Knowledge-Based Systems 24 (2011) 197–209

Contents lists available at ScienceDirect

Knowledge-Based Systems

journal homepage: www.elsevier .com/ locate /knosys

Induced generalized intuitionistic fuzzy operators

Zeshui Xu ⇑, Meimei XiaSchool of Economics and Management, Southeast University, Nanjing, Jiangsu 210096, China

a r t i c l e i n f o

Article history:Received 10 February 2010Received in revised form 16 April 2010Accepted 16 April 2010Available online 26 August 2010

Keywords:Fuzzy setsDecision makingAggregation operatorChoquet integralDempster–Shafer theory

0950-7051/$ - see front matter � 2010 Published bydoi:10.1016/j.knosys.2010.04.010

⇑ Corresponding author. Tel.: +86 25 84483382.E-mail addresses: xu_zeshui@263.net (Z. Xu), meim

a b s t r a c t

We study the induced generalized aggregation operators under intuitionistic fuzzy environments.Choquet integral and Dempster–Shafer theory of evidence are applied to aggregate inuitionistic fuzzyinformation and some new types of aggregation operators are developed, including the induced general-ized intuitionistic fuzzy Choquet integral operators and induced generalized intuitionistic fuzzy Demp-ster–Shafer operators. Then we investigate their various properties and some of their special cases.Additionally, we apply the developed operators to financial decision making under intuitionistic fuzzyenvironments. Some extensions in interval-valued intuitionistic fuzzy situations are also pointed out.

� 2010 Published by Elsevier B.V.

1. Introduction

Information aggregation is a pervasive activity in our daily life,for instance, as the exam results of a course of all students in a classare known, the teacher of the course usually needs to calculate thegrade point average and total score of the students; a companymay get the quarterly and annual profits by aggregating its dayprofits. In many practical situations, the input arguments may becorrelative, that is, it is necessary to consider the interrelation ofthese arguments, for example, ‘‘we are to evaluate a set of studentsin relation to three subjects: {mathematics, physics, literature}, wewant to give more importance to science-related subjects than toliterature, but on the other hand we want to give some advantageto students that are good both in literature and in any of the sci-ence-related subjects” [11,21]. In such cases, we may choose aproper operator or function which can capture the interaction ofthe arguments. Choquet integral [7] is just of this type, whichexplicitly models the importance of not only individual arguments,but of their subsets, as well as various interactions between thearguments [3].

The Choquet integral, originally introduced by Choquet [7], pro-vides a type of operator used to measure the expected utility of anuncertain event, and has been applied in many fields (e.g.,[38,13,22]). Yager [40,41] introduced the idea of order inducedaggregation to the Choquet aggregation operator and defined aninduced Choquet ordered averaging operator, which allows the

Elsevier B.V.

xia@163.com (M. Xia).

ordering of the arguments to be based upon some other associatedvariables instead of ordering the arguments based on their values.From another point of research, Yager [42] extended the Choquetintegral to a more general form by adding a parameter controllingthe power to which the argument values are raised. Tan and Chen[23] developed the induced Choquet ordered averaging operatorand applied it to aggregate fuzzy preference relations in groupdecision making.

Another important issue is the problem of decision makingunder uncertainty in which we are concerned with the selectionof decision alternatives. In uncertain environments, the uncer-tainty manifests itself in that a different payoff is obtained for dif-ferent states of nature. The Dempster–Shafer theory of evidence,which was developed by Dempster [8,9], plays a crucial role in pro-viding a unifying framework for representing the uncertainty, as itcan include situations of risk and ignorance in the same formula-tion, and has been used in a wide range of applications (e.g.,[18,19]). Yager [36] provided a methodology for selecting optimalalternatives in situations in which our knowledge about the uncer-tainty is contained in a Dempster–Shafer belief structure using theordered weighted aggregation (OWA) operators [35]. Then Merigóand Casanovas [17] suggested the use of the induced aggregationoperators in decision making with the Dempster–Shafer theoryfor the situations in which some experts prefer to aggregate thevariables with an inducing order instead of aggregating with thetraditional OWA operator.

In the above studies, all the input arguments aggregated byusing the usual Choquet integral, generalized Choquet integral,induced Choquet ordered averaging operator, Dempster–Shaferbelief structure, and induced Dempster–Shafer belief structure

198 Z. Xu, M. Xia / Knowledge-Based Systems 24 (2011) 197–209

operators are generally crisp numbers rather than other types ofarguments. However, many decision making problems cannot beassessed with crisp numbers because the knowledge of the deci-sion maker is vague or imprecise. Atanassov [1] introduced thenotion of intuitionistic fuzzy set, whose basic elements are intui-tionistic fuzzy values (IFVs) ([34,31]). Each IFV is characterizedby a membership degree and a non-membership degree, whichsatisfies the condition that their sum is smaller or equal to 1. TheIFV is highly useful in depicting uncertainty and vagueness of anobject, and thus can be used as a powerful tool to express datainformation under various different fuzzy environments whichhas attracted great attentions (e.g., [26,28,14,15,43], etc.).

In this paper, we extend the generalized Choquet integral andDempster–Shafer belief structure to aggregate intuitionistic fuzzyinformation, and develop some new types of aggregation opera-tors, including the induced generalized intuitionistic fuzzy Cho-quet integral operators and induced generalized intuitionisticfuzzy Dempster–Shafer operators. Then we give their applicationsin decision making under intuitionistic fuzzy environments. Fur-thermore, we extend them to accommodate interval-valued intui-tionistic fuzzy situations.

2. Basic knowledge review

2.1. Intuitionistic fuzzy sets

Let a set X = {x1,x2, . . . ,xn} be fixed. Atanossov [1] defined anintuitionistic fuzzy set (A-IFS) as A = {<xi, tA(xi), fA(xi) > jxi 2 X},which concludes two elements: a membership degree tA(xi) and anon-membership degree fA(xi) with the condition 0 6 tA(xi) + fA(xi)6 1, for all xi 2 X. The pair (tA(xi), fA(xi)) is called an intuitionisticfuzzy value (IFV) [34,31], and each IFV can be simply denoted asai ¼ ðtai

; faiÞ where tai

2 ½0;1�, fai2 ½0;1�, tai

þ fai6 1. Additionally,

sai¼ tai

� faiand hai

¼ taiþ fai

are called the score and accuracy de-gree of ai respectively.

For any three IFVs a = (ta, fa), a1 ¼ ðta1 ; fa1 Þ and a2 ¼ ðta2 ; fa2 Þ, thefollowing operational laws are valid.

(1) a1 � a2 ¼ ðta1 þ ta2 � ta1 ta2 ; fa1 fa2 Þ;(2) a1 � a2 ¼ ðta1 ta2 ; fa1 þ fa2 � fa1 fa2 Þ;(3) ka = (1 � (1 � ta)k, fa k), k > 0;(4) ak ¼ ðtk

a;1� ð1� faÞkÞ; k > 0.

To compare any two IFVs a1 and a2, Xu and Yager [34], and Xu[31] introduced a simple method as below:

(1) If sa1 < sa2 , then a1 < a2;(2) If sa1 ¼ sa2 , then

(a) If ha1 ¼ ha2 , then a1 = a2;(b) If ha1 < ha2 , then a1 < a2.

2.2. Interval-valued intuitionistic fuzzy sets

Atanassov and Gargov [2] further generalized A-IFS to interval-valued intuitionistic fuzzy set (A-IVIFS), and defined an A-IVIFS aseA ¼ f< xi;~teAðxiÞ;~feAðxiÞ >j xi 2 Xg, in which the membership degree

~teAðxiÞ � ½0;1� and the non-membership degree ~teAðxiÞ � ½0;1� are

intervals, which satisfy sup~teAðxiÞ þ sup ~feAðxiÞ 6 1, for every xi 2 X.

Xu and Chen [33] called 2-tuple ð~t~aðxiÞ;~f ~aðxiÞÞ an interval-valuedintuitionistic fuzzy value (IVIFV), and for convenience, denoted an

IVIFV by ~ai ¼ t�~ai; tþ~ai

h i; f�~ai

; fþ~ai

h i� �, where t�~ai

; tþ~ai

h i� ½0;1�, f�~ai

; fþ~ai

h i�

½0;1�, tþ~aiþ fþ~ai

6 1. For any three IVIFVs ~a ¼ t�~a ; tþ~a

� �; f�~a ; f

þ~a

� �� �; ~a1 ¼

t�~a1; tþ~a1

h i; f�~a1

; fþ~a1

h i� �and ~a2 ¼ t�~a2

; tþ~a2

h i; f�~a2

; fþ~a2

h i� �, the following

operations can be given:

(1) ~a1 � ~a2 ¼ t�~a1þ t�~a2

� t�~a1t�~a2; tþ~a1þ tþ~a2

� tþ~a1tþ~a2

h i; f�~a1

f�~a2;

h�

fþ~a1

fþ~a2; �Þ;h i h�

(2) ~a1 � ~a2 ¼ t�~a1t�~a2; tþ~a1

tþ~a2; f�~a1

þ f�~a2� f�~a1

f�~a2; fþ~a1þ fþ~a2

�fþ~a1fþ~a2�Þ;h i h i� �

(3) k~a ¼ 1� 1� t�~a� �k

;1� 1� tþ~a� �k

; f�~a� �k

; fþ~a� �k , k > 0;

(4) ~ak ¼ t�~a� �k

; tþ~a� �kh i

; 1� 1� f�~a� �k

;1� 1� fþ~a� �kh i� �

, k > 0.

Similar to the comparison method of IFVs, we introduce a meth-od for comparing any two IVIFVs ~a1 and ~a2 as follows [33]:

Let s~ai¼ t�~ai

� f�~aiþ tþ~ai

� fþ~ai

� �=2ði ¼ 1;2Þ be the scores of

~aiði ¼ 1;2Þ respectively, and h~ai¼ t�~ai

þ f�~aiþ tþ~ai

þ fþ~ai

� �=2ði ¼ 1;2Þ

be the accuracy degrees of ~aiði ¼ 1;2Þ, respectively, then,(1) If s~a1 > s~a2 , then ~a1 is larger than ~a2, denoted by ~a1 > ~a2;(2) If s~a1

¼ s~a2, then

(a) If h~a1 ¼ h~a2 , then there is no difference between ~a1 and~a2, denoted by ~a1 � ~a2;

(b) If h~a1 > h~a2 , then ~a1 is larger than ~a2, denoted by ~a1 > ~a2.

3. Operators based on the Choquet integral

Many aggregation operators for IFVs have been developed inwhich the weights of the corresponding elements are indepen-dent (e.g., [34,31]). However, in some actual situations, the aggre-gated arguments usually have some correlations with each other,the Choqut integral is an interesting and useful tool to deal withthis issue, which has attracted considerable research interestsin the last decades. Recently, some studies (e.g., [39,16,23]) havebeen done about a type of induced aggregation operators, whichtake as their argument pairs, in which one components calledorder-inducing variables are used to induce an ordering overthe second components which are the aggregated variables.In the existing studies, all the input arguments aggregated byusing the operators are generally crisp numbers. Consider thatthe IFVs are a powerful tool to depict uncertainty and vaguenessof objects under various different fuzzy environments. Xu [32],Tan and Chen [24] developed some inuitionistic fuzzy correlatedoperators based on Choquet integral. According to the idea of in-duced aggregation, in this section we develop some more generaloperators (i.e., the induced generalized intuitionistic fuzzy Cho-quet integral operators and induced generalized intuitionistic fuz-zy Dempster–Shafer operators) to aggregate intuitionistic fuzzyinformation.

3.1. Induced generalized intuitionistic fuzzy Choquet orderedaveraging operators

We first introduce the concept of fuzzy measure [20,25,10]which is the basis for further research.

Definition 1. A fuzzy measure m on the set X is a set functionm: #(X) ? [0,1] satisfying the following axioms:

(1) m(/) = 0, m(X) = 1;(2) B # C implies m(B) 6m(C), for all B,C # X;(3) m(B [ C) = m(B) + m(C) + qm(B)m(C), for all B,C # X and

B \ C = /, where q 2 (�1,1).

Especially, if q = 0, then (3) in Definition 1 reduces to theaxiom of additive measure m(B [ C) = m(B) + m(C), which indi-cates that there is no interaction between B and C; if q > 0, then

Z. Xu, M. Xia / Knowledge-Based Systems 24 (2011) 197–209 199

m(B [ C) > m(B) + m(C), which implies that the set {B,C} hasmultiplicative effect; if q < 0, then m(B [ C) < m(B) + m(C), whichimplies that the set {B,C} has substitutive effect, by parameter q,the interaction between sets or elements of set can berepresented.

Let X = {x1,x2, . . . ,xn} be a finite set, then [ni¼1xi ¼ X. To deter-

mine fuzzy measure on X avoiding the computational complexity,Sugeno [20] gave the following equation:

mðXÞ ¼ m [ni¼1xi

� �¼

1q

Qni¼1ð1þ qmðxiÞÞ � 1

� ; q – 0;

Pni¼1

mðxiÞ; q ¼ 0;

8>>><>>>: ð1Þ

the value of q can be uniquely determined from m(X) = 1, which canbe written as

qþ 1 ¼Yn

i¼1

ð1þ qmðxiÞÞ: ð2Þ

Especially, for every subset A # X, we have

mðAÞ ¼1q

Qxi2Að1þ qmðxiÞÞ � 1

!; q – 0;P

xi2AmðxiÞ; q ¼ 0:

8>>><>>>: ð3Þ

Let V be the set of all IFVs. Based on Definition 1 and the idea of or-der induced aggregation [40,41], in the following, we utilize theChoquet integral to develop some generalized aggregation opera-tors for IFVs.

Definition 2. Let X = {x1,x2, . . . ,xn} be a finite set, m be a fuzzy mea-sure on X defined by Eq. (3), and ai ¼ ðtai

; faiÞði ¼ 1; 2; . . . ; nÞ be a

collection of IFVs on X. An induced generalized intuitionistic fuzzyChoquet ordered averaging (I-GIFCOA) operator of dimension n is afunction I-GIFCOA: Vn ? V, which is defined to aggregate the set ofsecond arguments of a collection of 2-tuples (hu1,a1i, hu2,a2i,. . . , hun,ani) according to the following expression:

I� GIFCOAkðhu1;a1i; hu2;a2i; . . . ; hun;aniÞ

¼�n

i¼1 ðmðAðiÞÞ �mðAði�1ÞÞÞakðiÞ

� �� �1=k; k > 0;

�ni¼1 aðmðAðiÞÞ�mðAði�1ÞÞÞ

ðiÞ

� �; k ¼ 0;

8<: ð4Þ

where ui in 2-tuple hui,aii is referred to as the order-inducing vari-able and ai as the argument variable, (i): {1,2, . . . ,n} ? {1,2, . . . ,n} isa permutation such that u(1) P u(2) P � � �P u(n), A(i) = {x(1),x(2), . . . ,x(i)} when i P 1 and A(0) = ;.

According to Xu and Yager [34], and Zhao et al. [44], we can get

I�GIFCOAkðhu1;a1i; hu2;a2i; . . . ; hun;aniÞ

¼

1�Qni¼1

1� tkaðiÞ

� �mðAðiÞ Þ�mðAði�1ÞÞ� 1=k

;

1� 1�Qni¼1

1� ð1� faðiÞ Þk

� �mðAðiÞ Þ�mðAði�1ÞÞ� 1=k

!; k> 0;

Qni¼1

taðiÞ� �mðAðiÞÞ�mðAði�1Þ Þ

;1�Qni¼1

1� faðiÞ� �mðAðiÞÞ�mðAði�1Þ Þ

� ; k¼ 0:

8>>>>>>>>><>>>>>>>>>:ð5Þ

Especially, if there exist two 2-tuples hui,aii and huj,aji such thatui = uj, then we can replace the arguments of the tied 2-tuples bythe average of the arguments of the tied 2-tuples, i.e., replace ai

and aj by (ai � aj)/2. If k items are tied, then we replace these byk replica’s of their average.

In the case where u(1) P u(2) P � � �P u(n) and a(1) P a(2) P� � �P a(n), the I-GIFCOA operator becomes the generalized intui-tionistic fuzzy Choquet ordered averaging (GIFCOA) operator:

GIFCOAk hu1;a1i; hu2;a2i; . . . ; hun;anið Þ

¼ �n

j¼1ðmðAðjÞÞ�mðAðj�1ÞÞÞbk

ðjÞ

� �� 1=k

¼

1�Qnj¼1

1� tkbðjÞ

� �mðAðjÞÞ�mðAðj�1ÞÞ !1=k

;

0@1� 1�

Qnj¼1

1�ð1� fbðjÞ Þk

� �mðAðjÞÞ�mðAðj�1ÞÞ !1=k

1A; k> 0;

Qnj¼1ðtbðjÞ Þ

mðAðjÞÞ�mðAðj�1ÞÞ;1�Qni¼1ð1� fbðjÞ Þ

mðAðjÞÞ�mðAðj�1Þ Þ

; k¼ 0;

8>>>>>>>>>>>><>>>>>>>>>>>>:ð6Þ

where b(j) is the jth largest of ai (i = 1,2, . . . ,n).In the case where m is the additive measure (q = 0 in Eq. (1)),

then m(A(j)) �m(A(j�1)) = m(x(j)), and the GIFCOA operator (Eq. (6))reduces to the generalized intuitionistic fuzzy weighted orderedaveraging (GIFOWA) operator given by Zhao et al. [44] andintuitionistic fuzzy geometric aggregation (IFOWG) operator givenby Xu and Yager [34], respectively, for k > 0 and k = 0:

GIFOWAk hu1;a1i; hu2;a2i; . . . ; hun;anið Þ

¼ �nj¼1 mðxðjÞÞbk

ðjÞ

� �� �1=k¼ 1�Pn

j¼1ð1� tkbðjÞÞmðxðjÞÞ

� �1=k;

�1� 1�Pn

j¼1ð1� ð1� fbðjÞ ÞkÞmðxðjÞÞ

� �1=k; k > 0 ð7Þ

and

IFOWG hu1;a1i; hu2;a2i; . . . ; hun;anið Þ

¼ �nj¼1ðmðxðjÞÞbðjÞÞ ¼

Yn

j¼1

ðtbðjÞ ÞmðxðjÞÞ;1�

Yn

i¼1

ð1� fbðjÞ ÞmðxðjÞÞ

!; ð8Þ

where b(j) is the jth largest of ai (i = 1,2, . . . ,n).If the parameter k in the I-GIFCOA operator changes, then we

can also get some special operators defined as follows:

(1) If k = 1, then Eq. (5) reduces to

I� IFCOA hu1;a1i; hu2;a2i; . . . ; hun;anið Þ¼ �n

i¼1ððmðAðiÞÞ �mðAði�1ÞÞÞaðiÞÞ

¼ 1�Yn

i¼1

ð1� taðiÞ ÞmðAðiÞÞ�mðAði�1ÞÞ;

Yn

i¼1

ðfaðiÞ ÞmðAðiÞÞ�mðAði�1ÞÞ

!;

ð9Þ

which we call an induced intuitionistic fuzzy Choquet ordered aver-aging (I-IFCOA) operator.Especially, if u(1) P u(2) P � � �P u(n) and a(1) P a(2) P � � �P a(n),then the I-IFCOA operator (Eq. (9)) becomes the intuitionistic fuzzyChoquet ordered averaging (IFCOA) operator [32]:

IFCOA hu1;a1i; hu2;a2i; . . . ; hun;anið Þ¼ �n

i¼1ððmðAðjÞÞ �mðAðj�1ÞÞÞbðjÞÞ

¼ 1�Yn

j¼1

ð1� tbðjÞ ÞmðAjÞ�mðAði�1ÞÞ;

Yn

j¼1

ðfbðjÞ ÞmðAðjÞÞ�mðAðj�1ÞÞ

!;

ð10Þ

where b(j) is the jth largest of ai (i = 1,2, . . . ,n).

(2) If k = 0, then Eq. (5) becomes

I� IFCOG hu1;a1i; hu2;a2i; . . . ; hun;anið Þ

¼ �ni¼1a

mðAðiÞÞ�mðAði�1ÞÞðiÞ

¼Yn

i¼1

ðtaðiÞ ÞmðAðiÞÞ�mðAði�1ÞÞ;1�

Yn

i¼1

ð1� faðiÞ ÞmðAðiÞÞ�mðAði�1Þ

!;

ð11Þ

200 Z. Xu, M. Xia / Knowledge-Based Systems 24 (2011) 197–209

which we call an induced intuitionistic fuzzy Choquet ordered geo-metric (I-IFCOG) operator. Especially, if u(1) P u(2) P � � �P u(n) anda(1) P a(2) P � � �P a(n), then the I-IFCOG operator (Eq. (11)) reducesto the intuitionistic fuzzy Choquet ordered geometric (IFCOG) oper-ator [32]:

IFCOG hu1;a1i; hu2;a2i; . . . ; hun;anið Þ

¼ �n

j¼1b

mðAðiÞÞ�mðAði�1ÞÞj

¼Yn

j¼1

ðtbðjÞ ÞmðAðjÞÞ�mðAðj�1ÞÞ;1�

Yn

j¼1

ð1� fbðjÞ ÞmðAðjÞÞ�mðAðj�1ÞÞ

!;

ð12Þ

where b(j) is the jth largest of ai (i = 1,2, . . . ,n).

Furthermore, we can easily prove that the I-GIFCOA operator iscommutative, monotonic, bounded, and idempotent, which arepresented as follows:

Proposition 1 (Commutativity). If (hur(1),ar(1)i,hur(2),ar(2)i, . . . ,hur(n),ar(n)i) is any permutation of (hu1,a1i, hu2,a2i, . . . , hun,ani), then

I� GIFCOAk hu1;a1i; hu2;a2i; . . . ; hun;anið Þ¼ I� GIFCOAk hurð1Þ;arð1Þi; hurð2Þ;arð2Þi; . . . ; hurðnÞ;arðnÞi

� �: ð13Þ

Proposition 2 (Monotonicity). Let hu1;a1i; hu2;a2i; . . . ; hun;anið Þ;hu1; a1i; hu2; a2i; . . . ; hun; anið Þ be two collections of 2-tuples, such thatai 6 ai, i = 1,2, . . . ,n, then

I� GIFCOAk hu1;a1i; hu2;a2i; . . . ; hun;anið Þ6 I� GIFCOAk hu1; a1i; hu2; a2i; . . . ; hun; anið Þ: ð14Þ

Proposition 3 (Boundedness). Let a� ¼ ðminjðtajÞ;maxjðfaj

ÞÞ;aþ ¼ðmaxjðtaj

Þ;minjðfajÞÞ, then

a� 6 I� GIFCOAk hu1;a1i; hu2;a2i; . . . ; hun;anið Þ 6 aþ: ð15Þ

Proposition 4 (Idempotency). If ai = a, i = 1,2, . . . ,n, then

I� GIFCOAk hu1;a1i; hu2;a2i; . . . ; hun;anið Þ ¼ a: ð16Þ

Table 1Payoff matrix.

G1 G2 G3 G4

Y1 (0.5,0.3) (0.1,0.6) (0.5,0.4) (0.3,0.5)Y2 (0.6,0.1) (0.3,0.6) (0.4,0.3) (0.6,0.3)Y3 (0.5,0.1) (0.4,0.5) (0.3,0.2) (0.4,0.4)Y4 (0.8,0.1) (0.2,0.5) (0.7,0.1) (0.2,0.4)Y5 (0.6,0.3) (0.6,0.2) (0.5,0.3) (0.5,0.2)

3.2. Decision making based on the I-GIFCOA operator

With the above discussions, now we give a method for decisionmaking based on the I-GIFCOA operator.

Assume that there is a decision making problem with a collec-tion of alternatives Y = {Y1,Y2, . . . ,Ys} and the states of nature,G = {G1,G2, . . . ,Gt}. If the alternative Yi is selected under the stateGj, then aij is the payoff value which is denoted by IFVs. To getthe optimal alternative, the following steps are given:

(Method I).Step 1. Calculate the correlations between the states of natureusing the method given in Section 3.1 (Many methods has beendeveloped upon this issue, i.e., Tan and Chen used the fuzzymeasure given by Sugeno [20] to determine them (see Section3.1); Büyüközkan et al. [5,6] used the 2-additive measure[12,4] to determine them).Step 2. Calculate the inducing variables matrix U = (uij)st (Con-sidering that the fact that the attitudinal character is very com-plex because it involves the opinion of different members of theboard of directors, the experts use order-inducing variables torepresent it [16,17]. The inducing variables can also be givenby the experts using some methods, i.e., brainstorming sessions.

In most studies about the decision making, they are assumed.Perhaps, they should be determined according to the specificproblems concluding many human factors. It is worthy of inten-sive study in the future).Step 3. Utilize Eq. (5) to get the expected results Ci for the alter-native Yi (i = 1,2, . . . ,s):

Ci ¼ I� GIFCOAk hui1;ai2i; hui2;ai2i; . . . ; huit;aitið Þ

¼�t

j¼1ðmðGðjÞÞ �mðGðj�1ÞÞÞak

iðjÞ

� �� 1=k

; k > 0;

�n

j¼1aðmðAðiÞÞ�mðAði�1ÞÞÞ

iðjÞ

� �; k ¼ 0;

8>>><>>>: ð17Þ

where (j): {1,2, . . . , t} ? {1,2, . . . , t} is a permutation such thatui(1) P ui(2) P � � �P ui(t) .Step 4. Get the priority of Ci according to the comparisonmethod of IFVs, and yields the ranking of the alternativesYi(i = 1,2, . . . ,s). In Step 2, in the case where ui(1) P ui(2) P � � �Pui(t),ai(1) P ai(2) P � � �P ai(t), i = 1,2, . . . ,s, the I � GIFCOA opera-tor reduces to the IFCOA and IFCOG operators [32], respectively,for k = 1 and k = 0, which has been discussed in Section 3.1. Inthe following, we use the example given by Merigó and Gil-Laf-uente [16] to illustrate our method.

Example 1. Assume that an investor wants to invest some moneyin an enterprise in order to get the highest possible profits. Initially,he considers five possible alternatives: (1) Y1 is a computer com-pany, (2) Y2 is a chemical company, (3) Y3 is a food company, (4)Y4 is a car company, and (5) Y5 is a TV company. In order to evalu-ate these alternatives, the investor has brought together a group ofexperts. This group considers that the key factor is the economicenvironment in the global economy. After careful analysis, theyconsider four possible situations for the economic environment:(1) G1 = negative growth rate, (2) G2 = low growth rate, (3)G3 = medium growth rate, and (4) G4 = high growth rate.

The expected results of evaluations, depending on the situationGj that occurs and the alternative Yi that the investor chooses, aregiven as IFVs aij = (tij, fij), where tij denotes the degree of thealternative Yi satisfies the situation Gj, fij denotes the degree of thealternative Yi doesn’t satisfy the situation Gj. The results are shownin Table 1.

Next, we use the proposed decision making method to get theranking of the companies:

Step 1. Assume that the weights of the situations have correla-tions with each other and

mð/Þ ¼ 0; mðfG1gÞ ¼ 0:3; mðfG2gÞ ¼ 0:2;mðfG3gÞ ¼ 0:4; mðfG4gÞ ¼ 0:2:

By Eqs. (2) and (3), we have q = �0.2368, and

mðfG1;G2gÞ ¼ 0:4858; mðfG1;G3gÞ ¼ 0:6716;mðfG1;G4gÞ ¼ 0:4858; mðfG2;G3gÞ ¼ 0:5811

Table 2Inducing variables.

G1 G2 G3 G4

Y1 17 15 22 12Y2 15 22 25 13Y3 24 20 22 15Y4 16 21 25 28Y5 18 26 23 21

Table 3Aggrega

IFCOIFCOI-GIFI-GIFI-GIFI-GIFI-GIFI-GIFI-GIF

Table 4Rankings of the alternatives.

Rankings

IFCOG Y5 > Y4 > Y2Y > 3 > Y1

IFCOA Y4 > Y5 > Y2 > Y3 > Y1

I-GIFCOA0 Y5 > Y4 > Y2 > Y3 > Y1

I-GIFCOA1 Y4 > Y5 > Y2 > Y3 > Y1

I-GIFCOA2 Y4 > Y5 > Y2 > Y3 > Y1

I-GIFCOA5 Y4 > Y5 > Y2 > Y3 > Y1

I-GIFCOA10 Y4 > Y2 > Y5 > Y3 > Y1

I-GIFCOA20 Y4 > Y2 > Y5 > Y3 > Y1

I-GIFCOA50 Y4 > Y2 > Y5 > Y3 > Y1

Z. Xu, M. Xia / Knowledge-Based Systems 24 (2011) 197–209 201

mðfG2;G4gÞ ¼ 0:3905; mðfG3;G4gÞ ¼ 0:5811;mðfG1;G2;G3gÞ ¼ 0:8398; mðfG1;G2;G4gÞ ¼ 0:6628

mðfG1;G3;G4gÞ ¼ 0:8398; mðfG2;G3;G4gÞ ¼ 0:7535;mðfG1;G2;G3;G4gÞ ¼ 1:0

Step 2. The experts use order-inducing variables to representthe complex attitudinal character involving the opinions of dif-ferent members of the board of directors. The results are shownin Table 2 [16].Step 3. With this information, we can aggregate the expectedresults for each state of nature using Eq. (17). The IFCOA orIFCOG operator [32] can also be used in this step, but they don’tconsider the inducing invariables and ignore the subjectiveinformation. Table 3 shows the results when we use differentaggregation operators.Step 4. According to the comparison methods of IFVs in Sec-tion 2.1, we can get the rankings of the alternatives inTable 4.As we can see, the rankings of the alternatives maybe different with the change of the aggregation operators,which results in the indeterminacy of the final decision,because that each aggregation operator focuses on differentpoint of view. The I-GIFCOA operator is the generalizationof the IFCOG and IFCOA operators, while the former can pro-vide the decision maker more choices as the parameter kchanges, the latter only gives the decision maker one choice.When the I-GIFCOA operator is used, the ranking of the alter-natives is slightly different as k increases. It is very reason-able that k can be considered as the decision makers’ riskpreference similar to the parameters defined by Liu andWang (2007). As a result, the investor can properly selectthe desirable alternative in accordance with his/her interestand the actual needs.

3.3. Interval-valued generalized intuitionistic Choquet orderedaveraging operators

Let eV be the set of all interval-valued intuitionistic fuzzy values(IVIFVs). In this section, we extend the developed operators in

ted results.

C1 C2

G (0.3546,0.4289) (0.4651,0.3099)A (0.4186,0.4065) (0.4950,0.2411)COA0 (0.3514,0.4324) (0.4500,0.3250)COA1 (0.4175,0.4105) (0.4794,02560)COA2 (0.4352,0.4051) (0.4894,0.2466)COA5 (0.4639,0.3888) (0.5192,0.2218)COA10 (0.4806,0.3670) (0.5510,0.1916)COA20 (0.4901,0.3420) (0.5745,0.1563)COA50 (0.4953,0.3180) (0.5896,0.1240)

Section 3.1 to aggregate interval-valued intuitionistic fuzzyinformation.

Definition 3. Let X = {x1,x2, . . . ,xn}be a fixed set, m be a fuzzy

measure on X, and ~ai ¼ t�~ai; tþ~ai

h i; f�~ai

; fþ~ai

h i� �ði ¼ 1; 2; . . . ; nÞ be a

collection of IVIFVs on X. An induced generalized interval-valuedintuitionistic fuzzy Choquet ordered averaging ((I-GIVIFCOA))

operator of dimension n is a function I-GIVIFCOA: eV n ! eV , whichis defined to aggregate the set of second arguments of a collectionof 2-tuples hu1; ~a1i; hu2; ~a2i; . . . ; hun; ~anið Þ according to the followingexpression:

I� GIVIFCOAk hu1; ~a1i; hu2; ~a2i; . . . ; hun; ~anið Þ

¼�n

i¼1 ðmðAðiÞÞ �mðAði�1ÞÞÞ~akðiÞ

� �� �1=k; k > 0;

�ni¼1

~aðmðAðiÞÞ�mðAði�1ÞÞÞðiÞ

� �; k ¼ 0;

8<: ð18Þ

where ui in 2-tuple hui; ~aii is referred to as the order-inducing var-iable and ~ai as the argument variable, (i): {1,2, . . . ,n} ? {1,2, . . . ,n}is a permutation such that u(1) P u(2) P � � �P u(n),A(i) = {x(1),x(2), . . . ,x(i)} when i P 1 and A(0) = ;, using the operations for IVIFVs, we canget

I�GIVIFCOAk hu1; ~a1i; hu2; ~a2i; . . . ; hun; ~anið Þ

¼

1�Qni¼1ð1�ðt�~aðiÞ Þ

kÞmðAðiÞ Þ�mðAði�1Þ Þ� 1=k

; 1�Qni¼1

1� tþ~aðiÞ

� �k� mðAðiÞ Þ�mðAði�1Þ Þ

!1=k24 35;

0@1� 1�

Qni¼1

1� 1� f�~aðiÞ

� �k� mðAðiÞ Þ�mðAði�1Þ Þ

!1=k

;

241� 1�

Qni¼1

1� 1� fþ~aðiÞ

� �k� mðAðiÞ Þ�mðAði�1Þ Þ

!1=k351A; k> 0;

Qni¼1

t�~aðiÞ

� �mðAðiÞ Þ�mðAði�1Þ Þ;Qni¼1

tþ~aðiÞ

� �mðAðiÞ Þ�mðAði�1Þ Þ �

;

�1�

Qni¼1

1� f�~aðiÞ

� �mðAðiÞ Þ�mðAði�1Þ;1�

Qni¼1

1� fþ~aðiÞ

� �mðAðiÞ Þ�mðAði�1Þ �

; k¼ 0:

8>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>:

ð19Þ

In particular, if ui = uj in two 2-tuples hui; ~aii and huj; ~aji, then we re-place ~ai and ~aj by their average, i.e., ð~ai � ~ajÞ=2. If k items are tied,we replace these by k replica’s of their average.

C3 C4 C5

(0.3843,0.2676) (0.4829,0.2349) (0.5463,0.2939)(0.3984,0.2114) (0.6334,0.1634) (0.5514,0.2575)(0.3843,0.2687) (0.4537,0.2501) (0.5424,0.2643)(0.3984,0.2118) (0.6088,0.1741) (0.5474,0.2580)(0.4044,0.2055) (0.6385,0.1675) (0.5489,0.2570)(0.4241,0.1885) (0.6885,0.1510) (0.5542,0.2537)(0.4485,0.1685) (0.7214,0.1341) (0.5638,0.2471)(0.4711,0.1463) (0.7779,0.1083) (0.5772,0.2342)(0.4863,0.1213) (0.7779,0.1083) (0.5904, 0.2156)

202 Z. Xu, M. Xia / Knowledge-Based Systems 24 (2011) 197–209

In the case where u(1) P u(2) P � � �P u(n) and ~að1Þ P ~að2Þ P� � �P ~aðnÞ, the I-GIVIFCOA operator (Eq. (19)) becomes

GIVIFCOAk hu1; ~a1i;hu2; ~a2i; . . . ;hun; ~anið Þ

¼ �n

j¼1

ðmðAðjÞÞ�mðAðj�1ÞÞÞ~bðjÞ� �

¼

1�Qnj¼1

1� t�~bðjÞ

� �k� mðAðjÞ Þ�mðAðj�1Þ Þ

!1=k

; 1�Qnj¼1

1� tþ~bðjÞ

� �k� mðAðjÞ Þ�mðAðj�1Þ Þ

!1=k24 35;

0@1� 1�

Qnj¼1

1� 1� f�~bðjÞ

� �k� mðAðjÞ Þ�mðAðj�1Þ Þ

!1=k

;

241� 1�

Qnj¼1

1� 1� fþ~bðjÞ

� k !mðAðjÞ Þ�mðAðj�1Þ Þ

0@ 1A1=k3751CA; k>0;

Qnj¼1

t�~bðjÞ

� �mðAðjÞ Þ�mðAðj�1Þ Þ;Qnj¼1

tþ~bðjÞ

� �mðAðjÞ Þ�mðAðj�1Þ Þ" #

;

1�Qnj¼1

1� f�~bðjÞ

� �mðAðjÞ Þ�mðAðj�1Þ Þ;1�

Qnj¼1

1� fþ~bðjÞ

� mðAðjÞ Þ�mðAðj�1Þ Þ" #!

; k¼0;

8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:

ð20Þ

which we call a generalized interval-valued intuitionistic fuzzyChoquet ordered averaging (GIVIFCOA) operator, where ~bðjÞ is thejth largest of ~aiði ¼ 1;2; . . . ;nÞ.

Similar to Section 3.1, we can get some special operators as theparameter k changes in the I-GIVIFCOA operator.

(1) If k = 1, then Eq. (19) reduces to the induced interval-valuedintuitionistic fuzzy Choquet ordered averaging (I-IVIFCOA)operator:

I� IVIFCOA hu1; ~a1i; hu2; ~a2i; . . . ; hun; ~anið Þ¼ �n

i¼1ððmðAðiÞÞ �mðAði�1ÞÞÞ~aðiÞÞ

¼ 1�Yn

i¼1

1� t�~aðiÞ

� �mðAðiÞÞ�mðAði�1ÞÞ;

"

1�Yn

i¼1

1� tþ~aðiÞ

� �mðAðiÞÞ�mðAði�1ÞÞ#;Yn

i¼1

f�~aðiÞ

� �mðAðiÞÞ�mðAði�1ÞÞ;

"Yn

i¼1

fþ~aðiÞ

� �mðAðiÞÞ�mðAði�1ÞÞ#!

: ð21Þ

Especially, if u(1) P u(2) P � � �P u(n) and ~að1Þ P ~að2Þ P � � �P~aðnÞ, then the I-IVIFCOA operator (Eq. (21)) becomes the inter-val-valued intuitionistic fuzzy Choquet ordered averaging(IVIFCOA) operator [32]:

IVIFCOA hu1; ~a1i; hu2; ~a2i; . . . ; hun; ~anið Þ

¼ �n

j¼1ððmðAðjÞÞ �mðAðj�1ÞÞÞ~bðjÞÞ

¼ 1�Yn

j¼1

1� t�~bðjÞ

� �mðAðjÞÞ�mðAðj�1ÞÞ;

"

1�Yn

j¼1

1� tþ~bðjÞ

� �mðAðjÞÞ�mðAðj�1ÞÞ#;Yn

j¼1

f�~bðjÞ

� �mðAðjÞÞ�mðAðj�1ÞÞ;

"Yn

j¼1

fþ~bðjÞ

� �mðAðjÞÞ�mðAðj�1ÞÞ#!

; ð22Þ

where ~bðjÞ is the jth largest of ~aiði ¼ 1;2; . . . ;nÞ.

(2) If k = 0, then the I-GIVIFCOA operator (Eq. (19)) becomes

I� IVIFCOG hu1; ~a1i; hu2; ~a2i; . . . ; hun; ~anið Þ

¼ �ni¼1~amðAðiÞÞ�mðAði�1ÞÞ

ðiÞ ¼Yn

i¼1

t�~aðiÞ

� �mðAðiÞÞ�mðAði�1ÞÞ;

" Yn

i¼1

tþ~aðiÞ

� �mðAðiÞÞ�mðAði�1ÞÞ#; 1�

Yn

i¼1

1� f�~aðiÞ

� �mðAðiÞÞ�mðAði�1Þ;

"

1�Yn

i¼1

1� fþ~aðiÞ

� �mðAðiÞÞ�mðAði�1Þ

#!; ð23Þ

which we call an induced interval-valued intuitionistic fuzzyChoquet ordered geometric (I-IVIFCOG) operator. Especially,if u(1) P u(2) P � � �P u(n) and ~að1Þ P ~að2Þ P � � �P ~aðnÞ, thenthe I-IVIFCOG operator (Eq. (23)) reduces to the interval-val-ued intuitionistic fuzzy Choquet ordered geometric (IVIF-COG) operator [32]:

IVIFCOG hu1; ~a1i; hu2; ~a2i; . . . ; hun; ~anið Þ

¼ �n

j¼1

~bmðAðjÞÞ�mðAðj�1ÞÞðjÞ ¼

Yn

j¼1

t�~bðjÞ

� �mðAðjÞÞ�mðAðj�1ÞÞ;

" Yn

j¼1

tþ~bðjÞ

� �mðAðjÞÞ�mðAðj�1ÞÞ#; 1�

Yn

j¼1

1� f�bðjÞ

� �mðAðjÞÞ�mðAðj�1Þ;

"

1�Yn

j¼1

1� fþbðjÞ

� �mðAðjÞÞ�mðAðj�1Þ

#!; ð24Þ

where ~bðjÞ is the jth largest of ~aiði ¼ 1;2; . . . ;nÞ.

In Section 3.2, if each payoff value of the alternative Yi under thestate Gj is given by an IVIFV ~aij, then similar to Method I, we give amethod for decision making based on the I-GIVIFCOA operator.

(Method II).Step 1. See Method I.Step 2. See Method I.Step 3. Utilize Eq. (19) to get the expected results eCi for thealternatives Yi(i = 1,2, . . . ,s):

eCi ¼ I� GIVIFCOAk hui1; ~ai1i; hui2; ~ai2i; . . . ; huit; ~aitið Þ

¼�t

j¼1 ðmðAðjÞÞ �mðAðj�1ÞÞÞ~akiðjÞ

� �� �1=k; k > 0;

�nj¼1

~aðmðAðjÞÞ�mðAðj�1ÞÞðjÞ

� �; k ¼ 0;

8<: ð25Þ

where (j): {1,2, . . . , t} ? {1,2, . . . , t} is a permutation such thatui(1) P ui(2) P � � �P ui(t) .

Step 4. Get the priority of eCi according to the comparisonmethod of IVIFVs, and yields the ranking of the alternatives Yi

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

4. Aggregation operators based on the Dempster–Shafer beliefstructure

The Dempster–Shafer theory of evidence [8,9,18] is an usefultool for dealing with the problem of selecting an optimal alterna-tive in which there exists some uncertainty in our knowledge ofthe state of the world. It can provide an unifying framework forrepresenting various types of uncertainties including the situationsof risk and ignorance [36]. Various authors (e.g., [36,19,17]) hasstudied the decision making problem with Dempster–Shafer beliefstructures, in which the available information was supposed to benumerical. Sometimes, the available information is not clear and itis necessary to assess it with other forms such as IFVs. In this sec-tion, we give some induced intuitionistic fuzzy aggregation opera-tors based on the Dempster–Shafer belief structure, and applythem to decision making under intuitionistic fuzzy environments.The reason for using the induced idea is that the decision makers,sometimes, have an attitudinal character that differs from the val-ues of the arguments. Then, to aggregate the arguments, he/sheprefers to use another mechanism in the reordering step that iscloser in accordance with his/her interests [17].

4.1. Intuitionistic fuzzy Dempster–Shafer operators

The concept of Dempster–Shafer belief structure is given byDempster [8,9] and Shafer [18] as follows:

Z. Xu, M. Xia / Knowledge-Based Systems 24 (2011) 197–209 203

Definition 4. A Dempster–Shafer belief structure defined on aspace X = {x1,x2, . . . ,xn} consists of a collection of r non-null subsetsDj (j = 1,2, . . . ,r) of X, called focal elements, and a mapping p calledthe probability assignment, defined as p: 2X ? [0,1] such that:

(1) p(Dj) 2 [0,1];(2)

Prj¼1pðDjÞ ¼ 1;

(3) p(C) = 0, "C – Dj.

As mentioned above, a main characteristic of the Dempster–Shafer belief structure is that it can represent some traditionalcases of uncertainty. If it consists of n focal elements such thatDj = {xj} in which each focal elements is a singleton, then weevidently have to make decision under risk environment withDj = Pj = prob{xj}. Another special case is that when the beliefstructure consists of only one focal element D1 which com-prises all the states of nature (D1 = X = {x1,x2, . . . ,xn}), wherep(X) = 1. Then we have to make decision under ignoranceenvironment.

Definition 5. Let X = {x1,x2, . . . ,xn} be a fixed set, (hu1

,a1i, hu2,a2i, . . . , hun,ani) be a collection of 2-tuples on X, where ui

is the order-inducing variables and ai(i = 1,2, . . . ,n) be the aggre-gated arguments in the form of IFVs, and M ¼ MkjMk ¼ fhui;ðaiijxi 2 Dk; i ¼-be a collection of 2-tuples with r focal elements Dk (k = 1,2, . . . ,r). ABSI-GIFOA operator of dimension r is a function BSI-GIFOA: Vr ? Vdefined by

BSI�GIFOAk1 ;k2 ðMÞ ¼

�rk¼1 pðDkÞ �qk

j¼1 wjkbk1jk

� �� �k2=k1� � 1=k2

; k1 > 0;k2 > 0;

�rk¼1 pðDkÞ �qk

j¼1 bwjk

jk

� �� �k2� � 1=k2

; k1 ¼ 0;k2 > 0;

�rk¼1 �qk

j¼1 wjkbk1jk

� �� �1=k1� pðDk Þ

; k1 > 0;k2 ¼ 0;

�rk¼1 �qk

j¼1 bwjk

jk

� �� �pðDkÞ�

; k1 ¼ 0;k2 ¼ 0;

8>>>>>>>>>>>>><>>>>>>>>>>>>>:ð26Þ

where Wk ¼ ðw1k;w2k; . . . ;wqkkÞ is the weighting vector for the kthfocal element Dk such that

Pqkj¼1wjk ¼ 1 and wjk 2 [0,1], where qk

is the numbers of elements in Dk, bjk is the cik value of the pairhuik,ciki having the jth largest uik(i = 1,2, . . . ,qk), uik is the order-inducing variable, cik is the argument variable in the form of IFVs,and p(Dk) is the basic probability assignment.

By the operational laws of IFVs, we discuss some special cases ofthe BSI-GIFOA operator:

(1) If k1 > 0, k2 > 0, then Eq. (26) becomes

BSI�GIFOAk1 ;k2 ðMÞ

¼ �rk¼1 pðDkÞ �qk

j¼1 wjkbk1jk

� �� �k2=k1� � 1=k2

¼ 1�Prk¼1 1� 1�Pqk

j¼1ð1� tk1bjkÞwjk

� �k2=k1� pðDkÞ

!1=k2

;

0@1� 1�Pr

k¼1 1� 1�Pqkj¼1 1� 1� fbjk

� �k1� wjk

� k2=k1 !pðDkÞ

0@ 1A1=k21CA:

ð27Þ

(a) If Dk = {xk} k = 1,2, . . . ,n, then M = ({hu1,a1i}, {hu2,a2i}, . . .

, {hun,ani}) and Eq. (27) reduces to the generalized intui-tionistic fuzzy weighted averaging (GIFWA) operator[44].

GIFWAk2 ðMÞ ¼ �nk¼1 pðDkÞak2

k

� �� �1=k2

¼ 1�Pnk¼1 1� tk2

ak

� �pðDkÞ� 1=k2

;

1� 1�Pnk¼1 1� 1� fak

� �k2� �pðDkÞ

� 1=k2!;

ð28Þwhich is an extension of the intuitionsitic fuzzy weightedaveraging (IFWA) operator [31].(b) If the belief structure consists of only one focal element

which comprises all the states of nature (D1 = X = {x1,x2

, . . . ,xn}), where p(X) = 1, then M = ({hu1,a1i,hu2,a2i, . . . ,hun,ani}), and Eq. (27) reduces to the following operatorwhich we call an I-GIFOWA operator:� �� �1=k1

I�GIFOWAk1 ðMÞ ¼ �ni¼1 wjb

k1j

¼ 1�Pnj¼1ð1� tk1

bjÞwj

� �1=k1;

�1� 1�Pn

j¼1 1� ð1� fbjÞk1

� �wj� �1=k1

; ð29Þ

where bj is the ai value of the pair hui,aii having the jth largestui(i = 1,2, . . . ,n). If k = 1 in Eq. (29), then the I-GIFOWA opera-tor reduces to the induced intuitionisitc fuzzy orderedweighted averaging (I-IFOWA) operator [27] which is anextension of the induced ordered weighted averaging (IOWA)operator [39] and intuitionisitc fuzzy ordered weighted aver-aging (IFOWA) operator [31]:

I� IFOWAðMÞ ¼ �q1j¼1ðwjbjÞ

¼ 1�Pnj¼1ð1� tbj

Þwj ;Pnj¼1f

wjbj

� �; ð30Þ

where bj is the ai value of the pair hui,aii having the jth largestui(i = 1,2, . . . ,n). If ai P aj for ui P uj, i, j = 1,2, . . . ,n in the I-GIFOWA operator, then Eq. (29) reduces to the GIFOWA oper-ator [44] which is an extension of the IFWA and IFOWA oper-ators [31].

GIFOWAk1 ðMÞ ¼ �ni¼1 wðjÞb

k1ðjÞ

� �� �1=k1

¼ 1�Pnj¼1 1� tk1

bðjÞ

� �wðjÞ� �1=k1�

1� 1�Pnj¼1 1� ð1� fbðjÞ Þ

k1� �wðjÞ� �1=k1

; ð31Þ

where b(j) is the jth largest of ai (i = 1,2, . . . ,n).

(2) If k1 = k2 = k, then Eq. (26) yields

BSI� GIFOAkðMÞ ¼ �rk¼1 pðDkÞ�qk

j¼1 wjkbkjk

� �� �� �1=k

¼ 1�Prk¼1P

qkj¼1 1� tk

bjk

� �wjkpðDkÞ� 1=k

;

1� 1�Prk¼1P

qkj¼1 1� ð1� fbjk

Þk� �wjkpðDkÞ

� 1=k!:

ð32Þ

(3) If k1 = k2 = 1, then Eq. (26) becomes the BSI-IFOAA operator:

BSI� IFOAAðMÞ ¼ �rk¼1 pðDkÞ�qk

j¼1ðwjkbjkÞ� �

¼ 1�Yr

k¼1

Yqk

j¼1

1� tbjk

� �wjkpðDkÞ;

Yr

k¼1

Yqk

j¼1

ðfbjkÞwjkpðDkÞ

!: ð33Þ

(4) If k1 = 0, k2 > 0 then Eq. (26) yields

204 Z. Xu, M. Xia / Knowledge-Based Systems 24 (2011) 197–209

BSI�GIFOAk2 ðMÞ ¼ �rk¼1 pðDkÞ �qk

j¼1 bwjk

jk

� �� �k2� � 1=k2

¼ 1�Prk¼1 1�

Yqk

j¼1

twjkbjk

!k20@ 1ApðDkÞ

0B@1CA

1=k2

;

0B@

1� 1�Prk¼1 1�

Yqk

j¼1

1� fbjk

� �wjk

!k20@ 1ApðDkÞ

0B@1CA

1=k21CA:ð34Þ

If the belief structure consists of only one focal elementwhich comprises all the states of nature (D1 = X = {x1,x2

, . . . ,xn}), where p(X) = 1, the BSI-GIFOA operator becomesthe induced intuitionistic fuzzy ordered weighted geometric(I-IFOWG) operator [29], which is a generalization of theintuitionistic fuzzy ordered weighted geometric (IFWG) oper-ator and IFOWG operator [34]:

I� IFOWGðMÞ ¼ �nj¼1 b

wj

j

� �¼

Yn

j¼1

twjbj;1�

Yn

j¼1

ð1� fbjÞwj

!; ð35Þ

where bj is the ai value of the pair hui,aii having the jth largestui(i = 1,2, . . . ,n).

(5) If k1 = 0 and k2 = 1, then Eq. (26) reduces to

BSI� IFOAGðMÞ ¼ �rk¼1 pðDkÞ�qk

j¼1 bwjk

jk

� �� �¼ 1�

Yr

k¼1

1�Yqk

j¼1

tbjk

� �wjk

!pðDkÞ

;

0@Yr

k¼1

1�Yqk

j¼1

1� fbjk

� �wjk

!pðDkÞ1A; ð36Þ

which we call a BSI-IFOAG operator.

(6) If k1 > 0 and k2 = 0, then Eq. (26) reduces to

BSI�GIFOAk1 ðMÞ ¼ �rk¼1 �qk

j¼1 wjkbk1jk

� �� �1=k1� pðDkÞ

¼Yqk

j¼1

1�Prk¼1 1� tk1

bjk

� �wjk� �1=k1� pðDkÞ

;

1�Yqk

j¼1

1�Prk¼1 1� 1� fbjk

� �k1� wjk

� 1=k1 !pðDkÞ

1A:ð37Þ

(7) If k1 = 1 and k2 = 0, then Eq. (26) reduces to

BSI� IFOGAðMÞ ¼ �rk¼1 �

qkj¼1ðwjkbjkÞ

� �pðDkÞ

¼Yr

k¼1

1�Yqk

j¼1

1� tbjk

� �wjk

!pðDkÞ

;

0@1�

Yr

k¼1

1�Yqk

j¼1

fbjk

� �wjk

!pðDkÞ1A; ð38Þ

which we call a BSI-IFOGA operator.

(8) If k1 = 0 and k2 = 0, then Eq. (26) becomes

BSI� IFOGGðMÞ ¼ �rk¼1 �qk

j¼1 bwjk

jk

� �� �pðDkÞ�

¼Yr

k¼1

Yqk

j¼1

tbjk

� �wjkpðDkÞ;

1�Yr

k¼1

Yqk

j¼1

1� fbjk

� �wjkpðDkÞ!; ð39Þ

which we call a BSI-IFOGG operator.

Let M ¼ hu1k; c1ki; hu2k; c2ki; . . . ; huqkk; cqkkin o���k ¼ 1;2; . . . ; r� �

,Q = {i,kjk = 1,2, . . . ,r;i = 1,2, . . . ,qk}. Motivated by Merigó and Casa-novas [17] and Zhao et al. (2009), we can get that the BSI-GIFOAoperator has some good properties, such as commutativity, mono-tonicity, boundedness and idempotency shown as follows:

Proposition 5 (Commutativity). If M ¼ u1k; c1k

�; u2k; c

2k

�; . . . ;

��uqkk; c

qkk

D Egjk ¼ 1;2; . . . ; rÞ be any permutation of M then

BSI� GIFOAk1 ;k2 ðMÞ ¼ BSI� GIFOAk1 ;k2 ðMÞ: ð40Þ

Proposition 6 (Monotonicity). Let bM ¼ u1k; c1kh i; u2k; c2kh i; . . . ;fðuqkk; cqkk �

gjk ¼ 1;2; . . . ; rÞ. If 8 i; k 2 Q ; cik 6 cik, then

BSI� GIFOAk1 ;k2 ðMÞ 6 BSI� GIFOAk1 ;k2 ð bMÞ: ð41Þ

Proposition 7 (Boundedness). Let c� ¼ mini;k2Q ðtcikÞ;maxi;k2Q

�ðfcikÞÞ; cþ ¼ maxi;k2Q ðtcik

Þ;mini;k2Q ðfcikÞ

� �, then

c� 6 BSI� GIFOAk1 ;k2ðbMÞ6cþ

: ð42Þ

Proposition 8 (Idempotency). If "i, k 2 Q, cik = c, then

BSI� GIFOAk1 ;k2 ð bMÞ ¼ c: ð43Þ

4.2. Decision making based on the BSI-GIFCOA operator

Assume that there is a decision making problem with a collec-tion of alternatives Y = {Y1,Y2, . . . ,Ys} and the states of nature,G = {G1,G2, . . . ,Gt}. If the alternative Yi is selected under the stateGj, then aij is the payoff value which is denoted by IFVs. The knowl-edge of the states Gj(j = 1,2, . . . , t), is captured in terms of a beliefstructure p with the focal elements D1,D2, . . .,Dr, each of which isassociate with a weight p(Dk),

Prk¼1pðDkÞ ¼ 1. U = (uij)st is the

inducing values matrix. The problem is to select the alternativewith the best result. To do so, we propose a method based on theBSI-GIFCOA operator, which involves the following steps:

If the belief structure consists of only one focal element D whichcomprises all the states of nature G1,G2, . . . ,Gt, where p(D) = p({G1,G2, . . . ,Gt}) = 1. In such case, we can use the existing intuitionisticfuzzy aggregation operators (i.e., the I-IFOWA operator [27] to cal-culate the aggregated payoff Ci for each alternative Yi:

Ci ¼ GIFOWAk thui1;ai1i; hui2;ai2i; . . . ; huit ;aitið Þ

¼ �tj¼1ðwjak

iðjÞÞ� �1=k

; ð44Þ

where (j): {1,2, . . . , t} ? {1,2, . . . , t} is a permutation such thatui(1) P ui(2) P � � �P ui(t).

If the belief structure consists of t focal elements such thatDj = {Gj}(j = 1,2, . . . , t) in which each focal elements is a singleton,then we can also use some known operators (i.e., the IFWA opera-tor [31]) to calculate the aggregated payoff Ci for each alternativeYi:

Ci ¼ IFWAðai1;ai2; . . . ;aitÞ ¼ �tj¼1ðpðDjÞaijÞ: ð45Þ

If the belief structure doesn’t satisfy any one of the above two cases,the existing aggregation operators are ineffective, while the definedBSI-GIFCOA operator can work, which will be discussed in the fol-lowing method.

(Method III).Step 1. Calculate the attitudinal character of the decision makerto determine the inducing values matrix U = (uij)st (the methodused is similar to Step 1 in Method I).

Step 2. Find Mik ¼ uij;aij ���Gj 2 Dk; j ¼ 1;2; . . . ; t� �

¼ uð1Þik ;Dn

cð1Þik i; uð2Þik ; cð2Þik

D E; . . . ; uðqkÞ

ik ; cðqkÞik

D Eg the set of payoffs that are

TabPay

YYYYY

Z. Xu, M. Xia / Knowledge-Based Systems 24 (2011) 197–209 205

possible if we select the alternative Yi and the focal element Dk

occurs, where qk is the number of elements in Dk.Step 3. Utilize one of the existing methods (e.g., [37,30]) todetermine the weight vector Wqk

¼ wð1Þqk;wð2Þqk

; . . . ;wðqkÞqk

� �for

aggregating qk arguments in Mik.Step 4. Calculate the aggregated payoff, Vik, for Yi by the I-GIF-OWA operator (Eq. (29)) or I-IFOWG [29] operator when thefocal element Dk occurs:

Vik ¼�qk

j¼1 wðjÞqkbðjÞik

� �k1� � 1=k1

; k1 > 0;

�qkj¼1ðw

ðjÞqk

bðjÞik Þ; ; k1 ¼ 0;

8><>: ð46Þ

where bðjÞik is the cðlÞik value of the pair uðlÞik ; cðlÞik

D Ewith the

jth largest uðlÞik ðl ¼ 1;2; . . . ; qkÞ (In fact, we can aggregate

the values uðlÞik ; cðlÞik

D Eðl ¼ 1;2; . . . ; qkÞ using other operators,

such as the IFOWG operator [34], IFOWA operator [31] orI-IFOWA operator [27] instead of the I-GIFOWA operator orI-IFOWG operator [29].

Step 5. Utilize the BSI-GIFOA operator (22) to calculate theaggregated payoff Ci for each alternative Yi:

Ci ¼�r

k¼1 pðDkÞðVikÞk2� �� �1=k2

; k2 > 0;

�rk¼1 pðDkÞVikð Þ; k2 ¼ 0

8<:

¼

�rk¼1 pðDkÞ �qk

j¼1 wðjÞqkbðjÞik

� �k1� � k2=k1

! !1=k2

; k1 > 0;k2 > 0;

�rk¼1 pðDkÞ �qk

j¼1 wðjÞqkbðjÞik

� �� �k2� � 1=k2

; k1 ¼ 0;k2 > 0;

�rk¼1 pðDkÞ �qk

j¼1 wðjÞqkbðjÞik

� �k1� � 1=k1

!; k1 > 0;k2 ¼ 0;

�rk¼1 pðDkÞ�qk

j¼1 wðjÞqkbðjÞik

� �� �; k1 ¼ 0;k2 ¼ 0:

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð47Þ

(Actually, the GIFWA [44] operator or IFWG operator [34] isused to aggregate the values Vik (k = 1,2, . . . ,r) in Step 5).

Step 6. Select the alternative Yi with the largest Ci as the optimalone. Below we use the decision making problem of Section 3.2to illustrate the application of the BSI-GIFOA operator in deci-sion making.

Table 6Inducing variables.

G1 G2 G3 G4 G5 G6 G7 G8

Y1 25 16 24 18 20 13 19 14Y2 18 34 22 12 24 16 20 26Y3 13 21 28 22 19 25 16 26Y4 20 24 14 31 27 25 19 18Y5 25 16 23 30 15 21 18 26

Example 2. In Example 1, assume that the states of nature repre-senting the different economic situations are evaluated by the fol-lowing world growth rates:

G1 ¼ strong recession; G2 ¼ weak recession; G3

¼ growth rate near; G4 ¼ very low growth rate; G5

¼ low growth rate; G6 ¼medium growth rate; G7

¼ high growth rate; G8 ¼ very high growth

and the possible results, depending on the future of nature, are rep-resented as IFVs in Table 5.

The experts need to carefully analyze the problem so as toobtain the probabilistic information about the state of nature thatwill occur in the future which is represented by the values of belieffunction p:

pðD1Þ ¼ pðfG1;G5;G6;G7gÞ ¼ 0:4; pðD2Þ ¼ pðfG1;G3;G8gÞ¼ 0:3; pðD3Þ ¼ pðfG2;G3;G4gÞ ¼ 0:3:

le 5off matrix.

G1 G2 G3 G4

1 (0.3,0.4) (0.7,0.1) (0.4,0.4) (0.8,0.2)2 (0.5,0.3) (0.5,0.2) (0.7,0.1) (0.6,0.1)3 (0.4,0.5) (0.6,0.1) (0.5,0.1) (0.6,0.2)4 (0.3,0.4) (0.4,0.2) (0.8,0.1) (0.8,0.1)5 (0.5,0.3) (0.4,0.6) (0.6,0.3) (0.7,0.1)

Next, we use the developed decision making method to select thebest alternative:

Step 1. Suppose the induced aggregation variables are repre-sented in Table 6 [17].

Step 2. Find Mik ¼ huij;aijjGj 2 Dk; j ¼ 1;2; . . . ; t� �

¼ uð1Þ;cð1Þ

ikik

� �;

�uð2Þ;ik

Dcð2Þik i; . . . ; u

ðqkÞ;cðqk Þik

ik

� �g:

Y1 : M11 ¼ fh25; ð0:3; 0:4Þi; h20; ð0:4;0:5Þi; h13; ð0:5;0:2Þi; h19; ð0:1;0:5Þig;M12 ¼ fh25; ð0:3;0:4Þi; h24; ð0:4; 0:4Þi; h14; ð0:5;0:4Þig;M13 ¼ fh16; ð0:7;0:1Þi; h24; ð0:4; 0:4Þi; h18; ð0:8;0:2Þig:Y2 : M21 ¼ fh18; ð0:5; 0:3Þi; h24; ð0:6;0:2Þi; h16; ð0:7; 0:2Þi; h20; ð0:3;0:6Þig;M22 ¼ fh18; ð0:5;0:3Þi; h22; ð0:7; 0:1Þi; h26; ð0:4;0:3Þig;M23 ¼ fh34; ð0:5;0:2Þi; h22; ð0:7; 0:1Þi; h12; ð0:6;0:1Þig:Y3 : M31 ¼ fh20; ð0:4;0:5Þi; h19; ð0:7;0:3Þi; h25; ð0:8;0:1Þi; h13; ð0:4;0:5Þig;M32 ¼ fh13; ð0:4;0:5Þi; h28; ð0:5; 0:1Þi; h26; ð0:3;0:2Þig;M33 ¼ fh21; ð0:6;0:1Þi; h28; ð0:5; 0:1Þi; h22; ð0:6;0:2Þig:Y4 : M41 ¼ fh20; ð0:3;0:4Þi; h27; ð0:3;0:4Þi; h25; ð0:7;0:1Þi; h19; ð0:2;0:5Þig;M42 ¼ fh20; ð0:3; 0:4Þi; h14; ð0:8;0:1Þi; h18; ð0:7;0:1Þig;M43 ¼ fh24; ð0:4;0:2Þi; h14; ð0:8; 0:1Þi; h31; ð0:8;0:1Þig:Y5 : M51 ¼ fh25; ð0:5; 0:3Þi; h15; ð0:8;0:1Þi; h21; ð0:3; 0:4Þi; h18; ð0:6;0:2Þig;M52 ¼ hh25; ð0:5;0:3Þi; h23; ð0:6;0:3Þi; h26; ð0:5;0:3Þii;M53 ¼ fh16; ð0:4;0:6Þi; h23; ð0:6; 0:3Þi; h30; ð0:7;0:1Þig:

Step 3. Utilize the normal distribution based method [30] todetermine the weight vector Wqk

with qk elements:

W3 ¼ ð0:4;0:4;0:2Þ; W4 ¼ ð0:3;0:3;0:2;0:2Þ:

Step 4. Calculate the aggregated payoff, Vik, for Yi when the focalelement Dk occurs, using the IFOWG operator [34], IFOWA oper-ator [31], I-IFOWG operator [29], I-IFOWA operator [27], or I-GIFOWA operator (Eq. (29)), respectively. The results are pre-sented in Table 7.Step 5. If we calculate the expected values Vik (k = 1,2,3) of eachalternative Yiby the IFWG operator [34] or IFWA operator [31],the corresponding results are given in Tables 8 and 9.Step 6. Select the best alternative with the different operatorslisted in Table 10.

The operators in vertical line 1 of Table 10 are the ones used inStep 4, the operators in horizontal line 1 of Table 10 are the onesused in Step 5. As we can see from Table 10, with the change ofthe operators used in different steps, the rankings of thealternatives may be different, and thus, the investor canproperly select the desirable alternative according to hisinterest and the actual needs. In Step 4, the I-IFOWG, I-IFOWA

G5 G6 G7 G8

(0.4,0.5) (0.5,0.2) (0.1,0.5) (0.5,0.4)(0.6,0.2) (0.7,0.2) (0.3,0.6) (0.4,0.3)(0.7,0.3) (0.8,0.1) (0.4,0.5) (0.3,0.2)(0.3,0.4) (0.7,0.1) (0.2,0.5) (0.7,0.1)(0.8,0.1) (0.3,0.4) (0.6,0.2) (0.5,0.3)

Table 7Aggregated payoffs for all Vik.

IFOWG IFOWA I-IFOWG I-IFOWA I-GIFOWA5 I-GIFOWA10

V11 (0.3060,0.4029) (0.3647,0.3633) (0.2908,0.4198) (0.3429,0.3893) (0.3990,0.3538) (0.4324,0.3093)V12 (0.4129,0.4000) (0.4247,0.4000) (0.3728,0.4000) (0.3847,0.4000) (0.4085,0.4000) (0.4345,0.4000)V13 (0.6602,0.2083) (0.7070,0.1741) (0.5903,0.2700) (0.6634,0.2297) (0.7105,0.2093) (0.7410, 0.1877)V21 (0.5275,0.3219) (0.5709,0.2707) (0.4846,0.3673) (0.5329,0.3016) (0.5805,0.2666) (0.6148,0.2444)V22 (0.5471,0.2260) (0.5773,0.1933) (0.5232,0.2260) (0.5616,0.1933) (0.6044,0.1788) (0.6406, 0.1614)V23 (0.6153,0.1210) (0.6272,0.1149) (0.5933,0.1414) (0.6102,0.1320) (0.6277,0.1297) (0.6476,0.1256)V31 (0.5825,0.3402) (0.6495,0.2647) (0.5825,0.3402) (0.6495,0.2647) (0.6961,0.2219) (0.7280, 0.1855)V32 (0.3898,0.2366) (0.4067,0.1821) (0.3898,0.2366) (0.4067,0.1821) (0.4352,0.1646) (0.4589,0.1507)V33 (0.5785,0.1414) (0.5817,0.1320) (0.5578,0.1414) (0.5627,0.1320) (0.5686,0.1292) (0.5760, 0.1256)V41 (0.3567,0.3467) (0.4424,0.2759) (0.3567,0.3467) (0.4424,0.2759) (0.5597,0.2304) (0.6212,0.1887)V42 (0.6233,0.1701) (0.6978,0.1320) (0.5123,0.2347) (0.6118,0.1741) (0.6775,0.1540) (0.7123,0.1372)V43 (0.6964,0.1210) (0.7509,0.1149) (0.6063,0.1414) (0.6896,0.1320) (0.7355,0.1292) (0.7619,0.1256)V51 (0.5490,0.2382) (0.6200,0.2024) (0.4887,0.2782) (0.5596,0.2421) (0.6315,0.2212) (0.6885,0.1966)V52 (0.5378,0.3000) (0.5427,0.3000) (0.5186,0.3000) (0.5218,0.3000) (0.5272,0.3000) (0.5370, 0.3000)V53 (0.5885,0.3079) (0.6134,0.2221) (0.5885,0.3079) (0.6134,0.2221) (0.6340,0.1888) (0.6517,0.1638)

Table 8Aggregated payoffs for all Yi by the IFWA operator.

C1 C2 C3 C4 C5

IFOWG (0.4672,0.3298) (0.5614,0.2159) (0.5308,0.2344) (0.5626,0.2042) (0.5580, 0.2757)IFOWA (0.5111,0.2999) (0.5905,0.1892) (0.5672,0.1920) (0.6356,0.1700) (0.5962,0.2342)I-IFOWA (0.4729,0.3350) (0.5659,0.2060) (0.5614,0.1920) (0.5804,0.1926) (0.5659,0.2516)I-IFOWG (0.4202,0.3624) (0.5310,0.2384) (0.5240,0.2344) (0.4891,0.2357) (0.5295,0.2934)I-GIFOWA2 (0.4867,0.3301) (0.5754,0.2017) (0.5700,0.1868) (0.6030,0.1873) (0.5757,0.2462)I-GIFOWA5 (0.5196,0.3136) (0.6023,0.1903) (0.5934,0.1725) (0.6558,0.1716) (0.6037, 0.2311)I-GIFOWA10 (0.5519,0.2876) (0.6327,0.1767) (0.6180,0.1550) (0.6966,0.1518) (0.6372,0.2113)

Table 9Aggregated payoffs for all Yi by the IFWG operator.

C1 C2 C3 C4 C5

IFOWG (0.4216,0.3492) (0.5585,0.2373) (0.5153,0.2540) (0.5155,0.2327) (0.5571,0.2784)IFOWA (0.4656,0.3238) (0.5892,0.2033) (0.5460,0.2021) (0.5945,0.1880) (0.5938,0.2388)I-IFOWA (0.4326,0.3487) (0.5638,0.2216) (0.5406,0.2021) (0.5570,0.2047) (0.5633,0.2542)I-IFOWG (0.3874,0.3721) (0.5269,0.2634) (0.5097,0.2540) (0.4662,0.2564) (0.5260, 0.2938)I-GIFOWA2 (0.4465,0.3441) (0.5738,0.2158) (0.5482,0.1955) (0.5830,0.1978) (0.5728,0.2494)I-GIFOWA5 (0.4778,0.3286) (0.6015,0.2012) (0.5690,0.1778) (0.6433,0.1783) (0.5989,0.2364)I-GIFOWA10 (0.5090,0.3049) (0.6322,0.1855) (0.5909,0.1575) (0.6881,0.1548) (0.6286,0.2198)

Table 10Rankings of the alternatives.

IFWA IFWG

IFOWG Y4 > Y2 > Y3 > Y5 > Y1 Y2 > Y4 > Y5 > Y3 > Y1

IFOWA Y4 > Y2 > Y3 > Y5 > Y1 Y4 > Y2 > Y5 > Y3 > Y1

I-IFOWG Y3 > Y2 > Y4 > Y5 > Y1 Y2 > Y3 > Y5 > Y4 > Y1

I-IFOWA Y3 > Y4 > Y2 > Y5 > Y1 Y4 > Y2 > Y3 > Y5 > Y1

I-GIFOWA5 Y4 > Y3 > Y2 > Y5 > Y1 Y4 > Y2 > Y3 > Y5 > Y1

I-GIFOWA10 Y4 > Y3 > Y2 > Y5 > Y1 Y4 > Y2 > Y3 > Y5 > Y1

206 Z. Xu, M. Xia / Knowledge-Based Systems 24 (2011) 197–209

and I-GIFOWA operators consider the inducing informationgiven by the experts, thus can get more reasonable results thanthe IFOWG and IFOWA operators. However, each of the existingoperators doesn’t work by itself for the problem given inExample 2. Therefore, they need cooperation which is just theaim of the BSI-GIFOA operator given in Definition 5.

4.3. Interval-valued intuitionistic Dempster–Shafer operators

In the following, we extend the intuitionistic fuzzy Demp-ster–Shafer operators to interval-valued intuitionistic fuzzyenvironments, in which the available arguments are expressedin IVIFVs.

Definition 6. Let X = {x1,x2, . . . ,xn} be a fixed set, ðhu1; ~a1i;hu2; ~a2i; . . . ; hun; ~aniÞ be a collection of 2-tuples on X, whereui is the inducing variables and ~ai ði ¼ 1;2; . . . ;nÞ be theaggregated arguments in the form of IVIFVs. TheneM ¼ ð eMkj eMk ¼ fhui; ~aitijxi 2 Dk; i ¼ 1; 2; . . . ; ng; k ¼ 1; 2; . . . ; rÞ ¼ðfhu1k; ~c1ki; hu2k; ~c2ki; . . . ; huqkk; ~cqkkigjk ¼ 1; 2; . . . ; rÞ be a collec-tion of 2-tuple arguments with r focal elements Dk(k = 1,2, . . . ,r).A BSI-GIVIFOA operator of dimension r is a function BSI-GIVIFOA: eV r ! eV defined by

BSI�GIVIFOAk1 ;k2 ð eMÞ ¼�r

k¼1 pðDkÞ �qkj¼1 wjk

~bk1jk

� �� �k2=k1� � 1=k2

; k1 > 0;k2 > 0;

�rk¼1 pðDkÞ �qk

j¼1~b

wjk

jk

� �� �k2� � 1=k2

; k1 ¼ 0;k2 > 0;

�rk¼1 �qk

j¼1 wjk~bk1

jk

� �� �1=k1� pðDkÞ

; k1 > 0;k2 ¼ 0;

�rk¼1 �qk

j¼1~b

wjk

jk

� �� �pðDkÞ�

; k1 ¼ 0;k2 ¼ 0;

8>>>>>>>>>>>>><>>>>>>>>>>>>>:ð48Þ

where Wk ¼ ðw1k;w2k; . . . ;wqkkÞ is the weighting vector of the kthfocal element Dk such that

Pqkj¼1wjk ¼ 1 and wjk 2 [0,1], qk is the

number of elements in Dk, ~bjk is the ~cjk value of the pair huik; ~cikihaving the jth largest uik (i = 1,2, . . . ,qk), uik is the order-inducing

Z. Xu, M. Xia / Knowledge-Based Systems 24 (2011) 197–209 207

variable, ~cik is the argument variable, and p(Dk) is the basic proba-bility assignment.

By the operational laws of IVIFVs and mathematical inductionon n, some special cases of the BSI-GIVIFOA operator can be givenas follows:

(1) If k1 > 0, k2 > 0 then Eq. (48) reduces to

BSI�GIVIFOAk1 ;k2 ð eMÞ ¼ �rk¼1 pðDkÞ �qk

j¼1 wjk~bk1

jk

� �� �k2=k1� � 1=k2

¼ 1�Yr

k¼1

1� 1�Yqk

j¼1

ð1� ðt�~bjkÞk1 Þwjk

!k2=k10@ 1ApðDkÞ

0B@1CA

1=k2

;

2640B@1�

Yr

k¼1

1� 1�Yqk

j¼1

1� tþ~bjk

� �k1� wjk

!k2=k10@ 1ApðDkÞ

0B@1CA

1=k2375;

1� 1�Yr

k¼1

1� 1�Yqk

j¼1

1� 1� f�~bjk

� �k1� wjk

!k2=k10@ 1ApðDkÞ

0B@1CA

1=k2

;

2641� 1�

Yr

k¼1

1� 1�Yqk

j¼1

ð1� ð1� fþ~bjkÞk1 Þwjk

!k2=k10@ 1ApðDkÞ

0B@1CA

1=k23751CA:ð49Þ

(2) If k1 = k2 = k, then Eq. (48) reduces to

BSI�GIVIFOAkð eMÞ ¼ �rk¼1 pðDkÞ�qk

j¼1 wjk~bk

jk

� �� �� �1=k

¼ 1�Yr

k¼1

Yqk

j¼1

1� t�~bjk

� �k� wjkpðDkÞ

!1=k

;

240@1�

Yr

k¼1

Yqk

j¼1

1� tþ~bjk

� �k� wjk pðDkÞ

!1=k35;

1� 1�Yr

k¼1

Yqk

j¼1

1� 1� f�~bjk

� �k� wjk pðDkÞ

!1=k

;

241� 1�

Yr

k¼1

Yqk

j¼1

1� 1� fþ~bjk

� �k� wjk pðDkÞ

!1=k351A:ð50Þ

(3) If k1 = k2 = 1, then Eq. (48) becomes:

BSI� IVIFOAAð eMÞ ¼�rk¼1 pðDkÞ�qk

j¼1ðwjk~bjkÞ

� �¼ 1�

Yr

k¼1

Yqk

j¼1

1� t~bjk

� �wjkpðDkÞ;

"

1�Yr

k¼1

Yqk

jk¼1

1� t~bjk

� �wjkpðDkÞ#;Yr

k¼1

Yqk

j¼1

f�~bjk

� �wjkpðDkÞ;

"Yr

k¼1

Yqk

j¼1

fþ~bjk

� �wjkpðDkÞ#!

; ð51Þ

which we call a BSI-IVIFOAA operator.(4) If k1 = 0 and k2 > 0, then Eq. (48) reduces to:

BSI�GIVIFOAk2 ð eMÞ ¼ �rk¼1 pðDkÞ �qk

j¼1~b

wjk

jk

� �� �k2� � 1=k2

¼ 1�Yr

k¼1

1�Yqk

j¼1

t�~bjk

� �wjk

!pðDkÞ0@ 1A1=k2

;

2640B@1�

Yr

k¼1

1�Yqk

j¼1

tþ~bjk

� �wjk

!pðDk Þ0@ 1A1=k2

375;1� 1�

Yr

k¼1

1�Yqk

j¼1

1� f�~bjk

� �wjk

!k20@ 1ApðDkÞ

0B@1CA

1=k2

;

2641� 1�

Yr

k¼1

1�Yqk

j¼1

1� fþ~bjk

� �wjk

!k20@ 1ApðDk Þ

0B@1CA

1=k23751CA:ð52Þ

(5) If k1 = 0 and k2 = 1, then Eq. (48) reduces to the BSI-IVIFOAGoperator:

BSI� IVIFOAGð eMÞ ¼ �rk¼1 pðDkÞ�qk

j¼1~b

wjk

jk

� �� �¼ 1�

Yr

k¼1

1�Yqk

j¼1

t�~bjk

� �wjk

!pðDkÞ

;

240@1�

Yr

k¼1

1�Yqk

j¼1

tþ~bjk

� �wjk

!pðDkÞ35;

Yr

k¼1

1�Yqk

jk¼1

1� f�~bjk

� wjk

!pðDkÞ

;

24Yr

k¼1

1�Yqk

jk¼1

1� fþ~bjk

� wjk

!pðDkÞ351A: ð53Þ

(6) If k1 > 0 and k2 = 0, then Eq. (48) reduces to:

BSI�GIVIFOAk1 ð eMÞ ¼�rk¼1 �qk

j¼1 wjk~bk1

jk

� �� �1=k1� pðDkÞ

¼Yqk

j¼1

1�Yr

k¼1

1� t�~bjk

� �k1� wjk

!1=k10@ 1ApðDkÞ

;

2640B@Yqk

j¼1

1�Yr

k¼1

1� tþ~bjk

� �k1� wjk

!1=k10@ 1ApðDkÞ

375;1�

Yqk

j¼1

1�Yr

k¼1

1� 1� f�~bjk

� �k1� wjk

!1=k10@ 1ApðDkÞ

;

2641�

Yqk

j¼1

1�Yr

k¼1

1� 1� fþ~bjk

� �k1� wjk

!1=k10@ 1ApðDkÞ

3751CA:ð54Þ

(7) If k1 = 1 and k2 = 0, then from Eq. (48), we obtain the BSI-IVIFOGA operator:

BSI� IVIFOGAð eMÞ ¼ �rk¼1 �

qkj¼1 wjk

~bjk

� �� �pðDkÞ

¼Yr

k¼1

1�Yqk

j¼1

1� t�~bjk

� �wjk

!pðDkÞ

;

240@Yr

k¼1

1�Yqk

j¼1

1� tþ~bjk

� �wjk

!pðDkÞ35;

1�Yr

k¼1

1�Yqk

j¼1

f�bjk

� �wjk

!pðDkÞ

;

241�

Yr

k¼1

1�Yqk

j¼1

fþbjk

� �wjk

!pðDkÞ351A: ð55Þ

(8) If k1 = 0 and k2 = 0, then Eq. (48) becomes

BSI� IVIFOGGð eMÞ ¼ �rk¼1 �qk

j¼1~b

wjkjk

� �� �pðDkÞ�

¼Yr

k¼1

Yqk

j¼1

t�bjk

� �wjkpðDkÞ;Yr

k¼1

Yqk

j¼1

tþbjk

� �wjkpðDkÞ" #

;

1�Yr

k¼1

Yqk

j¼1

1� f�~bjk

� �wjkpðDkÞ;

"

1�Yr

k¼1

Yqk

j¼1

1� fþ~bjk

� �wjkpðDkÞ#!

; ð56Þ

which we call the BSI-IVIFOGG operator.

In Section 4.2, if each payoff value of the alternative Yi under thestate Gj is given by an IVIFV ~aij, then similar to Method III, in the

208 Z. Xu, M. Xia / Knowledge-Based Systems 24 (2011) 197–209

following we develop a method for decision making based on theBSI-GIVIFOA operator. The method can be described as follows:

(Method IV)

Step 1. See Method III.

Step 2. Find the set of payoffs: eMik ¼ uij; ~aij ���Gj 2 Dk; j ¼ 1;2;�

. . . ; tg ¼ uð1Þik ; ~cð1Þik

D E; uð2Þik ; ~c

ð2Þik

D E; . . . ; uðqkÞ

ik ; ~cðqkÞik

D En o.

Step 3. Determine the weight vector Wqk¼ wð1Þqk

;wð2Þqk; . . . ;wðqkÞ

qk

� �,

and calculate the aggregated payoff eV ik:

eV ik ¼�qk

j¼1 wðjÞqk~bðjÞik

� �k1� � 1=k1

; k1 > 0;

�qkj¼1 wðjÞqk

~bðjÞik

� �; k1 ¼ 0;

8>><>>: ð57Þ

where ~bðjÞik is the ~cðlÞik value of the pair uðlÞik ; ~cðlÞik

D Ewith the jth

largest uðlÞik ðl ¼ 1;2; . . . ; qkÞ.Step 4. Utilize the BSI-GIVIFOA operator (35) to calculate theaggregated payoff eCi for each alternative Yi:

eCi ¼�r

k¼1 pðDkÞ eV ik

� �k2� � 1=k2

; k2 > 0;

�rk¼1 p Dkð ÞeV ik

� �; k2 ¼ 0;

8>><>>:

¼

�rk¼1 pðDkÞ �qk

j¼1 wðjÞqk~bðjÞik

� �k1� � k2=k1

! !1=k2

; k1 > 0; k2 > 0;

�rk¼1 pðDkÞ �qk

j¼1 wðjÞqk~bðjÞik

� �� �k2� � 1=k2

; k1 ¼ 0; k2 > 0;

�rk¼1 pðDkÞ �qk

j¼1 wðjÞqk~bðjÞik

� �k1� � 1=k1

!; k1 > 0; k2 ¼ 0;

�rk¼1 pðDkÞ�qk

j¼1 wðjÞqk~bðjÞik

� �� �; k1 ¼ 0; k2 ¼ 0:

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð58Þ

Step 5. Select the best alternative Yi with the largest eCi.

5. Concluding remarks

In this paper, we have developed some induced generalizedaggregation operators under intuitionistic fuzzy environmentsbased on the Choquet integral and Dempster–Shafer belief struc-ture, whose fundamental characteristic is that they allow theordering of the available arguments to be based upon other associ-ated variables instead of ordering the arguments based on theirvalues. The induced operators based on the Choquet integral candeal with the situations in which the weights of the aggregatedarguments have some correlations with each other, while the in-duced operators based on the Dempster–Shafer theory of evidenceare very useful for the problem with uncertainty. Then their vari-ous properties and some of their special cases have been investi-gated. Furthermore, a financial decision making problem underintuitionistic fuzzy environments has been given to illustrate ouroperators. The numerical results showed that different optimalalternatives may be yielded by using different aggregation opera-tors, and thus, the decision maker can properly select the desirablealternative according to his interest and the actual needs. Addition-ally, further extensions of the developed operators and methods toaccommodate interval-valued intuitionistic fuzzy situations havealso been investigated in detail, which allows our results to havemore wide practical application potentials.

Acknowledgements

The work was supported by the National Science Fund for Dis-tinguished Young Scholars of China (No.70625005), the NationalNatural Science Foundation of China (No.71071161), the ProgramSponsored for Scientific Innovation Research of College Graduatein Jiangsu Province (No. CX10B_059Z) and the Scientific Research

Foundation of Graduate School of Southeast University. Theauthors are very grateful to the anonymous reviewers for theirconstructive comments and suggestions that have led to an im-proved version of this paper.

References

[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87–96.[2] K. Atanassov, G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets

and Systems 31 (1989) 343–349.[3] G. Beliakov, Learning weights in the generalized OWA operator, Fuzzy

Optimization and Decision Making 4 (2005) 119–130.[4] G. Büyüközkan, G. Mauris, L. Berrah, O. Feyzioglu, Providing elucidations of

website evaluation based on a multi-criteria aggregation, in: Proceedings ofthe International Fuzzy Systems Association World Congress (IFSA), Istanbul,Turkey, 2003, pp. 131–134.

[5] G. Büyüközkan, O. Feyzioglu, M.S. Ersoy, Evaluation of 4PL operating models: adecision making approach based on 2-additive Choquet integral, InternationalJournal of Production Economics 121 (2009) 112–120.

[6] G. Büyüközkan, D. Ruan, Coquet integral based aggregation approach to softwaredevelopment risk assessment, Information Sciences 180 (2010) 441–451.

[7] G. Choquet, Theory of capacities, Annales de l’Institut Fourier (Crenoble) 5(1953) 131–295.

[8] A.P. Dempster, Upper and lower probabilities induced by a multi-valuedmapping, Annals of Mathematical Statistics 38 (1967) 325–339.

[9] A.P. Dempster, A generalization of Bayesian inference, Journal of the RoyalStatistical Society 30 (1968) 205–247.

[10] D. Denneberg, Non-Additive Measure and Integral, Kluwer Academic Press,Boston, MA, 1994.

[11] M. Grabisch, Fuzzy integral in multicriteria decision making, Fuzzy Sets andSystems 69 (1995) 279–298.

[12] M. Grabisch, K-order additive discrete fuzzy measures and theirrepresentation, Fuzzy Sets and Systems 92 (1997) 167–189.

[13] M. Grabisch, T. Murofushi, M. Sugeno, Fuzzy Measures and Integrals, Physica-Verlag, Heidelberg, 1999.

[14] D.F. Li, Y.C. Wang, S. Liu, F. Shan, Fractional programming methodology formulti-attribute group decision-making using IFS, Applied Soft Computing 9(2009) 219–225.

[15] P.D. Liu, A novel method for hybrid multiple attribute decision making,Knowledge-Based Systems 22 (2009) 388–391.

[16] J.M. Merigó, A.M. Gil-Lafuente, The induced generalized OWA operator,Information Sciences 179 (2009) 729–741.

[17] J.M. Merigó, M. Casanovas, Induced aggregation operators in decision makingwith the Dempster–Shafer belief structure, International Journal of IntelligentSystems 24 (2009) 934–954.

[18] G. Shafer, Mathematical Theory of Evidence, Princeton University Press,Princeton, N J, 1976.

[19] R.P. Srivastava, T. Mock, Belief Functions in Business Decisions, Physica-Verlag,Heidelberg, 2002.

[20] M. Sugeno, Theory of Fuzzy Integral and its Application. Doctoral dissertation,Tokyo Institute of Technology, 1974.

[21] V. Torra, Information Fusion in Data Mining, Springer, Berlin, 2003.[22] V. Torra, Y. Narukawa, Modeling Decisions: Information Fusion and

Aggregation Operators, Springer, Berlin, 2007.[23] C.Q. Tan, X.H. Chen, Induced Choquet ordered averaging operator and its

application to group decision making, International Journal of IntelligentSystems 25 (2009) 59–82.

[24] C.Q. Tan, X.H. Chen, Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making, Expert Systems with Applications 37 (2010) 149–157.

[25] Z. Wang, G. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992.[26] G.W. Wei, Maximizing deviation method for multiple attribute decision

making in intuitionistic fuzzy setting, Knowledge-Based Systems 21 (2008)833–836.

[27] G.W. Wei, Induced intuitionistic fuzzy ordered weighted averaging operatorand its application to multiple attribute group decision making, Lecture Notesin Computer Science 5009 (2008) 124–131.

[28] G.W. Wei, Some induced geometric aggregation operators with intuitionisticfuzzy information and their application to group decision making, Applied SoftComputing 10 (2010) 423–431.

[29] G.W. Wei, GRA method for multiple attribute decision making withincomplete weight information in intuitionistic fuzzy setting, Knowledge-Based Systems 23 (2010) 243–247.

[30] Z.S. Xu, An overview of methods for determining OWA weights, InternationalJournal of Intelligent System 20 (2005) 843–865.

[31] Z.S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on FuzzySystems 15 (2007) 1179–1187.

[32] Z.S. Xu, Choquet integrals of weighted intuitionistic fuzzy information,Information Sciences 180 (2010) 726–736.

[33] Z.S. Xu, J. Chen, On geometric aggregation over interval-valued intuitionisticfuzzy information, in: The Fourth International Conference on Fuzzy Systemsand Knowledge Discovery (FSKD’07), Haikou, China, vol. 2, August, 24–27,2007, pp.466–471.

Z. Xu, M. Xia / Knowledge-Based Systems 24 (2011) 197–209 209

[34] Z.S. Xu, R.R. Yager, Some geometric aggregation operators based onintuitionistic fuzzy sets, International Journal of General Systems 35 (2006)417–433.

[35] R.R. Yager, On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE Transactions on Systems, Man, and Cybernetics18 (1988) 183–190.

[36] R.R. Yager, Decision making under Dempster–Shafer uncertainties,International Journal of General Systems 20 (1992) 233–245.

[37] R.R. Yager, Families of OWA operators, Fuzzy Sets and Systems 59 (1993) 125–148.

[38] R.R. Yager, J. Kacprzyk, The Ordered Weighted Averaging Operators: Theoryand Applications, Kluwer, Norwell, MA, 1997.

[39] R.R. Yager, D.P. Filev, Induced ordered weighted averaging operators, IEEETrans Systems, Man Cybernetics B 29 (1999) 141–150.

[40] R.R. Yager, Induced aggregation operators, Fuzzy Sets and Systems 137 (2003)59–69.

[41] R.R. Yager, Choquet aggregation using order inducing variables, InternationalJournal of Uncertainty, Fuzziness and Knowledge-Based Systems 12 (2004)69–88.

[42] R.R. Yager, Generalized OWA aggregation operators, Fuzzy Optimization andDecision Making 3 (2004) 93–107.

[43] J. Ye, Fuzzy decision-making method based on the weighted correlationcoefficient under intuitionistic fuzzy environment, European Journal ofOperational Research 205 (2010) 202–204.

[44] H. Zhao, Z.S. Xu, M.F. Ni, S.S. Liu, Generalized aggregation operators forintuitionistic fuzzy Sets, International Journal of Intelligent Systems 25 (2010)1–30.