0.4516, 0.5349) ⟩

x4 = ⟨(0.7449, 0.8064, 0.9390) , (0.5704, 0.6347, 0.7543) ,(0.4050, 0.5156, 0.6033) ⟩

x5 = ⟨(0.5351, 0.6188, 0.7876) , (0.4351, 0.4919, 0.5705) ,(0.2919, 0.3808, 0.4983) ⟩

Step 4: Calculate the value S (xi) , H (xi) (i = 1, 2,⋯ ,m) of thegreen suppliers after aggregation according toDefinition 3. And thecalculation results are shown in Table 4.

Step 5: Sort by the results of step 4.

Since S (x1) > S (x2) > S (x5) > S (x3) > S (x4), according to Defi-nition 4, we sort the green suppliers as x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4.Step 6: Replace 𝛾, p, q with different values and compare analysis.In order to consider the influence of parameters on the order-ing, this step analyzes the influence of the change of 𝛾 on theordering when p, q = 1, and the effect of the change of p, q on theordering when 𝛾 = 1. The effect of the change of 𝛾 is shown inTable 5, and the results of the change of p, q are shown in Table 6.Step 7: Comparative analysis

Compare the method proposed in this paper with the methods inother literatures, the results are shown in Table 7.

Table 1 Single-valued triangular Neutrosophic decision matrix.

c1 c2 c3 c4 c5

x1

x2

x3

x4

x5

Pdf_Folio:1098


Table 2 Score value.

c1 c2 c3 c4 c5x1 0.5858 0.800 0.5492 0.5767 0.5375x2 0.6450 0.6417 0.5125 0.6750 0.5625x3 0.5558 0.6617 0.5958 0.5442 0.5008x4 0.5067 0.5625 0.6283 0.6150 0.5858x5 0.5492 0.6350 0.5375 0.6417 0.6617

Table 3 Weight matrix.

c1 c2 c3 c4 c5x1 0.4065 0.2381 0.1905 0.1046 0.0603x2 0.4142 0.2672 0.1714 0.0879 0.0593x3 0.4421 0.2457 0.1626 0.0969 0.0527x4 0.4806 0.2435 0.1369 0.0861 0.0529x5 0.4534 0.2490 0.1581 0.0850 0.0545

Table 4 Score value and accuracy value.

x1 x2 x3 x4 x5S(xi)

0.6264 0.6130 0.5818 0.5553 0.5849H(xi)

0.3064 0.3375 0.2885 0.3143 0.2521

Table 5 The ranking orders of different 𝛾.

𝛾 Score Value Ranking Order1 S

(x1)= 0.6264, S

(x2)= 0.6130, S

(x3)= 0.5818, S

(x4

)= 0.5553, S

(x5)= 0.5849 x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4

2 S(x1)= 0.6194, S

(x2)= 0.6097, S

(x3)= 0.5776, S

(x4

)= 0.5628, S

(x5)= 0.5835 x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4

5 S(x1)= 0.6302, S

(x2)= 0.6275, S

(x3)= 0.5849, S

(x4

)= 0.5956, S

(x5)= 0.6001 x1 ≻ x2 ≻ x5 ≻ x4 ≻ x3

10 S(x1)= 0.6518, S

(x2)= 0.6436, S

(x3)= 0.6043, S

(x4

)= 0.6165, S

(x5)= 0.6272 x1 ≻ x2 ≻ x5 ≻ x4 ≻ x3

50 S(x1)= 0.6789, S

(x2)= 0.6610, S

(x3)= 0.6314, S

(x4

)= 0.6469, S

(x5)= 0.6654 x1 ≻ x2 ≻ x5 ≻ x4 ≻ x3

Table 6 The ranking orders of different p, q.

p, q Score Value Ranking Order0.001,1 S

(x1)= 0.6825, S

(x2)= 0.6518, S

(x3)= 0.6139, S

(x4

)= 0.6055, S

(x5)= 0.6198 x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4

0.01,1 S(x1)= 0.6800, S

(x2)= 0.6499, S

(x3)= 0.6126, S

(x4

)= 0.6032, S

(x5)= 0.6185 x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4

0.1,1 S(x1)= 0.6614, S

(x2)= 0.6363, S

(x3)= 0.6025, S

(x4

)= 0.5868, S

(x5)= 0.6080 x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4

1,1 S(x1)= 0.6264, S

(x2)= 0.6130, S

(x3)= 0.5818, S

(x4

)= 0.5553, S

(x5)= 0.5849 x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4

1,2 S(x1)= 0.6322, S

(x2)= 0.6159, S

(x3)= 0.5854, S

(x4

)= 0.5613, S

(x5)= 0.5896 x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4

1,5 S(x1)= 0.6489, S

(x2)= 0.6274, S

(x3)= 0.5954, S

(x4

)= 0.5759, S

(x5)= 0.6004 x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4

1,10 S(x1)= 0.6614, S

(x2)= 0.6363, S

(x3)= 0.6025, S

(x4

)= 0.5868, S

(x5)= 0.6080 x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4

0.1,0.1 S(x1)= 0.6264, S

(x2)= 0.6130, S

(x3)= 0.5818, S

(x4

)= 0.5553, S

(x5)= 0.5849 x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4

4,4 S(x1)= 0.6264, S

(x2)= 0.6130, S

(x3)= 0.5818, S

(x4

)= 0.5553, S

(x5)= 0.5849 x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4

10,10 S(x1)= 0.6264, S

(x2)= 0.6130, S

(x3)= 0.5818, S

(x4

)= 0.5553, S

(x5)= 0.5849 x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4

Table 7 The ranking orders of approaches.

Order

SVTNDPNBM x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4[13] TFNNWA x1 ≻ x2 ≻ x3 ≻ x5 ≻ x4[13] TFNNWG x1 ≻ x2 ≻ x5 ≻ x3 ≻ x4[14] VIKOR x1 ≻ x2 ≻ x3 ≻ x5 ≻ x4

As shown in Tables 5 and 6, the best green supplier is always x1, nomatter how the parameters change. As shown in Table 7, the bestgreen supplier is always x1, no matter which method is used. Theseresults demonstrate that the applicability and stability of the SVT-NDPNBM operator proposed in this paper.

6. CONCLUSION

In this paper, we use SVTNS to indicate the evaluation value, anduse the triangular fuzzy number to represent the truth-membershipfunction, the indeterminacy-membership function and thefalsity-membership function, which can retain more uncertaininformation of the object to be evaluated. The main contributionof this paper is to consider the advantages and flexibility of Dombioperations, PA operator, and BM operator, and combine them topropose the SVTNDPNBM operator. The new aggregate operatortakes into account the priority relationship and the interrelation-ship between the criteria. To make the new aggregate operatormore flexible, this paper introduces the Dombi operations. ThePdf_Folio:1099


feasibility of the SVTNDPNBM operator is verified by the examplechosen by the green supplier, and the stability of the SVTNDPNBMoperator is verified by the change of the parameters.

In the future, due to the variety of operators, we can combine dif-ferent operators and propose new operators. At the same time, wecan further explore the different properties and applications of theSVTNS.

CONFLICT OF INTEREST

We declare that we do not have conflicts of interest with the worksubmitted.

AUTHORS’ CONTRIBUTIONS

Funding acquisition was done by Mei Qin Wu; methodology pre-pared by Jian Ping Fan; original draft written, edited, and reviewedby Xuefei Jia.

Funding Statement

This workwas supported by the Fund for Shanxi “1331 Project” KeyInnovative Research Team (2017) and Shanxi “Disciplinary GroupConstruction Plan for Industry Innovation”: Management of Intel-ligent Logistics System (2018).

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Green Supplier Selection Based on Dombi Prioritized Bonferroni Mean Operator with Single-Valued Triangular Neutrosophic Sets1. INTRODUCTION2. PRELIMINARIES2.1. Single-Valued Triangular Neutrosophic Sets2.2. Dombi Operations2.3. PA Operator2.4. BM Operator

3. DOMBI PRIORITIZED NORMALIZED BM OPERATOR OF SVTNS3.1. Dombi Operations of SVTNS3.2. The SVTNDPNBM Operator

4. METHOD FOR SELECTING GREEN SUPPLIER5. NUMERICAL EXAMPLE6. CONCLUSION

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