Geometric Bonferroni means with their application in multi-criteria ...
Knowledge-Based Systems 40 (2013) 88–100
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Geometric Bonferroni means with their application in multi-criteria decision making Meimei Xia a,⇑, Zeshui Xu b, Bin Zhu c a
School of Economics and Management, Tsinghua University, Beijing 100084, China College of Sciences, PLA University of Science and Technology, Nanjing 210007, China c School of Economics and Management, Southeast University, Nanjing 210096, China b
a r t i c l e
i n f o
Article history: Received 23 March 2012 Received in revised form 25 October 2012 Accepted 27 November 2012 Available online 11 December 2012 Keywords: Bonferroni mean Geometric Bonferroni mean Intuitionstic fuzzy set Atanassov’s intuitionistic fuzzy geometric Bonferroni mean Weighted Atanassov’s intuitionistic fuzzy geometric Bonferroni mean Multi-criteria decision making
a b s t r a c t In this paper, we introduce the Bonferroni geometric mean, which is a generalization of the Bonferroni mean and geometric mean and can reflect the correlations of the aggregated arguments. To describe the uncertainty and fuzziness more objectively, intutionistic fuzzy set could be used for considering the membership, non-membership and uncertainty information. To aggregate the Atanassov’s intuitionistic fuzzy information, we further develop the Atanassov’s intuitionistic fuzzy geometric Bonferroni mean describing the interrelationship between arguments, and some properties and special cases of them are also discussed. Moreover, considering the importance of each argument, the weighted Atanassov’s intuitionistic fuzzy geometric Bonferroni mean is proposed and applied to multi-criteria decision making. An example is given to compare the proposed method with the existing ones. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Among the aggregation operators, the average mean (AM) and the geometric mean (GM) are two basic functions, based on which, a lot of extensions have been developed. For example, Yager [27] proposed the ordered weighted averaging (OWA) operator to reordering the arguments before being aggregated, motivated by which, some authors [7,22] investigated the ordered weighted geometric (OWG) operator. For the case that the given argument is a continuous interval valued rather than a finite set of arguments, Yager [29] developed a continuous ordered weighted averaging (C-OWA) operator, and Yager and Xu [32] further developed the continuous ordered weighted geometric (C-OWG) operator. For the linguistic fuzzy information, some aggregation operators were also developed based on the AM and the GM, such as the linguistic weighted averaging (LWA) operator [18], the linguistic ordered weighted averaging (LOWA) operator [16], the linguistic weighted geometric averaging (LWGA) operator [17] and the linguistic ordered weighted geometric averaging (LOWGA) operator [17]. It is noted that the above aggregation operators consider the aggregation arguments interdependent. However, the aggregated ⇑ Corresponding author. E-mail address: [email protected] (M. Xia). 0950-7051/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.knosys.2012.11.013
arguments are correlative, especially in multi-criteria decision making. To overcome this limitation, many aggregation operators have been developed to investigate the correlation among the arguments, Yager [28] introduced the power average (PA) to provide an aggregation operator which allows argument values to support each other in the aggregation process, based on which, Xu and Yager [25] developed a power geometric (PG) operator and its weighted form, developed a power ordered geometric (POG) operator and a power ordered weighted geometric (POWG) operator, and studied some of their properties. Xu [21] extended the PA and applied it to aggregate intuitionstic fuzzy information. Motivated by the Choquet integral [8], Yager [30] introduced the idea of order induced aggregation to the Choquet aggregation operator and defined an induced Choquet ordered averaging operator. Xu [20], Tan and Chen [15] developed some Atanassov’s intuitionistic fuzzy correlated operators based on Choquet integral. The Bonferroni mean (BM) originally introduced by Bonferroni [3] and then generalized by Yager [31]. The desirable characteristic of the BM is its capability to capture the interrelationship between input arguments. Xu [21] further applied the Bonferroni mean to Atanassov’s intuitionistic fuzzy environment and introduced the Atanassov’s intuitionistic fuzzy Bonferroni mean (IFBM) and the weighted Bonferroni mean (WIFBM). In this paper, we introduce a new Bonferroni mean based on the BM and the GM and further
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extend it to Atanassov’s intuitionistic fuzzy environment. To do this, the remainder of this paper is organized as follows. In Section 2, we briefly review some basic concepts and propose the definition of geometric Bonferroni mean (GBM). Section 3 extents the GBM to aggregate the Atanassov’s intuitionistic fuzzy information introducing the Atanassov’s intuitionistic fuzzy geometric Bonferroni mean (IFGBM), whose properties and special cases are also studied. In Section 4, we propose the weighted Atanassov’s intuitionistic fuzzy geometric Bonferroni mean (WIFGBM), and develop a method for multi-criteria decision making. Section 5 gives some concluding remarks. 2. Geometric Bonferroni mean The Bonferroni mean (BM), introduced by Bonferroni [3], is defined as follows: Definition 1 [3]. Let ai(i = 1, 2, . . . , n) be a collection of crisp data, where ai P 0, for all i, and p, q P 0, then we call 1 1pþq
0
C B n C B X 1 B p;q p qC ai aj C B ða1 ; a2 ; . . . ; an Þ ¼ B C Bnðn 1Þ A @ i; j ¼ 1
ð1Þ
i–j the Bonferroni mean (BM). Especially, if q = 0, then by Eq. (1), the BM reduces to the generalized mean operator [10] as follows:
0
1 11pþ0
0
j–i n 1X ap n i¼1 i
!1p ð2Þ
If p = 1 and q = 0, then Eq. (2) reduces to the well-known average mean (AM): n 1X B ða1 ; a2 ; . . . ; an Þ ¼ ai n i¼1 1;0
ð3Þ
Based on the usual geometric mean (GM) and the BM, we introduce the geometric Bonferroni mean such as: Definition 2. Let p, q > 0, and ai(i = 1, 2, . . . , n) be a collection of non-negative numbers. If
GBp;q ða1 ; a2 ; . . . ; an Þ ¼
n Y 1 1 ðpai þ qaj Þnðn1Þ pþq i; j ¼ 1 i–j
(1) (2) (3) (4)
a1 a2 ¼ ðla1 þ la2 la1 la2 ; v a1 v a2 Þ. a1 a2 ¼ ðla1 la2 ; v a1 þ v a2 v a1 v a2 Þ. ka ¼ 1 ð1 la Þk ; v ka ; k > 0. ak ¼ ðlka ; 1 ð1 v a Þk Þ; k > 0.
Moreover, the relations of these operational laws are given as:
a1 a2 = a2 a1. a1 a2 = a2 a1. k(a1 a2) = ka1 ka2. ða1 a2 Þk ¼ ak1 ak2 . k1a k2a = (k1 + k2)a. ak1 ak2 ¼ ak1 þk2 .
To rank any two AIFNs ai ¼ ðlai ; v ai Þ ði ¼ 1; 2Þ, Xu and Yager [24] gave a straightforward method: Definition 4 [24]. Let sai ¼ lai v ai ði ¼ 1; 2Þ be the scores of ai (i = 1,2) respectively, and hai ¼ lai þ v ai ði ¼ 1; 2Þ be the accuracy degrees of ai (i = 1,2) respectively, then If sa1 > sa2 , then a1 is larger than a2, denoted by a1 > a2; If sa1 ¼ sa2 , then (1) If ha1 ¼ ha2 , then a1 and a2 represent the same information, i.e., la1 ¼ la2 and v a1 ¼ v a2 , denoted by a1 = a2; (2) If ha1 > ha2 , then a1 is larger than a2, denoted by a1 > a2.
Definition 5. Let ai ¼ ðlai ; v ai Þ ði ¼ 1; 2; . . . ; nÞ be a collection of AIFNs. For any p, q > 0, if
0
(1) GBp,q(0, 0, . . . , 0) = 0. (2) GBp,q(a, a, . . . , a) = a, if ai = a, for all i. (3) GBp,q(a1, a2, . . . , an) P GBp,q(d1, d2, . . . , dn), i.e., GBp,q is monotonic, if ai P di, for all i. (4) mini{ai} 6 GBp,q(a1, a2, . . . , an) 6 maxi{ai}. Furthermore, if q = 0, then by Eq. (1), it reduces to the geometric mean: n n Y 1 1 1 Y ðpai Þnðn1Þ ¼ ðai Þn p i¼1 i; j ¼ 1 i–j
Definition 3 (24,19). Let ai ¼ ðlai ; v ai Þ ði ¼ 1; 2Þ and a = (la,va) be three AIFNs, then we have
ð4Þ
then we call GBp,q the geometric Bonferroni mean (GBM). Obviously, the GBM has the following properties:
GBp;0 ða1 ; a2 ; . . . ; an Þ ¼
The Atanassov’s intuitionistic fuzzy set (AIFS)Xu [1] (Atanossv, 1986) A = {hx, lA(x), vA(x)ijx 2 X} on the set X with the condition that lA(x), vA(x) P 0 and 0 6 lA(x) + vA(x) 6 1 is a useful tool to express the fuzziness and uncertainty, because that it contains three parts: the membership function lA(x), the non-membership function vA(x) and the hesitant function pA(x) = 1 lA(x) vA(x), which can reflect the decision makers’ preference more objectively. It is noted that the hesitant function pA(x) is determined by the membership function lA(x), the non-membership function vA(x), therefore, we only consider lA(x) and vA(x) in this paper. If the aggregation information in GBM is replaced by the Atanassov’s intuitionistic fuzzy numbers (AIFNs), which is the basic element of AIFS and denoted by a = (la, va), where la, va P 0, la + va 6 1, then we introduce a new aggregation operator in this section. Before doing this, we first introduce some basic operational laws for AIFNs:
(5) (6) (7) (8) (9) (10)
B CC B n n B1 X CC B 1 X B pB p;0 0 CC aj CC B ða1 ; a2 ; . . . ; an Þ ¼ B ai B Bn i¼1 Bðn 1Þ CC @ @ j ¼ 1 AA
¼
3. Atanassov’s intuitionistic fuzzy geometric Bonferroni mean
ð5Þ
1
C B 1 C nðn1Þ n 1 B C B p IFGBp;q ða1 ; a2 ; . . . ; an Þ ¼ a q a i j C p þ qB A @i; j ¼ 1 i–j
ð6Þ
then IFGBp,q is called the Atanassov’s intuitionistic fuzzy geometric Bonferroni mean (IFGBM). Based on the operational laws of the AIFNs given in Definition 3, we can derive the following theorem: Theorem 1. Let p, q > 0, and ai ¼ ðlai ; v ai Þ ði ¼ 1; 2; . . . ; nÞ be a collection of AIFNs, then the aggregated value by using the IFGBM is also an AIFN, and
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0
1 1pþq 0 B C B B n 1 C B nðn1Þ Y B C B IFGBp;q ða1 ; a2 ; . . . ; an Þ ¼B 1 1 ð1 lai Þp ð1 laj Þq C ; B1 B C B B A @ i; j ¼ 1 @ i–j 1 1 1pþq 0 C C B C n B 1 C nðn1Þ Y C C B ð7Þ 1 v pai v qaj C C B1 C C B A C @ i; j ¼ 1 A i–j
i.e., Eq. (7) holds. In addition, since
C B n C B Y 1 C B p q nðn1Þ 0 6 1 B1 ð1 ð1 lai Þ ð1 laj Þ Þ C C B A @ i; j ¼ 1
and then
then we have
pai qaj ¼ 1 ð1 lai Þp þ 1 ð1 laj Þq p
q
ð1 ð1 lai Þ Þð1 ð1 laj Þ Þ; vpai v qaj ¼ 1 ð1 lai Þp ð1 laj Þq ; v pai v qaj
ð9Þ
B C n 1 C 1 B B ðpai qaj Þnðn1Þ C IFGBp;q ða1 ; a2 ; . . . ; an Þ ¼ B C p þ q @i; j ¼ 1 A i–j 0 1 B C 1 C n 1 B B bnðn1Þ C ¼ ij C p þ qB @i; j ¼ 1 A i–j
0
i–j
1
1
n
ð11Þ
i–j
pai qaj i; j ¼ 1 i–j
1 nðn1Þ
B n B Y 1 nðn1Þ B 1 ð1 lai Þp ð1 laj Þq ¼B ;1 B @i; j ¼ 1 i–j
1 ð12Þ
i–j and then by Eq. (12) and the operational law (3), it yields
1 1pþq
C B n C B Y 1 C B p q nðn1Þ ð1 ð1 lai Þ ð1 laj Þ Þ 6 1 B1 C C B A @ i; j ¼ 1 i–j
0
1 1pþq
C B n C B Y 1 C B p q nðn1Þ ð1 ð1 lai Þ ð1 laj Þ Þ þ B1 C C B A @ i; j ¼ 1
i–j 0
1 1pþq
B C B B n C B Y B 1 C B p q nðn1Þ B ¼ B1 B1 ð1 ð1 lai Þ ð1 laj Þ Þ C ; C B B A @ i; j ¼ 1 @ i–j 111 0
ð16Þ
which completes the proof of Theorem 1. h In what follows, we investigate some desirable properties of the IFGBM:
IFGBp;q ða1 ; a2 ; . . . ; an Þ ¼ IFGBp;q ða; a; . . . ; aÞ 0
C C C C C C A
ð13Þ
1
B C n 1 C 1 B B ðpa qaÞnðn1Þ C ¼ B C p þ q @i; j ¼ 1 A i–j 0 1
pþq
C B n B 1 C nðn1Þ Y C B B1 1 v pai v qaj C C B A @ i; j ¼ 1 i–j
¼1
(1) (Idempotency) If all ai (i = 1, 2, . . . , n) are equal, i.e., ai = a = (la, va), for all i, then
n 1 1 ðpai qaj Þnðn1Þ p þ q i; j ¼ 1
0
C B n B 1 C Y B p q nðn1Þ C 1 v ai v aj þ B1 C C B A @ i; j ¼ 1
i–j
C 1 C p q nðn1Þ C 1 v ai v aj C C A i; j ¼ 1 n Y
IFBGp;q ða1 ; a2 ; . . . ; an Þ ¼
1 1pþq
0 i–j
which has been proven by Xu and Yager [24]. Then we replace bij ; lbij and mbij by pai qaj ; 1 ð1 lai Þp ð1 laj Þq and v pai v qaj in Eq. (11), respectively: 0
1 1pþq
C B n C B Y 1 C B ð1 ð1 lai Þp ð1 laj Þq Þnðn1Þ C ¼ 1 B1 C B A @ i; j ¼ 1 0
bijnðn1Þ i; j ¼ 1 i–j
1 1pþq
0 i–j
Since
n
C B n C B Y 1 C B ð1 ð1 lai Þp ð1 laj Þq Þnðn1Þ C 1 B1 C B A @ i; j ¼ 1 C B n B 1 C nðn1Þ Y C B 1 v pai v qaj þ B1 C C B A @ i; j ¼ 1
ð10Þ
C B n n C B Y 1 Y 1 C B nðn1Þ ¼B lbij ; 1 ð1 v bij Þnðn1Þ C C B A @i; j ¼ 1 i; j ¼ 1 i–j i–j
ð15Þ
1 1pþq
0
Let bij ¼ ðlbij ; mbij Þ ¼ pai qaj ¼ 1 ð1 lai Þp ð1 laj Þq ; v pai v qaj , then 0 1
0
61
i–j
ð8Þ
ð14Þ
1 1pþq
0
C B n B 1 C Y B p q nðn1Þ C 0 6 B1 1 v ai v aj C C B A @ i; j ¼ 1
qaj ¼ 1 ð1 laj Þq ; v qaj
61
i–j and
Proof. By the operational laws (1) and (3) described in Definition 3, we have
pai ¼ 1 ð1 lai Þp ; v pai ;
1 1pþq
0
¼
B C n 1 C 1 B B ððp þ qÞaÞnðn1Þ C B C p þ q @i; j ¼ 1 A i–j
nðn1Þ 1 ¼ ððp þ qÞaÞnðn1Þ ¼ a pþq
ð17Þ
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M. Xia et al. / Knowledge-Based Systems 40 (2013) 88–100
(2) (Commutativity) Let ai ¼ ðlai ; v ai Þ ði ¼ 1; 2; . . . ; nÞ be a collection of AIFNs. Then
IFGBp;q ða1 ; a2 ; . . . ; an Þ ¼ IFGBp;q ða_ 1 ; a_ 2 ; . . . ; a_ n Þ
(4) (Boundedness): Let ai ¼ ðlai ; v ai Þ ði ¼ 1; 2; . . . ; nÞ be a collection of AIFNs, and let
a ¼ ðminflai g; maxfv ai gÞ; aþ
ð18Þ
i
where ða_ 1 ; a_ 2 ; . . . ; a_ n Þ is any permutation of (a1, a2, . . . , an). Proof. Since ða_ 1 ; a_ 2 ; . . . ; a_ n Þ is any permutation of (a1, a2, . . . , an), then 1 0
C B n 1 C 1 B B ðpai qaj Þnðn1Þ C IFBGp;q ða1 ; a2 ; . . . ; an Þ ¼ C pþqB A @i; j ¼ 1 i–j 1 0
Case 1. If q ? 0, then by Eq. (7),0we have
IFGB ða1 ; a2 ; . . . ; an Þ 6 IFGB ðb1 ; b2 ; . . . ; bn Þ
i¼1
1
ð20Þ
Proof. Since lai 6 lbi and v ai P v bi , for all i, thenwhich completes the proof. h
1
C B n 1 C 1 B B ðpai qaj Þnðn1Þ C IFGBp;q ða1 ; a2 ; . . . ; an Þ ¼ C p þ qB A @i; j ¼ 1 i–j 1 1 n ¼ ðpai Þn p i¼1 0 !1 n 1 p Y p n @ ¼ 1 1 1 ð1 lai Þ ;
ð19Þ
p;q
ð23Þ
which can be obtained easily by the monotonicity.
(3) (Monotonicity) Let ai ¼ ðlai ; v ai Þ ði ¼ 1; 2; . . . ; nÞ and bi ¼ ðlbi ; v bi Þ ði ¼ 1; 2; . . . ; nÞ be two collections of AIFNs, if lai 6 lbi and v ai P v bi , for all i, then p;q
ð22Þ
i
a 6 IFGBp;q ða1 ; a2 ; . . . ; an Þ 6 aþ
i–j ¼ IFBG ða_ 1 ; a_ 2 ; . . . ; a_ n Þ
i
then
If the values of the parameters p and q change in the IFBGM, then some special cases can be obtained as follows:
C B n 1 C 1 B B ðpa_ i qa_ j Þnðn1Þ C ¼ C pþqB A @i; j ¼ 1 p;q
i
¼ ðmaxflai g; minfv ai gÞ
n Y
1 v ai p
1n
!1p 1 A
i¼1 p;0
ð24Þ ¼ IFBG ða1 ; a2 ; . . . ; an Þ which we call the generalized Atanassov’s intuitionistic fuzzy geomentric mean (GIFGM). Case 2. If p = 2 and q ? 0, then Eq. (7) is transformed as:
2
3 n n 1 nðn1Þ Y Y 1 ð1 ð1 lai Þp ð1 laj Þq Þnðn1Þ 6 1 ð1 lbi Þp ð1 lbj Þq 6 7 6 7 i; j ¼ 1 6 i; j ¼ 1 7 " 6 7 p q p q# ð1 lai Þ ð1 laj Þ P ð1 lbi Þ ð1 lbj Þ 6 i–j 7 i–j 6 7 )6 n p q p q n 1 7 Y Y v a i v a j P v bi v bj 1 6 7 p q nðn1Þ p q nðn1Þ 1 v ai v aj ð1 v bi v bj Þ 6 6 7 6 7 4 i; j ¼ 1 5 i; j ¼ 1 i–j i–j 20 3 1 1 1pþq 0 1pþq 6B 7 C B C 6B n n C B 1 C 7 Y Y 6B 7 1 C B p q nðn1Þ p q nðn1Þ C 6 B1 7 ð1 ð1 1 ð1 l Þ ð1 l Þ Þ P 1 l Þ ð1 l Þ C B C ai aj bi bj 6B C B C 7 6@ 7 A @ A i; j ¼ 1 i; j ¼ 1 6 7 7 )6 6 7 i–j i–j 6 7 n n 1 nðn1Þ 6 7 Y Y 1 p q p q nðn1Þ 61 7 P1 1 v ai v aj ð1 v bi v bj Þ 6 7 6 7 4 5 i; j ¼ 1 i; j ¼ 1 i–j i–j 3 2 1 1 0 1pþq 0 1pþq 6 B B C C 7 6 n n B C B 1 C 7 7 6 Y Y 1 C B p q nðn1Þ p q nðn1Þ C 61 B ð1 ð1 lai Þ ð1 laj Þ Þ 1 ð1 lbi Þ ð1 lbj Þ B1 C 6 1 B1 C 7 6 B C B C 7 6 @ A @ A 7 i; j ¼ 1 i; j ¼ 1 7 6 7 6 7 6 i–j i–j 7 )6 1 1 1pþq 1pþq 0 7 60 7 6 7 6B C C B 7 6B n n C C B 1 1 7 6B Y Y nðn1Þ nðn1Þ C C B p q p q 7 6 B1 1 1 v v P 1 v v C C B ai aj 7 6B bi bj C C B 7 6@ A A @ 5 4 i; j ¼ 1 i; j ¼ 1 i–j i–j 1 1 0 1pþq 0 1pþq B C n B C Y 1 C B ) 1 B1 ð1 ð1 lai Þp ð1 laj Þq Þnðn1Þ C B C @ A i; j ¼ 1 0
i–j
1 1pþq
B C n 1 C B nðn1Þ Y B C 1 ð1 lbi Þp ð1 lbj Þq 6 1 B1 C B C @ A i; j ¼ 1 i–j
B C n B 1 C nðn1Þ Y B C B1 1 v pai v qaj C B C @ A i; j ¼ 1 0
i–j
1 1pþq
B C n 1 C B nðn1Þ Y B C 1 v pbi v qbj B1 C B C @ A i; j ¼ 1 i–j
ð21Þ
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M. Xia et al. / Knowledge-Based Systems 40 (2013) 88–100
IFGB2;0 ða1 ; a2 ; . . . ; an Þ ¼
1 1 n ð2ai Þn 2 i¼1 0
¼ @1 1
Table 1 The Atanassov’s intuitionistic fuzzy decision matrix B.
n 1n Y 1 ð1 lai Þ2
!12 ;
i¼1 n 1n Y 1 v 2ai 1
!12 1 A
ð25Þ
y1 y2 y3 y4 y5
c1
c2
c3
(0.3, 0.4) (0.5, 0.2) (0.4, 0.5) (0.2, 0.6) (0.9, 0.1)
(0.7, 0.2) (0.4, 0.1) (0.7, 0.2) (0.8, 0.1) (0.6, 0.3)
(0.5, 0.3) (0.7, 0.1) (0.4, 0.4) (0.8, 0.2) (0.2, 0.5)
i¼1
which we call the Atanassov’s intuitionistic fuzzy square geometric mean (IFSGM).
Fig. 1. Scores for alternatives obtained by WIFGBM.
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M. Xia et al. / Knowledge-Based Systems 40 (2013) 88–100 n
Case 3. If p = 1 and q ? 0, then Eq. (7) reduces to the Atanassov’s intuitionistic fuzzy geometric [24]:
1
IFGB1;0 ða1 ; a2 ; . . . ; an Þ ¼ ain i¼1
¼
n Y i¼1
Fig. 2. Scores for alternatives obtained by WIFBM.
1
lnai ; 1
n Y i¼1
1
ð1 v ai Þn
!! ð26Þ
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M. Xia et al. / Knowledge-Based Systems 40 (2013) 88–100
Fig. 3. Scores for alternatives obtained by WIFGBM and WIFBM.
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M. Xia et al. / Knowledge-Based Systems 40 (2013) 88–100
0
Case 4. If p = q = 1, then Eq. (7) reduces to the following: 0
1
B C 1 C 1B n IFGB1;1 ða1 ; a2 ; . . . ; an Þ ¼ B ai aj nðn1Þ C B C 2 @i; j ¼ 1 A 0 i–j 0
112 B C B B n C B Y B 1 C B nðn1Þ ¼B 1 1 l Þð1 l ÞÞ ð1 ð1 C; B a a B i j C B B A @ i; j ¼ 1 @ i–j
112 1 CC B n CC B Y 1 C C B nðn1Þ 1 v v Þ ð1 CC B ai aj CC B AC @ i; j ¼ 1 A i–j 0
ð27Þ
which we call the Atanassov’s intuitionistic fuzzy interrelated square geometric mean (IGISGM).
1 0 1pþq B B C B 1 n B C p Y B w q nðn1Þ C B i B 1 1 lw IFBGp;q 1 lajj C ; w ða1 ; a2 ; . . . ; an Þ ¼ B1 B1 ai B C B @ A i; j ¼ 1 @ i–j 1 1 0 1pþq C B C C 1 C n B q nðn1Þ Y B C C wi p wj 1 ð1 ð1 v ai Þ Þ 1 ð1 v aj Þ B1 C C B C C @ A C i; j ¼ 1 A i–j
ð29Þ Based on Definition 6 and Theorem 2, we develop an approach for multi-criteria decision making under Atanassov’s intuitionistic fuzzy environment:
Step 1. Transform the decision matrix B = (bij)mn into the normalization matrix R = (rij)mn by the method given by Xu and Hu [23], where ( rij ¼ ðtij ; fij Þ ¼
4. The weighted Atanassov’s intuitionistic fuzzy geometric Bonferroni mean and its application in multi-criteria decision making For a multi-criteria decision making problem, let Y = {y1, y2, . . . , ym} be a set of m alternatives, C = {c1, c2 . . . , cn} be a set of n criteria, whose weight vector is w = (w1, w2, . . . , wn)T, satisfying P wi > 0 (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1. The performance of the alternative yi with respect to the criterion cj is measured by an AIFN bij = (lij, vij), where lij indicates the degree that the alternative yi satisfies the criterion cj, vij indicates the degree that the alternative yi does not satisfy the criterion cj, such that lij 2 [0, 1],vij 2 [0, 1], lij + vij 6 1, and all bij = (lij,vij) (i = 1, 2, . . . , m; j = 1, 2, . . . , n) are contained in the Atanassov’s intuitionistic fuzzy decision matrix B = (bij)mn. To get the priority of the alternatives, we should aggregate the performance of each alternative, however, it is noted that the IFGBM proposed in Section 3 does not consider the importance of the aggregated arguments, but in many practical problem, especially in multi-criteria decision making, the weight vector of the criteria is an important part in the aggregation, to avoid this issue, we introduce the following definition: Definition 6. Let ai ¼ ðlai ; v ai Þ ði ¼ 1; 2; . . . ; nÞ be a collection of AIFNs, w = (w1, w2, . . . , wn)T is the weight vector of ai (i = 1, 2, . . . , n), where wi indicates the importance degree of ai, P satisfying wi > 0 ði ¼ 1; 2; . . . ; nÞ; ni¼1 wi ¼ 1. If
0
1
B 1 C n 1 B wj nðn1Þ C wi C; B p IFGBp;q ða1 ; a2 ; . . . ; an Þ ¼ a q a i j C pþqB A @i; j ¼ 1 i–j
p; q
>0
bij ; for benefit criterion cj ij ; for cos t criterion cj ; b
i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n
ð30Þ is the complement of b , such that b ¼ ðv ; l Þ. where b ij ij ij ij ij Step 2. Aggregate all the performance values rij (j = 1, 2, . . ., n) of the ith line, and get the overall performance value ri corresponding to the alternative yi by the WIFGBM: r i ¼ ðti ; fi Þ ¼ WIFGBp;q w ðr i1 ; r i2 ; . . . ; r in Þ 0 1 0 1pþq B B C B n B 1 C Y B w p B w q nðn1Þ C 1 1 lij j 1 likk ¼B 1 C ; B1 B B C B @ A j; k ¼ 1 @ j–k 1 1 0 1pþq C B C C n 1 C B Y p q nðn1Þ B C C 1 1 ð1 v ij Þwj 1 ð1 v ik Þwk B1 C C B C C @ A C j; k ¼ 1 A j–k
ð31Þ
where p, q > 0. Step 3. Rank the overall performance values ri (i = 1, 2, . . . , m) according to Definition 3 and obtain the priority of the alternatives yi (i = 1, 2, . . . , m) according to ri (i = 1, 2, . . . , m). Especially, if we do not consider the non-membership information in Atanassov’s intuitionistic fuzzy decision making, then the usual fuzzy decision making method can be obtained as follows: The performance of the alternative yi with respect to the criterion cj is measured by a usual fuzzy number bij, where 0 6 bij 6 1, and all the values, bij = (lij,vij) (i = 1, 2, . . . , m; j = 1, 2, . . . , n), are contained in the fuzzy decision matrix B = (bij)mn. Then we can use the GBM or the BM to solve this problem:
ð28Þ IFBp;q w
then is called the weighted Atanassov’s intuitionistic fuzzy geometric Bonferroni mean (WIFGBM). Similar to Theorem 1, we have. Theorem 2. Let ai ¼ ðlai ; v ai Þ ði ¼ 1; 2; . . . ; nÞ be a collection of AIFNs, whose weight vector is w = (w1, w2, . . . , wn)T, which satisfies P wi > 0 (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1. Then the aggregated value by using the WIFGBM is also an AIFN, and
Table 2 The fuzzy decision matrix B.
y1 y2 y3 y4 y5
c1
c2
c3
0.3 0.5 0.4 0.2 0.9
0.7 0.4 0.7 0.8 0.6
0.5 0.7 0.4 0.8 0.2
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Step 1. Transform the decision matrix B = (bij)mn into the normalization matrix R = (rij)mn, where r ij ¼
for benefit criterion cj bij ; ; 1 bij ; for cos t criterion cj
i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n
ð32Þ
Step 2. Aggregate all the performance values rij (j = 1, 2, . . . , n) of the ith line, and get the overall performance value ri corresponding to the alternative yi by the GBM or the BM:
Fig. 4. Scores for alternatives obtained by GBM.
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ri ¼ GBp;q w ðr i1 ; r i2 ; . . . ; r in Þ ¼
1 pþq
n Y
j; k ¼ 1 j–k
1
ðpr ij þ qrik Þnðn1Þ
ð33Þ
1 1pþq
0
C B n C B X 1 B p q C p;q r i ¼ Bw ðri1 ; r i2 ; . . . ; r in Þ ¼ B rij rikj C C Bnðn 1Þ A @ j; k ¼ 1
or
j–k where p, q > 0.
Fig. 5. Scores for alternatives obtained by BM.
ð34Þ
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Step 3. Rank the overall performance values ri (i = 1, 2, . . . , m) and obtain the priority of the alternatives yi (i = 1, 2, . . . , m). Next, we give an example to illustrate the proposed method:
Example 1. A city is planning to build a municipal library. One of the problems facing the city development commissioner is to determine what kind of air-conditioning system should be installed in the library (adapted from Ref. [26]). The contractor offers five
Fig. 6. Scores for alternatives obtained by GBM and BM.
M. Xia et al. / Knowledge-Based Systems 40 (2013) 88–100
feasible alternatives yi (i = 1, 2, 3, 4, 5), which might be adapted to the physical structure of the library. Suppose that three criteria: (1) c1: economic, (2) c2: functional, and (3) c3: operational, are taken into consideration in the installation problem, the weight vector of the criteria cj (j = 1, 2, 3) is w = (0.3, 0.5, 0.2)T. Assume that the characteristics of the alternatives yi (i = 1, 2, 3, 4, 5) with respect to the criteria cj (j = 1, 2, 3) are represented by the AIFNs bij = (lij, vij), and all bij (i = 1, 2, 3, 4, 5; j = 1, 2, 3) are contained in the Atanassov’s intuitionistic fuzzy decision matrix B = (bij)53 (see Table 1). Step 1. Considering all the criteria cj (j = 1, 2, 3) are the benefit criteria, the performance values of the alternatives yi (i = 1, 2, 3, 4, 5) do not need normalization. Step 2. Utilize the WIFGBM to aggregate all the performance values bij (j = 1, 2, 3) of the ith line, and get the overall performance value bi corresponding to the alternative yi: b1 ¼ ð0:8084; 0:1034Þ; b4 ¼ ð0:8561; 0:0915Þ;
b2 ¼ ð0:8101; 0:0438Þ; b5 ¼ ð0:8381; 0:1006Þ
b3 ¼ ð0:8112; 0:1269Þ
Step 3. Calculate the scores of all the alternatives:
sb1 ¼ 0:7050;
sb2 ¼ 0:7664;
¼ 0:7647;
sb5 ¼ 0:7375
sb3 ¼ 0:6843;
sb4
Since sb2 > sb4 > sb5 > sb1 > sb3 , then the ranking of the alternatives yi (i = 1, 2, 3, 4, 5) is:
y2 y4 y5 y1 y3 If we take p = q = 2, then by bi ¼ ðli ; v i Þ ¼ IFGB2;2 w ðbi1 ; r i2 ; bi3 Þ, we get b1 ¼ ð0:8039; 0:1056Þ;
b2 ¼ ð0:7924; 0:0465Þ;
b4 ¼ ð0:8406; 0:0971Þ;
b5 ¼ ð0:8092; 0:1130Þ
b3 ¼ ð0:8100; 0:1290Þ
Then we calculate the scores of all the alternatives:
sb1 ¼ 0:6982;
sb2 ¼ 0:7459;
sb3 ¼ 0:6810;
sb4 ¼ 0:7435;
sb5 ¼ 0:6962 Since sb2 > sb4 > sb1 > sb5 > sb3 , then we have
y2 y4 y1 y5 y3 We can find that as the values of the parameters p and q change according to the decision maker’s subjective preferences, the rankings of the alternatives are slightly different, which can reflect the decision maker’s risk preferences. If we use the weighted Atanassov’s intuitionistic fuzzy Bonferroni mean (WIFBM) given by Xu and Yager [26] to aggregate the alternative performances, different results can be obtained. To give a detail comparison, we express the scores of alternatives by Figs. 1 and 2 as the parameters p and q change between 0 and 10. Fig. 1 describes the scores of alternatives obtained by our method, and Fig. 2 describes the ones obtained by Xu and Yager’s method. It is noted that most of the scores obtained by our method are bigger than 0 and most of the ones obtained by Xu and Yager’s method are smaller than 0, which indicates that our method can obtain more favorable (or optimistic) expectations, while the one given by Xu and Yager has more unfavorable (or pessimistic) expectations. It should be noted that concepts of optimism and pessimistic are the same as those defined in Chen [5,6]. Therefore, we can conclude that the WIFGBM can be considered as the optimistic one, while the WIFBM [26] can be considered as the pessimistic one and the values of the parameters can be considered as the optimistic or pessimistic levels. In order to obtain the more neutral results, we can use the arithmetic averages of the optimistic and pessimistic results, which can be found in Fig. 3.
99
If we only consider the membership of the Atanassov’s intuitionistic fuzzy set, then the Atanassov’s intuitionistic fuzzy set reduces to the usual fuzzy set, and the decision matrix in Example 1 reduces to that in the usual fuzzy decision making problem illustrated in Table 2. If we use the GBM or the BM to obtain the performance of each alternative, as the values of the parameters change, the performances of the alternatives can be illustrated by using the three-dimensional diagrams, see Figs. 4 and 5 for more details. Comparing Figs. 4 and 5, we can find that most of the values in Fig. 4 are smaller than those in Fig. 5, which implies that the GBM operator can obtain more pessimistic results, while the BM can obtain the more optimistic results. If we use the arithmetic average of the results obtained by these two results, more neutral results can be given in Fig. 6. Comparing Figs. 1 and 4, Figs. 2 and 5, Figs. 3 and 6, we can find that there is a big difference between each pair of figures, that is because that the former of each pair figures considers the non-membership information, while the latter does not, which illustrates that the non-membership information plays an important part in the overall performances of the alternatives. However, the WIFGBM is not a simple extension of the GBM, and the WIFBM is not a simple extension of t the BM, the operations for Atanassov’s intuitionistic fuzzy sets are very different from the ones of fuzzy sets, that is because the specific characteristic of intuitionistic fuzzy sets, each of which include two parts of information, the membership information and the non-membership information. In addition, the GBM and the BM do not consider the weights of the criteria, but in many practical problems, especially in multi-criteria decision making, the weight vector of the criteria is an important part in the aggregation. Therefore, it is necessary to introduce the WIFGBM and the WIFBM for Atanassov’s intuitionistic fuzzy sets and fuzzy sets, respectively. One aim of the proposed method is to provide more choices for the decision makers, and as for the choice of the parameter, we should fully respect the opinions of the decision makers. Maybe, as the values of the parameters change, the rankings of the alternatives may be different, but we do not say which is the most appropriate order, it just reflects the preferences of the decision makers, and depends on the optimistic or pessimistic nature of the decision makers, which has been investigated by many researchers [11,12,6]. Chen [4] denoted that ‘‘the specification of parameters can be assessed using the Life Orientation Test (LOT) devised by Sanz et al. [13], the Revised Life Orientation Test (LOT-R) devised by Scheier et al. [14], or the Extended Life Orientation Test (ELOT) devised by [3]Chang et al. (1997). These three scales have strong predictive and discriminative validity [9]. The values of the parameters and are set according to the normalized LOT score’’. Because the appropriateness of the optimistic and pessimistic point operators is generally context-dependent, we suggest that empirical studies be conducted in the future to determine parameter settings in the context of various specific applications [6].
5. Concluding remarks In this paper, we have applied the well-known geometric mean (GM) to Bonferroni mean (BM), and introduced the geometric Bonferroni mean (GBM), whose properties have been studied in detail. To aggregate the Atanassov’s intuitionistic fuzzy information, we have introduced the Atanassov’s intuitionistic fuzzy geometric Bonferroni mean (IFGBM), and discussed its special cases when the parameters are assigned different values. Considering the weight vector of the arguments, we have further developed the weighted Atanassov’s intuitionistic fuzzy geometric Bonferroni
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mean (IFGBM), and applied it to multi-criteria decision making. The proposed aggregation operators in this paper can reflect the interrelationship of the aggregated argument [31] and can provide the decision makers more choices by changing the values of the parameters determined by the preferences of the decision makers which is an issue to be further studied in the future. In the end, we compare the proposed aggregation method with the ones given by Xu and Yager [26], and find that our operator can give more optimistic results, while Xu and Yager’s can give more pessimistic ones. In the future, we will study the geometric Bonferroni mean under interval-valued Atanassov’s intuitionistic fuzzy environment [2]. It should be noted that the motivation of this paper is to investigate the Bonferroni means under intuitionistic fuzzy environment combing the geometric mean, other means can also be extended similarly. Bonferroni means was introduced in 1950, and have been investigated by many authors, although other extensions have been developed in the last decades, the main focus of them are invariable, that is to capture the expressed interrelationship between the criteria [31]. Our aim is to apply this core of Bonferroni mean to the intuitionistic fuzzy environment combing geometric mean, based on which, we can also investigate more general class of means including it similarly. We can find that the proposed formulas seem a little complex, and several parameters are provided for the decision makers, if we introduce other more general class of means including Boferroni means, more complex formulas and more parameters will be given, which will make the decision making too complex and too many parameters will make the decision makers be dazzled and they may do not know how to chose it. A saying may be suitable to express this situation, that is ‘‘too much is as bad as too little’’.
Acknowledgements The authors are very grateful to the anonymous reviewers for their insightful and constructive comments and suggestions that have led to an improved version of this paper. The work was supported in part by the National Natural Science Foundation of China (Nos. 71071161 and 61273209) and the China Postdoctoral Science Foundation (No. 2012M520311).
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[6] T.Y. Chen, Comparative analysis of SAW and TOPSIS based on interval-valued fuzzy sets: discussions on score functions and weight constraints, Expert Systems with Applications 39 (2012) 1848–1861. [7] F. Chiclana, F. Herrera, E. Herrera-Viedma, The ordered weighted geometric operator: properties and application, in: Proceedings of the Eighth International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Madrid, Spain, 2000, pp. 985–991. [8] G. Choquet, Theory of capacities, Annales de l’Institut Fourier (Crenoble) 5 (1953) 131–295. [9] F. Conway, C. Magai, C. Springer, S.C. Jones, Optimism and pessimism as predictors of physical and psychological health among grandmothers raising their grandchildren, Journal of Research in Personality 42 (2008) 1352–1357. [10] H. Dyckhoff, W. Pedrycz, Generalized means as model of compensative connectives, Fuzzy Sets and Systems 14 (1984) 143–154. [11] H.W. Liu, G.J. Wang, Multi-criteria decision making methods based on intuitionistic fuzzy sets, European Journal of Operations Research 197 (2007) 220–233. [12] J.M. Merigó, A.M. Gil-Lafuente, The induced generalized OWA operator, Information Sciences 179 (2009) 729–741. [13] J.A. Sanz, A. Fernandez, H. Bustince, F. Herrera, Improving the performance of fuzzy rule-based classification systems with interval-valued fuzzy sets and genetic amplitude tuning, Information Sciences 180 (2010) 3674–3685. [14] M.F. Scheier, C.S. Carver, M.W. Bridges, Distinguishing optimism from neuroticism (and trait anxiety, self-mastery, and self-esteem): a reevaluation of the life orientation test, Journal of Personality and Social Psychology 67 (1994) 1063–1078. [15] C.Q. Tan, X.H. Chen, Intuitionistic fuzzy Choquet integral operator for multicriteria decisionmaking, Expert Systems with Applications 37 (2010) 149–157. [16] Z.S. Xu, EOWA and EOWGoperators for aggregating linguistic labels based on linguistic reference relations, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 12 (2004) 791–810. [17] Z.S. Xu, A method based on linguistic aggregation operators for group decision making, Information Sciences 166 (2004) 19–30. [18] Z.S. Xu, On generalized induced linguistic aggregation operators, International Journal of General Systems 35 (2006) 17–28. [19] Z.S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems 15 (2007) 1179–1187. [20] Z.S. Xu, Choquet integrals of weighted intuitionistic fuzzy information, Information Sciences 180 (2010) 726–736. [21] Z.S. Xu, Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators, Knowledge-Based Systems (2011) doi:10.1016/j.knosys.2011.01.011. [22] Z.S. Xu, Q.L. Da, The ordered weighted geometric averaging operators, International Journal of Intelligent Systems 17 (2002) 709–716. [23] Z.S. Xu, H. Hu, Projection models for intuitionistic fuzzy multiple attribute decision making, International Journal of Information Technology and Decision Making 9 (2010) 267–280. [24] Z.S. Xu, R.R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General Systems 35 (2006) 417–433. [25] Z.S. Xu, R.R. Yager, Power-geometric operators and their use in group decision making, IEEE Transactions on Fuzzy Systems 18 (2010) 94–105. [26] Z.S. Xu, R.R. Yager, Intuitionistic fuzzy Bonferroni means, IEEE Transactions on Systems, Man and Cybernetics 41 (2011) 568–578. [27] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Transactions on Systems, Man, and Cybernetics 18 (1988) 183–190. [28] R.R. Yager, The power average operator, IEEE Transactions on Systems, Man and Cybernetics 31 (2001) 724–731. [29] R.R. Yager, OWA aggregation over a continuous interval argument with applications to decision making, IEEE Transactions on Systems, Man, and Cybernetics 34 (2004) 1952–1963. [30] R.R. Yager, Choquet aggregation using order inducing variables, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 12 (2004) 69–88. [31] R.R. Yager, On generalized Bonferroni mean operators for multi-criteria aggregation, International Journal of Approximate Reasoning 50 (2009) 1279–1286. [32] R.R. Yager, Z.S. Xu, The continuous ordered weighted geometric operator and its application to decision making, Fuzzy Sets and Systems 157 (2006) 1393– 1402.
Contents lists available at SciVerse ScienceDirect
Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
Geometric Bonferroni means with their application in multi-criteria decision making Meimei Xia a,⇑, Zeshui Xu b, Bin Zhu c a
School of Economics and Management, Tsinghua University, Beijing 100084, China College of Sciences, PLA University of Science and Technology, Nanjing 210007, China c School of Economics and Management, Southeast University, Nanjing 210096, China b
a r t i c l e
i n f o
Article history: Received 23 March 2012 Received in revised form 25 October 2012 Accepted 27 November 2012 Available online 11 December 2012 Keywords: Bonferroni mean Geometric Bonferroni mean Intuitionstic fuzzy set Atanassov’s intuitionistic fuzzy geometric Bonferroni mean Weighted Atanassov’s intuitionistic fuzzy geometric Bonferroni mean Multi-criteria decision making
a b s t r a c t In this paper, we introduce the Bonferroni geometric mean, which is a generalization of the Bonferroni mean and geometric mean and can reflect the correlations of the aggregated arguments. To describe the uncertainty and fuzziness more objectively, intutionistic fuzzy set could be used for considering the membership, non-membership and uncertainty information. To aggregate the Atanassov’s intuitionistic fuzzy information, we further develop the Atanassov’s intuitionistic fuzzy geometric Bonferroni mean describing the interrelationship between arguments, and some properties and special cases of them are also discussed. Moreover, considering the importance of each argument, the weighted Atanassov’s intuitionistic fuzzy geometric Bonferroni mean is proposed and applied to multi-criteria decision making. An example is given to compare the proposed method with the existing ones. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Among the aggregation operators, the average mean (AM) and the geometric mean (GM) are two basic functions, based on which, a lot of extensions have been developed. For example, Yager [27] proposed the ordered weighted averaging (OWA) operator to reordering the arguments before being aggregated, motivated by which, some authors [7,22] investigated the ordered weighted geometric (OWG) operator. For the case that the given argument is a continuous interval valued rather than a finite set of arguments, Yager [29] developed a continuous ordered weighted averaging (C-OWA) operator, and Yager and Xu [32] further developed the continuous ordered weighted geometric (C-OWG) operator. For the linguistic fuzzy information, some aggregation operators were also developed based on the AM and the GM, such as the linguistic weighted averaging (LWA) operator [18], the linguistic ordered weighted averaging (LOWA) operator [16], the linguistic weighted geometric averaging (LWGA) operator [17] and the linguistic ordered weighted geometric averaging (LOWGA) operator [17]. It is noted that the above aggregation operators consider the aggregation arguments interdependent. However, the aggregated ⇑ Corresponding author. E-mail address: [email protected] (M. Xia). 0950-7051/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.knosys.2012.11.013
arguments are correlative, especially in multi-criteria decision making. To overcome this limitation, many aggregation operators have been developed to investigate the correlation among the arguments, Yager [28] introduced the power average (PA) to provide an aggregation operator which allows argument values to support each other in the aggregation process, based on which, Xu and Yager [25] developed a power geometric (PG) operator and its weighted form, developed a power ordered geometric (POG) operator and a power ordered weighted geometric (POWG) operator, and studied some of their properties. Xu [21] extended the PA and applied it to aggregate intuitionstic fuzzy information. Motivated by the Choquet integral [8], Yager [30] introduced the idea of order induced aggregation to the Choquet aggregation operator and defined an induced Choquet ordered averaging operator. Xu [20], Tan and Chen [15] developed some Atanassov’s intuitionistic fuzzy correlated operators based on Choquet integral. The Bonferroni mean (BM) originally introduced by Bonferroni [3] and then generalized by Yager [31]. The desirable characteristic of the BM is its capability to capture the interrelationship between input arguments. Xu [21] further applied the Bonferroni mean to Atanassov’s intuitionistic fuzzy environment and introduced the Atanassov’s intuitionistic fuzzy Bonferroni mean (IFBM) and the weighted Bonferroni mean (WIFBM). In this paper, we introduce a new Bonferroni mean based on the BM and the GM and further
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extend it to Atanassov’s intuitionistic fuzzy environment. To do this, the remainder of this paper is organized as follows. In Section 2, we briefly review some basic concepts and propose the definition of geometric Bonferroni mean (GBM). Section 3 extents the GBM to aggregate the Atanassov’s intuitionistic fuzzy information introducing the Atanassov’s intuitionistic fuzzy geometric Bonferroni mean (IFGBM), whose properties and special cases are also studied. In Section 4, we propose the weighted Atanassov’s intuitionistic fuzzy geometric Bonferroni mean (WIFGBM), and develop a method for multi-criteria decision making. Section 5 gives some concluding remarks. 2. Geometric Bonferroni mean The Bonferroni mean (BM), introduced by Bonferroni [3], is defined as follows: Definition 1 [3]. Let ai(i = 1, 2, . . . , n) be a collection of crisp data, where ai P 0, for all i, and p, q P 0, then we call 1 1pþq
0
C B n C B X 1 B p;q p qC ai aj C B ða1 ; a2 ; . . . ; an Þ ¼ B C Bnðn 1Þ A @ i; j ¼ 1
ð1Þ
i–j the Bonferroni mean (BM). Especially, if q = 0, then by Eq. (1), the BM reduces to the generalized mean operator [10] as follows:
0
1 11pþ0
0
j–i n 1X ap n i¼1 i
!1p ð2Þ
If p = 1 and q = 0, then Eq. (2) reduces to the well-known average mean (AM): n 1X B ða1 ; a2 ; . . . ; an Þ ¼ ai n i¼1 1;0
ð3Þ
Based on the usual geometric mean (GM) and the BM, we introduce the geometric Bonferroni mean such as: Definition 2. Let p, q > 0, and ai(i = 1, 2, . . . , n) be a collection of non-negative numbers. If
GBp;q ða1 ; a2 ; . . . ; an Þ ¼
n Y 1 1 ðpai þ qaj Þnðn1Þ pþq i; j ¼ 1 i–j
(1) (2) (3) (4)
a1 a2 ¼ ðla1 þ la2 la1 la2 ; v a1 v a2 Þ. a1 a2 ¼ ðla1 la2 ; v a1 þ v a2 v a1 v a2 Þ. ka ¼ 1 ð1 la Þk ; v ka ; k > 0. ak ¼ ðlka ; 1 ð1 v a Þk Þ; k > 0.
Moreover, the relations of these operational laws are given as:
a1 a2 = a2 a1. a1 a2 = a2 a1. k(a1 a2) = ka1 ka2. ða1 a2 Þk ¼ ak1 ak2 . k1a k2a = (k1 + k2)a. ak1 ak2 ¼ ak1 þk2 .
To rank any two AIFNs ai ¼ ðlai ; v ai Þ ði ¼ 1; 2Þ, Xu and Yager [24] gave a straightforward method: Definition 4 [24]. Let sai ¼ lai v ai ði ¼ 1; 2Þ be the scores of ai (i = 1,2) respectively, and hai ¼ lai þ v ai ði ¼ 1; 2Þ be the accuracy degrees of ai (i = 1,2) respectively, then If sa1 > sa2 , then a1 is larger than a2, denoted by a1 > a2; If sa1 ¼ sa2 , then (1) If ha1 ¼ ha2 , then a1 and a2 represent the same information, i.e., la1 ¼ la2 and v a1 ¼ v a2 , denoted by a1 = a2; (2) If ha1 > ha2 , then a1 is larger than a2, denoted by a1 > a2.
Definition 5. Let ai ¼ ðlai ; v ai Þ ði ¼ 1; 2; . . . ; nÞ be a collection of AIFNs. For any p, q > 0, if
0
(1) GBp,q(0, 0, . . . , 0) = 0. (2) GBp,q(a, a, . . . , a) = a, if ai = a, for all i. (3) GBp,q(a1, a2, . . . , an) P GBp,q(d1, d2, . . . , dn), i.e., GBp,q is monotonic, if ai P di, for all i. (4) mini{ai} 6 GBp,q(a1, a2, . . . , an) 6 maxi{ai}. Furthermore, if q = 0, then by Eq. (1), it reduces to the geometric mean: n n Y 1 1 1 Y ðpai Þnðn1Þ ¼ ðai Þn p i¼1 i; j ¼ 1 i–j
Definition 3 (24,19). Let ai ¼ ðlai ; v ai Þ ði ¼ 1; 2Þ and a = (la,va) be three AIFNs, then we have
ð4Þ
then we call GBp,q the geometric Bonferroni mean (GBM). Obviously, the GBM has the following properties:
GBp;0 ða1 ; a2 ; . . . ; an Þ ¼
The Atanassov’s intuitionistic fuzzy set (AIFS)Xu [1] (Atanossv, 1986) A = {hx, lA(x), vA(x)ijx 2 X} on the set X with the condition that lA(x), vA(x) P 0 and 0 6 lA(x) + vA(x) 6 1 is a useful tool to express the fuzziness and uncertainty, because that it contains three parts: the membership function lA(x), the non-membership function vA(x) and the hesitant function pA(x) = 1 lA(x) vA(x), which can reflect the decision makers’ preference more objectively. It is noted that the hesitant function pA(x) is determined by the membership function lA(x), the non-membership function vA(x), therefore, we only consider lA(x) and vA(x) in this paper. If the aggregation information in GBM is replaced by the Atanassov’s intuitionistic fuzzy numbers (AIFNs), which is the basic element of AIFS and denoted by a = (la, va), where la, va P 0, la + va 6 1, then we introduce a new aggregation operator in this section. Before doing this, we first introduce some basic operational laws for AIFNs:
(5) (6) (7) (8) (9) (10)
B CC B n n B1 X CC B 1 X B pB p;0 0 CC aj CC B ða1 ; a2 ; . . . ; an Þ ¼ B ai B Bn i¼1 Bðn 1Þ CC @ @ j ¼ 1 AA
¼
3. Atanassov’s intuitionistic fuzzy geometric Bonferroni mean
ð5Þ
1
C B 1 C nðn1Þ n 1 B C B p IFGBp;q ða1 ; a2 ; . . . ; an Þ ¼ a q a i j C p þ qB A @i; j ¼ 1 i–j
ð6Þ
then IFGBp,q is called the Atanassov’s intuitionistic fuzzy geometric Bonferroni mean (IFGBM). Based on the operational laws of the AIFNs given in Definition 3, we can derive the following theorem: Theorem 1. Let p, q > 0, and ai ¼ ðlai ; v ai Þ ði ¼ 1; 2; . . . ; nÞ be a collection of AIFNs, then the aggregated value by using the IFGBM is also an AIFN, and
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0
1 1pþq 0 B C B B n 1 C B nðn1Þ Y B C B IFGBp;q ða1 ; a2 ; . . . ; an Þ ¼B 1 1 ð1 lai Þp ð1 laj Þq C ; B1 B C B B A @ i; j ¼ 1 @ i–j 1 1 1pþq 0 C C B C n B 1 C nðn1Þ Y C C B ð7Þ 1 v pai v qaj C C B1 C C B A C @ i; j ¼ 1 A i–j
i.e., Eq. (7) holds. In addition, since
C B n C B Y 1 C B p q nðn1Þ 0 6 1 B1 ð1 ð1 lai Þ ð1 laj Þ Þ C C B A @ i; j ¼ 1
and then
then we have
pai qaj ¼ 1 ð1 lai Þp þ 1 ð1 laj Þq p
q
ð1 ð1 lai Þ Þð1 ð1 laj Þ Þ; vpai v qaj ¼ 1 ð1 lai Þp ð1 laj Þq ; v pai v qaj
ð9Þ
B C n 1 C 1 B B ðpai qaj Þnðn1Þ C IFGBp;q ða1 ; a2 ; . . . ; an Þ ¼ B C p þ q @i; j ¼ 1 A i–j 0 1 B C 1 C n 1 B B bnðn1Þ C ¼ ij C p þ qB @i; j ¼ 1 A i–j
0
i–j
1
1
n
ð11Þ
i–j
pai qaj i; j ¼ 1 i–j
1 nðn1Þ
B n B Y 1 nðn1Þ B 1 ð1 lai Þp ð1 laj Þq ¼B ;1 B @i; j ¼ 1 i–j
1 ð12Þ
i–j and then by Eq. (12) and the operational law (3), it yields
1 1pþq
C B n C B Y 1 C B p q nðn1Þ ð1 ð1 lai Þ ð1 laj Þ Þ 6 1 B1 C C B A @ i; j ¼ 1 i–j
0
1 1pþq
C B n C B Y 1 C B p q nðn1Þ ð1 ð1 lai Þ ð1 laj Þ Þ þ B1 C C B A @ i; j ¼ 1
i–j 0
1 1pþq
B C B B n C B Y B 1 C B p q nðn1Þ B ¼ B1 B1 ð1 ð1 lai Þ ð1 laj Þ Þ C ; C B B A @ i; j ¼ 1 @ i–j 111 0
ð16Þ
which completes the proof of Theorem 1. h In what follows, we investigate some desirable properties of the IFGBM:
IFGBp;q ða1 ; a2 ; . . . ; an Þ ¼ IFGBp;q ða; a; . . . ; aÞ 0
C C C C C C A
ð13Þ
1
B C n 1 C 1 B B ðpa qaÞnðn1Þ C ¼ B C p þ q @i; j ¼ 1 A i–j 0 1
pþq
C B n B 1 C nðn1Þ Y C B B1 1 v pai v qaj C C B A @ i; j ¼ 1 i–j
¼1
(1) (Idempotency) If all ai (i = 1, 2, . . . , n) are equal, i.e., ai = a = (la, va), for all i, then
n 1 1 ðpai qaj Þnðn1Þ p þ q i; j ¼ 1
0
C B n B 1 C Y B p q nðn1Þ C 1 v ai v aj þ B1 C C B A @ i; j ¼ 1
i–j
C 1 C p q nðn1Þ C 1 v ai v aj C C A i; j ¼ 1 n Y
IFBGp;q ða1 ; a2 ; . . . ; an Þ ¼
1 1pþq
0 i–j
which has been proven by Xu and Yager [24]. Then we replace bij ; lbij and mbij by pai qaj ; 1 ð1 lai Þp ð1 laj Þq and v pai v qaj in Eq. (11), respectively: 0
1 1pþq
C B n C B Y 1 C B ð1 ð1 lai Þp ð1 laj Þq Þnðn1Þ C ¼ 1 B1 C B A @ i; j ¼ 1 0
bijnðn1Þ i; j ¼ 1 i–j
1 1pþq
0 i–j
Since
n
C B n C B Y 1 C B ð1 ð1 lai Þp ð1 laj Þq Þnðn1Þ C 1 B1 C B A @ i; j ¼ 1 C B n B 1 C nðn1Þ Y C B 1 v pai v qaj þ B1 C C B A @ i; j ¼ 1
ð10Þ
C B n n C B Y 1 Y 1 C B nðn1Þ ¼B lbij ; 1 ð1 v bij Þnðn1Þ C C B A @i; j ¼ 1 i; j ¼ 1 i–j i–j
ð15Þ
1 1pþq
0
Let bij ¼ ðlbij ; mbij Þ ¼ pai qaj ¼ 1 ð1 lai Þp ð1 laj Þq ; v pai v qaj , then 0 1
0
61
i–j
ð8Þ
ð14Þ
1 1pþq
0
C B n B 1 C Y B p q nðn1Þ C 0 6 B1 1 v ai v aj C C B A @ i; j ¼ 1
qaj ¼ 1 ð1 laj Þq ; v qaj
61
i–j and
Proof. By the operational laws (1) and (3) described in Definition 3, we have
pai ¼ 1 ð1 lai Þp ; v pai ;
1 1pþq
0
¼
B C n 1 C 1 B B ððp þ qÞaÞnðn1Þ C B C p þ q @i; j ¼ 1 A i–j
nðn1Þ 1 ¼ ððp þ qÞaÞnðn1Þ ¼ a pþq
ð17Þ
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(2) (Commutativity) Let ai ¼ ðlai ; v ai Þ ði ¼ 1; 2; . . . ; nÞ be a collection of AIFNs. Then
IFGBp;q ða1 ; a2 ; . . . ; an Þ ¼ IFGBp;q ða_ 1 ; a_ 2 ; . . . ; a_ n Þ
(4) (Boundedness): Let ai ¼ ðlai ; v ai Þ ði ¼ 1; 2; . . . ; nÞ be a collection of AIFNs, and let
a ¼ ðminflai g; maxfv ai gÞ; aþ
ð18Þ
i
where ða_ 1 ; a_ 2 ; . . . ; a_ n Þ is any permutation of (a1, a2, . . . , an). Proof. Since ða_ 1 ; a_ 2 ; . . . ; a_ n Þ is any permutation of (a1, a2, . . . , an), then 1 0
C B n 1 C 1 B B ðpai qaj Þnðn1Þ C IFBGp;q ða1 ; a2 ; . . . ; an Þ ¼ C pþqB A @i; j ¼ 1 i–j 1 0
Case 1. If q ? 0, then by Eq. (7),0we have
IFGB ða1 ; a2 ; . . . ; an Þ 6 IFGB ðb1 ; b2 ; . . . ; bn Þ
i¼1
1
ð20Þ
Proof. Since lai 6 lbi and v ai P v bi , for all i, thenwhich completes the proof. h
1
C B n 1 C 1 B B ðpai qaj Þnðn1Þ C IFGBp;q ða1 ; a2 ; . . . ; an Þ ¼ C p þ qB A @i; j ¼ 1 i–j 1 1 n ¼ ðpai Þn p i¼1 0 !1 n 1 p Y p n @ ¼ 1 1 1 ð1 lai Þ ;
ð19Þ
p;q
ð23Þ
which can be obtained easily by the monotonicity.
(3) (Monotonicity) Let ai ¼ ðlai ; v ai Þ ði ¼ 1; 2; . . . ; nÞ and bi ¼ ðlbi ; v bi Þ ði ¼ 1; 2; . . . ; nÞ be two collections of AIFNs, if lai 6 lbi and v ai P v bi , for all i, then p;q
ð22Þ
i
a 6 IFGBp;q ða1 ; a2 ; . . . ; an Þ 6 aþ
i–j ¼ IFBG ða_ 1 ; a_ 2 ; . . . ; a_ n Þ
i
then
If the values of the parameters p and q change in the IFBGM, then some special cases can be obtained as follows:
C B n 1 C 1 B B ðpa_ i qa_ j Þnðn1Þ C ¼ C pþqB A @i; j ¼ 1 p;q
i
¼ ðmaxflai g; minfv ai gÞ
n Y
1 v ai p
1n
!1p 1 A
i¼1 p;0
ð24Þ ¼ IFBG ða1 ; a2 ; . . . ; an Þ which we call the generalized Atanassov’s intuitionistic fuzzy geomentric mean (GIFGM). Case 2. If p = 2 and q ? 0, then Eq. (7) is transformed as:
2
3 n n 1 nðn1Þ Y Y 1 ð1 ð1 lai Þp ð1 laj Þq Þnðn1Þ 6 1 ð1 lbi Þp ð1 lbj Þq 6 7 6 7 i; j ¼ 1 6 i; j ¼ 1 7 " 6 7 p q p q# ð1 lai Þ ð1 laj Þ P ð1 lbi Þ ð1 lbj Þ 6 i–j 7 i–j 6 7 )6 n p q p q n 1 7 Y Y v a i v a j P v bi v bj 1 6 7 p q nðn1Þ p q nðn1Þ 1 v ai v aj ð1 v bi v bj Þ 6 6 7 6 7 4 i; j ¼ 1 5 i; j ¼ 1 i–j i–j 20 3 1 1 1pþq 0 1pþq 6B 7 C B C 6B n n C B 1 C 7 Y Y 6B 7 1 C B p q nðn1Þ p q nðn1Þ C 6 B1 7 ð1 ð1 1 ð1 l Þ ð1 l Þ Þ P 1 l Þ ð1 l Þ C B C ai aj bi bj 6B C B C 7 6@ 7 A @ A i; j ¼ 1 i; j ¼ 1 6 7 7 )6 6 7 i–j i–j 6 7 n n 1 nðn1Þ 6 7 Y Y 1 p q p q nðn1Þ 61 7 P1 1 v ai v aj ð1 v bi v bj Þ 6 7 6 7 4 5 i; j ¼ 1 i; j ¼ 1 i–j i–j 3 2 1 1 0 1pþq 0 1pþq 6 B B C C 7 6 n n B C B 1 C 7 7 6 Y Y 1 C B p q nðn1Þ p q nðn1Þ C 61 B ð1 ð1 lai Þ ð1 laj Þ Þ 1 ð1 lbi Þ ð1 lbj Þ B1 C 6 1 B1 C 7 6 B C B C 7 6 @ A @ A 7 i; j ¼ 1 i; j ¼ 1 7 6 7 6 7 6 i–j i–j 7 )6 1 1 1pþq 1pþq 0 7 60 7 6 7 6B C C B 7 6B n n C C B 1 1 7 6B Y Y nðn1Þ nðn1Þ C C B p q p q 7 6 B1 1 1 v v P 1 v v C C B ai aj 7 6B bi bj C C B 7 6@ A A @ 5 4 i; j ¼ 1 i; j ¼ 1 i–j i–j 1 1 0 1pþq 0 1pþq B C n B C Y 1 C B ) 1 B1 ð1 ð1 lai Þp ð1 laj Þq Þnðn1Þ C B C @ A i; j ¼ 1 0
i–j
1 1pþq
B C n 1 C B nðn1Þ Y B C 1 ð1 lbi Þp ð1 lbj Þq 6 1 B1 C B C @ A i; j ¼ 1 i–j
B C n B 1 C nðn1Þ Y B C B1 1 v pai v qaj C B C @ A i; j ¼ 1 0
i–j
1 1pþq
B C n 1 C B nðn1Þ Y B C 1 v pbi v qbj B1 C B C @ A i; j ¼ 1 i–j
ð21Þ
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IFGB2;0 ða1 ; a2 ; . . . ; an Þ ¼
1 1 n ð2ai Þn 2 i¼1 0
¼ @1 1
Table 1 The Atanassov’s intuitionistic fuzzy decision matrix B.
n 1n Y 1 ð1 lai Þ2
!12 ;
i¼1 n 1n Y 1 v 2ai 1
!12 1 A
ð25Þ
y1 y2 y3 y4 y5
c1
c2
c3
(0.3, 0.4) (0.5, 0.2) (0.4, 0.5) (0.2, 0.6) (0.9, 0.1)
(0.7, 0.2) (0.4, 0.1) (0.7, 0.2) (0.8, 0.1) (0.6, 0.3)
(0.5, 0.3) (0.7, 0.1) (0.4, 0.4) (0.8, 0.2) (0.2, 0.5)
i¼1
which we call the Atanassov’s intuitionistic fuzzy square geometric mean (IFSGM).
Fig. 1. Scores for alternatives obtained by WIFGBM.
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M. Xia et al. / Knowledge-Based Systems 40 (2013) 88–100 n
Case 3. If p = 1 and q ? 0, then Eq. (7) reduces to the Atanassov’s intuitionistic fuzzy geometric [24]:
1
IFGB1;0 ða1 ; a2 ; . . . ; an Þ ¼ ain i¼1
¼
n Y i¼1
Fig. 2. Scores for alternatives obtained by WIFBM.
1
lnai ; 1
n Y i¼1
1
ð1 v ai Þn
!! ð26Þ
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M. Xia et al. / Knowledge-Based Systems 40 (2013) 88–100
Fig. 3. Scores for alternatives obtained by WIFGBM and WIFBM.
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M. Xia et al. / Knowledge-Based Systems 40 (2013) 88–100
0
Case 4. If p = q = 1, then Eq. (7) reduces to the following: 0
1
B C 1 C 1B n IFGB1;1 ða1 ; a2 ; . . . ; an Þ ¼ B ai aj nðn1Þ C B C 2 @i; j ¼ 1 A 0 i–j 0
112 B C B B n C B Y B 1 C B nðn1Þ ¼B 1 1 l Þð1 l ÞÞ ð1 ð1 C; B a a B i j C B B A @ i; j ¼ 1 @ i–j
112 1 CC B n CC B Y 1 C C B nðn1Þ 1 v v Þ ð1 CC B ai aj CC B AC @ i; j ¼ 1 A i–j 0
ð27Þ
which we call the Atanassov’s intuitionistic fuzzy interrelated square geometric mean (IGISGM).
1 0 1pþq B B C B 1 n B C p Y B w q nðn1Þ C B i B 1 1 lw IFBGp;q 1 lajj C ; w ða1 ; a2 ; . . . ; an Þ ¼ B1 B1 ai B C B @ A i; j ¼ 1 @ i–j 1 1 0 1pþq C B C C 1 C n B q nðn1Þ Y B C C wi p wj 1 ð1 ð1 v ai Þ Þ 1 ð1 v aj Þ B1 C C B C C @ A C i; j ¼ 1 A i–j
ð29Þ Based on Definition 6 and Theorem 2, we develop an approach for multi-criteria decision making under Atanassov’s intuitionistic fuzzy environment:
Step 1. Transform the decision matrix B = (bij)mn into the normalization matrix R = (rij)mn by the method given by Xu and Hu [23], where ( rij ¼ ðtij ; fij Þ ¼
4. The weighted Atanassov’s intuitionistic fuzzy geometric Bonferroni mean and its application in multi-criteria decision making For a multi-criteria decision making problem, let Y = {y1, y2, . . . , ym} be a set of m alternatives, C = {c1, c2 . . . , cn} be a set of n criteria, whose weight vector is w = (w1, w2, . . . , wn)T, satisfying P wi > 0 (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1. The performance of the alternative yi with respect to the criterion cj is measured by an AIFN bij = (lij, vij), where lij indicates the degree that the alternative yi satisfies the criterion cj, vij indicates the degree that the alternative yi does not satisfy the criterion cj, such that lij 2 [0, 1],vij 2 [0, 1], lij + vij 6 1, and all bij = (lij,vij) (i = 1, 2, . . . , m; j = 1, 2, . . . , n) are contained in the Atanassov’s intuitionistic fuzzy decision matrix B = (bij)mn. To get the priority of the alternatives, we should aggregate the performance of each alternative, however, it is noted that the IFGBM proposed in Section 3 does not consider the importance of the aggregated arguments, but in many practical problem, especially in multi-criteria decision making, the weight vector of the criteria is an important part in the aggregation, to avoid this issue, we introduce the following definition: Definition 6. Let ai ¼ ðlai ; v ai Þ ði ¼ 1; 2; . . . ; nÞ be a collection of AIFNs, w = (w1, w2, . . . , wn)T is the weight vector of ai (i = 1, 2, . . . , n), where wi indicates the importance degree of ai, P satisfying wi > 0 ði ¼ 1; 2; . . . ; nÞ; ni¼1 wi ¼ 1. If
0
1
B 1 C n 1 B wj nðn1Þ C wi C; B p IFGBp;q ða1 ; a2 ; . . . ; an Þ ¼ a q a i j C pþqB A @i; j ¼ 1 i–j
p; q
>0
bij ; for benefit criterion cj ij ; for cos t criterion cj ; b
i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n
ð30Þ is the complement of b , such that b ¼ ðv ; l Þ. where b ij ij ij ij ij Step 2. Aggregate all the performance values rij (j = 1, 2, . . ., n) of the ith line, and get the overall performance value ri corresponding to the alternative yi by the WIFGBM: r i ¼ ðti ; fi Þ ¼ WIFGBp;q w ðr i1 ; r i2 ; . . . ; r in Þ 0 1 0 1pþq B B C B n B 1 C Y B w p B w q nðn1Þ C 1 1 lij j 1 likk ¼B 1 C ; B1 B B C B @ A j; k ¼ 1 @ j–k 1 1 0 1pþq C B C C n 1 C B Y p q nðn1Þ B C C 1 1 ð1 v ij Þwj 1 ð1 v ik Þwk B1 C C B C C @ A C j; k ¼ 1 A j–k
ð31Þ
where p, q > 0. Step 3. Rank the overall performance values ri (i = 1, 2, . . . , m) according to Definition 3 and obtain the priority of the alternatives yi (i = 1, 2, . . . , m) according to ri (i = 1, 2, . . . , m). Especially, if we do not consider the non-membership information in Atanassov’s intuitionistic fuzzy decision making, then the usual fuzzy decision making method can be obtained as follows: The performance of the alternative yi with respect to the criterion cj is measured by a usual fuzzy number bij, where 0 6 bij 6 1, and all the values, bij = (lij,vij) (i = 1, 2, . . . , m; j = 1, 2, . . . , n), are contained in the fuzzy decision matrix B = (bij)mn. Then we can use the GBM or the BM to solve this problem:
ð28Þ IFBp;q w
then is called the weighted Atanassov’s intuitionistic fuzzy geometric Bonferroni mean (WIFGBM). Similar to Theorem 1, we have. Theorem 2. Let ai ¼ ðlai ; v ai Þ ði ¼ 1; 2; . . . ; nÞ be a collection of AIFNs, whose weight vector is w = (w1, w2, . . . , wn)T, which satisfies P wi > 0 (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1. Then the aggregated value by using the WIFGBM is also an AIFN, and
Table 2 The fuzzy decision matrix B.
y1 y2 y3 y4 y5
c1
c2
c3
0.3 0.5 0.4 0.2 0.9
0.7 0.4 0.7 0.8 0.6
0.5 0.7 0.4 0.8 0.2
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Step 1. Transform the decision matrix B = (bij)mn into the normalization matrix R = (rij)mn, where r ij ¼
for benefit criterion cj bij ; ; 1 bij ; for cos t criterion cj
i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n
ð32Þ
Step 2. Aggregate all the performance values rij (j = 1, 2, . . . , n) of the ith line, and get the overall performance value ri corresponding to the alternative yi by the GBM or the BM:
Fig. 4. Scores for alternatives obtained by GBM.
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M. Xia et al. / Knowledge-Based Systems 40 (2013) 88–100
ri ¼ GBp;q w ðr i1 ; r i2 ; . . . ; r in Þ ¼
1 pþq
n Y
j; k ¼ 1 j–k
1
ðpr ij þ qrik Þnðn1Þ
ð33Þ
1 1pþq
0
C B n C B X 1 B p q C p;q r i ¼ Bw ðri1 ; r i2 ; . . . ; r in Þ ¼ B rij rikj C C Bnðn 1Þ A @ j; k ¼ 1
or
j–k where p, q > 0.
Fig. 5. Scores for alternatives obtained by BM.
ð34Þ
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M. Xia et al. / Knowledge-Based Systems 40 (2013) 88–100
Step 3. Rank the overall performance values ri (i = 1, 2, . . . , m) and obtain the priority of the alternatives yi (i = 1, 2, . . . , m). Next, we give an example to illustrate the proposed method:
Example 1. A city is planning to build a municipal library. One of the problems facing the city development commissioner is to determine what kind of air-conditioning system should be installed in the library (adapted from Ref. [26]). The contractor offers five
Fig. 6. Scores for alternatives obtained by GBM and BM.
M. Xia et al. / Knowledge-Based Systems 40 (2013) 88–100
feasible alternatives yi (i = 1, 2, 3, 4, 5), which might be adapted to the physical structure of the library. Suppose that three criteria: (1) c1: economic, (2) c2: functional, and (3) c3: operational, are taken into consideration in the installation problem, the weight vector of the criteria cj (j = 1, 2, 3) is w = (0.3, 0.5, 0.2)T. Assume that the characteristics of the alternatives yi (i = 1, 2, 3, 4, 5) with respect to the criteria cj (j = 1, 2, 3) are represented by the AIFNs bij = (lij, vij), and all bij (i = 1, 2, 3, 4, 5; j = 1, 2, 3) are contained in the Atanassov’s intuitionistic fuzzy decision matrix B = (bij)53 (see Table 1). Step 1. Considering all the criteria cj (j = 1, 2, 3) are the benefit criteria, the performance values of the alternatives yi (i = 1, 2, 3, 4, 5) do not need normalization. Step 2. Utilize the WIFGBM to aggregate all the performance values bij (j = 1, 2, 3) of the ith line, and get the overall performance value bi corresponding to the alternative yi: b1 ¼ ð0:8084; 0:1034Þ; b4 ¼ ð0:8561; 0:0915Þ;
b2 ¼ ð0:8101; 0:0438Þ; b5 ¼ ð0:8381; 0:1006Þ
b3 ¼ ð0:8112; 0:1269Þ
Step 3. Calculate the scores of all the alternatives:
sb1 ¼ 0:7050;
sb2 ¼ 0:7664;
¼ 0:7647;
sb5 ¼ 0:7375
sb3 ¼ 0:6843;
sb4
Since sb2 > sb4 > sb5 > sb1 > sb3 , then the ranking of the alternatives yi (i = 1, 2, 3, 4, 5) is:
y2 y4 y5 y1 y3 If we take p = q = 2, then by bi ¼ ðli ; v i Þ ¼ IFGB2;2 w ðbi1 ; r i2 ; bi3 Þ, we get b1 ¼ ð0:8039; 0:1056Þ;
b2 ¼ ð0:7924; 0:0465Þ;
b4 ¼ ð0:8406; 0:0971Þ;
b5 ¼ ð0:8092; 0:1130Þ
b3 ¼ ð0:8100; 0:1290Þ
Then we calculate the scores of all the alternatives:
sb1 ¼ 0:6982;
sb2 ¼ 0:7459;
sb3 ¼ 0:6810;
sb4 ¼ 0:7435;
sb5 ¼ 0:6962 Since sb2 > sb4 > sb1 > sb5 > sb3 , then we have
y2 y4 y1 y5 y3 We can find that as the values of the parameters p and q change according to the decision maker’s subjective preferences, the rankings of the alternatives are slightly different, which can reflect the decision maker’s risk preferences. If we use the weighted Atanassov’s intuitionistic fuzzy Bonferroni mean (WIFBM) given by Xu and Yager [26] to aggregate the alternative performances, different results can be obtained. To give a detail comparison, we express the scores of alternatives by Figs. 1 and 2 as the parameters p and q change between 0 and 10. Fig. 1 describes the scores of alternatives obtained by our method, and Fig. 2 describes the ones obtained by Xu and Yager’s method. It is noted that most of the scores obtained by our method are bigger than 0 and most of the ones obtained by Xu and Yager’s method are smaller than 0, which indicates that our method can obtain more favorable (or optimistic) expectations, while the one given by Xu and Yager has more unfavorable (or pessimistic) expectations. It should be noted that concepts of optimism and pessimistic are the same as those defined in Chen [5,6]. Therefore, we can conclude that the WIFGBM can be considered as the optimistic one, while the WIFBM [26] can be considered as the pessimistic one and the values of the parameters can be considered as the optimistic or pessimistic levels. In order to obtain the more neutral results, we can use the arithmetic averages of the optimistic and pessimistic results, which can be found in Fig. 3.
99
If we only consider the membership of the Atanassov’s intuitionistic fuzzy set, then the Atanassov’s intuitionistic fuzzy set reduces to the usual fuzzy set, and the decision matrix in Example 1 reduces to that in the usual fuzzy decision making problem illustrated in Table 2. If we use the GBM or the BM to obtain the performance of each alternative, as the values of the parameters change, the performances of the alternatives can be illustrated by using the three-dimensional diagrams, see Figs. 4 and 5 for more details. Comparing Figs. 4 and 5, we can find that most of the values in Fig. 4 are smaller than those in Fig. 5, which implies that the GBM operator can obtain more pessimistic results, while the BM can obtain the more optimistic results. If we use the arithmetic average of the results obtained by these two results, more neutral results can be given in Fig. 6. Comparing Figs. 1 and 4, Figs. 2 and 5, Figs. 3 and 6, we can find that there is a big difference between each pair of figures, that is because that the former of each pair figures considers the non-membership information, while the latter does not, which illustrates that the non-membership information plays an important part in the overall performances of the alternatives. However, the WIFGBM is not a simple extension of the GBM, and the WIFBM is not a simple extension of t the BM, the operations for Atanassov’s intuitionistic fuzzy sets are very different from the ones of fuzzy sets, that is because the specific characteristic of intuitionistic fuzzy sets, each of which include two parts of information, the membership information and the non-membership information. In addition, the GBM and the BM do not consider the weights of the criteria, but in many practical problems, especially in multi-criteria decision making, the weight vector of the criteria is an important part in the aggregation. Therefore, it is necessary to introduce the WIFGBM and the WIFBM for Atanassov’s intuitionistic fuzzy sets and fuzzy sets, respectively. One aim of the proposed method is to provide more choices for the decision makers, and as for the choice of the parameter, we should fully respect the opinions of the decision makers. Maybe, as the values of the parameters change, the rankings of the alternatives may be different, but we do not say which is the most appropriate order, it just reflects the preferences of the decision makers, and depends on the optimistic or pessimistic nature of the decision makers, which has been investigated by many researchers [11,12,6]. Chen [4] denoted that ‘‘the specification of parameters can be assessed using the Life Orientation Test (LOT) devised by Sanz et al. [13], the Revised Life Orientation Test (LOT-R) devised by Scheier et al. [14], or the Extended Life Orientation Test (ELOT) devised by [3]Chang et al. (1997). These three scales have strong predictive and discriminative validity [9]. The values of the parameters and are set according to the normalized LOT score’’. Because the appropriateness of the optimistic and pessimistic point operators is generally context-dependent, we suggest that empirical studies be conducted in the future to determine parameter settings in the context of various specific applications [6].
5. Concluding remarks In this paper, we have applied the well-known geometric mean (GM) to Bonferroni mean (BM), and introduced the geometric Bonferroni mean (GBM), whose properties have been studied in detail. To aggregate the Atanassov’s intuitionistic fuzzy information, we have introduced the Atanassov’s intuitionistic fuzzy geometric Bonferroni mean (IFGBM), and discussed its special cases when the parameters are assigned different values. Considering the weight vector of the arguments, we have further developed the weighted Atanassov’s intuitionistic fuzzy geometric Bonferroni
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mean (IFGBM), and applied it to multi-criteria decision making. The proposed aggregation operators in this paper can reflect the interrelationship of the aggregated argument [31] and can provide the decision makers more choices by changing the values of the parameters determined by the preferences of the decision makers which is an issue to be further studied in the future. In the end, we compare the proposed aggregation method with the ones given by Xu and Yager [26], and find that our operator can give more optimistic results, while Xu and Yager’s can give more pessimistic ones. In the future, we will study the geometric Bonferroni mean under interval-valued Atanassov’s intuitionistic fuzzy environment [2]. It should be noted that the motivation of this paper is to investigate the Bonferroni means under intuitionistic fuzzy environment combing the geometric mean, other means can also be extended similarly. Bonferroni means was introduced in 1950, and have been investigated by many authors, although other extensions have been developed in the last decades, the main focus of them are invariable, that is to capture the expressed interrelationship between the criteria [31]. Our aim is to apply this core of Bonferroni mean to the intuitionistic fuzzy environment combing geometric mean, based on which, we can also investigate more general class of means including it similarly. We can find that the proposed formulas seem a little complex, and several parameters are provided for the decision makers, if we introduce other more general class of means including Boferroni means, more complex formulas and more parameters will be given, which will make the decision making too complex and too many parameters will make the decision makers be dazzled and they may do not know how to chose it. A saying may be suitable to express this situation, that is ‘‘too much is as bad as too little’’.
Acknowledgements The authors are very grateful to the anonymous reviewers for their insightful and constructive comments and suggestions that have led to an improved version of this paper. The work was supported in part by the National Natural Science Foundation of China (Nos. 71071161 and 61273209) and the China Postdoctoral Science Foundation (No. 2012M520311).
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