Applied Mathematical Modelling 37 (2013) 5120–5133

Contents lists available at SciVerse ScienceDirect

Applied Mathematical Modelling

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

Group decision making based on intuitionistic multiplicativeaggregation operators

Meimei Xia a,⇑, Zeshui Xu b

a School of Economics and Management, Tsinghua University, Beijing 100084, Chinab Institute of Sciences, PLA University of Science and Technology, Nanjing 210007, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 July 2012Received in revised form 19 September 2012Accepted 15 October 2012Available online 29 October 2012

Keywords:Group decision makingIntuitionistic multiplicative preferencerelationAggregation operator

0307-904X/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.apm.2012.10.029

⇑ Corresponding author.E-mail address: meimxia@163.com (M. Xia).

Preference relations are the most common techniques to express decision maker’s prefer-ence information over alternatives or criteria. To consistent with the law of diminishingmarginal utility, we use the asymmetrical scale instead of the symmetrical one to expressthe information in intuitionistic fuzzy preference relations, and introduce a new kind ofpreference relation called the intuitionistic multiplicative preference relation, which con-tains two parts of information describing the intensity degrees that an alternative is ornot priority to another. Some basic operations are introduced, based on which, an aggrega-tion principle is proposed to aggregate the intuitionistic multiplicative preference informa-tion, the desirable properties and special cases are further discussed. Choquet Integral andpower average are also applied to the aggregation principle to produce the aggregationoperators to reflect the correlations of the intuitionistic multiplicative preference informa-tion. Finally, a method is given to deal with the group decision making based on intuition-istic multiplicative preference relations.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Preference relations are useful tools to express the preference information about a set of alternatives or criteria, and canbe roughly classified into two types: the fuzzy preference relations [1] (also called the reciprocal preference relations [2]) andthe multiplicative preference relations [3]. Due to the complexity of the decision making or the lack of knowledge about thebackground, it is difficult for the decision makers to use the exact values to express their preference information aboutalternatives or criteria, to deal with such cases, the interval-valued fuzzy preference relations [4,5] and the interval-valuedmultiplicative preference relations [6–8] are provided to allow the decision makers to use the interval-valued numbers [9] toexpress their preference information. The main differences between these two kinds of preference relations are on the rep-resented forms to express the decision makers’ preference information about the alternatives or criteria, the former is basedon the 0.1–0.9 scale, which is a symmetrical distribution around 0.5, while the latter is based on Saaty’s 1–9 scale which is anon-symmetrical distribution around 1 (see Table 1 for more details).

It is noted that the basic element in the above preference relations only provide the degree that an alternative is priorityto another, but sometimes, the decision makers may also provide the degree that an alternative is not priority to another, tosolve such issue, the intuitionitic fuzzy preference relation [10–12] is introduced, which is based on the intuitionistic fuzzyset [13] constructed by two functions, the membership function and the non-membership function, describing the fuzzinessand uncertainty more objectively than the usual fuzzy set [9]. By comparing the interval-valued fuzzy preference relation

. All rights reserved.

Table 1The comparison between the 0.1–0.9 scale and the 1–9 scale.

1–9 Scale 0.1–0.9 Scale Meaning

1/9 0.1 Extremely not preferred1/7 0.2 Very strongly not preferred1/5 0.3 Strongly not preferred1/3 0.4 Moderately not preferred1 0.5 Equally preferred3 0.6 Moderately preferred5 0.7 Strongly preferred7 0.8 Very strongly preferred9 0.9 Extremely preferredOther values between 1/9 and 9 Other values between 0 and 1 Intermediate values used to present compromise

M. Xia, Z. Xu / Applied Mathematical Modelling 37 (2013) 5120–5133 5121

and the intuitionitic fuzzy preference relation, we can find these two preference relations can be transformed between eachother, although the represented forms of them are the same, the meanings and the operations of them are different. It is easyto find that both of them use the balanced scale, i.e., 0.1–0.9 scale, to express the preference information, but in real-life, theinformation are usually distributed asymmetrically, the well-known law of diminishing marginal utility is the common phe-nomenon in economics even in our daily life.

Based on the above analysis, we can find that it is better to use the asymmetrical distributed scale to express the infor-mation, motivated by which, Xia et al. [14] used the 1–9 scale instead of the 0.1–0.9 scale in the intuitionistic fuzzy prefer-ence relation and introduced the new concept of intuitionistic multiplicative preference relation and proposed somecorresponding aggregation operators, which can avoid some disadvantages of the intuitionistic fuzzy aggregation operators.Up to now, a lot of work has been done about other preference relations, but little has been done about the intuitionisticmultiplicative preference relations. In this paper we give some aggregation techniques to aggregate the intuitionistic mul-tiplicative preference information based on the pseudo-multiplication, which can reduce to the aggregation operators givenby Xia et al. [14] when some specific forms are assigned. Based on the correlations between the aggregation arguments, weintroduce the Choquet Integral [15] and power average [16] to propose the aggregation operators to deal with the situationthat the intuitionistic multiplicative preference information is dependent with each other.

To do this, the reminder of this paper is constructed as follows: Section 2 mainly introduces the concept of intuitionisticmultiplicative preference relation. In Section 3, some operational laws are proposed for intuitionistic multiplicative informa-tion, the relationships and the correlations of them are also discussed. Section 4 gives an aggregation principle for the intui-tionistic multiplicative information, and gives some desirable properties and special cases. In Section 5, some aggregationoperators are developed to reflect the correlations of the aggregated arguments based on the well-known Choquet Integral[15] and power average [16]. Section 6 proposes an approach to deal with the group decision making under intuitionisticmultiplicative preference relations. Section 7 gives some discussions and limitations.

2. Intuitionistic multiplicative preference relation

For a set of alternatives X = {x1,x2, . . . ,xn}, Xu [11] defined the intuitionistic fuzzy preference relation B = (b ij)n�n, wherebij ¼ ðlbij

;vbijÞ is an intuitionistic fuzzy number (IFN; [17]), and lbij

indicates the degree that xi is preferred to xj, vbijindicates

the degree that xi is not preferred to xj, and both of them satisfy the condition that lbij¼ vbji

, vbij¼ lbji

, 0 6 lbijþ vbij

6 1 and

lbijþ vbij

6 1. It is noted that the IFN bij ¼ ðlbij;vbijÞ can be written as an interval-valued fuzzy number (IVFN)

bij ¼ b�ij ; bþij

h i¼ ½lbij

;1� vbij�, then the intuitionistic preference relation B = (bij)n�n can be written as an interval-valued fuzzy

preference relation ~B ¼ ðbijÞn�n ¼ ð½b�ij ; b

þij �Þn�n ¼ ð½lbij

;1� vbij�Þn�n. Conversely, an interval-valued fuzzy preference relation

~B ¼ ðbijÞn�n ¼ b�ij ; bþij

h i� �n�n

can also be written as an intuitionistic fuzzy preference relation B ¼ ðbijÞn�n ¼ ððlbij;vbijÞÞn�n ¼

b�ij ;1� bþij� �� �

n�n.

It is noted that each element bij in an intuitionistic fuzzy preference relation B is expressed by using the 0.1–0.9 scalewhich assumes that the grades between ‘‘Extremely not preferred’’ and ‘‘Extremely preferred’’ are distributed uniformlyand symmetrically (see Table 1 for more details), but in real-life, there exist the problems that need to assess their variableswith the grades that are not uniformly and symmetrically distributed [18,19]. The unbalanced distribution may appear dueto the nature of attributes of the problem, the law of diminishing marginal utility in economics is a good example. To investthe same resources to a company with bad performance and to a company with good performance, the former enhancesmore quickly than the latter. In other words, the gap between the grades expressing good information should be bigger thanthe one between the grades expressing bad information.

Based on the above analysis, Xia et al. [14] used the 1–9 scale instead the 0.1–0.9 scale to express the preference infor-mation in an intuitionistic fuzzy preference relation and introduced the intutionstic multiplicative preference relation de-fined as: A = (aij)n�n, where qaij

¼ raji, raij

¼ qaji, 0 6 qaij

raij6 1 and 1=9 6 qaij

;raij6 9. The basic element of an

5122 M. Xia, Z. Xu / Applied Mathematical Modelling 37 (2013) 5120–5133

intuitionistic multiplicative preference relation, i.e., aij ¼ ðqaij;raijÞ ¼ a�ij ;1=aþij

� �, is called the intuitionistic multiplicative

number (IMN; [14]), and qaijcan be considered as the intensity degree that xi is preferred to xj, raij

can be considered as

the intensity degree that xi is not preferred to xj. In fact, the intuitionistic multiplicative preference relation and the inter-val-valued multiplicative preference relation can be transformed between each other similar to the correlation betweenintuitionistic fuzzy preference relation and interval-valued preference relation. For example, the interval-valued multiplica-

tive preference relation ~A ¼ ð~aijÞn�n with the condition that aþij a�ji ¼ a�ij aþji ¼ 1, 1=9 6 a�ij 6 aþij 6 9, can be written as the intui-

tionistic multiplicative preference relation A = (aij)n�n, where aij ¼ ða�ij ;1=aþij Þ. Conversely, the intuitionistic multiplicativepreference relation A = (aij)n�n with aij ¼ ðqaij

;raijÞ can be written as an interval-valued multiplicative preference relation

~A ¼ ð~aijÞn�n, where ~aij ¼ ½qaij;1=raij

�. However, the operations on them are different, because the meanings of them are differ-

ent, the intuitionistic multiplicative preference relation contains two parts of information: the membership information andthe nonmembership information similar to the intuitionistic fuzzy preference relation, while the interval-valued multiplica-tive preference relation only contains the membership information similar to the interval-valued fuzzy preference relation.

Based on the above analysis, Xia et al. [14] gave the following definition:

Definition 1 ([14]). Let X = {x1,x2, . . . ,xn} be n alternatives, then the intuitionistic multiplicative preference relation is definedas A = (aij)n�n, where aij ¼ ðqaij

;raij Þ is an intuitionistic multiplicative number (IMN), and qaijindicates the intensity degree to

which xi is preferred to xj, raij indicates the intensity degree to which xi is not preferred to xj, and both of them should satisfythe condition that

qaij¼ raji

; raij¼ qaji

; 0 6 qaijraij6 1; 1=a 6 qaij

;raij6 a; a > 1: ð1Þ

It is noted that the fundamental element of an intuitionistic multiplicative preference relation is the IMN. To compare twoIMNs, the following comparison laws are also given:

Definition 2 ([14]). For an IMN a = (qa,ra), we call s(a) = qa/ra the score function of a, and h(a) = qara the accuracy functionof a. To compare two IMNs a1 and a2, the following laws can be given:

(1) If s(a1) > s(a2), then a1 > a2;(2) If s(a1) = s(a2), then

If h(a1) > h(a2), then a1 > a2;If h(a1) = h(a2), then a1 = a2.

3. Some extended intuitionistic multiplicative operations

Xia et al. [14] proposed some operations about intuitionistic fuzzy information, based on which, in this section, wepropose some extended operations about the intuitionistic multiplicative information, before doing this, we first give thedefinition as follows:

Definition 3 ([20]). The pseudo-multiplication � is defined as: x � y = g�1(g(x)g(y)), where g is a strictly decreasing functionsuch that g(t): (0,1) ? (0,1).

Based on the pseudo-multiplication, we can define some operations about IMNs expressed as follows:

Definition 4. Let ai ¼ ðqai;rai Þ ði ¼ 1;2Þ and a = (qa,ra) be three IMNs, then we have

(1) a1 þ a2 ¼ ðh�1ðhðqa1Þ � hðqa2

ÞÞ; g�1ðgðra1 Þ � gðra2 ÞÞÞ(2) a1 � a2 ¼ ðg�1ðgðqa1

Þ � gðqa2ÞÞ;h�1ðhðra1 Þ � hðra2 ÞÞÞ.

(3) ka = (h�1((h(qa))k),g�1 ((g(ra))k)), k > 0.(4) ak = (g�1((g(qa))k), h�1((h(ra))k)), k > 0.

where g is a strictly decreasing function such that g(t): (0,1) ? (0,1), and h(t) = g(1/t).Especially, if gðtÞ ¼ 1þt

t , then (1)–(4) in Definition 4 reduce to.

(1)0 a1 þ a2 ¼ qa1þ qa2

þ qa1qa2

;ra1 ra2

ra1þra2þ1

� �.

(2)0 a1 � a2 ¼qa1

qa2qa1þqa2

þ1 ;ra1 þ ra2 þ ra1ra2

� �.

(3)0 ka ¼ ð1þ qaÞk � 1; rk

að1þraÞk�rk

a

� �, k > 0.

(4)0 ak ¼ qka

ð1þqaÞk�qk

a; ð1þ raÞk � 1

� �, k > 0.

M. Xia, Z. Xu / Applied Mathematical Modelling 37 (2013) 5120–5133 5123

If gðtÞ ¼ 2þtt , then (1)–(4) in Definition 4 reduce to

(1)00 a1 þ a2 ¼ð1þ2qa1

Þð1þ2qa2Þ�1

2 ;2ra1 ra2

ð2þra1 Þð2þra2 Þ�ra1 ra2

� �.

(2)00 a1 � a2 ¼2qa1

qa2ð2þqa1

Þð2þqa2Þ�qa1

qa2;ð1þ2ra1 Þð1þ2ra2 Þ�1

2

� �.

(3)00 ka ¼ ð1þ2qaÞk�1

2 ;2rk

að2þraÞk�rk

a

� �, k > 0.

(4)00 ak ¼ 2qka

ð2þqaÞk�qk

a; ð1þ2raÞk�1

2

� �, k > 0.

which is given by Xia et al. [14].If gðtÞ ¼ cþt

t , c > 0, then (1)–(4) in Definition 4 reduce to.

(1)000 a1 þ a2 ¼ðcqa1

þ1Þðcqa2þ1Þ�1

c ;cra1 ra2

ðra1þcÞðra2þcÞ�ra1 ra2

� �.

(2)000 a1 � a2 ¼cqa1

qa2ðqa1

þcÞðqa2þcÞ�ra1 ra2

;ðcra1þ1Þðcra2þ1Þ�1

c

� �.

(3)000 ka ¼ ðcqaþ1Þk�1c ;

crka

ðraþcÞk�rka

� �, k > 0.

(4)000 ak ¼ cqka

ðqaþcÞk�qka; ðcraþ1Þk�1

c

� �, k > 0.

Especially, if c = 1, then (1)000–(4)000 reduce to (1)0–(4)0; if c = 2, then (1)000–(4)000 reduce to (1)00–(4)00.Moreover, some relationships of the operational laws can be discussed as follows:

Theorem 1. Let ai ¼ ðqai;rai Þ ði ¼ 1;2Þ and a = (qa,ra) be three IMNs, and k > 0, then the relations of these operational laws are

given as:

(1) a1 + a2 = a2 + a1.(2) a1 � a2 = a2 � a1.(3) k(a1 + a2) = ka1 + k a2.(4) ða1 � a2Þk ¼ ak

1 � ak2.

(5) k1a + k2a = (k1 + k2)a.(6) ak1 � ak2 ¼ ak1þk2 .

Proof. (1) and (2) are obvious, we prove the others:

ð3Þ kða1 þ a2Þ ¼ kðh�1ðhðqa1Þ � hðqa2

ÞÞ; g�1ðgðra1 Þ � gðra2 ÞÞÞ

¼ ðh�1ððhðh�1ðhðqa1Þ � hðqa2

ÞÞÞÞkÞ; g�1ððgðg�1ðgðra1 Þ � gðra2 ÞÞÞÞkÞÞ

¼ ðh�1ððhðqa1Þ � hðqa2

ÞÞkÞ; g�1ððgðra1 Þ � gðra2 ÞÞkÞÞ

ka1 þ ka2 ¼ ðh�1ððhðqa1ÞÞkÞ; g�1ððgðra1 ÞÞ

kÞÞ þ ðh�1ððhðqa2ÞÞkÞ; g�1ððgðra2 ÞÞ

kÞÞ

¼ ðh�1ðhðh�1ððhðqa1ÞÞkÞÞ � hðh�1ððhðqa2

ÞÞkÞÞÞ; g�1ðgðg�1ððgðra1 ÞÞkÞÞ � gðg�1ððgðra2 ÞÞ

kÞÞÞÞ

¼ ðh�1ððhðqa1ÞÞk � ðhðqa2

ÞÞkÞ; g�1ððgðra1 ÞÞk � ðgðra2 ÞÞ

kÞÞ ¼ kða1 þ a2Þ:

ð5Þ k1aþ k2a ¼ ðh�1ððhðqaÞÞk1 Þ; g�1ððgðraÞÞk1 ÞÞ þ ðh�1ððhðqaÞÞ

k2 Þ; g�1ððgðraÞÞk2 ÞÞ

¼ ðh�1ðhðh�1ððhðqaÞÞk1 ÞÞ � hðh�1ððhðqaÞÞ

k2 ÞÞÞ; g�1ðgðg�1ððgðraÞÞk1 ÞÞ � gðg�1ððgðraÞÞk2 ÞÞÞÞ

¼ ðh�1ððhðqaÞÞk1 � ðhðqaÞÞ

k2 Þ; g�1ððgðraÞÞk1 � ðgðraÞÞk2 ÞÞ ¼ ðk1 þ k2Þa

Similarly, (4) and (6) can be proven which completes the proof of the theorem. h

Theorem 2. Let ai ¼ ðqai;raiÞ ði ¼ 1;2Þ and a = (qa,ra) be three IMNs, and k > 0, then the followings are also valid:

(1) (ac)k = (ka)c.(2) k(ac) = (ak)c.(3) ac

1 þ ac2 ¼ ða1 � a2Þc.

(4) ac1 � ac

2 ¼ ða1 þ a2Þc ,

where ac = (ra,qa) denotes the complement of an IMN a.

5124 M. Xia, Z. Xu / Applied Mathematical Modelling 37 (2013) 5120–5133

Proof. Based on the operations defined in Definition 4, we have

(1) (ac)k = (g� 1((g(ra))k ),h�1((h(qa))k)) = (ka)c.(2) k (ac) = (h�1((h(ra))k),g�1((g(qa))k)) = (ak)c.(3) ac

1 þ ac2 ¼ ðh

�1ðhðra1 Þ � hðra2 ÞÞ; g�1ðgðqa1Þ � gðqa2

ÞÞÞ ¼ ða1 � a2Þc.(4) ac

1 � ac2 ¼ ðg�1ðgðra1 Þ � gðra2 ÞÞ;h

�1ðhðqa1Þ � gðqa2

ÞÞÞ ¼ ða1 þ a2Þc ,

which completes the proof. h

4. Some extended intuitionistic multiplicative aggregation operators

In this section, we mainly apply the operational laws defined in Section 3 to aggregate the intuitionistic multiplicativeinformation.

Definition 5. Let ai ¼ ðqai;rai Þ ði ¼ 1;2; . . . ;nÞ be a collection of IMNs, and w = (w1,w2, . . . ,wn)T be the weight vector of them,

where wi indicates the importance degree of ai, satisfying wi > 0 (i = 1,2, . . . ,n) andPn

i¼1wi ¼ 1, if

EIMWA ða1;a2; . . . ;anÞ ¼Xn

i¼1

wiai; ð2Þ

then EIMWA is called the extended intuitionistic multiplicative weighted averaging (EIMWA) operator.

Theorem 3. Let ai ¼ ðqai;raiÞ ði ¼ 1;2; . . . ;nÞ be a collection of IMNs, and w = (w1,w2, . . . ,wn)T be the weight vector of them,

where wi indicates the importance degree of ai, satisfying wi > 0 (i = 1,2, . . . ,n) andPn

i¼1wi ¼ 1, then the aggregated value by usingthe EIMWA operator is also an IMN, and

EIMWA ða1;a2; . . . ;anÞ ¼Xn

i¼1

wiai ¼ h�1Yn

i¼1

ðhðqaiÞÞwi

!; g�1

Yn

i¼1

ðgðraiÞÞwi

! !: ð3Þ

Proof. By using mathematical induction on n: For n = 2, we have

EIMWA ða1;a2Þ ¼X2

i¼1

wiai ¼ w1a1 þw2a2

¼ ðh�1ðhðh�1ððhðqa1ÞÞw1 ÞÞ � hðh�1ððhðqa2

ÞÞw2 ÞÞÞ; g�1ðgðg�1ððgðra1 ÞÞw1 ÞÞ � gðg�1ððgðra2 ÞÞ

w2 ÞÞÞÞ

¼ ðg�1ððgðqa1ÞÞw1 � ðgðqa2

ÞÞw2 Þ;h�1ððhðra1 ÞÞw1 � ððhðra2 ÞÞ

w2 ÞÞ: ð4Þ

Suppose Eq. (3) holds for n = k, that is

EIMWA ða1;a2; . . . ;akÞ ¼Xk

i¼1

wiai ¼ w1a1 þw2a2 þ � � � þwkak ¼ h�1Yk

i¼1

ðhðqaiÞÞwi

!; g�1

Yk

i¼1

ðgðraiÞÞwi

! !; ð5Þ

then

EIMWA ða1;a2; . . . ;ak;akþ1Þ ¼Xk

i¼1

wiai þwkþ1akþ1

¼ h�1Yk

i¼1

ðhðqaiÞÞwi

!; g�1

Yk

i¼1

ðgðraiÞÞwi

! !� ðh�1ððhðqakþ1

ÞÞwkþ1 Þ; g�1ððgðrakþ1ÞÞwkþ1 ÞÞ

¼ h�1 h h�1Yk

i¼1

ðhðqaiÞÞwi

! !� hðh�1ððhðqakþ1

ÞÞwkþ1 ÞÞ !

;

g�1 g g�1Yk

i¼1

ðgðraiÞÞwi

! !� gðg�1ððgðrakþ1

ÞÞwkþ1 ÞÞ !!

¼ h�1Yk

i¼1

ðhðqaiÞÞwi � ðhðqakþ1

ÞÞwkþ1

!; g�1

Yk

i¼1

ðgðraiÞÞwi � ðgðrakþ1

ÞÞwkþ1

! !

¼ h�1Ykþ1

i¼1

ðhðqaiÞÞwi

!; g�1

Ykþ1

i¼1

ðgðraiÞÞwi

! !; ð6Þ

i.e., Eq. (3) holds for n = k + 1. Thus Eq. (3) holds for all n.

M. Xia, Z. Xu / Applied Mathematical Modelling 37 (2013) 5120–5133 5125

In addition, we have known that h(t) = g(1/t), and g: (0, +1) ? (0, +1) is a strictly decreasing function, then h(t) is astrictly increasing function which indicates that

0 6 h�1Yn

i¼1

ðhðqaiÞÞwi

!; g�1

Yn

i¼1

ðgðraiÞÞwi

!61 ð7Þ

and

h�1Yn

i¼1

ðhðqaiÞÞwi

!� g�1

Yn

i¼1

ðgðraiÞÞwi

!6 h�1

Yn

i¼1

ðhðqaiÞÞwi

!� g�1

Yn

i¼1

ðgð1=qaiÞÞwi

!

¼ h�1Xn

i¼1

wihðqaiÞ

!� 1

h�1 Pni¼1wihðqai

Þ� � ¼ 1 ð8Þ

which completes the proof of Theorem 3. h

Then we can investigate some desirable properties of the EIMWA operator as follows:

Property 1. If all ai (i = 1,2, . . . ,n) are equal, i.e., ai = a = (qa,ra), for all i, then

EIMWA ða1;a2; . . . ;anÞ ¼ a: ð9Þ

Proof. Let ai = a = (qa,ra), we have

EIMWA ða1;a2; . . . ;anÞ ¼ EIMWA ða;a; . . . ;aÞ ¼Xn

i¼1

wia ¼ h�1Yn

i¼1

ðhðqaÞÞwi

!; g�1

Yn

i¼1

ðgðraÞÞwi

! !

¼ ðh�1ðhðqaÞÞ; g�1ðgðraÞÞÞ ¼ a: � ð10Þ

Property 2. Let ai ¼ ðqai;raiÞ and bi ¼ ðqbi

;rbiÞ ði ¼ 1;2; . . . ;nÞ be two collections of IMNs, if qai

6 qbiand rai

P rbi, for all i,

then

EIMWA ða1;a2; . . . ;anÞ 6 EIMWA ðb1;b2; . . . ;bnÞ: ð11Þ

Proof. We have known that h(t) = g(1/t), and g: (0, +1) ? (0, +1) is a strictly decreasing function, then h(t) is a strictlyincreasing function. Since qai

6 qbiand rai

P rbi, then we have

h�1Xn

i¼1

wihðqaiÞ

!6 h�1

Xn

i¼1

wihðqbiÞ

!; g�1

Xn

i¼1

wigðraiÞ

!P g�1

Xn

i¼1

wigðrbiÞ

!; ð12Þ

then

sðEIMWA ða1;a2; . . . ;anÞÞ 6 sðEIMWA ðb1;b2; . . . ; bnÞÞ; ð13Þ

which completes the proof. h

Based on Property 2, the following property can be obtained:

Property 3. Let ai ¼ ðqai;rai Þði ¼ 1;2; . . . ;nÞ be a collection of IMNs, and

a� ¼ ðminifqaig;maxifrai

gÞ; aþ ¼ ðmaxifqaig;minifrai

gÞ; ð14Þ

then

a� 6 EIMWA ða1;a2; . . . ;anÞ 6 aþ: ð15Þ

Property 4. Let ai ¼ ðqai;raiÞ ði ¼ 1;2; . . . ;nÞ be a collection of IMNs, w = (w1,w2, . . . ,wn)T be their weight vector such thatPn

i¼1wi ¼ 1, if b = (qb,rb) is an IMN, then

EIMWA ða1 � b;a2 � b; . . . ;an � bÞ ¼ EIMWA ða1;a2; . . . ;anÞ � b: ð16Þ

Proof. Since

ai þ b ¼ ðh�1ðhðqaiÞ � hðqbÞÞ; g�1ðgðrai

Þ � gðrbÞÞÞ; ð17Þ

5126 M. Xia, Z. Xu / Applied Mathematical Modelling 37 (2013) 5120–5133

we have

EIMWA ða1 þ b;a2 þ b; . . . ;an þ bÞ ¼ h�1Yn

i¼1

ðhðh�1ðhðqaiÞ � hðqbÞÞÞÞ

wi

!; g�1

Yn

i¼1

ðgðg�1ðgðraiÞ � gðrbÞÞÞÞwi

! !

¼ h�1Yn

i¼1

ðhðqaiÞ � hðqbÞÞ

wi

!; g�1

Yn

i¼1

ðgðraiÞ � gðrbÞÞwi

! !ð18Þ

and

EIMWA ða1;a1; . . . ;anÞ þ b ¼ h�1Yn

i¼1

ðhðqaiÞÞwi

!; g�1

Yn

i¼1

ðgðraiÞÞwi

! !þ ðqb;rbÞ

¼ h�1 h h�1Yn

i¼1

ðhðqaiÞÞwi

! !� hðqbÞ

!; g�1 g g�1

Yn

i¼1

ðgðraiÞÞwi

! !� gðrbÞ

! !

¼ h�1Yn

i¼1

ðhðqaiÞÞwi � hðqbÞ

!; g�1

Yn

i¼1

ðgðraiÞÞwi � gðrbÞ

! !

¼ h�1Yn

i¼1

ðhðqaiÞ � hðqbÞÞ

wi

!; g�1

Yn

i¼1

ðgðraiÞ � gðrbÞÞwi

! !; ð19Þ

which completes the proof. h

Property 5. Let ai ¼ ðqai;raiÞ ði ¼ 1;2; . . . ;nÞ be a collection of IMNs, and w = (w1,w2, . . . ,wn)T be their weight vector such thatPn

i¼1wi ¼ 1, if r > 0, then

EIMWA ðra1; ra2; . . . ; ranÞ ¼ r EIMWA ða1;a2; . . . ;anÞ: ð20Þ

Proof. According to Definition 4, we have

ra ¼ ðh�1ððhðqaiÞÞÞr ; g�1ððgðrai

ÞÞrÞÞ; ð21Þ

then

EIMWA ðra1; ra2; . . . ; ranÞ ¼ h�1Ykþ1

i¼1

ðhðh�1ððhðqaiÞÞrÞÞÞwi

!; g�1

Ykþ1

i¼1

ðgðg�1ððgðraiÞÞrÞÞÞwi

! !

¼ h�1Ykþ1

i¼1

ðhðqaiÞÞrwi

!; g�1

Ykþ1

i¼1

ðgðraiÞÞrwi

! !ð22Þ

and

r EIMWA ða1;a2; . . . ;anÞ ¼ h�1 h h�1Yn

i¼1

ðhðqaiÞÞwi

! ! !r !; g�1 g g�1

Yn

i¼1

ðgðraiÞÞwi

! ! !r ! !

¼ h�1Yn

i¼1

ðhðqaiÞÞwi

!r !; g�1

Yn

i¼1

ðgðraiÞÞwi

!r ! !: � ð23Þ

According to Properties 4 and 5, we can get Property 6 easily:

Property 6. Let ai ¼ ðqai;rai Þ ði ¼ 1;2; . . . ;nÞ be a collections of IMNs, and w = (w1,w2, . . . ,wn)T be the weight vector of them such

thatPn

i¼1wi ¼ 1, if r > 0, b = (qb,rb) is an IMN, then

EIMWA ðra1 � b; ra2 � b; . . . ; ran � bÞ ¼ r EFIMWA ða1;a2; . . . ;anÞ � b: ð24Þ

Property 7. Let ai ¼ ðqai;raiÞ and bi ¼ ðqbi

;rbiÞ ði ¼ 1;2; . . . ;nÞ be two collections of IMNs, and w = (w1,w2, . . . ,wn)T be the

weight vector of them such thatPn

i¼1wi ¼ 1, then

EIMWA ða1 þ b1;a2 þ b2; . . . ;an þ bnÞ ¼ EIMWA ða1;a2; . . . ;anÞ þ EIMWA ðb1;b2; . . . ;bnÞ: ð25Þ

Proof. According to Definition 4, we have

ai þ bi ¼ ðh�1ðhðqai

Þ � hðqbiÞÞ; g�1ðgðrai

Þ � gðrbiÞÞÞ; ð26Þ

M. Xia, Z. Xu / Applied Mathematical Modelling 37 (2013) 5120–5133 5127

then

EIMWA ða1 þ b1;a2 þ b2; . . . ;an þ bnÞ ¼ h�1Yn

i¼1

ðhðh�1ðhðqaiÞ � hðqbi

ÞÞÞÞwi

!; g�1

Yn

i¼1

ðgðg�1ðgðraiÞ þ gðrbi

ÞÞÞÞwi

! !

¼ h�1Yn

i¼1

ðhðqaiÞ � hðqbi

ÞÞwi

!; g�1

Yn

i¼1

ðgðraiÞ � gðrbi

ÞÞwi

! !ð27Þ

and

EIMWA ða1;a2; . . . ;anÞþEIMWAðb1;b2; . . . ;bnÞ

¼ h�1Yn

i¼1

ðhðqaiÞÞwi

!;g�1

Yn

i¼1

ðgðraiÞÞwi

! !þ h�1

Yn

i¼1

ðhðqbiÞÞwi

!;g�1

Yn

i¼1

ðgðrbiÞÞwi

! !

¼ h�1 h h�1Yn

i¼1

ðhðqaiÞÞwi

! !þh h�1

Yn

i¼1

ðhðqbiÞÞwi

! ! !;g�1 g g�1

Yn

i¼1

ðgðraiÞÞwi

! !þg g�1

Yn

i¼1

ðgðrbiÞÞwi

! ! ! !

¼ h�1Yn

i¼1

ðhðqaiÞÞwi �

Yn

i¼1

ðhðqbiÞÞwi

!;g�1

Yn

i¼1

ðgðraiÞÞwi �

Yn

i¼1

ðgðrbiÞÞwi

! !; ð28Þ

which completes the proof. h

If the multiplicative generator g is assigned different forms, then some specific intuitionistic multiplicative aggregationoperators can be obtained as follows:

Case 1. If gðtÞ ¼ 1þtt , then the EIMWA operator reduces to the following:

EIMWA ða1;a2; . . . ;anÞ ¼Yn

i¼1

ð1þ qaiÞwi � 1;

Qni¼1r

wiaiQn

i¼1ð1þ raiÞwi �

Qni¼1r

wiai

!: ð29Þ

Case 2. If gðtÞ ¼ 2þtt , then the EIMWA operator reduces to the following:

EIMWA ða1;a2; . . . ;anÞ ¼Qn

i¼1ð1þ 2qaiÞwi � 1

2;

2Qn

i¼1rwiaiQn

i¼1ð2þ raiÞwi �

Qni¼1r

wiai

!; ð30Þ

which was given by Xia et al. [14].Case 3. If gðtÞ ¼ cþt

t ; c > 0, then the EIMWA operator reduces to the following:

EIMWA ða1;a2; . . . ;anÞ ¼Qn

i¼1ð1þ cqaiÞwi � 1

c;

cQn

i¼1rwiaiQn

i¼1ðcþ raiÞwi �

Qni¼1r

wiai

!: ð31Þ

Especially, if c = 1, then the Eq. (31) reduces to Eq. (29); if c = 2, then Eq. (31) reduces to Eq. (30)

5. Some aggregation operators reflecting the correlations of the aggregated arguments

In this section, we mainly propose some aggregation operators to reflect the correlations or connections of the aggregatedarguments based on Choquet Integral [15,21] and power average [16], before doing this, some basic definitions are intro-duced firstly:

Definition 6 [22]. A normalized measure m on the set E is a function m:#(E) ? [0,1] satisfying the following axioms:

(1) m(/) = 0, m(E) = 1.(2) G # H implies m(G) 6m(H), for all B,C # E.(3) m(G [ H) = m(G) + m(H) + sm(G)m(H), for all G,H # E and G \ H = /, where s 2 (�1,1).

Especially, if s = 0, then (3) in Definition 6 reduces to the axiom of additive measure m(G [ H) = m(G) + m(H), which indi-cates that there is no interaction between G and H; if s > 0, then m(G [ H) > m(G) + m(H), which implies that the set {G,H} hasmultiplicative effect; if s < 0, then m(G [ H) < m(G) + m(H), which implies that the set {G,H} has substitutive effect, by param-eter s, the interaction between sets or elements of set can be represented.

Let E = {e1,e2, . . . ,ep} be a finite set, then [pk¼1ek ¼ E. To determine normalized measure on X avoiding the computational

complexity, Sugeno [22] gave the following equation:

5128 M. Xia, Z. Xu / Applied Mathematical Modelling 37 (2013) 5120–5133

mðEÞ ¼ m [pk¼1ek

� �¼

1sQp

k¼1ð1þ smðekÞÞ � 1� �

; s – 0;Xp

k¼1

mðekÞ; s ¼ 0

8><>: ð32Þ

and the value of s can be uniquely determined from m(E) = 1, which can be written as

sþ 1 ¼Yp

k¼1

ð1þ smðekÞÞ: ð33Þ

Especially, for every subset Ei # E, we have

mðEiÞ ¼1s

Yp

k¼1

ð1þ smðekÞÞ � 1

!; s – 0;

Xek2Ei

mðekÞ; s ¼ 0:

8>>><>>>:

ð34Þ

Definition 7. For two IMNs a1 and a2, we define the deviation between a1 and a2 as follows:

dða1;a2Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimax qa1

qa2;qa2

. .qa1

n o�max ra1 ra2 ;ra2

ra1

�r: ð35Þ

Definition 8. Let ai ¼ ðqai;raiÞ ði ¼ 1;2; . . . ;nÞ be a collection of IMNs, and then we define an extended intuitionistic multi-

plicative power averaging (EIMPA) operator as follows:

EIMPA ða1;a2; . . . ;anÞ ¼ h�1Yn

i¼1

ðhðqaiÞÞ

TðaiÞ=Xn

i¼1

TðaiÞ

0BBB@

1CCCA; g�1

Yn

i¼1

ðgðraiÞÞ

TðaiÞ=Xn

i¼1

TðaiÞ

0BBB@

1CCCA

0BBB@

1CCCA; ð36Þ

where TðaiÞ ¼Qn

j¼1Supðai;ajÞ, and Sup(ai,aj) is the support for ai from aj, with the conditions: (1) Sup(ai,aj) 2 [0,1]; (2)Sup(ai,aj) = Sup(aj,ai); (3) Sup(ai,aj) P Sup(as,at), if d(ai,aj) < d(as,at).

When the multiplicative generator is assigned different forms, some special cases can be obtained as follows:

Case 1. If gðtÞ ¼ 1þtt , then the EIMPA operator reduces to the following:

EIMPA ða1;a2; . . . ;anÞ ¼Yn

i¼1

ð1þ qaiÞTðaiÞ=

Pn

i¼1TðaiÞ � 1;

Qni¼1r

TðaiÞ=Pn

i¼1TðaiÞ

aiQni¼1ð1þ rai

ÞTðaiÞ=Pn

i¼1TðaiÞ �

Qni¼1r

TðaiÞ=Pn

i¼1TðaiÞ

ai

0@

1A: ð37Þ

Case 2. If gðtÞ ¼ 2þtt , then the EIMPA operator reduces to the following:

EIMPWA ða1;a2; . . . ;anÞ ¼Qn

i¼1ð1þ 2qaiÞTðaiÞ=

Pn

i¼1TðaiÞ � 1

2;

2Qn

i¼1rTðaiÞ=

Pn

i¼1TðaiÞ

aiQni¼1ð2þ rai

ÞTðaiÞ=Pn

i¼1TðaiÞ �

Qni¼1r

TðaiÞ=Pn

i¼1TðaiÞ

ai

0@

1A: ð38Þ

Case 3. If gðtÞ ¼ cþtt ; c > 0, then the EIMPA operator reduces to the following:

EIMPA ða1;a2; . . . ;anÞ ¼Qn

i¼1ð1þ cqaiÞTðaiÞ=

Pn

i¼1TðaiÞ � 1

c;

cQn

i¼1rTðaiÞ=

Pn

i¼1TðaiÞ

aiQni¼1ðcþ rai

ÞTðaiÞ=Pn

i¼1TðaiÞ �

Qni¼1r

TðaiÞ=Pn

i¼1TðaiÞ

ai

0@

1A: ð39Þ

Especially, if c = 1, then Eq. (39) reduces to Eq. (37); if c = 2, then Eq. (39) reduces to Eq. (38).

Based on Choquet Integral, we can let wi = m(Ei) �m(Ei�1), where Ei = {e1,e2, . . . ,ei}, i P 1 and E0 = ;, and in such case, wegive the following definition:

Definition 9. Let ai ¼ ðqai;rai Þ ði ¼ 1;2; . . . ;nÞ be a collection of IMNs defined on the set E = {e1,e2, . . . ,en}, and then we define

an extended intuitionistic multiplicative Choquet averaging (EIMCA) operator as follows:

EIMCA ða1;a2; . . . ;anÞ ¼ h�1Yn

i¼1

ðhðqaiÞÞmðXiÞ�mðXi�1Þ

!; g�1

Yn

i¼1

ðgðraiÞÞmðXiÞ�mðXi�1Þ

! !: ð40Þ

Some special cases can be discussed as follows:

M. Xia, Z. Xu / Applied Mathematical Modelling 37 (2013) 5120–5133 5129

Case 1. If gðtÞ ¼ 1þtt , then the EIMCA operator reduces to the following:

EIMCA ða1;a2; . . . ;anÞ ¼Yn

i¼1

ð1þ qaiÞmðXiÞ�mðXi�1Þ � 1;

Qni¼1r

mðXiÞ�mðXi�1ÞaiQn

i¼1ð1þ raiÞmðXiÞ�mðXi�1Þ �

Qni¼1r

mðXiÞ�mðXi�1Þai

!ð41Þ

Case 2. If gðtÞ ¼ 2þtt , then the EIMCA operator reduces to the following:

EIMCA ða1;a2; . . . ;anÞ ¼Qn

i¼1ð1þ 2qaiÞmðXiÞ�mðXi�1Þ � 12

;2Qn

i¼1rmðXiÞ�mðXi�1ÞaiQn

i¼1ð2þ raiÞmðXiÞ�mðXi�1Þ �

Qni¼1r

mðXiÞ�mðXi�1Þai

!: ð42Þ

Case 3. If gðtÞ ¼ cþtt ; c > 0, then the EIMCA operator reduces to the following:

HIMCAða1;a2; . . . ;anÞ ¼Qn

i¼1ð1þ cqaiÞmðXiÞ�mðXi�1Þ � 1c

;cQn

i¼1rmðXiÞ�mðXi�1ÞaiQn

i¼1ðcþ raiÞmðXiÞ�mðXi�1Þ �

Qni¼1r

mðXiÞ�mðXi�1Þai

!: ð43Þ

Especially, if c = 1, then Eq. (43) reduces to Eq. (41); if c = 2, then Eq. (43) reduces to Eq. (42).

Motivated by the ordered weighted averaging (OWA) operator [23], we can define the following definitions:

Definition 10. Let ai ¼ ðqai;rai Þ ði ¼ 1;2; . . . ;nÞ be a collection of IMNs, and then we define an extended intuitionistic

multiplicative power ordered averaging (EIMPOA) operator as follows:

EIMPOA ða1;a2; . . . ;anÞ ¼ h�1Yn

i¼1

h qaðiÞ

� �� �TðaðiÞÞPn

i¼1TðaðiÞÞ

!; g�1

Yn

i¼1

g raðiÞ

� �� �TðaðiÞÞPn

i¼1TðaðiÞÞ

! !; ð44Þ

where (i): {1,2, . . . ,n} ? {1,2, . . . ,n} is a permutation such that a(i) > a(i�1), i = 2,3, . . . ,n.By comparing Definitions 8 and 10, we can find that

EIMPOA ða1;a2; . . . ;anÞ ¼ EIMPA ða1;a2; . . . ;anÞ: ð45Þ

Definition 11. Let ai ¼ ðqai;raiÞ ði ¼ 1;2; . . . ;nÞ be a collection of IMNs defined on the set E = {e1,e2, . . . ,en}, and then we

define an extended intuitionistic multiplicative Choquet ordered averaging (EIMCOA) operator as follows:

EIMCOA ða1;a2; . . . ;anÞ ¼ h�1Yn

i¼1

h qaðiÞ

� �� �mðXðiÞÞ�mðXði�1ÞÞ !

; g�1Yn

i¼1

g raðiÞ

� �� �mðXðiÞÞ�mðXði�1ÞÞ ! !

; ð46Þ

where (i): {1,2, . . . ,n} ? {1,2, . . . ,n} is a permutation such that a(i) > a(i�1), i = 2,3, . . . ,n.

6. An approach to group decision making based on intuitionistic multiplicative preference relations

Suppose there are n alternatives x1,x2, . . . ,xn to be compared, there are p decision makers e1,e2, . . . ,ep to be authorized togive their preferences about these n alternatives, the decision maker ek uses the Saaty’s 1–9 scale to express their prefer-ences, and he/she not only provides the intensity degree qaij

that the alternative xi is priority to the alternative xj, but also

provide the intensity degree raijthat the alternative xi is not priority to the alternative xj, then the preference information

about alternatives xi and xj can be described by an IMN aðkÞij ¼ qaðkÞij;raðkÞ

ij

� �with the condition that qaðkÞ

ij¼ raðkÞ

ji, raðkÞ

ij¼ qaðkÞ

ji,

0 6 qaðkÞij; raðkÞ

ij6 1 and 1=9 6 qaðkÞ

ij; raðkÞ

ij6 9. When all the preferences about n alternatives are provided by the decision mak-

ers, then the intuitionistic multiplicative preference relations AðkÞ ¼ aðkÞij

� �n�n¼ qaðkÞ

ij;raðkÞ

ij

� �� �n�n

, k = 1,2, . . . ,p are

constructed.To get the ranking of the alternatives, the following steps are given as follows:

Step 1. Utilized the EIMPA operator to obtain the average value aðkÞi of the alternative xi for expert ek:

aðkÞi ¼ EIMPA aðkÞ1j ;aðkÞ2j ; . . . ;aðkÞnj

� �¼ h�1

Yn

j¼1

h qaðkÞij

� �� �TðaðkÞijÞ=Xn

i¼1

TðaðkÞijÞ

0BBB@

1CCCA; g�1

Yn

j¼1

g raðkÞij

� �� �TðaðkÞijÞ=Xn

i¼1

TðaðkÞijÞ

0BBB@

1CCCA

0BBB@

1CCCA ð47Þ

5130 M. Xia, Z. Xu / Applied Mathematical Modelling 37 (2013) 5120–5133

Step 2. Utilize the EIMCOA operator to obtain the average value ai of the alternative xi:

ai ¼ EIMCOA aðkÞ1 ;aðkÞ2 ; . . . ;aðkÞn

� �¼ h�1

Yp

k¼1

h qaðkÞi

� �� �mðEðkÞÞ�mðEðk�1ÞÞ !

; g�1Yp

k¼1

g raðkÞi

� �� �mðEðkÞÞ�mðEðk�1ÞÞ ! !

ð48Þ

Step 3. Calculate the score function s(ai) and the accuracy degree a(ai) of ai, and obtain the ranking of the alternativesaccording to s(ai) and a(ai).

Next, we use an example to illustrate the developed method:

Example 1. Four university students share a house, where they intend to have broadband Internet connection installedadapted from [24,25]. There are four options available to choose from, which are provided by three Internet-serviceproviders.

(1) x1: 1 Mbps broadband.(2) x2: 2 Mbps broadband.(3) x3: 3 Mbps broadband.(4) x4: 8 Mbps broadband.

Since the Internet service and its monthly bill will be shared among the four students ek (k = 1,2,3,4), they decide to per-form a group decision analysis. Suppose that the students reveal their preference relations for the options independently andanonymously, and construct the interval-valued multiplicative preference relations, for example, respectively:

~A1 ¼

½1;1� ½1=5;1=3� ½1=3;1� ½1=2;1�

½3;5� ½1;1� ½1=4;1=2� ½1=3;1=2�

½1;3� ½2;4� ½1;1� ½1=3;1�

½1;2� ½2;3� ½1;3� ½1;1�

0BBBBBBBBBB@

1CCCCCCCCCCA; ~A2 ¼

½1;1� ½1=3;1=2� ½1=4;1=2� ½1=3;1�

½2;3� ½1;1� ½1=5;1=3� ½1=4;1=2�

½2;4� ½3;5� ½1;1� ½1=2;1�

½1;3� ½2;4� ½1;2� ½1;1�

0BBBBBBBBBB@

1CCCCCCCCCCA;

~A3 ¼

½1;1� ½2;3� ½1=3;1=2� ½1;3�

½1=3;1=2� ½1;1� ½1=4;1=3� ½1=5;1=3�

½2;3� ½3;4� ½1;1� ½1;2�

½1=3;2� ½3;5� ½1=2;1� ½1;1�

0BBBBBBB@

1CCCCCCCA; ~A4 ¼

½1;1� ½1=3;1� ½1=2;1� ½1=2;2�

½1;3� ½1;1� ½1=5;1=4� ½1=4;1=3�

½1;2� ½4;5� ½1;1� ½1=2;1�

½1=2;2� ½3;4� ½1;2� ½1;1�

0BBBBBBB@

1CCCCCCCA:

Based on the above analysis, the interval-valued multiplicative preference relation can be transformed into the followingintuitionistic multiplicative preference relations:

A1 ¼

ð1;1Þ ð1=5;3Þ ð1=3;1Þ ð1=2;1Þð3;1=5Þ ð1;1Þ ð1=4;2Þ ð1=3;2Þð1;1=3Þ ð2;1=4Þ ð1;1Þ ð1=3;1Þð1;1=2Þ ð2;1=3Þ ð1;1=3Þ ð1;1Þ

0BBB@

1CCCA; A2 ¼

ð1;1Þ ð1=3;2Þ ð1=4;2Þ ð1=3;1Þð2;1=3Þ ð1;1Þ ð1=5;3Þ ð1=4;2Þð2;1=4Þ ð3;1=5Þ ð1;1Þ ð1=2;1Þð1;1=3Þ ð2;1=4Þ ð1;1=2Þ ð1;1Þ

0BBB@

1CCCA;

A3 ¼

ð1;1Þ ð2;1=3Þ ð1=3;2Þ ð1;1=3Þð1=3;2Þ ð1;1Þ ð1=4;3Þ ð1=5;3Þð2;1=3Þ ð3;1=4Þ ð1;1Þ ð1;1=2Þð1=3;1=2Þ ð3;1=5Þ ð1=2;1Þ ð1;1Þ

0BBB@

1CCCA; A4 ¼

ð1;1Þ ð1=3;1Þ ð1=2;1Þ ð1=2;1=2Þð1;1=3Þ ð1;1Þ ð1=5;4Þ ð1=4;3Þð1;1=2Þ ð4;1=5Þ ð1;1Þ ð1=2;1Þð1=2;1=2Þ ð3;1=4Þ ð1;1=2Þ ð1;1Þ

0BBB@

1CCCA:

Assume that the weights of the decision makers have correlations with each other and

mð/Þ ¼ 0; mðfe1gÞ ¼ 0:3; mðfe2gÞ ¼ 0:2; mðfe3gÞ ¼ 0:4; mðfe4gÞ ¼ 0:5:

By Eqs. (33) and (34), we have

mðfe1; e2gÞ ¼ 0:4610; mðfe2; e3gÞ ¼ 0:5480; mðfe1; e4gÞ ¼ 0:7024; mðfe1; e3gÞ ¼ 0:6219;mðfe2; e4gÞ ¼ 0:6350; mðfe3; e4gÞ ¼ 0:7699; mðfe1; e2; e3gÞ ¼ 0:7410;mðfe1; e2; e4gÞ ¼ 0:8110;mðfe2; e3; e4gÞ ¼ 0:8697; mðfe1; e3; e4gÞ ¼ 0:9197; mðfe1; e2; e3; e4gÞ ¼ 1:

Let gðtÞ ¼ 1þtt , to obtain the ranking of the alternative, we give the following steps:

M. Xia, Z. Xu / Applied Mathematical Modelling 37 (2013) 5120–5133 5131

Step 1. Let c = 1, and utilize the EIMPA operator to aggregate the average values aðkÞi of the alternative xi for expert ek:

að1Þ1 ¼ ð0:4897;1:0629Þ; að1Þ2 ¼ ð0:5534;1:3953Þ; að1Þ3 ¼ ð1:0125;0:5174Þ; að1Þ4 ¼ ð1:1398;0:4469Þ;að2Þ1 ¼ ð0:3714;1:4312Þ; að2Þ2 ¼ ð0:5577;1:3661Þ; að2Þ3 ¼ ð1:4627;0:4383Þ; að2Þ4 ¼ ð1:1086;0:4342Þ;að3Þ1 ¼ ð1:1040;0:5260Þ; að3Þ2 ¼ ð0:2973;2:3849Þ; að3Þ3 ¼ ð1:5698;0:4237Þ; að3Þ4 ¼ ð0:6936;0:7570Þ;að4Þ1 ¼ ð0:5372;0:8636Þ; að4Þ2 ¼ ð0:5348;1:4049Þ; að4Þ3 ¼ ð0:9602;0:6952Þ; að4Þ4 ¼ ð0:9740;0:5430Þ:

Step 2. Let c = 1, and utilize the EIMCOA operator to obtain the performance value ai for the alternative xi:

a1 ¼ ð1:0603;0:7520Þ; a2 ¼ ð0:4959;1:5213Þ; a3 ¼ ð1:2710;0:5945Þ; a4 ¼ ð0:9845;0:5219Þ:

Step 3. Calculate the score s(ai) of ai, we have

sða1Þ ¼ 0:7520; sða2Þ ¼ 0:3260; sða3Þ ¼ 2:1379; sða4Þ ¼ 1:8864

and the ranking of the alternatives is x3 � x4 � x1 � x2.

The multiplicative generator can be assigned other forms of functions. Here we will not enumerate them. By comparingthe proposed method and the one given by Xu and Yager [25], we can find that both of these two methods can get the sameresult, but the intutionistic multiplicative preference relation can express the decision makers’ preference more objectivelythan the interval-valued multiplicative preference relation for containing two information parts: the membership informa-tion and the non-membership information. Moreover, the proposed method utilizes the EIMPA operator to aggregate thepreference information provide by each decision maker for each alternative, which can reflect the objective correlations be-tween the aggregated arguments. And the proposed method utilizes the EIMCOA operator to aggregate the preference infor-mation provided by the decision makers, which can reflect the subjective correlations of the decision makers, thus, theproposed method can get more reasonable results than Xu and Yager’s method [25].

To give a further study about these two methods, we use the following interval-valued multiplicative preference relationsinstead of the ones in Example 1, then

~A1 ¼ ~A2 ¼ ~A3 ¼ ~A4 ¼

½1;1� ½1=3;1=2� ½1;2� ½3;4�½2;3� ½1;1� ½1=2;2=3� ½1;2�½1=2;1� ½3=2;2� ½1;1� ½4=3;2�½1=4;1=3� ½1=2;1� ½1=2;3=4� ½1;1�

0BBB@

1CCCA:

By using Xu and Yager’s method [25] (more detail steps can be found in the original paper), we can obtain the collectiveinterval-valued multiplicative preference relation:

~A ¼

½1;1� ½1=3;1=2� ½1;2� ½3;4�½2;3� ½1;1� ½1=2;2=3� ½1;2�½1=2;1� ½3=2;2� ½1;1� ½4=3;2�½1=4;1=3� ½1=2;1� ½1=2;3=4� ½1;1�

0BBB@

1CCCA

and the uncertain priority vector of ~A:

v1 ¼ ½0:2020;0:4041�; v2 ¼ ½0:2020;0:4041�; v3 ¼ ½0:2020;0:4041�; v4 ¼ ½0:1010;0:2020�:

Based on which, we can construct the possibility degree matrix:

P ¼

0:5000 0:5000 0:5000 1:00000:5000 0:5000 0:5000 1:00000:5000 0:5000 0:5000 1:00000:0000 0:0000 0:0000 0:5000

0BBB@

1CCCA

and we have p1 = 2.5, p2 = 2.5, p3 = 2.5, p4 = 0.5, which derives that x3 x2 x1 � x4.If we translate the interval-valued multiplicative preference relation into the intuitionistic multiplicative preference rela-

tions, we have

A1 ¼ A2 ¼ A3 ¼ A4 ¼

ð1;1Þ ð1=3;2Þ ð1;1=2Þ ð3;1=4Þð2;1=3Þ ð1;1Þ ð1=2;3=2Þ ð1;1=2Þð1=2;1Þ ð3=2;1=2Þ ð1;1Þ ð4=3;1=2Þð1=4;3Þ ð1=2;1Þ ð1=2;4=3Þ ð1;1Þ

0BBB@

1CCCA:

By using our method, let c = 1, and utilize the EIMPA operator to aggregate the average values aðkÞi of the alternative xi forexpert ek:

5132 M. Xia, Z. Xu / Applied Mathematical Modelling 37 (2013) 5120–5133

að1Þ1 ¼ að2Þ1 ¼ að3Þ1 ¼ að4Þ1 ¼ ð0:9601;0:7599Þ; að1Þ2 ¼ að2Þ2 ¼ að3Þ2 ¼ að4Þ2 ¼ ð0:9712;0:7550Þ;að1Þ3 ¼ að2Þ3 ¼ að3Þ3 ¼ að4Þ3 ¼ ð1:0571; 0:7037Þ; að1Þ4 ¼ að2Þ4 ¼ að3Þ4 ¼ að4Þ4 ¼ ð0:5254;1:2397Þ:

Let c = 1, and utilize the EIMCOA operator to obtain the performance value ai for the alternative xi:

a1 ¼ ð0:9601;0:7599Þ; a2 ¼ ð0:9712; 0:7550Þ; a3 ¼ ð1:0571;0:7037Þ; a4 ¼ ð0:5254;1:2397Þ:

Calculate the score s(ai) of ai, we have

sða1Þ ¼ 1:2635; sða2Þ ¼ 1:2863; sða3Þ ¼ 1:5023; sða4Þ ¼ 0:4238;

which derives that x3 � x2 � x1 � x4.Moreover, from the original interval-valued multiplicative preference relations ~A1, ~A2, ~A3 and ~A4, we can find that

~að1Þ32 ¼ ~að2Þ32 ¼ ~að3Þ32 ¼ ~að4Þ32 ¼ ½3=2;2�; ~að1Þ21 ¼ ~að2Þ21 ¼ ~að3Þ21 ¼ ~að4Þ21 ¼ ½2;3�;~að1Þ14 ¼ ~að2Þ14 ¼ ~að3Þ14 ¼ ~að4Þ14 ¼ ½3;4�;

which is consistent with the ranking obtained by our method.We can find that the in some situations Xu and Yager’s method [25] cannot give the ranking of the alternatives, our meth-

od can, that is because although the forms of the interval-valued multiplicative preference relation and the intuitionisticmultiplicative preference relation are the same, the meanings and the operations are different, which is similar to the rela-tionship between the interval-valued fuzzy preference relation and the intuitionistic fuzzy preference relation. The decisionmakers can choose the one they like.

7. Discuss and limitations

In this paper, we have introduced the intutitionistic multiplicative preference relation based on the interval-valued mul-tiplicative preference relation. Similar to the intuitionistic fuzzy preference relation, there are two information parts in anintuitionistic multiplicative preference relation expressing the intensity degree that an alternative is priority to anotherand the intensity degree that an alternative is not priority to another. But different to the intuitionistic fuzzy preference rela-tion, the intuitionistic multiplicative preference relation uses the Saaty’s scale to express the preference information which isa non-symmetric distribution around 1 describing the degrees between good and bad more objectively, and can avoid someunreasonable results, which have been given a detail discussion by Xia et al. [14].

We have introduced some extended operations on intuitionistic multiplicative preference information based on pseudo-multiplication, from which an aggregation principle has been developed, and the properties and special cases have been dis-cussed in details. Especially, when the pseudo-multiplication functions are assigned some specific forms, then the proposedaggregation operators can reduce to the ones given by Xia et al. [14]. Other aggregation operators have also been developedto reflect the correlations of the aggregated intuitionistic multiplicative information. An approach has been developed to dealwith the group decision making based on intuitionistic multiplicative preference relations, and an example has been given toillustrate the developed method.

It should be noted that the intuitionistic multiplicative preference relation and the interval-valued multiplicative prefer-ence relation can be transformed between each other, similar to the relationship between the inuitionistic fuzzy preferencerelation and the interval-valued fuzzy preference relation. We have given an example to compare the intuitioinstic multipli-cative preference relation and the interval-valued multiplicative preference relation, although the forms of them are thesame, but the meanings and the operations are different. By comparing the proposed method and the one given by Xuand Yager [25], we can find that our method can distinguish the good alternative from the bad, while Xu and Yager’s method[25] cannot at some situations. Of course, the proposed method has its limitations, for example, the operations proposed inthis paper seem a little complex. Every coin has two sides, the important is how to use its good side.

Acknowledgments

The authors are very grateful to the anonymous reviewers for their insightful and constructive comments and suggestionsthat have led to an improved version of this paper. The work was supported in part by the National Natural Science Foun-dation of China (Nos. 71071161 and 61273209) and the China Postdoctoral Science Foundation (No. 2012M520311).

References

[1] S.A. Orlovsky, Decision-making with a fuzzy preference relation, Fuzzy Sets Syst. 1 (1978) 155–167.[2] B. De Baets, H. De Meyer, B. De Schuymer, S. Jenei, Cyclic evaluation of transitivity of reciprocal relations, Soc. Choice Welfare 26 (2006) 217–238.[3] T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, NY, 1980.[4] Z.S. Xu, J. Chen, Some models for deriving the priority weights from interval reciprocal relations, Eur. J. Oper. Res. 184 (2008) 266–280.[5] S. Genç, F.E. Boran, D. Akay, Z.S. Xu, Interval multiplicative transitivity for consistency missing values and priority weights of interval reciprocal

relations, Inform. Sci. 180 (2010) 4877–4891.[6] Y.M. Wang, T.M.S. Elhag, A goal programming method for obtaining interval weights from an interval comparison matrix, Eur. J. Oper. Res. 177 (2007)

458–471.

M. Xia, Z. Xu / Applied Mathematical Modelling 37 (2013) 5120–5133 5133

[7] F. Liu, Acceptable consistency analysis of interval reciprocal comparison matrices, Fuzzy Sets Syst. 160 (2009) 2686–2700.[8] E. Conde, M.P.R. Pérez, A linear optimization problem to derive relative weights using an interval judgement matrix, Eur. J. Oper. Res. 201 (2010) 537–

544.[9] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Inform. Sci. 8 (1975) 199–249.

[10] E. Szmidt, J. Kacprzyk, A consensus-reaching process under intuitionistic fuzzy preference relations, Int. J. Intell. Syst. 18 (2003) 837–852.[11] Z.S. Xu, Intuitionistic preference relations and their application in group decision making, Inform. Sci. 177 (2007) 2363–2379.[12] Z.W. Gong, L.S. Li, J. Forrest, Y. Zhao, The optimal priority models of the intuitionistic fuzzy preference relation and their application in selecting

industries with higher meteorological sensitivity, Expert Syst. Appl. 38 (2011) 4394–4402.[13] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986) 87–96.[14] M.M. Xia, Z.S. Xu, H.C. Liao, Preference relations based on intuitionistic multiplicative information, IEEE Trans. Fuzzy Syst., http://dx.doi.org/10.1109/

TFUZZ.2012.2202907.[15] G. Choquet, Theory of capacities, Ann. Inst. Fourier (Crenoble) 5 (1953) 131–295.[16] R.R. Yager, The power average operator, IEEE Trans. Syst. Man Cybernet. 31 (2001) 724–731.[17] Z.S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Trans. Fuzzy Syst. 15 (2007) 1179–1187.[18] F. Herrera, E. Herrera-Viedma, Linguistic preference modeling: foundation models and new trends, in: Proceedings of 4th International Workshop on

Preferences and Decisions, Trento, Italy, 2003, pp. 47–51.[19] F. Herrera, E. Herrera-Viedma, L. Martínez, A fuzzy linguistic methodology to deal with unbalanced linguistic term sets, IEEE Trans. Fuzzy Syst. 16

(2008) 354–370.[20] E. Pap, N. Ralevic, Pseudo–Laplace transform, Nonlinear Anal. 33 (1998) 533–550.[21] R.R. Yager, Choquet aggregation using order inducing variables, Int. J. Uncertain. Fuzz. Knowl.-Based Syst. 12 (2004) 69–88.[22] M. Sugeno, Theory of fuzzy integral and its application, Doctoral dissertation, Tokyo Institute of Technology, 1974.[23] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Syst. Man Cybernet. 18 (1988) 183–190.[24] Y.M. Wang, C. Parkan, Optimal aggregation of fuzzy preference relations with an application to broadband internet service selection, Eur. J. Oper. Res.

187 (2008) 1476–1486.[25] Z.S. Xu, R.R. Yager, Power-geometric operators and their use in group decision making, IEEE Trans. Fuzzy Syst. 18 (2010) 94–105.