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International Journal of Systems Science

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The fuzzy cross-entropy for intuitionistic hesitantfuzzy sets and their application in multi-criteriadecision-making

Juan-juan Peng, Jian-qiang Wang, Xiao-hui Wu, Hong-yu Zhang & Xiao-hongChen

To cite this article: Juan-juan Peng, Jian-qiang Wang, Xiao-hui Wu, Hong-yu Zhang & Xiao-hongChen (2015) The fuzzy cross-entropy for intuitionistic hesitant fuzzy sets and their applicationin multi-criteria decision-making, International Journal of Systems Science, 46:13, 2335-2350,DOI: 10.1080/00207721.2014.993744

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International Journal of Systems Science, 2015Vol. 46, No. 13, 2335–2350, http://dx.doi.org/10.1080/00207721.2014.993744

The fuzzy cross-entropy for intuitionistic hesitant fuzzy sets and their applicationin multi-criteria decision-making

Juan-juan Penga,b, Jian-qiang Wanga,∗, Xiao-hui Wua,b, Hong-yu Zhanga and Xiao-hong Chena

aSchool of Business, Central South University, Changsha, China; bSchool of Economics and Management, Hubei University ofAutomotive Technology, Shiyan, China

(Received 14 May 2014; accepted 27 November 2014)

In this paper, the cross-entropy of intuitionistic hesitant fuzzy sets (IHFSs) is developed by integrating the cross-entropy ofintuitionistic fuzzy sets (IFSs) and hesitant fuzzy sets (HFSs). First, several measurement formulae are discussed and theirproperties are studied. Then, two approaches, which are based on the developed intuitionistic hesitant fuzzy cross-entropy, areproposed for solving multi-criteria decision-making (MCDM) problems within an intuitionistic hesitant fuzzy environment.For both methods, an optimisation model is established in order to determine the weight vector for MCDM problemswith incomplete information on criteria weights. Finally, an example is provided in order to illustrate the practicality andeffectiveness of the proposed approaches.

Keywords: multi-criteria decision-making; intuitionistic fuzzy sets; hesitant fuzzy sets; intuitionistic hesitant fuzzy sets;intuitionistic hesitant fuzzy cross-entropy

1. Introduction

In fuzzy sets proposed by Zadeh (1965), the membershipdegree of each element in a universe is a real number be-tween 0 and 1. Such fuzzy sets are regarded as an importanttool to solve multi-criteria decision-making (MCDM) prob-lems (Bellman & Zadeh, 1970; Yager, 1997) and can also beapplied to fuzzy logic and approximate reasoning (Zadeh,1975), and pattern recognition (Pedrycz, 1990). However,the information regarding alternatives that correspond toa fuzzy concept may be incomplete. It is possible that thesum of the membership degree and non-membership de-gree of an element in the universe corresponding to thefuzzy concept is less than1. Subsequently, as an extensionof fuzzy sets, intuitionistic fuzzy sets (IFSs) were intro-duced by Atanassov (1986, 1999). It is characterised bytwo functions expressing the degree of membership and thedegree of non-membership, respectively. Then Atanassovand Gargov (1989) proposed interval-valued intuitionisticfuzzy sets (IVIFSs) that are the extension of IFSs, wherethe membership degree and non-membership degree of anelement in an IVIFS are represented by intervals in [0, 1]rather than crisp values between 0 and 1. To date, IFSsand IVFSs have been widely investigated and applied toMCDM problems (Liu, 2014; Szmidt & Kacprzyk, 2002;Tan, Jiang, & Chen, 2013; Wan & Dong, 2014; Wang, Han,& Zhang, 2014; Wang, Nie, Zhang, & Chen, 2013a, 2013b;Wang & Zhang, 2013).

∗Corresponding author. Email: jqwang@csu.edu.cn

More recently, due to the advantage of measuring fuzzi-ness and discrimination information, the entropy measuresand cross-entropy measures of IFSs and IVIFSs have beenproposed and used to solve fuzzy MCDM problems. For ex-ample, Szmidt and Kacprzyk (2001) introduced the entropymeasure and distance measures of IFSs, and applied themto MCDM problems. Furthermore, Li, Deng, Li, and Zeng(2012) investigated the relationship between the similaritymeasure and the entropy of IFSs and proposed new en-tropy of IFSs. Chen and Li (2011) suggested an alternativeobjective weighting method to generate objective weights,which was based on intuitionistic fuzzy entropy measures.Ye (2009a) proposed a method of fault diagnosis based onthe vague cross-entropy. Vlachos and Sergiadis (2007) de-fined the cross-entropy of IFSs and applied it to patternrecognition, medical diagnosis and image segmentation. Ye(2009b, 2010, 2011) introduced the cross-entropy of IFSsand IVIFSs and utilised them to solve MCDM problems.Wang and Li (2011) provided two improved methods forsolving MCDM problems, which were based on the cross-entropy of IFSs. Hung, Lin, and Weng (2011) introduced thediscrimination information and cross-entropy of IFSs andalso used them to improve the fault diagnosis of turbineproblems. Wu and Zhang (2011) proposed the intuition-istic fuzzy weighted entropy, and presented a method todeal with intuitionistic fuzzy MCDM problems. Mao, Yao,and Wang (2013) introduced the cross-entropy and entropy

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2336 J.-j. Peng et al.

measures of IFSs. Zhang and Yu (2012) constructed a seriesof mathematical programming models, which were basedon an interval-valued intuitionistic fuzzy cross-entropy, inorder to determine the criteria weights and applied them toMCDM problems. Xia and Xu (2012) proposed two meth-ods for determining the optimal weights of criteria anddeveloped two pairs of entropy and cross-entropy measuresfor intuitionistic fuzzy values. However, in actual decision-making, single values in IFSs and IVIFSs cannot conveyinformation precisely.

Hesitant fuzzy sets (HFSs), which are another extensionof fuzzy sets, are highly useful in handling situations wherepeople are hesitant in providing their preferences with re-gard to objects in a decision-making process, and have pro-vided a theory for solving MCDM problems in certain sit-uations. They were first introduced by Torra and Narukawa(2009) and Torra (2010), and permit the membership degreeof an element to be a set of several possible values between0 and 1. To date, HFSs have been the subject of a great dealof research. For example, Zhang and Xu (2014) defined anacronym in Portuguese of the interactive and multi-criteriadecision-making (TODIM) method, which was based onthe measurement functions of HFSs. Furthermore, Wang,Wang, et al. (2014) proposed an outranking approach withHFSs. Having reviewed the extant research, Rodriguez,Martinez, Torra, Xu, and Herrera (2014) summarised thecurrent state of, and suggested future directions for, HFSs.Chen, Xu, and Xia (2013) generalised the concept of HFSsto that of interval-valued hesitant fuzzy sets (IVHFSs), inwhich the membership degrees of an element to a given setare not exactly defined, but denoted by several possible in-terval values. Xu and Xia (2012) introduced the concepts ofentropy and cross-entropy for hesitant fuzzy information,and developed two MCDM methods based on these con-cepts. However, these cross-entropy measures were definedunder the assumption that any two hesitant fuzzy numbers(HFNs) are of equal length when compared. If this is notthe case, then the shorter one should be extended by re-peating its maximum value until it is the same length as theother HFN. Indeed, according to the decision-makers’ pref-erences, it is possible to extend the shorter HFN by addingany of its values until it is equal in length to the longerone. Therefore, due to the varied preferences of decision-makers, this may lead to a different optimal alternative.

Furthermore, HFSs and IVHFSs do not consider thenon-membership degree of an element in a set; therefore,the combination of IFSs and HFSs is also a research is-sue of interest. Qian, Wang, and Feng (2013) extendedHFSs with IFSs and referred to them as generalised HFSs,which, in essence, extended the element of HFSs from areal number to intuitionistic fuzzy numbers (IFNs). Zhu,Xu, and Xia (2012) defined dual HFSs, and also discussedtheir basic operations and properties. Moreover, Farha-dinia (2014) discussed the correlation for dual IVHFSs.Peng, Wang, Wang, and Chen (2014) introduced an MCDM

approach with hesitant interval-valued intuitionistic fuzzysets (HIVIFSs), which are an extension of dual IVHFSs.However, dual HFSs are defined in terms of sets of values,as opposed to precise numbers, for the membership degreesand non-membership degrees of IFSs. Moreover, dual HFSsrequire the sum of the maximum membership degree andmaximum non-membership degree to be no more than 1,which limits their application in certain cases. For example,decision-makers may deem that the possible membershipdegrees of an alternative against the criterion ‘excellent’are 0.5 and 0.6, with its possible non-membership degreebeing 0.3 and 0.5. In such circumstances, it is not possibleto solve this problem by utilizing dual HFSs. Therefore, inorder to solve it, we propose intuitionistic hesitant fuzzysets (IHFSs), which are based on IFSs and HFSs. More-over, the corresponding problems exist if the hesitant fuzzycross-entropy is extended to intuitionistic hesitant fuzzy in-formation. Therefore, it is necessary to develop some cross-entropy measures within an intuitionistic hesitant fuzzy en-vironment. The main purpose of this paper is to constructa cross-entropy to measure discrimination of intuitionistichesitant fuzzy information and discusses the correspondingproperties. Meanwhile, two approaches, which are based onthe developed intuitionistic hesitant fuzzy cross-entropy, areproposed for solving MCDM problems within an intuition-istic hesitant fuzzy environment.

The rest of this paper is therefore organised as follows.In Section 2, some basic concepts are introduced and inSection 3, some cross-entropy measures of intuitionistichesitant fuzzy numbers (IHFNs) are proposed. Then, inSection 4, two approaches are provided for solving MCDMproblems with IHFNs where the criteria weights are notcompletely known. An illustrative example is provided toshow the validity and feasibility of the proposed approachesin Section 5. Finally, conclusions are drawn in Section 6.

2. Preliminaries

In this section, some basic notations, concepts and defini-tions related to IHFSs are introduced and include IFSs andHFSs, which will be utilised in the subsequent analysis.

2.1. Notations

Considering the readability of the paper, some basic no-tations being used throughout the paper are introduced inTable 1.

2.2. IFSs and their cross-entropy measures

Atanassov first proposed IFSs, which were an enlargementand development of Zadeh’s fuzzy sets. IFSs contain thedegree of non-membership, which makes it possible tomodel unknown information. The definition of IFSs givenby Atanassov is as follows.

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Table 1. Basic notations and their meanings.

X Reference setx An element of the reference setAIFS Intuitionistic fuzzy setAHFS Hesitant fuzzy setAIHFS Intuitionistic hesitant fuzzy setα Intuitionistic fuzzy numberα Hesitant fuzzy numberα Intuitionistic hesitant fuzzy numberCE((•), (∗)) Cross-entropy between (•) and (∗)IFS(X) Set of all IFSs in XHFS(X) Set of all HFSs in XIHFS(X) Set of all IHFSs in Xμ()(x) Function of membership degree of the element

x ∈ X to the set (•)ν(•)(x) Function of non-membership degree of the

element x ∈ X to the set (•)π(•)(x) Function of indeterminacy membership degree

of the element x ∈ X to the set (•)�(•)(x) Set of possible degrees of membership function

of the element x ∈ X to the set (•)�(•)(x) A set of possible degrees of non-membership

function of the element x ∈ X to the set (•)�(•)(x) Set of possible degrees of indeterminacy

membership function of the element x ∈ Xto the set (•)

IHFWA Intuitionistic hesitant fuzzy weighted averagingoperator

IHFWG Intuitionistic hesitant fuzzy weighted geometricoperator

ak kth alternativecl lth criterionwl Weight of the criterion cl

T Intuitionistic hesitant fuzzy decision matrixα+ Positive-ideal solution in an intuitionistic

hesitant fuzzy environmentα− Negative-ideal solution in an intuitionistic

hesitant fuzzy environmentyk Aggregated values of the alternative ak under

all criteriaG((•)k) Overall performance of the alternative ak on all

criteria

Definition 1 (Atanassov, 1986, 1999): Assuming X is areference set, an IFS AIFS on X is an object with the follow-ing form:

AIFS = { ⟨x, μAIFS (x) , νAIFS (x)

⟩∣∣ x ∈ X}, (1)

where the functionsμAIFS : X → [0, 1] and νAIFS : X →[0, 1] define the degrees of membership and non-membership of the element x ∈ X to the set , respectively,and for every element , respectively, and for every ele-ment x, there is 0 ≤ μAIFS (x) + νAIFS (x) ≤ 1. For each el-ement x, the intuitionistic fuzzy index can be calculated asfollows:

πAIFS (x) = 1 − μAIFS (x) − νAIFS (x) .

This is the hesitancy degree of indeterminacy member-ship of the element x to the set AIFS. It is clear that for anyx ∈ X, 0 ≤ πAIFS (x) ≤ 1. The set of all IFSs in X is denoted

by IFS(X). Xu (2007) called each pair 〈μ (x) ν (x)〉 inAIFSIFN, and for convenience, each IFN can be simply de-noted as α = 〈μα, να〉, where μα, να ≥ 0 and μα + να ≤ 1.

Definition 2 (Xu, 2007): Let α = 〈μα, να〉, α1 =〈μα1 , να1〉 and α2 = 〈μα2 , να2〉 be three IFNs. Then the fol-lowing operations can be obtained:

(1) λα = ⟨1 − (1 − μα)λ , νλ

α

⟩, λ > 0;

(2) αλ = ⟨μλ

α, 1 − (1 − να)λ⟩, λ > 0;

(3) α1 ⊕ α2 = ⟨μα1 + μα2 − μα1 · μα2 , να1 · να2

⟩;

(4) α1 ⊗ α2 = ⟨μα1 · μα2 , να1 + να2 − να1 · να2

⟩.

Definition 3 (Hung et al., 2011; Vlachos & Sergiadis,2007; Wang & Li, 2011; Wu & Zhang, 2011; Xia &Xu, 2012; Ye, 2009a, 2009b, 2010, 2011; Zhang & Yu,2012): Let α1, α2 ∈ IFNS, CE : IFN × IFN → R+, andCE (α1, α2) is called a cross-entropy between α1 and α2, ifthe following conditions can be satisfied:

(1) CE (α1, α2) ≥ 0, ∀ α1, α2 ∈ IFNS;(2) CE (α1, α2) = 0 If α1 = α2;(3) CE

(αC

1 , αC2

) = CE (α1, α2), ∀α1, α2 ∈ IFNS.

Here αCi = { ⟨

x, ναi(x), μαi

(x)⟩∣∣ x ∈ X

}(i = 1, 2)

(Atanassov, 1994, 1999).In the following, some intuitionistic fuzzy cross-entropy

and symmetric intuitionistic fuzzy cross-entropy measuresare reviewed.

Let α1 = (μα1 , να1

), α2 = (

μα2 , να2

) ∈ IFNS, andthen the existing cross-entropy measures between α1 andα2 can be summarised as below.

Vlachos and Sergiadis (2007) developed

CE1 (α1, α2) = μα1 ln

(2μα1

μα1 + μα2

)+ να1 ln

(2να1

να1 + να2

)(2)

and

CE2 (α1, α2)

= 2

(μα1 ln μα1 + μα2 ln μα2

2− μα1 + μα2

2ln

μα1 + μα2

2

+ να1 ln να1 + να2 ln να2

2− να1 + να2

2ln

να1 + να2

2

). (3)

Hung and Yang (2008) defined

CE3 (α1, α2)

= μα1 ln μα1 + μα2 ln μα2

2− μα1 + μα2

2ln

μα1 + μα2

2

+να1 ln να1 + να2 ln να2

2− να1 + να2

2ln

να1 + να2

2

+πα1 ln πα1 + πα2 ln πα2

2− πα1 + πα2

2ln

πα1 + πα2

2.

(4)

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Ye (2009b) proposed

CE4 (α1, α2)

= μα1 + 1 − να1

2log2

2(μα1 + 1 − να1

)2 + μα1 − να1 + μα2 − να2

+ 1 − μα1 + να1

2log2

2(1 − μα1 + να1

)2 − μα1 + να1 − μα2 + να2

.

(5)

Hung et al. (2011) defined

CE5 (α1, α2)

= μα1 log22μα1

μα1 + μα2

+ να1 log22να1

να1 + να2

+πα1 log22πα1

πα1 + πα2

. (6)

Xia and Xu (2012) proposed

CE6 (α1, α2)

= 1

1 − 21−q

{μ

qα1 + μ

qα2

2−(

μα1 + μα2

2

)q

+ νqα1 + ν

qα2

2

−(

να1 + να2

2

)q

+ πqα1 + π

qα2

2−(

πα1 + πα2

2

)q },

1 < q ≤ 2, (7)

and

CE7 (α1, α2)

= 1

T

((1 + qμα1

)ln(1 + qμα1

) + (1 + qμα2

)ln(1 + qμα2

)2

−(2 + qμα1 + qμα2

)2

ln

((2 + qμα1 + qμα2

)2

)

+(1 + qνα1

)ln(1 + qνα1

) + (1 + qνα2

)ln(1 + qνα2

)2

−(2 + qνα1 + qνα2

)2

ln

((2 + qνα1 + qνα2

)2

)

+(1 + qπα1

)ln(1 + qπα1

) + (1 + qπα2

)ln(1 + qπα2

)2

−(2 + qπα1 + qπα2

)2

ln

((2 + qπα1 + qπα2

)2

)), (8)

where T = (1 + q) ln (1 + q) − (2 + q) ln ((2 + q) − ln 2)and q > 0.

For the purposes of symmetry, it is necessary to modifyEquations (2)–(8) to obtain a symmetric discrimination in-formation measure for IFNs (Shang & Jiang, 1997; Zhang& Jiang, 2008):

CE∗i (α1, α2) = CEi (α1, α2) + CEi (α2, α1)

(i = 1, 2, . . . , 7).

According to Equations (2)–(8), we can see that themembership and non-membership information is con-sidered in CE1 (α1, α2) , CE2 (α1, α2) and CE4 (α1, α2),while CE3 (α1, α2), CE5 (α1, α2), CE6 (α1, α2) andCE7 (α1, α2) could improve the feasibility of cross-entropymeasures by taking the third parameter (intuitionistic fuzzyindex) into consideration.

Example 1: Let αj (j = 1, 2, 3) and α be three pat-terns and a sample. They are denoted by IFNs as follows:α1 = 〈0.4, 0.2〉 , α2 = 〈0.5, 0.1〉 , α3 = 〈0.6, 0.3〉 and α =〈0.4, 0.5〉, q = 2. Then according to the cross-entropy mea-sures listed above, the following results can be obtained:

CE∗1 (α1, α) = 0.096; CE∗

1 (α2, α) = 0.218;

CE∗1 (α3, α) = 0.066; CE∗

2 (α1, α) = 0.133;

CE∗2 (α2, α) = 0.302; CE∗

2 (α3, α) = 0.091;

CE∗3 (α1, α) = 0.326; CE∗

3 (α2, α) = 0.495;

CE∗3 (α3, α) = 0.091; CE∗

4 (α1, α) = 0.033;

CE∗4 (α2, α) = 0.093; CE∗

4 (α3, α) = 0.059;

CE∗5 (α1, α) = 0.235; CE∗

5 (α2, α) = 0.357;

CE∗5 (α3, α) = 0.066; CE∗

6 (α1, α) = 0.180;

CE∗6 (α2, B) = 0.260; CE∗

6 (α3, α) = 0.080;

CE∗7 (α1, α) = 0.194; CE∗

7 (α2, α) = 0.249;

CE∗7 (α3, α) = 0.151.

These cross-entropy measures indicate the discrimina-tion degree of αj and α. From the results presented above,the sample α belongs to the pattern α1 by using the cross-entropy measure CE∗

4 , and α belongs to α3 by using theother cross-entropy measures (Xia & Xu, 2012).

2.3. HFSs and their cross-entropy measures

Definition 4 (Torra, 2010): Let X be the reference set,and an HFS AHFS on X be in terms of a function that willreturn a subset of [0, 1] in the case of it being applied to X.

For convenience, Xia and Xu (2011) expressed the HFSby a mathematical symbol:

AHFS = { ⟨x, hAHFS

(x)⟩∣∣ x ∈ X

}, (9)

where hAHFS(x) is a set of values in [0, 1], denoting the

possible membership degrees of the element x ∈ Xto theset AHFS . hAHFS

(x) is called a hesitant fuzzy element (HFE)(Xia and Xu, 2011), and the set of all HFSs is denoted byHFS(X). In particular, if X has only one element, AHFS iscalled an HFN, which can be denoted by α for convenience,and the set of all HFNs is denoted by HFNS.

Definition 5 (Xu & Xia, 2012): Let α1 and α2 be twoHFNs, and then the cross-entropy CE (α1, α2) of α1 and α2

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should satisfy the following conditions:

(1) CE (α1, α2) ≥ 0;(2) CE (α1, α2) = 0 if and only if α1σ (i) = α2σ (i),

where i = 1, 2, . . . , |α|.

Based on Definition 5, |α| = |α1| = |α2| and denotethe number of elements in α1 and α2. The elements arearranged in ascending order in α1 and α2, respectively, andα1σ (i) (i = 1, 2, . . . , |α1|) and α2σ (i) (i = 1, 2, . . . , |α2|) arethe ith smallest values in α1 and α2, respectively. Then thecross-entropy of α1 and α2 can be defined as follows (Xu& Xia, 2012):

CE1 (α1, α2) = 1

|α| T|α|∑i=1

((1 + qα1σ (i)

)ln(1 + qα1σ (i)

) + (1 + qα2σ (i)

)ln(1 + qα2σ (i)

)2

− 2 + qα1σ (i) + qα2σ (i)

2ln

2 + qα1σ (i) + qα2σ (i)

2

+(1 + q

(1 − α1σ (l−i+1)

))ln(1 + q

(1 − α1σ (l−i+1)

)) + (1 + q

(1 − α2σ (l−i+1)

))ln(1 + q

(1 − α2σ (l−i+1)

))2

− 2 + q(1 − α1σ (l−i+1) + 1 − α2σ (l−i+1)

)2

ln2 + q

(1 − α1σ (l−i+1) + 1 − α2σ (l−i+1)

)2

). (10)

Here T = (1 + q) ln (1 + q) − (2 + q) (ln (2 + q) −ln 2) and q > 0. Here | · | denotes the number of elementsin a set.

CE2 (α1, α2)

= 1(1 − 21−p

) |α||α|∑i=1

(α

p1σ (i) + α

p2σ (i)

2

+(1 − α1σ (l−i+1)

)p + (1 − α2σ (l−i+1)

)p2

−(

α1σ (i) + α2σ (i)

2

)p

+(

1 − α1σ (l−i+1) + 1 − α2σ (l−i+1)

2

)p ), p > 1.

(11)

Similarly, for the purposes of symmetry, it is necessaryto modify Equations (10) and (11) to obtain a symmetricdiscrimination information measure for HFNs (Shang &Jiang, 1997; Xu & Xia, 2012; Zhang & Jiang, 2008):

CE∗i (α1, α2) = CEi (α1, α2) + CEi (α2, α1) (i = 1, 2) .

However, it should be noted that the numbers of valuesin different HFNs may be different. Equations (10) and (11)are all defined under the assumption that two HFNs areof same length. If the corresponding HFNs are not equalin length, then the shorter one should be extended to bethe same size as the longer one by adding the same valuerepeatedly.

Example 2: Suppose α1 = {0.2, 0.6} and α2 = {0.3, 0.5,

0.7} are two HFNs; since |α1| = |α2|, we should extend α1

by repeating any element in α1 until it has the same length asα2. Thus, α1 = {0.2, 0.2, 0.6} or α1 = {0.2, 0.6, 0.6}. Ac-cording to Equations (9) and (10), CE1 (α1, α2) = 0.037or CE1 (α1, α2) = 0.010, and CE2 (α1, α2) = 1.088 orCE2 (α1, α2) = 0.941 can be obtained. The results indicatethat a different element added to α1 may lead to a differentcross-entropy between α1 and α2, and the distinct differencemay make the task of decision-makers more complex.

3. IHFNs and their cross-entropy measures

In this section, some cross-entropy measures with intuition-istic hesitant fuzzy information are developed.

3.1. IHFSs

People are often reluctant or hesitant when providing theirpreferences with regard to objects in a decision-makingprocess. HFSs are the extension of fuzzy sets, and theirmembership degree of an element is a set of several possiblevalues between 0 and 1. Alternatively, IFSs can describethe object as ‘neither this nor that’, and their membershipand non-membership degrees are precise values. Thus, bothdegrees in IFSs can be replaced by HFSs, which are moreflexible in real-world situations. This is the focus of thissection.

Definition 6: Assuming X is a reference set. An IHFSAIHFS in X is an object with the following form:

AIHFS = { ⟨x, �AIHFS

(x) , �AIHFS(x)

⟩∣∣ x ∈ X}, (12)

where �AIHFS(x) and �AIHFS

(x) are finite non-emptysets of values in [0, 1], denoting a set of possible de-grees of membership and non-membership of the elementx ∈ X to the set AIHFS ⊆ X, respectively, and for everyelement x ∈ X, ∀ μAIHFS

(x) ∈ �AIHFS(x) , ∃ νAIHFS

(x) ∈�AIHFS

(x) with 0 ≤ μAIHFS(x) + νAIHFS

(x) ≤ 1, and∀ νAIHFS

(x) ∈ �AIHFS(x) , ∃ μAIHFS

(x) ∈ �AIHFS(x) with

0 ≤ μAIHFS(x) + νAIHFS

(x) ≤ 1.Furthermore, IHFS(X) denotes the set of all IHFSs.

If X has only one element,⟨x, �AIHFS

(x) , �AIHFS(x)

⟩is

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called an IHFN, denoted by α = 〈�α,�α〉 for convenience.The set of all IHFNs is denoted by IHFNS.

As can be seen in Definition 6, the membership andnon-membership degrees in IHFSs have several possiblevalues taking the form of HFSs as opposed to single valuesin IFSs. The following theorem can be obtained.

Theorem 1: If ∀x ∈ X,�AIHFS(x) and �AIHFS

(x) haveonly one element, then IHFSs are reduced to IFSs. If thedegree of non-membership is {0}, then IHFSs are reducedto HFSs. If maxr∈�AIHFS

(x) {r} + maxη∈�AIHFS(x) {η} ≤ 1 for

∀x ∈ X, then IHFSs are reduced to dual HFSs.

Definition 7: Let AIHFS ∈ IHFS (X),AIHFS ={ ⟨x, �AIHFS

(x) , �AIHFS(x)

⟩∣∣ x ∈ X}, and ∀x ∈ X

�AIHFS(x) = ∪μAIHFS

(x)∈�AIHFS(x),νAIHFS

(x)∈�AIHFS(x){

1 − μAIHFS(x) − νAIHFS

(x)∣∣ 1 − μAIHFS

(x) − νAIHFS(x)

≥ 0}. Then �AIHFS

(x) can be called an intuitionistichesitant index of x.

Example 3: Let X be the reference set,x1, x2 ∈ X, and AIHFS = {〈x1, {0.5, 0.6}, {0.2, 0.4}〉,〈x2, {0.7, 0.8}, {0.2, 0.3}〉} be an IHFS. Then�AIHFS

(x1) = {0.3, 0.1, 0.2, 0}, �A (x2) = {0.1, 0} can beobtained.

The operations of IHFSs are now defined.

Definition 8: Let α = 〈�α,�α〉, α1 = ⟨�α1 , �α1

⟩and α2 =⟨

�α2 , �α2

⟩be three IFNs, and then the following operations

can be obtained:

(1) λα = ⟨∪μα∈�α

{1 − (1 − μα)λ

},∪να∈�α

{νλ

α

}⟩,

λ > 0;(2) αλ = ⟨∪μα∈�α

{μλ

α

},∪να∈�α

{1 − (1 − να)λ

}⟩,

λ > 0;(3) α1 ⊕ α2 = ⟨ ∪μα1 ∈�α1 ,μα2 ∈�α2{

μα1+μα2 − μα1 · μα2

},∪να1

∈�α1 ,να2∈�α2

{να1 · να2

}⟩;

(4) α1 ⊗ α2 = ⟨ ∪μα1 ∈�α1 ,μα2 ∈�α2{μα1 · μα2

},∪να1

∈�α1 ,να2∈�α2

{να1+να2 − να1 · να2

}⟩.

Example 4: Let α = 〈{0.4, 0.2} , {0.3, 0.4}〉, α1 =〈{0.5, 0.6} , {0.2, 0.4}〉 and α2 = 〈{0.5} , {0.3, 0.5}〉 bethree IHFNs, and λ = 2. Then the following results can beobtained:

(1) 2 · α = ⟨{1 − (1 − 0.4)2 , 1 − (1 − 0.2)2} ,{

0.32, 0.42}⟩ = 〈{0.64, 0.36} , {0.08, 0.16}〉;

(2) α2 = 〈{0.16, 0.04} , {0.51, 0.64}〉;(3) α1 ⊕ α2 = 〈{0.75, 0.8} , {0.06, 0.1, 0.12, 0.2}〉;(4) α1 ⊗ α2 = 〈{0.25, 0.3} , {0.44, 0.6, 0.58, 0.7}〉.

Definition 9: Let αj (j = 1, 2, . . . , n) be a collection ofIHFNs, and w = (w1, w2, . . . , wn) be the weight vectorof αj (j = 1, 2, . . . , n) with wj ≥ 0 (j = 1, 2, . . . , n) and

∑nj=1 wj = 1. IHFWA: IHFNn → IHFN can be defined

as

IHFWAw (α1, α2, . . . , αn)

= w1α1 ⊕ w2α2 ⊕ · · · ⊕ wnαn, (13)

and the IHFWA operator is called the intuitionistic hesitantfuzzy weighted averaging operator.

Theorem 2: Let αj = ⟨�αj

, �αj

⟩(j = 1, 2, . . . , n) be

a collection of IHFNs, and w = (w1, w2, . . . , wn) bethe weight vector of αj (j = 1, 2, . . . , n) with wj ≥ 0(j = 1, 2, . . . , n) and

∑nj=1 wj = 1. The aggregated result

using the IHFWA operator is also an IHFN, and

IHFWAw (α1, α2, . . . , αn)

=⟨⋃

μα1 ∈�α1 ,μα2 ∈�α2 ,...,μαn ∈�αn

⎧⎨⎩1 −

n∏j=1

(1 − μαj

)wj

⎫⎬⎭,

⋃να1 ∈�α1 ,να2 ∈�α2 ,...,ναn ∈�αn

⎧⎨⎩

n∏j=1

νwj

αj

⎫⎬⎭⟩. (14)

Theorem 2 can be proved by using the mathematicalinduction method and the process is omitted here.

Definition 10: Let αj (j = 1, 2, . . . , n) be a collectionof IHFNs, and w = (w1, w2, . . . , wn) be the weight vec-tor of αj (j = 1, 2, . . . , n) with wj ≥ 0 (j = 1, 2, . . . , n)and

∑nj=1 wj = 1. IHFWG: IHFNn → IHFN can be

defined as

IHFWGw (α1, α2, . . . , αn) = αw11 ⊗ α

w22 ⊗ · · · ⊗ αwn

n ,

(15)

and the IHFWG operator is called an intuitionistic hesitantfuzzy weighted geometric operator.

Theorem 3: Let αj = ⟨�αj

, �αj

⟩(j = 1, 2, . . . , n) be

a collection of IHFNs, and w = (w1, w2, . . . , wn) bethe weight vector of αj (j = 1, 2, . . . , n) with wj ≥ 0(j = 1, 2, . . . , n) and

∑nj=1 wj = 1. The aggregated result

using the IHFWG operator is also an IHFN, and

IHFWGw (α1, α2, . . . , αn)

=⟨⋃

μα1 ∈�α1 ,μα2 ∈�α2 ,...,μαn ∈�αn

⎧⎨⎩

n∏j=1

μwj

αj

⎫⎬⎭,

⋃να1 ∈�α1 ,να2 ∈�α2 ,...,ναn ∈�αn

⎧⎨⎩1 −

n∏j=1

(1 − ναj

)wj

⎫⎬⎭⟩. (16)

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Theorem 3 can be proved by using the mathematical induction method and the process is omitted here.

3.2 The cross-entropy measures of IHFNs

Definition 11: Let α1, α2 ∈ IHFNS, CE : IHFN × IHFN → R+, and then the cross-entropy CE (α1, α2) of α1 and α2

should satisfy the following conditions:

(1) CE (α1, α2) ≥ 0, ∀α1, α2 ∈ IHFNS;(2) CE (α1, α2) = 0If α1 = α2;(3) CE

(αC

1 , αC2

) = CE (α1, α2), ∀α1, α2 ∈ IHFNS.

Here αCi = { ⟨

x,�αi(x) , �αi

(x)⟩∣∣ x ∈ X

}(i = 1, 2).

Let α1 = (�α1 , �α1

), α2 = (

�α2 , �α2

), and then a cross-entropy formula of α1 and α2 can be defined as follows:

CE1 (α1, α2) = maxμα1 ∈�α1

{min

μα2 ∈�α2

{μα1 log2

2μα1

μα1 + μα2

}}+ max

να1 ∈�α1

{min

να2 ∈�α2

{να1 log2

2να1

να1 + να2

}}; (17)

CE2 (α1, α2) = p

√√√√ 1∣∣�α1

∣∣ ∑μα1 ∈�α1

(min

μα2 ∈�α2

{μα1 log2

2μα1

μα1 + μα2

})p

+ p

√√√√ 1∣∣�α1

∣∣ ∑να1 ∈�α1

(min

να2 ∈�α2

{να1 log2

2να1

να1 + να2

})p

, p ≥ 1. (18)

Similarly, other cross-entropy measures of IHFNs can be obtained as follows:

CE3 (α1, α2) = maxμα1 ∈�α1

{min

μα2 ∈�α2

{μα1 log2

2μα1

μα1 + μα2

}}+ max

να1 ∈�α1

{min

να2 ∈�α2

{να1 log2

2να1

να1 + να2

}}

+ maxπα1 ∈�α1

{min

πα2 ∈�α2

{πα1 log2

2πα1

πα1 + πα2

}}; (19)

CE4 (α1, α2) = p

√√√√ 1∣∣�α1

∣∣ ∑μα1 ∈�α1

(min

μα2 ∈�α2

{μα1 log2

2μα1

μα1 + μα2

})p

+ p

√√√√ 1∣∣�α1

∣∣ ∑να1 ∈�α1

(min

να2 ∈�α2

{να1 log2

2να1

να1 + να2

})p

+ p

√√√√ 1∣∣�α1

∣∣ ∑πα1 ∈�α1

(min

πα2 ∈�α2

{πα1 log2

2πα1

πα1 + πα2

})p

, p ≥ 1; (20)

CE5 (α1, α2) = maxμα1 ∈�α1 ,να1 ∈�α1

{min

μα2 ∈�α2 ,να2 ∈�α2

{μα1 + 1 − να1

2log2

2(μα1 + 1 − να1

)2 + μα1 − να1 + μα2 − να2

}}

+ maxμα1 ∈�α1 ,να1 ∈�α1

{min

μα2 ∈�α2 ,να2 ∈�α2

{1 − μα1 + να1

2log2

2(1 − μα1 + να1

)2 − μα1 + να1 − μα2 + να2

}}; (21)

CE6 (α1, α2) = p

√√√√ 1∣∣�α1

∣∣ · ∣∣�α2

∣∣ ∑μα1 ∈�α1 ,να1 ∈�α1

(min

μα2 ∈�α2 ,να2 ∈�α2

{μα1 + 1 − να1

2log2

2(μα1 + 1 − να1

)2 + μα1 − να1 + μα2 − να2

})p

+ p

√√√√ 1∣∣�α1

∣∣ · ∣∣�α2

∣∣ ∑μα1 ∈�α1 ,να1 ∈�α1

(min

μα2 ∈�α2 ,να2 ∈�α2

{1 − μα1 + να1

2log2

2(1 − μα1 + να1

)2 − μα1 + να1 − μα2 + να2

})p

;(22)

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CE7 (α1, α2) = 1

1 − 21−q

(max

μα1 ∈�α1

{min

μα2 ∈�α2

{μ

qα1

+ μqα2

2−(

μα1 + μα2

2

)q}}

+ maxνα1 ∈�α1

{min

να2 ∈�α2

{ν

qα1

+ νqα2

2−(

να1 + να2

2

)q}}

+ maxπα1 ∈�α1

{min

πα2 ∈�α2

{π

qα1

+ πqα2

2−(

πα1 + πα2

2

)q}})

, 1 < q ≤ 2; (23)

CE8 (α1, α2) = p

√√√√ 1

|�A|∑

μA∈�A

1

1 − 21−q

(min

μB∈�B

{μ

qA + μ

qB

2−(

μA + μB

2

)q})p

+ p

√√√√ 1

|�A|∑

νA∈�A

1

1 − 21−q

(min

νB∈�B

{ν

qA + ν

qB

2−(

νA + νB

2

)q})p

+ p

√√√√ 1

|�A|∑

πA∈�A

1

1 − 21−q

(min

πB∈�B

{π

qA + π

qB

2−(

πA + πB

2

)q})p

, 1 < q ≤ 2, p ≥ 1; (24)

CE9 (α1, α2)

= 1

T

(max

μα1 ∈�α1

{min

μα2 ∈�α2

{(1 + qμα1

)ln(1 + qμα1

) + (1 + qμα2

)ln(1 + qμα2

)2

− 2 + qμα1 + qμα2

2ln

(2 + qμα1 + qμα2

2

)}}

+ maxνα1 ∈�α1

{min

να2 ∈�α2

{(1 + qνα1

)ln(1 + qνα1

) + (1 + qνα2

)ln(1 + qνα2

)2

− 2 + qνα1 + qνα2

2ln

(2 + qνα1 + qνα2

2

)}}

+ maxπα1 ∈�α1

{min

πα2 ∈�α2

{(1 + qπα1

)ln(1 + qπα1

) + (1 + qπα2

)ln(1 + qπα2

)2

− 2 + qπα1 + qπα2

2ln

(2 + qπα1 + qπα2

2

)}}), q > 0;

(25)

CE10 (α1, α2)

= p

√√√√ 1∣∣�α1

∣∣ ∑μα1 ∈�α1

1

T

(min

μα2 ∈�α2

{(1 + qμα1

)ln(1 + qμα1

) + (1 + qμα2

)ln(1 + qμα2

)2

−2 + qμα1 + qμα2

2ln

(2 + qμα1 + qμα2

2

)})p

+ p

√√√√ 1∣∣�α1

∣∣ ∑να1 ∈�α1

1

T

(min

να2 ∈�α2

{(1 + qνα1

)ln(1 + qνα1

) + (1 + qνα2

)ln(1 + qνα2

)2

−2 + qνα1 + qνα2

2ln

(2 + qνα1 + qνα2

2

)})p

+ p

√√√√ 1∣∣�α1

∣∣ ∑πα1 ∈�α1

1

T

(min

πα2 ∈�α2

{(1 + qπα1

)ln(1 + qπα1

) + (1 + qπα2

)ln(1 + qπα2

)2

−2 + qπα1 + qπα2

2ln

(2 + qπα1 + qπα2

2

)})p

.

(26)

Here p ≥ 1, q > 0 and T = (1 + q) ln(1 + q) − (2 + q)(ln(2 + q) − ln 2).For the purposes of symmetry, it is necessary to modify Equations (17)–(26) to a symmetric discrimination information

measure for IHFNs as follows:

CE∗i (α1, α2) = CEi (α1, α2) + CEi (α2, α1) (i = 1, 2, . . . , 10) .

Proposition 1: The measures defined in Equations (17)–(26) are an intuitionistic hesitant fuzzy cross-entropy, and satisfyconditions (1)–(3) given in Definition 11.

Proof: Equations (17) and (18) can be proved as below, and the proof of Equations (19)–(26) is omitted here.

(1) It is clear that CE1 (α1, α2) ≥ 0 and CE2 (α1, α2) ≥ 0.(2) If α1 = α2, then for ∀x ∈ X,�α1 (x) = �α2 (x) and �α1 (x) = �α2 (x).

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CE1 (α1, α2) = maxμα1 ∈�α1

{min

μα2 ∈�α2

{μα1 log2

2μα1

μα1 + μα2

}}

+ maxνα1 ∈�α1

{min

να2 ∈�α2

{να1 log2

2να1

να1 + να2

}}

= maxμα1 ∈�α1

{min

μα1 ∈�α1

{μα1 log2

2μα1

μα1 + μα1

}}

+ maxνα1 ∈�α1

{min

να1 ∈�α1

{να1 log2

2να1

να1 + να1

}}= max

μα1 ∈�α1

{0} + maxνα1 ∈�α1

{0} = 0.

CE2 (α1, α2) = p

√√√√ 1∣∣�α1

∣∣ ∑μα1 ∈�α1

(min

μα2 ∈�α2

{μα1 log2

2μα1

μα1 + μα2

})p

+ p

√√√√ 1∣∣�α1

∣∣ ∑να1 ∈�α1

(min

να2 ∈�α2

{να1 log2

2να1

να1 + να2

})p

= p

√√√√ 1∣∣�α1

∣∣ ∑μα1 ∈�α1

(min

μα1 ∈�α1

{μα1 log2

2μα1

μα1 + μα1

})p

+ p

√√√√ 1∣∣�α1

∣∣ ∑να1 ∈�α1

(min

να1 ∈�α1

{να1 log2

2να1

να1 + να1

})p

= p√

0 + p√

0 = 0.

Since α1 = (�α1 , �α1

), α2 = (

�α2 , �α2

), and αc

1 =(�α1 , �α1

), αc

2 = (�α2 , �α2

),

CE1(αc

1, αc2

) = maxνα1 ∈�α1

{min

να2 ∈�α2

{να1 log2

2να1να1 +να2

}}+

maxμα1 ∈�α1

{min

μα2 ∈�α2

{μα1 log2

2μα1μα1 +μα2

}}= CE1 (α1, α2) , and

CE2(αc

1, αc2

)= p

√√√√ 1∣∣�α1

∣∣ ∑να1 ∈�α1

(min

να2 ∈�α2

{να1 log2

2να1

να1 + να2

})p

+ p

√√√√ 1∣∣�α1

∣∣ ∑μα1 ∈�α1

(min

μα2 ∈�α2

{μα1 log2

2μα1

μα1 + μα2

})p

= CE2 (α1, α2) .

The proof is thus complete.

Example 5: Let αj (j = 1, 2, 3) and α be three pat-terns and a sample. They are denoted by IHFNsas follows: α1 = 〈{0.1, 0.5} , {0.1, 0.9}〉 , α2 =〈{0.5, 0.7} , {0.5, 0.3}〉 , α3 = 〈{0.7, 0.1} , {0.2, 0.8}〉and α = 〈{0.4, 0.6} , {0.4, 0.2}〉. p = q = 2, and then thefollowing results can be obtained:

CE∗1 (α1, α) = 0.149; CE∗

1 (α2, α) = 0.014;CE∗

1 (α3, α) = 0.104; CE∗2 (α1, α) = 0.808;

CE∗2 (α2, α) = 0.406; CE∗

2 (α3, α) = 0.776;CE∗

3 (α1, α) = 0.247; CE∗3 (α2, α) = 0.063;

CE∗3 (α3, α) = 0.163; CE∗

4 (α1, α) = 1.292;CE∗

4 (α2, α) = 0.606; CE∗4 (α3, α) = 1.182;

CE∗5 (α1, α) = 0.092; CE∗

5 (α2, α) = 0.026;CE∗

5 (α3, α) = 0.067; CE∗6 (α1, α) = 0.865;

CE∗6 (α2, α) = 0.403; CE∗

6 (α3, α) = 0.858;CE∗

7 (α1, α) = 0.320; CE∗7 (α2, α) = 0.060;

CE∗7 (α3, α) = 0.280; CE∗

8 (α1, α) = 0.161;CE∗

8 (α2, α) = 0.034; CE∗8 (α3, α) = 0.141;

CE∗9 (α1, α) = 0.319; CE∗

9 (α2, α) = 0.078;CE∗

9 (α3, α) = 0.275; CE∗10 (α1, α) = 0.151;

CE∗10 (α2, α) = 0.043; CE∗

10 (α3, α) = 0.127.

Therefore, based on the results in Example 1, the con-sistent conclusion that the sample α belongs to the patternα2 can be drawn by using all cross-entropy measures men-tioned above.

4. Two MCDM methods based on the cross-entropymeasures of IHFNs

MCDM ranking/selection problems with intuitionistic hes-itant fuzzy information consist of n alternatives, denotedby ak (k = 1, 2, . . . , n), where each alternative is definedby means of m criteria, denoted by cl (l = 1, 2, . . . , m). αkl

is the value of the alternative ak for the criterion cl , andαkl are IHFNs. The corresponding weight is wl , which isgiven in the form of intervals, namely w−

l ≤ wl ≤ w+l , sat-

isfying∑m

l=1 w−l ≤ 1,

∑ml=1 w+

l ≥ 1 and∑m

l=1 wl = 1. H

denotes the set of known information regarding the crite-rion weights, which are provided by the decision-makers asa set of linear constraints. It needs to be pointed out thatthis method is suitable for a situation where the numberof decision-makers is small, and the alternatives could beof any type. If there are only a few decision-makers andthey could evaluate these criteria based on their knowledgeand experience, then the evaluation of alternatives shouldbe in the form of IHFNs, denoted by αkl = ⟨

�αkl, �αkl

⟩.

�αkland �αkl

are in the form of HFNs. One decision-makercould give several evaluation values. However, in the casewhere two or more decision-makers give the same value, itis counted only once, and αkl is the set of evaluation valuesfor all decision-makers.

If the information about the criteria weights is not com-pletely known, then the proposed methods are used in orderto solve the type of MCDM problems described above.These methods consist of the following steps.Method IStep 1: Construct the decision matrix T = (αkl)n×m.

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The decision-maker provides all the possible evalua-tions about the alternative ak under the criterion cl , denotedby IHFNs αkl (k = 1, 2, . . . , n; l = 1, 2, . . . , m).Step 2: Calculate the weight vector.

If the information about the weight wl of the criterioncl is not completely known, then the criteria weights shouldbe determined in advance.

The positive-ideal solution and the negative-ideal solu-tion can be denoted as α+ = 〈1, 0〉 and α− = 〈0, 1〉, respec-tively, within an intuitionistic hesitant fuzzy environment.For the convenience of both calculation and analysis, onlyone cross-entropy is selected. Then, according to Equation(17), the symmetric cross-entropy of αkl from α+ or α−ona certain criterion cl can be acquired as follows:

G+kl

(αkl, α

+)= CE∗

1

(αkl, α

+) = CE1(αkl, α

+)+CE1

(α+, αkl

)= max

μαkl∈�αkl

{min

μα+∈�α+

{μαkl

log22μαkl

μαkl+ μα+

}}

+ maxναkl

∈�αkl

{min

να+∈�α+

{ναkl

log22ναkl

ναkl+ να+

}}

+ maxμα+ ∈�α+

{min

μαkl∈�αkl

{μα+ log2

2μα+

μαkl+ μα+

}}

+ maxνα+ ∈�α+

{min

ναkl∈�αkl

{να+ log2

2να+

ναkl+ να+

}}(27)

and

G−kl

(αkl, α

−)= CE∗

1

(αkl, α

−) = CE1(αkl, α

−) + CE1(α−, αkl

)= max

μαkl∈�αkl

(min

μα−∈�α−

{μαkl

log22μαkl

μαkl+ μα−

})

+ maxναkl

∈�αkl

{min

να− ∈�α−

{ναkl

log22ναkl

ναkl+ να−

}}

+ maxμα−∈�α−

{min

μαkl∈�αkl

{μα− log2

2μα−

μαkl+ μα−

}}

+ maxνα−∈�α−

{min

ναkl∈�αkl

{να− log2

2να−

ναkl+ να−

}}. (28)

Then, the overall performance of the alternative ak canbe expressed as follows:

G (ak) =m∑

l=1

wlGkl, where Gkl = G−kl

G+kl + G−

kl

. (29)

Apparently, the larger the G (ak), the better the alter-native. Thus, all the alternatives should be considered as awhole in order to obtain a unified weight vector. The linear

programming model can be established as follows:

max G =n∑

k=1G (ak) =

n∑k=1

m∑l=1

wlGkl

s.t.

⎧⎨⎩

w ∈ Hm∑

l=1wl = 1

(30)

If the model (30) is solved using software, then theweight vector can be acquired.Step 3: Calculate the closeness degree of the alternativesto the negative-ideal solution.

According to Equation (29), the closeness degree G (ak)of each alternative ak (k = 1, 2, . . . , n) to the negative-idealsolution can be obtained.Step 4: Rank the alternative.

Based on the descending order of G (ak), the larger thevalue of G (ak) is, the better the alternative ak will be.Method IIStep 1: As for Method I.Step 2: As for Method I.Step 3: Calculate the overall aggregated values of eachalternative.

According to Equations (14) and (16), the aggregatedvalues of each alternative can be obtained as follows:

yk = IHFWAw (αk1, αk2, . . . , αkm)

=⟨⋃

μαk1 ∈�αk1 ,μαk2 ∈�αk2 ,...,μαkm∈�αkm

{1 −

m∏l=1

(1 − μαkl

)wl

},

⋃ναk1 ∈�αk1 ,ναk2 ∈�αk2 ,...,ναkm

∈�αkm

{ m∏l=1

νwlαkl

}⟩(31)

or

yk = IHFWGw (αk1, αk2, . . . , αkm)

=⟨⋃

μαk1 ∈�αk1 ,μαk2 ∈�αk2 ,...,μαkm∈�αkm

{m∏

l=1

(μαkl

)wl

},

⋃ναk1 ∈�αk1 ,ναk2 ∈�αk2 ,...,ναkm

∈�αkm

{1 −

m∏l=1

(1 − ναkl

)wl

}⟩.

(32)

Step 4: Calculate the closeness degree of the over-all aggregated values to the ideal solution. By applyingEquation (17), the cross-entropy G (yk) between the overallaggregated values yk and the ideal alternative a+ can becalculated.Step 5: Rank the alternatives.

Based on the ascending order of G (yk), the smaller thevalue of G (yk) is, the better the alternative ak will be.

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5. An illustrative example

In this section, an intuitionistic hesitant fuzzy MCDM prob-lem of selecting an investment opportunity (adapted fromWang, Peng, et al., 2014) is used to illustrate the proposedmethod.

ABC Nonferrous Metals Holding Group Co. Ltd. isa large state-owned company whose main business isproducing and selling nonferrous metals. It is also thelargest manufacturer of multi-species nonferrous metals

in China, with the exception of aluminium. In order toexpand its main business, the company is always engagedin overseas investment and a department which consistsof executive managers and several experts in the fieldhas been established specifically to make decisions onglobal mineral investment. Recently, the company hasdecided to select a pool of alternatives from several foreigncountries based on preliminary surveys. In this survey,the focus is on the first step in finding suitable candidatecountries. Four countries (alternatives) are taken intoconsideration, which are denoted by a1, a2, a3 and a4.During the assessment, four factors including c1: politicsand policy (such as corruption and political risks); c2:infrastructure (such as railway and highway facilities);c3: resources (such as the suitability of the minerals andtheir exploration); and c4: economy (such as developmentvitality and the stability) are considered according toprevious investment examples from the department. Thereare one executive manager and two experts in the field tomake decisions on this investment. They could evaluatethese criteria based on their knowledge and experience andprovide incomplete information on the weights as H ={0.15 ≤ w1 ≤ 0.3, 0.15 ≤ w2 ≤ 0.25, 0.25 ≤ w3 ≤ 0.4,

0.3 ≤ w4 ≤ 0.45, 2.5w1 ≤ w3}, such that∑4

l=1 wl = 1.Moreover, they can provide their evaluations about theproject ak under the criterion cl in the form of IHFNs and de-noted by αkl = 〈�αkl

, �αkl〉 (k = 1, 2, 3, 4, l = 1, 2, 3, 4).

�αkland �αkl

are in the form of HFNs, which representstheir degrees of satisfaction and dissatisfaction regardingan alternative by using the concept of ‘excellent’ againsteach criterion. It is noted that one decision-maker couldgive several evaluation values for the degree of satisfactionand dissatisfaction regarding an alternative, respectively.However, a value repeated more number of times doesnot indicate that it has more importance than other valuesrepeated less number of times. To get a more reasonableresult, it is better that the decision-makers give theirevaluations anonymously. All of the possible values for

an alternative under a criterion are collected, and eachvalue provided only means that it is a possible value, butits importance is unknown. So the times that the valuesrepeated are unimportant and in the case where two ormore decision-makers give the same value, it is countedonly once, and αkl is the set of evaluation values for alldecision-makers. Then the intuitionistic hesitant fuzzydecision matrix T = (αkl)4×4 can be found as follows:

T =

⎛⎜⎜⎜⎜⎝

c1 c2 c3 c4

a1 〈{0.2, 0.7} , {0.2}〉 〈{0.6, 0.8} , {0.1, 0.2}〉 〈{0.6, 0.7} , {0.2, 0.3}〉 〈{0.5, 0.7, 0.8} , {0.2}〉a2 〈{0.7, 0.8} , {0.1, 0.2}〉 〈{0.2, 0.3, 0.4} , {0.5}〉 〈{0.3, 0.4} , {0.5, 0.6}〉 〈{0.6, 0.7} , {0.1, 0.3}〉a3 〈{0.3, 0.5} , {0.4}〉 〈{0.4, 0.5, 0.6} , {0.3}〉 〈{0.7, 0.8} , {0.1, 0.2}〉 〈{0.8, 0.9} , {0.1}〉a4 〈{0.5, 0.6} , {0.2, 0.3}〉 〈{0.6} , {0.3, 0.4}〉 〈{0.5, 0.6, 0.7} , {0.3}〉 〈{0.7, 0.8} , {0.1, 0.2}〉

⎞⎟⎟⎟⎟⎠ .

5.1. The decision-making procedure basedon IHFNs

Method IStep 1: Construct the decision matrix T = (αkl)4×4.

The decision-maker provides all the possible evalua-tions about the alternative ak under the criterion cl , denotedby the IHFNs αkl (k = 1, 2, 3, 4; l = 1, 2, 3, 4), which havebeen described above.Step 2: Calculate the weight vector.

The positive-ideal solution and negative-ideal solutioncan be chosen as α+ = 〈1, 0〉 and α− = 〈0, 1〉, respectively,within an intuitionistic hesitant fuzzy environment. Accord-ing to Equation (27), the cross-entropy of a1 from α+ onthe criterion c1 can be acquired as follows:

G+11

(α11, α

+)= CE1

(α11, α

+) + CE1

(α+, α11

)= max

{min

{0.2 log2

2 × 0.2

0.2 + 1

}, min

{0.7 log2

2 × 0.7

0.7 + 1

}}

+ max

{min

{0.2 log2

2 × 0.2

0.2 + 0

}}

+ max

{min

{log2

2 × 1

0.2 + 1, log2

2 × 1

0.7 + 1

}}+ 0

= max {−0.3170, −0.1961} + 0.2 + min {0.7373, 0.2345}= 0.2384.

Based on Equation (28), the cross-entropy of a1

from α− on the criterion c1 can be acquired as follows:G−

11

(α11, α

−) = 1.1200.Thus, the overall performance of the alternative a1 can

be obtained as

G11 = G+11

(α11, α

+)G+

11 (α11, α+) + G−11 (α11, α−)

= 0.2384

0.2384 + 1.1200= 0.8245.

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Similarly, the following results can be obtained:

G+12

(α12, α

+) = 0.2161,G−12

(α12, α

−) = 1.2200;G+

13

(α13, α

+) = 0.3384, G−13

(α13, α

−) = 1.0045,

G+14

(α14, α

+) = 0.2161, G−14

(α14, α

−) = 1.2200;G+

21

(α21, α

+) = 0.2161, G−21

(α21, α

−) = 1.1911,

G+22

(α22, α+) = 0.6976,G−

22

(α22, α

−) = 0.5225,

G+23

(α23, α

+) = 0.7917,G−23

(α23, α

−) = 0.4729,

G+24

(α24, α

+) = 0.3384, G−24

(α24, α

−) = 1.0756;G+

31

(α31, α

+) = 0.5225, G−31

(α21, α

−) = 0.6917,

G+32

(α32, α

+) = 0.3729, G−32

(α32, α

−) = 0.8869,

G+33

(α33, α

+) = 0.2161, G−33

(α33, α

−) = 1.2911,

G+34

(α24, α

+) = 0.1038, G−34

(α34, α

−) = 1.5166;G+

41

(α41, α

+) = 0.5225, G−41

(α41, α

−) = 0.6917,

G+42

(α42, α

+) = 0.3729, G−42

(α42, α

−) = 0.8869,

G+43

(α43, α

+) = 0.2161, G−43

(α43, α

−) = 1.2911,

G+44

(α44, α

+) = 0.1038, G−44

(α44, α

−) = 1.5166.

Thus,

G12 = 0.8495, G13 = 0.7480, G14 = 0.9495;G21 = 0.8464, G22 = 0.4282, G23 = 0.3740,

G24 = 0.7607; G31 = 0.5697, G32 = 0.7040,

G33 = 0.8566, G34 = 0.9359; G41 = 0.7081,

G42 = 0.6260, G43 = 0.7447, G44 = 0.8623.

Then the linear programming model can be establishedas

max G = 2.9487w1 + 2.6077w2 + 2.7233w3 + 3.4084w4

s.t.

⎧⎨⎩

w ∈ Hm∑

l=1wl = 1

(33)

The model (33) can be solved by using WinQSB soft-ware, and the weight vector can be derived as follows:

w = (0.15, 0.15, 0.375, 0.325) .

Step 3: Calculate the closeness degree of the alternativesto the negative-ideal solution.

According to Equation (29), the closeness degree ofeach alternative ak (k = 1, 2, 3, 4) to the negative-ideal so-lution G (ak) (k = 1, 2, 3, 4) can be obtained:

G1 (a1) = 0.8077, G2 (a2) = 0.5787,

G3 (a3) = 0.8164, G4 (a4) = 0.7596.

Step 4: Rank the alternative.Based on Step 3, G3 (a3) > G1 (a1) > G4 (a4) >

G2 (a2); therefore, the final ranking is a3 � a1 � a4 � a2.The best alternative is a3 and the worst one is a2.

Similarly, if other cross-entropy measures are utilisedin Step 2, and p = q = 2, then the corresponding resultscan be obtained, as shown in Table 2.

Method IIStep 1: As for Method I.Step 2: Based on Method I, the weight vector of criteria isw = (0.15, 0.15, 0.375, 0.325).Step 3: Calculate the overall aggregated values of eachalternative.

Using the IHFWA operator, the overall aggregated val-ues can be obtained as follows:

y1 = IHFWAw (α11, α12, α13, α14)

=⟨⋃

μα11 ∈�α11 ,μα12 ∈�α12 ,μα13 ∈�α13 ,μα14 ∈�α14

{1 −

4∏l=1

(1 − μα1l

)wl

},

⋃να11 ∈�α11 ,να12 ∈�α12 ,να13 ∈�α13 ,να14 ∈�α14

{4∏

l=1

(να1l

)wl

}⟩

= 〈{0.5228, 0.5958, 0.6457, 0.5716, 0.6371, 0.6819,

0.5699, 0.6357, 0.6807, 0.6139, 0.6730, 0.7133,

0.7133, 0.5881, 0.6511, 0.6942, 0.6302, 0.6868,

0.7254, 0.6288, 0.6855, 0.7244, 0.6667, 0.7177,

0.7256, {0.1803, 0.2099, 0.2000, 0.2328〉;

y2 = 〈{0.4757, 0.5225, 0.5051, 0.5493, 0.4861, 0.5319,

0.5149, 0.5582, 0.4978, 0.5426, 0.5260, 0.5683,

0.5066, 0.5506, 0.5343, 0.5759, 0.5164, 0.5596,

0.5435, 0.5843, 0.5274, 0.5696, 0.5540, 0.5938},{0.2328, 0.3327, 0.2493, 0.3562, 0.2583, 0.3691,

0.2766, 0.3952}〉;

y3 = 〈{0.6687, 0.7355, 0.7154, 0.7728, 0.6776, 0.7426,

0.7231, 0.7789, 0.6882, 0.7511, 0.7322, 0.7862,

0.6850, 0.7485, 0.7294, 0.7840, 0.6935, 0.7553,

0.7367, 0.7898, 0.7036, 0.7634, 0.7454, 0.7967},{0.1452, 0.1883}〉;

y4 = 〈{0.5904, 0.6410, 0.6233, 0.6698, 0.6618, 0.7036,

0.6039, 0.6528, 0.6357, 0.6807, 0.6730, 0.7133},{0.1975, 0.2474, 0.2062, 0.2584, 0.2099, 0.2630,

0.2192, 0.2746}〉

Step 4: Calculate the closeness degree of the overall aggre-gated values to the positive-ideal solution.

By applying Equation (27), the cross-entropy G (yk)between the overall aggregated values yk and the positive-ideal alternative a+ can be calculated:

G (y1) = CE∗1

(y1, α

+) = 0.4204,G (y2) = 0.4707,

G (y3) = 0.2049,G (y4) = 0.3094.

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Table 2. Rankings obtained by different cross-entropy measures.

Closeness degree of each alternative

Cross-entropy a1 a2 a3 a4 Final ranking Best alternative Worst alternative

CE∗2 0.7907 0.5803 0.8155 0.7354 a3 � a1 � a4 � a2 a3 a2

CE∗3 0.6397 0.5568 0.7595 0.6996 a3 � a4 � a1 � a2 a3 a2

CE∗4 0.7579 0.5581 0.7615 0.6980 a3 � a1 � a4 � a2 a3 a2

CE∗5 0.6347 0.5673 0.7250 0.7182 a3 � a4 � a1 � a2 a3 a2

CE∗6 0.7086 0.5637 0.7470 0.6933 a3 � a1 � a4 � a2 a3 a2

CE∗7 0.8101 0.5793 0.8377 0.7708 a3 � a1 � a4 � a2 a3 a2

CE∗8 0.9237 0.5588 0.8957 0.9306 a4 � a1 � a3 � a2 a4 a2

CE∗9 0.6265 0.5306 0.6801 0.4790 a3 � a4 � a1 � a2 a3 a2

CE∗10 0.6037 0.5924 0.7002 0.6851 a3 � a4 � a1 � a2 a3 a2

Step 5: Rank the alternatives.Based on Step 4, G (y2) > G (y1) > G (y4) > G (y3) is

obtained, and the final ranking is a3 � a4 � a1 � a2. Thus,the best alternative is a3 and the worst alternative is a2.

Similarly, if the IHFWG operator is utilised in Step 3,then the cross-entropy G (yk) between the overall aggre-gated values yk and the ideal alternative a+ can be calcu-lated:

G (y1) = CE∗1

(y1, α

+) = 0.2659,

G (y2) = 0.5541,G (y3) = 0.2474,G (y4) = 0.3247.

Then G (y2) > G (y4) > G (y1) > G (y3), and the finalranking is a3 � a1 � a4 � a2. Again the best alternative isa3 and the worst alternative is a2.

If the IHFWA operator is utilised in Step 3, other cross-entropy measures are utilised in Step 4, and p = q = 2,then the corresponding results can be obtained, as shown inTable 3.

If the IHFWG operator is utilised in Step 3, other cross-entropy measures are utilised in Step 4, and p = q = 2,then the corresponding results are obtained, as shown inTable 4.

5.2. Analysis and discussion

In Table 3, it can be seen that if the cross-entropy measureCE∗

8 is utilised, then the final ranking is a4 � a1 � a3 � a2,which is quite different from the other results. However,in most cases, the final ranking is a3 � a4 � a1 � a2 ora3 � a1 � a4 � a2. The best alternative is a3, and the worstalternative is always a2; with the cross-entropy measuresCE∗

1 , CE∗2 , CE∗

4 , CE∗6 and CE∗

7 , a1 is ranked first, whereasa4 is superior to a1 by using the remaining cross-entropymeasures. The subtle differences in ranking are simply dueto the distinct information fusion mechanisms of thosecross-entropy measures. The results of Method II, whichare shown in Tables 3 and4, remain consistent with theranking results obtained in Method I. In order to calculatethe actual aggregation values of the alternatives, differentaggregation operators can be utilised. At the same time, twoaggregation operators are all based on different t-conormsand t-norms (Beliakov, Bustince, Goswami, Mukherjee, &Pal, 2011) and are used to deal with the different rela-tionships of the aggregated arguments, which can providemore choices for the decision-makers. In general, decision-makers can choose the IHFWA operator to calculate theactual aggregation values of the alternatives, because thiscalculation is simpler than that of the other aggregationoperators.

Table 3. Rankings obtained by the IHFWA operator.

Closeness degree of each alternative

Cross-entropy a1 a2 a3 a4 Final ranking Best alternative Worst alternative

CE∗2 0.7081 0.9615 0.5487 0.7340 a3 � a1 � a4 � a2 a3 a2

CE∗3 0.7231 0.8070 0.4628 0.6249 a3 � a4 � a1 � a2 a3 a2

CE∗4 0.5736 0.6584 0.4327 0.6440 a3 � a1 � a4 � a2 a3 a2

CE∗5 0.3684 0.5979 0.3238 0.4829 a3 � a1 � a4 � a2 a3 a2

CE∗6 0.4901 0.5667 0.4329 0.5535 a3 � a1 � a4 � a2 a3 a2

CE∗7 0.4452 0.6239 0.3598 0.3850 a3 � a4 � a1 � a2 a3 a2

CE∗8 0.2509 0.4541 0.2259 0.3998 a3 � a1 � a4 � a2 a3 a2

CE∗9 0.4205 0.5606 0.3411 0.3705 a3 � a4 � a1 � a2 a3 a2

CE∗10 0.4531 0.5434 0.4022 0.5171 a3 � a1 � a4 � a2 a3 a2

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Table 4. Rankings obtained by the IHFWG operator.

Closeness degree of each alternative

Cross-entropy a1 a2 a3 a4 Final ranking Best alternative Worst alternative

CE∗2 0.4590 0.4988 0.3340 0.3729 a3 � a4 � a1 � a2 a3 a2

CE∗3 0.3215 0.5021 0.4244 0.2008 a4 � a1 � a3 � a2 a4 a2

CE∗4 0.3859 0.5574 0.3367 0.5112 a3 � a1 � a4 � a2 a3 a2

CE∗5 0.6389 0.6955 0.5383 0.5850 a3 � a4 � a1 � a2 a3 a2

CE∗6 0.5122 0.5670 0.4453 0.4689 a3 � a4 � a1 � a2 a3 a2

CE∗7 0.4574 0.5912 0.3880 0.4056 a3 � a4 � a1 � a2 a3 a2

CE∗8 0.3589 0.5067 0.4105 0.3340 a4 � a1 � a3 � a2 a4 a2

CE∗9 0.5003 0.5416 0.4711 0.4415 a4 � a3 � a1 � a2 a4 a2

CE∗10 0.3598 0.5784 0.3327 0.4101 a3 � a1 � a4 � a2 a3 a2

From the analyses presented above, the proposedMCDM approaches based on cross-entropy measures ofIHFNs have the following advantages.

First, IHFNs used in this paper can express the evalua-tion information more flexibly. They can embed several val-ues in membership degrees and non-membership degreesof an IFN, and can retain the completeness of original dataor the inherent thoughts of decision-makers, which is theprerequisite of guaranteeing accuracy of final outcomes.

Second, the proposed intuitionistic hesitant cross-entropy measures are different from the existing hesitantcross-entropy measures that always involve the extensionswhose impact on the final solution may be considerable,because the proposed cross-entropy measures can includethe advantages of intuitionistic cross-entropy and hesitantcross-entropy, and overcome these shortcoming. This canavoid losing and distorting the preference information pro-vided, which makes the final results better correspond withreal decision-making problems.

Finally, the proposed approaches can provide a usefuland flexible way to efficiently facilitate the decision-makingprocess within an intuitionistic hesitant fuzzy environment.Moreover, the first approach could handle some specialcases where the weight information is not always availableand instead only partial knowledge of criteria weights maybe obtained as a group of linear constraints.

6. Conclusion

In this paper, MCDM problems where criteria weights areunknown and the evaluation values of criteria for an alter-native are given in the form of IHFNs were investigated.Besides, some cross-entropy measures of IHFNs were pro-posed, and one approach was developed in order to generatean optimal weight vector of criteria based on the cross-entropy. This approach utilised the criteria weights to getthe overall intuitionistic hesitant fuzzy values of alterna-tives and their ranking. Moreover, the approach using theproposed aggregation operators was developed for solvingMCDM problems with intuitionistic hesitant fuzzy infor-

mation, which was based on the determined weights of cri-teria. Furthermore, the illustrative example demonstratedthe validity and practicability of the proposed approaches.The prominent feature of two proposed methods is that theycan provide a useful and flexible way to efficiently facili-tate the decision-making process within an intuitionistichesitant fuzzy environment. In particular, the cross-entropymeasures of IHFNs that were introduced in the paper canovercome the shortcomings discussed in Section 2.2. Inthe future, the study of distance and similarity measure ofIHFSs will be continued.

AcknowledgementsThe authors thank the editors and anonymous reviewers for theirhelpful comments and suggestions.

FundingThis work was supported by the National Natural ScienceFoundation of China [grant number 71271218], [grant number71431006], [grant number 71221061]; the Humanities and So-cial Sciences Foundation of Ministry of Education of China[grant number14YJA630079]; the Science Foundation for Doc-tors of Hubei University of Automotive Technology [grant numberBK201405].

Notes on contributors

Juan-juan Peng received her M.Sc. degree in computationalmathematics from Wuhan University of Technology, China, in2007. She is currently a Ph.D. student in Business School, CentralSouth University; she is also a lecturer in School of Economicsand Management, Hubei University of Automotive Technology,China. Her current research focuses on decision-making theoryand application, risk management and control, and informationmanagement.

Jian-qiang Wang received the Ph.D. degree in management sci-ence and engineering from Central South University, Changsha,China, in 2005. He is currently a professor in Business School,Central South University. His current research interests includedecision-making theory and application, risk management andcontrol, and information management.

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International Journal of Systems Science 2349

Xiao-hui Wu received his M.Sc. degree in computing sciencefrom University of Portsmouth, UK, in 2006. He is currentlya Ph.D. student in Business School, Central South University;he is also a lecturer in School of Econmics and Management,Hubei University of Automotive Technology, China. His currentresearch focuses on decision-making theory and application, riskmanagement and control, and big data mining.

Hong-Yu Zhang received her Ph.D. degree in management sci-ence and engineering from Business School, Central South Uni-versity, Changsha, China, in 2009. She is currently an associatedprofessor in Business School, Central South University. Her re-search interests include the area of information management andits applications in production operations. Her current research fo-cuses on remanufacturing production management and decision-making theory.

Xiao-hong Chen received her Ph.D in Tokyo University of Tech-nology, Japan, in 1999. She is currently a professor at the School ofBusiness, Central South University, Changsha, China. Her currentresearch interests lie in the field of decision theory & method, de-cision support system, resource-saving and environment-friendlysociety. He has published in several journals, including DecisionSupport system, Expert Systems with Applications, etc.

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