Some issues on intuitionistic fuzzy aggregation operators based on ...
Knowledge-Based Systems 31 (2012) 78–88
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Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm Meimei Xia a, Zeshui Xu a,b,⇑, Bin Zhu a a b
School of Economics and Management, Southeast University, Nanjing, Jiangsu 211189, China Institute of Sciences, PLA University of Science and Technology, Nanjing, Jiangsu 210007, China
a r t i c l e
i n f o
Article history: Received 18 November 2011 Received in revised form 19 January 2012 Accepted 7 February 2012 Available online 15 February 2012 Keywords: Multi-criteria decision making Archimedean t-conorm Archimedean t-norm Intuitionistic fuzzy set Aggregation operator
a b s t r a c t Archimedean t-conorm and t-norm are generalizations of a lot of other t-conorms and t-norms, such as Algebraic, Einstein, Hamacher and Frank t-conorms and t-norms or others, and some of them have been applied to intuitionistic fuzzy set, which contains three functions: the membership function, the nonmembership function and the hesitancy function describing uncertainty and fuzziness more objectively. Recently, Beliakov et al. [3] constructed some operations about intuitionistic fuzzy sets based on Archimedean t-conorm and t-norm, from which an aggregation principle is proposed for intuitionistic fuzzy information. In this paper, we propose some other operations on intuitionistic fuzzy sets, study their properties and relationships, and based on which, we study the properties of the aggregation principle proposed by Beliakov et al. [3], and give some specific intuitionistic fuzzy aggregation operators, which can be considered as the extensions of the known ones. In the end, we develop an approach for multi-criteria decision making under intuitionistic fuzzy environment, and illustrate an example to show the behavior of the proposed operators. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction An intuitionistic fuzzy set (IFS) [1] A on a fixed set X is defined as A = {hx, lA(x), vA(x)ijx 2 X} with the condition that 0 6 lA(x) + vA(x) 6 1, lA(x) and vA(x) P 0. We can find that an IFS is constructed by two information functions, which not only describe the membership degree lA(x) of x 2 X in A, but also describe the non-membership degree vA(x) of x 2 X in A. Moreover, the hesitancy information of x 2 X in A can be denoted by pA(x) = 1 lA(x) vA(x) which is called the hesitant index, and therefore IFS can describe the uncertainty and fuzziness more objectively than the usual fuzzy set (FS) [40]. For convenience, the pair a = (la, va) is called an intuitionistic fuzzy number (IFN) [29], where la, va P 0 and la + va 6 1. Since it was introduced, IFS has attracted more and more attentions from researchers and has been used to deal with many problems, especially the multi-criteria decision making problem which can be roughly described as: Let y = {y1, y2, . . . , ym} be the set of alternatives, c = {c1, c2, . . . , cn} be the set of criteria, the degree that alternative yi satisfies to the criterion cj can be denoted as lij, the degree that the alternative yi does not satisfy the criterion cj can be denoted as vij, then the performance of the alternative yi under the criteria cj can be described as an IFN aij = (lij, vij) with the ⇑ Corresponding author at: School of Economics and Management, Southeast University, Nanjing, Jiangsu 211189, China. E-mail addresses: [email protected] (M. Xia), [email protected] (Z. Xu), [email protected] (B. Zhu). 0950-7051/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2012.02.004
condition that 0 6 lij, vij 6 1 and lij + vij 6 1. When all the performances of the alternatives are provided, the intuitionistic fuzzy decision matrix D = (aij)nn = ((lij, vij))mn is constructed. Up to now, many methods have been proposed to deal with the multicriteria decision making under intuitionistic fuzzy environment, which fall into two groups, the first group is to calculate the relative values of the alternatives and the second group is to calculate the actual aggregation values of the alternatives. By comparing these two types of methods for obtaining the ranking of the alternatives, the second one can reflect the actual results of the alternatives more objectively, while the first one can only obtain the relative results of the alternatives to the ideal alternative or others. To calculate the relative values of the alternatives, many classical methods have been extended to intuitionistic fuzzy environment, such as the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method [5,15,19,21], the gray relational analysis (GRA) method [11,26,43], the Elimination et Choice Translating Reality (ELECTRE) method [28], the Vlse Kriterijumska Optimizacija Kompromisno Resenje (VIKOR) method [12,23], the maximizing deviation method [25,31] and the entropy method [9,28,37,38] and so on. To calculate the actual aggregation values of the alternatives, a lot of aggregation operators have been developed. Xu [29] proposed some operational laws for IFNs based on Algebraic t-conorm and t-norm, and developed the intuitionistic fuzzy weighed averaging operator, the intuitionistic fuzzy ordered weighted averaging operator and the intuitionistic fuzzy hybrid averaging operator,
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based on which, Xu and Yager [34] gave some other aggregation operators combining the geometric mean. Based on the generalized ordered weighted aggregation operator proposed by Yager [35], Zhao et al. [44] and Li [20] introduced the generalized intuitionistic fuzzy aggregation operator, which gives a mapping to the arguments before aggregation, and an inverse mapping to the aggregated results at the end. Yang and Chen [36] proposed the quasi-arithmetic intuitionistic fuzzy ordered weighted averaging operator, the quasi-intuitionistic fuzzy aggregation operator based on the Choquet integral and the Dempster–Shafer belief structure. Chen et al. [10] adopted three families of parametric fuzzy unions and intersections of fuzzy operations, to create other aggregation operators for interval-valued intuitionistic fuzzy sets (IVIFSs) [2], which are the generalizations of IFSs allowing the membership and non-membership degrees represented by interval-valued fuzzy numbers (IVFNs) [41]. Wang and Liu [24] introduced some operations on IFSs, such as Einstein sum, Einstein product, Einstein exponentiation, etc., and developed some new geometric aggregation operators, such as the intuitionistic fuzzy Einstein weighted geometric operator and the intuitionistic fuzzy Einstein ordered weighted geometric operator. Moreover, a lot of work [18,27,30, 32,33,36,39] has been done about the weight vector of the intuitionistic fuzzy aggregation operators or others [6,42]. Based on Archimedean t-conorm and t-norm [17,22], and the aggregation functions for the classical FSs [4,13,14], Beliakov et al. [3] gave some operations about intuitionistic fuzzy sets, proposed two general concepts for constructing other types of aggregation operators for IFSs extending the existing methods and showed that the operators obtained by using the Łukasiewicz t-norm are consistent with the ones on ordinary FSs [40]. We can find that the above aggregation operators are all based on different t-conorms and t-norms and are used to deal with different relationships of the aggregated arguments, which can provide more choices for the decision makers. In this paper, we first review the intuitionistic fuzzy operations proposed by Beliakov et al. [3], and then give some other ones based on Archimedean t-conorm and t-norm. We further give an in-depth study on the aggregation principle proposed by Beliakov et al. [3], obtain some important conclusions, and propose some specific intuitionistic fuzzy aggregation operators. The remainder of this paper is organized as follows: Section 2 introduces some new intuitionistic fuzzy operational laws based on Archimedean t-conorm and t-norm. Section 3 investigates the properties of aggregation principle proposed by Beliakov et al. [3], and develops some specific intuitionistic fuzzy aggregation operators. Section 4 proposes an approach to multi-criteria decision making. Section 5 gives some conclusions. 2. Some intuitionistic fuzzy operations based on t-conorm and t-norm Definition 1 (17,22). A function T: [0, 1] [0, 1] ? [0, 1] is called a t-norm if it satisfies the following four conditions: (1) (2) (3) (4)
T(1,x) = x, for all x. T(x, y) = T(y, x), for all x and y. T(x, T(y, z)) = T(T(x, y), z), for all x, y and z. If x 6 x0 and y 6 y0 , then T(x, y) 6 T(x0 , y0 ).
(3) S(x, S(y, z)) = S(S(x, y), z), for all x, y and z. (4) If x 6 x0 and y 6 y0 , then S(x, y) 6 S(x0 , y0 ). It is also usually required that a t-norm and a t-conorm are continuous functions. Definition 3 (17,22). A t-norm function T(x, y) is called Archimedean t-norm if it is continuous and T(x, x) < x for all x 2 (0, 1). An Archimedean t-norm is called strictly Archimedean t-norm if it is strictly increasing in each variable for x, y 2 (0, 1). Definition 4 (17,22). A t-conorm function S(x, y) is called Archimedean t-conorm if it is continuous and S(x, x) > x for all x 2 (0, 1). An Archimedean t-conorm is called strictly Archimedean t-conorm if it is strictly increasing in each variable for x, y 2 (0, 1). It is well known [16] that a strict Archimedean t-norm is expressed via its additive generator g as T(x, y) = g1(g(x) + g(y)), and similarly, applied to its dual t-conorm S(x, y) = h1(h(x) + h(y)) with h(t) = g(1 t). We notice that an additive generator of a continuous Archimedean t-norm is a strictly decreasing function g: [0, 1] ? [0, 1] such that g(1) = 0. If we assign specific forms to the function g, then some well-known t-conorms and t-norms can be obtained: (1) Let g(t) = logt, then h(t) = log (1 t), g1(t) = et, h1(t) = 1 et, and Algebraic t-conorm and t-norm [4] are obtained as follows:
SA ðx; yÞ ¼ x þ y xy;
T A ðx; yÞ ¼ xy
ð1Þ
; g 1 ðtÞ ¼ et 2þ1 ; (2) Let gðtÞ ¼ log 2t , then hðtÞ ¼ log 2ð1tÞ 1t t 1
h ðtÞ ¼ 1 et 2þ1, and we can get Einstein t-conorm and tnorm [4]:
SE ðx; yÞ ¼ (3) Let log
xþy ; 1 þ xy
T E ðx; yÞ ¼
xy 1 þ ð1 xÞð1 yÞ
ð2Þ
cÞt ; c > 0, then we have hðtÞ ¼ gðtÞ ¼ log cþð1 t 1 1 ; g ðtÞ ¼ et þcc1 ; h ðtÞ ¼ 1 et þcc1, and Ham1t
cþð1cÞð1tÞ
acher t-conorm and t-norm [4] are obtained as follows:
x þ y xy ð1 cÞxy ; c>0 1 ð1 cÞxy xy T Hc ðx; yÞ ¼ ; c>0 c þ ð1 cÞðx þ y xyÞ
SHc ðx; yÞ ¼
ð3Þ ð4Þ
Especially, if c = 1, then Hamacher t-conorm and t-norm reduce to the Algebraic t-conorm and t-norm respectively; if c = 2, then Hamacher t-conorm and t-norm reduce to the Einstein t-conorm and t-norm respectively. c1 ; c > 1, then hðtÞ ¼ log c1t ; g 1 ðtÞ ¼ (4) Let gðtÞ ¼ log cct1 1 1 c1þec c1þec log log 1 ec ec ; h ðtÞ ¼ 1 g 1 ðtÞ ¼ , and we have Frank log c log c t-conorm and t-norm [4] as follows:
ðc1x 1Þðc1y 1Þ ; SFc ðx; yÞ ¼ 1 logc 1 þ c1 ðcx 1Þðcy 1Þ T Fc ðx; yÞ ¼ logc 1 þ ; c>1 c1
c>1
ð5Þ ð6Þ
Especially, if c ? 1, then we have Definition 2 (17,22). A function S: [0, 1] [0, 1] ? [0, 1] is called a t-conorm if it satisfies the following four conditions: (1) S(0,x) = x, for all x. (2) S(x, y) = S(y, x), for all x and y.
limc!1 gðtÞ ¼ limc!1 log ¼ limc!1 log
c1 ct 1
1
tct1 1
¼ log t
ð7Þ
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which indicates that limc!1 SFc ðx; yÞ ¼ SAc ðx; yÞ and limc!1 T Fc ðx; yÞ ¼ T Ac ðx; yÞ. Based on Archimedean t-norm and t-conorm [17], Beliakov al. [3] defined the sum operation on two IFNs ai ¼ ðlai ; v ai Þði ¼ 1; 2Þ as a1 a2 ¼ Sðla1 ; la2 Þ; Tðv a1 ; v a2 Þ , which can be expressed by the following:
et
a1 a2 ¼ S la1 ; la2 ; T v a1 ; v a2
1 ¼ h h la1 þ h la2 ; g 1 g v a1 þ g v a2
ð8Þ
1la 1la 1 1Þðc 2 1Þ (17) a1 a2 ¼ 1 logc 1 þ ðc ; c1 v v ðc a1 1Þðc a2 1Þ Þ; c > 1. logc 1 þ c1 l l ðc a1 1Þðc a2 1Þ ; (18) a1 a2 ¼ logc 1 þ c1 1v a 1v a 1 1Þðc 2 1Þ ðc Þ; c > 1. 1 logc 1 þ c1 k k 1 l va a ; logc 1 þ ððcc1Þ1Þ ; k > 0; c > 1. (19) ka ¼ 1 logc 1 þ ðcðc1Þ1Þ k1 k1 k la ðc1v a 1Þk ; k > 0; c > 1. (20) ak ¼ logc 1 þ ððcc1Þ1Þ k1 ; 1 logc 1 þ ðc1Þk1 which are the ones defined based on Frank t-conorm and tnorm. Especially, if c = 1, then (17)–(20) reduce to (5)–(8).
Beliakov et al. [3] also mentioned that for an IFN a = (la, va), let k a = (lka, vka), then g(vka) = kg(va) and h(lka) = k h(la). With the above analysis, the operations about IFNs based on Archimedean t-norm and Archimedean t-conorm [17] can be also expressed as follows:
Moreover, some relations of the operational laws can be discussed as follows:
Definition 5. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2Þ and a = (la, va) be three IFNs, then we have
Theorem 1. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2Þ and a = (la, va) be three IFNs, then the relations of these operational laws are given as:
(1) a1 a2 ¼ Sðla1 ; la2 Þ; Tðv a1 ; v a2 Þ 1 ¼ h ðhðla1 Þ þ hðla2 ÞÞ; g 1 ðgðv a1 Þ þ gðv a2 ÞÞ . (2) a1 a2 ¼ Tðla1 ; la2 Þ; Sðv a1 ; v a2 Þ ¼ g 1 ðgðla1 Þ þ gðla2 ÞÞ; 1
h ðhðv a1 Þ þ hðv a2 ÞÞÞ. 1 (2) ka ¼ h ðkhðla ÞÞ; g 1 ðkgðv a ÞÞ ; k > 0. 1 (3) ak ¼ g 1 ðkgðla ÞÞ; h ðkhðv a ÞÞ ; k > 0.
(1) (2) (3) (4) (5) (6)
Proof. (1) and (2) are obvious, we prove the others:
Especially, if g(t) = log (t), then we have: (5) a1 a2 ¼ ðla1 þ la2 la1 la2 ; v a1 v a2 Þ. (6) a1 a2 ¼ ðla1 la2 ; v a1 þ v a2 v a1 v a2 Þ. (7) ka ¼ ð1 ð1 la Þk ; v ka Þ; k > 0. (8) ak ¼ lka ; 1 ð1 v a Þk ; k > 0
1
(3) kða1 a2 Þ ¼ kðh ðhðla1 Þ þ hðla2 ÞÞ; g 1 ðgðv a1 Þ þ gðv a2 ÞÞÞ ¼ 1
1
¼ ðh ðkðhðla1 Þ þ hðla2 ÞÞÞ; g 1 ðkðgðv a1 Þ þ gðv a2 ÞÞÞÞ. 1 1 (4) ka1 ka2 ¼ ðh ðkhðla1 ÞÞ; g 1 ðkgðv a1 ÞÞÞ ðh ðkhðla2 ÞÞ; g 1 1
a1
a2
1
a
1
g ðkgðv a1 Þ þ kgðv a2 ÞÞÞ ¼ kða1 a2 Þ. 1 1 (5) k1 a k2 a ¼ ðh ðk1 hðla ÞÞ; g 1 ðk1 gðv a ÞÞÞ ðh ðk2 hðla ÞÞ; g 1 1
(13) a1 a2 ¼ (14) a1 a2 ¼ (15) ka ¼ (16) a ¼ k
cþð1cÞt
t
1
2
Similarly, (4) and (6) can be proven which completes the proof of the theorem. h
a
[24] based on Einstein t-conorm and t-norm.
1
1
ðg ðk1 gðv a ÞÞÞ þ gðg ðk2 gðv a ÞÞÞÞÞ ¼ ðh ðk1 hðla Þ þ k2 hðla ÞÞ; g 1 ðk1 gðv a Þ þk2 gðv a ÞÞÞ ¼ ðk1 þ k2 Þa. 1
where (9) and (10) are the ones defined by Wang and Liu
If gðtÞ ¼ log
1
ðk2 gðv a ÞÞÞ ¼ ðh ðhðh ðk1 hðla ÞÞÞ þ hðh ðk2 hðla ÞÞÞÞ; g 1 ðg
2v k ; ð2v ÞÞak þv k a a
a
1
1
(11) ka ¼ ; k > 0. 2lk v a Þk ð1v a Þk (12) ak ¼ ð2l Þak þlk ; ð1þ ;k > 0 ð1þv Þk þð1v Þk a
1
ðkgðv a2 ÞÞÞ ¼ ðh ðhðh ðkhðla1 ÞÞÞ þ hðh ðkhðla2 ÞÞÞÞ; g 1 ðg ðg 1 ðkgðv a1 ÞÞÞ þ gðg 1 ðkgðv a2 ÞÞÞÞÞ ¼ ðh ðkhðla1 Þ þ khðla2 ÞÞ;
conorm and t-norm. If gðtÞ ¼ log 2t , then we have: t la þla v a v a2 (9) a1 a2 ¼ 1þl1 l 2 ; 1þð1v a1 Þð1 . v Þ a a1 a2 1 2 la la2 v a1 þv a2 (10) a1 a2 ¼ 1þð1l 1 Þð1 l Þ ; 1þv a v a . ð1þla Þk ð1la Þk ð1þla Þk þð1la Þk
1
ðh ðkhðh ðhðla1 Þ þ hðla2 ÞÞÞÞ;g 1 ðkgðg 1 ðgðv a1 Þ þ gðv a2 ÞÞÞÞÞ
which are the ones defined by Xu [29] based on Algebraic t-
a1 a2 = a2 a1. a1 a2 = a2 a1. k(a1 a2) = ka1 ka2. ða1 a2 Þk ¼ ak1 ak2 . k1a k2 a = (k1 + k2)a. ak1 ak2 ¼ ak1 þk2 .
; c > 0, then we have: .
la1 þla2 la1 la2 ð1cÞla1 la2 va va ; cþð1cÞðv a 1þv a2 v a v a Þ 1ð1cÞla la 1 2 1 2 1 2
la1 la2 v a1 þv a2 v a1 v a2 ð1cÞv a1 v a2 cþð1cÞðla1 þla2 la1 la2 Þ ; 1ð1cÞv a1 v a2
ð1þðc1Þla Þk ð1la Þk ð1þðc1Þla Þk þðc1Þð1la Þk
.
cv k ; ð1þðc1Þð1v aÞÞk þðc1Þv k ; k > 0. a
a
k k clka ; ð1þðc1Þv a Þ ð1v a Þ ð1þðc1Þð1la ÞÞk þðc1Þlka ð1þðc1Þv a Þk þðc1Þð1v a Þk
; k > 0.
which are the ones defined based on Hammer t-conorm and t-norm. Especially, if c = 1, then (13)–(16) reduce to (5)–(8); if c = 2, then (13)–(16) reduce to (9)–(12). ; c > 1, then we have: If gðtÞ ¼ log cct1 1
Theorem 2. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2Þ and a = (la, va) be three IFNs, and k > 0, then the following are also valid: (1) (2) (3) (4)
(ac)k = (ka)c. k(ac) = (ak)c. ac1 ac2 ¼ ða1 a2 Þc . ac1 ac2 ¼ ða1 a2 Þc .
where ac = (va, la) denotes the complement of an IFN a. Proof. Based on the operations defined in Definition 5, we have: 1 (1) ðac Þk ¼ g 1 ðkgðv a ÞÞ; h khðla Þ ¼ ðkaÞc . 1 (2) kðac Þ ¼ h ðkhðv a ÞÞ; g 1 ðkgðv a ÞÞ ¼ ðak Þc . 1 (3) ac1 ac2 ¼ h hðv a1 Þ þ hðv a2 Þ ; g 1 gðla1 Þ þ gðla2 Þ ¼ ða1 a2 Þc .
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M. Xia et al. / Knowledge-Based Systems 31 (2012) 78–88
(4) ac1 ac2 ¼ g
1
1 gðv a1 Þ þ gðv a2 Þ ; h hðla1 Þ þgðla2 ÞÞÞ ¼ ða1
c
a2 Þ . which completes the proof. h 3. Intuitionistic fuzzy aggregation operators based on t-conorm and t-norm The operational laws defined in Section 2 can be used to aggregate the intuitionistic fuzzy information, which is the focus of this section. Definition 6 (3). Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collection of IFNs, and w = (w1, w2, . . . , wn)T the weight vector of ai(i = 1, 2, . . . , n), where wi indicates the importance degree of ai, satisfying wi P 0 P (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1, if: n
ATS-IFWAða1 ; a2 ; . . . ; an Þ ¼ wi ai
i.e., Eq. (10) holds for n = k + 1. Thus Eq. (10) holds for all n. In addition, we have known that h(t) = g(1 t), and g: [0, 1] ? [0, 1] is a strictly decreasing function, then h(t) is a strictly increasing function which indicates that
06h
n X
! wi hðlai Þ ; g
n X
1
i¼1
! wi gðv ai Þ
1
h
n X i¼1 1
6h
then ATS-IFWA is called the Archimedean t-conorm and t-norm based intuitionistic fuzzy weighted averaging (ATS-IFWA) operator.
n X
!
!
wi h
lai
¼h
1
n X
wi h
lai
i¼1
þg
1
! wi g 1 lai
n X i¼1
þ1h
n X
1
i¼1
ATS-IFWAða1 ; a2 ; . . . ; an Þ n
¼ wi ai ¼ i¼1
h
1
n X
! wi hðlai Þ ; g
1
i¼1
n X
!! wi gðv ai Þ
:
lai
!
¼1
ð15Þ
i¼1
which completes the proof of Theorem 3.
h
Then we can investigate some desirable properties of the ATSIFWA operator, before doing this, a definition is given firstly: Definition 7 (7). Let sðai Þ ¼ lai v ai ði ¼ 1; 2Þ be the scores of ai(i = 1,2) respectively, if s(a1) > s(a2), then a1 is larger than a2, denoted by a1 > a2.
ð10Þ
Proof. By using mathematical induction on n: For n = 2, we have: 2
ATS-IFWA ða1 ; a2 Þ ¼ wi ai ¼ w1 a1 w2 a2 i¼1 1 1 1 ¼ h h h w1 h la1 w2 h la2 þh h ; g 1 g g 1 w1 g v a1 þ g g 1 w2 g v a2 1 ¼ g 1 w1 g la1 þ w2 g la2 ; h w1 h v a1 þ w2 h v a2
Property 1. If all ai(i = 1, 2, . . . , n) are equal, i.e., ai = a = (la, va), for all i, then
ATS-IFWA ða1 ; a2 ; . . . ; an Þ ¼ a
ATS-IFWA ða1 ; a2 ; . . . ; an Þ ¼ ATS-IFWA ða; a; . . . ; aÞ n
¼ wi a ¼
Suppose Eq. (10) holds for n = k, that is
i¼1
k
i¼1
ð12Þ
i¼1
then k
ATS-IFWA ða1 ; a2 ; . . . ; ak ; akþ1 Þ ¼ wi ai wkþ1 akþ1 i¼1 ! !! k k X X 1 1 ¼ h wi h lai ; g wi g v ai i¼1 i¼1 1 wkþ1 h lakþ1 ; g 1 wkþ1 g v akþ1 h !! ! k X 1 1 1 h h wi h lai wkþ1 h lakþ1 þh h ; ¼ h i¼1 !! !! k X wi g v ai þ g g 1 wkþ1 g v akþ1 g 1 g g 1 i¼1 ! k X 1 wi h lai þ wkþ1 h lakþ1 ; ¼ h i¼1 !! k X 1 wi g v ai þ wkþ1 g v akþ1 g i¼1 ! !! kþ1 kþ1 X X 1 ð13Þ wi h lai ; g 1 wi g v ai ¼ h i¼1
h
1
n X
!
wi hðla Þ ; g
n X
1
i¼1
!! wi gðv a Þ
i¼1
1 ¼ h hðla Þ ; g 1 ðgðv a ÞÞ ¼ a
ATS-IFWA ða1 ; a2 ; . . . ; ak Þ ¼ wi ai ¼ w1 a1 w2 a2 wk ak ! !! k k X X 1 1 ¼ h wi hðlai Þ ; g wi hgðv ai Þ
ð16Þ
Proof. Let ai = a = (la, va), we have
ð11Þ
i¼1
wi h
i¼1
Next we give a further study:
i¼1
ð14Þ
! ! n X wi h lai wi g v ai þ g 1
i¼1
Theorem 3 (3). Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collection of IFNs, and w = (w1, w2, . . . , wn)T be the weight vector of ai(i = 1, 2, . . . , n), where wi indicates the importance degree of ai, satisfying wi P 0 Pn (i = 1, 2, . . . , n) and i¼1 wi ¼ 1, then the aggregated value by using the ATS-IFWA operator is also an IFN, and
61
i¼1
and
ð9Þ
i¼1
1
ð17Þ
Property 2. Let ai ¼ ðlai ; v ai Þ and bi ¼ ðlbi ; v bi Þði ¼ 1; 2; . . . ; nÞ be two collections of IFNs, if lai 6 lbi and v ai P v bi , for all i, then
sðATS-IFWA ða1 ; a2 ; . . . ; an ÞÞ 6 sðATS-IFWA ðb1 ; b2 ; . . . ; bn ÞÞ
ð18Þ
Proof. We have known that h(t) = g(1 t), and g: [0, 1] ? 0, 1] is a strictly decreasing function, then h(t) is a strictly increasing function. Since lai 6 lbi and v ai P v bi , then we have 1
h
n X
! ! ! n n X X 1 1 wi h lai wi h lbi ; g wi g v ai 6h
i¼1
i¼1
P g 1
n X
wi g
vb
!
i¼1
i
ð19Þ
i¼1
then
sðATS-IFWA ða1 ; a2 ; . . . ; an ÞÞ 6 sðATS-IFWA ðb1 ; b2 ; . . . ; bn ÞÞ which completes the proof.
ð20Þ
h
Based on Property 2, the following property can be obtained:
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M. Xia et al. / Knowledge-Based Systems 31 (2012) 78–88
Property 3. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collection of IFNs, and
a ¼ mini flai g; maxi fv ai g ; aþ ¼ maxi flai g; mini fv ai g
ð21Þ
then
ATS-IFWA ðr a1 ; r a2 ; .. .; r an Þ !! ! kþ1 kþ1 X X 1 1 1 1 ¼ h wi h h rh lai wi g g rg v ai ;g i¼1
then
sða Þ 6 sðATS IFWA ða1 ; a2 ; . . . ; an ÞÞ 6 sðaþ Þ
ð22Þ
¼ h
1
i¼1
kþ1 X
! !! kþ1 X ; g 1 wi rh lai wi rg v ai
i¼1
Property 4. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collections of IFNs, w = (w1, w2, . . . , wn)T be their weight vector such that wi P 0 P (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1, if b = (lb, vb) is an IFN, then
and r ATS-IFWA ða1 ; a2 ;... ; an Þ ¼ h
1
1
rh h
ATS-IFWA ða1 b; a2 b; . .. ; an bÞ ¼ ATS-IFWA ða1 ; a2 ;. .. ; an Þ b ð23Þ
n X
w i h l ai
i¼1
¼ h
1
r
n X
!!!
!
wi h lai ;g 1 r
aj b ¼ h1 ðhðlai Þ þ hðlb ÞÞ; g 1 gðv ai Þ þ gðv b Þ
¼
n X
wi g g 1 g v ai þ g v b
i¼1
! ; wi h lai þ h lb
1
n X
1
n X
h
i¼1
g
wi g v ai þ g v b
va
va
!!!!
i
i¼1
!!
ð30Þ
i
i¼1
Property 6. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collections of IFNs, and w = (w1, w2, . . . , wn)T be the weight vector of them such that P wi P 0 (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1, if r > 0, b = (lb, vb) is an IFN, then
ATS-IFWA ða1 b; a2 b; . . . ; an bÞ ! n X 1 1 ; wi h h h lai þ h lb ¼ h g 1
wi g
ð24Þ
we have
i¼1
n X
wi g
According to Properties 4 and 5, we can get Property 6 easily:
n X
;g 1 rg g 1
i¼1
Proof. Since
ð29Þ
i¼1
ATS-IFWA ðra1 b; r a2 b; . . . ; r an bÞ
!!
¼ r ATS-IFWA ða1 ; a2 ; . . . ; an Þ b
ð31Þ
Property 7. Let ai ¼ ðlai ; v ai Þ and bi ¼ ðlbi ; v bi Þði ¼ 1; 2; . . . ; nÞ be two collections of IFNs, and w = (w1, w2, . . . , wn)T be the weight vector P of them such that wi P 0 (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1, then
!! ð25Þ
i¼1
ATS-IFWA ða1 b1 ; a2 b2 ; . . . ; an bn Þ ¼ ATS-IFWA ða1 ; a2 ; . . . ; an Þ ATS-IFWA ðb1 ; b2 ; . . . ; bn Þ
ð32Þ
and
ATS-IFWA ða1 ; a1 ; .. .; an Þ b ! !! n n X X 1 1 ¼ h wi h lai ; g wi g v ai lb ; v b i¼1
¼ h
1
n X
h h
!! ! þ h lb ; wi h lai
i¼1
g 1 g g 1
n X
wi g
va
!! þg
i
i¼1
¼ h
1
wi h
lai þ h lb
i¼1
¼ h
1
n X
wi h lai þ h lb
vb
! ; g 1 !
va
i
þg
vb
;g
i¼1
n X
g 1
wi g v ai þ g v b
!! wi g g 1 g v ai þ g v bi
i¼1
!!
1
¼ h
!! ! n X 1 wi h lai þ h lbi wi g v ai þ g v bi ;g
n X i¼1
i¼1
i¼1
ð34Þ
ð26Þ which completes the proof.
and
h
Property 5. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collections of IFNs, and w = (w1, w2, . . . , wn)T be their weight vector such that wi P 0 P (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1, if r > 0, then
ATS-IFWA ðr a1 ; ra2 ; . . . ; ran Þ ¼ r ATS-IFWA ða1 ; a2 ; . . . ; an Þ
ð27Þ
ATS-IFWA ða1 ; a2 ; ...; an Þ ATS-IFWA ðb1 ;b2 ;...;bn Þ !! ! n n X X 1 1 ¼ h wi h lai ; g wi g v ai i¼1
h
1
n X
i¼1
!! ! n X wi h lbi ; g 1 wi g v bi
i¼1
Proof. According to Definition 5, we have
1 r a ¼ h rhðla Þ ; g 1 ðrgðv a ÞÞ
ð33Þ
i¼1
!!
i¼1 1
ATS-IFWA ða1 b1 ; a2 b2 ;...; an bn Þ ! n X 1 1 ¼ h wi h h h lai þ h lbi ;
wi g
n X
then
!!
n X
ai bi ¼ h1 hðlai Þ þ hðlbi Þ ; g 1 g v ai þ gðv bi Þ
i¼1
1
n X
Proof. According to Definition 5, we have
1
¼ h
i¼1
1
h h
n X
!! !!! n X 1 wi h lai wi h lbi þh h ;
i¼1
ð28Þ g
1
g g
1
n X i¼1
wi g
!!
va
i
i¼1
þg g
1
n X i¼1
wi g
!!!!
vb
i
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M. Xia et al. / Knowledge-Based Systems 31 (2012) 78–88
¼ h
1
! n n X X wi h lai þ wi h lbi ;
wi
ATS-IFWG ða1 ; a2 ; . . . ; an Þ ¼ ni¼1 ai
ð40Þ
ð35Þ
then ATS-IFWG is called the Archimedean t-cornorm and t-norm based the intuitionistic fuzzy geometric (ATS-IFWG) operator. Based on the operational laws of the IFNs given in Definition 5, we can derive the following theorem:
If the additive generator g is assigned different forms, then some specific intuitionistic fuzzy aggregation operators can be obtained as follows:
Theorem 4. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collection of IFNs, and w = (w1, w2, . . . , wn)T be the weight vector of ai(i = 1, 2, . . . , n), where wi indicates the importance degree of ai, satisfying wi > 0 P (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1, then the aggregated value by using the ATS-IFWG operator is also an IFN, and
i¼1
g 1
n X
i¼1
wi g
i¼1
va
i
þ
n X
wi g
!!
vb
i
i¼1
which completes the proof. h
Case 1. If g(t) = log (t), then the ATS-IFWA operator reduces to the following:
IFWA ða1 ; a2 ; . . . ; an Þ ¼
1
n Y i¼1
ð1 lai Þwi ;
n Y
!
v wa
i i
ð36Þ
i¼1
the following: EIFWA ða1 ; a2 ; ...; an Þ wi Q wi Qn ! n Qn w i¼1 1 lai i¼1 1 þ lai 2 i¼1 v aii ¼ Q wi Q wi ; Qn wi Qn wi n n þ i¼1 v ai i¼1 2 v ai þ i¼1 1 lai i¼1 1 þ lai ð37Þ
which is called the Einstein intuitionistic fuzzy weighted averaging (EIFWA) operator. cÞt ; c > 0, then the ATS-IFWA operator Case 3. If gðtÞ ¼ log cþð1 t reduces to the following:
HIFWA ða1 ; a2 ; . . . ; an Þ wi Q wi Qn ni¼1 1 lai i¼1 1 þ ðc 1Þlai ¼ Q wi wi ; Q n þ ðc 1Þ ni¼1 1 lai i¼1 1 þ ðc 1Þlai Qn i¼1
c
i¼1
v wa
1 þ ðc 1Þ 1 v ai
Qn w þ ðc 1Þ i¼1 v aii
g 1
n X
! wi gðlai Þ ; h
1
n X
!! wi hðv ai Þ
ð41Þ
i¼1
Similarly, we can prove the ATS-IFWG operator also satisfies the properties that the ATS-IFWA operator has, here we will not repeat them. Moreover, if the additive generator g is assigned different forms, then the following intuitionistic fuzzy aggregation operators can be obtained: Case 1. If g(t) = log (t), then the ATS-IFWG operator reduces to:
IFWG ða1 ; a2 ; . . . ; an Þ ¼
n Y
n Y lai ; 1 ð1 v ai Þwi wi
i¼1
! ð42Þ
i¼1
which is the intuitionistic fuzzy weighted geometric (IFWG) operator defined by Xu and Yager [34]. Case 2. If gðtÞ ¼ log 2t , then the ATS-IFWG operator reduces t to: EIFWG ða1 ; a2 ;...; an Þ wi Qn w ! Qn Qn w i¼1 1 v ai i 2 i¼1 laii i¼1 1 þ v ai ¼ Q ; Qn wi Q wi Qn wi w n þ ni¼1 laii i¼1 1 þ v ai þ i¼1 1 v ai i¼1 2 lai ð43Þ
!
i
iwi
w
¼ ni¼1 ai i ¼
i¼1
which is the intuitionistic fuzzy weighted averaging (IFWA) operator defined by Xu [29]. Case 2. If gðtÞ ¼ log 2t , then the ATS-IFWA operator reduces to t
Qn
ATS-IFWG ða1 ; a2 ; . . . ; an Þ
ð38Þ
which is called the Hammer intuitionistic fuzzy weighted averaging (HIFWA) operator. Especially, if c = 1, then the HIFWA operator reduces to the IFWA operator; if c = 2, then the HIFWA reduces to the EIFWA operator. operator ; Case 4. If gðtÞ ¼ log cct1 c > 1, then the ATS-IFWA operator 1 reduces to the following: FIFWA ða1 ; a1 ;. .. ; an Þ ! Qn Qn ! 1l ðc ai 1Þwi ðcv ai 1Þwi ¼ 1 logc 1 þ i¼1 ;logc 1 þ i¼1 c1 c1 ð39Þ
which is called the Frank intuitionistic fuzzy weighted averaging (FIFWA) operator. Especially, if c ? 1, then the FIFWA operator reduces to the IFWA operator. Motivated by the geometric mean, the following definition is given: Definition 8. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collection of IFNs, and w = (w1, w2, . . . , wn)T be the weight vector of ai(i = 1, 2, . . . , n), where wi indicatesPthe importance degree of ai, satisfying wi > 0 (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1, if
which is called the Einstein intuitionistic fuzzy weighted geometric (EIFWG) operator defined by Wang and Liu [24]. cÞt ; c > 0, then the ATS-IFWG operator Case 3. If gðtÞ ¼ log cþð1 t reduces to:
HIFWG ða1 ; a2 ; . . . ; an Þ n
¼
w
crodi¼1 laii ; wi Q Qn w þ ðc 1Þ ni¼1 laii i¼1 1 þ ðc 1Þ 1 lai ! wi Qn w Qn i¼1 1 v ai i i¼1 1 þ ðc 1Þv ai w i w ð44Þ Qn Q þ ðc 1Þ ni¼1 1 v ai i i¼1 1 þ ðc 1Þv ai
which is called the Hammer intuitionistic fuzzy weighted geometric (HIFWG) operator. Especially, if c = 1, then the HIFWG operator reduces to the IFWA operator; if c = 2, then the HIFWG operator reduces to the EIFWG operator. ; c > 1, then the ATS-IFWG operator Case 4. If gðtÞ ¼ log cct1 1 reduces to: FIFWG ða1 ; a1 ;...; an Þ Qn Qn l ðc ai 1Þwi ðc1v ai 1Þwi ¼ logc 1 þ i¼1 ;1 logc 1 þ i¼1 c1 c1 ð45Þ
which is called the Frank intuitionistic fuzzy weighted geometric (FIFWG) operator. Especially, if c ? 1, then the FIFWG operator reduces to the IFWG operator.
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4. An approach to intuitionistic fuzzy multi-criteria decision making For a multi-criteria decision making under intuitionistic fuzzy environment, let y = {y1, y2, . . . , ym} be a set of alternatives to be selected, and c = {c1, c2, . . . , cn} be a set of criteria to be evaluated. To evaluate the performance of the alternative yi under the criterion cj, the decision maker is required to provide not only the information that the alternative yi satisfies the criterion cj, but also the information that the alternative yi does not satisfies the criterion cj. These two part information can be expressed by lij and vij which denote the degrees that the alternative yi satisfy the criterion cj and does not satisfy the criterion cj, then the performance of the alternative yi under the criteria cj can be expressed by an IFN aij = (lij, vij) with the condition that 0 6 lij, vij 6 1 and lij + vij 6 1. When all the performances of the alternatives are provided, the intuitionistic fuzzy decision matrix D ¼ ðaij Þmn ¼ ððlij ; v ij ÞÞmn can be constructed. To obtain the ranking of the alternatives, the following steps are given: Step 1. Transform the intuitionistic fuzzy decision matrix D = (aij)nn into the normalized intuitionistic fuzzy decision matrix B = (bij)nn, where
(
bij ¼
aij ; for benefit attribute xi ; i ¼ 1;2;...;m; j ¼ 1;2;...;n acij ; for cost attribute xi ð46Þ
Step 2. Aggregate the IFNs bi of the alternative yi(i = 1, 2, . . . , m), by the ATS-IFWA operator or the ATS-IFWG operator: n
bi ¼ ATS-IFWA ðbi1 ; bi2 ; . . . ; bin Þ ¼ wj bij ; j¼1
i ¼ 1; 2; . . . ; m ð47Þ
or w
bi ¼ ATS-IFWGðbi1 ; bi2 ; . . . ; bin Þ ¼ nj¼1 bij j ;
i ¼ 1; 2; . . . ; m ð48Þ
Table 1 The intuitionistic fuzzy decision matrix D.
y1 y2 y3 y4 y5 y6
c1
c2
c3
c4
(0.60, 0.18) (0.41, 0.25) (0.62, 0.18) (0.21, 0.58) (0.38, 0.19) (0.56, 0.12)
(0.24, 0.44) (0.49, 0.09) (0.67, 0.28) (0.76, 0.22) (0.65, 0.32) (0.50, 0.41)
(0.10, 0.54) (0.10, 0.39) (0.36, 0.42) (0.48, 0.34) (0.06, 0.29) (0.21, 0.07)
(0.45, 0.23) (0.52, 0.45) (0.12, 0.67) (0.15, 0.53) (0.24, 0.39) (0.06, 0.28)
Step 3. Calculate the scores s(bi) of bi by Definition 3, and obtain the priority of the alternatives according to the ranking of bi (i = 1, 2, . . . , m), the bigger the bi, the better the alternative yi. To illustrate the proposed method, we give an example adapted from Ref. [8] as follows: Example 1. The purchasing manager in a small enterprise considers various criteria involving c1: financial factors (e.g., economic performance, financial stability), c2: performance (e.g., delivery, quality, price), c3: technology (e.g., manufacturing capability, design capability, ability to cope with technology changes), and c4: organizational culture and strategy (e.g., feeling of trust, internal and external integration of suppliers, compatibility across levels and functions of the buyer and supplier). The set of evaluative criteria is denoted by C = {c1, c2, c3, c4}, whose weight vector is w = (0.34, 0.23, 0.22, 0.21)T. There are six suppliers available, and the set of all alternatives is denoted by Y = {y1, y2, . . . , y6}. The characteristics of the supplier yi (i = 1, 2, . . . , 6) in terms of the criteria in C are expressed by the following decision matrix (see Table 1): To obtain the alternative(s), the following steps are given: Step 1. Considering all the criteria cj(j = 1, 2, 3, 4) are the benefit criteria, the performance values of the alternatives yi(i = 1, 2, . . . , 6) do not need normalization.
Fig. 1. Scores for alternatives obtained by the HIFWA operator.
M. Xia et al. / Knowledge-Based Systems 31 (2012) 78–88
85
Fig. 2. Scores for alternatives obtained by the HIFWG operator.
Fig. 3. Deviation values for alternatives between the HIFWA and HIFWG operators.
Step 2. Aggregate the intuitionistic fuzzy values ai of the alternative ci by the HIFWA operator (without loss of generality, let c = 1):
a1 ¼ ð0:4075; 0:2964Þ; a2 ¼ ð0:4005; 0:2466Þ; a3 ¼ ð0:5079; 0:3163Þ; a4 ¼ ð0:4437; 0:4049Þ; a5 ¼ ð0:3783; 0:2734Þ; a2 ¼ ð0:3955; 0:1689Þ Step 3. Calculate the scores s(ai) of ai by Definition 7:
sða1 Þ ¼ 0:1111;
sða2 Þ ¼ 0:1539;
sða3 Þ ¼ 0:1915;
sða4 Þ ¼ 0:0388;
sða5 Þ ¼ 0:1049;
sða6 Þ ¼ 0:2266
Since s(a6) > s(a3) > s(a2) > s(a1) > s(a5) > s(a4), we can obtain the priority of the alternatives yi (i = 1, 2, . . . , 6):
y6 y3 y2 y1 y5 y4 :
To investigate the variation trend of the scores and the rankings of the alternatives with the change of the value of the parameter c, we use figures to illustrate these issues. Fig. 1 gives the scores of the alternatives obtained by the HIFWA operator as the parameter c is assigned different values, from which we can find that the scores of the alternatives decrease as the value of the parameter c increases from 0 to 10. Fig. 2 shows the scores of the alternatives obtained by the HIFWG operator, and as the value of c increases from 0 to 10, we can find that the scores of alternatives increase. Fig. 3 illustrates the deviation values between the scores obtained by the HIFWA operator and the ones obtained by the HIFWG operator. It is noted that the scores obtained by the HIFWA operator are bigger than the ones obtained by the HIFWG operator, and as the value of the parameter c increases, the deviation decreases. Moreover, if c = 1, then the scores and rankings of the alternatives obtained in Fig. 1 are the ones obtained by the IFWA operator [29],
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M. Xia et al. / Knowledge-Based Systems 31 (2012) 78–88
Fig. 4. Scores for alternatives obtained by the FIFWA operator.
Fig. 5. Scores for alternatives obtained by the FIFWG operator.
and the results obtained in Fig. 2 are just the ones obtained by the IFWG operator [30]. If we use the FIFWA or FIFWG operator instead of the HIFWA or HIFWG operator to aggregate the attribute values of alternatives, then the scores of alternatives can be found in Figs. 4 and 5, respectively. Fig. 4 gives the scores of the alternatives obtained by the FIFWA operator as the parameter c is assigned different values, from which we can find that the scores of the alternatives decrease as the value of the parameter c increases from 1 to 100. Fig. 5 shows the scores of the alternatives obtained by the FIFWG operator, and as the value of the parameter c increase from 1 to 100, we can find that the scores of the alternatives increase. Fig. 6 illustrates the
deviation values between the scores obtained by FIFWA operator and the ones obtained by the FIFWG operator, it is noted that the scores obtained by the HIFWA operator are bigger than the ones obtained by the HIFWG operator, and as the value of c increases, the deviation decreases. From the above analysis, we can find that the parameter c can be considered as a reflection of the decision makers’ preferences, as the parameter c is assigned different values, the scores of the alternatives are different, and the rankings of the alternatives are also different. Therefore, the proposed aggregation operators with parameters can provide the decision makers more choices and thus the proposed method is more flexible than the existing ones,
M. Xia et al. / Knowledge-Based Systems 31 (2012) 78–88
87
Fig. 6. Deviation values for alternatives between the FIFWA and FIFWG operators.
because we can choose different values of the parameter according to the different situations, which is an interesting topic and is worthy to be further studied in the future. 5. Concluding remarks In this paper, we have given a further study about the application of Archimedean t-conorm and t-norm under intuitionistic fuzzy environment, and given some new operational laws for IFNs, studied their properties and correlations, based on which, some properties of the aggregation principle given by Beliakov et al. [3] have been further discussed, and some important conclusions have been obtained. Some specific intuitionistic fuzzy aggregation operators have been developed including the Hamacher intuitionistic fuzzy aggregation operator and the Frank intuitionistic fuzzy aggregation operator. As the parameter changes in the proposed aggregation operators, some existing ones can be obtained, moreover, the proposed aggregation operators also satisfy all the properties that the existing ones have. An approach for multi-criteria decision making has been developed based on the proposed aggregation operators, and a detailed discussion has been given about the variation trend of the scores and rankings of the alternatives as the parameter changes in the aggregation operators. Acknowledgments 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 partly supported by the National Natural Science Foundation of China (No. 71071161), the National Science Fund for Distinguished Young Scholars of China (No. 70625005), the Ministry of Education Foundation of Humanities and Social Sciences (No. 10YJC630269) and the Pre-Research Foundation of PLA University of Science and Technology (No. 20110511).
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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm Meimei Xia a, Zeshui Xu a,b,⇑, Bin Zhu a a b
School of Economics and Management, Southeast University, Nanjing, Jiangsu 211189, China Institute of Sciences, PLA University of Science and Technology, Nanjing, Jiangsu 210007, China
a r t i c l e
i n f o
Article history: Received 18 November 2011 Received in revised form 19 January 2012 Accepted 7 February 2012 Available online 15 February 2012 Keywords: Multi-criteria decision making Archimedean t-conorm Archimedean t-norm Intuitionistic fuzzy set Aggregation operator
a b s t r a c t Archimedean t-conorm and t-norm are generalizations of a lot of other t-conorms and t-norms, such as Algebraic, Einstein, Hamacher and Frank t-conorms and t-norms or others, and some of them have been applied to intuitionistic fuzzy set, which contains three functions: the membership function, the nonmembership function and the hesitancy function describing uncertainty and fuzziness more objectively. Recently, Beliakov et al. [3] constructed some operations about intuitionistic fuzzy sets based on Archimedean t-conorm and t-norm, from which an aggregation principle is proposed for intuitionistic fuzzy information. In this paper, we propose some other operations on intuitionistic fuzzy sets, study their properties and relationships, and based on which, we study the properties of the aggregation principle proposed by Beliakov et al. [3], and give some specific intuitionistic fuzzy aggregation operators, which can be considered as the extensions of the known ones. In the end, we develop an approach for multi-criteria decision making under intuitionistic fuzzy environment, and illustrate an example to show the behavior of the proposed operators. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction An intuitionistic fuzzy set (IFS) [1] A on a fixed set X is defined as A = {hx, lA(x), vA(x)ijx 2 X} with the condition that 0 6 lA(x) + vA(x) 6 1, lA(x) and vA(x) P 0. We can find that an IFS is constructed by two information functions, which not only describe the membership degree lA(x) of x 2 X in A, but also describe the non-membership degree vA(x) of x 2 X in A. Moreover, the hesitancy information of x 2 X in A can be denoted by pA(x) = 1 lA(x) vA(x) which is called the hesitant index, and therefore IFS can describe the uncertainty and fuzziness more objectively than the usual fuzzy set (FS) [40]. For convenience, the pair a = (la, va) is called an intuitionistic fuzzy number (IFN) [29], where la, va P 0 and la + va 6 1. Since it was introduced, IFS has attracted more and more attentions from researchers and has been used to deal with many problems, especially the multi-criteria decision making problem which can be roughly described as: Let y = {y1, y2, . . . , ym} be the set of alternatives, c = {c1, c2, . . . , cn} be the set of criteria, the degree that alternative yi satisfies to the criterion cj can be denoted as lij, the degree that the alternative yi does not satisfy the criterion cj can be denoted as vij, then the performance of the alternative yi under the criteria cj can be described as an IFN aij = (lij, vij) with the ⇑ Corresponding author at: School of Economics and Management, Southeast University, Nanjing, Jiangsu 211189, China. E-mail addresses: [email protected] (M. Xia), [email protected] (Z. Xu), [email protected] (B. Zhu). 0950-7051/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2012.02.004
condition that 0 6 lij, vij 6 1 and lij + vij 6 1. When all the performances of the alternatives are provided, the intuitionistic fuzzy decision matrix D = (aij)nn = ((lij, vij))mn is constructed. Up to now, many methods have been proposed to deal with the multicriteria decision making under intuitionistic fuzzy environment, which fall into two groups, the first group is to calculate the relative values of the alternatives and the second group is to calculate the actual aggregation values of the alternatives. By comparing these two types of methods for obtaining the ranking of the alternatives, the second one can reflect the actual results of the alternatives more objectively, while the first one can only obtain the relative results of the alternatives to the ideal alternative or others. To calculate the relative values of the alternatives, many classical methods have been extended to intuitionistic fuzzy environment, such as the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method [5,15,19,21], the gray relational analysis (GRA) method [11,26,43], the Elimination et Choice Translating Reality (ELECTRE) method [28], the Vlse Kriterijumska Optimizacija Kompromisno Resenje (VIKOR) method [12,23], the maximizing deviation method [25,31] and the entropy method [9,28,37,38] and so on. To calculate the actual aggregation values of the alternatives, a lot of aggregation operators have been developed. Xu [29] proposed some operational laws for IFNs based on Algebraic t-conorm and t-norm, and developed the intuitionistic fuzzy weighed averaging operator, the intuitionistic fuzzy ordered weighted averaging operator and the intuitionistic fuzzy hybrid averaging operator,
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based on which, Xu and Yager [34] gave some other aggregation operators combining the geometric mean. Based on the generalized ordered weighted aggregation operator proposed by Yager [35], Zhao et al. [44] and Li [20] introduced the generalized intuitionistic fuzzy aggregation operator, which gives a mapping to the arguments before aggregation, and an inverse mapping to the aggregated results at the end. Yang and Chen [36] proposed the quasi-arithmetic intuitionistic fuzzy ordered weighted averaging operator, the quasi-intuitionistic fuzzy aggregation operator based on the Choquet integral and the Dempster–Shafer belief structure. Chen et al. [10] adopted three families of parametric fuzzy unions and intersections of fuzzy operations, to create other aggregation operators for interval-valued intuitionistic fuzzy sets (IVIFSs) [2], which are the generalizations of IFSs allowing the membership and non-membership degrees represented by interval-valued fuzzy numbers (IVFNs) [41]. Wang and Liu [24] introduced some operations on IFSs, such as Einstein sum, Einstein product, Einstein exponentiation, etc., and developed some new geometric aggregation operators, such as the intuitionistic fuzzy Einstein weighted geometric operator and the intuitionistic fuzzy Einstein ordered weighted geometric operator. Moreover, a lot of work [18,27,30, 32,33,36,39] has been done about the weight vector of the intuitionistic fuzzy aggregation operators or others [6,42]. Based on Archimedean t-conorm and t-norm [17,22], and the aggregation functions for the classical FSs [4,13,14], Beliakov et al. [3] gave some operations about intuitionistic fuzzy sets, proposed two general concepts for constructing other types of aggregation operators for IFSs extending the existing methods and showed that the operators obtained by using the Łukasiewicz t-norm are consistent with the ones on ordinary FSs [40]. We can find that the above aggregation operators are all based on different t-conorms and t-norms and are used to deal with different relationships of the aggregated arguments, which can provide more choices for the decision makers. In this paper, we first review the intuitionistic fuzzy operations proposed by Beliakov et al. [3], and then give some other ones based on Archimedean t-conorm and t-norm. We further give an in-depth study on the aggregation principle proposed by Beliakov et al. [3], obtain some important conclusions, and propose some specific intuitionistic fuzzy aggregation operators. The remainder of this paper is organized as follows: Section 2 introduces some new intuitionistic fuzzy operational laws based on Archimedean t-conorm and t-norm. Section 3 investigates the properties of aggregation principle proposed by Beliakov et al. [3], and develops some specific intuitionistic fuzzy aggregation operators. Section 4 proposes an approach to multi-criteria decision making. Section 5 gives some conclusions. 2. Some intuitionistic fuzzy operations based on t-conorm and t-norm Definition 1 (17,22). A function T: [0, 1] [0, 1] ? [0, 1] is called a t-norm if it satisfies the following four conditions: (1) (2) (3) (4)
T(1,x) = x, for all x. T(x, y) = T(y, x), for all x and y. T(x, T(y, z)) = T(T(x, y), z), for all x, y and z. If x 6 x0 and y 6 y0 , then T(x, y) 6 T(x0 , y0 ).
(3) S(x, S(y, z)) = S(S(x, y), z), for all x, y and z. (4) If x 6 x0 and y 6 y0 , then S(x, y) 6 S(x0 , y0 ). It is also usually required that a t-norm and a t-conorm are continuous functions. Definition 3 (17,22). A t-norm function T(x, y) is called Archimedean t-norm if it is continuous and T(x, x) < x for all x 2 (0, 1). An Archimedean t-norm is called strictly Archimedean t-norm if it is strictly increasing in each variable for x, y 2 (0, 1). Definition 4 (17,22). A t-conorm function S(x, y) is called Archimedean t-conorm if it is continuous and S(x, x) > x for all x 2 (0, 1). An Archimedean t-conorm is called strictly Archimedean t-conorm if it is strictly increasing in each variable for x, y 2 (0, 1). It is well known [16] that a strict Archimedean t-norm is expressed via its additive generator g as T(x, y) = g1(g(x) + g(y)), and similarly, applied to its dual t-conorm S(x, y) = h1(h(x) + h(y)) with h(t) = g(1 t). We notice that an additive generator of a continuous Archimedean t-norm is a strictly decreasing function g: [0, 1] ? [0, 1] such that g(1) = 0. If we assign specific forms to the function g, then some well-known t-conorms and t-norms can be obtained: (1) Let g(t) = logt, then h(t) = log (1 t), g1(t) = et, h1(t) = 1 et, and Algebraic t-conorm and t-norm [4] are obtained as follows:
SA ðx; yÞ ¼ x þ y xy;
T A ðx; yÞ ¼ xy
ð1Þ
; g 1 ðtÞ ¼ et 2þ1 ; (2) Let gðtÞ ¼ log 2t , then hðtÞ ¼ log 2ð1tÞ 1t t 1
h ðtÞ ¼ 1 et 2þ1, and we can get Einstein t-conorm and tnorm [4]:
SE ðx; yÞ ¼ (3) Let log
xþy ; 1 þ xy
T E ðx; yÞ ¼
xy 1 þ ð1 xÞð1 yÞ
ð2Þ
cÞt ; c > 0, then we have hðtÞ ¼ gðtÞ ¼ log cþð1 t 1 1 ; g ðtÞ ¼ et þcc1 ; h ðtÞ ¼ 1 et þcc1, and Ham1t
cþð1cÞð1tÞ
acher t-conorm and t-norm [4] are obtained as follows:
x þ y xy ð1 cÞxy ; c>0 1 ð1 cÞxy xy T Hc ðx; yÞ ¼ ; c>0 c þ ð1 cÞðx þ y xyÞ
SHc ðx; yÞ ¼
ð3Þ ð4Þ
Especially, if c = 1, then Hamacher t-conorm and t-norm reduce to the Algebraic t-conorm and t-norm respectively; if c = 2, then Hamacher t-conorm and t-norm reduce to the Einstein t-conorm and t-norm respectively. c1 ; c > 1, then hðtÞ ¼ log c1t ; g 1 ðtÞ ¼ (4) Let gðtÞ ¼ log cct1 1 1 c1þec c1þec log log 1 ec ec ; h ðtÞ ¼ 1 g 1 ðtÞ ¼ , and we have Frank log c log c t-conorm and t-norm [4] as follows:
ðc1x 1Þðc1y 1Þ ; SFc ðx; yÞ ¼ 1 logc 1 þ c1 ðcx 1Þðcy 1Þ T Fc ðx; yÞ ¼ logc 1 þ ; c>1 c1
c>1
ð5Þ ð6Þ
Especially, if c ? 1, then we have Definition 2 (17,22). A function S: [0, 1] [0, 1] ? [0, 1] is called a t-conorm if it satisfies the following four conditions: (1) S(0,x) = x, for all x. (2) S(x, y) = S(y, x), for all x and y.
limc!1 gðtÞ ¼ limc!1 log ¼ limc!1 log
c1 ct 1
1
tct1 1
¼ log t
ð7Þ
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which indicates that limc!1 SFc ðx; yÞ ¼ SAc ðx; yÞ and limc!1 T Fc ðx; yÞ ¼ T Ac ðx; yÞ. Based on Archimedean t-norm and t-conorm [17], Beliakov al. [3] defined the sum operation on two IFNs ai ¼ ðlai ; v ai Þði ¼ 1; 2Þ as a1 a2 ¼ Sðla1 ; la2 Þ; Tðv a1 ; v a2 Þ , which can be expressed by the following:
et
a1 a2 ¼ S la1 ; la2 ; T v a1 ; v a2
1 ¼ h h la1 þ h la2 ; g 1 g v a1 þ g v a2
ð8Þ
1la 1la 1 1Þðc 2 1Þ (17) a1 a2 ¼ 1 logc 1 þ ðc ; c1 v v ðc a1 1Þðc a2 1Þ Þ; c > 1. logc 1 þ c1 l l ðc a1 1Þðc a2 1Þ ; (18) a1 a2 ¼ logc 1 þ c1 1v a 1v a 1 1Þðc 2 1Þ ðc Þ; c > 1. 1 logc 1 þ c1 k k 1 l va a ; logc 1 þ ððcc1Þ1Þ ; k > 0; c > 1. (19) ka ¼ 1 logc 1 þ ðcðc1Þ1Þ k1 k1 k la ðc1v a 1Þk ; k > 0; c > 1. (20) ak ¼ logc 1 þ ððcc1Þ1Þ k1 ; 1 logc 1 þ ðc1Þk1 which are the ones defined based on Frank t-conorm and tnorm. Especially, if c = 1, then (17)–(20) reduce to (5)–(8).
Beliakov et al. [3] also mentioned that for an IFN a = (la, va), let k a = (lka, vka), then g(vka) = kg(va) and h(lka) = k h(la). With the above analysis, the operations about IFNs based on Archimedean t-norm and Archimedean t-conorm [17] can be also expressed as follows:
Moreover, some relations of the operational laws can be discussed as follows:
Definition 5. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2Þ and a = (la, va) be three IFNs, then we have
Theorem 1. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2Þ and a = (la, va) be three IFNs, then the relations of these operational laws are given as:
(1) a1 a2 ¼ Sðla1 ; la2 Þ; Tðv a1 ; v a2 Þ 1 ¼ h ðhðla1 Þ þ hðla2 ÞÞ; g 1 ðgðv a1 Þ þ gðv a2 ÞÞ . (2) a1 a2 ¼ Tðla1 ; la2 Þ; Sðv a1 ; v a2 Þ ¼ g 1 ðgðla1 Þ þ gðla2 ÞÞ; 1
h ðhðv a1 Þ þ hðv a2 ÞÞÞ. 1 (2) ka ¼ h ðkhðla ÞÞ; g 1 ðkgðv a ÞÞ ; k > 0. 1 (3) ak ¼ g 1 ðkgðla ÞÞ; h ðkhðv a ÞÞ ; k > 0.
(1) (2) (3) (4) (5) (6)
Proof. (1) and (2) are obvious, we prove the others:
Especially, if g(t) = log (t), then we have: (5) a1 a2 ¼ ðla1 þ la2 la1 la2 ; v a1 v a2 Þ. (6) a1 a2 ¼ ðla1 la2 ; v a1 þ v a2 v a1 v a2 Þ. (7) ka ¼ ð1 ð1 la Þk ; v ka Þ; k > 0. (8) ak ¼ lka ; 1 ð1 v a Þk ; k > 0
1
(3) kða1 a2 Þ ¼ kðh ðhðla1 Þ þ hðla2 ÞÞ; g 1 ðgðv a1 Þ þ gðv a2 ÞÞÞ ¼ 1
1
¼ ðh ðkðhðla1 Þ þ hðla2 ÞÞÞ; g 1 ðkðgðv a1 Þ þ gðv a2 ÞÞÞÞ. 1 1 (4) ka1 ka2 ¼ ðh ðkhðla1 ÞÞ; g 1 ðkgðv a1 ÞÞÞ ðh ðkhðla2 ÞÞ; g 1 1
a1
a2
1
a
1
g ðkgðv a1 Þ þ kgðv a2 ÞÞÞ ¼ kða1 a2 Þ. 1 1 (5) k1 a k2 a ¼ ðh ðk1 hðla ÞÞ; g 1 ðk1 gðv a ÞÞÞ ðh ðk2 hðla ÞÞ; g 1 1
(13) a1 a2 ¼ (14) a1 a2 ¼ (15) ka ¼ (16) a ¼ k
cþð1cÞt
t
1
2
Similarly, (4) and (6) can be proven which completes the proof of the theorem. h
a
[24] based on Einstein t-conorm and t-norm.
1
1
ðg ðk1 gðv a ÞÞÞ þ gðg ðk2 gðv a ÞÞÞÞÞ ¼ ðh ðk1 hðla Þ þ k2 hðla ÞÞ; g 1 ðk1 gðv a Þ þk2 gðv a ÞÞÞ ¼ ðk1 þ k2 Þa. 1
where (9) and (10) are the ones defined by Wang and Liu
If gðtÞ ¼ log
1
ðk2 gðv a ÞÞÞ ¼ ðh ðhðh ðk1 hðla ÞÞÞ þ hðh ðk2 hðla ÞÞÞÞ; g 1 ðg
2v k ; ð2v ÞÞak þv k a a
a
1
1
(11) ka ¼ ; k > 0. 2lk v a Þk ð1v a Þk (12) ak ¼ ð2l Þak þlk ; ð1þ ;k > 0 ð1þv Þk þð1v Þk a
1
ðkgðv a2 ÞÞÞ ¼ ðh ðhðh ðkhðla1 ÞÞÞ þ hðh ðkhðla2 ÞÞÞÞ; g 1 ðg ðg 1 ðkgðv a1 ÞÞÞ þ gðg 1 ðkgðv a2 ÞÞÞÞÞ ¼ ðh ðkhðla1 Þ þ khðla2 ÞÞ;
conorm and t-norm. If gðtÞ ¼ log 2t , then we have: t la þla v a v a2 (9) a1 a2 ¼ 1þl1 l 2 ; 1þð1v a1 Þð1 . v Þ a a1 a2 1 2 la la2 v a1 þv a2 (10) a1 a2 ¼ 1þð1l 1 Þð1 l Þ ; 1þv a v a . ð1þla Þk ð1la Þk ð1þla Þk þð1la Þk
1
ðh ðkhðh ðhðla1 Þ þ hðla2 ÞÞÞÞ;g 1 ðkgðg 1 ðgðv a1 Þ þ gðv a2 ÞÞÞÞÞ
which are the ones defined by Xu [29] based on Algebraic t-
a1 a2 = a2 a1. a1 a2 = a2 a1. k(a1 a2) = ka1 ka2. ða1 a2 Þk ¼ ak1 ak2 . k1a k2 a = (k1 + k2)a. ak1 ak2 ¼ ak1 þk2 .
; c > 0, then we have: .
la1 þla2 la1 la2 ð1cÞla1 la2 va va ; cþð1cÞðv a 1þv a2 v a v a Þ 1ð1cÞla la 1 2 1 2 1 2
la1 la2 v a1 þv a2 v a1 v a2 ð1cÞv a1 v a2 cþð1cÞðla1 þla2 la1 la2 Þ ; 1ð1cÞv a1 v a2
ð1þðc1Þla Þk ð1la Þk ð1þðc1Þla Þk þðc1Þð1la Þk
.
cv k ; ð1þðc1Þð1v aÞÞk þðc1Þv k ; k > 0. a
a
k k clka ; ð1þðc1Þv a Þ ð1v a Þ ð1þðc1Þð1la ÞÞk þðc1Þlka ð1þðc1Þv a Þk þðc1Þð1v a Þk
; k > 0.
which are the ones defined based on Hammer t-conorm and t-norm. Especially, if c = 1, then (13)–(16) reduce to (5)–(8); if c = 2, then (13)–(16) reduce to (9)–(12). ; c > 1, then we have: If gðtÞ ¼ log cct1 1
Theorem 2. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2Þ and a = (la, va) be three IFNs, and k > 0, then the following are also valid: (1) (2) (3) (4)
(ac)k = (ka)c. k(ac) = (ak)c. ac1 ac2 ¼ ða1 a2 Þc . ac1 ac2 ¼ ða1 a2 Þc .
where ac = (va, la) denotes the complement of an IFN a. Proof. Based on the operations defined in Definition 5, we have: 1 (1) ðac Þk ¼ g 1 ðkgðv a ÞÞ; h khðla Þ ¼ ðkaÞc . 1 (2) kðac Þ ¼ h ðkhðv a ÞÞ; g 1 ðkgðv a ÞÞ ¼ ðak Þc . 1 (3) ac1 ac2 ¼ h hðv a1 Þ þ hðv a2 Þ ; g 1 gðla1 Þ þ gðla2 Þ ¼ ða1 a2 Þc .
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M. Xia et al. / Knowledge-Based Systems 31 (2012) 78–88
(4) ac1 ac2 ¼ g
1
1 gðv a1 Þ þ gðv a2 Þ ; h hðla1 Þ þgðla2 ÞÞÞ ¼ ða1
c
a2 Þ . which completes the proof. h 3. Intuitionistic fuzzy aggregation operators based on t-conorm and t-norm The operational laws defined in Section 2 can be used to aggregate the intuitionistic fuzzy information, which is the focus of this section. Definition 6 (3). Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collection of IFNs, and w = (w1, w2, . . . , wn)T the weight vector of ai(i = 1, 2, . . . , n), where wi indicates the importance degree of ai, satisfying wi P 0 P (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1, if: n
ATS-IFWAða1 ; a2 ; . . . ; an Þ ¼ wi ai
i.e., Eq. (10) holds for n = k + 1. Thus Eq. (10) holds for all n. In addition, we have known that h(t) = g(1 t), and g: [0, 1] ? [0, 1] is a strictly decreasing function, then h(t) is a strictly increasing function which indicates that
06h
n X
! wi hðlai Þ ; g
n X
1
i¼1
! wi gðv ai Þ
1
h
n X i¼1 1
6h
then ATS-IFWA is called the Archimedean t-conorm and t-norm based intuitionistic fuzzy weighted averaging (ATS-IFWA) operator.
n X
!
!
wi h
lai
¼h
1
n X
wi h
lai
i¼1
þg
1
! wi g 1 lai
n X i¼1
þ1h
n X
1
i¼1
ATS-IFWAða1 ; a2 ; . . . ; an Þ n
¼ wi ai ¼ i¼1
h
1
n X
! wi hðlai Þ ; g
1
i¼1
n X
!! wi gðv ai Þ
:
lai
!
¼1
ð15Þ
i¼1
which completes the proof of Theorem 3.
h
Then we can investigate some desirable properties of the ATSIFWA operator, before doing this, a definition is given firstly: Definition 7 (7). Let sðai Þ ¼ lai v ai ði ¼ 1; 2Þ be the scores of ai(i = 1,2) respectively, if s(a1) > s(a2), then a1 is larger than a2, denoted by a1 > a2.
ð10Þ
Proof. By using mathematical induction on n: For n = 2, we have: 2
ATS-IFWA ða1 ; a2 Þ ¼ wi ai ¼ w1 a1 w2 a2 i¼1 1 1 1 ¼ h h h w1 h la1 w2 h la2 þh h ; g 1 g g 1 w1 g v a1 þ g g 1 w2 g v a2 1 ¼ g 1 w1 g la1 þ w2 g la2 ; h w1 h v a1 þ w2 h v a2
Property 1. If all ai(i = 1, 2, . . . , n) are equal, i.e., ai = a = (la, va), for all i, then
ATS-IFWA ða1 ; a2 ; . . . ; an Þ ¼ a
ATS-IFWA ða1 ; a2 ; . . . ; an Þ ¼ ATS-IFWA ða; a; . . . ; aÞ n
¼ wi a ¼
Suppose Eq. (10) holds for n = k, that is
i¼1
k
i¼1
ð12Þ
i¼1
then k
ATS-IFWA ða1 ; a2 ; . . . ; ak ; akþ1 Þ ¼ wi ai wkþ1 akþ1 i¼1 ! !! k k X X 1 1 ¼ h wi h lai ; g wi g v ai i¼1 i¼1 1 wkþ1 h lakþ1 ; g 1 wkþ1 g v akþ1 h !! ! k X 1 1 1 h h wi h lai wkþ1 h lakþ1 þh h ; ¼ h i¼1 !! !! k X wi g v ai þ g g 1 wkþ1 g v akþ1 g 1 g g 1 i¼1 ! k X 1 wi h lai þ wkþ1 h lakþ1 ; ¼ h i¼1 !! k X 1 wi g v ai þ wkþ1 g v akþ1 g i¼1 ! !! kþ1 kþ1 X X 1 ð13Þ wi h lai ; g 1 wi g v ai ¼ h i¼1
h
1
n X
!
wi hðla Þ ; g
n X
1
i¼1
!! wi gðv a Þ
i¼1
1 ¼ h hðla Þ ; g 1 ðgðv a ÞÞ ¼ a
ATS-IFWA ða1 ; a2 ; . . . ; ak Þ ¼ wi ai ¼ w1 a1 w2 a2 wk ak ! !! k k X X 1 1 ¼ h wi hðlai Þ ; g wi hgðv ai Þ
ð16Þ
Proof. Let ai = a = (la, va), we have
ð11Þ
i¼1
wi h
i¼1
Next we give a further study:
i¼1
ð14Þ
! ! n X wi h lai wi g v ai þ g 1
i¼1
Theorem 3 (3). Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collection of IFNs, and w = (w1, w2, . . . , wn)T be the weight vector of ai(i = 1, 2, . . . , n), where wi indicates the importance degree of ai, satisfying wi P 0 Pn (i = 1, 2, . . . , n) and i¼1 wi ¼ 1, then the aggregated value by using the ATS-IFWA operator is also an IFN, and
61
i¼1
and
ð9Þ
i¼1
1
ð17Þ
Property 2. Let ai ¼ ðlai ; v ai Þ and bi ¼ ðlbi ; v bi Þði ¼ 1; 2; . . . ; nÞ be two collections of IFNs, if lai 6 lbi and v ai P v bi , for all i, then
sðATS-IFWA ða1 ; a2 ; . . . ; an ÞÞ 6 sðATS-IFWA ðb1 ; b2 ; . . . ; bn ÞÞ
ð18Þ
Proof. We have known that h(t) = g(1 t), and g: [0, 1] ? 0, 1] is a strictly decreasing function, then h(t) is a strictly increasing function. Since lai 6 lbi and v ai P v bi , then we have 1
h
n X
! ! ! n n X X 1 1 wi h lai wi h lbi ; g wi g v ai 6h
i¼1
i¼1
P g 1
n X
wi g
vb
!
i¼1
i
ð19Þ
i¼1
then
sðATS-IFWA ða1 ; a2 ; . . . ; an ÞÞ 6 sðATS-IFWA ðb1 ; b2 ; . . . ; bn ÞÞ which completes the proof.
ð20Þ
h
Based on Property 2, the following property can be obtained:
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M. Xia et al. / Knowledge-Based Systems 31 (2012) 78–88
Property 3. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collection of IFNs, and
a ¼ mini flai g; maxi fv ai g ; aþ ¼ maxi flai g; mini fv ai g
ð21Þ
then
ATS-IFWA ðr a1 ; r a2 ; .. .; r an Þ !! ! kþ1 kþ1 X X 1 1 1 1 ¼ h wi h h rh lai wi g g rg v ai ;g i¼1
then
sða Þ 6 sðATS IFWA ða1 ; a2 ; . . . ; an ÞÞ 6 sðaþ Þ
ð22Þ
¼ h
1
i¼1
kþ1 X
! !! kþ1 X ; g 1 wi rh lai wi rg v ai
i¼1
Property 4. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collections of IFNs, w = (w1, w2, . . . , wn)T be their weight vector such that wi P 0 P (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1, if b = (lb, vb) is an IFN, then
and r ATS-IFWA ða1 ; a2 ;... ; an Þ ¼ h
1
1
rh h
ATS-IFWA ða1 b; a2 b; . .. ; an bÞ ¼ ATS-IFWA ða1 ; a2 ;. .. ; an Þ b ð23Þ
n X
w i h l ai
i¼1
¼ h
1
r
n X
!!!
!
wi h lai ;g 1 r
aj b ¼ h1 ðhðlai Þ þ hðlb ÞÞ; g 1 gðv ai Þ þ gðv b Þ
¼
n X
wi g g 1 g v ai þ g v b
i¼1
! ; wi h lai þ h lb
1
n X
1
n X
h
i¼1
g
wi g v ai þ g v b
va
va
!!!!
i
i¼1
!!
ð30Þ
i
i¼1
Property 6. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collections of IFNs, and w = (w1, w2, . . . , wn)T be the weight vector of them such that P wi P 0 (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1, if r > 0, b = (lb, vb) is an IFN, then
ATS-IFWA ða1 b; a2 b; . . . ; an bÞ ! n X 1 1 ; wi h h h lai þ h lb ¼ h g 1
wi g
ð24Þ
we have
i¼1
n X
wi g
According to Properties 4 and 5, we can get Property 6 easily:
n X
;g 1 rg g 1
i¼1
Proof. Since
ð29Þ
i¼1
ATS-IFWA ðra1 b; r a2 b; . . . ; r an bÞ
!!
¼ r ATS-IFWA ða1 ; a2 ; . . . ; an Þ b
ð31Þ
Property 7. Let ai ¼ ðlai ; v ai Þ and bi ¼ ðlbi ; v bi Þði ¼ 1; 2; . . . ; nÞ be two collections of IFNs, and w = (w1, w2, . . . , wn)T be the weight vector P of them such that wi P 0 (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1, then
!! ð25Þ
i¼1
ATS-IFWA ða1 b1 ; a2 b2 ; . . . ; an bn Þ ¼ ATS-IFWA ða1 ; a2 ; . . . ; an Þ ATS-IFWA ðb1 ; b2 ; . . . ; bn Þ
ð32Þ
and
ATS-IFWA ða1 ; a1 ; .. .; an Þ b ! !! n n X X 1 1 ¼ h wi h lai ; g wi g v ai lb ; v b i¼1
¼ h
1
n X
h h
!! ! þ h lb ; wi h lai
i¼1
g 1 g g 1
n X
wi g
va
!! þg
i
i¼1
¼ h
1
wi h
lai þ h lb
i¼1
¼ h
1
n X
wi h lai þ h lb
vb
! ; g 1 !
va
i
þg
vb
;g
i¼1
n X
g 1
wi g v ai þ g v b
!! wi g g 1 g v ai þ g v bi
i¼1
!!
1
¼ h
!! ! n X 1 wi h lai þ h lbi wi g v ai þ g v bi ;g
n X i¼1
i¼1
i¼1
ð34Þ
ð26Þ which completes the proof.
and
h
Property 5. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collections of IFNs, and w = (w1, w2, . . . , wn)T be their weight vector such that wi P 0 P (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1, if r > 0, then
ATS-IFWA ðr a1 ; ra2 ; . . . ; ran Þ ¼ r ATS-IFWA ða1 ; a2 ; . . . ; an Þ
ð27Þ
ATS-IFWA ða1 ; a2 ; ...; an Þ ATS-IFWA ðb1 ;b2 ;...;bn Þ !! ! n n X X 1 1 ¼ h wi h lai ; g wi g v ai i¼1
h
1
n X
i¼1
!! ! n X wi h lbi ; g 1 wi g v bi
i¼1
Proof. According to Definition 5, we have
1 r a ¼ h rhðla Þ ; g 1 ðrgðv a ÞÞ
ð33Þ
i¼1
!!
i¼1 1
ATS-IFWA ða1 b1 ; a2 b2 ;...; an bn Þ ! n X 1 1 ¼ h wi h h h lai þ h lbi ;
wi g
n X
then
!!
n X
ai bi ¼ h1 hðlai Þ þ hðlbi Þ ; g 1 g v ai þ gðv bi Þ
i¼1
1
n X
Proof. According to Definition 5, we have
1
¼ h
i¼1
1
h h
n X
!! !!! n X 1 wi h lai wi h lbi þh h ;
i¼1
ð28Þ g
1
g g
1
n X i¼1
wi g
!!
va
i
i¼1
þg g
1
n X i¼1
wi g
!!!!
vb
i
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M. Xia et al. / Knowledge-Based Systems 31 (2012) 78–88
¼ h
1
! n n X X wi h lai þ wi h lbi ;
wi
ATS-IFWG ða1 ; a2 ; . . . ; an Þ ¼ ni¼1 ai
ð40Þ
ð35Þ
then ATS-IFWG is called the Archimedean t-cornorm and t-norm based the intuitionistic fuzzy geometric (ATS-IFWG) operator. Based on the operational laws of the IFNs given in Definition 5, we can derive the following theorem:
If the additive generator g is assigned different forms, then some specific intuitionistic fuzzy aggregation operators can be obtained as follows:
Theorem 4. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collection of IFNs, and w = (w1, w2, . . . , wn)T be the weight vector of ai(i = 1, 2, . . . , n), where wi indicates the importance degree of ai, satisfying wi > 0 P (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1, then the aggregated value by using the ATS-IFWG operator is also an IFN, and
i¼1
g 1
n X
i¼1
wi g
i¼1
va
i
þ
n X
wi g
!!
vb
i
i¼1
which completes the proof. h
Case 1. If g(t) = log (t), then the ATS-IFWA operator reduces to the following:
IFWA ða1 ; a2 ; . . . ; an Þ ¼
1
n Y i¼1
ð1 lai Þwi ;
n Y
!
v wa
i i
ð36Þ
i¼1
the following: EIFWA ða1 ; a2 ; ...; an Þ wi Q wi Qn ! n Qn w i¼1 1 lai i¼1 1 þ lai 2 i¼1 v aii ¼ Q wi Q wi ; Qn wi Qn wi n n þ i¼1 v ai i¼1 2 v ai þ i¼1 1 lai i¼1 1 þ lai ð37Þ
which is called the Einstein intuitionistic fuzzy weighted averaging (EIFWA) operator. cÞt ; c > 0, then the ATS-IFWA operator Case 3. If gðtÞ ¼ log cþð1 t reduces to the following:
HIFWA ða1 ; a2 ; . . . ; an Þ wi Q wi Qn ni¼1 1 lai i¼1 1 þ ðc 1Þlai ¼ Q wi wi ; Q n þ ðc 1Þ ni¼1 1 lai i¼1 1 þ ðc 1Þlai Qn i¼1
c
i¼1
v wa
1 þ ðc 1Þ 1 v ai
Qn w þ ðc 1Þ i¼1 v aii
g 1
n X
! wi gðlai Þ ; h
1
n X
!! wi hðv ai Þ
ð41Þ
i¼1
Similarly, we can prove the ATS-IFWG operator also satisfies the properties that the ATS-IFWA operator has, here we will not repeat them. Moreover, if the additive generator g is assigned different forms, then the following intuitionistic fuzzy aggregation operators can be obtained: Case 1. If g(t) = log (t), then the ATS-IFWG operator reduces to:
IFWG ða1 ; a2 ; . . . ; an Þ ¼
n Y
n Y lai ; 1 ð1 v ai Þwi wi
i¼1
! ð42Þ
i¼1
which is the intuitionistic fuzzy weighted geometric (IFWG) operator defined by Xu and Yager [34]. Case 2. If gðtÞ ¼ log 2t , then the ATS-IFWG operator reduces t to: EIFWG ða1 ; a2 ;...; an Þ wi Qn w ! Qn Qn w i¼1 1 v ai i 2 i¼1 laii i¼1 1 þ v ai ¼ Q ; Qn wi Q wi Qn wi w n þ ni¼1 laii i¼1 1 þ v ai þ i¼1 1 v ai i¼1 2 lai ð43Þ
!
i
iwi
w
¼ ni¼1 ai i ¼
i¼1
which is the intuitionistic fuzzy weighted averaging (IFWA) operator defined by Xu [29]. Case 2. If gðtÞ ¼ log 2t , then the ATS-IFWA operator reduces to t
Qn
ATS-IFWG ða1 ; a2 ; . . . ; an Þ
ð38Þ
which is called the Hammer intuitionistic fuzzy weighted averaging (HIFWA) operator. Especially, if c = 1, then the HIFWA operator reduces to the IFWA operator; if c = 2, then the HIFWA reduces to the EIFWA operator. operator ; Case 4. If gðtÞ ¼ log cct1 c > 1, then the ATS-IFWA operator 1 reduces to the following: FIFWA ða1 ; a1 ;. .. ; an Þ ! Qn Qn ! 1l ðc ai 1Þwi ðcv ai 1Þwi ¼ 1 logc 1 þ i¼1 ;logc 1 þ i¼1 c1 c1 ð39Þ
which is called the Frank intuitionistic fuzzy weighted averaging (FIFWA) operator. Especially, if c ? 1, then the FIFWA operator reduces to the IFWA operator. Motivated by the geometric mean, the following definition is given: Definition 8. Let ai ¼ ðlai ; v ai Þði ¼ 1; 2; . . . ; nÞ be a collection of IFNs, and w = (w1, w2, . . . , wn)T be the weight vector of ai(i = 1, 2, . . . , n), where wi indicatesPthe importance degree of ai, satisfying wi > 0 (i = 1, 2, . . . , n) and ni¼1 wi ¼ 1, if
which is called the Einstein intuitionistic fuzzy weighted geometric (EIFWG) operator defined by Wang and Liu [24]. cÞt ; c > 0, then the ATS-IFWG operator Case 3. If gðtÞ ¼ log cþð1 t reduces to:
HIFWG ða1 ; a2 ; . . . ; an Þ n
¼
w
crodi¼1 laii ; wi Q Qn w þ ðc 1Þ ni¼1 laii i¼1 1 þ ðc 1Þ 1 lai ! wi Qn w Qn i¼1 1 v ai i i¼1 1 þ ðc 1Þv ai w i w ð44Þ Qn Q þ ðc 1Þ ni¼1 1 v ai i i¼1 1 þ ðc 1Þv ai
which is called the Hammer intuitionistic fuzzy weighted geometric (HIFWG) operator. Especially, if c = 1, then the HIFWG operator reduces to the IFWA operator; if c = 2, then the HIFWG operator reduces to the EIFWG operator. ; c > 1, then the ATS-IFWG operator Case 4. If gðtÞ ¼ log cct1 1 reduces to: FIFWG ða1 ; a1 ;...; an Þ Qn Qn l ðc ai 1Þwi ðc1v ai 1Þwi ¼ logc 1 þ i¼1 ;1 logc 1 þ i¼1 c1 c1 ð45Þ
which is called the Frank intuitionistic fuzzy weighted geometric (FIFWG) operator. Especially, if c ? 1, then the FIFWG operator reduces to the IFWG operator.
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M. Xia et al. / Knowledge-Based Systems 31 (2012) 78–88
4. An approach to intuitionistic fuzzy multi-criteria decision making For a multi-criteria decision making under intuitionistic fuzzy environment, let y = {y1, y2, . . . , ym} be a set of alternatives to be selected, and c = {c1, c2, . . . , cn} be a set of criteria to be evaluated. To evaluate the performance of the alternative yi under the criterion cj, the decision maker is required to provide not only the information that the alternative yi satisfies the criterion cj, but also the information that the alternative yi does not satisfies the criterion cj. These two part information can be expressed by lij and vij which denote the degrees that the alternative yi satisfy the criterion cj and does not satisfy the criterion cj, then the performance of the alternative yi under the criteria cj can be expressed by an IFN aij = (lij, vij) with the condition that 0 6 lij, vij 6 1 and lij + vij 6 1. When all the performances of the alternatives are provided, the intuitionistic fuzzy decision matrix D ¼ ðaij Þmn ¼ ððlij ; v ij ÞÞmn can be constructed. To obtain the ranking of the alternatives, the following steps are given: Step 1. Transform the intuitionistic fuzzy decision matrix D = (aij)nn into the normalized intuitionistic fuzzy decision matrix B = (bij)nn, where
(
bij ¼
aij ; for benefit attribute xi ; i ¼ 1;2;...;m; j ¼ 1;2;...;n acij ; for cost attribute xi ð46Þ
Step 2. Aggregate the IFNs bi of the alternative yi(i = 1, 2, . . . , m), by the ATS-IFWA operator or the ATS-IFWG operator: n
bi ¼ ATS-IFWA ðbi1 ; bi2 ; . . . ; bin Þ ¼ wj bij ; j¼1
i ¼ 1; 2; . . . ; m ð47Þ
or w
bi ¼ ATS-IFWGðbi1 ; bi2 ; . . . ; bin Þ ¼ nj¼1 bij j ;
i ¼ 1; 2; . . . ; m ð48Þ
Table 1 The intuitionistic fuzzy decision matrix D.
y1 y2 y3 y4 y5 y6
c1
c2
c3
c4
(0.60, 0.18) (0.41, 0.25) (0.62, 0.18) (0.21, 0.58) (0.38, 0.19) (0.56, 0.12)
(0.24, 0.44) (0.49, 0.09) (0.67, 0.28) (0.76, 0.22) (0.65, 0.32) (0.50, 0.41)
(0.10, 0.54) (0.10, 0.39) (0.36, 0.42) (0.48, 0.34) (0.06, 0.29) (0.21, 0.07)
(0.45, 0.23) (0.52, 0.45) (0.12, 0.67) (0.15, 0.53) (0.24, 0.39) (0.06, 0.28)
Step 3. Calculate the scores s(bi) of bi by Definition 3, and obtain the priority of the alternatives according to the ranking of bi (i = 1, 2, . . . , m), the bigger the bi, the better the alternative yi. To illustrate the proposed method, we give an example adapted from Ref. [8] as follows: Example 1. The purchasing manager in a small enterprise considers various criteria involving c1: financial factors (e.g., economic performance, financial stability), c2: performance (e.g., delivery, quality, price), c3: technology (e.g., manufacturing capability, design capability, ability to cope with technology changes), and c4: organizational culture and strategy (e.g., feeling of trust, internal and external integration of suppliers, compatibility across levels and functions of the buyer and supplier). The set of evaluative criteria is denoted by C = {c1, c2, c3, c4}, whose weight vector is w = (0.34, 0.23, 0.22, 0.21)T. There are six suppliers available, and the set of all alternatives is denoted by Y = {y1, y2, . . . , y6}. The characteristics of the supplier yi (i = 1, 2, . . . , 6) in terms of the criteria in C are expressed by the following decision matrix (see Table 1): To obtain the alternative(s), the following steps are given: Step 1. Considering all the criteria cj(j = 1, 2, 3, 4) are the benefit criteria, the performance values of the alternatives yi(i = 1, 2, . . . , 6) do not need normalization.
Fig. 1. Scores for alternatives obtained by the HIFWA operator.
M. Xia et al. / Knowledge-Based Systems 31 (2012) 78–88
85
Fig. 2. Scores for alternatives obtained by the HIFWG operator.
Fig. 3. Deviation values for alternatives between the HIFWA and HIFWG operators.
Step 2. Aggregate the intuitionistic fuzzy values ai of the alternative ci by the HIFWA operator (without loss of generality, let c = 1):
a1 ¼ ð0:4075; 0:2964Þ; a2 ¼ ð0:4005; 0:2466Þ; a3 ¼ ð0:5079; 0:3163Þ; a4 ¼ ð0:4437; 0:4049Þ; a5 ¼ ð0:3783; 0:2734Þ; a2 ¼ ð0:3955; 0:1689Þ Step 3. Calculate the scores s(ai) of ai by Definition 7:
sða1 Þ ¼ 0:1111;
sða2 Þ ¼ 0:1539;
sða3 Þ ¼ 0:1915;
sða4 Þ ¼ 0:0388;
sða5 Þ ¼ 0:1049;
sða6 Þ ¼ 0:2266
Since s(a6) > s(a3) > s(a2) > s(a1) > s(a5) > s(a4), we can obtain the priority of the alternatives yi (i = 1, 2, . . . , 6):
y6 y3 y2 y1 y5 y4 :
To investigate the variation trend of the scores and the rankings of the alternatives with the change of the value of the parameter c, we use figures to illustrate these issues. Fig. 1 gives the scores of the alternatives obtained by the HIFWA operator as the parameter c is assigned different values, from which we can find that the scores of the alternatives decrease as the value of the parameter c increases from 0 to 10. Fig. 2 shows the scores of the alternatives obtained by the HIFWG operator, and as the value of c increases from 0 to 10, we can find that the scores of alternatives increase. Fig. 3 illustrates the deviation values between the scores obtained by the HIFWA operator and the ones obtained by the HIFWG operator. It is noted that the scores obtained by the HIFWA operator are bigger than the ones obtained by the HIFWG operator, and as the value of the parameter c increases, the deviation decreases. Moreover, if c = 1, then the scores and rankings of the alternatives obtained in Fig. 1 are the ones obtained by the IFWA operator [29],
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M. Xia et al. / Knowledge-Based Systems 31 (2012) 78–88
Fig. 4. Scores for alternatives obtained by the FIFWA operator.
Fig. 5. Scores for alternatives obtained by the FIFWG operator.
and the results obtained in Fig. 2 are just the ones obtained by the IFWG operator [30]. If we use the FIFWA or FIFWG operator instead of the HIFWA or HIFWG operator to aggregate the attribute values of alternatives, then the scores of alternatives can be found in Figs. 4 and 5, respectively. Fig. 4 gives the scores of the alternatives obtained by the FIFWA operator as the parameter c is assigned different values, from which we can find that the scores of the alternatives decrease as the value of the parameter c increases from 1 to 100. Fig. 5 shows the scores of the alternatives obtained by the FIFWG operator, and as the value of the parameter c increase from 1 to 100, we can find that the scores of the alternatives increase. Fig. 6 illustrates the
deviation values between the scores obtained by FIFWA operator and the ones obtained by the FIFWG operator, it is noted that the scores obtained by the HIFWA operator are bigger than the ones obtained by the HIFWG operator, and as the value of c increases, the deviation decreases. From the above analysis, we can find that the parameter c can be considered as a reflection of the decision makers’ preferences, as the parameter c is assigned different values, the scores of the alternatives are different, and the rankings of the alternatives are also different. Therefore, the proposed aggregation operators with parameters can provide the decision makers more choices and thus the proposed method is more flexible than the existing ones,
M. Xia et al. / Knowledge-Based Systems 31 (2012) 78–88
87
Fig. 6. Deviation values for alternatives between the FIFWA and FIFWG operators.
because we can choose different values of the parameter according to the different situations, which is an interesting topic and is worthy to be further studied in the future. 5. Concluding remarks In this paper, we have given a further study about the application of Archimedean t-conorm and t-norm under intuitionistic fuzzy environment, and given some new operational laws for IFNs, studied their properties and correlations, based on which, some properties of the aggregation principle given by Beliakov et al. [3] have been further discussed, and some important conclusions have been obtained. Some specific intuitionistic fuzzy aggregation operators have been developed including the Hamacher intuitionistic fuzzy aggregation operator and the Frank intuitionistic fuzzy aggregation operator. As the parameter changes in the proposed aggregation operators, some existing ones can be obtained, moreover, the proposed aggregation operators also satisfy all the properties that the existing ones have. An approach for multi-criteria decision making has been developed based on the proposed aggregation operators, and a detailed discussion has been given about the variation trend of the scores and rankings of the alternatives as the parameter changes in the aggregation operators. Acknowledgments 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 partly supported by the National Natural Science Foundation of China (No. 71071161), the National Science Fund for Distinguished Young Scholars of China (No. 70625005), the Ministry of Education Foundation of Humanities and Social Sciences (No. 10YJC630269) and the Pre-Research Foundation of PLA University of Science and Technology (No. 20110511).
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