Some geometric aggregation operators based on ...

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Some geometric aggregation operators based on intuitionistic fuzzy sets a

Zeshui Xu & Ronald R. Yager

b

a

School of Economics and Management, Tsinghua University, Department of Management Science and Engineering, Beijing, 100084, People's Republic of China b

Machine Intelligence Institute, Iona College, New Rochelle, NY, 10801, USA

Available online: 29 Nov 2006

To cite this article: Zeshui Xu & Ronald R. Yager (2006): Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General Systems, 35:4, 417-433 To link to this article: http://dx.doi.org/10.1080/03081070600574353

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International Journal of General Systems, Vol. 35, No. 4, August 2006, 417–433

Some geometric aggregation operators based on intuitionistic fuzzy sets ZESHUI XU†* and RONALD R. YAGER‡{ †Department of Management Science and Engineering, School of Economics and Management, Tsinghua University, Beijing 100084, People’s Republic of China ‡Machine Intelligence Institute, Iona College, New Rochelle, NY 10801, USA (Received 12 October 2005; in final form 26 December 2005) The weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator are two common aggregation operators in the field of information fusion. But these two aggregation operators are usually used in situations where the given arguments are expressed as crisp numbers or linguistic values. In this paper, we develop some new geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator, which extend the WG and OWG operators to accommodate the environment in which the given arguments are intuitionistic fuzzy sets which are characterized by a membership function and a non-membership function. Some numerical examples are given to illustrate the developed operators. Finally, we give an application of the IFHG operator to multiple attribute decision making based on intuitionistic fuzzy sets. Keywords: Information fusion; Intuitionistic fuzzy set; Intuitionistic fuzzy weighted geometric (IFWG) operator; Intuitionistic fuzzy ordered weighted geometric (IFOWG) operator; Intuitionistic fuzzy hybrid geometric (IFHG) operator

1. Introduction Atanassov (1986, 1989, 1994a, b, 1999, 2000) and Atanassov and Gargov (1989) introduced the concept of intuitionistic fuzzy set, which is a generalization of the concept of fuzzy set (Zadeh 1965). The intuitionistic fuzzy set has received more and more attention since its appearance. Gau and Buehrer (1993) introduced the concept of vague set. Chen and Tan (1994) and Hong and Choi (2000) presented some techniques for handling multicriteria fuzzy decision making problems based on vague sets. But Bustince and Burillo (1996) showed that vague sets are intuitionistic fuzzy sets. De et al. (2000) defined concentration, dilation and normalization of intuitionistic fuzzy sets, and proved some propositions in this context. Bustince et al. (2000) presented the concepts of intuitionistic fuzzy generators and studied the complementary of an intuitionistic set from the intuitionistic fuzzy generators. Szmidt and Kacprzyk (2000) defined the four basic distances between intuitionistic fuzzy sets: the Hamming distance, the normalized Hamming distance, the Euclidean distance, and the normalized Euclidean distance. Szmidt *Corresponding author. Email: [email protected] {Email: [email protected] International Journal of General Systems ISSN 0308-1079 print/ISSN 1563-5104 online q 2006 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/03081070600574353


418

Z. Xu and R. R. Yager

and Kacprzyk (2001) proposed a non-probabilistic-type entropy measure for intuitionistic fuzzy sets. Mondal and Samanta (2001) defined topology of interval-valued intuitionistic fuzzy sets, and studied some of its properties. Mondal and Samanta (2002) introduced a concept of intuitionistic gradation of openness on fuzzy subsets of a nonempty set, and defined an intuitionistic fuzzy topological space. They proved that the category of intuitionistic fuzzy topological spaces and gradation preserving mappings is a topological category, and studied compactness of intuitionistic fuzzy topological spaces. Deschrijver and Kerre (2003) investigated the composition of intuitionistic fuzzy relations. Li (2004) defined two dissimilarity measures between intuitionistic fuzzy sets, and generalized these measures to intuitionistic fuzzy structures. Grzegorzewski (2004) suggested some methods for measuring distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets, based on the Hausdorff metric. The proposed new distances are generalizations of the Hamming distance, the Euclidean distance and their normalized counterparts. However, it seems that in the literature there is no investigation on aggregation operators for aggregating a collection of intuitionistic fuzzy sets, except of some operations on intuitionistic fuzzy sets (Atanassov 1986, De et al. 2000). In this paper, we shall develop some geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator. To do so, this paper is structured as follows. In Section 2, we review the weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator. In Section 3, we develop the IFWG operator, the IFOWG operator, and the IFHG operator, and study their various properties. In Section 4, we give an application of the IFHG operator to multiple attribute decision making with intuitionistic fuzzy information. Concluding remarks are made in Section 5.

2. The WG and OWG operators Let aj( j ¼ 1, 2, . . . , n) be a collection of non-negative real numbers, v ¼ (v1, v2, . . . ,vn)T be P the weight vector of aj( j ¼ 1, 2, . . . , n), with vj . 0 and nj¼1 vj ¼ 1, then a WG operator is defined as (Saaty 1980, Acze´l and Saaty 1983, Willet and Sharda 1991, Benjamin et al.1992, Xu 2000, Xu and Da 2003): WGv ða1 ; a2 ; . . .; an Þ ¼

n Y

v

aj j :

ð1Þ

j¼1

Another geometric aggregation operator called the OWG operator based on the OWA operator (Yager 1988, Yager 1993, Yager and Kacprzyk 1997, Yager 2004) and the geometric mean, was defined as follows (Xu and Da 2003, Yager 2004, Herrera et al. 2001, Chiclana et al. 2001, Xu and Da 2002, Herrera et al. 2003, Xu 2004a). n An OWG operator of dimension n is a mapping OWG : R þ ! R þ which has associated P T with it a weighting vector w ¼ (w1, w2, . . . ,wn) , with wj . 0 and nj¼1 wj ¼ 1, such that OWGw ða1 ; a2 ; . . .; an Þ ¼

n Y j¼1

where bj is the jth largest of aj( j ¼ 1, 2, . . . , n).

w

bj j

ð2Þ

Geometric aggregation operators

419


Central to this operator is the reordering of the arguments, in particular an argument ai is not associated with a particular weight wi but rather a weight wi is associated with a particular ordered position i of the argument. One important issue in the OWG operator is to determine its associated weights. Many methods can be used to determine the OWG weights (please see Xu (2005) for more details). Especially, Xu (2005) developed a normal distribution based method, which is defined as follows: 2

wj ¼

e

2ð j2m2n Þ

Pn

i¼1

2sn

e

2ði2m2n Þ

2

;

j ¼ 1; 2; . . .; n

ð3Þ

2sn

where mn is the mean of the collection of 1, 2, . . . , n, and sn(sn . 0) is the standard deviation of the collection of 1, 2, . . . , n, i.e.

mn ¼

1 nð1 þ nÞ 1 þ n ¼ n 2 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1X sn ¼ ði 2 mn Þ2 : n i¼1 The prominent characteristic of the method is that it can relieve the influence of unfair arguments on the final results by assigning low weights to those “false” or “biased” ones. The WG and OWG operators, however, have usually been used in situations where the input arguments are the exact values. Xu (2004b, c) extended the WG and OWG operators to accommodate linguistic environment. In the following section, we shall extend the WG and OWG operators to accommodate the situations where the input arguments are intuitionistic fuzzy values.

3. The IFWG and IFOWG operators 3.1 Operational laws and relations Atanassov (1986) generalized the concept of fuzzy set (Zadeh 1965), and defined the concept of intuitionistic fuzzy set as follows. Let a set X be fixed. An intuitionistic fuzzy set A in X is an object having the form: A ¼ {, x; mA ðxÞ; vA ðxÞ . jx [ X}

ð4Þ

where the functions mA: X ! [0, 1] and vA: X ! [0, 1] define the degree of membership and the degree of non-membership of the element x [ X to A , X, respectively, and for every x [ X:

mA ðxÞ þ vA ðxÞ # 1: If we use a membership function tA and a non-membership function fA to denote the lower bounds on mA, then, the degree of membership of x in the intuitionistic fuzzy set A is bounded to a subinterval [tA(x), 1 2 fA(x)] of [0,1]. Gau and Buehrer (1993) called the interval [tA(x), 1 2 fA(x)] a vague value. However, Bustince and Burillo (1996) showed that vague sets are

420


intuitionistic fuzzy sets. For computational convenience, in this paper, we call the interval [tA(x), 1 2 fA(x)] an intuitionistic fuzzy value, and replace equation (4) with A ¼ {, x; ½tA ðxÞ; 1 2 f A ðxÞ . jx [ X}

ð5Þ


correspondingly. The intuitionistic fuzzy value [tA(x), 1 2 fA(x)] indicates that the exact degree of membership mF(x) of x may be unknown. But it is bounded by tA ðxÞ # mA ðxÞ # 1 2 f A ðxÞ where tA(x) þ fA(x) # 1. The interval [tA(x), 1 2 fA(x)] has a physical interpretation, for example, if [tA(x), 1 2 fA(x)] ¼ [0.4, 0.7], then we can see that tA(x) ¼ 0.4, 1 2 fA(x) ¼ 0.7; fA(x) ¼ 0.3. It can be interpreted as “the vote for resolution is 4 in favor, 3 against, and 3 abstentions” (1993). Let a~ ¼ ½ta~ ; 1 2 f a~ and b~ ¼ ½tb~ ; 1 2 f b~ be two intuitionistic fuzzy values, we introduce two operational laws of a~ and b~ as follows: (1) a~ ^b~ ¼ ½ta~ tb~ ; ð1 2 f a~ Þð1 2 f b~ Þ; (2) a~ l ¼ ½tla~ ; ð1 2 f a~ Þl ; l . 0: Theorem 1. Let a~ ¼ ½ta~ ; 1 2 f a~ and b~ ¼ ½tb~ ; 1 2 f b~ be two intuitionistic fuzzy values, and let c~ ¼ a~ ^b~ and d~ ¼ a~ l (l . 0), then both c~ and d~ are also intuitionistic fuzzy values. Proof. Since a~ ¼ ½ta~ ; 1 2 f a~ and b~ ¼ ½tb~ ; 1 2 f b~ are two intuitionistic fuzzy values, we have ta~ [ ½0; 1;

f a~ [ ½0; 1;

tb~ [ ½0; 1;

f b~ [ ½0; 1;

ta~ þ f a~ # 1;

tb~ þ f b~ # 1

then by the operational law (1), we have 0 # ta~ tb~ # 1;

0 # ð1 2 f a~ Þð1 2 f b~ Þ # 1

ta~ tb~ þ 1 2 ð1 2 f a~ Þð1 2 f b~ Þ # ð1 2 f a~ Þð1 2 f b~ Þ þ 1 2 ð1 2 f a~ Þð1 2 f b~ Þ ¼ 1 thus c~ is an intuitionistic fuzzy value. Also since tal~ $ 0;

1 2 ð1 2 f a~ Þl $ 0

and tal~ þ 1 2 ð1 2 f a~ Þl # ð1 2 f a~ Þl þ 1 2 ð1 2 f a~ Þl ¼ 1 thus d~ is also an intuitionistic fuzzy value. By the operational laws (1) and (2), we have. Theorem 2. Let a~ ¼ ½ta~ ; 1 2 f a~ and b~ ¼ ½tb~ ; 1 2 f b~ be two intuitionistic fuzzy values, l, l1, l2 . 0, then ~ a; (1) a~ ^b~ ¼ b^~ l ~ (2) ð~a^bÞ ¼ a~ l ^b~ l ; (3) a~ l1 ^~a l2 ¼ a~ l1 þl2 :


421

Proof (1) By the operational law (1), we have ~ a: a~ ^b~ ¼ ½ta~ tb~ ; ð1 2 f a~ Þð1 2 f b~ Þ ¼ ½tb~ ta~ ; ð1 2 f b~ Þð1 2 f a~ Þ ¼ b^~ (2) Since a~ ^b~ ¼ ½ta~ tb~ ; ð1 2 f a~ Þð1 2 f b~ Þ


then, by the operational law (2), it follows that ~ l ¼ ½ðta~ t ~ Þl ; ðð1 2 f a~ Þð1 2 f ~ ÞÞl ¼ ½tl~ tl~ ; ð1 2 f a~ Þl ð1 2 f ~ Þl ð~a^bÞ a b b b b Also since a~ l ¼ ½tla~ ; ð1 2 f a~ Þl ;

b~ l ¼ ½tlb~ ; ð1 2 f b~ Þl

then a~ l ^b~ l ¼ ½tal~ tlb~ ; ð1 2 f a~ Þl ð1 2 f b~ Þl hence, ~ l ¼ a~ l ^b~ l : ð~a^bÞ (3) Since a~ l1 ¼ ½tal~ 1 ; ð1 2 f a~ Þl1 ;

a~ l2 ¼ ½tla~ 2 ; ð1 2 f a~ Þl2

then a~ l1 ^~a l2 ¼ ½tal~ 1 tla~ 2 ; ð1 2 f a~ Þl1 ð1 2 f a~ Þl2 ¼ ½ðta~ ta~ Þl1 þl2 ; ð1 2 f a~ Þl1 þl2 ¼ a~ l1 þl2 : Chen and Tan (1994) introduced a score function S of an intuitionistic fuzzy value, which is represented as follows. Let a~ ¼ ½ta~ ; 1 2 f a~ be an intuitionistic fuzzy value, where ta~ [ ½0; 1;

f a~ [ ½0; 1;

ta~ þ f a~ # 1:

The score of a~ can be evaluated by the score function S shown as: Sð~aÞ ¼ ta~ 2 f a~

ð6Þ

where Sð~aÞ [ ½21; 1. The larger the score Sð~aÞ, the greater the intuitionistic fuzzy value a~ . Hong and Choi (2000) defined an accuracy function H: Hð~aÞ ¼ ta~ þ f a~

ð7Þ

to evaluate the degree of accuracy of the intuitionistic fuzzy value a~ ¼ ½ta~ ; 1 2 f a~ , where Hð~aÞ [ ½0; 1. The larger the value of Hð~aÞ, the more the degree of accuracy of the degree of membership of the intuitionistic fuzzy value a~ . As presented above, the score function S and the accuracy function H are, respectively, defined as the difference and the sum of the membership function t and the nonmembership function f. Hong and Choi (2000) showed that the relation between the score function S and the accuracy function H is similar to the relation between mean and variance in statistics. Based on the score function S and the accuracy function H, in the following, we give an order relation between two intuitionistic fuzzy values, which is defined as follows.

422


Definition 1. Let a~ ¼ ½ta~ ; 1 2 f a~ and b~ ¼ ½tb~ ; 1 2 f b~ be two intuitionistic fuzzy values, ~ ¼ t ~ 2 f ~ be the scores of a~ and b, ~ respectively, and let Hð~aÞ ¼ Sð~aÞ ¼ ta~ 2 f a~ and SðbÞ b b ~ ~ respectively, then ta~ þ f a~ and HðbÞ ¼ tb~ þ f b~ be the accuracy degrees of a~ and b,


~ then a~ is smaller than b, ~ denoted by a~ , b; ~ . If Sð~aÞ , SðbÞ, ~ then . If Sð~aÞ , SðbÞ, ~ then a~ and b~ represent the same information, denoted by a~ ¼ b; ~ (1) If Hð~aÞ ¼ HðbÞ, ~ then a~ is smaller than b, ~ denoted by a~ , b. ~ (2) If Hð~aÞ , HðbÞ, 3.2 The IFWG operator For convenience, let V be the set of all intuitionistic fuzzy values. Definition 2. Let a~ j ¼ ½ta~ j ; 1 2 f a~ j values, and let IFWG: Vn ! V, if

ð j ¼ 1; 2; . . .; nÞ be a collection of intuitionistic fuzzy

IFWGv ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ a~ v1 1 ^~av2 2 ^· · ·^~avn n

ð8Þ

where v ¼ (v1, v2, . . . , vn)T be the weight vector of a~ j ( j ¼ 1, 2, . . . ,n), and vj . 0, Pn T j¼1 vj ¼ 1, then IFWG is called the IFWG operator. Especially, if v ¼ (1/n, 1/n, . . . ,1/n) , then the IFWG operator is reduced to the intuitionistic fuzzy geometric (IFG) operator, which is defined as follows: 1

IFGð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ ð~a1 ^~a2 ^· · ·^~an Þn :

Theorem 3. Let a~ j ¼ ½ta~ j ; 1 2 f a~ j ð j ¼ 1; 2; . . .; nÞ be a collection of intuitionistic fuzzy values, then their aggregated value by using the IFWG operator is also an intuitionistic fuzzy value, and " # n n Y Y vj vj IFGWv ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ ta~ j ; ð1 2 f a~ j Þ ð9Þ j¼1

j¼1

T

where v ¼ (v1, v2, . . . ,vn) be the weight vector of a~ j ( j ¼ 1, 2, . . . , n), and vj . 0, Pn j¼1 vj ¼ 1. Proof. The first result follows quickly from Definition 1 and Theorem 1. In the following, we prove equation (9) by using mathematical induction on n: (1) For n ¼ 2: Since

h i a~ l1 1 ¼ tla~ 11 ; ð1 2 f a~ 1 Þl1 ;

h i a~ l2 2 ¼ tal~ 22 ; ð1 2 f a~ 2 Þl2

then h i IFWGv ð~a1 ; a~ 2 Þ ¼ a~ v1 1 ^~av2 2 ¼ tal~ 11 tal~ 22 ; ð1 2 f a~ 1 Þl1 ð1 2 f a~ 2 Þl2 thus, equation (9) holds.


423

(2) If equation (9) holds for n ¼ k, that is " IFWGv ð~a1 ; a~ 2 ; . . .; a~ k Þ ¼

k Y

v ta~ jj ;

k Y


j¼1

# ð1 2 f a~ j Þ

vj

j¼1

then, when n ¼ k þ 1, by the operational laws (1) and (2), we have " ! ! # k k Y Y v vj kþ1 IFWGv ð~a1 ; a~ 2 ; . . .; a~ kþ1 Þ ¼ ta~ j ·tva~ kþ1 ; ð1 2 f a~ j Þvj · 1 2 f a~ kþ1 kþ1 j¼1

¼

" kþ1 Y

v ta~ jj ;

j¼1

j¼1 kþ1 Y

# vj

ð1 2 f a~ j Þ

j¼1

i.e. equation (9) holds for n ¼ k þ 1. Thus, equation (9) holds for all n. Example 1. Let a~ 1 ¼ ½0:1; 0:3; a~ 2 ¼ ½0:4; 0:7; a~ 3 ¼ ½0:6; 0:9, and a~ 4 ¼ ½0:2; 0:5 be four intuitionistic fuzzy values, and v ¼ (0.2, 0.3, 0.1, 0.4)T be the weight vector of a~ j ( j ¼ 1, 2, 3, 4), then ta~ 1 ¼ 0:1;

ta~ 2 ¼ 0:4;

f a~ 2 ¼ 1 2 0:7 ¼ 0:3;

ta~ 3 ¼ 0:6;

ta~ 4 ¼ 0:2;

f a~ 3 ¼ 1 2 0:9 ¼ 0:1;

f a~ 1 ¼ 1 2 0:3 ¼ 0:7 f a~ 4 ¼ 1 2 0:5 ¼ 0:5

Thus " IFWGv ð~a1 ; a~ 2 ; a~ 3 ; a~ 4 Þ ¼

4 Y j¼1

v ta~ jj ;

4 Y

# ð1 2 f a~ j Þ

vj

j¼1

¼ ½0:10:2 £ 0:40:3 £ 0:60:1 £ 0:20:4 ; 0:30:3 £ 0:70:4 £ 0:90:2 £ 0:50:1 ¼ ½0:2392; 0:5296: Theorem 4. Let a~ j ¼ ½ta~ j ; 1 2 f a~ j ð j ¼ 1; 2; . . .; nÞ be a collection of intuitionistic fuzzy values, and v ¼ (v1, v2, . . . , vn)T be the weight vector of a~ j ( j ¼ 1, 2, . . . , n), with vj . 0 P and nj¼1 vj ¼ 1. If all a~ j ( j ¼ 1, 2, . . . , n) are equal, i.e. a~ j ¼ a~ , for all j, then IFWGv ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ a~ : Proof. By Theorem 2, we have IFWGv ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ a~ v1 1 ^~av2 2 ^· · ·^~avn n ¼ a~ v1 ^~a v2 ^· · ·^~a vn ¼ a~

Pn j¼1

vj

¼ a~ :

Theorem 5. Let a~ j ¼ ½ta~ j ; 1 2 f a~ j ð j ¼ 1; 2; . . .; nÞ be a collection of intuitionistic fuzzy values, and let 2 þ a~ ¼ minðta~ j Þ; 1 2 maxð f a~ j Þ ; a~ ¼ maxðta~ j Þ; 1 2 minð f a~ j Þ j

j

j

j

424


then


a~ 2 # IFWGv ð~a1 ; a~ 2 ; . . .; a~ n Þ # a~ þ :

ð10Þ

Proof. Since minðta~ j Þ # ta~ j # maxðta~ j Þ, minðf a~ j Þ # f a~ j # maxðf a~ j Þ, for all j, then j j j j Pn v n n Y v Y j¼1 j ta~ jj $ minðta~ j Þvj ¼ minðta~ j Þ ¼ minðta~ j Þ j¼1

n Y

j

j

j¼1

n Y

v ta~ jj

ð1 2 f a~ j Þvj #

j

j

vj P n vj n Y j¼1 # maxðta~ j Þ ¼ maxðta~ j Þ ¼ maxðta~ j Þ

j¼1

j¼1

j

vj Pn vj n Y j¼1 ð1 2 f a~ j Þvj $ 1 2 maxð f a~ j Þ ¼ 1 2 maxð f a~ j Þ ¼ 1 2 maxð f a~ j Þ

j¼1

n Y

j

j¼1

j

j¼1 n Y j¼1

j

vj 1 2 minð f a~ j Þ

¼

j

ð12Þ

ð13Þ

j

Pn v j j¼1 1 2 minð f a~ j Þ ¼ 1 2 minð f a~ j Þ: j

ð11Þ

j

ð14Þ

Let IFWGv ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ ½ta~ ; 1 2 f a~ , then Sð~aÞ ¼ ta~ 2 f a~ # maxðta~ j Þ 2 minð f a~ j Þ ¼ Sð~a þ Þ j

j

Sð~aÞ ¼ ta~ 2 f a~ $ minðta~ j Þ 2 maxð f a~ j Þ ¼ Sð~a 2 Þ: j

j

If Sð~aÞ , Sð~a þ Þ and Sð~aÞ . Sð~a 2 Þ then by Definition 1, we have a~ 2 , IFWGv ð~a1 ; a~ 2 ; . . .; a~ n Þ , a~ þ :

ð15Þ

If Sð~aÞ ¼ Sð~a þ Þ, i.e. ta~ 2 f a~ ¼ maxðta~ j Þ 2 minðf a~ j Þ, then by equations (13) and (14), we j j have ta~ ¼ maxðta~ j Þ; j

f a~ ¼ minð f a~ j Þ j

thus Hð~aÞ ¼ ta~ þ f a~ ¼ maxðta~ j Þ þ minð f a~ j Þ ¼ Hð~a þ Þ j

j

in this case, from Definition 1, it follows that IFWGv ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ a~ þ :

ð16Þ

If Sð~aÞ ¼ Sð~a 2 Þ, i.e. ta~ 2 f a~ ¼ minðta~ j Þ 2 maxðf a~ j Þ, then by equations (11) and (12), we j j have ta~ ¼ minðta~ j Þ; j

f a~ ¼ maxð f a~ j Þ j


425

thus Hð~aÞ ¼ ta~ þ f a~ ¼ minðta~ j Þ þ maxð f a~ j Þ ¼ Hð~a 2 Þ j

j

in this case, from Definition 3, it follows that


IFWGv ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ a~ 2

ð17Þ

From equations (15) –(17), we know that equation (10) always holds. Theorem 6. Let a~ j ¼ ½ta~ j ; 1 2 f a~ j ( j ¼ 1, 2, . . . , n) and a~ *j ¼ ½ta~ *j ; 1 2 f a~ *j ( j ¼ 1, 2, . . . , n) be two collections of intuitionistic fuzzy values, If ta~ j # ta~ *j and f a~ j $ f a~ *j , for all j, then IFWGv ð~a1 ; a~ 2 ; . . .; a~ n Þ # IFWGv ð~a*1 ; a~ *2 ; . . .; a~ *n Þ:

ð18Þ

Proof. Since ta~ j # ta~ *j and f a~ j $ f a~ *j , for all j, then n Y

v

ta~ jj #

j¼1

n Y

n Y

v

ta~ *j j

j¼1

ð1 2 f a~ j Þvj #

j¼1

n Y

ð1 2 f a~ *j Þvj

j¼1

hence n Y

v ta~ jj

2

12

j¼1

n Y

! vj

ð1 2 f a~ j Þ

#

j¼1

n Y j¼1

v ta~ *j j

2

12

n Y

! ð1 2 f a~ *j Þ

vj

:

ð19Þ

j¼1

Let a~ ¼ IFWGv ð~a1 ; a~ 2 ; . . .; a~ n Þ and a~ * ¼ IFWGv ð~a*1 ; a~ *2 ; . . .; a~ *n Þ, then by equation (19), we have Sð~aÞ # Sð~a * Þ: If Sð~aÞ , Sð~a * Þ, then by Definition 1, we have IFWGv ð~a1 ; a~ 2 ; . . .; a~ n Þ , IFWGv ð~a*1 ; a~ *2 ; . . .; a~ *n Þ:

ð20Þ

If Sð~aÞ ¼ Sð~a * Þ, i.e. n Y

v ta~ jj

2

12

j¼1

n Y

! ð1 2 f a~ j Þ

vj

j¼1

¼

n Y j¼1

v ta~ *j j

2

12

n Y

! ð1 2 f a~ *j Þ

j¼1

then, by the conditions ta~ j # ta~ *j and f a~ j $ f a~ *j , for all j, we have n Y j¼1

v

ta~ jj ¼

n Y j¼1

v

ta~ *j ; j

12

n Y j¼1

ð1 2 f a~ j Þvj ¼ 1 2

n Y j¼1

vj

ð1 2 f a~ *j Þvj

426


thus Hð~aÞ ¼

n Y

v

ta~ jj þ 1 2

j¼1

n Y

ð1 2 f a~ j Þvj ¼

j¼1

n Y

v

ta~ *j þ 1 2

j¼1

j

n Y

ð1 2 f a~ *j Þvj ¼ Sð~a * Þ:

ð21Þ

j¼1


From equations (20) and (21), we know that equation (18) always holds. 3.3 The IFOWG operator Definition 3 Let a~ j ¼ ½ta~ j ; 1 2 f a~ j ð j ¼ 1; 2; . . .; nÞ be a collection of intuitionistic fuzzy values. An intuitionistic fuzzy OWG (IFOWG) operator of dimension n is a mapping IFOWG : Vn ! V, that has an associated vector w ¼ ðw1 ; w2 ; . . .; wn ÞT such that wj . 0 and Pn j¼1 wj ¼ 1. Furthermore, IFOWGw ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ ð~asð1Þ Þw1 ^ð~asð2Þ Þw2 ^· · ·^ð~asðnÞ Þwn

ð22Þ

where ðsð1Þ; sð2Þ; . . .; sðnÞÞ is a permutation of ð1; 2; . . .; nÞ such that a~ sð j21Þ $ a~ sð jÞ for all j. Especially, if w ¼ ð1=n; 1=n; . . .; 1=nÞT , then the IFOWG operator is reduced to the intuitionistic fuzzy geometric (IFG) operator. Similar to Theorem 3, we have; Theorem 7. Let a~ j ¼ ½ta~ j ; 1 2 f a~ j ð j ¼ 1; 2; . . .; nÞ be a collection of intuitionistic fuzzy values, then their aggregated value by using the IFOWG operator is also an intuitionistic fuzzy value, and " # n n Y Y vj vj IFOWGw ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ ðta~ sð jÞ Þ ; ð1 2 f a~ sð jÞ Þ ð23Þ j¼1

j¼1

where w ¼ ðw1 ; w2 ; . . .; wn ÞT is the weighting vector of the IFOWG operator, with wj . 0 and Pn j¼1 wj ¼ 1, which can also be determined by using the normal distribution based method (Xu 2005). Example 2. Let a~ 1 ¼ ½0:3; 0:4, a~ 2 ¼ ½0:4; 0:5, a~ 3 ¼ ½0:6; 0:7, a~ 4 ¼ ½0:7; 0:9, and a~ 5 ¼ ½0:1; 0:6 be five intuitionistic fuzzy values, then ta~ 1 ¼ 0:3;

ta~ 2 ¼ 0:4;

f a~ 1 ¼ 1 2 0:4 ¼ 0:6;

ta~ 3 ¼ 0:6;

ta~ 4 ¼ 0:7;

f a~ 2 ¼ 1 2 0:5 ¼ 0:5;

f a~ 4 ¼ 1 2 0:9 ¼ 0:1;

ta~ 5 ¼ 0:1

f a~ 3 ¼ 1 2 0:7 ¼ 0:3

f a~ 5 ¼ 1 2 0:4 ¼ 0:6:

By equation (6), we calculate the scores of a~ j ð j ¼ 1; 2; 3; 4; 5Þ: Sð~a1 Þ ¼ 0:3 2 0:6 ¼ 20:3;

Sð~a2 Þ ¼ 0:4 2 0:5 ¼ 20:1;

Sð~a4 Þ ¼ 0:7 2 0:1 ¼ 0:6;

Sð~a5 Þ ¼ 0:1 2 0:6 ¼ 20:5

Sð~a3 Þ ¼ 0:6 2 0:3 ¼ 0:3;

Since Sð~a4 Þ . Sð~a3 Þ . Sð~a2 Þ . Sð~a1 Þ . Sð~a5 Þ


427

thus a~ sð1Þ ¼ ½0:7; 0:9;

a~ sð2Þ ¼ ½0:6; 0:7;


a~ sð4Þ ¼ ½0:3; 0:4;

a~ sð3Þ ¼ ½0:4; 0:5;

a~ sð5Þ ¼ ½0:1; 0:6:

Suppose that w ¼ ð0:1117; 0:2365; 0:3036; 0:2365; 0:1117ÞT (derived from equation (3)) is the weighting vector of the IFOWG operator. Then, by equation (23), it follows that " # 5 5 Y Y vj vj ðta~ sð jÞ Þ ; ð1 2 f a~ sð jÞ Þ IFOWGw ð~a1 ; a~ 2 ; a~ 3 ; a~ 4 ; a~ 5 Þ ¼ j¼1

j¼1

¼ ½0:70:1117 £ 0:60:2365 £ 0:40:3036 £ 0:30:2365 £ 0:10:1117 ; 0:90:1117 £ 0:70:2365 £ 0:50:3036 £ 0:40:2365 £ 0:60:1117 ¼ ½0:3750; 0:5597: Similar to the IFWG operator, the IFOWG operator also has the following properties. Theorem 8. Let a~ j ¼ ½ta~ j ; 1 2 f a~ j ð j ¼ 1; 2; . . .; nÞ be a collection of intuitionistic fuzzy values, and w ¼ ðw1 ; w2 ; . . .; wn ÞT be the weighting vector of the IFOWG operator, with P wj . 0 and nj¼1 wj ¼ 1. If all a~ j ð j ¼ 1; 2; . . .; nÞ are equal, i.e. a~ j ¼ a~ , for all j, then IFOWGw ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ a~ :

Theorem 9. Let a~ j ¼ ½ta~ j ; 1 2 f a~ j ð j ¼ 1; 2; . . .; nÞ be a collection of intuitionistic fuzzy values, and let 2 þ a~ ¼ minðta~ j Þ; 1 2 maxð f a~ j Þ ; a~ ¼ maxðta~ j Þ; 1 2 minð f a~ j Þ j

j

j

j

then a~ 2 # IFOWGw ð~a1 ; a~ 2 ; . . .; a~ n Þ # a~ þ :

Theorem 10. Let a~ j ¼ ½ta~ j ; 1 2 f a~ j ð j ¼ 1; 2; :::; nÞ and a~ *j ¼ ½ta~ *j ; 1 2 f a~ *j ð j ¼ 1; 2; :::; nÞ be two collections of intuitionistic fuzzy values, If ta~ j # ta~ *j and f a~ j $ f a~ *j , for all j, then IFOWGw ð~a1 ; a~ 2 ; . . .; a~ n Þ # IFOWGw ð~a*1 ; a~ *2 ; . . .; a~ *n Þ: The IFOWG operator has also the following properties. Theorem 11. (Commutativity) Let a~ j ¼ ½ta~ j ; 1 2 f a~ j ð j ¼ 1; 2; . . .; nÞ and a~ 0j ¼ ½ta~ 0j ; 1 2 f a~ 0j ð j ¼ 1; 2; . . .; nÞ be two collections of intuitionistic fuzzy values, then IFOWGw ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ IFOWGw ð~a01 ; a~ 02 ; . . .; a~ 0n Þ where ð~a01 ; a~ 02 ; . . .; a~ 0n Þ is any permutation of ð~a1 ; a~ 2 ; . . .; a~ n Þ.

ð24Þ

428


Proof. Let IFOWGð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ ð~asð1Þ Þw1 ^ð~asð2Þ Þw2 ^· · ·^ð~asðnÞ Þwn IFOWGð~a01 ; a~ 02 ; . . .; a~ 0n Þ ¼ ð~a0sð1Þ Þw1 ^ð~a0sð2Þ Þw2 ^· · ·^ð~a0sðnÞ Þwn


Since ð~a01 ; a~ 02 ; . . .; a~ 0n Þ is a permutation of ð~a1 ; a~ 2 ; . . .; a~ n Þ, then we have a~ sð jÞ ¼ a~ 0sð jÞ ;

j ¼ 1; 2; . . .; n

then, equation (24) holds. Theorem 12. Let a~ j ¼ ½ta~ j ; 1 2 f a~ j ð j ¼ 1; 2; . . .; nÞ be a collection of intuitionistic fuzzy values, and w ¼ ðw1 ; w2 ; . . .; wn ÞT be the weighting vector of the IFOWG operator, with P wj . 0 and nj¼1 wj ¼ 1, then (1) If w1 ! 1, then IFOWGð~a1 ; a~ 2 ; . . .; a~ n Þ ! maxð~aj Þ; j (2) If wn ! 1, then IFOWGð~a1 ; a~ 2 ; . . .; a~ n Þ ! minð~aj Þ; (3) If wj ! 1, then IFOWGð~a1 ; a~ 2 ; . . .; a~ n Þ ! ja~ sð jÞ , where a~ sð jÞ is the jth largest of a~ i ði ¼ 1; 2; . . .; nÞ. From Definitions 2 and 3, we know that the IFWG operator weights only the intuitionistic fuzzy values, while the IFOWG operator weights only the ordered positions of the intuitionistic fuzzy values instead of weighting the intuitionistic fuzzy values themselves. In the following subsection, we develop an IFHG operator, which weights both the given intuitionistic fuzzy value and its ordered position. 3.4 The IFHG operator Definition 4. An IFHG operator is a mapping IFHG : Vn ! V, which has an associated P vector w ¼ ðw1 ; w2 ; . . .; wn ÞT with wj . 0; nj¼1 wj ¼ 1 such that IFHGv;w ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ ð~a_sð1Þ Þw1 ^ð~a_sð2Þ Þw2 ^· · ·^ð~a_sðnÞ Þwn

ð25Þ

where a~_sð jÞ is the jth largest of the weighted intuitionistic fuzzy values nv a~_ j ð~a_ j ¼ a~ j j ; 1; 2; . . .; nÞ, v ¼ ðv1 ; v2 ; . . .; vn ÞT is the weight vector of a~ j ð j ¼ 1; 2; . . .; nÞ P with vj . 0, nj¼1 vj ¼ 1, and n is the balancing coefficient, which plays a role of balance, (in this case, if the vector ðv1 ; v2 ; . . .; vn ÞT goes to ð1=n; 1=n; . . .; 1=nÞT , then the vector ð~an1v1 ; a~ n2v2 ; . . .; a~ nnvn ÞT goes to ð~a1 ; a~ 2 ; . . .; a~ n ÞT ). Let a~_sð jÞ ¼ ½ta_~sð jÞ ; 1 2 ta_~sð jÞ ( j ¼ 1, 2, . . . , n) then, similar to Theorem 3, we have " # n n Y Y wj vj IFHGv;w ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ ta_~sð jÞ ; ð1 2 f a_~sð jÞ Þ ð26Þ j¼1

j¼1

and the aggregated value derived by using the IFHG operator is also an intuitionistic fuzzy value. Example 3. Let a~ 1 ¼ ½0:3; 0:6, a~ 2 ¼ ½0:5; 0:7, a~ 3 ¼ ½0:5; 0:6, a~ 4 ¼ ½0:4; 0:9, and a~ 5 ¼ ½0:7; 0:8 be five intuitionistic fuzzy values, and let v ¼ ð0:18; 0:22; 0:16; 0:21; 0:23ÞT be the


429

weight vector of a~ j ð j ¼ 1; 2; 3; 4; 5Þ. Then ta~ 1 ¼ 0:3;

ta~ 2 ¼ 0:5;

1 2 f a~ 1 ¼ 0:6;

ta~ 3 ¼ 0:5;

ta~ 4 ¼ 0:4;

1 2 f a~ 2 ¼ 0:7;

1 2 f a~ 4 ¼ 0:9;

ta~ 5 ¼ 0:7

1 2 f a~ 3 ¼ 0:6

1 2 f a~ 5 ¼ 0:8:


By the operational law (2), we get the weighted intuitionistic fuzzy values as follows: a~_1 ¼ ½0:35£0:18 ; 0:65£0:18 ¼ ½0:338; 0:631 a~_2 ¼ ½0:55£0:22 ; 0:75£0:22 ¼ ½0:467; 0:675 a~_3 ¼ ½0:55£0:16 ; 0:65£0:16 ¼ ½0:574; 0:665 a~_4 ¼ ½0:45£0:21 ; 0:95£0:21 ¼ ½0:382; 0:895 a~_5 ¼ ½0:75£0:23 ; 0:85£0:23 ¼ ½0:664; 0:774: By equation (6), we calculate the scores of a~_j ð j ¼ 1; 2; 3; 4; 5Þ: Sð~a_1 Þ ¼ 0:338 2 ð1 2 0:631Þ ¼ 20:031;

Sð~a_2 Þ ¼ 0:467 2 ð1 2 0:675Þ ¼ 0:142

Sð~a_3 Þ ¼ 0:574 2 ð1 2 0:665Þ ¼ 0:239;

Sð~a_4 Þ ¼ 0:382 2 ð1 2 0:895Þ ¼ 0:277

Sð~a_5 Þ ¼ 0:664 2 ð1 2 0:774Þ ¼ 0:438 Since Sð~a_5 Þ . Sð~a_4 Þ . Sð~a_3 Þ . Sð~a_2 Þ . Sð~a_1 Þ thus a~_sð1Þ ¼ ½0:664; 0:774;

a~_sð2Þ ¼ ½0:382; 0:895;

a~_sð3Þ ¼ ½0:574; 0:665;

a~_sð5Þ ¼ ½0:338; 0:631

a~_sð4Þ ¼ ½0:467; 0:675; where ta_~sð1Þ ¼ 0:667;

ta_~sð2Þ ¼ 0:382;

1 2 f a_~sð1Þ ¼ 0:774;

ta_~sð3Þ ¼ 0:574; 1 2 f a_~sð2Þ ¼ 0:895;

1 2 f a_~sð4Þ ¼ 0:675;

ta_~sð4Þ ¼ 0:467;

ta_~sð5Þ ¼ 0:338

1 2 f a_~sð3Þ ¼ 0:665

1 2 f a_~sð5Þ ¼ 0:631:

Suppose that w ¼ ð0:1117; 0:2365; 0:3036; 0:2365; 0:1117ÞT (derived from equation (3)) is the weighting vector of the IFHG operator. Then, by equation (26), it follows that

IFHGv;w ð~a1 ; a~ 2 ; a~ 3 ; a~ 4 ; a~ 5 Þ ¼

" 5 Y j¼1

w ta_~sjð jÞ ;

5 Y

# ð1 2 f a_~sð jÞ Þvj

j¼1

¼ ½0:6670:1117 £ 0:3820:2365 £ 0:5740:3036 £ 0:4670:2365 £ 0:3380:1117 ; 0:7740:1117 £ 0:8950:2365 £ 0:6650:3036 £ 0:6750:2365 £ 0:6310:1117 ¼ ½0:4759; 0:7239:

430


Theorem 13. The IFWG operator is a special case of the IFHG operator. Proof. Let w ¼ ð1=n; 1=n; . . .; 1=nÞT , then IFHGv;w ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ ð~a_sð1Þ Þw1 ^ð~a_sð2Þ Þw2 ^· · ·^ð~a_sðnÞ Þwn ¼ ð~a_sð1Þ ^~a_sð2Þ ^· · ·^~a_sðnÞ Þ1=n


¼ a~ v1 1 ^~av2 2 ^· · ·^~avn n ¼ IFWGv ð~a1 ; a~ 2 ; . . .; a~ n Þ:

Theorem 14. The IFOWG operator is a special case of the IFHG operator. Proof. Let v ¼ ð1=n; 1=n; . . .; 1=nÞT , then a~_j ¼ a~ j , i ¼ 1; 2; . . .; n, thus IFHGv;w ð~a1 ; a~ 2 ; . . .; a~ n Þ ¼ ð~a_sð1Þ Þw1 ^ð~a_sð2Þ Þw2 ^· · ·^ð~a_sðnÞ Þwn ¼ ð~asð1Þ Þw1 ^ð~asð2Þ Þw2 ^· · ·^ð~asðnÞ Þwn ¼ IFOWGw ð~a1 ; a~ 2 ; . . .; a~ n Þ: From Theorems 13 and 14, it can be known clearly that the IFHG operator generalizes both the IFOWG and IFWG operators, and reflects the importance degrees of both the given intuitionistic fuzzy argument and the ordered position of the argument.

4. An application of the IFHG operator to multiple attribute decision making In the following, we apply the IFHG operator to multiple attribute decision making based on intuitionistic fuzzy information. Let A ¼ {A1 ; A2 ; . . .; Am } be a set of alternatives, and let B ¼ {B1 ; B2 ; . . .; Bn } be a set of attributes. v ¼ ðv1 ; v2 ; . . .; vn ÞT is the weight vector of Bj ð j ¼ 1; 2; . . .; nÞ, with vj . 0 and Pn j¼1 vj ¼ 1. Assume that the characteristics of the alternatives Ai ði ¼ 1; 2; . . .; mÞ is represented by the intuitionistic fuzzy set: Ai ¼ { , B1 ; ½ti1 ; ð1 2 f i1 Þ .; , B2 ; ½ti2 ; ð1 2 f i2 Þ .; · · ·; , Bn ; ½tin ; ð1 2 f in Þ . } ð27Þ where tij indicates the degree that the alternative Ai satisfies the attribute Bj , f ij indicates the degree that the alternative Ai does not satisfy the attribute Bj , tij [ ½0; 1, f ij [ ½0; 1, tij þ f ij # 1, 1 # i # m, 1 # j # n. Let a~ ij ¼ ½tij ; ð1 2 f ij Þ, for all i, j, then equation (27) can rewritten as Ai ¼ { , Bj ; a~ ij . jBj [ B}

ð28Þ

To get the best alternative(s), we can utilize the IFHG operator: a~ i ¼ IFHGv;w ð~ai1 ; a~ i2 ; . . .; a~ in Þ;

i ¼ 1; 2; . . .; m

ð29Þ

to derive the overall values a~ i ¼ ½ti ; 1 2 f i ði ¼ 1; 2; . . .; mÞ of the alternatives Ai ði ¼ 1; 2; . . .; mÞ, where w ¼ ðw1 ; w2 ; . . .; wn ÞT is the weighting vector of the IFHG P operator, with wj . 0 and nj¼1 wj ¼ 1, which can be determined by the normal distribution based method (Xu 2005).


431


Then by equation (2), we calculate the scores Sð~ai Þði ¼ 1; 2; . . .; mÞ of the overall values a~ i ði ¼ 1; 2; . . .; mÞ, and utilize the scores Sð~ai Þði ¼ 1; 2; . . .; mÞ to rank the alternatives Ai ði ¼ 1; 2; . . .; nÞ, and then to select the best one(s) (if there is no difference between two scores Sð~ai Þ and Sð~aj Þ, then we need to calculate the accuracy degrees Hð~ai Þ and Hð~aj Þ of the overall values a~ i and a~ j , respectively, and then rank the alternatives Ai and Aj in accordance with the accuracy degrees Hð~ai Þ and Hð~aj Þ.

5. Concluding remarks In this paper, we have introduced two operational laws of intuitionistic fuzzy values, and developed some new geometric aggregation operators, including the IFWG operator, the IFOWG operator, and the IFHG operator, which extend the WG operator and the OWG operator to accommodate the situations where the given arguments are intuitionistic fuzzy sets. We have investigated various properties of these operators. Both the OWG weights and the IFHG weights can be derived from the normal distribution based method, which can relieve the influence of unfair arguments on the final results by assigning low weights to those unduly high or unduly low ones. We have applied the IFHG operator to multiple attribute decision making based on intuitionistic fuzzy sets, which develops the theories both the geometric aggregation operators and the intuitionistic fuzzy sets.

Acknowledgements The authors are very grateful to the anonymous referees for their insightful and constructive comments and suggestions that led to an improved version of this paper. The work was supported by the National Natural Science Foundation of China under Grant 70571087.

References J. Acze´l and T.L. Saaty, “Procedures for synthesizing ratio judgments”, J. Math. Psych., 27, pp. 93–102, 1983. K. Atanassov, “Intuitionistic fuzzy sets”, Fuzzy Sets Syst., 20, pp. 87–96, 1986. K. Atanassov, “More on intuitionistic fuzzy sets”, Fuzzy Sets Syst., 33, pp. 37–46, 1989. K. Atanassov, “New operations defined over the intuitionistic fuzzy sets”, Fuzzy Sets Syst., 61, pp. 137–142, 1994a. K. Atanassov, “Operators over interval valued intuitionistic fuzzy sets”, Fuzzy Sets Syst., 64, pp. 159 –174, 1994b. K. Atanassov, Intuitionistic Fuzzy Sets: Theory and Applications, Heidelberg: Physica-Verlag, 1999. K. Atanassov, “Two theorems for intuitionistic fuzzy sets”, Fuzzy Sets Syst., 110, pp. 267– 269, 2000. K. Atanassov and G. Gargov, “Interval valued intuitionistic fuzzy sets”, Fuzzy Sets Syst., 31, pp. 343–349, 1989. C.O. Benjamin, L.C. Ehie and Y. Omurtag, “Planning facilities at the university of missourirolla”, Interface, 4, pp. 95–105, 1992. H. Bustince and P. Burillo, “Vague sets are intuitionistic fuzzy sets”, Fuzzy Sets Syst., 79, pp. 403–405, 1996. H. Bustince, J. Kacprzyk and V. Mohedano, “Intuitionistic fuzzy generators: application to intuitionistic fuzzy complementation”, Fuzzy Sets Syst., 114, pp. 485–504, 2000. S.M. Chen and J.M. Tan, “Handling multicriteria fuzzy decision-making problems based on vague set theory”, Fuzzy Sets Syst., 67, pp. 163– 172, 1994. F. Chiclana, F. Herrera and E. Herrera-Viedma, “Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations”, Fuzzy Sets Syst., 122, pp. 277 –291, 2001. S.K. De, R. Biswas and A.R. Roy, “Some operations on intuitionistic fuzzy sets”, Fuzzy Sets Syst., 114, pp. 477 –484, 2000. G. Deschrijver and E. Kerre, “On the composition of intuitionistic fuzzy relations”, Fuzzy Sets Syst., 136, pp. 333–361, 2003. W. Gau and D.J. Buehrer, “Vague sets”, IEEE Trans. Syst. Man Cybern., 23, pp. 610–614, 1993.


432


P. Grzegorzewski, “Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric”, Fuzzy Sets Syst., 148, pp. 319 –328, 2004. F. Herrera, E. Herrera-Viedma and F. Chiclana, “Multiperson decision-making based on multiplicative preference relations”, European Journal of Operational Research, 129, pp. 372 –385, 2001. F. Herrera, E. Herrera-Viedma and F. Chiclana, “A study of the origin and uses of the ordered weighted geometric operator in multicriteria decision making”, International Journal of Intelligent Systems, 18, pp. 689–707, 2003. D.H. Hong and C.H. Choi, “Multicriteria fuzzy decision-making problems based on vague set theory”, Fuzzy Sets Syst., 114, pp. 103–113, 2000. D.F. Li, “Some measures of dissimilarity in intuitionistic fuzzy structures”, Journal of Computer and System Sciences, 68, pp. 115–122, 2004. T.K. Mondal and S.K. Samanta, “Topology of interval-valued intuitionistic fuzzy sets”, Fuzzy Sets Syst., 119, pp. 483–494, 2001. T.K. Mondal and S.K. Samanta, “On intuitionistic gradation of openness”, Fuzzy Sets Syst., 131, pp. 323–336, 2002. T.L. Saaty, The Analytic Hierarchy Process, New York: McGraw-Hill, 1980. E. Szmidt and J. Kacprzyk, “Distances between intuitionistic fuzzy sets”, Fuzzy Sets Syst., 114, pp. 505–518, 2000. E. Szmidt and J. Kacprzyk, “Entropy for intuitionistic fuzzy sets”, Fuzzy Sets Syst., 118, pp. 467 –477, 2001. K. Willet and R. Sharda, “Using the analytic hierarchy process in water resources planning: selection of flood control projects”, Socio-Economic Planning Sciences, 2, pp. 103 –112, 1991. Z.S. Xu, “On consistency of the weighted geometric mean complex judgment matrix in AHP”, European Journal of Operational Research, 126, pp. 683 –687, 2000. Z.S. Xu, Uncertain Multiple Attribute Decision Making: Methods and Applications, Beijing: Tsinghua University Press, 2004a. Z.S. Xu, “A method based on linguistic aggregation operators for group decision making with linguistic preference relations”, Information Sciences, 166, pp. 19 –30, 2004b. Z.S. Xu, “EOWA and EOWG operators for aggregating linguistic labels based on linguistic preference relations”, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 12, pp. 791–810, 2004c. Z.S. Xu, “An overview of methods for determining OWA weights”, International Journal of Intelligent Systems, 20, pp. 843–865, 2005. Z.S. Xu and Q.L. Da, “The ordered weighted geometric averaging operators”, International Journal of Intelligent Systems, 17, pp. 709– 716, 2002. Z.S. Xu and Q.L. Da, “An overview of operators for aggregating information”, International Journal of Intelligent Systems, 18, pp. 953– 969, 2003. R.R. Yager, “On ordered weighted averaging aggregation operators in multicriteria decisionmaking”, IEEE Trans. Syst. Man Cybern., 18, pp. 183 –190, 1988. R.R. Yager, “Families of OWA operators”, Fuzzy Sets Syst., 59, pp. 125 –166, 1993. R.R. Yager and J. Kacprzyk, The Ordered Weighted Averaging Operator: Theory and Application, Norwell, MA: Kluwer, 1997. R.R. Yager, “Generalized OWA aggregation operators”, Fuzzy Optimization and Decision Making, 3, pp. 93 –107, 2004. L.A. Zadeh, “Fuzzy sets”, Information and Control, 8, pp. 338 –353, 1965.

Zeshui Xu is a Professor at Sciences Institute, PLA University of Science and Technology, Nanjing, China. He received a PhD degree in management science and engineering from Southeast University, Nanjing, China in 2003. From April 2003 to May 2005, he was a postdoctoral researcher at School of Economics and Management, Southeast University. He is a postdoctoral researcher at School of Economics and Management, Tsinghua University since October of 2005. He services on the Editorial Board of Information: An International Journal. He has authored a book, Uncertain Multiple Attribute Decision Making: Methods and Applications (Tsinghua University Press, Beijing, 2004) and has contributed over 150 journal articles to professional journal such as European Journal of Operational Research, Journal of Optimization Theory and Applications, International Journal of Intelligent Systems, International Journal of Approximate Reasoning, Information Sciences, Omega, Decision Support Systems, International Journal of General Systems, Group Decision and Negotiation, Fuzzy Sets and Systems, Information Fusion, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. He is also a paper reviewer of many professional journals such as IEEE Transactions on Systems, Man, and Cybernetics, IEEE Transactions on Fuzzy Systems, Information


433


Sciences, European Journal of Operational Research, International Journal of Intelligent Systems, Decision Support Systems, International Journal of Approximate Reasoning, International Journal of Information Technology and Decision Making, Fuzzy Optimization and Decision Making, Mathware and Soft Computing. His current research interests include information fusion, group decision making, computing with words, and aggregation operators. Ronald R. Yager has worked in the area of fuzzy sets and related disciplines of computational intelligence for over twenty-five years. He has published over 500 papers and fifteen books. He is considered one the worlds leading experts in fuzzy sets technology. He was the recipient of the IEEE Computational Intelligence Society Pioneer award in Fuzzy Systems. Dr. Yager is a fellow of the IEEE, the New York Academy of Sciences and the Fuzzy Systems Association. He was recently given an award by the Polish Academy of Sciences for his contributions. He served at the National Science Foundation as program director in the Information Sciences program. He was a NASA/Stanford visiting fellow and a research associate at the University of California, Berkeley. He has been a lecturer at NATO Advanced Study Institutes. He received his undergraduate degree from the City College of New York and his Ph. D. from the Polytechnic University of New York. Currently, he is Director of the Machine Intelligence Institute and Professor of Information and Decision Technologies at Iona College. He is editor and chief of the International Journal of Intelligent Systems. He serves on the editorial board of a number of journals including the IEEE Transactions on Fuzzy Systems, Neural Networks, Data Mining and Knowledge Discovery, IEEE Intelligent Systems, Fuzzy Sets and Systems, the Journal of Approximate Reasoning and the International Journal of General Systems. In addition to his pioneering work in the area of fuzzy logic he has made fundamental contributions in decision making under uncertainty and the fusion of information. Among his current interests is the development of intelligent semantic web technology, communication for cooperating autonomous systems, information aggregation and decision making in adversarial and uncertain environments.

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