Intuitionistic fuzzy Choquet aggregation operator based on...

Scientia Iranica E (2013) 20(6), 2109{2122

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringwww.scientiairanica.com

Intuitionistic fuzzy Choquet aggregation operator basedon Einstein operation laws

Dejian Yu�

School of Information, Zhejiang University of Finance and Economics, Hangzhou, 310018, China.

Received 29 October 2011; received in revised form 8 August 2012; accepted 14 May 2013

KEYWORDSMulti-criteria decisionmaking;Intuitionistic fuzzyset;Aggregation operator;Choquet integral;Einstein operationlaws.

Abstract. In this paper, we study the intuitionistic fuzzy information aggregationoperators based on Einstein operation laws under the condition in which the aggregatedarguments are independent. The Einstein-based Intuitionistic Fuzzy Choquet Averaging(EIFCA) operator is proposed. Furthermore, the relationship between the EIFCA operatorand the IFCA operator is investigated. The desirable properties of the EIFCA operator,such as boundeness, monotonicity, shift-invariance and homogeneity are discussed. Amulti-criteria decision making approach, based on the EIFCA operator is proposed underintuitionistic fuzzy environment. A comparative example is given for demonstrating theapplicability of the proposed decision procedure and for �nding links with other operators-based decision approach.c 2013 Sharif University of Technology. All rights reserved.

1. Introduction

Intuitionistic Fuzzy Sets (IFS) [1-2], introduced byAtanassov, is a generalization of the concept of fuzzyset. IFS is characterized by three functions, ex-pressing degrees of membership, non-membership andindeterminacy, so it is more powerful to deal withuncertainty and vagueness in real applications thantype-2 fuzzy set [3-4], type-fuzzy set [3], fuzzy multiset[5-6] and hesitant fuzzy set [7-8] which only considerthe membership degree. For example, if a boy wantsto �nd a girlfriend and evaluates the girl from tenaspects, there are six aspects which satis�es the boy,three aspects do not satisfy, and he is uncertain withone aspect of the girl. In such a case, other types offuzzy set can only re ect the satis�ed aspects whichlose some uncertain information, while intuitionisticfuzzy set can describe all the satis�ed, unsatis�ed anduncertain information. Because of its appearance, IFShas attracted much attention [9-34].

Research on the intuitionistic fuzzy information

*. E-mail address: [email protected]

aggregation method is one of the hot topics of the IFStheory. Based on the famous OWA operator [35-36]and the GOWA operator [37] proposed by Americanscholar Yager, many extended operators have beenappeared. For example, Xu [15] proposed the Intu-itionistic Fuzzy Weighted Averaging (IFWA) operator,ordered IFWA (IFOWA) operator and the intuition-istic hybrid aggregation (IFHA) operator. From thegeometric point of view, Xu and Yager [38] introducedthe Intuitionistic Fuzzy Weighted Geometric (IFWG)operator, ordered IFWG (IFOWG) operator and theIntuitionistic Fuzzy Hybrid Geometric (IFHG) opera-tor. Zhao et al. [39] proposed the generalized formsof the IFWA, IFOWA and IFHA operators, and theyproofed that the operators proposed by Xu [15] arespecial cases of the operators. The above aggregationoperators for intuitionistic fuzzy information are underthe condition in which the aggregated arguments areindependent. However, it cannot meet the requirementof real situations. Choquet integral [40,41] is a pow-erful tool to deal with this situation, based on whichXu [42], Tan and Chen [10] and Tan [43] developedsome intuitionistic fuzzy Choquet operators such as

2110 Dejian Yu/Scientia Iranica, Transactions E: Industrial Engineering 20 (2013) 2109{2122

the Intuitionistic Fuzzy Choquet Average (IFCA) op-erator and the Intuitionistic Fuzzy Choquet Geometric(IFCG) operator. Tan and Chen [11] proposed theinduced intuitionistic fuzzy Choquet integral operatorand applied it to decision making. The prominentcharacteristic of those operators is that they cannotonly adapt to the intuitionistic fuzzy environment,but also re ect the interrelationship of the individualcriteria.

It needs to be pointed out that the basic opera-tional laws of IFS for the above aggregation operatorsare the algebraic operational laws. Einstein productand Einstein sum are good alternatives for the algebraicproduct and algebraic sum, respectively [44]. Thepurpose of this paper is to investigate intuitionisticfuzzy information aggregation methods based on theEinstein product and Einstein sum under the assump-tion that the aggregated arguments are correlative. Todo this, the remainder of this paper is constructed asfollows: Section 2 brie y reviews some basic concepts.In Section 3, we propose the Einstein-based Intuition-istic Fuzzy Choquet Averaging (EIFCA) operator; thedesirable properties of the EIFCA are also studied inthis section. An approach to multi-criteria decisionmaking based on the proposed operator is proposed inSection 4; a comparative example is also illustrated inthis section. S ection 5 gives some conclusion remarks.

2. Some basic concepts

As a generalization of fuzzy set, Intuitionistic FuzzySet (IFS) assigns to each element a membership degreeand a non-membership degree. Atanassov [1] gave thede�nition of Intuitionistic Fuzzy Set (IFS) as follows.

De�nition 1. If a set X be �xed, the concept ofIntuitionistic Fuzzy Set (IFS) A on X is de�ned asfollows:

A = f< x; �A(x); vA(x) > jx 2 X g ; (1)

where the functions �A(x) and vA(x) denote thedegrees of membership and non-membership of theelement x 2 X to the set A, respectively, with thecondition that 0 � �A(x) � 1, 0 � vA(x) � 1 and0 � �A(x)+vA(x) � 1. For convenience, Xu [15] named� = (��; v�) an Intuitionistic Fuzzy Value (IFV). Inthis paper, we let V be the set of all IFVs.

For three IFVs �, �1, �2 2 V , some Einsteinoperational laws were given as follows [44]:

1. �1 �" �2 =��1+��21+��1��2

; v�1v�21+(1�v�1 )(1�v�2 )

�,

2. �1 " �2 =�

��1��21+(1��1 )(1��2 ) ;

v�1+v�21+v�1v�2

�,

3. �� =�

2��(2��)�+��

; (1+v�)��(1�v�)�

(1+v�)�+(1�v�)�

�,

4. �� =�

(1+��)��(1��)�

(1+��)�+(1��)� ;2v��

(2�v�)�+v��

�.

Chen and Tan [45] introduced the score function s(�) =�� v� to get the score of �, then Hong and Choi [46]de�ned the accuracy function h(�) = �� + v� toevaluate the accuracy degree of �. Based on thescore function s and the accuracy function h, Xu andYager [38] gave an order relation between two IFVs �1and �2:

1. If s(�1) < s(�2), then �1 < �2;2. If s(�1) = s(�2), then:

i) If h(�1) = h(�2), then �1 = �2;ii) If h(�1) < h(�2), then �1 < �2.

Let �(fxig) (i = 1; 2; � � � ; n) be the weights of theelements xi 2 X (i = 1; 2; � � � ; n) where � is afuzzy measure; Sugeno [47], Wang and Klir [48] andDenneberg [49] de�ned a fuzzy measure as follows.

De�nition 2. A fuzzy measure � on the set X isa function � : �(X) ! [0; 1] satisfying the followingaxioms:

1. �(;) = 0, �(X) = 1;2. B � C implies �(B) � �(C), for all B;C � X;3. �(B[C) = �(B)+�(C)+��(B)�(C), for all B;C �

X and B \ C = ;, where � > �1.

Based on De�nition 2, Xu [42], Tan and Chen [10]and Tan [43] de�ned Intuitionistic Fuzzy ChoquetAveraging (IFCA) and Intuitionistic Fuzzy ChoquetGeometric (IFCG) operators as follows.

De�nition 3. Let � be a fuzzy measure on X,and �(xj) = (��(xj); v�(xj)) (j = 1; 2; � � � ; n) be acollection of intuitionistic fuzzy sets, then:

IFCA(�(x1); �(x2); � � � ; �(xn))

=n�j=1

��(B�(j) �B�(j�1))�(x�(j))

�=�

1�Yn

j=1(1� ��(j))

�(B�(j)�B�(j�1)) ;Yn

j=1(v��(j))

�(B�(j)�B�(j�1))�; (2)

IFCG(�(x1); �(x2); � � � ; �(xn))

=nj=1

��(x�(j))

��(B�(j)�B�(j�1))

=�Yn

j=1(��(j))

�(B�(j)�B�(j�1));

1�Yn

j=1(1� v��(j))

�(B�(j)�B�(j�1))�; (3)

Dejian Yu/Scientia Iranica, Transactions E: Industrial Engineering 20 (2013) 2109{2122 2111

are called the Intuitionistic Fuzzy Choquet Av-eraging (IFCA) and Intuitionistic Fuzzy ChoquetGeometric (IFCG) operators, respectively, where(�(1); �(2); � � � ; �(n)) is a permutation of (1; 2; � � � ; n),such that �(x�(1)) � �(x�(2)) � � � ��(x�(n)), B�(k) =fx�(j)jj � kg, for k � 1, and B�(0) = ;.

3. Intuitionistic fuzzy Choquet operator basedon Einstein operation laws

In this section, we shall investigate the intuitionisticfuzzy information aggregation operator combined withChoquet integral. Based on the Einstein operationallaws described by De�nition 2, we give the de�nitionof the Einstein-based Intuitionistic Fuzzy ChoquetAveraging (EIFCA) operator as follows:

De�nition 4. Let � be a fuzzy measure on Xand �j(j = 1; 2; � � � ; n) be a collection of IFVs, anEinstein-based Intuitionistic Fuzzy Choquet Averaging(EIFCA) operator is a mapping V n ! V , and:

E(C)Z�d� = EIFCA(�(x1); �(x2); � � � ; �(xn))

=n�j=1

��(B�(j) �B�(j�1))�(x�(j))

�; (4)

where (C)R�d� denotes the Choquet integral and

(�(1); �(2); � � � ; �(n)) is a permutation of (1; 2; � � � ; n),such that �(x�(1)) � �(x�(2)) � � � ��(x�(n)), B�(k) =fx�(j)jj � kg, for k � 1, and B�(0) = ;.

Based on the operational laws of the IFVs de-scribed in Section 2, we can derive Theorem 1 easily.

Theorem 1. Let � be a fuzzy measure on X, �j =(��j ; v�j ) (j = 1; 2; � � � ; n) be a collection of IFVs and��(j) be the jth largest of them, then their aggregatedvalue by using the EIFCA operator is also an IFVs,and is de�ned in Eq. (5) which is shown in Box I.

Proof. The proof including Eqs. (6) to (13) is shownin Box II. It should be noted that the proof ofTheorem 1 was done by many references such as [15,39-44].

In order to analyze the relationship between theEIFCA and IFCA operators proposed by Tan and

Chen [10] and Xu [42], we introduce the followinglemma [50].

Lemma 1. Let xj > 0, !j > 0, j = 1; 2; � � � ; n andPnj=1 !j = 1, then:

Yn

j=1x!jj �

nXj=1

!jxj ; (14)

with equality if and only if x1 = x2 = � � � = xn.

Theorem 2. Let � be a fuzzy measure on X, �j =(��j ; v�j ) (j = 1; 2; � � � ; n) be a collection of IFVs and��(j) be the jth largest of them, then:

EIFCA(�1; �2; � � � ; �n) � IFCA(�1; �2; � � � ; �n):(15)

Proof. On one hand, since �(B�(j)�B�(j�1)) � 0 forall j and

Pnj=1 �(B�(j) � B�(j�1)) = 1, then based on

Lemma 1, we have:Yn

j=1

�1 + ��(j)

��(B�(j)�B�(j�1))

+Yn

j=1

�1� ��(j)

��(B�(j)�B�(j�1))

�nXj=1

�(B�(j) �B�(j�1))�1 + ��(j)

�+

nXj=1

�(B�(j) �B�(j�1))�1� ��(j)

�=

nXj=1

�(B�(j) �B�(j�1))�1 + ��(j)

�+

nXj=1

�(B�(j) �B�(j�1))�1� ��(j)�

=nXj=1

�(B�(j) �B�(j�1))

EIFCA (�1; �2; � � � ; �n) =�Qn

j=1(1+��(j) )�(B�(j)�B�(j�1))�Qnj=1(1��(j) )�(B�(j)�B�(j�1))Qn

j=1(1+��(j) )�(B�(j)�B�(j�1))+Qnj=1(1��(j) )�(B�(j)�B�(j�1)) ;

2Qnj=1 v

�(B�(j)�B�(j�1))��(j)Qn

j=1(2�v��(j) )�(B�(j)�B�(j�1))+Qnj=1 v

�(B�(j)�B�(j�1))��(j)

!: (5)

Box I.


EIFCA (�1; �2; � � � ; �n) =�Qn


j=1(1+��(j) )�(B�(j)�B�(j�1))+Qnj=1(1��(j) )�(B�(j)�B�(j�1)) ;

2Qnj=1 v

�(B�(j)�B�(j�1))��(j)Qn

j=1(2�v��(j) )�(B�(j)�B�(j�1))+Qnj=1 v

�(B�(j)�B�(j�1))��(j)

!; (6)

by using mathematical induction on n:For n = 2: since:

�(B�(1) �B�(1�1))�(x�(1))

=

(1+��(1))

�(B�(1)�B�(1�1))�(1��(1))�(B�(1)�B�(1�1))

(1+��(1))�(B�(1)�B�(1�1))+(1��(1))

�(B�(1)�B�(1�1)) ;2v�(B�(1)�B�(1�1))

�(1)

(2�v�(1))�(B�(1)�B�(1�1))+v

�(B�(1)�B�(1�1))

�(1)

!; (7)

�(B�(2) �B�(2�1))�(x�(2))

=

(1+��(2))

�(B�(2)�B�(2�1))�(1��(2))�(B�(2)�B�(2�1))

(1+��(2))�(B�(2)�B�(2�1))+(1��(2))

�(B�(2)�B�(2�1)) ;2v�(B�(2)�B�(2�1))

�(2)

(2�v�(2))�(B�(2)�B�(2�1))+v

�(B�(2)�B�(2�1))

�(2)

!: (8)

Then:

�(B�(1) �B�(1�1))�(x�(1))� �(B�(2) �B�(2�1))�(x�(2))

=

0B@ (1+��(1))�(B�(1)�B�(1�1))�(1��(1))

�(B�(1)�B�(1�1))

(1+��(1))�(B�(1)�B�(1�1))

+(1��(1))�(B�(1)�B�(1�1)) +

(1+��(2))�(B�(2)�B�(2�1))�(1��(2))

�(B�(2)�B�(2�1))

(1+��(2))�(B�(2)�B�(2�1))

+(1��(2))�(B�(2)�B�(2�1))

1+(1+��(1))

�(B�(1)�B�(1�1))�(1��(1))�(B�(1)�B�(1�1))

(1+��(1))�(B�(1)�B�(1�1))

+(1��(1))�(B�(1)�B�(1�1))

(1+��(2))�(B�(2)�B�(2�1))�(1��(2))

�(B�(2)�B�(2�1))

(1+��(2))�(B�(2)�B�(2�1))

+(1��(2))�(B�(2)�B�(2�1))

2v�(B�(1)�B�(1�1))�(1)

(2�v�(1))�(B�(1)�B�(1�1))

+v�(B�(1)�B�(1�1))�(1)

2v�(B�(2)�B�(2�1))�(2)

(2�v�(2))�(B�(2)�B�(2�1))

+v�(B�(2)�B�(2�1))�(2)

1+

0@1� 2v�(B�(1)�B�(1�1))�(1)

(2�v�(1))�(B�(1)�B�(1�1))

+v�(B�(1)�B�(1�1))�(1)

1A0@1� 2v�(B�(2)�B�(2�1))�(2)

(2�v�(2))�(B�(2)�B�(2�1))

+v�(B�(2)�B�(2�1))�(2)

1A1CCCA (9)

=�Q2

j=1(1+��(j) )�(B�(j)�B�(j�1))�Q2j=1(1��(j) )�(B�(j)�B�(j�1))Q2

j=1(1+��(j) )�(B�(j)�B�(j�1))+Q2j=1(1��(j) )�(B�(j)�B�(j�1)) ;

2Q2j=1 v

�(B�(j)�B�(j�1))��(j)Q2

j=1(2�v��(j) )�(B�(j)�B�(j�1))+Q2j=1 v

�(B�(j)�B�(j�1))��(j)

!. (10)

Box II. Continued on the next page.

+nXj=1

�(B�(j) �B�(j�1))

+nXj=1

��(j)�(B�(j) �B�(j�1))

�nXj=1

��(j)�(B�(j) �B�(j�1)) = 2: (16)

Therefore, Eq. (17) is given in Box III.On the other hand, since:

Yn

j=1(2� v��(j))

�(B�(j)�B�(j�1))

+Yn

j=1v�(B�(j)�B�(j�1))��(j)

�nXj=1

�(B�(j)�B�(j�1))(2� v��(j))

+nXj=1

�(B�(j) �B�(j�1))v��(j) = 2; (18)

then Eq. (19) is given in Box IV. Let EIFCA(�1; �2;


If Eq. (6) holds for n = k, that is:

k�j=1

��(B�(j) �B�(j�1))�(x�(j))

�=�Qk

j=1(1+��(j) )�(B�(j)�B�(j�1))�Qkj=1(1��(j) )�(B�(j)�B�(j�1))Qk

j=1(1+��(j) )�(B�(j)�B�(j�1))+Qkj=1(1��(j) )�(B�(j)�B�(j�1)) ;

2Qkj=1 v

�(B�(j)�B�(j�1))��(j)Qk

j=1(2�v��(j) )�(B�(j)�B�(j�1))+Qkj=1 v

�(B�(j)�B�(j�1))��(j)

!; (11)

then, when n = k + 1, by the operational laws of IFVs, we have:

k+1�j=1

��(B�(j) �B�(j�1))�(x�(j))

�=

k�j=1

��(B�(j) �B�(j�1))�(x�(j))

�� (B�(k+1) �B�(k))�(x�(k+1))�

=�Qk+1

j=1 (1+��(j) )�(B�(j)�B�(j�1))�Qk+1j=1 (1��(j) )�(B�(j)�B�(j�1))Qk+1

j=1 (1+��(j) )�(B�(j)�B�(j�1))+Qk+1j=1 (1��(j) )�(B�(j)�B�(j�1)) ;

2Qk+1j=1 v

�(B�(j)�B�(j�1))��(j)Qk+1

j=1 (2�v��(j) )�(B�(j)�B�(j�1))+Qk+1j=1 v

�(B�(j)�B�(j�1))��(j)

!; (12)

i.e. Eq. (6) holds for n = k + 1. Thus, Eq. (6) holds for all n. Then:

EIFCA (�1; �2; � � � ; �n) =�Qn


j=1(1+��(j) )�(B�(j)�B�(j�1))+Qnj=1(1��(j) )�(B�(j)�B�(j�1)) ;

2Qnj=1 v

�(B�(j)�B�(j�1))��(j)Qn

j=1(2�v��(j) )�(B�(j)�B�(j�1))+Qnj=1 v

�(B�(j)�B�(j�1))��(j)

!; (13)

which completes the proof of Theorem 1.�

Box II. Continued.

Qnj=1

�1+��(j)

��(B�(j)�B�(j�1))�Qnj=1

�1��(j)

��(B�(j)�B�(j�1))Qnj=1

�1+��(j)

��(B�(j)�B�(j�1))+Qnj=1

�1��(j)

��(B�(j)�B�(j�1))

= 1� 2Qnj=1

�1��(j)

��(B�(j)�B�(j�1))Qnj=1

�1+��(j)

��(B�(j)�B�(j�1))+Qnj=1

�1��(j)

��(B�(j)�B�(j�1)) � 1�Qnj=1

�1� ��(j)

��(B�(j)�B�(j�1)) ; (17)

where the equality holds if and only if ��(j) (j = 1; 2; � � � ; n) are equal.

Box III.

� � � ; �n) = (��; v�) = � and IFCA(�1; �2; � � � ; �n) =(�0�; v0�) = �0, then Eq. (19) can be transformed to:

�� 0�; and v� � v0�: (20)

Based on Eq. (20), we have:

s(�) = �� v� � �0� � v0� = s(�0): (21)

If s(�) < s(�+), then:

EIFCA(�1; �2;� � �; �n)< IFCA(�1; �2;� � �; �n): (22)

If s(�) = �� v� = �0� � v0� = s(�0), i.e. Eqs. (23)to (25) are given in Box V. Therefore:

h(�) = �� + v� = h0(�) = �0� + v0�: (26)

Thus, is follows that:

EIFCA(�1; �2; � � � ; �n) = IFCA(�1; �2; � � � ; �n):(27)

From Eqs. (22) and (27), we know that Eq. (15) alwaysholds.


2Qnj=1 v

�(B�(j)�B�(j�1))��(j)Qn

j=1(2�v��(j) )�(B�(j)�B�(j�1))+Qnj=1 v

�(B�(j)�B�(j�1))��(j)

�Qnj=1 v

�(B�(j)�B�(j�1))��(j) : (19)

Box IV.

Qnj=1(1+��(j) )�(B�(j)�B�(j�1))�Qn

j=1(1��(j) )�(B�(j)�B�(j�1))Qnj=1(1+��(j) )�(B�(j)�B�(j�1))+

Qnj=1(1��(j) )�(B�(j)�B�(j�1)) � 2

Qnj=1 v

�(B�(j)�B�(j�1))��(j)Qn

j=1(2�v��(j) )�(B�(j)�B�(j�1))+Qnj=1 v

�(B�(j)�B�(j�1))��(j)

=�

1�Qnj=1

�1� ��(j)

��(B�(j)�B�(j�1))��Qn

j=1 v�(B�(j)�B�(j�1))��(j) j = 1; 2; � � � ; n (23)

Then we have:

Qnj=1(1+��(j) )�(B�(j)�B�(j�1))�Qn

j=1(1��(j) )�(B�(j)�B�(j�1))Qnj=1(1+��(j) )�(B�(j)�B�(j�1))+

Qnj=1(1��(j) )�(B�(j)�B�(j�1)) =

�1�Qn

j=1�1� ��(j)

��(B�(j)�B�(j�1))�

(24)

2Qnj=1 v

�(B�(j)�B�(j�1))��(j)Qn

j=1(2�v��(j) )�(B�(j)�B�(j�1))+Qnj=1 v

�(B�(j)�B�(j�1))��(j)

=Qnj=1 v

�(B�(j)�B�(j�1))��(j) (25)

Box V.

Theorem 3 (Idempotency). Let � be a fuzzymeasure on X, �j = (��j ; v�j ) (j = 1; 2; � � � ; n) be acollection of IFVs and ��(j) be the jth largest of them.If all �j (j = 1; 2; � � � ; n) are equal, i.e. �j = �, for allj, then:

EIFCA(�1; �2; � � � ; �n) = �: (28)

Proof. By De�nition 4, we have:

EIFCA(�1; �2; � � � ; �n)

=n�j=1

��(B�(j) �B�(j�1))�(x�(j))

�=��(B�(1) �B�(1�1))�(x�(1))

�� (B�(2) �B�(2�1))�(x�(2))

�� (B�(n) �B�(n�1))�(x�(n))

�=��(B�(1) �B�(1�1))�

�� (B�(2) �B�(2�1))�

� � � �� (B�(n) �B�(n�1))�

�=

0@ nXj=1

��(B�(j) �B�(j�1))

��

1A=��(B�(n) �B�(1�1))�

�= �: (29)

Theorem 4 (Boundeness). Let � be a fuzzy mea-sure on X, �j = (��j ; v�j ) (j = 1; 2; � � � ; n) be acollection of IFVs and ��(j) be the jth largest of them,and also let:

�� =�

minj

(��(j));maxj

(v��(j))�;

�+ =�

maxj

(��(j));minj

(v��(j))�:

Then:

�� EIFCA (�1; �2; � � � ; �n) � �+: (30)

Proof. Let f(x) = 1�x1+x , x 2 [0; 1], then f 0(x) =

�2(1+x)2 < 0, i.e. f(x) is a decreasing function. Since,minj

(��(j)) � ��j � maxj

(��(j)) for all j, then:

f(maxj

(��(j))) � f(��(j)) � f(minj

(��(j))); (31)

i.e.:

1�maxj

(��(j))

1 + maxj

(��(j))� 1� (��(j))

1 + (��(j))�

1�minj

(��(j))

1 + minj

(��(j));

j = 1; 2; � � � ; n: (32)


Since �(B�(j) �B�(j�1)) � 0 for all j, then we have:

0@1�maxj

(��(j))

1 + maxj

(��(j))

1A�(B�(j)�B�(j�1))

�

1� (��(j))1 + (��(j))

!�(B�(j)�B�(j�1)

�0@1�min

j(��(j))

1 + minj

(��(j))

1A�(B�(j)�B�(j�1))

;

j = 1; 2; � � � ; n: (33)

Thus:

Yn

j=1

0@1�maxj

(��(j))

1 + maxj

(��(j))

1A�(B�(j)�B�(j�1))

�Yn

j=1

1� (��(j))1 + (��(j))

!�(B�(j)�B�(j�1))

�Yn

j=1

0@1�minj

(��(j))

1 + minj

(��(j))

1A�(B�(j)�B�(j�1))

;(34)

)0@1�max

j(��(j))

1 + maxj

(��(j))

1A nPj=1

�(B�(j)�B�(j�1))

�Yn

j=1

1� (��(j))1 + (��(j))

!�(B�(j)�B�(j�1))

�0@1�min

j(��(j))

1 + minj

(��(j))

1A nPj=1

�(B�(j)�B�(j�1))

; (35)

)1�max

j(��(j))

1 + maxj

(��(j))

�Yn

j=1

1� (��(j))1 + (��(j))

!�(B�(j)�B�(j�1))

�1�min

j(��(j))

1 + minj

(��(j)); (36)

) 21 + max

j(��(j))

� 1 +Yn

j=1

1� (��(j))1 + (��(j))

!�(B�(j)�B�(j�1))

� 21 + min

j(��(j))

; (37)

)1 + min

j(��(j))

2

� 1

1 +Qnj=1

�1�(��(j) )1+(��(j) )

��(B�(j)�B�(j�1))

�1 + max

j(��(j))

2; (38)

)1 + minj

(��(j))

� 2

1 +Qnj=1

�1�(��(j) )1+(��(j) )

��(B�(j)�B�(j�1))

� 1 + maxj

(��(j)); (39)

)minj

(��(j))

� 2

1 +Qnj=1

�1�(��(j) )1+(��(j) )

��(B�(j)�B�(j�1))� 1

� maxj

(��(j)): (40)

And Eq. (41) is shown in Box VI.Let g(y) = 2�y

y , y 2 (0; 1], then g0(y) = �2y2 < 0,

i.e. g(y) is a decreasing function. Since minj

(v��(j)) �v�j � max

j(v��(j)), for all j, then:

f�

maxj

(v��(j))�� f �v��(j)

� � f �minj

(v��(j))�;(42)

i.e.:

2�maxj

(v��(j))

maxj

(v��(j))� 2� v��(j)

v��(j)

�2�min

j(v��(j))

minj

(v��(j));

j = 1; 2; � � � ; n: (43)


) minj(��(j)) �Qnj=1 (1+��(j) )�(B�(j)�B�(j�1))�Qn

j=1 (1��(j) )�(B�(j)�B�(j�1))Qnj=1 (1+��(j) )�(B�(j)�B�(j�1))+

Qnj=1 (1��(j) )�(B�(j)�B�(j�1)) � maxj(��(j)): (41)

Box VI.

Since �(B�(j) �B�(j�1)) � 0 for all j, then we have:0@2�maxj

(v��(j))

maxj

(v��(j))

1A�(B�(j)�B�(j�1))

�

2� v��(j)

v��(j)

!�(B�(j)�B�(j�1))

�0@2�min

j(v��(j))

minj

(v��(j))

1A�(B�(j)�B�(j�1))

; (44)

)Yn

j=1

0@2�maxj

(v��(j))

maxj

(v��(j))

1A�(B�(j)�B�(j�1))

�Yn

j=1

2� v��(j)v��(j)

!�(B�(j)�B�(j�1))

�Yn

j=1

0@2�minj

(v��(j))

minj

(v��(j))

1A�(B�(j)�B�(j�1))

;(45)

)0@2�max

j(v��(j))

maxj

(v��(j))

1A nPj=1

�(B�(j)�B�(j�1))

�

2� v��(j)

v��(j)

! nPj=1

�(B�(j)�B�(j�1))

�0@2�min

j(v��(j))

minj

(v��(j))

1A nPj=1

�(B�(j)�B�(j�1))

; (46)

)2�max

j(v��(j))

maxj

(v��(j))

�

2� v��(j)

v��(j)

! nPj=1

�(B�(j)�B�(j�1))

�2�min

j(v��(j))

minj

(v��(j)); (47)

) 2maxj

(v��(j))

�

2� v��(j)

v��(j)

! nPj=1

�(B�(j)�B�(j�1))

+ 1

� 2minj

(v��(j)); (48)

)minj

(v��(j))

2

� 1�2�v��(j)v��(j)

� nPj=1

�(B�(j)�B�(j�1))

+ 1

�maxj

(v��(j))

2; (49)

)minj

(v��(j))

� 2�2�v��(j)v��(j)

� nPj=1

�(B�(j)�B�(j�1))

+ 1

� maxj

(v��(j)); (50)

and Eq. (51) is shown in Box VII. Let:

EIFCA(�1; �2; � � � ; �n) = � = (��; v�); (52)

we have:

minj

(��(j)) � �� maxj

(��(j)); (53)

minj

(v��(j)) � v� � maxj

(v��(j)); (54)

s(�)=��v��maxj

(��(j))�minj

(v��(j))=s(�+);(55)

s(�)=��v� � minj

(��(j))�maxj

(v��(j))=s(��):(56)


) minj

(v��(j)) � 2Qnj=1 v

�(B�(j)�B�(j�1))��(j)Qn

j=1(2�v��(j) )�(B�(j)�B�(j�1))+Qnj=1 v

�(B�(j)�B�(j�1))��(j)

� maxj

(v��(j)): (51)

Box VII.

If s(�) < s(�+) and s(�) > s(��), then by orderrelation between two IFVs which was described inSection 2, we have:

�� < EIFCA(�1; �2; � � � ; �n) < �+: (57)

If s(�) = s(�+), i.e. �� v� = maxj

(��(j)) �minj

(v��(j)), we have:

�� = maxj

(��(j)); v� = minj

(v��(j)): (58)

Therefore:

h(�)=��+v�=maxj

(��(j))+minj

(v��(j))=h(�+);(59)

then:

EIFCA(�1; �2; � � � ; �n) = �+: (60)

If s(�)=s(��), i.e. ��v�=minj

(��(j))�maxj

(v��(j)),

then we have:

�� = minj

(��(j)); v� = maxj

(v��(j)): (61)

Therefore:

h(�)= ��+��=minj

(��(j))+maxj

(��(j))=h(��):(62)

Thus, it follows that:

EIFCA(�1; �2; � � � ; �n) = ��: (63)

From Eqs. (57), (60) and (63), we know that Eq. (30)always holds. It should be noted that the proof ofTheorems 2 and 4 are referred to the proved methodsprovided by Wang and Liu [44].

The Monotonicity of the EIFCA operator can beobtained by a similar proving method.

Theorem 5 (Monotonicity). Let � be a fuzzymeasure on X, �j = (��j ; v�j ) (j = 1; 2; � � � ; n) and��j = (��j ; v��j ) (j = 1; 2; � � � ; n) be two collections ofIFVs, ��(j) be the jth largest of �j(j = 1; 2; � � � ; n)and ��(j) be the jth largest of ��j (j = 1; 2; � � � ; n),� > 0. If ��(j) � ��(j)

and v��(j) � v��(j), for all j,

then:

EIFCA(�1; �2; � � � ; �n) � EIFCA(��1; ��2; � � � ; ��n):(64)

Theorem 6. Let � be a fuzzy measure on X, �j =(��j ; v�j ) (j = 1; 2; � � � ; n) be a collection of IFVs and��(j) be the jth largest of them. If(�01; �02; � � � ; �0n) isany permutation of �j (j = 1; 2; � � � ; n), then:

EIFCA(�1; �2; � � � ; �n) = EIFCA(�01; �02; � � � ; �0n):(65)

Proof. According to De�nition 4, we have:

EIFCA(�(x1); �(x2); � � � ; �(xn))

=n�j=1

��(B�(j) �B�(j�1))�(x�(j))

�; (66)

and:

EIFCA(�0(x1); �0(x2); � � � ; �0(xn))

=n�j=1

��(B�(j) �B�(j�1))�0(x�(j))

�: (67)

Since (�01; �02; � � � ; �0n) is any permutation of �j , wehave:

�(��(i)) = �0(��(i)i = 1; 2; � � � ; n: (68)

Therefore, EIFCA(�1; �2; � � � ; �n) = EIFCA(�01; �02;� � � ; �0n), which completes the proof of Theorem 6.From De�nition 4, the following property of the

EIFCA operator can be obtained easily.

Property 1 (Shift-invariance). Let � be a fuzzymeasure on X and �j = (��j ; v�j ) (j = 1; 2; � � � ; n) bea collection of IFVs. If � = (�� ; ��) is an intuitionisticfuzzy value on X, then:

EIFCA(�1 � �; �2 � �; � � � ; �n � �)

= EIFCA(�1; �2; � � � ; �n)� �: (69)

Property 2 (Homogeneity). Let � be a fuzzymeasure on X and �j = (��j ; v�j ) (j = 1; 2; � � � ; n)be a collection of IFVs. If � > 0, then:

EIFCA(��1; ��2; � � � ; ��n)

= � EIFCA(�1; �2; � � � ; �n): (70)


Property 3. Let � be a fuzzy measure onX and �j =(��j ; v�j ) (j = 1; 2; � � � ; n) be a collection of IFVs. If� = (�� ; ��) is an intuitionistic fuzzy value on X andif � > 0, then:

EIFCA(��1 � �; ��2 � �; � � � ; ��n � �)

= � EIFCA(�1; �2; � � � ; �n)� �: (71)

Property 4. Let � be a fuzzy measure on X, �j =(��j ; v�j ) and �j = (��j ; v�j ) (j = 1; 2; � � � ; n) be twocollections of IFVs, then:

EIFCA(�1 � �1; �2 � �2; � � � ; �n � �n)

= EIFCA(�1; �2; � � � ; �n)

� EIFCA(�1; �2; � � � ; �n): (72)

4. An approach to multi-criteria decisionmaking under intuitionistic fuzzyenvironment

The multi-criteria decision making is a very practicalmethod in the real world, and have very signi�cante�ect on both theory and practical. It aims to �ndthe best alternatives from a �nite number of options.Multi-criteria decision making, using IFVs, is an impor-tant variety of decision making theory [15-18]. In thefollowing, we utilize the EIFCA operator to develop anapproach for multi-criteria decision making using IFVs,which includes the following steps:

Step 1. The intuitionistic fuzzy decision makingmatric B = (bij)m�n always needs to be standardized,since there are cost attributes (the smaller the attributevalues the better). Xu and Hu [51] have proposed astandardization method.

rij = (�ij ; vij) =

(bij ; for bene�t attribute Cj�bij ; for cos t attribute Cj

i = 1; 2; � � � ;m j = 1; 2; � � � ; n; (73)

where �bij is the complement of bij such that:

�bij = (fij ; tij);

i = 1; 2; � � � ;m; j = 1; 2; � � � ; n: (74)

Based on Eqs. (94) and (95), the standardized decisionmaking matric R = (rij)m�n can be obtained.

Step 2. By the order relation between IFVs, rij isreordered such that ri(1) � ri(2) � � � � � ri(n).

Step 3. Based on Choquet integral, calculate thecorrelations between the criteria, using the methodgiven in Section 2 [10,11].

Step 4. Utilize the EIFCA operator to aggregateall the performance values rij(j = 1; 2; � � � ; n) of theith line, and get the overall performance value ricorresponding to the alternative Ai.

Step 5. Calculate the score function and accuracyfunction of the overall values ri (i = 1; 2; � � � ;m), thenutilize order relation between IFVs described in Section2 to rank the overall performance values ri and selectthe best one.

Example 1. Suppose a multinational corporationin China is planning its �nancial strategy for thenext year, according to the group strategy objective.After preliminary screening, the four alternatives areproduced.

A1: To invest in the Southeast Asian markets;

A2: To invest in the Eastern European market;

A3: To invest in the North American market;

A4: To invest in the local market.

This evaluation proceeds from the following four as-pects, such as the growth analysis (c1), the riskanalysis (c2), the sociopolitical impact analysis (c3)and the environmental impact analysis (c4). The fouralternatives Ai (i = 1; 2; � � � ; 4) are to be evaluatedby corresponding experts, using the IFVs, as shown inTable 1.

In order to choose the most appropriate invest-ment program, the main steps are as follows (based onthe EIFCA operator):

Step 1: Since the criteria c2 and c4 are the costcriteria, the decision matrix need normalization. Nor-malized decision matrix is shown in Table 2.

Step 2: According to Table 2, by the order relationbetween IFVs, the evaluation rij of the candidate Aisuch that ri(1) � ri(2) � � � � � ri(n) (i = 1; 2; � � � ; 4), is

Table 1. The evaluation information on the projects.

c1 c2 c3 c4

A1 (0.6,0.2) (0.2,0.8) (0.8,0.1) (0.5,0.3)A2 (0.4,0.1) (0.1,0.6) (0.5,0.2) (0.2,0.8)A3 (0.7,0.3) (0.2,0.8) (0.6,0.3) (0.1,0.4)A4 (0.5,0.5) (0.1,0.9) (0.8,0.1) (0.2,0.7)


Table 2. The normalized evaluation information on theprojects.

c1 c2 c3 c4

A1 (0.6,0.2) (0.8,0.2) (0.8,0.1) (0.3,0.5)A2 (0.4,0.1) (0.6,0.1) (0.5,0.2) (0.8,0.2)A3 (0.7,0.3) (0.8,0.2) (0.6,0.3) (0.4,0.1)A4 (0.5,0.5) (0.9,0.1) (0.8,0.1) (0.7,0.2)

reordered as follows:

r1(1) = (0:8; 0:1); r1(2) = (0:8; 0:2);

r1(3) = (0:6; 0:2); r1(4) = (0:3; 0:5);

r2(1) = (0:8; 0:2); r2(2) = (0:6; 0:1);

r2(3) = (0:5; 0:2); r2(4) = (0:4; 0:1);

r3(1) = (0:8; 0:2); r3(2) = (0:7; 0:3);

r3(3) = (0:6; 0:3); r3(4) = (0:4; 0:1);

r4(1) = (0:9; 0:1); r4(2) = (0:8; 0:1);

r4(3) = (0:7; 0:2); r4(4) = (0:5; 0:5):

Step 3. Suppose the fuzzy measures of criteria of Cand criteria sets of C are as follows:

�(;) = 0; �(c1) = 0:38; �(c2) = 0:27;

�(c3) = 0:36; �(c4) = 0:21;

�(c1; c2) = 0:77; �(c1; c3) = 0:64;

�(c1; c4) = 0:51; �(c2; c3) = 0:44;

�(c2; c4) = 0:31; �(c3; c4) = 0:45;

�(c1; c2; c3) = 0:83; �(c1; c2; c4) = 0:69;

�(c1; c3; c4) = 0:78; �(c2; c3; c4) = 0:55;

�(c1; c2; c3; c4) = 1:

Step 4. Utilize the EIFCA operator to aggregateall the performance values rij(j = 1; 2; � � � ; 4) of theith line, and get the overall performance value ricorresponding to the alternative Ai (i = 1; 2; 3; 4).r1is calculated and shown in Box VIII. Similarly:

r2 = (0:5804; 0:1372); r3 = (0:6879; 0:2251);

r4 = (0:7291; 0:2318);

Step 5. Calculate the scores of ri (i = 1; 2; 3; 4),respectively:

S1 = 0:4819; S2 = 0:4432;

S3 = 0:4628; S4 = 0:4973:

Since:

S4 > S1 > S3 > S2;

we have:

A4 � A1 � A3 � A2:

Hence, the best �nancial strategy is A4, i.e. to investin the local market.

If we use the IFCA operator proposed by Xu [42]and Tan and Chen [10] (i.e., Eq. (2) described inSection 2) to get the overall values r00i of the optionsAi (i = 1; 2; 3; 4) then:

r001 = (0:6757; 0:1821); r002 = (0:5915; 0:1366);

r003 = (0:6922; 0:2231); r004 = (0:7381; 0:2227):

According to the overall values r00i (i = 1; 2; 3; 4), bythe order relations of IFVs, we can obtain that:

r004 > r001 > r003 > r002 ;

i.e.:

A4 � A1 � A3 � A2:

Hence, the best �nancial strategy is A4, i.e. to investin the local market. The optimal �nancial strategy andthe ranking of the �nancial strategies are the same asthe ones obtained by EIFCA Operator. Furthermore,we �nd that:

r1 = (0:6676; 0:1857) < (0:6757; 0:1821) = r001 ;

r2 = (0:5804; 0:1372) < (0:5915; 0:1366) = r002 ;

r3 = (0:6879; 0:2251) < (0:6922; 0:2231) = r003 ;

r4 = (0:7291; 0:2318) < (0:7381; 0:2227) = r004 :

These results satisfy Theorem 2.

5. Concluding remarks

In this paper, we have proposed the Einstein-basedIntuitionistic Fuzzy Choquet Averaging (EIFCA) oper-ator. Various properties of EIFCA operator have beenstudied in this paper. Furthermore, the relationshipsbetween EIFCA operator and some existing operatorsuch as IFCA have been discussed. The EIFCAoperator were distinguished from the existing operators


r1 =�

(1+0:8)0:36(1+0:8)0:44�0:36(1+0:6)0:83�0:44(1+0:3)1�0:83�(1�0:8)0:36(1�0:8)0:44�0:36(1�0:6)0:83�0:44(1�0:3)1�0:83

(1+0:8)0:36(1+0:8)0:44�0:36(1+0:6)0:83�0:44(1+0:3)1�0:83+(1�0:8)0:36(1�0:8)0:44�0:36(1�0:6)0:83�0:44(1�0:3)1�0:83 ,

2�0:10:360:2(0:44�0:36)0:2(0:83�0:44)0:5(1�0:83)

(2�0:1)0:36(2�0:2)(0:44�0:36)(2�0:2)(0:83�0:44)(2�0:5)(1�0:83)+0:10:360:2(0:44�0:36)0:2(0:83�0:44)0:5(1�0:83)

�= (0:6676; 0:1857)

Box VIII.

not only due to the EIFCA operator, using the Einsteinoperations, but also due to the consideration of theinter-dependent phenomena among the evaluated cri-teria, which makes the method proposed in this paperto have more wide practical application potentials.The results of this paper can be extended to the dualhesitant fuzzy environment and applied to supplierselection, personnel evaluation and so on.

Acknowledgments

This work has been supported by China NationalNatural Science Foundation (No.71301142) and Zhe-jiang Natural Science Foundation of China (No.LQ13G010004).

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Biography

Dejian Yu received the BS degree in Information Man-agement and Information System from Heilongjianginstitute of Science and Technology, Harbin, China,

in 2006. The MS degree in System Engineering fromNanjing University of Aeronautics and Astronautics,Nanjing, China, in 2010, and the PhD degree inManagement Science and Engineering from SoutheastUniversity, Nanjing, China, in 2012. He is currently aLecturer with the School of Information, Zhejiang Uni-versity of Finance and Economics, Hangzhou, China.His current research interests include aggregation op-erators, information fusion, and multi-criteria decisionmaking.

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