THE ELECTRE I MULTI-CRITERIA DECISION-MAKING METHOD BASED ON HESITANT FUZZY SETS

The ELECTRE I Multi-Criteria Decision-Making Method

Based on Hesitant Fuzzy Sets

Na Chen

School of Economics and Management

Southeast University, Nanjing, Jiangsu 211189, China

School of Applied Mathematics

Nanjing University of Finance and Economics

Nanjing, Jiangsu 210023, Chinachenna [email protected]

Zeshui Xu*

Business School, Sichuan University

Chengdu 610064, P. R. [email protected]

Meimei Xia

School of Economics and Management

Tsinghua University

Beijing 100084, [email protected]

Published 28 November 2013

Hesitant fuzzy set (HFS), which allows the membership degree of an element to a set represented

by several possible values, is considered as a powerful tool to express uncertain information in

the process of multi-criteria decision making (MCDM) problems. In this paper, we develop ahesitant fuzzy ELECTRE I (HF-ELECTRE I) method and apply it to solve the MCDMproblem

under hesitant fuzzy environments. The new method is formulated using the concepts of hesi-

tant fuzzy concordance and hesitant fuzzy discordance which are based on the given score

function and deviation function, and employed to determine the preferable alternative.Numerical examples are provided to demonstrate the application of the proposed method, and

the in°uence of di®erent numbers of alternatives on outranking relations is analyzed based on a

derived sensitive parameter interval in which a change in the parameters has no e®ects on the setof the nonoutranked alternatives. The randomly generated numerical cases are also investigated

in the framework of the HF-ELECTRE I method. Furthermore, the outranking relations

obtained in the HF-ELECTRE I method with those derived from the aggregation operator-

based approach and the ELECTRE III and ELECTRE IV methods are discussed.

Keywords: Hesitant fuzzy set; ELECTRE; multi-criteria decision making; hesitant fuzzy

ELECTRE I; hesitant fuzzy concordance.

2000 Mathematics Subject Classi¯cation: 90B50, 91B06

*Corresponding author.

International Journal of Information Technology & Decision Making

Vol. 14, No. 3 (2015) 621–657

°c World Scienti¯c Publishing CompanyDOI: 10.1142/S0219622014500187

621

http://dx.doi.org/10.1142/S0219622014500187

1. Introduction

Multi-criteria decision making (MCDM) is an e®ective framework that can be used in

a systematic and quantitative way to deal with the decision-making problems in the

presence of a number of alternatives and several (con°icting) criteria.1–4 Outranking

as a major category of MCDM can be used to select which alternative is preferable,

incomparable or indi®erent by comparing an alternative with another one under each

criterion.5–8 The advantage of the outranking method9–16 lies in that it is able to take

purely ordinal scales into account, and in that indi®erence and preference thresholds

can be considered when modeling the imperfect knowledge of data.

The ELECTRE (ELimination Et Choix Traduisant la Realit�e) method plays a

signi¯cant role in the group of outranking. It was ¯rst set forth by Benayoun

et al.,17,18 and referred as ELECTRE-I, which, along with its improvements,

ELECTRE Iv and ELECTRE Is,19,20 constitute the so-called current version of the

ELECTRE methods for choice problems. The approach is further developed as

the ELECTRE-II, III and IV methods aiming at dealing with ranking problems, and

as the ELECTRE-A and ELECTRE-TRI methods, devised to devote to sorting

problems. See Figueira et al.21 for more details on these developments. Various

forms of the ELECTRE methods have been widely applied to many ¯elds,21,22 for

example, project selection,23,24 transportation,25 environment or water manage-

ment,26 energy,27,28 agriculture and forest management.29,30

Furthermore, in the ¯eld of group decision making22,31–45 the ELECTRE method

has been applied to it by many authors (see, e.g., Refs. 46–51). Dias and Clímaco33

computed ELECTRE's credibility indices under partial information corresponding

to the case when the decision makers are unsure which values each parameter should

take, which may result from insu±cient, imprecise or contradictory information, as

well as from di®erent preferences among a group of decision makers. Mousseau

et al.40 proposed new techniques to solve inconsistencies among constraints on the

parameters and presents an application to the aggregation/disaggregation procedure

for the ELECTRE TRI method. Leyva and Fernandez49 presented an extension

of the ELECTRE III multi-criteria outranking methodology to assist a group of

decision makers with di®erent value systems to achieve a consensus on a set of

possible alternatives. Fernandez and Olmedo46 proposed an agent model based on

ideas of concordance and discordance for group ranking problems. Kaya and Kah-

raman22 illustrated an environmental impact assessment methodology based on an

integrated fuzzy AHP–ELECTRE approach in the context of urban industrial

planning. In addition, the ELECTRE method has recently been employed52–60 to

handle the case that the evaluation information of the decision-making problems

may be uncertain61 and fuzzy, which is caused by the limited knowledge of decision

makers and the fuzzy nature of the real world. For instance, Hatami-Marbini and

Tavana53 proposed the extended ELECTRE I method to account for the uncertain,

imprecise and linguistic assessments provided by a group of decision makers and used

the mean value in aggregating the opinions of all decision makers. Wu and Chen59

utilized the approach to solve the MCDM problems in Atanassov's intuitionistic

622 N. Chen, Z. Xu & M. Xia

fuzzy (A-IF) environments. Vahdani et al.60 applied it to assimilate the concepts of

interval weights and data.

Torra and Narukawa have recently put forward the concept of hesitant fuzzy sets

(HFSs),61,62 which is an extension of fuzzy set,63 to describe the situation that permits

the membership of an element to a given set having a few di®erent values, which can

arise in a group decision-making problem. For example, two decision makers discuss

themembership degree of an element x to a setA; one wants to assign 0.3 and the other

0.6. They cannot reach consistency each other. For such a circumstance, the degrees

can be represented by a hesitant fuzzy element (HFE) f0.3, 0.6g. Because of the

importance of HFS theory in applications, recent work has been done about the

aggregation operators of HFSs,64–66 the distance and entropy measures for HFSs,67–69

and hesitant fuzzy preference relations.70 The similarities and di®erences between

HFSs and other generalized fuzzy sets including intuitionistic fuzzy sets,71 type-2 fuzzy

sets,72 type-n fuzzy sets72 and fuzzy multisets73 have been discussed in Refs. 61 and 62.

When performing group decision-making, in frequently used approaches the

opinions of the decision makers for each criterion and alternative are ¯rst aggregated

and only a set of average criteria can be obtained, which implies a valid common

decision. Thus these aggregation methods do not re°ect di®erences between indi-

vidual decision makers. Bana e Costa31 and Kim et al.36,37 corrected the average sum

aggregation of the performances in the framework of additive value functions. Dias

and Clímaco35 designed a decision support system aiming at not imposing an

aggregated model from the individual ones, but to re°ect to each member the con-

sequences of his/her inputs. Speci¯cally, they dealt with the aggregation of multi-

criteria performances by means of an additive value function under imprecise

information using VIP analysis. The ELECTRE TRI methods14,34,40,74 were also

employed for groups with imprecise information on parameter values.

As has been pointed out, the HFS introduced can incorporate all possible opinions

of the group members and thus provides an intuitionistic description on the di®er-

ences among group members. In other words, when seeking a consensual solution in

the process of group decision making, introducing HFSs can avoid aggregation,61

that is, it does not need to force average preference on group members. The aim of

the present work is to investigate the ELECTRE I method under hesitant fuzzy

environments called hesitant fuzzy ELECTRE I (HF-ELECTRE I) method, which is

primarily introduced to aid the process of group decision making. Moreover, in our

method, we de¯ne the concept of deviation function for HFEs to take the di®erences

of opinions among group members into account.

The formulation of the HF-ELECTRE I method contains a construction of out-

ranking relations between actions. The construction is based on a classi¯cation of

di®erent types of hesitant fuzzy concordance andhesitant fuzzy discordance sets, which

is completed byde¯ning the concepts of score function anddeviation function forHFEs.

This article is organized as follows. Section 2 brie°y describes the concepts related to

HFSs and the ELECTRE method. The HF-ELECTRE I method is presented and

discussed in Sec. 3. Numerical examples are given in Sec. 4. A sensitive analysis for our

The ELECTRE I MCDM Method Based on Hesitant Fuzzy Sets 623

numerical example and a comparison with the aggregation operator approach and the

ELECTRE III and ELECTRE IV methods for MCDM are demonstrated in Sec. 5.

Conclusions and future research challenges are summarized in Sec. 6.

2. Preliminaries

In this section, we introduce some basic concepts related to HFSs, including the

distance measure, the score function and the deviation function of HFSs, and the

ELECTRE method.

2.1. Hesitant fuzzy sets

De¯nition 1. 61,62 Let X be a reference set, a HFS A on X is de¯ned in terms of a

function hAðxiÞ when applied to X returns a subset of ½0; 1�. To be easily understood,

Xia and Xu64 expressed the HFS by a mathematical symbol:

A ¼ f< xi; hAðxiÞ > jxi 2 X ; i ¼ 1; . . . ; ng; ð1Þwhere hAðxiÞ is a set of some di®erent values in ½0; 1�, representing the possible

membership degrees of the element xi 2 X to A. For convenience, we call hAðxiÞa HFE.64

Example 1. Let X ¼ fx1; x2; x3g be a reference set, hAðx1Þ ¼ f0:2; 0:4; 0:5g,hAðx2Þ ¼ f0:3; 0:4g and hAðx3Þ ¼ f0:3; 0:2; 0:5; 0:6g be the HFEs of xiði ¼ 1; 2; 3Þto a set A, respectively. Then A can be considered as a HFS, i.e.,

A ¼ f< x1; f0:2; 0:4; 0:5g >;< x2; f0:3; 0:4g >;< x3; f0:3; 0:2; 0:5; 0:6g >g:To compare di®erent HFEs, in our previous work,66 we de¯ned the concept of the

score function.

De¯nition 2. 66 For a HFE h, sðhÞ ¼ 1lh

P�2h� is called the score function of h,

where lh is the number of the elements in h. For two HFEs h1 and h2, if sðh1Þ > sðh2Þ,then h1 is superior to h2, denoted by h1 � h2; if sðh1Þ ¼ sðh2Þ, then h1 is indi®erent to

h2, denoted by h1 � h2.

It is noted that sðhÞ is directly related to the average value of all elements in h

expressing the average opinion of decision makers. The higher the average value, the

bigger the score sðhÞ, and thus, the better the HFE h. But De¯nition 2 does not take

into account the situation that two HFEs h1 and h2 have the same score, but their

deviation degrees may be di®erent. The deviation degree of all elements with respect

to the average value in a HFE re°ects how these elements agree with each other; that

is, they have a higher consistency. To better represent this issue, below we de¯ne the

concept of deviation function:

De¯nition 3. For a HFE h, we de¯ne the deviation degree �ðhÞ of h as follows:

�ðhÞ ¼ 1

lh

X�2h

ð� � sðhÞÞ2" #1

2

: ð2Þ


As it can be seen that �ðhÞ is just conventional standard variance in statistics, which

re°ects the deviation degree between all values in a HFE and their average value.

Based on sðhÞ and �ðhÞ, we can give a comparison law between HFEs.

De¯nition 4. Let h1 and h2 be two HFEs, sðh1Þ and sðh2Þ the scores of h1 and h2,

respectively, and �ðh1Þ and �ðh2Þ the deviation degrees of h1 and h2, respectively.

Then

. If sðh1Þ < sðh2Þ, then h1 � h2;

. If sðh1Þ ¼ sðh2Þ, then:(i) If �ðh1Þ ¼ �ðh2Þ, then h1 � h2;

(ii) If �ðh1Þ < �ðh2Þ, then h1 � h2;

(iii) If �ðh1Þ > �ðh2Þ, then h1 � h2.

Example 2. Let h1 ¼ f0:2; 0:3; 0:5; 0:8g, h2 ¼ f0:4; 0:6; 0:8g and h3 ¼ f0:3; 0:45;0:6g be three HFEs, respectively. Since sðh1Þ ¼ 0:45, sðh2Þ ¼ 0:6 and sðh3Þ ¼ 0:45,

then sðh1Þ ¼ sðh3Þ < sðh2Þ, and we get h1 � h2 and h3 � h2. Furthermore, �ðh1Þ ¼0:2291 and �ðh3Þ ¼ 0:1225, thus, �ðh1Þ > �ðh3Þ, and we have h1 � h3. Therefore,

h1 � h3 � h2.

2.2. Distance measures for HFEs

It is noted that the number of values for di®erent HFEs may be di®erent, and the

values are usually out of order, then we can arrange them in any order for conven-

ience. Suppose that we arrange the values of a HFE h in an increasing order, and let

h�ðiÞði ¼ 1; 2; . . . ; nÞ be the ith smallest value in h. To calculate the distance between

two HFEs � and �, let l ¼ maxfl�; l�g, where l� and l� are respectively the numbers

of values in HFEs � and �. To operate correctly, we give the following regulation.

When l� 6¼ l�, we can make them equivalent through adding values to the HFE that

has less number of elements. In terms of pessimistic principles, the smallest element

can be added while the opposite case will be adopted following optimistic principles.

In our work, we adopt the latter. Speci¯cally, if l� < l�, then � should be extended by

adding the maximum value in it until it has the same length with �; if l� > l�, then �

should be extended by adding the maximum value in it until it has the same length

with �.

Based on the above operational principles, we de¯ne the distance measure, i.e.,

Eq.(3), in previous work66 and Eq. (4) in the present work:

d1ð�; �Þ ¼1

l

Xl

i¼1

��ðiÞ � ��ðiÞ�� ; ð3Þ

d2ð�; �Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

l

Xl

i¼1

��ðiÞ � ��ðiÞ�� 2

vuut : ð4Þ


These two formulas can be considered as the extensions of the well-known Hamming

distance and Euclidean distance under hesitant fuzzy environments. They satisfy the

following properties:

(i) 0 � dð�; �Þ � 1;

(ii) dð�; �Þ ¼ 0 if and only if ��ðiÞ ¼ ��ðiÞ, for each i, i ¼ 1; 2; . . . ; l;

(iii) dð�; �Þ ¼ dð�; �Þ;(iv) For three HFEs �, � and �, which have the same length l, if ��ðiÞ � ��ðiÞ � ��ðiÞ

for each i, i ¼ 1; 2; . . . ; l, then dð�; �Þ � dð�; �Þ and dð�; �Þ � dð�; �Þ.

2.3. The main features of the ELECTRE method

The prominent features of the ELECTRE method include four binary relations, the

preference modeling through outranking relations, and the concepts of concordance

and discordance.

De¯nition 5.7 To compare two actions a and b, binary relations are de¯ned on the

set A. For a pair ða; bÞ 2 A� A, let:

. P denotes the strict preference relation; and aPbmeans that \a is strictly preferred

to b";

. I denotes the indi®erence relation, and aIb means that \a is indi®erent to b";

. Q denotes the weak preference relation, and aQbmeans that \a is weakly preferred

to b, which expresses hesitation between indi®erence (I Þ and preference (P);

. R denotes the incomparability relation, and aRb means that \a is not comparable

to b". It corresponds to an absence of clear and positive reasons that would justify

any of the three preceding relations.

De¯nition 6.7 Modeling preference in the ELECTRE method is via the

comprehensive binary outranking relation S , whose meaning is \at least as good

as"; in general, S ¼ P [Q [ I . Considering two actions ða; bÞ 2 A� A, four cases

appear:

(i) aSb and not bSa, i.e., aPb ða is strictly preferred to bÞ;(ii) bSa and not aSb, i.e., bPa ðb is strictly preferred to aÞ;(iii) aSb and bSa, i.e., aIb ða is indi®erent to bÞ;(iv) Not aSb and not bSa, i.e., aRb ða is incomparable to bÞ.De¯nition 7.7 All outranking-based methods rely on the concepts of concordance

and discordance which represent, in a certain sense, the reasons for and against an

outranking situation:

(i) The concordance concept: To validate an outranking aSb, a su±cient majority

of criteria in favor of this assertion must occur;

(ii) The discordance concept: The assertion aSb cannot be validated if a minority of

criteria is strongly against this assertion.


3. ELECTRE I Method Based on HFSs

In this section, we extend the ELECTRE I method to develop a new method in order

to solve the MCDM problems under hesitant fuzzy environments.

3.1. Construction of hesitant fuzzy decision matrix

AMCDM problem can be expressed as a decision matrix whose elements indicate the

evaluation information of all alternatives with respect to each criterion. We con-

struct here a hesitant fuzzy decision matrix, whose elements are HFEs, which are

given due to the fact that the membership degree of the alternative Ai satisfying the

criterion xj may originate from a doubt between a few di®erent values.

Consider a MCDM problem where there is a discrete set of m alternatives,

A ¼ fA1;A2; . . . ;Amg. Let X ¼ fx1; x2; . . . ; xng be the set of all criteria. A HFS Ai

de¯ned on X is given by Ai ¼ f< xj ; hAiðxjÞ > jxj 2 Xg, where hAi

ðxjÞ ¼f�j� 2 hAi

ðxjÞ; 0 � � � 1g, i ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; n. hAiðxjÞ indicates possible

membership degrees of the ith alternative Ai under the jth criterion xj and can be

expressed as a HFE hij . The hesitant fuzzy decision matrix, H , can be written as:

H ¼

h11 h12 � � � h1nh21 h22 � � � h2n

..

. ... . .

. ...

hm1 hm2 � � � hmn

266664

377775: ð5Þ

Considering that each criterion has di®erent importance, the weight vector of all

criteria, W , given by the decision makers, is de¯ned via:

W ¼ ðw1;w2; . . . ;wnÞT; ð6Þwhere 0 � wj � 1,

Pnj¼1 wj ¼ 1, and wj is the importance degree of the criterion xj .

3.2. Concordance and discordance sets

The ELECTRE method is composed of the construction and exploitation of one or

several outranking relation(s).21 The construction is based on a comparison of each

pair of actions on the criteria, through which concordance and discordance indices

are obtained and they are further used to analyze the outranking relations among

di®erent alternatives.

In the traditional ELECTRE methods, each criterion in di®erent alternatives

can be divided into two di®erent subsets: concordance set and discordance set. The

former is composed of all criteria for which Ak is preferred to Al , and the latter is

the complementary subset. However, in hesitant fuzzy environments, according to

the concepts of score function and deviation function, we can compare di®erent

alternatives on the criteria and classify di®erent types of hesitant fuzzy con-

cordance (discordance) sets as the hesitant fuzzy concordance (discordance) set and

the weak hesitant fuzzy concordance (discordance) set. A better alternative has the


higher score or the lower deviation degree in cases where the alternatives have the

same score.

As mentioned before, the set of all criteria is denoted as X ¼ fx1; x2; . . . ; xng and a

HFS Ai on X is given by Ai ¼ f< xj ; hAiðxjÞ > jxj 2 Xg, where hAi

ðxjÞ ¼ f�j� 2hAi

ðxjÞ; 0 � � � 1g ði ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; nÞ indicating all possible member-

ship degrees of the ith alternative Ai under the jth criterion xj , expressed by a

HFE hij .

For each pair of the alternatives Ak and Alðk; l ¼ 1; 2; . . . ;m and k 6¼ lÞ, thehesitant fuzzy concordance set JCkl

of Ak and Al is the sum of all those criteria where

the performance of Ak is superior to Al . The hesitant fuzzy concordance set JCklcan

be formulated as follows:

JCkl¼ fj sðhkjÞ

�� sðhljÞ and �ðhkjÞ < �ðhljÞg; ð7Þ

where J ¼ fjjj ¼ 1; 2; . . . ; ng represents a set of subscripts of all criteria. The weak

hesitant fuzzy concordance set JC 0klis de¯ned by

JC 0kl¼ fj sðhkjÞ

�� sðhljÞ and �ðhkjÞ �ðhljÞg: ð8Þ

The main di®erence between JCkland JC 0

kllies in the deviation function. The lower

deviation values re°ect that the opinions of decision makers have a higher consist-

ency degree. So JCklis more concordant than JC 0

kl.

Similarly, the hesitant fuzzy discordance set JDkl, which is composed of all criteria

for which Ak is inferior to Al , can be formulated as follows:

JDkl¼ fj sðhkjÞ

�� < sðhljÞ and �ðhkjÞ �ðhljÞg: ð9ÞIf sðhkjÞ < sðhljÞ and the deviation value �ðhkjÞ < �ðhljÞ, then we de¯ne this cir-

cumstance as the weak hesitant fuzzy discordance set JD 0kl, which is expressed by

JD 0kl¼ fj sðhkjÞ

�� < sðhljÞ and �ðhkjÞ < �ðhljÞg: ð10ÞObviously, JDkl

is more discordant than JD 0kl.

3.3. HF-ELECTRE I method

In this subsection, we construct the corresponding matrices for di®erent types of the

hesitant fuzzy concordance sets and the hesitant fuzzy discordance sets, and propose

the hesitant fuzzy ELECTRE I (HF-ELECTRE I) method. The hesitant fuzzy

concordance index is the ratio of the sum of the weights related to criteria in the

hesitant fuzzy concordance sets to that of all criteria. It can be computed by the

values of the hesitant fuzzy concordance set in the HF-ELECTRE I method. The

concordance index ckl of Ak and Al in the HF-ELECTRE I method is de¯ned as:

ckl ¼!C �P

j2JCklwj þ !C 0 �P

j2JC 0kl

wjPnj¼1 wj

¼ !C �Xj2JCkl

wj þ !C 0 �Xj2JC 0

kl

wj ; ð11Þ


where wj is the weight of criteria andPn

j¼1 wj ¼ 1 for the normalized weight vector of

all criteria. !C and !C 0 are respectively the weights of the hesitant fuzzy concordance

sets and the weak hesitant fuzzy concordance sets depending on the attitudes

of the decision makers. The ckl re°ects the relative importance of Ak with respect to

Al . Obviously, 0 � ckl � 1. A large value of ckl indicates that the alternative Ak

is superior to the alternative Al . We can thus construct the asymmetrical

hesitant fuzzy concordance matrix C using the obtained value of the indices

cklðk; l ¼ 1; 2; . . . ;m; l 6¼ kÞ, namely,

C ¼

� � � � c1l � � � c1ðm�1Þ c1m

..

. . .. ..

. . .. ..

. ...

ck1 � � � ckl � � � ckðm�1Þ ckm

..

. . .. ..

. . .. ..

. ...

cm1 � � � cml � � � cmðm�1Þ �

266666664

377777775: ð12Þ

Di®erent from the hesitant fuzzy concordance index, the hesitant fuzzy discordance

index is re°ective of relative di®erence of Ak with respect to Al in terms of discor-

dance criteria. The discordance index is de¯ned via:

dkl ¼maxj2JDkl

[JD 0kl

f!D � dðwjhkj ;wjhljÞ; !D 0 � dðwjhkj ;wjhljÞgmaxj2Jdðwjhkj ;wjhljÞ

; ð13Þ

where !D and !D 0 denote the weights of the hesitant fuzzy discordance set and the

weak discordance set respectively depending on the decision makers' attitudes, and

dðwjhkj ;wjhljÞ is the distance measure de¯ned in Eqs. (3) or (4) .

Hesitant fuzzy discordance matrix is established by the hesitant fuzzy discordance

index for all pairwise comparisons of alternatives:

D ¼

� � � � d1l � � � d1ðm�1Þ d1m

..

. . .. ..

. . .. ..

. ...

dk1 � � � dkl � � � dkðm�1Þ dkm

..

. . .. ..

. . .. ..

. ...

dm1 � � � dml � � � dmðm�1Þ �

2666666664

3777777775: ð14Þ

As seen in Eqs. (11) and (13), the elements of C di®er substantially from those of D,

making the two matrices have complementary relationship; that is, the matrix C

represents the weights resulted from hesitant fuzzy concordance indices, whereas the

asymmetrical matrix D re°ects the relative di®erence of wjhij for all hesitant fuzzy

discordance indices. Note that discordance matrix re°ects limited compensation

between alternatives, that is, when the di®erence of two alternatives on a criterion

arrives at a certain extent, compensation of the loss on a given criterion by a gain on

another one may not be acceptable for the decision makers.21 Because of this reason


discordance matrix is established di®erently from the case of the establishment of

concordance matrix.

The hesitant fuzzy concordance dominance matrix can be calculated according to

the cut-level of hesitant fuzzy concordance indices. If the hesitant fuzzy concordance

index ckl of Ak relative to Al is over a minimum level, then the superiority degree of

Ak to Al increases. The hesitant fuzzy concordance level can be de¯ned as the

average of all hesitant fuzzy concordance indices in the following manner:

�c ¼Xmk¼1

Xml¼1;l 6¼k

cklmðm � 1Þ : ð15Þ

Based on the concordance level, the concordance dominance matrix F (i.e., a Boolean

matrix) can be expressed as:

F ¼

� � � � f1l � � � f1ðm�1Þ f1m

..

. . .. ..

. . .. ..

. ...

fk1 � � � fkl � � � fkðm�1Þ fkm

..

. . .. ..

. . .. ..

. ...

fm1 � � � fml � � � fmðm�1Þ �

26666666664

37777777775

ð16Þ

whose elements satisfy

fkl ¼ 1; if ckl �c

fkl ¼ 0; if ckl < �c

(; ð17Þ

where each element 1 in the matrix F indicates that an alternative is preferable to the

other one.

Likewise, the elements of the hesitant fuzzy discordance matrix are also measured

by the discordance level �d, which can be de¯ned as the average of the elements in the

hesitant fuzzy discordance matrix:

�d ¼Xmk¼1

Xml¼1;l 6¼k

dklmðm � 1Þ : ð18Þ

Then, based on the discordance level, the discordance dominance matrix G can be

constructed as:

G ¼

� � � � g1l � � � g1ðm�1Þ g1m

..

. . .. ..

. . .. ..

. ...

gk1 � � � gkl � � � gkðm�1Þ gkm

..

. . .. ..

. . .. ..

. ...

gm1 � � � gml � � � gmðm�1Þ �

26666666664

37777777775; ð19Þ


where

gkl ¼ 1; if dkl � �d

gkl ¼ 0; if dkl > �d:

(ð20Þ

The elements of the matrix G measure the degree of the discordance. Hence, the

discordant statement would be no longer valid if the element value dkl � �d. That is to

say, the elements of the matrix G, whose values are 1, show the dominant relations

among the alternatives.

Then the aggregation dominance matrix E is constructed from the elements of the

matrix F and the matrix G through the following formula:

E ¼ F G; ð21Þwhere each element ekl of E is derived with

ekl ¼ fkl � gkl : ð22ÞFinally, we exploit the outranking relations aiming at elaborating recommendations

from the results obtained in previous construction of the outranking relations.21 By

means of binary relations presented in Sec. 2, we can construct a graph, from which

the preferable alternative is selected. Speci¯c details are depicted in Fig. 1. If ekl ¼ 1,

then Ak is strictly preferred to Al or Ak is weakly preferred to Al . If ekl ¼ 1

and elk ¼ 1, then Ak is indi®erent to Al . If ekl ¼ 0 and elk ¼ 0, then Ak is incom-

parable to Al .

We summarize the proposed HF-ELECTRE I method in the following steps and

also display it in Fig. 2.

Step 1. Construct the hesitant fuzzy decision matrix. A group of decision makers

determine the relevant criteria of the potential alternatives and give the evaluation

information in the form of HFEs of the alternative with respect to the criteria. They

also determine the importance vector W ¼ ðw1;w2; . . . ;wnÞT for the relevant cri-

teria, and the relative weight vector ! ¼ ð!C ; !C 0 ; !D; !D 0 Þ of di®erent types of thehesitant fuzzy concordance sets and the hesitant fuzzy discordance sets.

Step 2. Calculate the score function and the deviation function of each evaluation

information of alternatives on the criteria in the form of HFEs according to De¯-

nition 2 and Eq. (2).

Step 3. Construct the hesitant fuzzy concordance sets and the weak hesitant fuzzy

concordance sets using Eqs. (7) and (8).

lAk lA RA

kl kle andekA

lAkAk lA IA

kl lke andelAkAk l k lA PA or A QA

1kle =

Fig. 1. A graphical representation of the binary relations.


Step 4. Construct the hesitant fuzzy discordance sets and the weak hesitant fuzzy

discordance sets using Eqs. (9) and (10).

Step 5. Calculate the hesitant fuzzy concordance indexes using Eq. (11) and obtain

the hesitant fuzzy concordance matrix using Eq. (12).

Step 6. Calculate the hesitant fuzzy discordance indexes using Eq. (13) and obtain

the hesitant fuzzy discordance matrix using Eq. (14).

Step 7. Identify the concordance dominance matrix using Eqs. (15)–(17).

Step 8. Identify the discordance dominance matrix using Eqs. (18)–(20).

Step 9. Construct the aggregation dominance matrix using Eqs. (21) and (22).

Step 10. Draw a decision graph and choose the preferable alternative.

Step 11. End.

Step 1Construct the hesitant fuzzy decision matrix

Step 2Calculate the score function and deviation function

Step 3Construct the hesitant fuzzy concordance sets and the weak

hesitant fuzzy concordance sets

Step 4Construct the hesitant fuzzy discordance sets and the weak

hesitant fuzzy discordance sets

Step 5Calculate the hesitant fuzzy concordance indexes and obtain the hesitant fuzzy

concordance matrix

Step 6Calculate the hesitant fuzzy discordance indexes and obtain the hesitant fuzzy

discordance matrix

Step 7 Identify the concordance dominance matrix

Step 8Identify the discordance dominance matrix

Step 9Construct the aggregation dominance matrix

Step 10Draw a decision graph and choose the preferable alternative

Fig. 2. The procedure of the HF-ELECTRE I method.


4. Numerical Example

In this section, we use a numerical example to illustrate the details of the proposed

HF-ELECTRE I method.

Example 3.64,75 The enterprise's board of directors, which includes ¯ve members, is

to plan the development of large projects (strategy initiatives) for the following ¯ve

years. Four possible projects Aiði ¼ 1; 2; 3; 4Þ have been marked. It is necessary to

compare these projects to select themost important of them aswell as order them from

the point of view of their importance, taking into account four attributes suggested by

the balanced scorecard methodology76 (it should be noted that all of them are of the

maximization type): x1: ¯nancial perspective, x2: the customer satisfaction, x3:

internal business process perspective, and x4: learning and growth perspective.

In order to avoid psychic contagion, the decision makers are required to provide

their preferences in anonymity. Suppose that the weight vector of the attributes is

w ¼ ð0:2; 0:3; 0:15; 0:35ÞT, and the hesitant fuzzy decision matrix is presented in the

form of HFEs in Table 1.

Step 1. The hesitant fuzzy decision matrix and the weight vector of the attributes

have been given above. The decision makers also give the relative weights of the

hesitant fuzzy concordance sets (weak hesitant fuzzy concordance sets) and the

hesitant fuzzy discordance sets (weak hesitant fuzzy discordance sets) respectively as

follows:

! ¼ ð!C ; !C 0 ; !D; !D 0 Þ ¼ 1;2

3; 1;

2

3

� �:

Step 2. Calculate the score and the deviation values of each evaluation information

of alternatives on the criteria in the form of HFEs according to De¯nition 2 and

Eq. (2) represented in Tables 2 and 3:

Table 1. Hesitant fuzzy decision matrix for Example 3.

x1 x2 x3 x4

A1 f0.2,0.4,0.7g f0.2,0.6,0.8g f0.2,0.3,0.6,0.7,0.9g f0.3,0.4,0.5,0.7,0.8gA2 f0.2,0.4,0.7,0.9g f0.1,0.2,0.4,0.5g f0.3,0.4,0.6,0.9g f0.5,0.6,0.8,0.9gA3 f0.3,0.5,0.6,0.7g f0.2,0.4,0.5,0.6g f0.3,0.5,0.7,0.8g f0.2,0.5,0.6,0.7gA4 f0.3,0.5,0.6g f0.2,0.4g f0.5,0.6,0.7g f0.8,0.9g

Table 2. Score values obtained by the

score function.

x1 x2 x3 x4

A1 0.4333 0.5333 0.54 0.54A2 0.55 0.3 0.55 0.7

A3 0.525 0.425 0.575 0.5

A4 0.4667 0.3 0.6 0.85


The data in Tables 2 and 3 are obtained from the data of Table 1. For example,

sðh21Þ ¼0:2þ 0:4þ 0:7þ 0:9

4¼ 0:55;

�ðh21Þ ¼1

4ðð0:2� 0:55Þ2 þ ð0:4� 0:55Þ2 þ ð0:7� 0:55Þ2

�

þð0:9� 0:55Þ2Þ�1

2 ¼ 0:2693:

Step 3. Construct the hesitant fuzzy concordance sets and the weak hesitant fuzzy

concordance sets:

JC ¼� � 4 �3; 4 � 4 �1; 3 2; 3 � �1; 3; 4 2; 3; 4 3; 4 �

2664

3775; JC 0 ¼

� 2 2 2

1 � 1 1; 2

� � � 1; 2� � � �

2664

3775:

For example, since

sðh14Þ ¼ 0:54 > sðh34Þ ¼ 0:5 and �ðh14Þ ¼ 0:1855 < �ðh34Þ ¼ 0:1871

then JC13¼ f4g. Since

sðh12Þ ¼ 0:5333 > sðh32Þ ¼ 0:425 and �ðh12Þ ¼ 0:2494 > �ðh32Þ ¼ 0:1479

then JC 013¼ f2g.

Step 4. Construct the hesitant fuzzy discordance sets and the weak hesitant fuzzy

discordance sets:

JD ¼� 3; 4 1; 3 1; 3; 4

� � 2; 3 3; 4

4 4 � 3; 4� � � �

2664

3775; JD 0 ¼

� 1 � �2 � � �2 1 � �2 1 1; 2 �

2664

3775:

For example, since

sðh13Þ ¼ 0:54 < sðh33Þ ¼ 0:575; �ðh13Þ ¼ 0:2577 > �ðh33Þ ¼ 0:1920;

sðh11Þ ¼ 0:4333 < sðh31Þ ¼ 0:525; �ðh11Þ ¼ 0:2055 > �ðh31Þ ¼ 0:1479

then JD13¼ f1; 3g.

Table 3. Deviation values obtained by

the deviation function.

x1 x2 x3 x4

A1 0.2055 0.2494 0.2577 0.1855

A2 0.2693 0.1581 0.2291 0.1581

A3 0.1479 0.1479 0.1920 0.1871

A4 0.1247 0.1 0.0816 0.05


Step 5. Calculate the hesitant fuzzy concordance indexes and the hesitant fuzzy

concordance matrix:

C ¼ ðcklÞ4�4 ¼

� 0:2 0:55 0:2

0:6333 � 0:4833 0:3333

0:35 0:45 � 0:3333

0:7 0:8 0:5 �

266664

377775:

For example, c13 ¼ !C � w4 þ !C 0 � w2 ¼ 1� 0:35þ 23 � 0:3 ¼ 0:55.

Step 6. Calculate the hesitant fuzzy discordance indexes and the hesitant fuzzy

discordance matrix:

D ¼ ðdklÞ4�4 ¼

� 0:7778 0:3429 1

0:6667 � 0:5357 1

0:6667 1 � 1

0:3361 0:3265 0:1143 �

266664

377775:

For instance,

d31 ¼maxj2JD31

[JD 031

f!D � dðwjh3j ;wjh1jÞ; !D 0 � dðwjh3j ;wjh1jÞgmaxj2Jdðwjh3j ;wjh1jÞ

¼ maxf1� dðw4h34;w4h14Þ; 23 � dðw2h32;w2h12Þg0:0525

¼ maxf1� 0:028; 23 � 0:0525g0:0525

¼ 0:035

0:0525¼ 0:6667;

where

dðw1h31;w1h11Þ ¼1

4� 0:2� j0:2� 0:3j þ j0:4� 0:5j þ j0:7� 0:6jð

þj0:7� 0:7jÞ ¼ 0:015;

dðw2h32;w2h12Þ ¼ 0:0525; dðw3h33;w3h13Þ ¼ 0:018; dðw4h34;w4h14Þ¼ 0:028:

Step 7. Calculate the hesitant fuzzy concordance level and identify the concordance

dominance matrix, respectively:

�c ¼Xmk¼1

Xml¼1;l 6¼k

cklmðm � 1Þ ¼ 0:4611; F ¼

� 0 1 0

1 � 1 0

0 0 � 0

1 1 1 �

266664

377775:


Step 8. Calculate the hesitant fuzzy discordance level and identify the discordance

dominance matrix, respectively:

�d ¼Xmk¼1

Xml¼1;l 6¼k

dklmðm � 1Þ ¼ 0:6472; G ¼

� 0 1 0

0 � 1 0

0 0 � 0

1 1 1 �

2664

3775:

Step 9. Construct the aggregation dominance matrix:

E ¼ F G ¼� 0 1 0

0 � 1 0

0 0 � 0

1 1 1 �

2664

3775:

Step 10. As it can be seen from the aggregation dominance matrix, A1 is preferred

to A3, A2 is preferred to A3 and A4 is preferred to A1, A2 and A3. Hence, A4 is the

best alternative. The results are depicted in Fig. 3.

5. Discussion

5.1. Comparison with the aggregation operator-based approach

The numerical example used here was also considered by Xia and Xu,64 who

suggested the aggregation operators to fuse the hesitant fuzzy information and made

the ranking of projects also by carrying out score functions for the same numerical

example as ours. We ¯nd that both the average aggregation operators and the HF-

ELECTRE I method give a consistent result, which clearly illustrates the validity of

our proposed approach. These two proposed approaches are complementary when

applied to solve di®erent types of the MCDM problems. When the number of criteria

in a MCDM problem is not larger than 4, the aggregation operator-based approach is

a suitable tool because of the simple solving steps. However, when the number of

criteria exceeds 4 for hesitant fuzzy information, which often appears in some actual

MCDM problems, the approach might encounter a barrier in applications because of

the need for tremendous computation. For such cases, the proposed HF-ELECTRE I

3A

4A

1A

2A

Fig. 3. Decision graph of Example 3.


method is particularly useful, which is logically simple and demands less compu-

tational e®orts. In what follows, we shall illustrate its application for the selection of

investments where the number of criteria arrives at 6.

Example 4.77 Assume that an enterprise wants to invest money in another country.

A group composing of four decision makers in the enterprise considers ¯ve possible

investments:A1: invest in the Asian market, A2: invest in the South American

market, A3: invest in the African market, A4: invest in the three continents and A5:

do not invest in any continent.

When analyzing the investments, the decisionmakers have considered the following

general characteristics: x1: risks of the investment, x2: bene¯ts in the short term, x3:

bene¯ts in the midterm, x4: bene¯ts in the long term, x5: di±culty of the investment,

and x6: other aspects. After a careful analysis of these characteristics, the decision

makers have given the following information in the form of HFEs shown in Table 4.

Suppose that the weight vector of the criteria is w ¼ ð0:25; 0:2; 0:15; 0:1; 0:2; 0:1ÞT,the decision makers have also given the relative weights of the hesitant fuzzy con-

cordance (weak hesitant fuzzy concordance) sets and the hesitant fuzzy discordance

(weak hesitant fuzzy discordance) sets as ! ¼ ð!C ; !C 0 ; !D; !D 0 Þ ¼ ð1; 34 ; 1; 34Þ.Similar to the solving procedure used in the above numerical example, we obtain the

aggregation dominance matrix:

E ¼ F G ¼

� 1 0 1 0

0 � 0 1 0

0 0 � 1 0

0 0 0 � 0

0 1 0 1 �

2666664

3777775

and a decision graph is constructed in Fig. 4.

We can see from the graph that three preference relations are obtained. That is:

(1)A1 � A2 � A4; (2)A5 � A2 � A4; (3)A3 � A4. In contrast, if adopting the hesi-

tant fuzzy average operators to aggregate the present hesitant information, the

amount of data is extremely huge. For example, the number of computed data after

aggregating A1 reaches 3� 45 ¼ 3072. If aggregating all Aiði ¼ 1; 2; . . . ; 5Þ, then the

corresponding number of computed data is 6677. The number will grow rapidly with

increasing the number of alternatives and criteria. In our HF-ELECTR I method, the

Table 4. Hesitant fuzzy decision matrix for Example 4.

A1 A2 A3 A4 A5

x1 f0.5,0.6,0.7g f0,2,0.3,0.6,0.7g f0.3,0.4,0.6g f0.2,0.3,0,6g f0.4,0.5,0.6,0.7gx2 f0.4,0.6,0.7,0.9g f0.5,0.6,0.8g f0.7,0.8g f0.3,0.4,0.6,0.7g f0.7,0.9gx3 f0.3,0.6,0.8,0.9g f0.2,0.4,0.6g f0.3,0.6,0.7,0.9g f0.2,0.4,0.5,0.6g f0.5,0.7,0.8,0.9gx4 f0.4,0.5,0.6,0.8g f0.3,0.6,0.7,0.8g f0.2,0.3,0.5,0.6g f0.1,0.3,0.4g f0.3,0.5,0.6gx5 f0.2,0.5,0.6,0.7g f0.3,0.6,0.8g f0.3,0.4,0.7g f0.2,0.3,0.7g f0.4,0.5gx6 f0.2,0.3,0.4,0.6g f0.1,0.4g f0.4,0.5,0.7g f0.2,0.5,0.6g f0.2,0.3,0.7g


value for the number of calculation is 462. To determine the trends of the compu-

tation complexity for the two methods, we generate a great number of n � n hesitant

fuzzy decision matrices H ¼ ðhijÞn�n (as an example, here we take the number of

values for each hij to be 4) by the Matlab Optimization Toolbox. The calculation

times for the HF-ELECTRE I method and the aggregation operator methods are

respectively ð5n3 þ 9n2 � 10n þ 4Þ=2 and nð4n þ 1Þ. To be clearer, we choose the

cases of n ¼4, 5, 10, 15, 20 to demonstrate the trends of the computation complexity

with increasing n for the two methods. The results are given in Table 5.

5.2. Outranking relations for di®erent number of alternatives

and criteria

In Example 3, only four alternatives are considered. To check possible in°uence

arising from a change in the number of alternatives, we compare the outranking

relation for di®erent number of alternatives (i.e., N ¼ 4; 5; 6 and 7) under the same

criteria. For this purpose, we perform a calculation with the Matlab Optimization

Toolbox based on the HF-ELECTRE I method:

(i) N ¼ 5. We use the same calculation procedure as that in Example 3:

Step 1. Give the hesitant fuzzy decision matrix for the case of N ¼ 5 (Table 6).

Note that the former four alternatives are the same as those in Table 1 which

corresponds to N ¼ 4.

Step 2. Calculate the score values and the deviation values of evaluation infor-

mation of the alternative A5 on the criteria in the form of HFEs listed in Table 7.

Note that for the former four alternatives (i.e.,A1;A2,A3,A4), their scores and

deviations of evaluation information are the same as those in Example 3, and thus,

they are not repeated in Table 7.

5A

4A1A

2A 3A

Fig. 4. Decision graph of Example 4.

Table 5. Calculation times for the HF-ELECTRE I method and the aggregation operator

method.

Method n ¼ 4 n ¼ 5 n ¼ 10 n ¼ 15 n ¼ 20

The HF-ELECTRE I method 214 402 2902 9377 21702

The aggregation operator method 1028 5125 1:05� 107 1:61� 1010 2:19� 1013


Step 3. Construct the hesitant fuzzy concordance sets and the weak hesitant

fuzzy concordance sets:

JC ¼

� � 4 � �3; 4 � 4 � �1; 3 2; 3 � � �1; 3; 4 2; 3; 4 3; 4 � 3; 4

1; 4 1; 2; 4 2; 4 � �

2666664

3777775;

JC 0 ¼

� 2 2 2 2; 3

1 � 1 1; 2 3

� � � 1; 2 3� � � � �� 1 1; 2 �

266664

377775:

Step 4. Construct the hesitant fuzzy discordance sets and the weak hesitant

fuzzy discordance sets:

JD ¼

� 3; 4 1; 3 1; 3; 4 1; 4

� � 2; 3 3; 4 1; 2; 4

4 4 � 3; 4 2; 4� � � � �� 3; 4 �

266664

377775;

JD 0 ¼

� 1 � � �2 � � � �2 1 � � 1

2 1 1; 2 � 1; 2

2; 3 3 3 � �

2666664

3777775:

Table 6. Hesitant fuzzy decision matrix for N ¼ 5.

x1 x2 x3 x4

A1 f0.2,0.4,0.7g f0.2,0.6,0.8g f0.2,0.3,0.6,0.7,0.9g f0.3,0.4,0.5,0.7,0.8gA2 f0.2,0.4,0.7,0.9g f0.1,0.2,0.4,0.5g f0.3,0.4,0.6,0.9g f0.5,0.6,0.8,0.9gA3 f0.3,0.5,0.6,0.7g f0.2,0.4,0.5,0.6g f0.3,0.5,0.7,0.8g f0.2,0.5,0.6,0.7gA4 f0.3,0.5,0.6g f0.2,0.4g f0.5,0.6,0.7g f0.8,0.9gA5 f0.4,0.6,0.7,0.9g f0.3,0.4,0.6g f0.3,0.4,0.5,0.8g f0.6,0.7,0.9g

Table 7. Score values and deviation values of A5.

A5 x1 x2 x3 x4

Score values 0.65 0.4333 0.5 0.7333

Deviation values 0.1803 0.1247 0.1871 0.1247


Step 5. Calculate the hesitant fuzzy concordance indices and the hesitant fuzzy

concordance matrix:

C ¼ ðcklÞ5�5 ¼

� 0:2 0:55 0:2 0:3

0:6333 � 0:4833 0:3333 0:1

0:35 0:45 � 0:3333 0:1

0:7 0:8 0:5 � 0:5

0:55 0:85 0:7833 0:3333 �

26666664

37777775:

Step 6. Calculate the hesitant fuzzy discordance indices and the hesitant fuzzy

discordance matrix:

D ¼ ðdklÞ5�5 ¼

� 0:7778 0:3429 1 1

0:6667 � 0:5357 1 1

0:6667 1 � 1 1

0:3361 0:3265 0:1143 � 0:4285

0:3663 0:0952 0:0779 1 �

26666664

37777775:

As seen from Step 3 to Step 6, adding the alternative A5 does not change the

hesitant fuzzy concordance (discordance) sets, the weak hesitant fuzzy con-

cordance (discordance) sets, the hesitant fuzzy concordance (discordance) indi-

ces, and the hesitant fuzzy concordance (discordance) matrix of the former four

alternatives. Its role is to add the last column and the last line of the resulting

corresponding matrices. Similar situations also appear when the number of the

alternatives is N ¼ 6 and N ¼ 7.

Step 7. Calculate the hesitant fuzzy concordance level and identify the con-

cordance dominance matrix, respectively:

�c ¼Xmk¼1

Xml¼1;l 6¼k

cklmðm � 1Þ ¼ 0:4525; F ¼

� 0 1 0 0

1 � 1 0 0

0 0 � 0 0

1 1 1 � 1

1 1 1 0 �

26666664

37777775:

Step 8. Calculate the hesitant fuzzy discordance level and identify the discor-

dance dominance matrix, respectively:

�d ¼Xmk¼1

Xml¼1;l 6¼k

dklmðm � 1Þ ¼ 0:6367; G ¼

� 0 1 0 0

0 � 1 0 0

0 0 � 0 0

1 1 1 � 1

1 1 1 0 �

26666664

37777775:



E ¼ F G ¼

� 0 1 0 0

0 � 1 0 0

0 0 � 0 0

1 1 1 � 1

1 1 1 0 �

2666664

3777775:

Step 10. As it can be seen from the aggregation dominance matrix that A1 is

preferred to A3; A2 is preferred to A3; A4 is preferred to A1 A2, A3 and A5; A5 is

preferred to A1, A2 and A3. The results are depicted in Fig. 5.

(ii) N ¼ 6. Results involving only the alternative A6 obtained in Steps 1 and 2 are

listed in Tables 8 and 9, respectively.

When the number of alternatives arrives at 6, the results obtained from Step 3 to

Step 6 can be expressed with the 6� 6 matrix in which except the data of the 6th line

and the 6th column, other data are the same as those in 5� 5 matrix when the

number of alternatives is 5. Thus, we only list in Table 10 the data of the 6th line and

the 6th column of the 6� 6 matrix, which are the hesitant fuzzy concordance set JCkl,

the weak hesitant fuzzy concordance set JC 0kl, the weak hesitant fuzzy discordance set

1A

3A

5A

4A

2A

Fig. 5. Decision graph of ¯ve alternatives.

Table 8. Hesitant fuzzy decision matrix for A6.

x1 x2 x3 x4

A6 f0.5,0.7,0.9g f0.4,0.5,0.7,0.9g f0.5,0.8g f0.4,0.5,0.8g

Table 9. Score values and deviation values of A6.

A6 x1 x2 x3 x4

Score values 0.7 0.625 0.65 0.5667



JDkl, the hesitant fuzzy discordance set JD 0

kl, the hesitant fuzzy concordance indexes

ckl , and the hesitant fuzzy discordance indexes dkl , respectively.

Steps 7 and 8. Calculate the hesitant fuzzy concordance level and discordance

level:

�c ¼Xmk¼1

Xml¼1;l 6¼k

cklmðm � 1Þ ¼ 0:4489;

�d ¼Xmk¼1

Xml¼1;l 6¼k

dklmðm � 1Þ ¼ 0:6167:


E ¼

� 0 1 0 0 0

0 � 1 0 0 0

0 0 � 0 0 0

1 1 1 � 1 0

1 1 1 0 � 0

1 1 1 0 0 �

266666664

377777775:


preferred toA3; A2 is preferred to A3; A4 is preferred to A1, A2, A3 and A5; A5 is

preferred to A1, A2 and A3; A6 is preferred to A1, A2 and A3 (see Fig. 6).

(iii) N ¼ 7. Analogous to the case of N ¼ 6, main results obtained from Step 1 to

Step 6 are summarized in Tables 11–13.

Steps 7 and 8. Calculate the hesitant fuzzy concordance level and discordance

level:

�c ¼Xmk¼1

Xml¼1;l 6¼k

cklmðm � 1Þ ¼ 0:4258;

Table 10. Data of JCkl; JC 0

kl; JDkl

; JD 0kl; ckl and dkl of the 6th line and

the 6th column for N ¼ 6.

JCklJC 0

klJDkl

JD 0kl

ckl dkl

k ¼ 1; l ¼6 �� 1,2,3,4 �� 0 1

k ¼ 2; l ¼ 6 4 �� 1,3 2 0.35 0.6667

k ¼ 3; l ¼ 6 �� 3,4 1,2 0 0.7292

k ¼ 4; l ¼ 6 4 �� 1,2,3 0.35 0.5238k ¼ 5; l ¼ 6 4 �� 1,3 2 0.35 0.5786

k ¼ 6; l ¼ 1 1, 2, 3, 4 �� 1 0

k ¼ 6; l ¼ 2 1, 3 2 4 �� 0.55 0.2692

k ¼ 6; l ¼ 3 3, 4 1,2 �� 0.8333 0k ¼ 6; l ¼ 4 �� 1,2,3 4 �� 0.4333 1

k ¼ 6; l ¼ 5 1, 3 2 4 �� 0.55 1


1A

3A

5A

4A

2A

6A

Fig. 6. Decision graph of six alternatives.

Table 11. Hesitant fuzzy decision matrix for A7.

x1 x2 x3 x4

A7 f0.2,0.3,0.4g f0.2,0.3,0.4,0.7g f0.3,0.5,0.6,0.7g f0.2,0.3,0.4,0.5g

Table 12. Scores and deviations of A7.

A7 x1 x2 x3 x4

Score values 0.3 0.4 0.525 0.35


Table 13. Data of JCkl; JC 0

kl; JDkl

; JD 0kl; ckl and dkl of the 7th line

and the 7th column for N ¼ 7.

JCklJC 0

klJDkl

JD 0kl

ckl dkl

k ¼ 1; l ¼ 7 � 1,2,3,4 � � 0.6667 0

k ¼ 2; l ¼ 7 � 1,3,4 � 2 0.4667 0.1633k ¼ 3; l ¼ 7 2 1,3,4 � � 0.7667 0

k ¼ 4; l ¼ 7 3,4 1 � 2 0.6333 0.1088

k ¼ 5; l ¼ 7 2 1,4 3 � 0.6667 0.0756

k ¼ 6; l ¼ 7 � 1,2,3,4 � � 0.6667 0k ¼ 7; l ¼ 1 � � � 1,2,3,4 0 0.6667

k ¼ 7; l ¼ 2 � 2 � 1,3,4 0.2 0.6667

k ¼ 7; l ¼ 3 � � 2 1,3,4 0 0.6667k ¼ 7; l ¼ 4 � 2 3,4 1 0.2 1

k ¼ 7; l ¼ 5 3 � 2 1,4 0.15 0.6667

k ¼ 7; l ¼ 6 � � � 1,2,3,4 0 0.6667


�d ¼Xmk¼1

Xml¼1;l 6¼k

dklmðm � 1Þ ¼ 0:5520:


E ¼

� 0 1 0 0 0 1

0 � 1 0 0 0 1

0 0 � 0 0 0 1

1 1 1 � 1 0 1

1 1 1 0 � 0 1

1 1 1 0 0 � 1

0 0 0 0 0 0 �

26666666664

37777777775:


preferred to A3 and A7; A2 is preferred to A3 and A7; A3 is preferred to A7; A4 is

preferred to A1, A2, A3, A5 and A7; A5 is preferred to A1;A2A3 and A7; A6

is preferred to A1;A2A3 and A7 (see Fig. 7).

To facilitate to see the possible in°uence due to a change in the number of

alternatives, the outranking relations obtained for the cases of N ¼ 4; 5; 6 and 7 are

listed in Table 14.

It can be seen from Table 14 that varying the number of alternatives does not

change the outranking relations in Example 3 and the result that A4 is a non-

outranked alternative. We explain it in the following way: First, when the number of

alternatives is respectively 5, 6 and 7, the hesitant fuzzy concordance (discordance)

indices given in Example 3 do not be changed. Second, although a variation in the

number of alternatives slightly modi¯es the hesitant fuzzy concordance level �c and

discordance level �d (see Table 14), but the changes in �c and �d are still within the

sensitivity range of Example 3 in which the parameter changes will not a®ect the set

of the nonoutranked alternatives. Speci¯cally, in Example 3, �c ¼ 0:4611 and�d ¼ 0:6472. A decrease (increase) in �c (�d) cannot bring about a change in the set of

1A

3A

5A

4A

2A

6A

7A

Fig. 7. Decision graph of seven alternatives.


alternatives that are not outranked by other alternatives.78 Thus, �c could be lowered

to zero and �d could be raised to 1. Moreover, we note from the hesitant fuzzy

concordance (discordance) matrices (given in Step 5 (Step 6) in Example 3) that

when �c is increased above 0.55, the alternative A3 is no longer outranked by any

other alternative. For the alternative A1 (A2), the corresponding value is 0.7 (0.8), as

for �c < 0:7 (�c < 0:8Þ it remains outranked by the alternative A4. The ¯rst change in

the set of the nonoutranked alternatives occurs when �c is increased to 0.55. A similar

analysis can be performed for the discordance level. When �d is lowered below 0.3361,

A1 is no longer outranked. The corresponding value for A2ðA3Þ is 0.3265 (0.1143), so

the lower bound for �d is thus given by 0.3361. Taken together, we ¯nd in Example 3

that when the parameters satisfy 0 < �c < 0:55 and 0:3361 < �d < 1, the set of the

nonoutranked alternatives will not be a®ected.

In the following, we survey outranking relations for di®erent number of criteria.

To this end we increase the number of criteria by 5 and 6 (see Table 15).

Because the weights of all criteria satisfy the normalization constraintPnj¼1 wj ¼ 1, varying the number of criteria inevitably changes the original weights

of criteria. This will a®ect outranking relations, as have been pointed out in many

previous works (see, e.g., Refs. 78 and 79). As an illustration, we list in Table 16 the

corresponding results for outranking relations within the HF-ELECTRE I frame-

work and the comparison with the case that the number of criteria is 4. As it can be

seen in the table, the results of outranking relations for di®erent number of criteria

calculated with the HF-ELECTRE I method are consistent with the general

expectation.

In order to further evaluate the HF-ELECTRE I method, a simulation with

randomly generated cases is made in a direct and transparent way. Random data are

Table 14. Comparison of the outranking relations for di®erent number of alternatives.

The number of

alternatives

�c �d Results

N ¼ 4(Example 3) 0.4611 0.6472 A1 � A3; A2 � A3; A4 � A1;A2;A3

N ¼ 5 0.4525 0.6367 A1 � A3; A2 � A3; A4 � A1;A2;A3;A5;

A5 � A1;A2;A3

N ¼ 6 0.4489 0.6167 A1 � A3; A2 � A3; A4 � A1;A2;A3;A5;

A5 � A1;A2;A3; A6 � A1;A2;A3

N ¼ 7 0.4258 0.5520 A1 � A3;A7; A2 � A3;A7; A3 � A7; A4 � A1;A2;A3;A5;A7;

A5 � A1;A2;A3;A7; A6 � A1;A2;A3;A7

Table 15. Hesitant fuzzy information of the 5th and 6th criterion for each

alternative.

A1 A2 A3 A4

x5 f0.3,0.6,0.7g f0.2,0.4,0.5,0.6g f0.3,0.7,0.8g f0.4,0.5,0.7,0.9gx6 f0.4,0.5,0.7,0.8g f0.5,0.7,0.9g f0.3,0.4,0.6,0.9g f0.5,0.6,0.8g


generated to form the MCDM problems with all possible combinations of 4, 6, 8, 10

alternatives and 4, 6, 8, 10 criteria. So, 16ð¼ 4� 4Þ di®erent instances are examined

in this study. We ¯nd within the framework of HF-ELECTRE I that the preferred

choice of alternatives in each instance is alternative A1 (see below for details).

In the following, the cases corresponding to the combinations of four alternatives

(i.e., A ¼ fA1;A2;A3;A4gÞ with 4, 6, 8, 10 criteria are employed to illustrate our

simulation process. Under the environment of group decision making, the perform-

ance of each alternative on each criterion can be considered as a HFS, represented by

Ai ¼ f< xj ; hAiðxjÞ > jxj 2 Xg; i ¼ 1; . . . ; 4, where hAi

ðxjÞð¼ f�j� 2 hAiðxjÞ; 0 � �

� 1gÞ indicates possible satisfaction degrees of the criterion xj to the alternative Ai.

The characteristic of satisfaction degrees assigned to these four alternatives is

as follows: For � 2 hA1ðxjÞ; � � Uð0:8; 1Þ; for � 2 hA2

ðxjÞ; � � Uð0:5; 0:8Þ; for � 2hA3

ðxjÞ; � � U ð0:3; 0:5Þ; for � 2 hA4ðxjÞ; � � Uð0; 0:3Þ, where Uða; bÞ means the

uniform distribution on the interval ½a; b�.In terms of the above-described way, we have simulated 100 times corresponding

to each instance with the number of criteria being 4, 6, 8 and 10, respectively. Besides

the � values, in each simulation, the number of � in each hAiðxjÞ and the weights of

all criteria (which satisfy 0 � wj � 1 andPn

j¼1 wj ¼ 1Þ are also randomly generated.

Consequently, all quantities calculated in the HF-ELECTRTE I °uctuate in each

simulation. As an example, Fig. 8 displays the scores of hAiðxjÞ of four alternatives on

a criterion, which shows an obvious variation for di®erent simulation.

Following the steps of the HF-ELECTRE I method outlined previously, our

simulation results for the randomly generated instances demonstrate that out-

ranking relations are consistent with the expectation, that is, alternative A1 is the

best choice due to a larger role of the performance of the alternatives in in°uencing

outranking relations as compared to the weights and threshold values.80 This con-

sistency further indicates the validity of the HF-ELECTRE I method.

5.3. Comparison with ELECTRE III and ELECTRE IV methods

As one knows, there exist several other outranking ELECTRE type methods, for

example, ELECTRE III21,80 and ELECTRE IV81 methods, which are able to deal

with the ranking problems, that is, it is concerned with the ranking of all the actions

belonging to a given set of actions from the best to the worst.21 Although our HF-

ELECTRE I method is used to construct a partial prioritization and to choose a set

Table 16. Comparison of outranking relations for di®erent number of criteria.

The number

of criteria

Weight Results

4 (Example 3) w ¼ ð0:2; 0:3; 0:15; 0:35ÞT A1 � A3; A2 � A3; A4 � A1;A2;A3

5 w ¼ ð0:2; 0:3; 0:15; 0:2; 0:15ÞT A2 � A1; A3 � A1; A4 � A1;A2;A3

6 w ¼ ð0:2; 0:15; 0:15; 0:2; 0:15; 0:15ÞT A1 � A3;A2 � A1;A3;A3 � A1;A4 � A1;A2;A3


of preferable actions, which is somewhat di®erent from the ELECTRE III and

ELECTRE IV methods, it is interesting to compare the results obtained using these

di®erent methods aiming at seeing the ranking di®erences among them. For calcu-

lation details and main results, see Appendix A.

One can see from Appendix A that the conclusions obtained with the ELECTRE

III and ELECTRE IV methods are partially consistent with that derived with the

HF-ELECTRE I methods, whereas a di®erence in the outranking relations is also

noticed. We explain it in the following way: In the ELECTRE III and ELECTRE IV

group outranking methods, because group decision information is obtained through

averaging for all decision makers, so only information involving the average opinion

of all decision makers is considered, whereas in the HF-ELECTRE I method, in

addition to that consideration, the deviation degrees which re°ect the di®erence in

opinions between the individual decision makers and their averages are also

accounted for. It should be pointed out that the ELECTRE III and ELECTRE IV

methods can give the ranking of all alternatives, which is di®erent from the HF-

ELECTRE I method that gives the partial outranking relations, thus, these two

kinds of group decision-making approaches are complementary.

6. Conclusion

We have investigated the MCDM problems under hesitant fuzzy environments in

which uncertainties arising from the decision makers are expressed by several poss-

ible values. By de¯ning the concepts of hesitant fuzzy concordance and hesitant

fuzzy discordance based on the given score function and deviation function, we have

developed a hesitant fuzzy ELECTRE I method and applied it to MCDM. We have

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

times

scor

e v

alue

s

Fig. 8. (Color online) Score values of hAiðxjÞ of four alternatives on a criterion in 100 times simulations.

From top to bottom, they correspond to A1 (yellow line), A2 (blue line), A3 (green line) and A4 (red line),

respectively.


performed numerical calculations and a comparison with the previous work, which

shows that the HF-ELECTRE I method can not only predict the consistent results

with the aggregation operator based method, but also signi¯cantly reduce the

computational e®orts, in particular when the number of criteria in a MCDM is larger

than 5. We have further surveyed the in°uence of the number of alternatives on

the outranking relations and given the numerical example a sensitive range for the

parameters; that is, the changes of the parameters will not a®ect the set of the

nonoutranked alternatives. We have made a study with randomly generated cases,

which also indicate the validity of the HF-ELETTRE I method. In addition, we have

worked out the outranking relations with the ELECTRE III and ELECTRE IV

methods and compared them with those obtained with the HF-ELECTRE I method.

When employing the proposed HF-ELECTRE I method to a general MCDM,

further work is still needed. First, the strong and weak preferences are not clearly

discriminated. Accounting for both preferences simultaneously will have some

positive implications on the richness of the outranking information that is available

to the decision makers. To this end, in the future, we will infer the weak preferences

among alternatives from the weak hesitant fuzzy concordance and the discordance

sets, and develop the corresponding ELECTRE type methods that incorporate

hesitant fuzzy information. Second, to further extend the application range of the

HF-ELECTRE I method, in particular for the case that all decision makers assign

di®erent weights for a criterion, it also will be necessary to generalize the original

de¯nition of HFSs61,62 which assumes that the decision makers in the group setting

unanimously agree on the weight of a criterion, as has recently been attempted by

Zhu82 who generalized the concept of HFSs so that di®erent weights for a criterion

can be taken into account. In the future, we will study new HF-ELECTRE type

methods to MCDM based on the generalized HFSs. Third, in the present work, we

have used the sensitivity analysis, which is a traditional answer to the di±culties in

setting the parameter values,78,79 as an important way to evaluate the HF-ELEC-

TRE I method. However, a more complete evaluation for a MCDM approach

including our HF-ELECTRE I needs to consider other quality factors, such as

convergence of opinions, e±ciency of the selected nondominated alternatives, etc.,

besides the sensitivity analysis. We will make such a study in further work.

Given that HFSs are a suitable technique of denoting uncertain information that

is widely encountered in daily life and that MCDM aids the analysis in ¯nance,

energy planning, telecommunication network design, etc.,21 the HF-ELECTRE I

method developed here, as a kind of approach to MCDM, is therefore of considerable

practicality in these diverse ¯elds and consequently, it constitutes a potentially

useful tool to handle those issues involving hesitant fuzzy information.

Acknowledgments

The authors thank the anonymous reviewers for their comments and suggestions,

which have considerably led to an improved version of this paper. The work was


supported by the National Natural Science Foundation of China (Nos. 71071161 and

61273209).

Appendix A

To facilitate a comparison, the same example (Example 3) used to illustrate the HF-

ELECTRE I method is also considered for the ELECTRE III and ELECTRE IV

methods, which is composed of the construction and the exploitation of the out-

ranking relations.

We ¯rst brie°y discuss the ELECTRE III method, for more details see Refs. 21

and 80.

We introduce the construction of the outranking relations. Let two alternatives a

and b belong to a given set of actions, and yjðaÞ and yjðbÞ be the performance of a and

b in terms of the criterion xj . We denote indi®erence, preference and veto thresholds

on a criterion j introduced by the decision makers with qj ; pj and vj , respectively,

where j ¼ 1; 2; . . . ; n.

A partial concordance index cjða; bÞ is de¯ned as follows for each criterion j:

cjða; bÞ ¼1 if yjðaÞ þ qj yjðbÞ;0 if yjðaÞ þ pj � yjðbÞ; j ¼ 1; . . . ; n

ðpj þ yjðaÞ � yjðbÞÞ=ðpj � qjÞ otherwise:

8><>:

ðA:1Þ

Let wj be the importance coe±cient for the criterion j. C ða; bÞ is an overall

concordance index and de¯ned as:

C ða; bÞ ¼ 1

w

Xnj¼1

wjcjða; bÞ; where w ¼Xnj¼1

wj : ðA:2Þ

The discordance index for each criterion j, djða; bÞ is calculated as:

djða; bÞ ¼0 if yjðaÞ þ pj yjðbÞ;1 if yjðaÞ þ vj � yjðbÞ; j ¼ 1; . . . ; n

ðyjðbÞ � yjðaÞ � pjÞ=ðvj � pjÞ otherwise:

8><>:

ðA:3ÞThe credibility degree Sða; bÞ for each pair ða; bÞ is de¯ned as:

Sða; bÞ ¼C ða; bÞ if djða; bÞ � C ða; bÞ 8j

C ða; bÞ � Qj2J :dj ða;bÞ>Cða;bÞ

1� djða; bÞ1� C ða; bÞ otherwise:

8><>: ðA:4Þ

The ELECTRE model is \exploited" to produce a ranking of alternatives from

the credibility matrix. We follow the standard implementation presented in Refs. 21

and 80.


In what follows, we present the process that the ELECTRE III method is utilized

to tackle Example 3.

Step 1. Construction of fuzzy group decision matrix:

As mentioned previously, in the hesitant fuzzy decision matrix H ¼ ðhijÞ4�4 (i.e.,

Eq. (5)), each element hij ¼ f�j� 2 hij ; 0 � � � 1g indicates the possible member-

ship degrees of the ith alternative Ai under the jth criterion by the decision makers.

To derive the group decision matrix, we aggregate the decision makers' individual

decision information with the averaging operator, which is de¯ned by ~h ij ¼1lhij

P�2hij�. As it can be easily seen that the results obtained from the formula are

just those of the score function mentioned earlier, which has been given in Table 2.

So, the fuzzy group decision matrix ~H ¼ ð~h ijÞ4�4 is obtained.

Step 2. Indi®erence, preference and veto threshold values on a criterion j are

introduced by the decision makers and given in Table A.1.

Step 3. Calculate the partial concordance index cjða; bÞ with Eq. (A.1) and sum-

marize the results in Table A.2.

Step 4. Calculate the overall concordance index Cða; bÞ with Eq. (A.2) and sum-

marize the results in Table A.3.

Step 5. Calculate the discordance index djða; bÞ with Eq. (A.3) and summarize the

results in Table A.4.

Step 6. Calculate the credibility degree Sða; bÞ with Eq. (A.4) and summarize the

results in Table A.5.

Step 7. Perform the exploiting procedure with the credibility matrix. Two pre-

orders are obtained via a descending and ascending distillation process respectively,

see Fig. A.1.

Based on these two pre-orders, the result of outranking is A4 � A2 � A1 � A3.

We next analyze the calculations with the ELECTRE IV method.

Let two alternatives a and b belong to a given set of actions, and yjðaÞ and yjðbÞ bethe performance of a and b in terms of the criterion xj . We denote indi®erence,

preference and veto thresholds on the criterion j introduced by the decision makers

with qj , pj and vj , respectively, where j ¼ 1; 2; . . . ; n. Jþða; bÞ and J�ða; bÞ respec-tively represent the sums of all those criteria where the performance of a is superior

Table A.1. Three threshold values.

Threshold values x1 x2 x3 x4

qj 0.02 0.05 0.02 0.02

pj 0.1 0.1 0.05 0.1

vj 0.3 0.3 0.2 0.3


Table A.3. Concordance matrix.

A1 A2 A3 A4

A1 1 0.45 0.7458 0.4665

A2 0.7 1 0.6750 0.5

A3 0.6125 0.6375 1 0.6250A4 0.7 0.8418 0.6043 1

Table A.2. Partial concordance index for each criterion j.

A1 A2 A3 A4

c1ðAi ;AjÞ A1 1 0 0.1038 0.8325

A2 1 1 1 1

A3 1 0.9375 1 1A4 1 0.2088 0.5213 1

c2ðAi ;AjÞ A1 1 1 1 1

A2 0 1 0 1

A3 0 1 1 1

A4 0 1 0 1

c3ðAi ;AjÞ A1 1 1 0.5 0

A2 1 1 0.8333 0

A3 1 1 1 0.8333

A4 1 1 1 1

c4ðAi ;AjÞ A1 1 0 1 0

A2 1 1 1 0A3 0.75 0 1 0

A4 1 1 1 1

Table A.4. Discordance index for each criterion j.

A1 A2 A3 A4

d1ðAi ;AjÞ A1 0 0.0835 0 0

A2 0 0 0 0

A3 0 0 0 0

A4 0 0 0 0

d2ðAi ;AjÞ A1 0 0 0 0

A2 0.6665 0 0.125 0

A3 0.0415 0 0 0

A4 0.6665 0 0.125 0

d3ðAi ;AjÞ A1 0 0 0 0.0667

A2 0 0 0 0

A3 0 0 0 0

A4 0 0 0 0

d4ðAi ;AjÞ A1 0 0.3 0 1

A2 0 0 0 0.25

A3 0 0.5 0 1

A4 0 0 0 0


and inferior to b, that is,

Jþða; bÞ ¼ fj 2 J j : yjðbÞ þ qjðyjðbÞÞ <�� yjðaÞg; ðA:5Þ

J�ða; bÞ ¼ fj 2 J j : yjðaÞ þ qjðyjðaÞÞ <�� yjðbÞg: ðA:6Þ

The ELECTRE IV method contains two levels of the outranking relations: the

strong outranking relation (SF) and the weak outranking relation (Sf ), which are

de¯ned by

aSFb () 8j; yjðaÞ þ pjðyjðaÞÞ yjðbÞ and jjJþða; bÞjj > jjJ�ða; bÞjj; ðA:7ÞaSf b () 8j; yjðaÞ þ pjðyjðaÞÞ yjðbÞ

or9l; ylðaÞ þ vlðylðaÞÞ ylðbÞ > ylðaÞ þ plðylðaÞÞand jjj : yjðaÞ � yjðbÞ > pjðyjðaÞÞjj n=2;

( ðA:8Þ

where jjJ jj denotes the number of elements in the set J .

The ELECTRE IV method exploiting procedure is the same as in the ELECTRE

III method.21,80 The details of dealing with Example 3 with the ELECTRE IV

method are given.

Step 1. The process is the same as that with the ELECTRE III method.

Step 2. Indi®erence, preference and veto threshold values on the criterion j are the

same as those given in Table A.1.

Step 3. Calculate Jþða; bÞ and J�ða; bÞ with Eqs. (A.5) and (A.6). The results are

summarized in Table A.6.

Table A.5. Credibility matrix.

A1 A2 A3 A4

A1 1 0.45 0.7458 0A2 0.7 1 0.6750 0.5

A3 0.6125 0.6375 1 0

A4 0.7 0.8418 0.6043 1

2A

1A

3A

4A

(a)

4A 2A 1A 3A

(b)

Fig. A.1. (a) Descending distillation and (b) ascending distillation.


Step 4. Calculate the outranking relation S between alternatives with Eqs. (A.7)

and (A.8). The results are also displayed in Table A.6.

We can see from Table A.6 that A1�Sf A3, A2�Sf A1, A4�Sf A1, and A4�SFA2.

Step 5. Draw the strong and weak outranking graphs in Fig. A.2.

With the exploiting procedure, the resulting outranking is A4 � A2 � A1 � A3.

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2A4A

(a)

1A

4A

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THE ELECTRE I MULTI-CRITERIA DECISION-MAKING METHOD BASED ON HESITANT FUZZY SETS

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