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Multi-criteria decision-making method based on a cross-entropy with intervalneutrosophic sets

Zhang-peng Tian , Hong-yu Zhang , Jing Wang , Jian-qiang Wang and Xiao-hong ChenSchool of Business, Central South University, Changsha, PR China

ARTICLE HISTORY

Received May Accepted September

KEYWORDS

Multi-criteriadecision-making; intervalneutrosophic sets;cross-entropy; mathematicalprogramming; TOPSIS

ABSTRACT

In this paper, two optimisation models are established to determine the criterion weights in multi-criteriadecision-making situations where knowledge regarding the weight information is incomplete and the cri-terion values are interval neutrosophic numbers. The proposed approach combines interval neutrosophicsets and TOPSIS, and the closeness coefficients are expressedas interval numbers. Furthermore, the relativelikelihood-based comparison relations are constructed to determine the ranking of alternatives. A fuzzycross-entropy approach is proposed to calculate the discrimination measure between alternatives and theabsolute ideal solutions, after a transformation operator has been developed to convert interval neutro-sophic numbers into simpli ed neutrosophic numbers. Finally, an illustrative example is provided, and acomparative analysis is conducted between theapproachdeveloped in this paper andotherexisting meth-ods, to verify the feasibility and effectiveness of the proposed approach.

. IntroductionFuzzy sets (FSs), which were proposed by Zadeh (1965), areregarded as a comprehensive tool for solving multi-criteriadecision-making (MCDM) problems (Bellman & Zadeh, 1970).In order to resolve the uncertainty of non-membership degrees,Atanassov (1986) introduced intuitionistic fuzzy sets (IFSs),which are an extension of Zadeh’s FSs. IFSs have been widely applied in solving MCDM problems to date (Chen & Chang,2015;Chen,Cheng, & Chiou, 2016; Wang etal., 2014; Yue,2014).

Moreover, interval-valued intuitionistic fuzzy sets (IVIFSs)(Atanassov & Gargov, 1989) were proposed, which are an exten-sion of FSs and IFSs. In recent years, MCDM problems withIVIFSs have attracted much attention from researchers (Chen,2014; Liu, Shen, Zhang, Chen, & Wang, 2015; Tan et al., 2014;Wan & Dong, 2014). Furthermore, the TOPSIS method, pro-posed byHwang and Yoon (1981), has also beenused for solvingMCDM problems (Cao, Wu, & Liang, 2015; Yue, 2014; Zhang& Yu, 2012). Moreover, fuzzy linear programming models havebeen constructed to address MCDM problems with incompletecriterion weight information (Chen, 2014; Dubey, Chandra, & Mehra, 2012; Wan, Wang, Lin& Dong, 2015; Wan, Xu, Wang, & Dong, 2015; Zhang & Yu, 2012).

Although the theories of FSs and IFSs have been developedand generalised, they cannot deal with all types of uncertain-ties in real problems. Indeed certain types of uncertainties, suchas indeterminate and inconsistent information, cannot be man-aged.For example, FSsand IFSs cannot effectivelydeal with a sit-uationwherea paper is sent toa reviewer, and heor she says itis70% acceptable and 60% unacceptable, and his or her statementis 20% uncertain; therefore, some new theories are required.

Since neutrosophic sets (NSs) (Smarandache, 1999) con-sider the truth membership, indeterminacy membership and

CONTACT Jian-qiang Wang jqwang@csu.edu.cn

falsity membership simultaneously, it is more practical and ex-ible than FSs and IFSs in dealing with uncertain, incompleteand inconsistent information. For the aforementioned exam-ple, the reviewer’s opinion can be presented as x (0.7, 0.2, 0.6)by means of NSs. However, without a speci c description, itis hard to apply NSs in actual scienti c and engineering situ-ations. Hence, single-valued neutrosophic sets (SVNSs), whichare an extension of NSs, were introduced by Wang et al. (2010).Subsequently, the similarity and entropy measures (Majum-

dar & Samant, 2014), the correlation coefficient (Ye, 2013)and the cross-entropy (Ye, 2014c) of SVNSs have been devel-oped. Additionally, Ye ( 2014a) introduced simpli ed neutro-sophic sets (SNSs), and Peng, Wang, Wang, Zhang, and Chen(2015) and Peng, Wang, Zhang, and Chen ( 2014) de ned theirnovel operations and aggregation operators. Finally, furtherextensions of NSs, such as interval neutrosophic sets (INSs)(Wang et al., 2005), bipolar neutrosophic sets (BNSs) (Deli,Ali, & Smarandache, 2015) and multivalued neutrosophic sets(MVNSs) (Peng, Wang, Wu, Wang, & Chen, 2015; Wang & Li,2015), have also been proposed.

In certain real-life situations, the INS, as a particular exten-sion of an NS, can be more exible in assessing objections thanan SNS. Recently, the studies relating to INSs have been focusedon particular areas, which can be roughly classi ed into twogroups.

The rst group is based on interval neutrosophic aggregationoperators, such as interval neutrosophic number weighted aver-aging (INNWA) and interval neutrosophic number weightedgeometric (INNWG) operators (Zhang, Wang, & Chen, 2014),interval neutrosophic number Choquet integral (INNCI) oper-ator (Sun et al., 2015), interval neutrosophic number orderedweighted averaging (INNOWA) and interval neutrosophic

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number ordered weighted geometric (INNWG) operators (Ye,2015) and interval neutrosophic prioritised ordered weightedaggregation (INPOWA) operator (Liu & Wang, 2015).

The second group is based on interval neutrosophic mea-sures (Broumi & Smarandache, 2014a, 2014b; Chi & Liu, 2013;Ye, 2014b; Zhang, Ji, Wang, & Chen 2015a; Zhang et al. 2015b).Speci cally, Chi and Liu (2013) extended TOPSIS to an INSbased on a distance measure. Broumi and Smarandache (2014a,

2014b) de ned a new cosine similarity measure and a correla-tion coefficient of an INS. Moreover, Ye (2014b) de ned twointerval neutrosophic similarity measures based on the Ham-ming and Euclidean distances. Zhang et al. (2015b) de ned sev-eral interval neutrosophic outranking relations based on ELEC-TRE IV. Finally, Zhang et al. (2015a) proposed an improvedweighted correlation coefficient based on the integrated weightfor an INS.

The aforementioned methods are effective when managinginterval neutrosophic MCDM problems; however, they havesome drawbacks which are outlined below.

(1)TheexistingMCDM methods require thecriterionweightinformation to be completely known (Broumi & Smarandache,2014b; Ye, 2014b, 2015; Zhang et al., 2014, 2015a, 2015b), or aresupported by fuzzy measures of criteria (Sun et al., 2015). How-ever, due to the increasing complexity of MCDM problems, it isdifficult and subjective to provide exact criterion weight infor-mation or fuzzy measures. (2) The MCDM methods (Broumi & Smarandache, 2014b; Chi & Liu, 2013; Ye, 2014b; Zhang et al.,2015a, 2015b) utilise the weighted measures, such as distance,correlation coefficient and similarity measures, to rank alter-natives; in these processes, the interval neutrosophic informa-tion is measured with crisp real numbers. However, the meansof aggregating interval numbers into real numbers may lead tosome operational de ciencies and a large amount of informa-tion loss. (3) All of the developed aggregation operators (Liu & Wang, 2015;Sunet al., 2015; Ye,2015; Zhang etal., 2014) directly process the assessment information with INNs. However, theseapproaches are complex and tedious, and not enough attentionis given to reducing the computational complexity when pro-cessing the evaluation information.

To overcome these disadvantages, in this paper, a novel andcomprehensive approach for managing MCDM with INNs isdeveloped. The main novelties and the signi cant contributionsare summarised below.

(1) In order to derive the criterion weights, two mathemati-calprogramming models are constructed to determine theopti-mal weight values, and decision-makers (DMs) may provide

incomplete or inconsistent opinions on the weight information.(2) A novel form of the closeness coefficient with interval val-ues is derived on the basis of TOPSIS and a cross-entropy of INSs. Furthermore, the weighted interval closeness coefficientsare employed to determine the ranking of alternatives. (3) Tolower the computational complexity, a modi ed transforma-tion operator is developed for converting INNs into simpli edneutrosophic numbers (SNNs). Nevertheless, the parameters of the transformation operator are not arti cially set, but obtainedthrough mathematical derivation.

The rest of the paper is organised as follows. In Section 2,interval numbers, as well as the concepts ofNSs, SNSs and INSs,arebrie y reviewed.In Section3 , onthe basis ofan operator that

can transform each INN into an SNN, a cross-entropy for SNNsis proposed. Subsequently, a TOPSIS approach in the context of INSsforsolving MCDMproblemswithincompletelyknown cri-terion information is developed in Section 4. In Section 5, anillustrative example is provided and a comparative analysis isconducted between the proposed approach and other existingmethods. Finally, conclusions are drawn in Section 6.

. PreliminariesIn this section, some basic concepts and de nitions related toINSs, including interval numbers, de nitions and operationallaws of NSs, SNSs and INSs are introduced; these will be utilisedin the latter analysis.

. . Interval numbersIntervalnumbersandtheir operations areof utmost signi cancewhen exploring the operations of INSs. In the following para-graphs, some de nitions and operational laws of interval num-bers are provided.

De nition 1 (Sengupta & Pal, 2000; Xu, 2008): Let a =[aL, aU ] = {x |aL ≤ x ≤ aU }, and then a is called an intervalnumber. In particular, a = [aL, aU ] will be degenerated to a realnumber, if aL = aU .

Consider two non-negative interval numbers a = [aL, aU ]and b = [bL, bU ], where 0 ≤ aL ≤ x ≤ aU and 0 ≤ bL ≤ x ≤bU . Subsequently, their operations are de ned as follows (Sen-gupta & Pal, 2000; Xu, 2008):

(1) a + b = [aL + bL, aU + bU ],(2) λa = [λ aL, λ aU ], λ > 0.

De nition 2 (Xu & Da, 2003): For any two interval numbersa = [aL, aU ] and b = [bL, bU ], the possibility of a ≥ b is for-

mulated by p( a ≥ b) = max{1 − max{ bU

− aL

L( a )+ L( b) , 0}, 0}, whereL( a ) = aU − aL and L( b) = bU − bL.

The possibility degree of a ≥ b has the following properties(Xu & Da, 2003):

(1) 0 ≤ p( a ≥ b) ≤ 1;(2) p( a ≥ b) = p( b ≥ a ) = 0.5, if p( a ≥ b) = p( b ≥ a ) ;(3) p( a ≥ b) + p( b ≥ a ) = 1;(4) p( a ≥ b) = 0 if aU ≤ bL, p( a ≥ b) = 1, if aL ≥ bU ;(5) For any interval numbers a, b and c, p( a ≥ c) ≥ 0.5, if

p( a ≥ b) ≥ 0.5 and p( b ≥ c) ≥ 0.5; p( a ≥ c) = 0.5, if and only if p( a ≥ b) = p( b ≥ c) = 0.5.

. . NSs and SNSsDue to the in uence of subjective factors, it is difficult for DMsto explicitly express preferences if indeterminacy and inconsis-tency exist. NS can effectively capture such information (Guo& Sengür, 2014; Mohan, Krishnaveni, & Guo, 2013; Solis & Panoutsos, 2013).

De nition 3 (Smarandache, 1999): Let X be a space of points(objects) with a generic element in X , denoted by x . An NS Ain X is characterised by a truth-membership function t A(x ),an indeterminacy-membership function i A(x ) and a falsity-membership function f A(x ) . t A(x ) , i A(x ) and f A(x ) are real

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standard or non-standard subsets of ]0 − , 1+ [; that is, t A(x ) : X → ]0− , 1+ [, i A(x ) : X → ]0− , 1+ [, and f A(x ) : X → ]0− ,1+ [. There is no restriction on the sum of t A(x ) , i A(x ) and f A(x ), thus 0− ≤ sup t A(x ) + sup i A(x ) + sup f A(x ) ≤ 3+ .

Since it is difficult to apply NSs to practical problems, Ye(2014a) reduced NSs of non-standard interval numbers into atype of SNS of standard interval numbers.

De nition 4 (Rivieccio, 2008; Ye, 2014a): Let an NS A in X becharacterised by t A(x ), i A(x ) and f A(x ), which are single subin-tervals/subsets in the real standard [0 , 1]; that is, t A(x ) : X →[0, 1], i A(x ) : X → [0, 1], and f A(x ) : X → [0, 1]. Also, thesumof t A(x ) , i A(x ) and f A(x ) satis esthecondition 0 ≤ t A(x ) +i A(x ) + f A(x ) ≤ 3. Then, a simpli cation of A is denoted by A = { (x , t A(x ), i A(x ), f A(x )) |x ∈ X }, which is called an SNSand is a subclass of NSs. If X = 1, an SNS will be degeneratedto an SNN.De nition 5 (Rivieccio, 2008; Ye, 2014a): An SNS A is con-tained in the other SNS B, denoted by A ⊆ B, if and only if t A(x ) ≤ t B(x ), i A(x ) ≥ iB(x ) and f A(x ) ≥ f B(x ) , for any x ∈ X .

The complement set of A, denoted by Ac

, is de ned as Ac

={(x , f A(x ), i A(x ), t A(x )) |x ∈ X }.

. . INSsIn actual applications, sometimes it is not easy to expressthe truth membership, indeterminacy membership and falsity membership by crisp values, but they may be easily describedby interval numbers. Wang et al. ( 2005) further de ned INSs.

De nition 6 (Rivieccio, 2008; Wang et al., 2005): Let X be aspace of points (objects) with generic elements in X , denotedby x . An INS ˜ A in X is characterised by a truth-membershipfunction t ˜ A(x ), an indeterminacy-membership function i ˜ A(x )

and a falsity-membership function f ˜ A(x ) . For each point x in X , t A(x ) = [t L˜ A

, t U ˜ A

], i ˜ A(x ) = [iL˜ A, iU

˜ A] and f ˜ A(x ) = [ f L˜ A

, f U ˜ A

], and0 ≤ t U

˜ A+ iU

˜ A+ f U

˜ A≤ 3. In particular, an INS will be reduced to

an SNS, if t L˜ A = t U

˜ A, iL˜ A

= iU ˜ A

and f L˜ A = f U

˜ A. In addition, if X =

1, an INS will be degenerated to an INN.

De nition 7 (Rivieccio, 2008; Wang et al., 2005): An INS ˜ Ais contained in the other INS B, denoted by ˜ A ⊆ B, if andonly if t L˜ A

≤ t LB, t U ˜ A

≤ t U B , iL˜ A

≥ iLB, iU ˜ A

≥ iU B

, f L˜ A ≥ f LB and f U

˜ A≥

f U B , for any x ∈ X . In particular, ˜ A = B, if ˜ A ⊆ B and ˜ A ⊇ B.

The complement set of ˜ A, denoted by ˜ Ac, is de ned as ˜ Ac ={(x , [ f L˜ A

, f U ˜ A

], [iL˜ A, iU

˜ A], [t L˜ A

, t U ˜ A

]) |x ∈ X }.

. A transformation operator and a cross-entropymeasure of SNNsIn this section, a transformation operator is developed to con- vert INNs intoSNNs.Moreover,a simpli edneutrosophiccross-entropy measure is de ned.

. . A transformation operator between INNs and SNNsBustince and Burillo (1995) proposed an operator H p,q, whichcan transform each IVFS into an IFS. As an improved extensionof H p,q, an operator H p,q, r is de ned for converting each INNinto an SNN.

De nition8 Let p, q, r ∈ [0, 1]be three xednumbers;anyINNcan be transformed into an SNN through the operator H p, q, r :

H p, q, r ( ˜ A) = t L˜ A + pW t A , iL˜ A

+ qW i ˜ A, f L˜ A

+ rW f ˜ A,

where W t A = t U ˜ A

− t L˜ A, W i ˜ A

= iU ˜ A

− iL˜ A and W f ˜ A

= f U ˜ A

− f L˜ A.

Obviously, H p,q, r ( ˜ A) is an SNN, determined with respect to p, qand r ; that is, H p,q, r ( ˜ A) is well de ned in all value ranges of p, qand r .

Example 1: Assume two INNs ˜ A = ([0.7, 0.8], [0, 0.1],[0.1, 0.2]) and B = ( [0.4, 0.5], [0.2, 0.3], [0.3, 0.4]). Then, thefollowing transformation results can be obtained utilising theoperator H p, q, r :

H p,q, r ( ˜ A) = (0.7 + p × 0.1, 0 + q × 0.1, 0.1 + r × 0.1) and

H p,q, r (B) = (0.4 + p × 0.1, 0.2 + q × 0.1, 0.3 + r × 0.1).

It is shown that H p,q, r ( ˜ A) and H p,q, r (B) will be two spe-ci c SNNs if the values of p, q and r are given; the method todetermine the parameter values will be discussed in detail inSection 4.

. . A cross-entropy measure of SNNsShang and Jiang (1997) proposed a fuzzy cross-entropy and asymmetric discrimination information measure between twoFSs. Vlachos and Sergiadis (2007) then proposed an intuition-istic fuzzy cross-entropy for IFSs, and Ye (2011) de ned afuzzy cross-entropy for IVIFSs. Furthermore, the fuzzy cross-entropy has been employed for various purposes includingderiving criterion weights by constructing mathematical pro-

gramming models (Zhang & Yu 2012); technical efficiency analysis (Macedo & Scotto, 2014); multi-objective optimisa-tion(Caballero, Hernández-Díaz, Laguna, & Molina, 2015); andMCDM problems (Meng & Chen, 2015; Peng, Wang, Wu, et al.,2014; Ye, 2014c; Zhao et al., 2013).

In a similar manner to the proposals of Ye ( 2014c) and Vla-chos and Sergiadis (2007), the following de nition of a fuzzy cross-entropy for SNNs is proposed.

De nition 9 Let A, B ∈ SNS, and then the cross-entropy I NS( A, B) between A and B should satisfy the following condi-tions:

(1) I NS( A, B) = I NS(B, A) ;(2) I NS( A, B) = I NS( Ac, Bc) , where Ac and Bc are the com-

plement sets of A and B, respectively, as de ned in De -nition 5; and

(3) I NS( A, B) ≥ 0 and I NS( A, B) = 0, if and only if A = B.

De nition 10 Let A = (t A, i A, f A) and B = (t B, iB, f B) be twoSNNs, and then the cross-entropy of A and B can be de ned asfollows:

I NS( A, B) = t Aln 2t At A + t B

+ i Aln 2i Ai A + iB

+ f Aln 2 f A

f A + f B. (1)

Equation (1) can indicate the degree of discrimination of A from B. It is obvious that I NS( A, B) is not symmetrical with

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respect to its arguments. Therefore, a modi ed symmetric dis-crimination information measure based on I NS( A, B) can bede ned as

DNS( A, B) = I NS( A, B) + I NS(B, A). (2)

The larger DNS( A, B) is, the larger the difference between Aand B will be, and vice versa.

Proposition 1: Themeasures de ned in Equations (1) and ( 2 ) arethe simpli ed neutrosophic cross-entropy, and satisfy conditions( 1 )–(3) given in De nition 9.

Proof: Clearly, conditions (1) and (2) are obvious. The proof of condition ( 3) is shown below.

Consider the function f (x ) = x ln x , where x ∈ (0, 1].Then, f (x ) = 1 + ln x and f (x ) = 1/x > 0, where x ∈ (0, 1].Accordingly, f (x ) = x ln x is a convex function. Therefore,for any two points x 1, x 2 ∈ (0, 1], the inequality f (x 1 )+ f (x 2 )

2 ≥ f ( x 1+ x 2

2 ) holds.Utilise f (x ) = x ln x in the above inequality and x 1 ln x 1 +

x 2 ln x 2 − (x 1 + x 2) ln x 1+ x 2

2 ≥ 0 can be obtained; in this case,the equality holds only if x 1 = x 2. Similarly, the following equa-tion can be obtained:

DNS( A, B) = I NS( A, B) + I NS(B, A)

= (t A ln t A+ t B ln t B) − (t A+ t B) ln t A+ t B

2

T

+ (i A ln i A+ iB ln iB) − ( i A+ iB) ln i A+ iB

2

I

+ ( f A ln f A+ f B ln f B) − ( f A+ f B) ln f A+ f B

2 F

.

Because T ≥ 0, I ≥ 0 and F ≥ 0, DNS( A, B) ≥ 0 holds;DNS( A, B) = 0 holds only if t A = t B, i A = iB and f A = f B,namely A = B.

Example 2: Assume two SNNs A = (0.8, 0.1, 0.2) and B =(0.5, 0.3, 0.4). Then, the following result can be obtained by applying Equations (1) and (2):

DNS( A, B) = I NS( A, B) + I NS(B, A) = 0.1212.

. An MCDM approach based on the cross-entropyand TOPSISThis section presents an approach that is based on the cross-entropy and TOPSIS for solving interval neutrosophic MCDMproblems with incomplete weight information.

For an MCDM problem, let A = {a1, a2, . . . , am}be a set consisting of m alternatives, and let C ={c1, c2, . . . , cn} be a set consisting of n criteria. Assumethat w = ( w 1, w 2, . . . , w n ) is the weight vector of crite-ria, where w j ∈ [0, 1] and n

j= 1 w j = 1. Let B = [bi j] =[([t Li j, t U

i j ], [iLi j, iU i j ], [ f Li j, f U

i j ])]m× n be the decision matrix, where[bi j] = [([t Li j, t U

i j ], [iLi j, iU i j ], [ f Li j, f U

i j ])]m× n is the evaluation

information of each alternative ai( i = 1, 2, . . . , m) on thecriterion c j( j = 1, 2, . . . , n) in the form of INNs.

In general, there are two types of criterion, namely maximis-ing and minimising criteria. Inorder to make thecriterion typesconstant, the minimising criteria need to be transformed intomaximising ones. Suppose the standardised matrix is expressedas R = [r i j]. The original decision matrix B can then be con- verted into R based on the primary transformation principle of Xu and Hu (2010), where

r i j =

bi j = t Li j, t U i j , iLi j, iU

i j , f Li j , f U i j ,

for maximising criterion c j

bci j = f Li j , f U

i j , iLi j, iU i j , t Li j, t U

i j ,for minimising criterion c j

, (3)

in which bci j is thecomplement setof bi j, de nedin De nition 7.

. . A fuzzy cross-entropy based on TOPSISThe absolute positive ideal solution (PIS) and theabsolute nega-tive ideal solution (NIS) of INSs are, respectively, denoted by a+

and a− , and can be expressed as follows (Chi & Liu, 2013):

a+ = ( [1, 1], [0, 0], [0, 0]) and a− = ( [0, 0], [1, 1], [1, 1]) .

In order to obtain the cross-entropy or degree of discrimina-tion between ai( i = 1, 2, . . . , m) and the ideal solutions, eachINN is transformed into an SNN based on the operator H p,q, r ,which was introduced in De nition 8. Let pi j, qi j, r i j ∈ [0, 1],and then any INN, denoted by ([t Li j, t U

i j ], [iLi j, iU i j ], [ f Li j , f U

i j ]), canbe transformed into the following form:

H p,q, r ( r i j ) = t Li j + pi jW t ij , iLi j + qi jW iij , f Li j + r i jW f ij , (4)

where W t ij = t U i j − t Li j, W iij = iU

i j − iLi j and W f ij = f U i j − f Li j .

Using Equations (1) , (2)and (4), thedegreeof discriminationof ai( i = 1, 2, . . . , m) from the ideal solutions a+ and a_ withrespect to c j( j = 1, 2, . . . , n) can be, respectively, calculated asfollows:

D+i j = (t Li j + pi jW t ij )ln

2(t Li j + pi jW t ij )

(t Li j + pi jW t ij + 1)

+ ln 2

(t Li j + pi jW t ij + 1) + ( iLi j + qi jW iij )ln2

+ ( f Li j + r i jW f ij )ln2, (5)

D−i j = ( t Li j + pi jW t ij ) ln 2 + ( iLi j + qi jW iij ) ln

2( iLi j + qi jW ii j )

( iLi j + qi jW iij + 1)

+ ln 2

( iLi j + qi jW iij + 1)

+ ( f Li j + r i jW f ij ) ln2( f Li j + r i jW f ij )

( f Li j + r i jW f ij + 1)

+ ln 2

( f Li j + r i jW f ij + 1)

. (6)

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In the TOPSIS method, the closeness coefficientDi j = D−

i j /( D−i j + D+

i j ) denotes the performance of ai( i = 1, 2,. . . , m) on c j ( j = 1, 2, · · · , n). Then, the performance of ai,denoted by Di, can be obtained by using Equations (5) and ( 6):

Di =n

j= 1

w jDi j =n

j= 1

w jD−

i j

D−i j + D+

i j, (7)

where w j represents the weight of c j. Therefore, the larger Di is,the better ai will be.

In view of the fact that an INS is characterised by a truth-membership function, a indeterminacy-membership functionand a falsity-membership function, whose values are intervalsrather than speci c numbers, it is infeasible to designate anSNNfor thegivenINN byarti ciallychoosingonlycertain pi j, qi j andr i j in Equation (4) to indicate the evaluation information. Thisis because it may lead to the distortion or loss of the originalinformation (Zhang & Yu, 2012).

By combining Equations (5) and (6), Di j becomes a compos-ite function and may take different values as the numerical vari-ations pi j, qi j and r i j change. In order to avoid information loss,an interval Di j = [DL

i j, DU i j ] is applied to represent the perfor-

mance of ai( i = 1, 2, . . . , m) on c j( j = 1, 2, . . . , n) , where DLi j

and DU i j are the lower and upper bounds, respectively.

In the following paragraphs, DLi j and DU

i j are determinedthrough mathematical derivation.

As shown in Equations (5) and (6), both D+i j and D−

i j are mul-tivariate continuous functions with respect to pi j, qi j and r i j.They canalso reach themaximum and minimum in thedomain pi j, qi j, r i j ∈ [0, 1]. Then, calculate the partial derivative of D+

i j

and D−i j on pi j, qi j and r i j:

∂D+i j

∂ pi j= W t ij ln

2(t Li j + pi jW t ij )

(t Li j + pi jW t ij + 1),

∂D+i j

∂qi j= W iij ln 2 ≥ 0, and

∂D+i j

∂ r i j= W f i j ln 2 ≥ 0.

Since 2(t Li j + pi jW t ij ) ≤ t Li j + pi jW t ij + 1, ∂D+

ij

∂ pij≤ 0.

Similarly, ∂D−

ij

∂ pij= W t ij ln 2 ≥ 0,

∂D−ij

∂qij= W iij ln

2( iLij + qij W ii j )

(iLi j+ qij W iij + 1) ≤

0, and ∂D−

ij

∂ r ij= W f ij ln

2( f Lij + r ij W f ij )( f Li j+ r ij W f ij + 1) ≤ 0.

This means that D+i j is a monotone decreasing function

with respect to pi j and a monotone increasing function withrespect to qi j or r i j. Moreover, D−

i j is a monotone increasingfunction with respect to p

i j and a monotone decreasing func-

tion with respect to qi j or r i j. Thus, D+i j can reach its max-

imum and D−i j can reach the minimum if pi j = 0 and qi j =

r i j = 1. Likewise, D+i j reaches its minimum and D−

i j reachesthe maximum if pi j = 1 and qi j = r i j = 0. As a result, it iseasy to understand that Di j reaches its minimum when pi j = 0and qi j = r i j = 1, and reaches its maximum when pi j = 1 andqi j = r i j = 0.

Then, DLi j and DU

i j can be obtained by integrating Equa-tions (5) and (6) into Di j, respectively:

DLi j =

t Li j ln 2 + iU i j ln

2iU ij

( iU ij + 1) + f U

i j ln 2 f U

ij

( f U ij + 1) + ln 2

( iU ij + 1) + ln 2

( f U ij + 1)

(t Li j + iU i j + f U

i j ) ln 2 + t Li j ln 2t Li j

(t Li j+ 1) + iU i j ln

2iU ij

(iU ij + 1) + f U

i j ln 2 f U

ij

( f U ij + 1) + ln 2

(t Lij + 1) + ln 2(iU

ij + 1) + ln 2( f U

ij + 1)

, (8)

DU i j =

t U i j ln 2 + iLi j ln

2iLij( iLij + 1) + f Li j ln

2 f Lij( f Lij + 1) + ln 2

( iLij + 1) + ln 2( f Lij + 1)

(t U i j + iLi j + f Li j ) ln 2 + t U

i j ln 2t U

ij

(t U ij + 1) + iLi j ln

2iLij( iLij + 1) + f Li j ln

2 f Lij( f Lij + 1) + ln 2

(t U ij + 1) + ln 2

( iLij + 1) + ln 2( f Lij + 1)

, (9)

where DL

i j and DU

i j are also theminimum and maximum of D

i j =

D−i j /( D−

i j + D+i j ), respectively.

Based on the analysis above, Equation (7) can be rewrittenas

Di =n

j= 1

w jDi j =n

j= 1

w j DLi j, DU

i j , (10)

where Di ≥ D j means that ai( i = 1, 2, . . . , m) is not inferior toa j( j = 1, 2, . . . , m) , andtheweight informationis incompletely known. Obviously, Di represents the comprehensive evaluation values, and stands for the preference of ai; that is, the larger Di

is, the better ai will be.

. . Fuzzy linear programmingmodels for determiningcriterion weightsIn the decision-making process, the importance of different cri-teriashould be taken into consideration. Suppose 0 denotes theset of all the weight vectors, and

0 = (w 1, w 2, . . . , w n ) w j ≥ 0( j = 1, 2, . . . , n),n

j= 1

w j = 1 .

In some actual decision-making situations, the incompleteinformation regarding the criterion weights provided by DMscan usually be constructed using several basic ranking forms(Dubey et al., 2012; Li, 2011). These weight information struc-tures may be expressed in the following ve basic relations,which are denoted by the subsets s(s = 1, 2, . . . , 5) in 0,respectively (Chen, 2014).

(1) A weak ranking:

1 = (w 1, w 2, . . . , w n ) ∈ 0 w j1 ≥ w j2 forall j1 ∈ γ 1 and j2 ∈ 1 ,

where γ 1 and 1 are two disjoint subsets of the subscript indexset N = {1, 2, . . . , n} of all criteria.

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(2) A strict ranking:

2 = { (w 1, w 2, . . . , w n ) ∈ 0| w j1 − w j2 ≥ δ j1 j2 for all j1 ∈ γ 2 and j2 ∈ 2},

where δ j1 j2 > 0 is a constant, and γ 2 and 2 are twodisjoint sub-sets of N .

(3) A ranking of differences:

3 = { ( w 1, w 2, . . . , w n ) ∈ 0| w j1 − w j2 ≥ w j3 − w j4 for all j1 ∈ γ 3, j2 ∈ 3, j3 ∈ 3 and j4 ∈ 3},

where γ 3, 3, 3 and 3 are four disjoint subsets of N .(4) An interval form:

4 = (w 1, w 2, . . . , w n ) ∈ 0 η j1 ≥ w j1 ≥ δ j1 for all j1 ∈ γ 4 ,

where η j1 > 0 and δ j1 > 0 are constants, satisfying η j1 > δ j1 ,and γ 4 is a subset of N .

(5) A ranking with multiples:

5 = { (w 1, w 2, . . . , w n ) ∈ 0| w j1 ≥ δ j1 j2 w j2 for all j1 ∈ γ 5 and j2 ∈ 5},

where δ j1 j2 > 0 is a constant, and γ 5 and 5 are twodisjoint sub-sets of N .

Let denote a set of the known information on the crite-rion weights, and = 1 ∪ 2 ∪ 3 ∪ 4 ∪ 5. Given the con-ditions in , theoptimalweight valuesof thecriteria in Equation(10) can be determined via the following linear programmingmodels:

max D = ni= 1

n j= 1 w jD M

i js.t. (w 1, w 2, . . . , w n ) ∈

, (M1)

where D M i j = (DL

i j + DU i j )/ 2.

TheDMs might express inconsistent opinions about thepref-erences and weight information in the case of contingency.Under these circumstances, a multi-objective nonlinear pro-gramming model using a goal programming technique can beused to tackle problems that involve inconsistent weight infor-mation. For j1 = j2 = j3 = j4, is revised as by introducingseveral non-negative deviation variables:

= { (w 1, w 2, . . . , w n ) ∈ 0| w j1 + e−(1) j1 j2 ≥ w j2 for all

j1 ∈ γ 1 and j2 ∈ 1;w j1 − w j2 + e−

(2) j1 j2 ≥ δ j1 j2 for all j1 ∈ γ 2 and j2 ∈ 2;

w j1 − w j2 − w j3 + w j4 + e−(3) j1 j2 j3 j4 ≥ 0 for all

j1 ∈ γ 3, j2 ∈ 3, j3 ∈ 3 and j4 ∈ 3;w j1 + e−

(4) j1 ≥ δ j1 , w j1 − e+(4) j1 ≤ η j1 for all j1 ∈ γ 4;

w j1 / w j2 + e−(5) j1 j2 ≥ δ j1 j2 for all j1 ∈ γ 5 and j2 ∈ 5}.

Furthermore, based on Model ( M1), the following bi-objective nonlinear programming model can be established in

case of inconsistent preference information:

max D =n

i= 1

n

j= 1

w jD M i j

min E = j1 , j2 , j3 , j4∈N

e−(1) j1 j2 + e−

(2) j1 j2 + e−(3) j1 j2 j3 j4 + e−

(4) j1 + e+(4) j1 + e−

(5) j1 j2

s.t .

( w 1, w 2, . . . , w n ) ∈ e−

(1) j1 j2 ≥ 0 j1 ∈ γ 1 and j2 ∈ 1

e−(2) j1 j2 ≥ 0 j1 ∈ γ 2 and j2 ∈ 2

e−(3) j1 j2 j3 j4 ≥ 0 j1 ∈ γ 3, j2 ∈ 3, j3 ∈ 3 and j4 ∈ 3

e−(4) j1 ≥ 0, e+

(4) j1 ≥ 0 j1 ∈ γ 4e−

(5) j1 j2 ≥ 0 j1 ∈ γ 5 and j2 ∈ 5

(M2)

where D M i j = (DL

i j + DU i j )/ 2. Bysolving Model (M2), theoptimal

weight vector w = ( w 1, w 2, · · · , w n ) and the optimal deviation values e−

(1) j1 j2 , e−(2) j1 j2 , e−

(3) j1 j2 j3 j4 , e−(4) j1 , e+

(4) j1 and e−(5) j1 j2 can be

derived.Considering an MCDM problem that contains incomplete

and consistent preference information, Model ( M1) can beapplied to obtain the best weight vector, and the optimisationmodel can be easily solved by using the simplex method. Foran MCDM problem that contains incomplete and inconsistentpreference information, Model (M2) can be employed to deter-mine the optimal solution.

. . The proposed algorithm with incompletecriteriainformationIn view of the determination of the weight vector of criteria, thecomprehensive preference of each alternative ai can be denotedby an interval value; that is, Equation (10) is revised as Di =

n j= 1 w jDi j = [ n

j= 1 w jDLi j,

n j= 1 w jDU

i j ] by using the opera-tional laws of interval values given in De nition 1. Then, apairwise comparison must be made between the alternatives,and subsequently the pairwise comparison matrix (likelihoodmatrix) P can be constructed as follows:

P = pi j m× m =

p11 p12 · · · p1m p21 p22 · · · p2m

......

......

pm1 pm2 · · · pmm

, (11)

where pi j = p(ai ≥ a j) = p(Di ≥ D j) = max

{1 − max{ DU

j − DLi

L(Di )+ L(D j ), 0}, 0}, and L(Di) = DU

i − DLi .

According to Xu and Da (2003), the ranking vector of thelikelihood matrix can be de ned as follows:

ω i =n j= 1 pi j + m

2 − 1

m(m − 1), i = 1, 2, . . . , m. (12)

Consequently, the ranking of all alternatives can be obtainedaccording to the descending order of ω i (i = 1, 2, · · · , m). Thatis, the larger ωi is, the better the alternative ai will be.

Based on the above analysis, an approach can be developedfor an MCDM problem that contains three key stages: (1) col-lection and normalisation stage, (2)determination stage and (3)selection stage.A conceptual model of theproposed approach isshown in Figure 1.

The main steps are outlined as follows.

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Collect the criterion weight informationand normalise the decision matrix

Collection and normalisation stage (Step 1)

Determination stage (Steps 2-4)

Calculate the lowerand upper bounds

Construct the programming model andidentify the optimal weight vector

Calculate the comprehensive performance and obtain theranking vector (Using the likelihood-based comparison method)

Rank all alternatives

Selection stage (Step 5)

Decision-makersEvaluation values

Normalised evaluation values

Rankingvector

Interval values

Interval values Optimal weights

Figure . A ow chart of the proposed approach.

Step 1: Normalise the decision matrix.Use Equation (3) to transform B into R. For convenience, thenormalised values of ai( i = 1, 2, . . . , m) with respect to c j( j =1, 2, . . . , n) are also expressed as ([t Li j, t U

i j ], [iLi j, iU i j ], [ f Li j, f U

i j ]).Step 2: Calculate the lower and upper bounds of Di j.Use Equations (8) and (9) to derive the lower bound DL

i j andthe upper bound DU

i j , respectively.Step 3: Identify the optimal weight vector and calculate the

preference of each alternative.Solve Model (M1) or (M2) to identify the optimal weight

vector, and calculate the comprehensive performance Di( i =1, 2, . . . , m) by using the operational laws of the interval valuesin De nition 1.

Step 4: Construct the likelihood matrix and obtain the rank-ing vector.

Construct the likelihood matrix P by using Equation (11)and obtain the ranking vector ω = (ω 1, ω 2, . . . , ω m ) based onEquation (12) .

Step 5: Determine the ranking of all alternatives.Determine the ranking of all alternatives according to the

descending order of ω i = (1, 2, . . . , m) and select the optimalone(s).

. An illustrative exampleIn this section, the example of an investment appraisal project

is used to demonstrate the application of the proposed MCDMapproach; then its validity and effectiveness will be testedthrough a comparative analysis.

The following case is adapted from Wang et al. (2015).ABC Nonferrous Metals Co. Ltd. is a large state-owned com-

pany whose main business is thedeep processing of non-ferrous

metals. It is also the largest manufacturer of multi-species non-ferrous metals, with the exception of aluminium, in China. Toexpand its main business, the company regularly engages inoverseas investment and a department consisting of executivemanagersand several experts in the eld has been established tomake decisions regarding global mineral investment. This over-seas investment department recently decided to select a poolof alternatives from several foreign countries based on prelimi-nary surveys. After a thorough investigation, ve countries weretaken into consideration, denoted by {a1, a2, · · · , a5}. There aremany factors that affect the investment environment, but fourwere chosen based on the experience of the department’s per-sonnel, namely c1: resources; c2 : politics and policy; c3 : econ-omy; and c4 : infrastructure.

The members of the overseas investment departmenthave met to determine the evaluation information. Con-sequently, following a heated discussion, they came to aconsensus on the nal evaluations which were expressedby INNs shown in Table 1 Moreover, they were only ableto provide incomplete information on the weights, that is,

= {0.15 ≤ w 1 ≤ 0.3, 0.15 ≤ w 2 ≤ 0.25, 0.25 ≤ w 3 ≤ 0.4,0.3 ≤ w 4 ≤ 0.45, 2.5w 1 ≤ w 3}.

. . Illustration of the proposed approachIn the following steps, the main procedures of obtaining the

optimal ranking of alternatives are presented.Step 1: Normalise the decision matrix.As all the criteria are maximising type, the matrix does not

need to be normalised, i.e., R = B.Step 2: Calculate the lower and upper bounds of Di j.Derive thelower bound DL

i j andthe upperbound DU i j by using

Equations ( 8) and,(9), respectively:

Di j5× 4

=

[0.5525, 0.7141][0.4966, 0.7571][0.5610, 0.7528][0.3580, 0.4433][0.4406, 0.5637]

[0.5363, 0.6410][0.4964, 0.7853][0.4462, 0.6148][0.4433, 0.5825][0.6962, 0.8171]

[0.6130, 0.7571] [0.6726, 0.7613][0.5363, 0.6429] [0.5297, 0.6962][0.5525, 0.6753] [0.4568, 0.5965][0.4350, 0.5525] [0.6962, 0.8171][0.5873, 0.7141] [0.4964, 0.6272]

.

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Table . The evaluation information.

c c c c

a ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ .a ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ .a ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ .a ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ .a ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ . , . ],[ . , . ]) ([ . , . ],[ .

Step 3: Identify the optimal weight vector and calculate thepreference of each alternative.

Because no inconsistent weight information exists in theevaluation, Model ( M1) can be applied to identify the optimalweight vector:

max D = 2.8198w 1 + 3.0295w 2 + 3.033w 3 + 3.175w 4

s.t.

0.15 ≤ w 1 ≤ 0.30.15 ≤ w 2 ≤ 0.240.25 ≤ w 3 ≤ 0.40.3 ≤ w 4 ≤ 0.452.5w 1 ≤ w 3

4 j= 1 w j = 1

The optimal weight vector can be obtained as w =(0.15, 0.15, 0.375, 0.325) . Then, Di (i = 1, 2, · · · , 5) can becalculated by referring to the operational laws of interval valuesin De nition 1:

D1 = [0.6118, 0.7346], D2 = [0.5222, 0.6987],

D3 = [0.5067, 0.6522],

D4 = [0.5096, 0.6266] and D5 = [0.5521, 0.6788].

Step 4: Construct the likelihood matrix and obtain the rank-ing vector.

Use Equation (11) to construct the likelihood matrix P and obtain the ranking vector ω = (ω 1, ω 2, . . . , ω 5) based onEquation (12) :

P = pi j 5× 5

=

0.5 0.7096 0.8493 0.9384 0.73160.2904 0.5 0.5962 0.6444 0.48360.1507 0.4038 0.5 0.5434 0.36790.06160.2684

0.35560.5164

0.45660.6321

0.5 0.30570.6943 0.5

,

ω1 = 0.2614, ω 2 = 0.2007, ω 3 = 0.1733,

ω4 = 0.1590 and ω5 = 0.2056.

Step 5: Determine the ranking of all alternatives.According to the descending order of ωi = (1, 2, . . . , 5) , the

ranking of all alternatives is a1 a5 a2 a3 a4 and thebest one is a1.

. . Comparative analysis and discussionIn order to further verify the feasibility and effectiveness of theproposed approach, a comparative analysis is now conductedusing six existing methods with the analysis being based on thesame illustrative example.

(1) Chi and Liu’s method (2013) contains two major phases(criterion weights determination and ranking obtain-ment with TOPSIS). First, the maximising deviationmethod is developed to determine the criterion weightsand then an extended TOPSIS method is employed torank the alternatives.

(2) In Ye’s method (2014b), the distance-based similarity measures are employed, which involves aggregating theweighted similarity measures between each alternativeand the PIS.

(3) In Zhang et al.’s methods (2014), rst, the comprehensiveINNs are aggregated by using the INNWA or INNWGoperators, and then the ranking vectors can be obtainedby constructing the likelihood matrices based on thescore function. Moreover, Zhang et al. (2015b) devel-oped an outranking method for MCDM on the basis of the score function constructing the outranking relationmatrix. Since Zhanget al.’s method( 2015b) does not takethe criterion weights into consideration but the illustra-tive example does, it is necessary to construct the out-ranking relation matrix after calculating the weightedevaluation values.

Considering the criterion weights obtained using the pro-posed programming model, the results obtained by differentmethods are summarised in Table 2.

It can be seen that there are some differences between them.The reasons for the inconsistency of the rankings are explainedas follows.

(1) The difference in the ranking results of the proposedapproach and that of Chi and Liu ( 2013) is the sequence

Table . The ranking results of the different methods.

Methods Ranking results

Chi and Liu’s method ( ) a a a a a

Ye’s methods ( b)Similarity measure based on the Hamming

distancea a a a a

Similarity measure based on the Euclideandistance

a a a a a

Zhang et al.’s methods ( )Method based on the INNWA operator a a a a a

Method based on the INNWG operator a a a a a

Zhang et al.’s method ( b) ( p = . andq = . )

a a {a , a , a }

The proposed approach a a a a a

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Table . The ranking results obtained by revising the methods of Ye ( b).

Methods Ranking results

Similarity measure based on the Hammingdistance

a a a a a

Similarity measure based on the Euclideandistance

a a a a a

of a3 and a4. In Chi and Liu’s method (2013), the rela-tive closeness coefficients (performance of each alterna-tive) are conducted based on the relative ideal solutions,which have certain drawbacks. First, it is not easy tochoose the appropriate PIS and NIS with INNs becauseeach INN has three interval elements. Second, the PISand NIS are closely related to the number of alterna-tives aswell as theevaluationvalues. Thus, they may vary as the original information changes. Third, the relativecloseness coefficients are in the form of crisp real num-bers, which may cause information loss and affect theranking results. Finally, suppose that there are m alter-natives and n criteria to be evaluated. In order to deter-mine the criterion weights, Chi and Liu’s method ( 2013)needs to derive m × m × n distance measures, whereasthe proposed approach only needs to calculate m × ncross-entropy measures.Thus, it takes less time than thatof Chi and Liu (2013).

(2) Similarly, the order of a3 and a4 is the only differencebetween the proposed approach and the rst method of Ye (2014b). This is because the comparison methods inYe (2014b) only consider the weighted similarity mea-suresbetweeneach alternativeand thePIS.If only thePISis taken into account and the NIS is ignored, the rank-ing of alternatives may be incorrectly reversed and this

may be ampli ed in the nal results. However, the rank-ing result in this method will be identical to that of theproposed approach in a situation where, simultaneously,the PIS is replaced with the absolute one and the abso-lute NIS is not ignored; moreover, the closeness coeffi-cient would have to be utilised to determine the rankingof alternatives. The updated results are shown in Table 3.Therefore, the methods in Ye ( 2014b) are not reliableenough.

(3) Thepositions of a3 and a4 obtainedby themethod basedon the INNWA operator (Zhang et al., 2014) are notconsistent with either those obtained by the methodbased on the INNWG operator (Zhang et al., 2014)

or the proposed approach. This may be caused by theinherent characteristics of aggregation operators, as theINNWA operator focuses on the impact of the overallcriterion values, while the INNWG operator emphasisesthe impact of a single item. Additionally, the outrank-ing method of Zhang et al. ( 2015b) can only yield partialorders of alternatives, in which a3, a4 and a5 are indis-tinguishable. This method has to convert the INNs intoreal numbers and arti cially setboth the threshold p andindifference threshold q before constructing the domi-nance relations. It is by nature inappropriate to replacethe INNs with real numbers, and it may lead to infor-mation loss in the transformation process. Furthermore,

when manually providing the parameters p and q, it isdifficult to avoid subjective randomness. Therefore, theresult obtained by the outranking method is not alwaysreliable.

According to the comparative analysis, the following advan-tages over the other methods can be outlined.

(1) The calculations required for the proposed approach arerelatively straightforward and time-saving, and the bur-

den of computation can be greatly decreased with thehelp of the proven mathematical derivation.

(2) In the proposed approach, the interval closeness coeffi-cients are conducted to rank alternatives. In this way, thefuzziness of the original information can be maintainedand fully utilised. Therefore, the proposed approach ismore competent in interval neutrosophic MCDM thanthe other methods considered.

(3) In the proposed approach, the transformation opera-tor is employed to convert INNs into SNNs, which canavoid various kinds of aggregation operators process-ing directly with INNs. Furthermore, the parametersof the transformation operator are determined throughmathematical derivation and not arti cially produced.Thereby, the nal ranking obtained by the proposedapproach is more conclusive than those produced by the other methods, and it is evident that the proposedapproach is accurate and reliable.

. ConclusionsINSs are exible at expressing the uncertain, imprecise, incom-plete andinconsistent informationthat is verycommonin scien-ti c and engineering situations; therefore, the study of MCDMmethods with INSs is highly signi cant. In this paper, a trans-

formation operator and cross-entropy were de ned. Conse-quently, an MCDM method was established based on cross-entropy and TOPSIS, which calculated the cross-entropy aftertransforming INNs into SNNs on the basis of the transforma-tion operator. Furthermore, it aggregated the performances of alternatives into interval numbers, from which two mathemati-cal programming models were constructed to identify the crite-rion weights. Finally, a ranking result was obtained by compar-ing these weighted interval numbers with a possibility degreemethod.

The advantages of this study are that the approach is bothsimple and convenient to compute and effective at decreasingthe loss of evaluation information. The feasibility and validity of

the proposed approach have been veri ed through the illustra-tive example and comparative analysis. The comparison resultsdemonstrated that the proposed approach can provide morereliable and precise outcomes than other methods. Therefore,this approach has great application potential in solving MCDMproblemsin anintervalneutrosophicenvironment, inwhichcri-terion values with respect to alternatives are evaluated by theform of INNs and the criterion weights are incomplete.

AcknowledgementsThe authors thank the editors and anonymous reviewers for their helpfulcomments and suggestions.

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Disclosure statementNo potential con ict of interest was reported by the authors.

FundingThis work was supported by the National Natural Science Foundation of China [grant number 71571193], [grant number 71431006].

Notes on contributors

Zhang-pengTian receivedhisBS degreein electroniccommercefrom Tian- jin Chengjian University, Tianjin, China, in 2012. He is currently workingtowards the MS degree with the Business School, Central South University,Changsha, China.

Hong-yu Zhang received her MS degree in computer software and the-ory and PhD degree in management science and engineering from CentralSouth University, Changsha, China, in 2005 and 2009, respectively. She iscurrently an associate professor with the Business School, Central SouthUniversity. Her research interests include information management andits applications in production operations. Her current research focuses onremanufacturing production management and decision-making theory.

Jing Wang received her MSc degree in information engineering from theUniversity of Osnabrueck, Osnabrueck, Germany, in 2006. She is currently working towards the PhD degree with the Business School, Central South

University, Changsha, China. She is also a lecturer with International Col-lege, Central South University of Forestry and Technology. Her currentresearch interests include decision-making theory and application, risk management and control, and information management.

Jian-qiang Wang received his PhD degree in management science andengineering from Central South University, Changsha, China, in 2005. Heis currently a professor with the Business School, Central South University.His current research interests include decision-making theory and applica-tion, risk management and control, and information management.

Xiao-hong Chen received her PhD degree in management from the TokyoUniversity of Technology, Tokyo, Japan, in 1999. She is currently a profes-sor with the Business School, Central South University, Changsha, China.Her current research interests include decision theory and methods, deci-sion support systems, resource-saving, and the environment-friendly soci-ety. Shehas publishedacademicpapersin manyjournals,including DecisionSupport System and Expert Systems with Applications.

ORCIDZhang-peng Tian http://orcid.org/0000-0002-0825-9194Hong-yu Zhang http://orcid.org/0000-0001-5142-5277 Jing Wang http://orcid.org/0000-0002-2407-5985 Jian-qiang Wang http://orcid.org/0000-0001-7668-4881 Xiao-hong Chen http://orcid.org/0000-0003-3919-5215

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