Applied Soft Computing 25 (2014) 336346Contents lists available at ScienceDirectAppliedSoftComputingj our nal homepage: www. el sevi er . com/ l ocat e/ asocAnoutrankingapproachformulti-criteriadecision-makingproblemswithsimpliedneutrosophicsetsJuan-juanPenga,b,Jian-qiangWanga,,Hong-yuZhanga,Xiao-hongChenaaSchool of Business, Central South University, Changsha 410083, ChinabSchool of Economics and Management, Hubei University of Automotive Technology, Shiyan 442002, Chinaarticl einfoArticle history:Received 30 September 2013Received in revised form27 June 2014Accepted 13 August 2014Available online 10 September 2014Keywords:Multi-criteria decision-makingSimplied neutrosophic setsOutranking relationsabstractIn thispaper,a newoutrankingapproachformulti-criteriadecision-making(MCDM)problemsisdevel-oped inthecontextofa simpliedneutrosophicenvironment,wherethetruth-membershipdegree,indeterminacy-membershipdegreeandfalsity-membershipdegreeforeachelementare singletonsub-sets in[0,1].Firstly,thenoveloperationsofsimpliedneutrosophicsets (SNSs)and relationalpropertiesaredeveloped.Thensomeoutrankingrelationsforsimpliedneutrosophicnumber(SNNs)aredened,basedon ELECTRE,andthepropertieswithintheoutrankingrelationsarefurtherdiscussedindetail.Additionally,basedon theoutrankingrelationsofSNNs,a rankingapproachis developedinordertosolveMCDMproblems.Finally,twopracticalexamplesareprovidedtoillustratethepracticalityandeffectivenessoftheproposedapproach.Moreover,acomparisonanalysisbasedon thesameexampleisalso conducted. 2014ElsevierB.V.Allrightsreserved.1. IntroductionIn many cases, it is difcult for decision-makers to precisely express a preference when solving MCDM problems with inaccurate,uncertain or incomplete information. Under these circumstances, Zadehs fuzzy sets (FSs) [1], where the membership degree is representedby a real number between zero and one, are regarded as an important tool for solving MCDMproblems [2,3], fuzzy logic and approximatereasoning [4], and pattern recognition [5].However, FSs cannot handle certain cases where it is hard to dene the membership degree using one specic value. In order toovercome the lack of knowledge of non-membership degrees, Atanassov [6] introduced intuitionistic fuzzy sets (IFSs), an extension ofZadehs FSs. Furthermore, Gau and Buehrer [7] dened vague sets and subsequently, Bustince [8] pointed out those vague sets and IFSsare mathematically equivalent objects. To date, IFSs have been widely applied in solving MCDMproblems [916], neural networks [17,18],medical diagnosis [19], color region extraction [20,21], and market prediction [22]. IFSs simultaneously take into account the membershipdegree, non-membership degree and hesitation degree. Therefore, they are more exible and practical when addressing fuzziness anduncertainty than traditional FSs. Moreover, in some actual cases, the membership degree, non-membership degree and hesitation degreeof an element in IFSs maynot be a specic number; hence, they were extended to interval-valued intuitionistic fuzzy sets (IVIFSs) [23]. Inorder to handle situations where people are hesitant inexpressing their preference regarding objects ina decision-making process, hesitantfuzzy sets (HFSs) were introduced by Torra and Narukawa [24,25]. Furthermore, Zhu et al. [26,27] proposed dual HFSs and outlined theiroperations and properties. Chen et al. [28] proposed interval-valued hesitant fuzzy sets (IVHFSs) and applied them to MCDM problems.Farhadinia [29] discussed the correlation for dual IVHFSs and Peng et al. [30] introduced a MCDMapproach with hesitant interval-valuedintuitionistic fuzzy sets (HIVIFSs), which is an extension of dual IVHFSs.Although the theory of FSs has been developed and generalized, it cannot deal with all types of uncertainty in different real-worldproblems. Types of uncertainty, such as indeterminate and inconsistent information, cannot be managed. For example, when an expert isasked for their opinion about a certain statement, he or she maysay the possibility that the statement is true is 0.5, the possibility that theCorresponding author. Tel.: +86 73188830594; fax: +86 7318710006.E-mail address: jqwang@csu.edu.cn (J.-q. Wang).http://dx.doi.org/10.1016/j.asoc.2014.08.0701568-4946/ 2014 Elsevier B.V. All rights reserved.J.-j. Peng et al. / Applied Soft Computing 25 (2014) 336346 337statement is false is 0.6 and the degree that he or she is not sure is 0.2 [31]. This issue is beyond the scope of FSs and IFSs and thereforesome newtheories are required.Smarandache [32,33] proposed neutrosophic logic and neutrosophic sets (NSs) and subsequently Rivieccio [34] pointed out that a NS isa set where each element of the universe has a degree of truth, indeterminacy and falsity and it lies within ]0, 1+[, the non-standard unitinterval. Clearly this is the extension of the standard interval [0,1]. Furthermore, the uncertainty presented here, i.e. the indeterminacyfactor, is dependent on the truth and falsity values, whereas the incorporated uncertainty is dependent on the degrees of belongingnessand non-belongingness of IFSs [35]. Additionally, the aforementioned example of NSs can be expressed as x(0.5, 0.2, 0.6). However, withouta specic description, NSs are difcult to apply to real-life situations. Therefore, single-valued neutrosophic sets (SVNSs) were proposed,which are an extension of NSs [31,35]. Majumdar et al. [35] introduced a measure of entropy of SVNSs. Furthermore, the correlationcoefcients of SVNSs as well as a decision-making method using SVNSs were introduced [36]. In addition, Ye [37] also introduced theconcept of simplied neutrosophic sets (SNSs), which can be described by three real numbers in the real unit interval [0,1], and proposedanMCDMmethodusing the aggregationoperators of SNSs. Wang et al. [38] andLupi nez [39] proposedthe concept of interval neutrosophicsets (INSs) and provided the set-theoretic operators of INSs. Broumi and Smarandache [40] discussed the correlation coefcient of INSs andZhang et al. [41] developed a MCDMmethod based on aggregation operators within an interval neutrosophic environment. Furthermore,Ye [42,43] proposed the similarity measures between SVNSs and INSs, which were based on the relationship between similarity measuresand distances. However, in certain cases, the operations of SNSs provided by Ye [37] maybe impractical. For example, the sum of anyelement and the maximum value should be equal to the maximum value, but this is not always the case during operations [37]. Thesimilarity measures and distances of SVNSs that are based on those operations mayalso be unrealistic. Peng et al. [44] dened the noveloperations and aggregation operators, which were based on the operations in Ye [37], and applied themto MCDMproblems.Furthermore, the aforementioned methods using SNSs always involve operations and measures whose inuence on the nal solutionmaybe signicant. However, there is another method that can overcome these shortcomings, namely the relation model. Relation modelsutilize outranking relations or priority functions for ranking the alternatives in terms of their priority according to the criteria. Recentlyrelation models have been widely acknowledged as they are more related to the actual decision-making process. The ELECTRE methods,which were originally developed by Benayoun and Roy [45,46] are typical of this eld. Subsequently, ELECTRE I, II, III, IV, IS, and TRI [4548]were developed, which are extensions of ELECTRE. To date, the ELECTRE methods have been successfully and widely used in various elds,including biological engineering [49,50], energy sources [51,52], environmental studies [53,54], economics [55], value engineering [56],medical science [57], communication and transportation [58], and location selection problems [59,60]. For example, Devi and Yadav [59]developed an MCDMmethod based on ELECTRE in order to solve plant location problems where the evaluation values are IFSs. Moreover,Hatami-Marbini and Tavana [61] proposed an extension of the ELECTRE I method to solve group decision-making problems within a fuzzyenvironment. Vahdani et al. [62] proposed a novel ELECTRE method to solve MCDM problems with interval weights and data. Vahdaniand Hadipour [63] developed a novel ELECTRE method to solve MCDMproblems with interval-valued fuzzy information. Yang et al. [64]proposed an outranking method for MCDMproblems with duplex linguistic information. Yang and Wang [65] developed a decision aidingtechnique based on incomplete preference information and applied it to solve MCDMproblems. Wang et al. [66,67] proposed outrankingapproaches tosolve MCDMproblems withHFSs andhesitant fuzzy linguistic termsets. Chen[68] developedanELECTREoutranking methodto solve multiple criteria group decision-making problems with interval type-2 fuzzy sets.In this paper, the novel operations of SNSs are developed, some outranking relations of simplied neutrosophic numbers (SNNs) thatare based on ELECTRE are proposed, and the rational properties of SNNs are discussed. Furthermore, an outranking approach for MCDMproblems with SNNs is developed, which could overcome the drawbacks that were discussed earlier in the paper.The paper is structured as follows. In Section 2, the denition and novel operations of SNSs are provided. In Section 3, some outrankingrelations of SNNs that are based on ELECTRE are dened and some properties are discussed. In Section 4, an outranking approach for MCDMproblems with SNNs is developed. Two worked examples appear in Section 5 and conclusions are drawn in Section 6.2. Preliminaries2.1. NSs and SNSsIn this section, the denitions of NSs and SNSs are introduced in order to assist with the latter analysis.Denition 1 ([32]). Let X be a space of points (objects), with a generic element in X denoted by x. An NS A in X is characterized by atruth-membership function TA(x), a indeterminacy-membership function IA(x) and a falsity-membership function FA(x). TA(x), IA(x) andFA(x) are real standard or nonstandard subsets of ]0, 1+[, that is, TA(x) : X ]0, 1+[, IA(x) : X ]0, 1+[, and FA(x) : X ]0, 1+[. Thereis no restriction on the sumof TA(x), IA(x) and FA(x), therefore 0 supTA(x) +supIA(x) +supFA(x) 3+.Denition 2 ([32]). An NS A is contained in another NS B, denoted by A B, if and only if inf TA(x) inf TB(x), supTA(x) supTB(x),inf IA(x)inf IB(x), supIA(x)supIB(x), inf FA(x)inf FB(x) and supFA(x)supFB(x) for any x X.Since it is difcult to apply NSs to practical problems, Ye [37] reduced NSs of nonstandard intervals into the SNSs of standard intervals,which preserves the operations of NSs.Denition 3 ([37]). Let X be a space of points (objects), with a generic element in X denoted by x. An NS A in X is characterized byTA(x), IA(x) and FA(x), which are singleton subintervals/subsets in the real standard [0,1], that is TA(x) : X [0,1], IA(x) : X [0, 1], andFA(x) : X [0,1]. Then, a simplication of A is denoted by:A = {|x X} (1)which is called an SNS and is a subclass of NSs. The set of all SNSs in X is represented as SNSS(X). If, in particular, X has only one element, Ais called a simplied neutrosophic number (SNN), which can be denoted by A =.The set of all SNNs is represented as SNNS.The operations of SNSs are also dened by Ye [37].Denition 4 ([37]). Let A and B be two SNSs, the following operations can be true.338 J.-j. Peng et al. / Applied Soft Computing 25 (2014) 336346(1) A +B = {|x X};(2) A B = {|x X};(3) A = {|x X},>0;(4) A

= {|x X},>0.Denition 4 has certain limitations and these are nowoutlined.(1) In some situations, operations such as A +B and A B might be impractical. This is demonstrated in Example 1.Example 1. Let A = {}and B = {}be two SNSs. Clearly, B = {} is the larger of these SNSs.Theoretically, the sumof anynumber andthe maximumnumber shouldbe equal tothe maximumvalue. However, accordingtoDenition4,A +B = {}/ = B, thereforetheoperation+ cannot beaccepted. Similar contradictions exist inother operations of Denition4, and thus those dened above are incorrect.(2) The correlation coefcient of SNSs [36], which is based on the operations of Denition 4, cannot be accepted in some specic cases.Example 2. Let A1 = {}andA2 = {}be two SNSs, andB = {}be the largest SNS. According to thecorrelation coefcient of SNSs [36], W1(A1, B) = W2(A2, B) = 1 can be obtained, but this does not indicate which one is the best. However,it is clear that A1 is superior to A2.(3) In addition, the cross-entropy measure for SNSs [42], which is based on the operations of Denition 4, cannot be accepted in speciccases.Example 3. Let A1 = {}and A2 = {}be twoSNSs, and B = {}be the largest SNS According tothe cross-entropy measure for SNSs [42], S1(A1, B) = S2(A2, B) = 1 can be obtained, which indicates that A1 is equal to A2. However, it isnot possible to discern which one is the best, but as TA2(x) >TA1(x), IA2(x) >IA1(x) and FA2(x) >FA1(x) for any x in X, it is clear that A2 issuperior to A1.(4) If IA(x) = IB(x) for any x in X, then A and B are both reduced to twoIFSs. However, the operations presented in Denition 4 are not inaccordance with the operations of two IFSs [922].Denition5 ([37]). Let X = {x1, x2, . . .,xn} and A and B be two SNSs, then A is contained in B, i.e. A B if and only if TA(x) TB(x), IA(x)IB(x)and FA(x)FB(x) for any x X.Clearly, if the equalities are not accepted, then A B is obtained.2.2. The novel operations of SNNsSome novel operations of SNNs are nowdened.Denition 6. Let A and B be two SNNs, the operations of SNNs can be dened as follows:(1) A =, >0;(2) A

=, >0;(3) A +B =;(4) A B =.Theorem1. Let A, B and C be three SNNs, then the following equations can be true.(1) A +B = B +A;(2) A B = B A;(3) (A +B) = A +B,>0;(4) (A B)

= A

+B

,>0;(5) 1A +2A = (1+2)A, 1 >0, 2 >0;(6) A

1 A

2 = A(1+2), 1 >0, 2 >0;(7) (A +B) +C = A +(B +C);(8) (A B) C = A (B C).Example 4. Let A =and B =be two SNNs, and= 2, then the following results can be achieved:(1) 2 A ==;(2) A2==;(3) A +B ==;(4) A B ==.Denition 7. Let ASNNS, then the complement of an SNN A is denoted by AC, which is dened by AC=.Denition 8. Let A and B be two SNNs, then A = B if and only if A B and B A.Denition 9 ([35]). Let A and B be any two SNNs, then the normalized Euclidean distance between A and B can be dened as follows:d(A, B) =_13n(|TA TB|2+|IAIB|2+|FA FB|2).(2)J.-j. Peng et al. / Applied Soft Computing 25 (2014) 336346 3393. The outranking relations of SNNsThe binary relations between two SNNs that are based on ELECTRE are nowdened.Denition 10. Let A and B be two SNNs, then the strong dominance relation, weak dominance relation, and indifference relation of SNNscan be dened as follows.(1) If TATB, IA