Intuitionistic fuzzy geometric interaction averaging ... - Semantic Scholar
Information Sciences 259 (2014) 142–159
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Intuitionistic fuzzy geometric interaction averaging operators and their application to multi-criteria decision making Yingdong He a, Huayou Chen a,⇑, Ligang Zhou a, Jinpei Liu b, Zhifu Tao a a b
School of Mathematical Science, Anhui University, Hefei Anhui 230601, China School of Business, Anhui University, Hefei Anhui 230601, China
a r t i c l e
i n f o
Article history: Received 7 November 2012 Received in revised form 30 June 2013 Accepted 5 August 2013 Available online 20 August 2013 Keywords: Intuitionistic fuzzy set Probability non-membership (PN) function operator Probability hetergeneous (PH) operator Intuitionistic fuzzy geometric interaction averaging operator Decision making
a b s t r a c t This paper proposes some new geometric operations on intuitionistic fuzzy sets (IFSs) based on probability non-membership (PN) function operator, probability membership (PM) function operator and probability hetergeneous (PH) operator, which are constructed from the probability point of view. The geometric interpretations of these operations are given. Moreover, we develop some intuitionistic fuzzy geometric interaction averaging (IFGIA) operators. The properties of these aggregation operators are investigated. The key advantage of the IFGIA operators is that the interactions between non-membership function and membership function of different IFSs are considered. Finally, an approach to multiple attributes decision making is given based on the proposed aggregation operators under intuitionistic fuzzy environment, and an example is illustrated to show the validity and feasibility of the proposed approach. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Intuitionistic fuzzy sets (IFSs), as a generalization of fuzzy sets [42], developed by Atanassov [3], is a powerful tool to deal with vagueness. Since information aggregation is a pervasive activity in daily life, many researches had been done on this issue [7,15,24,27,32,37,34,35,30,39,38,18,26]. Among them, the weighted geometric averaging (WGA) operator [24] and the ordered weighted geometric averaging (OWGA) operator [34] are the most common operators. On a basis of the multiplication operation by Atanassov [4] and power operation by De and Biswas [12] on intuitionistic fuzzy sets, Xu and Yager [37] proposed some intutionistic fuzzy geometric aggregation operators and applied them to multi-attribute decision making problems. After these pioneering works, more attentions have been paid to intuitionistic fuzzy multi-criteria decision making problems [1,2,5,6,8,10,11,13,14,19–23,25,29,33,36,41,43–51]. Dymova and Sevastjinov [13] presented a method to deal with the intuitionistic fuzzy multi-criteria decision making problems based on Dempster–Shefer theory of evidence. Ye [40] used entropy weight to get criteria weights and rank alternatives according to the correlation coefficients. Xu [33] developed intuitionistic fuzzy power aggregation operators. Xu and Xia [36] presented the induced generalized aggregation operators under interval-valued intuitionistic fuzzy environments. Zhu et al. [51] proposed hesitant fuzzy geometric Bonferroni means. Li et al. [19] investigated the relationship between the similarity measure and the entropy of IFSs. Zhang et al. [45] presented
⇑ Corresponding author. Tel.: +86 13615690958. E-mail addresses: [email protected] (Y. He), [email protected] (H. Chen), [email protected] (L. Zhou), [email protected] (J. Liu), [email protected] (Z. Tao). 0020-0255/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2013.08.018
Y. He et al. / Information Sciences 259 (2014) 142–159
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the intuitionistic fuzzy rough approximation operators and discussed their connections with special intuitionistic fuzzy relations. However, it is found that the operational laws and geometric aggregation operators on intuitionistic fuzzy sets in [4,37] are not suitable to be used in the special circumstances. For example, suppose that A and B are two intuitionistic fuzzy sets, A = huA, vAi and uA = 0, B = huB, vBi and uB – 0, then according to the multiplication operation by Atanassov [4], we have uAB = 0. Obviously, uB is not accounted for at all, which is an undesirable feature of an averaging operation. Furthermore, the IFHGA operator [37] has similar problems. For example, if Ai ¼ huAi ; v Ai i; i ¼ 1; 2; . . . ; n; i – k are a collection of intutionitisc fuzzy sets, Ak ¼ h0; v Ak i, and v Ak – 0, then we have uIFHGAx;w ðA1 ;...;An Þ ¼ 0 by using aggregation law in [37]. It is obvious that uAi ði ¼ 1; 2; . . . ; n; i – kÞ have no effects on the aggregation result. Motivated by the works of [4,37] and the idea of interactions between non-membership function and membership function of different intuitionistic fuzzy sets, we focus on developing some new geometric operations on intuitionistic fuzzy sets (IFSs) and giving the geometric interpretations of these operations. Based on the new operations, we propose some intuitionistic fuzzy geometric interaction aggregation operators, including the IFWGIA operator, the IFOWGIA operator and the IFHGIA operator, which are more practical for an averaging operator. By the comparison with the existing method, it is concluded that the method proposed in this paper is a good complement to the existing works on IFSs, especially when one of the membership degrees of intutionitisc fuzzy sets is zero. The rest of the paper is organized as follows. Section 2 reviews some basic concepts. Section 3 introduces new geometric operations on intuitionistic fuzzy sets and gives the geometric interpretations of these operations. In Section 4, we develop the intuitionistic fuzzy geometric interaction averaging (IFGIA) operators, and investigate their properties. In Section 5, an approach to intuitionistic fuzzy multi-criteria decision making is given based on the proposed IFHGIA operator. In Section 6, a numerical example is illustrated to show the feasibility and validity of the new approach, and the comparison between the work of this paper and other corresponding works is presented systematically. Finally, Section 7 concludes the paper. 2. Preliminaries The concept of fuzzy sets (FSs) was introduced by Zadeh [42]. Let X be a universe of discourse in the following. Definition 1 [42]. A fuzzy set F in X is defined as follows: F = {hx, uF(x)ijx 2 X}, where uF:X ? [0, 1] is the membership function of the fuzzy set F, and 0 6 uF(x) 6 1. Atanassov [3] generalized the fuzzy set to intuitionistic fuzzy set (IFS) by adding an hesitation degree. Definition 2 [3]. An intuitionistic fuzzy set in X is an expression: A = {hx, uA(x), vA(x)ijx 2 X}, where the functions uA:X ? [0, 1] and vA:X ? [0, 1] define the degree of membership and the degree of nonmembership of the element x 2 X to A, and for every x 2 X, 0 6 uA(x) + vA(x) 6 1. For each IFS A in X, if pA(x) = 1 uA(x) vA(x), for all x 2 X, then pA(x) is called the degree of indeterminacy of the element x to the set A. In practice, intuitionistic fuzzy numbers can be denoted as A = hu, vi [32,37]. For convenience, the sets of all the intuitionistic fuzzy numbers are denoted by IFNs. Some basic operations, such as multiplication operation [4] and power operation [12], were introduced under intuitionistic fuzzy environment. Definition 3 (4,12). Suppose that A = huA, vAi and B = huB, vBi are two intuitionistic fuzzy sets, then
ð1Þ A B ¼ huA uB ; v A þ v B v A v B i D E ð2Þ Ak ¼ ukA ; 1 ð1 v A Þk ; k > 0
ð1Þ ð2Þ
Chen and Tan [9] proposed a score function S(A) = uA vA to evaluate the degree of suitability that an alternative satisfies a decision make’s requirement under intuitionistic fuzzy environment, where A is an intuitionistic fuzzy set, and A = huA, vAi. The score of A is directly related to the deviation between uA and vA, i.e., the bigger the score of intuitionistic fuzzy set A, the larger the intuitionistic fuzzy set A. Hong and Choi [16] presented an accuracy function H(A) = uA + vA to evaluate the accuracy degree of the intuitionistic fuzzy set A = huA, vAi, where 0 6 H(A) 6 1. The larger the value of H(A), the higher the accuracy degree of intuitionistic fuzzy set A [32]. Based on score function [9] and accuracy function [16], Xu [32,37] gave the comparison law for intuitionistic fuzzy sets as follows. Definition 4 (32,37). Let A = huA, vAi and B = huB, vBi be two intuitionistic fuzzy sets. Then A < B if and only if (i) S(A) < S(B). or (ii) S(A) = S(B) and H(A) < H(B).
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3. New geometric operational laws on intuitionistic fuzzy sets 3.1. New geometric operational laws and corresponding geometric interpretations Considering that the existing multiplication operation [4] on IFSs cannot be used in some cases from averaging point of view, we introduce new operational laws on intuitionistic fuzzy sets, including multiplication operation and power operation. The feature of new geometric operational laws is that we take the interactions into consideration between non-membership function and membership function of different intuitionistic fuzzy sets. Definition 5. Suppose that A = huA, vAi and B = huB, vBi are two intuitionistic fuzzy numbers, then
^ ¼ hð1 v A Þð1 v B Þ ð1 ðuA þ v A ÞÞð1 ðuB þ v B ÞÞ; 1 ð1 v A Þð1 v B Þi 1Þ AB
ð3Þ
2Þ Ak ¼ hð1 v A Þk ð1 ðuA þ v A ÞÞk ; 1 ð1 v A Þk i; k > 0
ð4Þ
The geometric meaning of new multiplication operation on intuitionistic fuzzy sets can be interpreted from three aspects as follows. The operational rule between non-membership function and non-membership function of different IFSs. The operational rule between membership function and membership function of different IFSs. The operational rule between non-membership function and membership function of different IFSs. Let A = huA, vAi and B = huB, vBi be two intuitionistic fuzzy numbers. (1) The operational rule between non-membership function and non-membership function can be explained geometrically as follows in Fig. 1. vA and vB denote the non-membership degree of A and B, respectively. From the probability point of view, we may as well regard vA and vB as two independent events, E(vA, vB) represents the probability of vA and vB occurring simultaneously. Thus, E(vA, vB) = vA vB. Therefore, v AB ^ ¼ v A þ v B v A v B . v AB ^ is called the probability non-membership (PN) function operator, i.e.,
PNðv A ; v B Þ ¼ v A þ v B v A v B
ð5Þ
(2) The operational rule between membership function and membership function can be explained geometrically as follows in Fig. 2. uA and uB denote the membership degree of A and B, respectively. We may as well regard uA and uB as two independent events, E(uA,uB) represents the probability of uA and uB occurring simultaneously. Thus, E(uA, uB) = uA uB. Therefore, uAB ^ ¼ uA þ uB uA uB . uAB ^ is called the probability membership (PM) function operator, i.e.,
PMðuA ; uB Þ ¼ uA þ uB uA uB :
ð6Þ
(3) The operational rule between non-membership function and membership function can be explained geometrically as follows in Figs. 3 and 4. vA denotes the non-membership degree of A, uB denotes the membership degree ofB. Similarly, I(vA, uB) represents the probability of two independent events vA and lB occurring simultaneously. Thus, I(vA,uB) = vA uB. I(vA, uB) is called probability hetergeneous (PH) operator, i.e.,
PHðv A ; uB Þ ¼ uB v A :
ð7Þ
In similar way, I(vB, uA) denotes the probability of two independent events vB and uA occurring simultaneously. Thus, I(vB, uA) = vB uA. I(vB, uA) is called the probability hetergeneous (PH) operator, i.e.
PHðv B ; uA Þ ¼ uA v B :
ð8Þ
(4) The interaction between different intuitionistic fuzzy sets can be explained geometrically as follows in Fig. 5.
Fig. 1. Operational rule of non-membership function of different IFSs.
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Fig. 2. Operational rule of membership function of different IFSs.
Fig. 3. Operational rule of non-membership function and membership function of different IFSs.
Fig. 4. Operational rule of non-membership function and membership function of different IFSs.
Fig. 5. Operational rule of different IFSs.
From Fig. 5 and Eqs. (5)–(8), the multiplication of different IFNs can be rewritten as follows.
^ ¼ hPMðuA ; uB Þ PHðuA ; v B Þ PHðuB ; v A Þ; PNðv A ; v B Þi AB
ð9Þ
^ contains PH(vA, uB) and PH(vB, uA), while the From Fig. 5 and Eq. (9), it is obvious that the non-membership function of AB ^ does not contain PH(vA, uB) and PH(vB, uA). membership function of AB In fact, if A, B 2 IFNs, A = huA, vAi, vA – 1 and uA = 0, B = huB, vBi and uB – 0, then according to Eq. (9), we have
PMðuA ; uB Þ PHðuA ; v B Þ PHðuB ; v A Þ ¼ uB uB v A – 0; which shows that uA cannot play a decisive role. As a result, the proposed Definition 5 is more practical for an averaging operator, which can be seen visually by the following example. Example 1. Suppose that A = huA, vAi = h0, 0.6i and B = huB, vBi = h0.6, 0.2i are two intuitionistic fuzzy numbers, then by Definition 3, we have
A B ¼ h0 0:6; 0:6 þ 0:2 0:6 0:2i ¼ h0; 0:68i: According to Definition 5, we get that
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^ ¼ h0 þ 0:6 0 0:2 0 0:6 0:6 0:6; 0:6 þ 0:2 0:6 0:2i ¼ h0:24; 0:68i: AB i.e., uAB ^ ¼ 0:24, which indicates that uA dose not play a decisive role. It is noted that PN(vA, vB) = vA + vB vA vB = 1 (1 vA) (1 vB), and
PMðuA ; uB Þ PHðuA ; v B Þ PHðuB ; v A Þ ¼ ðPMðuA ; uB Þ þ PNðv A ; v B Þ PHðuA ; v B Þ PHðuB ; v A ÞÞ PNðv A ; v B Þ ¼ ððuA þ uB uA uB Þ þ ðv A þ v B v A v B Þ ðuA v B Þ ðuB v A ÞÞ PNðv A ; v B Þ ¼ ððuA þ v A Þ þ ðuB þ v B Þ ðuA þ v A Þ ðuB þ v B ÞÞ PNðv A ; v B Þ ¼ ð1 ð1 ðuA þ v A ÞÞ ð1 ðuB þ v B ÞÞÞ ð1 ð1 v A Þ ð1 v B ÞÞ ¼ ð1 v A Þ ð1 v B Þ ð1 ðuA þ v A ÞÞ ð1 ðuB þ v B ÞÞ: Therefore, Eq. (9) is equivalent to Eq. (3). Next, we explain the origin of the power operation. By Eq. (3), we have
^ ¼ hð1 v A Þ2 ð1 ðuA þ v A ÞÞ2 ; 1 ð1 v A Þ2 i: A2 ¼ AA Similarly, we have
^ ¼ hð1 v A Þ3 ð1 ðuA þ v A ÞÞ3 ; 1 ð1 v A Þ3 i: A3 ¼ A2 A By analogy, for any positive integer n, we have
^ ¼ hð1 v A Þn ð1 ðuA þ v A ÞÞn ; 1 ð1 v A Þn i: An ¼ An1 A We extend positive integer n to any nonnegative real number k, and have
Ak ¼ hð1 v A Þk ð1 ðuA ðxÞ þ v A ðxÞÞÞk ; 1 ð1 v A Þk i; k > 0: 3.2. The properties of the new geometric operations In this subsection, we investigate some properties of the PN function operator, the PM function operator, the PH function operator, the new multiplication operation and power operation under intuitionistic fuzzy environment. Theorem 1. Let A = huA, vAi, B = huB, vBi, C = huC, vCi, A0 ¼ huA0 ; v A0 i and B0 ¼ huB0 ; v B0 i be five intuitionistic fuzzy numbers, then (1) (2) (3) (4) (5)
Boundedness: PN(1, 1) = 1, PN(0, 0) = 0, 0 6 PN(vA, vB) 6 1. Monotonicity: if v A 6 v A0 and v B 6 v 0B , then PNðv A ; v B Þ 6 PN v 0A ; v 0B . Commutativity: PN(vA, vB) = PN(vB, vA). Associativity: PN(vA, PN(vB, vC)) = PN(PN(vA, vB), vC). Zero element: PN(0, vA) = PN(vA, 0) = vA.
Proof. (1) According to Eq. (5), we have
PNðv A ; v B Þ ¼ v A þ v B v A v B ¼ 1 ð1 v A Þð1 v B Þ
ð10Þ
Therefore,
PNð1; 1Þ ¼ 1 þ 1 1 1 ¼ 1;
PNð0; 0Þ ¼ 1 1 ¼ 0;
It is noted that 0 6 vA 6 1, 0 6 vB 6 1. Thus, 0 6 (1 vA)(1 vB) 6 1, then according to Eq. (10), we have 0 6 PN(vA, vB) 6 1. (2) Because v A 6 v 0A and v B 6 v 0B , we have 1 v A P 1 v 0A ; 1 v B P 1 v 0B , According to Eq. (10), we obtain
PNðv A ; v B Þ 6 PN
v 0A ; v 0B
(4) According to Eq. (10), we get
PNðv A ; PNðv B ; v C ÞÞ ¼ 1 ð1 v A Þð1 PNðv B ; v C ÞÞ and
1 PNðv B ; v C Þ ¼ ð1 v B Þð1 v C Þ; Thus, PN(vA, PN(vB, vC)) = 1 (1 vA)(1 vB)(1 vC).
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Similarly, PN(PN(vA, vB), vC) = 1 (1 vA)(1 vB)(1 vC). Therefore, PN(vA, PN(vB, vC)) = PN(PN(vA, vB), vC). According to Eq. (5), we can get the results of (3) and (5) easily. So the proofs are omitted here.
h
Theorem 2. Let A = huA, vAi, B = huB, vBi, C = huC, vCi, A0 ¼ huA0 ; v A0 i and B0 ¼ huB0 ; v B0 i be five intuitionistic fuzzy numbers, then we have (1) Boundedness: PH(0, 0) = 0, PH(1, 1) = 1, 0 6 PH(vA, vB) 6 1. (2) Monotonicity: If uA 6 u0A and v B 6 v 0B , then PHðuA ; v B Þ 6 PH u0A ; v 0B ; 0 0 PHðv A ; uB Þ 6 PH v A ; uB . (3) Commutativity: PH(uA, vB) = PH(vB, uA), PH(vA, uB) = PH(uB, vA). (4) Associativity: PH(uA, PH(uB, uC)) = PH(PH(uA, uB), uC). (5) Identify element: PH(1, uA) = PH(uA, 1) = uA.
and
if
v A 6 v 0A
and
uB 6 u0B ,
then,
Proof. According to Eq. (8), we can get Theorem 2 easily, so it is omitted here. h In practice, the operational results of geometric operational laws on intuitionistic fuzzy numbers are also intuitionistic fuzzy numbers. ^ ¼ huC ; v C i; D ¼ Ak ¼ huD ; v D i, then C, D 2 IFNs. Theorem 3. If A = huA, vAi 2 IFNs and B ¼ huB ; v B i 2 IFNs; k > 0; C ¼ AB The proof of theorem 3 is similar to that in [37], so it is omitted here. Theorem 4. Let A = huA, vAi and B = huB, vBi be two intuitionistic fuzzy numbers, k, k1, k2 > 0. Then, we have ^ ¼ BA; ^ (1) AB ^ k ¼ Ak B ^ k; (2) ðABÞ k1 k2 ðk1 þk2 Þ ^ (3) A A ¼ A . The proof of theorem 4 is similar to that in [37], so it is omitted here. 4. Intuitionistic fuzzy geometric interaction averaging operators 4.1. Intuitionistic fuzzy weighted geometric averaging (IFWGA) operator Xu and Yager [37] generalized the WGA operators and the OWGA operators to intuitionistic fuzzy environment and proposed the intuitionistic fuzzy weighted geometric averaging (IFWGA) operator and the intuitionistic fuzzy ordered weighted geometric averaging (IFOWGA) operator. Definition 6 [37]. Let Ai ¼ huAi ; v Ai iði ¼ 1; 2; . . . ; nÞ be a collection of intuitionistic fuzzy numbers. If the mapping
IFWGAw ðA1 ; A2 . . . ; An Þ ¼
1 Aw 1
2 Aw 2
...
n Aw n
¼
* n Y
w uAii ; 1
i¼1
n Y
+
ð1 v Ai Þ
wi
;
i¼1
then IFWGAw is called intuitionistic fuzzy weighted geometric averaging operator with respect to a weighting vector w, P where w = (w1, w2, . . . , wn)T is the weighting vector of Ai with wi 2 [0, 1] and ni¼1 wi ¼ 1. Definition 7 [37]. Let Ai ¼ huAi ; v Ai iði ¼ 1; 2; . . . ; nÞ be a collection of intuitionistic fuzzy numbers. If the mapping w1
IFOWGAw ðA1 ; . . . ; An Þ ¼ ðArð1Þ Þ
. . . ðArðnÞ Þ
wn
¼
* n Y i¼1
w uAri ðiÞ ; 1
+ n Y wi ; ð1 v ArðiÞ Þ
ð11Þ
i¼1
then IFOWGAw is called intuitionistic fuzzy ordered weighted geometric averaging operator with respect to a weighting vector w, where Ar(i) is the ith largest value of Ai (i = 1, 2, . . . , n) according to Definition 4, w = (w1, w2, . . . , wn)T is the weighting P vector with wi 2 [0, 1] and ni¼1 wi ¼ 1. 4.2. Intuitionistic fuzzy weighted geometric interaction averaging (IFWGIA) operator Motivated by the WGA operator [24] and the IFWGA operator [37], we develop intuitionistic fuzzy weighted geometric interaction averaging (IFWGIA) operator based on the proposed operational laws.
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Definition 8. Let Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ be a collection of intuitionistic fuzzy numbers. If the mapping n
w IFWGIAw ðA1 ; . . . ; An Þ ¼ ^ Ai i ; i¼1
then IFWGIAw is called intuitionistic fuzzy weighted geometric interaction averaging (IFWGIA) operator, where w = (w1, w2, P . . . , wn)T is the weighting vector of Ai with wi 2 [0, 1] and ni¼1 wi ¼ 1. Theorem 5. Let Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ be a collection of intuitionistic fuzzy numbers. Then,
IFWGIAw ðA1 ; . . . ; An Þ ¼
* n Y
wi
ð1 v Ai Þ
i¼1
n Y
+ n Y wi : ð1 ðuAi þ v Ai ÞÞ ; 1 ð1 v Ai Þ wi
i¼1
ð12Þ
i¼1
Proof. We prove Eq. (12) by using mathematical induction on n.
D E 1 1 1 1 (1) When n = 1, w1 = 1, we have IFWGIAw ðA1 Þ ¼ Aw . 1 ¼ huA1 ; v A1 i ¼ ð1 v A1 Þ ð1 ðuA1 þ v A1 ÞÞ ; 1 ð1 v A1 Þ Thus, Eq. (12) holds for n = 1. (2) If Eq. (12) holds for n = k, i.e.,
IFWGIAw ðA1 ; A2 ; . . . ; Ak Þ ¼
* k Y
+ k k Y Y wi wi : ð1 v Ai Þ ð1 ðuAi þ v Ai ÞÞ ; 1 ð1 v Ai Þ wi
i¼1
i¼1
i¼1
Then, when n = k + 1, by the operational laws in Definition 5, we have kþ1
w IFWGIAw ðA1 ; A2 ; . . . Akþ1 Þ ¼ ^ Ai i i¼1
^ kþ1 Þwkþ1 ¼ IFWGIAw ðA1 ; A2 ; . . . Ak ÞðA * + k k k Y Y Y wi wi wi ¼ ð1 v Ai Þ ð1 ðuAi þ v Ai ÞÞ ; 1 ð1 v Ai Þ i¼1
i¼1
i¼1
^ hð1 v Ai Þwi ð1 ðuAi þ v Ai ÞÞwi ; 1 ð1 v Ai Þwi i * + kþ1 kþ1 kþ1 Y Y Y ð1 v Ai Þwi ð1 ðuAi þ v Ai ÞÞwi ; 1 ð1 v Ai Þwi ; ¼ i¼1
i¼1
i¼1
i.e. Eq. (12) holds for n = k + 1. Therefore, by using mathematical induction on n, Eq. (12) holds for all n. h Theorem 6. If Ai ¼ huAi ; v Ai i 2 IFNs, i = 1, 2, . . . , n, then the aggregated value by using the IFWGIA operator is also an intuitionistic fuzzy number, i.e. IFWGIA(A1, . . . , An) 2 IFNs. Proof. Since Ai ¼ huAi ; v Ai i 2 IFNs; i ¼ 1; 2; . . . ; n, by Definition 2, we have
0 6 uAi ; v Ai 6 1 and 0 6 uAi þ v Ai 6 1; then
061
n Y ð1 v Ai Þwi 6 1;
06
i¼1
n Y
ð1 v Ai Þwi
i¼1
n Y ð1 ðuAi þ v Ai ÞÞwi 6 1 i¼1
and
1
n Y
! ð1 v Ai Þ
wi
þ
n odi¼1 ð1
i¼1
wi
v Ai Þ
n Y i¼1
! wi
ð1 ðuAi þ v Ai ÞÞ
¼1
n Y
ð1 ðuAi þ v Ai ÞÞwi 2 ½0; 1:
i¼1
Thus,
IFWGIAðA1 ; . . . ; An Þ 2 IFNs:
Theorem 7. Let Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ and Bi ¼ huBi ; v Bi iði ¼ 1; . . . ; nÞ be two collections of intuitionistic fuzzy numbers and P w = (w1, w2, . . . , wn)T is the associated weighting vector satisfying wi 2 [0, 1] and ni¼1 wi ¼ 1.
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(1) Idempotency: If Ai ¼ A0 ¼ huA0 ; v A0 i for all i, then IFWGIAw(A1, A2, . . . , An) = A0. (2) Boundedness: Let A ¼ hmaxf0; ðminðuAi þ v Ai Þ maxðv Ai ÞÞg; maxðv Ai Þi,
Aþ ¼ hmaxðuAi þ v Ai Þ minðv Ai Þ; minðv Ai Þi: Then we have A 6 IFWGIAw(A1, A2, . . . , An) 6 A+. (3) Monotonicity: When v Ai P v Bi , uAi þ v Ai 6 uBi þ v Bi for all i, we have IFWGIAw(A1, A2, . . . , An) 6 IFWGIAw(B1, B2, . . . , Bn).
Proof. (1) Since Ai ¼ A0 ¼ huA0 ; v A0 iði ¼ 1; . . . ; nÞ and
IFWGIAw ðA1 ; A2 ; . . . An Þ ¼
* n Y
wi
ð1 v A0 Þ
i¼1
* ¼
n Y
Pn
i¼1 wi
¼ 1, by Theorem 5, we have
ð1 ðuA0
n Y þ v A0 ÞÞ ; 1 ð1 v A0 Þwi
i¼1
i¼1
n X
n X
wi
ð1 v A0 Þ i¼1
+
wi
n X
wi
wi
+
ð1 ðuA0 þ v A0 ÞÞ i¼1 ; 1 ð1 v A0 Þ i¼1
¼ huA0 ; v A0 i ¼ A0 :
Pn Q (2) Since maxðv Ai Þ ¼ 1 ð1 maxðv Ai ÞÞ i¼1 wi P 1 ni¼1 ð1 v Ai Þwi , and Pn Q Q 1 ni¼1 ð1 v Ai Þwi P 1 ni¼1 ð1 minðv Ai ÞÞwi ¼ 1 ð1 minðv Ai ÞÞ i¼1 wi ¼ minðv Ai Þ, thus, n X
n X
maxðuAi þ v Ai Þ minðv Ai Þ ¼ ð1 minðv A1 ÞÞ i¼1
ð1 maxðuAi þ v Ai ÞÞ i¼1 n X
wi
P
n Y
n Y
i¼1
i¼1
ð1 v Ai Þwi
wi
wi
ð1 ðuAi þ v Ai ÞÞwi P ð1 maxðv Ai ÞÞ i¼1
ð1 minðuAi þ v Ai ÞÞ i¼1
¼ minðuAi þ v Ai Þ maxðv Ai Þ: According to Theorem 6, IFWGIA(A1, . . . , An) 2 IFNs, we have n n Y Y ð1 v Ai Þwi ð1 ðuAi þ v Ai ÞÞwi P 0: i¼1
i¼1
Therefore, n n Y Y ð1 v Ai Þwi ð1 ðuAi þ v Ai ÞÞwi P maxf0; ðminðuAi þ v Ai Þ maxðv Ai ÞÞg: i¼1
i¼1
Then, according to Definition 4, we obtain that
A 6 IFWGIAw ðA1 ; A2 . . . An Þ 6 Aþ : Q Q (3) Since v Ai P v Bi , we have 1 ni¼1 ð1 v Ai Þwi 6 1 ni¼1 ð1 v Bi Þwi . Because v Ai P v Bi ði ¼ 1; 2; . . . ; nÞ and uAi þ v Ai 6 uBi þ v Bi ði ¼ 1; 2; . . . ; nÞ, we have n n n n Y Y Y Y ð1 v Ai Þwi ð1 ðuAi þ v Ai ÞÞwi 6 ð1 v Bi Þwi ð1 ðuBi þ v Bi ÞÞwi : i¼1
i¼1
i¼1
n X
i¼1
Therefore, according to Definition 4, we get
IFWGIAw ðA1 ; A2 ; . . . An Þ 6 IFWGIAw ðB1 ; B2 ; . . . Bn Þ: In some special cases, the IFWGIA operator reduces to the IFWGA operator. h Remark 1. If w = (1, 0, . . . , 0)T, then IFWGIAw(A1, A2, . . . , An) = IFWGAw(A1, A2, . . . , An). Remark 2. If w = (0, 0, . . . , 1)T, then IFWGIAw(A1, A2, . . . , An) = IFWGAw(A1, A2, . . . , An).
wi
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Remark 3. If wi = 1, wj = 0, and j – i, then we have IFWGIAw(A1, A2, . . . , An) = IFWGAw(A1, A2, . . . , An). 4.3. Intuitionistic fuzzy ordered weighted geometric interaction averaging (IFOWGIA) operator Inspired by the OWGA operator [34] and the IFOWGA operator [37], we propose intuitionistic fuzzy ordered weighted geometric interaction averaging (IFOWGIA) operator. Definition 9. Let Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ be a collection of intuitionistic fuzzy numbers. If
IFOWGIAw ðA1 ; . . . ; An Þ ¼ ^ni¼1 ðArðiÞ Þwi ; then IFOWGIAw is called intuitionistic fuzzy ordered weighted geometric interaction averaging operator, where Ar(i) is the ith largest value of Ai(i = 1, . . . , n), w = (w1, w2, . . . , wn)T is the weighting vector of the IFOWGIA operator, with wi 2 [0, 1] and Pn i¼1 wi ¼ 1. Theorem 8. Let Ai ¼ huAi ; v Ai i, i = 1, 2, . . . , n be a collection of intuitionistic fuzzy numbers. Then,
IFOWGIAw ðA1 ; . . . ; An Þ ¼
* n Y
wi
ð1 v ArðiÞ Þ
i¼1
n Y
ð1 ðuArðiÞ
+ n wi Y ; þ v ArðiÞ ÞÞ ; 1 1 v ArðiÞ wi
i¼1
ð13Þ
i¼1
The proof is similar to Theorem 5, so it is omitted here. Theorem 9. Let Ai ¼ huAi ; v Ai i 2 IFNs, i = 1, 2, . . . , n. Then the aggregated value by using the IFOWGIA operator is also an intuitionitic fuzzy number, i.e.,
IFOWGIAðA1 ; . . . ; An Þ 2 IFNs: Proof. Since ArðiÞ ¼ huArðiÞ ; v ArðiÞ i 2 IFN, i = 1, 2, . . . , n, according to Definition 2, we have
0 6 uArðiÞ ; v ArðiÞ 6 1 and 0 6 uArðiÞ þ v ArðiÞ 6 1; Therefore,
061
n Y ð1 v ArðiÞ Þwi 6 1;
06
n Y
i¼1
ð1 v ArðiÞ Þwi
i¼1
n Y
1 ðuArðiÞ þ v ArðiÞ Þ
wi
61
i¼1
and
1
n Y
! ð1 v ArðiÞ Þwi
þ
i¼1
¼1
n Y
n Y
ð1 v ArðiÞ Þwi
i¼1 wi
ð1 ðuArðiÞ þ v ArðiÞ ÞÞ
n wi Y 1 uArðiÞ þ v ArðiÞ
!
i¼1
2 ½0; 1:
i¼1
Thus, IFOWGIA(A1, . . . , An) 2 IFNs.
h
Theorem 10. Suppose that Ai ¼ huAi ; v Ai i and Bi ¼ huBi ; v Bi i are two intuitionistic fuzzy numbers, i = 1, . . . , n. And w = (w1, w2, P = (w1, w2, . . . , wn)T is the weighting vector of the IFOWGIA operator, satisfying wi 2 [0, 1] and ni¼1 wi ¼ 1. (1) Idempotency: If Ai ¼ A0 ¼ huA0 ; v A0 i for all i, then IFOWGIAw(A1, A2, . . . , An) = A0. (2) Boundedness: If A ¼ hmaxf0; ðminðuAi þ v Ai Þ maxðv Ai ÞÞg; maxðv Ai Þi, Aþ ¼ hmaxðuAi þ v Ai Þ minðv Ai Þ; minðv Ai Þi, then
A 6 IFOWGIAw ðA1 ; A2 . . . An Þ 6 Aþ : (3) Commutativity: Suppose that A0i ¼ huA0i ; v A0i iði ¼ 1; . . . nÞ is any permutation of Ai ¼ huAi ; v Ai iði ¼ 1; . . . nÞ, then
IFOWGIAw A01 ; A02 . . . A0n ¼ IFOWGIAw ðA1 ; A2 . . . An Þ:
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Proof. (1) Since Ai = A0 for all i, and
IFOWGIAw ðA1 ; . . . ; An Þ ¼ ¼
* n Y
i¼1 wi
¼ 1, according to Theorem 8, we have
ð1 v ArðiÞ Þwi
i¼1
n n Y Y ð1 ðuArðiÞ þ v ArðiÞ ÞÞwi ; 1 ð1 v ArðiÞ Þwi i¼1
n n Y Y ð1 v A0 Þwi ð1 ðuA0 þ v A0 ÞÞwi ; 1 ð1 v A0 Þwi
i¼1
i¼1 n X
wi
ð1 v A0 Þ i¼1
+
+
i¼1
i¼1
n X
* ¼
* n Y
Pn
n X
wi
wi
+
ð1 ðuA0 þ v A0 ÞÞ i¼1 ; 1 ð1 v A0 Þ i¼1
¼ huA0 ; v A0 i ¼ A0 : (2) Obviously, n X
wi
maxðv Ai Þ ¼ 1 ð1 maxðv ArðiÞ ÞÞ i¼1
P1
n Y ð1 v ArðiÞ Þwi i¼1
¼1
n Y
ð1 v Ai Þwi P 1
i¼1
n Y
n X
wi
ð1 minðv ArðiÞ ÞÞ
¼ 1 ð1 minðv ArðiÞ ÞÞ i¼1
wi
¼ minðv Ai Þ;
i¼1
i.e.,
maxðv Ai Þ P 1
n Y ð1 v ArðiÞ Þwi P minðv Ai Þ: i¼1
Similarly, we have n X
n X
maxðuAi þ v Ai Þ minðv Ai Þ ¼ ð1 minðv ArðiÞ ÞÞ i¼1
ð1 maxðuArðiÞ þ v ArðiÞ ÞÞ i¼1 n X
wi
wi
n X wi wi n n Y Y P ð1 v ArðiÞ Þwi ð1 ðuArðiÞ þ v ArðiÞ ÞÞwi P ð1 maxðv ArðiÞ ÞÞ i¼1 ð1 minðuArðiÞ þ v ArðiÞ ÞÞ i¼1 i¼1
i¼1
¼ minðuAi þ v Ai Þ maxðv Ai Þ; i.e.,
maxðuAi þ v Ai Þ minðv Ai Þ P
n Y
ð1 v ArðiÞ Þwi
i¼1
n Y ð1 ðuArðiÞ þ v ArðiÞ ÞÞwi P minðuAi þ v Ai Þ maxðv Ai Þ i¼1
By Theorem 9, IFOWGIA(A1, . . . , An) 2 IFNs, we have n n Y Y ð1 v ArðiÞ Þwi ð1 ðuArðiÞ þ v ArðiÞ ÞÞwi P 0: i¼1
i¼1
Qn
Q Therefore, i¼1 ð1 v ArðiÞ Þwi ni¼1 ð1 ðuArðiÞ þ v ArðiÞ ÞÞwi P maxf0; ðminðuAi þ v Ai Þ maxðv Ai ÞÞg. Then, according to Definition 4, we have
A 6 IFOWGIAw ðA1 ; A2 . . . An Þ 6 Aþ : (3) According to Eq. (13), we get
IFOWGIAw A01 ; . . . ; A0n
* + n n n wi Y wi wi Y Y ; 1 v A0rðiÞ 1 uA0rðiÞ þ v A0rðiÞ ;1 1 v A0rðiÞ ¼ i¼1
IFOWGIAw ðA1 ; A2 . . . An Þ ¼
* n Y i¼1
Since
A0i
i¼1
i¼1
+ n n Y Y wi wi wi : ð1 v ArðiÞ Þ ð1 ðuArðiÞ þ v ArðiÞ ÞÞ ; 1 ð1 v ArðiÞ Þ i¼1
i¼1
¼ huA0i ; v A0i iði ¼ 1; . . . nÞ is any permutation of Ai ¼ huAi ; v Ai iði ¼ 1; . . . nÞ, then we have A0rðiÞ ¼ ArðiÞ ði ¼ 1; . . . nÞ. Thus,
IFOWGIAw A01 ; A02 . . . A0n ¼ IFOWGIAw ðA1 ; A2 . . . An Þ:
The IFOWGIA operator reduces to the IFOWGA operator when we take special cases of the weighting vector w. h Remark 4. If w = (1, 0, . . . , 0)T, then IFOWGIAw(A1, A2, . . . , An) = IFOWGAw(A1, A2, . . . , An).
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Remark 5. If w = (0, 0, . . . , 1)T, then IFOWGIAw(A1, A2, . . . , An) = IFOWGAw(A1, A2, . . . , An). Remark 6. If wi = 1, wj = 0, and j – i, then
IFOWGAw ðA1 ; A2 . . . An Þ ¼ IFOWGIAw ðA1 ; A2 . . . An Þ ¼ ArðiÞ ; where Ar(i) is the ith largest of Ai(i = 1, . . . , n). 4.4. Intuitionistic fuzzy hybrid geometric interaction averaging Operator Considering both the given intuitionistic fuzzy value and its ordered position, Xu and Yager [37] developed the intuitionistic fuzzy hybrid geometric averaging (IFHA) operator. Definition 10 37. Let Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ be a collection of intuitionistic fuzzy numbers. The IFHGA operator with respect to weighting vector w (IFHGAw) is defined as
e rð1Þ Þwi . . . ð A e rðnÞ Þwn ¼ IFHGAx;w ðA1 ; . . . ; An Þ ¼ ð A
* n Y i¼1
wi
u ;1 eA rðiÞ
n Y i¼1
+
wi
ð1 v e Þ A rðiÞ
;
ð14Þ
e rðiÞ is theith largest of the intuitionistic fuzzy values A e i ¼ Anxi ; i ¼ 1; . . . ; n. x = (x1, . . . , xn)T is the weight vector of where A i Pn Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ, satisfying xi 2 [0, 1] and i¼1 xi ¼ 1, n is the balancing coefficient. Suppose that Ai ¼ huAi ; v Ai i 2 IFNs, i = 1, 2, . . . , n. If uAk ¼ 0 and uAi – 0ði – k; Þ, then according to Definition 10, we have uIFHGIAx;w ðA1 ;...;An Þ ¼ 0. Obviously, uAi ði – k; i ¼ 1; 2; . . . ; nÞ are not accounted for at all, which is an undesirable feature of an averaging operation. As a result, we propose the intuitionistic fuzzy hybrid geometric interaction averaging operator, taking interactions into consideration between membership function and non-membership function of different intuitionistic fuzzy sets. Definition 11. Let Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ be a collection of intuitionistic fuzzy numbers, X be the set of all intutionistic fuzzy numbers. The intuitionistic fuzzy hybrid geometric interaction averaging (IFHGIA) operator of dimension n is a P mapping IFHGIA:Xn ? X, which has an associated vector w = (w1, w2, . . . , wn)T, satisfying wi 2 [0, 1] and ni¼1 wi ¼ 1 such that
e rðiÞ Þwi ; IFHGIAx;w ðA1 ; . . . ; An Þ ¼ ^ni¼1 ð A e rðiÞ is the ith largest of the intuitionistic fuzzy values A e i ¼ Anxi ; i ¼ 1; . . . ; n. x = (x1, . . . , xn)T is the weight vector of where A i Pn Ai, i = 1, . . . , n, satisfying xi 2 [0, 1] and i¼1 xi ¼ 1, n is the balancing coefficient.. Theorem 11. Let Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ be a collection of intuitionistic fuzzy numbers. Then
IFHGIAx;w ðA1 ; . . . ; An Þ ¼
* n Y i¼1
wi
½1 v e A rðiÞ
n Y i¼1
½1 ðue
A rðiÞ
+ n Y wi : þ v e Þ ; 1 ð1 v e Þ A rðiÞ A rðiÞ wi
ð15Þ
i¼1
The proof is similar to Theorem 5, so it is omitted here. The IFHGIA operator can be interpreted from four aspects as follows. (1) It not only includes the interactions of non-membership function of different IFSs, and that of membership function of different IFSs, but also the interactions are involved between non-membership function and membership function of different IFSs. (2) It weights the intuitionistic fuzzy numbers Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ by the associated weights x = (x1, . . . , xn)T and nx a balancing coefficient n, and then we can get the weighted intuitionistic fuzzy numbers Ai i ði ¼ 1; . . . ; nÞ. nx e rð1Þ ; A e rð2Þ ; . . . ; A e rðnÞ Þ, where (3) It reorders the weighted intuitionistic fuzzy values Ai i ði ¼ 1; . . . ; nÞ in descending order ð A e rðiÞ is the ith largest of the weighted intuitionistic fuzzy values A e i ¼ Anxi ; i ¼ 1; . . . ; n. A i (4) It considers the given intuitionistic fuzzy numbers and the ordered positions, and then the intuitionistic fuzzy nume rðiÞ Þwi ði ¼ 1; 2; . . . ; nÞ are aggregated into a collective one. bers ð A Suppose that Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ is a collection of IFNs. If uAk ¼ 0; uAi – 0ði – k; i ¼ 1; 2; . . . ; nÞ, then by Definition 11, we have uIFHGIAx;w ðA1 ;...;An Þ – 0. Thus, uAk does not play a decisive role. Therefore, the proposed IFHGIA operator can be seen as a good complement to the existing IFHGA operator, which can be seen clearly in the following example. Example 2. Let A1 = h0, 0.5i, A2 = h0.4, 0.2i, A3 = h0.5, 0.4i, A4 = h0.3, 0.3i, A5 = h0.7, 0.1i be five intuitionistic fuzzy numbers. x = (0.25, 0.20, 0.15, 0.18, 0.22)T is the weight vector of Ai (i = 1, 2, . . . , 5). Then according to the operational law of Definition 3, we have
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Y. He et al. / Information Sciences 259 (2014) 142–159
e 1 ¼ h050:25 ; 1 ð1 0:5Þ50:25 i ¼ h0; 0:5796i; A e 2 ¼ h0:450:20 ; 1 ð1 0:2Þ50:20 i ¼ h0:4000; 0:2000i; A e 4 ¼ h0:350:18 ; 1 ð1 0:3Þ50:18 i ¼ h0:287; 0:275i; e 3 ¼ h0:550:15 ; 1 ð1 0:4Þ50:15 i ¼ h0:504; 0:318i; A A 50:22 50:22 e 5 ¼ h1 ð1 0:1Þ ; 0:7 i ¼ h0:7203; 0:1094i: A According to Definition 3, we obtain that
e 1 Þ ¼ 0:5796; Sð A
e 2 Þ ¼ 0:2000; Sð A
e 3 Þ ¼ 0:2763; Sð A
e 4 Þ ¼ 0:0638; Sð A
e 5 Þ ¼ 0:5660: Sð A
Obviously,
e 5 Þ > Sð A e 3 Þ > Sð A e 2 Þ > Sð A e 4 Þ > Sð A e 1 Þ: Sð A Therefore,
e rð1Þ ¼ h0:6755; 0:1094i; A e rð5Þ ¼ h0; 0:5796i: A
e rð2Þ ¼ h0:5946; 0:3183i; A
e rð3Þ ¼ h0:4000; 0:2000i; A
Supposing that w = (w1, w2, . . . , w5)T is determined by w = (0.112, 0.236, 0.304, 0.236, 0.112). By Definition 10, we get
A ¼ IFHGAw;x ðA1 ; A2 ; A3; A4 ; A5 Þ ¼
* 5 Y j¼1
D
wj
ðue Þ ; 1 A rðjÞ
5 Y j¼1
the
normal
e rð4Þ ¼ h0:3384; 0:2746i; A
distribution
based
method
[31],
+ ð1 v e Þ
wj
A rðjÞ
¼ 0:67550:112 0:59460:236 0:40:304 0:33840:236 0:00000:112 ; 1 ð1 0:1094Þ0:112 ð1 0:3183Þ0:236 ð1 0:2Þ0:304 ð1 0:2746Þ0:236 ð1 0:5796Þ0:112
E
¼ h0; 0:2911i: While, by Definition 11 and Theorem 11, we have
D E e 1 ¼ ð1 0:5Þ50:25 ð1 0:5Þ50:25 ; 1 ð1 0:5Þ50:25 ¼ h0; 0:5796i; A D E e 2 ¼ ð1 0:2Þ50:20 ð1 0:6Þ50:20 ; 1 ð1 0:2Þ50:20 ¼ h0:4000; 0:2000i; A D E e 3 ¼ ð1 0:4Þ50:15 ð1 0:9Þ50:15 ; 1 ð1 0:4Þ50:15 ¼ h0:5039; 0:3183i; A D E e 4 ¼ ð1 0:3Þ50:18 ð1 0:6Þ50:18 ; 1 ð1 0:3Þ50:18 ¼ h0:2870; 0:2746i; A D E e 5 ¼ ð1 0:1Þ50:22 ð1 0:8Þ50:22 1 ð1 0:1Þ50:22 ¼ h0:7203; 0:1094i: A According to Definition 5, we have
e 1 Þ ¼ 0:5796; Sð A
e 2 Þ ¼ 0:2000; Sð A
e 3 Þ ¼ 0:1856; Sð A e 4 Þ ¼ 0:0125; Sð A e 5 Þ ¼ 0:6109: Sð A
Obviously,
e 5 Þ > Sð A e 2 Þ > Sð A e 3 Þ > Sð A e 4 Þ > Sð A e 1 Þ: Sð A As a result, we have
e rð1Þ ¼ h0:7203; 0:1094i; A e rð5Þ ¼ h0; 0:5796i: A
e rð2Þ ¼ h0:4000; 0:2000i; A
e rð3Þ ¼ h0:5039; 0:3183i; A
e rð4Þ ¼ h0:2870; 0:2746i; A
By Theorem 11, it follows that
A ¼ IFHGIAw;x ðA1 ; A2 ; A3; A4 ; A5 Þ * + 5 5 5 Y Y Y wj wj wj ¼ h0:4093; 0:2919i: ½1 v e ½1 ðue þ v e Þ ; 1 ð1 v e Þ ¼ A rð1jÞ A rð1jÞ A rð1jÞ A rð1jÞ j¼1
j¼1
j¼1
It is evident that uIFHGIAx;w ðA1 ;...;A5 Þ ¼ 0:4093 – 0. Therefore, uA1 does not play a decisive role. Practically, the IFHGIA operator is a generalization of the IFWGIA operator or the IFOWGIA operator. Remark 7. If w ¼ ðw1 ; w2 ; . . . ; wn ÞT ¼
Remark 8. If x ¼ ðx1 ; x2 ; . . . ; xn ÞT ¼
1
1 T n;...n ,
1 n
the IFHGIA operator reduces to the IFWGIA operator.
T ; . . . 1n , the IFHGIA operator reduces to the IFOWGIA operator.
then
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Y. He et al. / Information Sciences 259 (2014) 142–159
5. Multi-criteria decision making with the IFHGIA operator For a multi-criteria decision making problem, it is assumed that X = {x1, x2, . . . , xn} is a set of alternatives, G = {G1, G2, P . . . , Gm} is a set of attributes with the associated weighting vector w = (w1, w2, . . . , wm)T, satisfying wi 2 [0, 1] and m i¼1 wi ¼ 1. Suppose that the characteristics of the alternatives xi (i = 1, 2, . . . , n) are represented by intuitionistic fuzzy sets Aij ¼ huAij ; v Aij iði ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; mÞ in Table 1, where uAij indicates the degree that the alternative xi satisfies the attribute Gj, v Aij denotes the degree that the alternative xi does not satisfy the attribute Gj, i = 1, 2, . . . , n; j = 1, 2, . . . , m. Then we have the following decision making method. Step 1: Assume that x = (x1, x2, . . . , xm)T is the weighting vector of Ai1, Ai2, . . . , Aim, i = 1, 2, . . . , n, satisfying xj 2 [0, 1] and Pm mxj e ði ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; mÞ in Table 2. j¼1 xj ¼ 1, m is the balancing coefficient. By Definition 5, we get A ij ¼ Aij e e e e e i1 ; A e i2 ; . . . A e im , Step 2: By Definition 4, we get A rði1Þ ; A rði2Þ ; . . . A rðimÞ , i = 1, 2, . . . , n, where A rðijÞ is the jth largest of A i = 1, 2, . . . , n. Step 3: According to the aggregation operators in Definition 11, we get the final intuitionistic fuzzy sets Ai (i = 1, 2, . . . , n) by using the normal distribution based method [31] to determine weights of the IFOWGIA operator. Step 4: According to Definition 4, we get the order of the final intuitionistic fuzzy sets Ai(i = 1, 2, . . . , n). Step 5: Rank all the alternatives xi(i = 1, 2, . . . , n) and select the best one (s).
6. Numerical example and systematic comparison 6.1. Numerical example Assume that an investment company wants to invest a sum of money in the best option. Three possible alternatives are to be considered by analyzing the market. x1 is a car company. x2 is a food company. x3 a computer company. In order to assess these alternatives, the investors have brought panel data. After careful review of the information, they summarize the ability of companies with five attributes G = {G1, G2, G3, G4, G5}.
G1: G2: G3: G4: G5:
The The The The The
risk analysis. growth analysis. social-political impact analysis. environmental impact analysis. development of the society.
The three possible alternatives are to be evaluated by the intuitionistic fuzzy information in Table 3 under above five attributes.
Table 1 Intuitionistic fuzzy matrix (Aij)nm. G1
G2
...
Gm1
Gm
A11 A21 ... An1
A12 A22 ... An2
... ... ... ...
A1,m1 A2,m1 ... An,m1
A1,m A2,m ... An,m
G1
G2
...
Gm1
Gm
e 12 A e 22 A
...
x2
e 11 A e 21 A
...
e 1;m1 A e 2;m1 A
e 1;m A e 2;m A
... xn
... e n;1 A
... e n;2 A
... ...
... e n;m1 A
... e n;m A
x1 x2 ... xn
Table 2 e ij Þ Intuitionistic fuzzy matrix ð A . nm
x1
155
Y. He et al. / Information Sciences 259 (2014) 142–159 Table 3 Intuitionistic fuzzy matrix (Aij)35.
x1 x2 x3
G1
G2
G3
G4
G5
h0.2, 0.5i h0.2, 0.7i h0.2, 0.7i
h0.4, 0.2i h0.6, 0.3i h0.5, 0.3i
h0.5, 0.4i h0.4, 0.3i h0.4, 0.5i
h0.3, 0.3i h0.4, 0.4i h0.3, 0.4i
h0.7, 0.1i h0.6, 0.1i h0.6, 0.2i
Table 4 e ij ÞÞ . Score matrix ðSð A 35
i=1 i=2 i=3
j=1
j=2
j=3
j=4
j=5
0.3811 0.6122 0.6122
0.2000 0.3000 0.2000
0.1856 0.1252 0.0114
0.0125 0.0280 0.0755
0.6109 0.5152 0.3944
Table 5 e rðijÞ Þ . Intuitionistic fuzzy ordered matrix ð A 35
i=1 i=2 i=3
j=1
j=2
j=3
j=4
j=5
h0.7203, 0.1094i h0.6246, 0.1094i h0.6120, 0.2176i
h0.4000, 0.2000i h0.6000, 0.3000i h0.5000, 0.3000i
h0.5039, 0.3183i h0.3599, 0.2347i h0.4168, 0.4054i
h0.2870, 0.2746i h0.3965, 0.3686i h0.2930, 0.3686i
h0.1984, 0.5796i h0.1658, 0.7780i h0.1658, 0.7780i
e ij ¼ Anxj , where the balancing coefficient n = 5, Step 1: According to operational law in Definition 5, we obtain A ij T x = {0.25, 0.20, 0.15, 0.18, 0.22} is the weighting vector of Ai1, Ai2, . . . , Ai5, i = 1, 2, 3; j = 1, 2, . . . , 5. e ij ÞÞ Step 2: By Definition 4, we get the score matrix ðSð A 35 in Table 4. Obviously,
e 15 Þ > Sð A e 12 Þ > Sð A e 13 Þ > Sð A e 14 Þ > Sð A e 11 Þ; Sð A e 25 Þ > Sð A e 22 Þ > Sð A e 23 Þ > Sð A e 24 Þ > Sð A e 21 Þ Sð A and
e 35 Þ > Sð A e 32 Þ > Sð A e 33 Þ > Sð A e 34 Þ > Sð A e 31 Þ: Sð A e rðijÞ Þ e rðijÞ ði ¼ 1; 2; 3; j ¼ 1; 2; . . . ; 5Þ is the jth largThen, we get the intuitionistic fuzzy ordered matrix ð A in Table 5, where A 35 e i1 ; A e i2 ; A e i3 ; A e i4 ; A e i5 ; i ¼ 1; 2; 3. est of A Step 3: The weights w = (0.112, 0.236, 0.304, 0.236, 0.112) are determined by the normal distribution based method [31]. According to aggregation operators in Definition 9, we obtain the final intuitionistic fuzzy sets Ai(i = 1, 2, 3):
A1 ¼ IFHGIAw;x ðA11 ; A12 ; A13; A14 ; A15 Þ ¼
* 5 Y j¼1
1 ve
wj
A rð1jÞ
5 Y 1 ðue
þ ve
wj 5 Y Þ ; 1 ð1 v e
5 Y 1 ðue
5 Y 1 ðue
j¼1
A rð1jÞ
+
Þ
wj
þ ve
wj 5 Y Þ ; 1 ð1 v e
Þ
wj
þ ve
wj 5 Y Þ ; 1 ð1 v e
Þwj
A rð1jÞ
j¼1
A rð1jÞ
¼ h0:4294; 0:2988i; A2 ¼ IFHGIAw;x ðA21 ; A22 ; A23; A24 ; A25 Þ ¼
* 5 Y j¼1
1 ve
wj
A rð2jÞ
j¼1
A rð2jÞ
A rð2jÞ
j¼1
A rð2jÞ
+
¼ h0:4383; 0:3659i; A3 ¼ IFHGIAw;x ðA31 ; A32 ; A33; A34 ; A35 Þ ¼
* 5 Y j¼1
1 ve
¼ h0:3927; 0:4212i: Step 4: According to Definition 4, we have
SðA1 Þ ¼ 0:1306; SðA2 Þ ¼ 0:0724; SðA3 Þ ¼ 0:0285: Thus, S(A1) > S(A2) > S(A3). Step 5: According to the scores in Step 4, we have
A rð3jÞ
wj
j¼1
A rð3jÞ
A rð3jÞ
j¼1
A rð3jÞ
+
156
Y. He et al. / Information Sciences 259 (2014) 142–159
Table 6 e ij ÞÞ . Score matrix ðSð A 35 j=1
j=2
j=3
j=4
j=5
0.5796 0.6442 0.6442
0.2000 0.3000 0.2000
0.2763 0.2683 0.0976
0.0638 0.0698 0.0302
0.5660 0.4607 0.3525
j=1
j=2
j=3
j=4
j=5
h0.6755, 0.1094i h0.5701, 0.1094i h0.5701, 0.2176i
h0.5946, 0.3183i h0.6000, 0.3000i h0.5000, 0.3000i
h0.4000, 0.2000i h0.5030, 0.2347i h0.5030, 0.4054i
h0.3384, 0.2746i h0.4384, 0.3686i h0.3384, 0.3686i
h0, 0.5796i h0.1337, 0.7780i h0.1337, 0.7780i
i=1 i=2 i=3
Table 7 e rðijÞ Þ . Intuitionistic fuzzy ordered matrix ð A 35
i=1 i=2 i=3
Table 8 e ij ÞÞ . Score matrix ðSð A 35
i=1 i=2 i=3
j=1
j=2
j=3
j=4
j=5
0.5796 0.6122 0.6122
0.2000 0.3000 0.2000
0.1856 0.1252 0.0144
0.0125 0.0280 0.0755
0.6109 0.5152 0.3944
x1 x2 x3 : Therefore, the optimal alternative is x1. If we use the method with the operator proposed in [37], we also have x1 x2 x3. i.e., the best alternative is x1. Thus, the results of the above two methods are same. Therefore, the method proposed in this paper is effective and valid. 6.2. Systematic comparison with corresponding work of other papers However, the method developed in [37] has some differences from the proposed method in this paper. For example, if A11 ¼ A011 ¼ h0; 0:5i, we have the following steps using the method in [37]. e ij ¼ A5xj i ¼ 1; 2; 3; j ¼ 1; 2; . . . ; 5, where x = {0.25, 0.20, 0.15, 0.18, 0.22}T is Step 1: By the operational law of Definition 5, A ij the weighting vector of Ai1, Ai2, . . . , Ai5, i = 1, 2, 3. e ij ÞÞ Step 2: By Definition 4 we obtain the score matrix ðSð A in Table 6. 35 Obviously,
e 15 Þ > Sð A e 13 Þ > Sð A e 12 Þ > Sð A e 14 Þ > Sð A e 11 Þ; Sð A e 25 Þ > Sð A e 22 Þ > Sð A e 24 Þ > Sð A e 23 Þ > Sð A e 21 Þ Sð A and
e 35 Þ > Sð A e 32 Þ > Sð A e 34 Þ > Sð A e 33 Þ > Sð A e 31 Þ: Sð A e rðijÞ Þ e rðijÞ is the jth largest of Then, we get the intuitionistic fuzzy ordered matrix ð A in Table 7, where A 35 e i1 ; A e i2 ; . . . A e i5 ; i ¼ 1; 2; 3. A Step 3: By the aggregation operator in Definition 8, and w = (0.112, 0.236, 0.304, 0.236, 0.112). It follows that
A1 ¼ IFHGAw;x ðA11 ; A12 ; A13; A14 ; A15 Þ ¼
* 5 Y j¼1
wj
ðue Þ ; 1 A rð1jÞ
5 Y j¼1
+ ð1 v e
A rð1jÞ
Similarly, A2 = h0.4438, 0.3659i, A3 = h0.3999, 0.4211i. Step 4: By Definition 4, S(A1) = 0.2911, S(A2) = 0.0779, S(A3) = 0.0212. Thus, S(A2) > S(A3) > S(A1). Step 5: Therefore, x2 x3 x1 Therefore, the optimal alternative is x2.
wj
Þ
¼ h0; 0:2911i;
157
Y. He et al. / Information Sciences 259 (2014) 142–159 Table 9 e rðijÞ Þ . Intuitionistic fuzzy ordered matrix ð A 35
i=1 i=2 i=3
j=1
j=2
j=3
j=4
j=5
h0.7203, 0.1094i h0.6246, 0.1094i h0.6120, 0.2176i
h0.4000, 0.2000i h0.6000, 0.3000i h0.5000, 0.3000i
h0.5039, 0.3183i h0.3599, 0.2347i h0.4168, 0.4054i
h0.2870, 0.2746i h0.3965, 0.3686i h0.2930, 0.3686i
h0, 0.5796i h0.1658, 0.7780i h0.1658, 0.7780i
e 1j Þ > Sð A e 3j Þðj ¼ 1; 2; . . . ; 5Þ. By Table 7 and Definition 4, we have that By Table 6, we know that Sð A e e Sð A rð1jÞ Þ P Sð A rð3jÞ Þ; j ¼ 1; . . . ; 5. Thus, it is more acceptable for us to get the result that x1 x3 from monotonicity point of view. Therefore, the method proposed in [37] is not very stable. However, if we use the method developed by this paper, we have following steps. e ij ¼ A5xj i ¼ 1; 2; 3; j ¼ 1; 2; . . . ; 5, Step 1: According to operational law in Definition 5, we get A ij T x = {0.25, 0.20, 0.15, 0.18, 0.22} is the weighting vector of Ai1, Ai2, . . . , Ai5, i = 1, 2, 3. e ij ÞÞ Step 2: By Definition 4, we get the score matrix ðSð A in Table 8. 35 Obviously,
where
e 15 Þ > Sð A e 12 Þ > Sð A e 13 Þ > Sð A e 14 Þ > Sð A e 11 Þ; Sð A e 25 Þ > Sð A e 22 Þ > Sð A e 23 Þ > Sð A e 24 Þ > Sð A e 21 Þ; Sð A and
e 35 Þ > Sð A e 32 Þ > Sð A e 33 Þ > Sð A e 34 Þ > Sð A e 31 Þ: Sð A e rðijÞ Þ e rðijÞ is the jth largest of Then, we get the intuitionistic fuzzy ordered matrix ð A in Table 9, where A 35 e e e A i1 ; A i2 ; . . . A i5 ; i ¼ 1; 2; 3. Step 3: By the aggregation operator in Definition 11 and Theorem 11, and w = (0.112, 0.236, 0.304, 0.236, 0.112), we get that
A1 ¼ IFHGIAw;x ðA11 ; A12 ; A13; A14 ; A15 Þ ¼
* 5 Y j¼1
1 ve
wi
A rð1jÞ
5 Y j¼1
1 ðue
A rð1jÞ
þ ve
A rð1jÞ
wi Þ
;1
5 Y ð1 v e j¼1
A rð1jÞ
+ Þwj
¼ h0:4093; 0:2919i; Similarly, A2 = h0.4383, 0.3659i, A3 = h 0.3927, 0.4212i. Step 4: By Definition 4, we have
SðA1 Þ ¼ 0:1105;
SðA2 Þ ¼ 0:0724 and SðA3 Þ ¼ 0:0285:
Therefore, S(A1) > S(A2) > S(A3). Step 5: According to the scores S(Ai)(i = 1, 2, 3), we rank all the alternatives xi(i = 1, 2, 3) and obtain that
x1 x2 x3 : Therefore, the optimal alternative is x1. By Table 8, we have
e 1j Þ P Sð A e 3j Þ; j ¼ 1; . . . ; 5: Sð A By Table 9 and Definition 4, we have
e rð1jÞ Þ P Sð A e rð3jÞ Þ; j ¼ 1; . . . ; 5: Sð A Thus, it is acceptable for us to get the result that x1 x3 from monotonicity point of view. That is to say, the result of this paper is acceptable, which means the proposed method is a good complement to the existing works on IFSs in the case that one of the membership degree is zero. We have summarized the systematic comparison in four aspects: (1) Atanassov [4] originally defined the multiplication operation on IFSs, De et al. [12] defined the power operation by reasoning from the multiplication operation. In this paper, we define the new geometric operations of IFSs from the probability point of view and give the geometric interpretations of these geometric operations. We propose the new multiplication operation on IFSs based on probability non-membership (PN) function operator, probability membership (PM) function operator and probability hetergeneous (PH) operator. And the geometric meanings of these new operations are interpreted in Figs. 1–5. (2) The existing geometric operations just consider the effects of membership degree or non-membership degree of different IFSs, while the new geometric operations also take the interactions of non-membership degree and membership de-
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Y. He et al. / Information Sciences 259 (2014) 142–159
gree of different IFSs into consideration, which could be used and explained reasonably in more case, especially when one of the membership degree is zero. It can be seen clearly in Example 1. (3) By the aggregation operators in [37], when there exists only one membership degree of IFS equals to zero, the membership degree of aggregation result of n IFSs is zero even if the membership degrees of n 1 IFSs are not zero, which is the weakness of the aggregation operators in [37]. In the above cases, if we use the operators presented in this paper, the aggregation result can be explained reasonably, which could be seen clearly by Example 2. (4) In the process of intuitionistic fuzzy multi-criteria decision making, the evaluated values of candidate alternatives under attributes are IFSs. In Section 6.1, we know that if none of the membership degree of IFSs is zero, we can get the same optimal alternative by the approaches developed in [37], and the ranking result is same. But when only one membership degree of IFSs is zero, the result gotten by the proposed approach in this paper is acceptable from monotonicity point of view. 7. Conclusions Intuitionistic fuzzy information aggregation is an interesting research field of the IFS theory. The rational operational laws on IFS are very important to aggregate the intuitionistic fuzzy information. In this paper, we propose the new operational laws on IFSs and develop the intuitionistic fuzzy geometric interaction averaging (IFGIA) operator, which provide a good complement to the existing works on IFSs. In the succeeding work, we plan to extend our research to the interval-valued intuitionistic fuzzy environment, we will develop some new operational laws on interval-valued intuitionistic fuzzy sets and propose the corresponding interval-valued intuitionistic fuzzy geometric interaction averaging (IVIFGIA) operators. We intend to apply these approaches to multiattribute decision making, pattern recognition, data mining, clustering and medical diagnosis. In decision-making, the generalization capability of extracting fuzzy rules is the key index. Recently, some new refinement techniques [28,44] related to maximize the uncertainty or to combine multiple classifiers have been proposed to improve the generalization of the decision rules. The fuzzy information systems is related to extracting rules because it contains conditional attributes and decision attributes. We will conduct some numerical experiments on comparing with some existing fuzzy rule extraction methods [17,43] in the future research under the intuitionistic fuzzy environment. Acknowledgments This work was supported by National Natural Science Foundation of China (71071002, 71371011, 71301001), Higher School Specialized Research Fund for the Doctoral Program (20123401110001), The Scientific Research Foundation of the Returned Overseas Chinese Scholars, Anhui Provincial Natural Science Foundation (1308085QG127), Provincial Natural Science Research Project of Anhui Colleges (KJ2012A026), Humanity and Social Science Youth foundation of Ministry of Education (13YJC630092), Humanities and social science Research Project of Department of Education of Anhui Province (SK2013B041), Natural Science Foundation of Anhui Provincial Higher School and Foundation for the Young Scholar of Anhui University (2009QN022B). The authors are thankful to the anonymous reviewers and the editor for their valuable comments and constructive suggestions that have led to an improved version of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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Information Sciences journal homepage: www.elsevier.com/locate/ins
Intuitionistic fuzzy geometric interaction averaging operators and their application to multi-criteria decision making Yingdong He a, Huayou Chen a,⇑, Ligang Zhou a, Jinpei Liu b, Zhifu Tao a a b
School of Mathematical Science, Anhui University, Hefei Anhui 230601, China School of Business, Anhui University, Hefei Anhui 230601, China
a r t i c l e
i n f o
Article history: Received 7 November 2012 Received in revised form 30 June 2013 Accepted 5 August 2013 Available online 20 August 2013 Keywords: Intuitionistic fuzzy set Probability non-membership (PN) function operator Probability hetergeneous (PH) operator Intuitionistic fuzzy geometric interaction averaging operator Decision making
a b s t r a c t This paper proposes some new geometric operations on intuitionistic fuzzy sets (IFSs) based on probability non-membership (PN) function operator, probability membership (PM) function operator and probability hetergeneous (PH) operator, which are constructed from the probability point of view. The geometric interpretations of these operations are given. Moreover, we develop some intuitionistic fuzzy geometric interaction averaging (IFGIA) operators. The properties of these aggregation operators are investigated. The key advantage of the IFGIA operators is that the interactions between non-membership function and membership function of different IFSs are considered. Finally, an approach to multiple attributes decision making is given based on the proposed aggregation operators under intuitionistic fuzzy environment, and an example is illustrated to show the validity and feasibility of the proposed approach. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Intuitionistic fuzzy sets (IFSs), as a generalization of fuzzy sets [42], developed by Atanassov [3], is a powerful tool to deal with vagueness. Since information aggregation is a pervasive activity in daily life, many researches had been done on this issue [7,15,24,27,32,37,34,35,30,39,38,18,26]. Among them, the weighted geometric averaging (WGA) operator [24] and the ordered weighted geometric averaging (OWGA) operator [34] are the most common operators. On a basis of the multiplication operation by Atanassov [4] and power operation by De and Biswas [12] on intuitionistic fuzzy sets, Xu and Yager [37] proposed some intutionistic fuzzy geometric aggregation operators and applied them to multi-attribute decision making problems. After these pioneering works, more attentions have been paid to intuitionistic fuzzy multi-criteria decision making problems [1,2,5,6,8,10,11,13,14,19–23,25,29,33,36,41,43–51]. Dymova and Sevastjinov [13] presented a method to deal with the intuitionistic fuzzy multi-criteria decision making problems based on Dempster–Shefer theory of evidence. Ye [40] used entropy weight to get criteria weights and rank alternatives according to the correlation coefficients. Xu [33] developed intuitionistic fuzzy power aggregation operators. Xu and Xia [36] presented the induced generalized aggregation operators under interval-valued intuitionistic fuzzy environments. Zhu et al. [51] proposed hesitant fuzzy geometric Bonferroni means. Li et al. [19] investigated the relationship between the similarity measure and the entropy of IFSs. Zhang et al. [45] presented
⇑ Corresponding author. Tel.: +86 13615690958. E-mail addresses: [email protected] (Y. He), [email protected] (H. Chen), [email protected] (L. Zhou), [email protected] (J. Liu), [email protected] (Z. Tao). 0020-0255/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2013.08.018
Y. He et al. / Information Sciences 259 (2014) 142–159
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the intuitionistic fuzzy rough approximation operators and discussed their connections with special intuitionistic fuzzy relations. However, it is found that the operational laws and geometric aggregation operators on intuitionistic fuzzy sets in [4,37] are not suitable to be used in the special circumstances. For example, suppose that A and B are two intuitionistic fuzzy sets, A = huA, vAi and uA = 0, B = huB, vBi and uB – 0, then according to the multiplication operation by Atanassov [4], we have uAB = 0. Obviously, uB is not accounted for at all, which is an undesirable feature of an averaging operation. Furthermore, the IFHGA operator [37] has similar problems. For example, if Ai ¼ huAi ; v Ai i; i ¼ 1; 2; . . . ; n; i – k are a collection of intutionitisc fuzzy sets, Ak ¼ h0; v Ak i, and v Ak – 0, then we have uIFHGAx;w ðA1 ;...;An Þ ¼ 0 by using aggregation law in [37]. It is obvious that uAi ði ¼ 1; 2; . . . ; n; i – kÞ have no effects on the aggregation result. Motivated by the works of [4,37] and the idea of interactions between non-membership function and membership function of different intuitionistic fuzzy sets, we focus on developing some new geometric operations on intuitionistic fuzzy sets (IFSs) and giving the geometric interpretations of these operations. Based on the new operations, we propose some intuitionistic fuzzy geometric interaction aggregation operators, including the IFWGIA operator, the IFOWGIA operator and the IFHGIA operator, which are more practical for an averaging operator. By the comparison with the existing method, it is concluded that the method proposed in this paper is a good complement to the existing works on IFSs, especially when one of the membership degrees of intutionitisc fuzzy sets is zero. The rest of the paper is organized as follows. Section 2 reviews some basic concepts. Section 3 introduces new geometric operations on intuitionistic fuzzy sets and gives the geometric interpretations of these operations. In Section 4, we develop the intuitionistic fuzzy geometric interaction averaging (IFGIA) operators, and investigate their properties. In Section 5, an approach to intuitionistic fuzzy multi-criteria decision making is given based on the proposed IFHGIA operator. In Section 6, a numerical example is illustrated to show the feasibility and validity of the new approach, and the comparison between the work of this paper and other corresponding works is presented systematically. Finally, Section 7 concludes the paper. 2. Preliminaries The concept of fuzzy sets (FSs) was introduced by Zadeh [42]. Let X be a universe of discourse in the following. Definition 1 [42]. A fuzzy set F in X is defined as follows: F = {hx, uF(x)ijx 2 X}, where uF:X ? [0, 1] is the membership function of the fuzzy set F, and 0 6 uF(x) 6 1. Atanassov [3] generalized the fuzzy set to intuitionistic fuzzy set (IFS) by adding an hesitation degree. Definition 2 [3]. An intuitionistic fuzzy set in X is an expression: A = {hx, uA(x), vA(x)ijx 2 X}, where the functions uA:X ? [0, 1] and vA:X ? [0, 1] define the degree of membership and the degree of nonmembership of the element x 2 X to A, and for every x 2 X, 0 6 uA(x) + vA(x) 6 1. For each IFS A in X, if pA(x) = 1 uA(x) vA(x), for all x 2 X, then pA(x) is called the degree of indeterminacy of the element x to the set A. In practice, intuitionistic fuzzy numbers can be denoted as A = hu, vi [32,37]. For convenience, the sets of all the intuitionistic fuzzy numbers are denoted by IFNs. Some basic operations, such as multiplication operation [4] and power operation [12], were introduced under intuitionistic fuzzy environment. Definition 3 (4,12). Suppose that A = huA, vAi and B = huB, vBi are two intuitionistic fuzzy sets, then
ð1Þ A B ¼ huA uB ; v A þ v B v A v B i D E ð2Þ Ak ¼ ukA ; 1 ð1 v A Þk ; k > 0
ð1Þ ð2Þ
Chen and Tan [9] proposed a score function S(A) = uA vA to evaluate the degree of suitability that an alternative satisfies a decision make’s requirement under intuitionistic fuzzy environment, where A is an intuitionistic fuzzy set, and A = huA, vAi. The score of A is directly related to the deviation between uA and vA, i.e., the bigger the score of intuitionistic fuzzy set A, the larger the intuitionistic fuzzy set A. Hong and Choi [16] presented an accuracy function H(A) = uA + vA to evaluate the accuracy degree of the intuitionistic fuzzy set A = huA, vAi, where 0 6 H(A) 6 1. The larger the value of H(A), the higher the accuracy degree of intuitionistic fuzzy set A [32]. Based on score function [9] and accuracy function [16], Xu [32,37] gave the comparison law for intuitionistic fuzzy sets as follows. Definition 4 (32,37). Let A = huA, vAi and B = huB, vBi be two intuitionistic fuzzy sets. Then A < B if and only if (i) S(A) < S(B). or (ii) S(A) = S(B) and H(A) < H(B).
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3. New geometric operational laws on intuitionistic fuzzy sets 3.1. New geometric operational laws and corresponding geometric interpretations Considering that the existing multiplication operation [4] on IFSs cannot be used in some cases from averaging point of view, we introduce new operational laws on intuitionistic fuzzy sets, including multiplication operation and power operation. The feature of new geometric operational laws is that we take the interactions into consideration between non-membership function and membership function of different intuitionistic fuzzy sets. Definition 5. Suppose that A = huA, vAi and B = huB, vBi are two intuitionistic fuzzy numbers, then
^ ¼ hð1 v A Þð1 v B Þ ð1 ðuA þ v A ÞÞð1 ðuB þ v B ÞÞ; 1 ð1 v A Þð1 v B Þi 1Þ AB
ð3Þ
2Þ Ak ¼ hð1 v A Þk ð1 ðuA þ v A ÞÞk ; 1 ð1 v A Þk i; k > 0
ð4Þ
The geometric meaning of new multiplication operation on intuitionistic fuzzy sets can be interpreted from three aspects as follows. The operational rule between non-membership function and non-membership function of different IFSs. The operational rule between membership function and membership function of different IFSs. The operational rule between non-membership function and membership function of different IFSs. Let A = huA, vAi and B = huB, vBi be two intuitionistic fuzzy numbers. (1) The operational rule between non-membership function and non-membership function can be explained geometrically as follows in Fig. 1. vA and vB denote the non-membership degree of A and B, respectively. From the probability point of view, we may as well regard vA and vB as two independent events, E(vA, vB) represents the probability of vA and vB occurring simultaneously. Thus, E(vA, vB) = vA vB. Therefore, v AB ^ ¼ v A þ v B v A v B . v AB ^ is called the probability non-membership (PN) function operator, i.e.,
PNðv A ; v B Þ ¼ v A þ v B v A v B
ð5Þ
(2) The operational rule between membership function and membership function can be explained geometrically as follows in Fig. 2. uA and uB denote the membership degree of A and B, respectively. We may as well regard uA and uB as two independent events, E(uA,uB) represents the probability of uA and uB occurring simultaneously. Thus, E(uA, uB) = uA uB. Therefore, uAB ^ ¼ uA þ uB uA uB . uAB ^ is called the probability membership (PM) function operator, i.e.,
PMðuA ; uB Þ ¼ uA þ uB uA uB :
ð6Þ
(3) The operational rule between non-membership function and membership function can be explained geometrically as follows in Figs. 3 and 4. vA denotes the non-membership degree of A, uB denotes the membership degree ofB. Similarly, I(vA, uB) represents the probability of two independent events vA and lB occurring simultaneously. Thus, I(vA,uB) = vA uB. I(vA, uB) is called probability hetergeneous (PH) operator, i.e.,
PHðv A ; uB Þ ¼ uB v A :
ð7Þ
In similar way, I(vB, uA) denotes the probability of two independent events vB and uA occurring simultaneously. Thus, I(vB, uA) = vB uA. I(vB, uA) is called the probability hetergeneous (PH) operator, i.e.
PHðv B ; uA Þ ¼ uA v B :
ð8Þ
(4) The interaction between different intuitionistic fuzzy sets can be explained geometrically as follows in Fig. 5.
Fig. 1. Operational rule of non-membership function of different IFSs.
Y. He et al. / Information Sciences 259 (2014) 142–159
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Fig. 2. Operational rule of membership function of different IFSs.
Fig. 3. Operational rule of non-membership function and membership function of different IFSs.
Fig. 4. Operational rule of non-membership function and membership function of different IFSs.
Fig. 5. Operational rule of different IFSs.
From Fig. 5 and Eqs. (5)–(8), the multiplication of different IFNs can be rewritten as follows.
^ ¼ hPMðuA ; uB Þ PHðuA ; v B Þ PHðuB ; v A Þ; PNðv A ; v B Þi AB
ð9Þ
^ contains PH(vA, uB) and PH(vB, uA), while the From Fig. 5 and Eq. (9), it is obvious that the non-membership function of AB ^ does not contain PH(vA, uB) and PH(vB, uA). membership function of AB In fact, if A, B 2 IFNs, A = huA, vAi, vA – 1 and uA = 0, B = huB, vBi and uB – 0, then according to Eq. (9), we have
PMðuA ; uB Þ PHðuA ; v B Þ PHðuB ; v A Þ ¼ uB uB v A – 0; which shows that uA cannot play a decisive role. As a result, the proposed Definition 5 is more practical for an averaging operator, which can be seen visually by the following example. Example 1. Suppose that A = huA, vAi = h0, 0.6i and B = huB, vBi = h0.6, 0.2i are two intuitionistic fuzzy numbers, then by Definition 3, we have
A B ¼ h0 0:6; 0:6 þ 0:2 0:6 0:2i ¼ h0; 0:68i: According to Definition 5, we get that
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^ ¼ h0 þ 0:6 0 0:2 0 0:6 0:6 0:6; 0:6 þ 0:2 0:6 0:2i ¼ h0:24; 0:68i: AB i.e., uAB ^ ¼ 0:24, which indicates that uA dose not play a decisive role. It is noted that PN(vA, vB) = vA + vB vA vB = 1 (1 vA) (1 vB), and
PMðuA ; uB Þ PHðuA ; v B Þ PHðuB ; v A Þ ¼ ðPMðuA ; uB Þ þ PNðv A ; v B Þ PHðuA ; v B Þ PHðuB ; v A ÞÞ PNðv A ; v B Þ ¼ ððuA þ uB uA uB Þ þ ðv A þ v B v A v B Þ ðuA v B Þ ðuB v A ÞÞ PNðv A ; v B Þ ¼ ððuA þ v A Þ þ ðuB þ v B Þ ðuA þ v A Þ ðuB þ v B ÞÞ PNðv A ; v B Þ ¼ ð1 ð1 ðuA þ v A ÞÞ ð1 ðuB þ v B ÞÞÞ ð1 ð1 v A Þ ð1 v B ÞÞ ¼ ð1 v A Þ ð1 v B Þ ð1 ðuA þ v A ÞÞ ð1 ðuB þ v B ÞÞ: Therefore, Eq. (9) is equivalent to Eq. (3). Next, we explain the origin of the power operation. By Eq. (3), we have
^ ¼ hð1 v A Þ2 ð1 ðuA þ v A ÞÞ2 ; 1 ð1 v A Þ2 i: A2 ¼ AA Similarly, we have
^ ¼ hð1 v A Þ3 ð1 ðuA þ v A ÞÞ3 ; 1 ð1 v A Þ3 i: A3 ¼ A2 A By analogy, for any positive integer n, we have
^ ¼ hð1 v A Þn ð1 ðuA þ v A ÞÞn ; 1 ð1 v A Þn i: An ¼ An1 A We extend positive integer n to any nonnegative real number k, and have
Ak ¼ hð1 v A Þk ð1 ðuA ðxÞ þ v A ðxÞÞÞk ; 1 ð1 v A Þk i; k > 0: 3.2. The properties of the new geometric operations In this subsection, we investigate some properties of the PN function operator, the PM function operator, the PH function operator, the new multiplication operation and power operation under intuitionistic fuzzy environment. Theorem 1. Let A = huA, vAi, B = huB, vBi, C = huC, vCi, A0 ¼ huA0 ; v A0 i and B0 ¼ huB0 ; v B0 i be five intuitionistic fuzzy numbers, then (1) (2) (3) (4) (5)
Boundedness: PN(1, 1) = 1, PN(0, 0) = 0, 0 6 PN(vA, vB) 6 1. Monotonicity: if v A 6 v A0 and v B 6 v 0B , then PNðv A ; v B Þ 6 PN v 0A ; v 0B . Commutativity: PN(vA, vB) = PN(vB, vA). Associativity: PN(vA, PN(vB, vC)) = PN(PN(vA, vB), vC). Zero element: PN(0, vA) = PN(vA, 0) = vA.
Proof. (1) According to Eq. (5), we have
PNðv A ; v B Þ ¼ v A þ v B v A v B ¼ 1 ð1 v A Þð1 v B Þ
ð10Þ
Therefore,
PNð1; 1Þ ¼ 1 þ 1 1 1 ¼ 1;
PNð0; 0Þ ¼ 1 1 ¼ 0;
It is noted that 0 6 vA 6 1, 0 6 vB 6 1. Thus, 0 6 (1 vA)(1 vB) 6 1, then according to Eq. (10), we have 0 6 PN(vA, vB) 6 1. (2) Because v A 6 v 0A and v B 6 v 0B , we have 1 v A P 1 v 0A ; 1 v B P 1 v 0B , According to Eq. (10), we obtain
PNðv A ; v B Þ 6 PN
v 0A ; v 0B
(4) According to Eq. (10), we get
PNðv A ; PNðv B ; v C ÞÞ ¼ 1 ð1 v A Þð1 PNðv B ; v C ÞÞ and
1 PNðv B ; v C Þ ¼ ð1 v B Þð1 v C Þ; Thus, PN(vA, PN(vB, vC)) = 1 (1 vA)(1 vB)(1 vC).
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Similarly, PN(PN(vA, vB), vC) = 1 (1 vA)(1 vB)(1 vC). Therefore, PN(vA, PN(vB, vC)) = PN(PN(vA, vB), vC). According to Eq. (5), we can get the results of (3) and (5) easily. So the proofs are omitted here.
h
Theorem 2. Let A = huA, vAi, B = huB, vBi, C = huC, vCi, A0 ¼ huA0 ; v A0 i and B0 ¼ huB0 ; v B0 i be five intuitionistic fuzzy numbers, then we have (1) Boundedness: PH(0, 0) = 0, PH(1, 1) = 1, 0 6 PH(vA, vB) 6 1. (2) Monotonicity: If uA 6 u0A and v B 6 v 0B , then PHðuA ; v B Þ 6 PH u0A ; v 0B ; 0 0 PHðv A ; uB Þ 6 PH v A ; uB . (3) Commutativity: PH(uA, vB) = PH(vB, uA), PH(vA, uB) = PH(uB, vA). (4) Associativity: PH(uA, PH(uB, uC)) = PH(PH(uA, uB), uC). (5) Identify element: PH(1, uA) = PH(uA, 1) = uA.
and
if
v A 6 v 0A
and
uB 6 u0B ,
then,
Proof. According to Eq. (8), we can get Theorem 2 easily, so it is omitted here. h In practice, the operational results of geometric operational laws on intuitionistic fuzzy numbers are also intuitionistic fuzzy numbers. ^ ¼ huC ; v C i; D ¼ Ak ¼ huD ; v D i, then C, D 2 IFNs. Theorem 3. If A = huA, vAi 2 IFNs and B ¼ huB ; v B i 2 IFNs; k > 0; C ¼ AB The proof of theorem 3 is similar to that in [37], so it is omitted here. Theorem 4. Let A = huA, vAi and B = huB, vBi be two intuitionistic fuzzy numbers, k, k1, k2 > 0. Then, we have ^ ¼ BA; ^ (1) AB ^ k ¼ Ak B ^ k; (2) ðABÞ k1 k2 ðk1 þk2 Þ ^ (3) A A ¼ A . The proof of theorem 4 is similar to that in [37], so it is omitted here. 4. Intuitionistic fuzzy geometric interaction averaging operators 4.1. Intuitionistic fuzzy weighted geometric averaging (IFWGA) operator Xu and Yager [37] generalized the WGA operators and the OWGA operators to intuitionistic fuzzy environment and proposed the intuitionistic fuzzy weighted geometric averaging (IFWGA) operator and the intuitionistic fuzzy ordered weighted geometric averaging (IFOWGA) operator. Definition 6 [37]. Let Ai ¼ huAi ; v Ai iði ¼ 1; 2; . . . ; nÞ be a collection of intuitionistic fuzzy numbers. If the mapping
IFWGAw ðA1 ; A2 . . . ; An Þ ¼
1 Aw 1
2 Aw 2
...
n Aw n
¼
* n Y
w uAii ; 1
i¼1
n Y
+
ð1 v Ai Þ
wi
;
i¼1
then IFWGAw is called intuitionistic fuzzy weighted geometric averaging operator with respect to a weighting vector w, P where w = (w1, w2, . . . , wn)T is the weighting vector of Ai with wi 2 [0, 1] and ni¼1 wi ¼ 1. Definition 7 [37]. Let Ai ¼ huAi ; v Ai iði ¼ 1; 2; . . . ; nÞ be a collection of intuitionistic fuzzy numbers. If the mapping w1
IFOWGAw ðA1 ; . . . ; An Þ ¼ ðArð1Þ Þ
. . . ðArðnÞ Þ
wn
¼
* n Y i¼1
w uAri ðiÞ ; 1
+ n Y wi ; ð1 v ArðiÞ Þ
ð11Þ
i¼1
then IFOWGAw is called intuitionistic fuzzy ordered weighted geometric averaging operator with respect to a weighting vector w, where Ar(i) is the ith largest value of Ai (i = 1, 2, . . . , n) according to Definition 4, w = (w1, w2, . . . , wn)T is the weighting P vector with wi 2 [0, 1] and ni¼1 wi ¼ 1. 4.2. Intuitionistic fuzzy weighted geometric interaction averaging (IFWGIA) operator Motivated by the WGA operator [24] and the IFWGA operator [37], we develop intuitionistic fuzzy weighted geometric interaction averaging (IFWGIA) operator based on the proposed operational laws.
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Definition 8. Let Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ be a collection of intuitionistic fuzzy numbers. If the mapping n
w IFWGIAw ðA1 ; . . . ; An Þ ¼ ^ Ai i ; i¼1
then IFWGIAw is called intuitionistic fuzzy weighted geometric interaction averaging (IFWGIA) operator, where w = (w1, w2, P . . . , wn)T is the weighting vector of Ai with wi 2 [0, 1] and ni¼1 wi ¼ 1. Theorem 5. Let Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ be a collection of intuitionistic fuzzy numbers. Then,
IFWGIAw ðA1 ; . . . ; An Þ ¼
* n Y
wi
ð1 v Ai Þ
i¼1
n Y
+ n Y wi : ð1 ðuAi þ v Ai ÞÞ ; 1 ð1 v Ai Þ wi
i¼1
ð12Þ
i¼1
Proof. We prove Eq. (12) by using mathematical induction on n.
D E 1 1 1 1 (1) When n = 1, w1 = 1, we have IFWGIAw ðA1 Þ ¼ Aw . 1 ¼ huA1 ; v A1 i ¼ ð1 v A1 Þ ð1 ðuA1 þ v A1 ÞÞ ; 1 ð1 v A1 Þ Thus, Eq. (12) holds for n = 1. (2) If Eq. (12) holds for n = k, i.e.,
IFWGIAw ðA1 ; A2 ; . . . ; Ak Þ ¼
* k Y
+ k k Y Y wi wi : ð1 v Ai Þ ð1 ðuAi þ v Ai ÞÞ ; 1 ð1 v Ai Þ wi
i¼1
i¼1
i¼1
Then, when n = k + 1, by the operational laws in Definition 5, we have kþ1
w IFWGIAw ðA1 ; A2 ; . . . Akþ1 Þ ¼ ^ Ai i i¼1
^ kþ1 Þwkþ1 ¼ IFWGIAw ðA1 ; A2 ; . . . Ak ÞðA * + k k k Y Y Y wi wi wi ¼ ð1 v Ai Þ ð1 ðuAi þ v Ai ÞÞ ; 1 ð1 v Ai Þ i¼1
i¼1
i¼1
^ hð1 v Ai Þwi ð1 ðuAi þ v Ai ÞÞwi ; 1 ð1 v Ai Þwi i * + kþ1 kþ1 kþ1 Y Y Y ð1 v Ai Þwi ð1 ðuAi þ v Ai ÞÞwi ; 1 ð1 v Ai Þwi ; ¼ i¼1
i¼1
i¼1
i.e. Eq. (12) holds for n = k + 1. Therefore, by using mathematical induction on n, Eq. (12) holds for all n. h Theorem 6. If Ai ¼ huAi ; v Ai i 2 IFNs, i = 1, 2, . . . , n, then the aggregated value by using the IFWGIA operator is also an intuitionistic fuzzy number, i.e. IFWGIA(A1, . . . , An) 2 IFNs. Proof. Since Ai ¼ huAi ; v Ai i 2 IFNs; i ¼ 1; 2; . . . ; n, by Definition 2, we have
0 6 uAi ; v Ai 6 1 and 0 6 uAi þ v Ai 6 1; then
061
n Y ð1 v Ai Þwi 6 1;
06
i¼1
n Y
ð1 v Ai Þwi
i¼1
n Y ð1 ðuAi þ v Ai ÞÞwi 6 1 i¼1
and
1
n Y
! ð1 v Ai Þ
wi
þ
n odi¼1 ð1
i¼1
wi
v Ai Þ
n Y i¼1
! wi
ð1 ðuAi þ v Ai ÞÞ
¼1
n Y
ð1 ðuAi þ v Ai ÞÞwi 2 ½0; 1:
i¼1
Thus,
IFWGIAðA1 ; . . . ; An Þ 2 IFNs:
Theorem 7. Let Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ and Bi ¼ huBi ; v Bi iði ¼ 1; . . . ; nÞ be two collections of intuitionistic fuzzy numbers and P w = (w1, w2, . . . , wn)T is the associated weighting vector satisfying wi 2 [0, 1] and ni¼1 wi ¼ 1.
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(1) Idempotency: If Ai ¼ A0 ¼ huA0 ; v A0 i for all i, then IFWGIAw(A1, A2, . . . , An) = A0. (2) Boundedness: Let A ¼ hmaxf0; ðminðuAi þ v Ai Þ maxðv Ai ÞÞg; maxðv Ai Þi,
Aþ ¼ hmaxðuAi þ v Ai Þ minðv Ai Þ; minðv Ai Þi: Then we have A 6 IFWGIAw(A1, A2, . . . , An) 6 A+. (3) Monotonicity: When v Ai P v Bi , uAi þ v Ai 6 uBi þ v Bi for all i, we have IFWGIAw(A1, A2, . . . , An) 6 IFWGIAw(B1, B2, . . . , Bn).
Proof. (1) Since Ai ¼ A0 ¼ huA0 ; v A0 iði ¼ 1; . . . ; nÞ and
IFWGIAw ðA1 ; A2 ; . . . An Þ ¼
* n Y
wi
ð1 v A0 Þ
i¼1
* ¼
n Y
Pn
i¼1 wi
¼ 1, by Theorem 5, we have
ð1 ðuA0
n Y þ v A0 ÞÞ ; 1 ð1 v A0 Þwi
i¼1
i¼1
n X
n X
wi
ð1 v A0 Þ i¼1
+
wi
n X
wi
wi
+
ð1 ðuA0 þ v A0 ÞÞ i¼1 ; 1 ð1 v A0 Þ i¼1
¼ huA0 ; v A0 i ¼ A0 :
Pn Q (2) Since maxðv Ai Þ ¼ 1 ð1 maxðv Ai ÞÞ i¼1 wi P 1 ni¼1 ð1 v Ai Þwi , and Pn Q Q 1 ni¼1 ð1 v Ai Þwi P 1 ni¼1 ð1 minðv Ai ÞÞwi ¼ 1 ð1 minðv Ai ÞÞ i¼1 wi ¼ minðv Ai Þ, thus, n X
n X
maxðuAi þ v Ai Þ minðv Ai Þ ¼ ð1 minðv A1 ÞÞ i¼1
ð1 maxðuAi þ v Ai ÞÞ i¼1 n X
wi
P
n Y
n Y
i¼1
i¼1
ð1 v Ai Þwi
wi
wi
ð1 ðuAi þ v Ai ÞÞwi P ð1 maxðv Ai ÞÞ i¼1
ð1 minðuAi þ v Ai ÞÞ i¼1
¼ minðuAi þ v Ai Þ maxðv Ai Þ: According to Theorem 6, IFWGIA(A1, . . . , An) 2 IFNs, we have n n Y Y ð1 v Ai Þwi ð1 ðuAi þ v Ai ÞÞwi P 0: i¼1
i¼1
Therefore, n n Y Y ð1 v Ai Þwi ð1 ðuAi þ v Ai ÞÞwi P maxf0; ðminðuAi þ v Ai Þ maxðv Ai ÞÞg: i¼1
i¼1
Then, according to Definition 4, we obtain that
A 6 IFWGIAw ðA1 ; A2 . . . An Þ 6 Aþ : Q Q (3) Since v Ai P v Bi , we have 1 ni¼1 ð1 v Ai Þwi 6 1 ni¼1 ð1 v Bi Þwi . Because v Ai P v Bi ði ¼ 1; 2; . . . ; nÞ and uAi þ v Ai 6 uBi þ v Bi ði ¼ 1; 2; . . . ; nÞ, we have n n n n Y Y Y Y ð1 v Ai Þwi ð1 ðuAi þ v Ai ÞÞwi 6 ð1 v Bi Þwi ð1 ðuBi þ v Bi ÞÞwi : i¼1
i¼1
i¼1
n X
i¼1
Therefore, according to Definition 4, we get
IFWGIAw ðA1 ; A2 ; . . . An Þ 6 IFWGIAw ðB1 ; B2 ; . . . Bn Þ: In some special cases, the IFWGIA operator reduces to the IFWGA operator. h Remark 1. If w = (1, 0, . . . , 0)T, then IFWGIAw(A1, A2, . . . , An) = IFWGAw(A1, A2, . . . , An). Remark 2. If w = (0, 0, . . . , 1)T, then IFWGIAw(A1, A2, . . . , An) = IFWGAw(A1, A2, . . . , An).
wi
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Remark 3. If wi = 1, wj = 0, and j – i, then we have IFWGIAw(A1, A2, . . . , An) = IFWGAw(A1, A2, . . . , An). 4.3. Intuitionistic fuzzy ordered weighted geometric interaction averaging (IFOWGIA) operator Inspired by the OWGA operator [34] and the IFOWGA operator [37], we propose intuitionistic fuzzy ordered weighted geometric interaction averaging (IFOWGIA) operator. Definition 9. Let Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ be a collection of intuitionistic fuzzy numbers. If
IFOWGIAw ðA1 ; . . . ; An Þ ¼ ^ni¼1 ðArðiÞ Þwi ; then IFOWGIAw is called intuitionistic fuzzy ordered weighted geometric interaction averaging operator, where Ar(i) is the ith largest value of Ai(i = 1, . . . , n), w = (w1, w2, . . . , wn)T is the weighting vector of the IFOWGIA operator, with wi 2 [0, 1] and Pn i¼1 wi ¼ 1. Theorem 8. Let Ai ¼ huAi ; v Ai i, i = 1, 2, . . . , n be a collection of intuitionistic fuzzy numbers. Then,
IFOWGIAw ðA1 ; . . . ; An Þ ¼
* n Y
wi
ð1 v ArðiÞ Þ
i¼1
n Y
ð1 ðuArðiÞ
+ n wi Y ; þ v ArðiÞ ÞÞ ; 1 1 v ArðiÞ wi
i¼1
ð13Þ
i¼1
The proof is similar to Theorem 5, so it is omitted here. Theorem 9. Let Ai ¼ huAi ; v Ai i 2 IFNs, i = 1, 2, . . . , n. Then the aggregated value by using the IFOWGIA operator is also an intuitionitic fuzzy number, i.e.,
IFOWGIAðA1 ; . . . ; An Þ 2 IFNs: Proof. Since ArðiÞ ¼ huArðiÞ ; v ArðiÞ i 2 IFN, i = 1, 2, . . . , n, according to Definition 2, we have
0 6 uArðiÞ ; v ArðiÞ 6 1 and 0 6 uArðiÞ þ v ArðiÞ 6 1; Therefore,
061
n Y ð1 v ArðiÞ Þwi 6 1;
06
n Y
i¼1
ð1 v ArðiÞ Þwi
i¼1
n Y
1 ðuArðiÞ þ v ArðiÞ Þ
wi
61
i¼1
and
1
n Y
! ð1 v ArðiÞ Þwi
þ
i¼1
¼1
n Y
n Y
ð1 v ArðiÞ Þwi
i¼1 wi
ð1 ðuArðiÞ þ v ArðiÞ ÞÞ
n wi Y 1 uArðiÞ þ v ArðiÞ
!
i¼1
2 ½0; 1:
i¼1
Thus, IFOWGIA(A1, . . . , An) 2 IFNs.
h
Theorem 10. Suppose that Ai ¼ huAi ; v Ai i and Bi ¼ huBi ; v Bi i are two intuitionistic fuzzy numbers, i = 1, . . . , n. And w = (w1, w2, P = (w1, w2, . . . , wn)T is the weighting vector of the IFOWGIA operator, satisfying wi 2 [0, 1] and ni¼1 wi ¼ 1. (1) Idempotency: If Ai ¼ A0 ¼ huA0 ; v A0 i for all i, then IFOWGIAw(A1, A2, . . . , An) = A0. (2) Boundedness: If A ¼ hmaxf0; ðminðuAi þ v Ai Þ maxðv Ai ÞÞg; maxðv Ai Þi, Aþ ¼ hmaxðuAi þ v Ai Þ minðv Ai Þ; minðv Ai Þi, then
A 6 IFOWGIAw ðA1 ; A2 . . . An Þ 6 Aþ : (3) Commutativity: Suppose that A0i ¼ huA0i ; v A0i iði ¼ 1; . . . nÞ is any permutation of Ai ¼ huAi ; v Ai iði ¼ 1; . . . nÞ, then
IFOWGIAw A01 ; A02 . . . A0n ¼ IFOWGIAw ðA1 ; A2 . . . An Þ:
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Proof. (1) Since Ai = A0 for all i, and
IFOWGIAw ðA1 ; . . . ; An Þ ¼ ¼
* n Y
i¼1 wi
¼ 1, according to Theorem 8, we have
ð1 v ArðiÞ Þwi
i¼1
n n Y Y ð1 ðuArðiÞ þ v ArðiÞ ÞÞwi ; 1 ð1 v ArðiÞ Þwi i¼1
n n Y Y ð1 v A0 Þwi ð1 ðuA0 þ v A0 ÞÞwi ; 1 ð1 v A0 Þwi
i¼1
i¼1 n X
wi
ð1 v A0 Þ i¼1
+
+
i¼1
i¼1
n X
* ¼
* n Y
Pn
n X
wi
wi
+
ð1 ðuA0 þ v A0 ÞÞ i¼1 ; 1 ð1 v A0 Þ i¼1
¼ huA0 ; v A0 i ¼ A0 : (2) Obviously, n X
wi
maxðv Ai Þ ¼ 1 ð1 maxðv ArðiÞ ÞÞ i¼1
P1
n Y ð1 v ArðiÞ Þwi i¼1
¼1
n Y
ð1 v Ai Þwi P 1
i¼1
n Y
n X
wi
ð1 minðv ArðiÞ ÞÞ
¼ 1 ð1 minðv ArðiÞ ÞÞ i¼1
wi
¼ minðv Ai Þ;
i¼1
i.e.,
maxðv Ai Þ P 1
n Y ð1 v ArðiÞ Þwi P minðv Ai Þ: i¼1
Similarly, we have n X
n X
maxðuAi þ v Ai Þ minðv Ai Þ ¼ ð1 minðv ArðiÞ ÞÞ i¼1
ð1 maxðuArðiÞ þ v ArðiÞ ÞÞ i¼1 n X
wi
wi
n X wi wi n n Y Y P ð1 v ArðiÞ Þwi ð1 ðuArðiÞ þ v ArðiÞ ÞÞwi P ð1 maxðv ArðiÞ ÞÞ i¼1 ð1 minðuArðiÞ þ v ArðiÞ ÞÞ i¼1 i¼1
i¼1
¼ minðuAi þ v Ai Þ maxðv Ai Þ; i.e.,
maxðuAi þ v Ai Þ minðv Ai Þ P
n Y
ð1 v ArðiÞ Þwi
i¼1
n Y ð1 ðuArðiÞ þ v ArðiÞ ÞÞwi P minðuAi þ v Ai Þ maxðv Ai Þ i¼1
By Theorem 9, IFOWGIA(A1, . . . , An) 2 IFNs, we have n n Y Y ð1 v ArðiÞ Þwi ð1 ðuArðiÞ þ v ArðiÞ ÞÞwi P 0: i¼1
i¼1
Qn
Q Therefore, i¼1 ð1 v ArðiÞ Þwi ni¼1 ð1 ðuArðiÞ þ v ArðiÞ ÞÞwi P maxf0; ðminðuAi þ v Ai Þ maxðv Ai ÞÞg. Then, according to Definition 4, we have
A 6 IFOWGIAw ðA1 ; A2 . . . An Þ 6 Aþ : (3) According to Eq. (13), we get
IFOWGIAw A01 ; . . . ; A0n
* + n n n wi Y wi wi Y Y ; 1 v A0rðiÞ 1 uA0rðiÞ þ v A0rðiÞ ;1 1 v A0rðiÞ ¼ i¼1
IFOWGIAw ðA1 ; A2 . . . An Þ ¼
* n Y i¼1
Since
A0i
i¼1
i¼1
+ n n Y Y wi wi wi : ð1 v ArðiÞ Þ ð1 ðuArðiÞ þ v ArðiÞ ÞÞ ; 1 ð1 v ArðiÞ Þ i¼1
i¼1
¼ huA0i ; v A0i iði ¼ 1; . . . nÞ is any permutation of Ai ¼ huAi ; v Ai iði ¼ 1; . . . nÞ, then we have A0rðiÞ ¼ ArðiÞ ði ¼ 1; . . . nÞ. Thus,
IFOWGIAw A01 ; A02 . . . A0n ¼ IFOWGIAw ðA1 ; A2 . . . An Þ:
The IFOWGIA operator reduces to the IFOWGA operator when we take special cases of the weighting vector w. h Remark 4. If w = (1, 0, . . . , 0)T, then IFOWGIAw(A1, A2, . . . , An) = IFOWGAw(A1, A2, . . . , An).
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Remark 5. If w = (0, 0, . . . , 1)T, then IFOWGIAw(A1, A2, . . . , An) = IFOWGAw(A1, A2, . . . , An). Remark 6. If wi = 1, wj = 0, and j – i, then
IFOWGAw ðA1 ; A2 . . . An Þ ¼ IFOWGIAw ðA1 ; A2 . . . An Þ ¼ ArðiÞ ; where Ar(i) is the ith largest of Ai(i = 1, . . . , n). 4.4. Intuitionistic fuzzy hybrid geometric interaction averaging Operator Considering both the given intuitionistic fuzzy value and its ordered position, Xu and Yager [37] developed the intuitionistic fuzzy hybrid geometric averaging (IFHA) operator. Definition 10 37. Let Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ be a collection of intuitionistic fuzzy numbers. The IFHGA operator with respect to weighting vector w (IFHGAw) is defined as
e rð1Þ Þwi . . . ð A e rðnÞ Þwn ¼ IFHGAx;w ðA1 ; . . . ; An Þ ¼ ð A
* n Y i¼1
wi
u ;1 eA rðiÞ
n Y i¼1
+
wi
ð1 v e Þ A rðiÞ
;
ð14Þ
e rðiÞ is theith largest of the intuitionistic fuzzy values A e i ¼ Anxi ; i ¼ 1; . . . ; n. x = (x1, . . . , xn)T is the weight vector of where A i Pn Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ, satisfying xi 2 [0, 1] and i¼1 xi ¼ 1, n is the balancing coefficient. Suppose that Ai ¼ huAi ; v Ai i 2 IFNs, i = 1, 2, . . . , n. If uAk ¼ 0 and uAi – 0ði – k; Þ, then according to Definition 10, we have uIFHGIAx;w ðA1 ;...;An Þ ¼ 0. Obviously, uAi ði – k; i ¼ 1; 2; . . . ; nÞ are not accounted for at all, which is an undesirable feature of an averaging operation. As a result, we propose the intuitionistic fuzzy hybrid geometric interaction averaging operator, taking interactions into consideration between membership function and non-membership function of different intuitionistic fuzzy sets. Definition 11. Let Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ be a collection of intuitionistic fuzzy numbers, X be the set of all intutionistic fuzzy numbers. The intuitionistic fuzzy hybrid geometric interaction averaging (IFHGIA) operator of dimension n is a P mapping IFHGIA:Xn ? X, which has an associated vector w = (w1, w2, . . . , wn)T, satisfying wi 2 [0, 1] and ni¼1 wi ¼ 1 such that
e rðiÞ Þwi ; IFHGIAx;w ðA1 ; . . . ; An Þ ¼ ^ni¼1 ð A e rðiÞ is the ith largest of the intuitionistic fuzzy values A e i ¼ Anxi ; i ¼ 1; . . . ; n. x = (x1, . . . , xn)T is the weight vector of where A i Pn Ai, i = 1, . . . , n, satisfying xi 2 [0, 1] and i¼1 xi ¼ 1, n is the balancing coefficient.. Theorem 11. Let Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ be a collection of intuitionistic fuzzy numbers. Then
IFHGIAx;w ðA1 ; . . . ; An Þ ¼
* n Y i¼1
wi
½1 v e A rðiÞ
n Y i¼1
½1 ðue
A rðiÞ
+ n Y wi : þ v e Þ ; 1 ð1 v e Þ A rðiÞ A rðiÞ wi
ð15Þ
i¼1
The proof is similar to Theorem 5, so it is omitted here. The IFHGIA operator can be interpreted from four aspects as follows. (1) It not only includes the interactions of non-membership function of different IFSs, and that of membership function of different IFSs, but also the interactions are involved between non-membership function and membership function of different IFSs. (2) It weights the intuitionistic fuzzy numbers Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ by the associated weights x = (x1, . . . , xn)T and nx a balancing coefficient n, and then we can get the weighted intuitionistic fuzzy numbers Ai i ði ¼ 1; . . . ; nÞ. nx e rð1Þ ; A e rð2Þ ; . . . ; A e rðnÞ Þ, where (3) It reorders the weighted intuitionistic fuzzy values Ai i ði ¼ 1; . . . ; nÞ in descending order ð A e rðiÞ is the ith largest of the weighted intuitionistic fuzzy values A e i ¼ Anxi ; i ¼ 1; . . . ; n. A i (4) It considers the given intuitionistic fuzzy numbers and the ordered positions, and then the intuitionistic fuzzy nume rðiÞ Þwi ði ¼ 1; 2; . . . ; nÞ are aggregated into a collective one. bers ð A Suppose that Ai ¼ huAi ; v Ai iði ¼ 1; . . . ; nÞ is a collection of IFNs. If uAk ¼ 0; uAi – 0ði – k; i ¼ 1; 2; . . . ; nÞ, then by Definition 11, we have uIFHGIAx;w ðA1 ;...;An Þ – 0. Thus, uAk does not play a decisive role. Therefore, the proposed IFHGIA operator can be seen as a good complement to the existing IFHGA operator, which can be seen clearly in the following example. Example 2. Let A1 = h0, 0.5i, A2 = h0.4, 0.2i, A3 = h0.5, 0.4i, A4 = h0.3, 0.3i, A5 = h0.7, 0.1i be five intuitionistic fuzzy numbers. x = (0.25, 0.20, 0.15, 0.18, 0.22)T is the weight vector of Ai (i = 1, 2, . . . , 5). Then according to the operational law of Definition 3, we have
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Y. He et al. / Information Sciences 259 (2014) 142–159
e 1 ¼ h050:25 ; 1 ð1 0:5Þ50:25 i ¼ h0; 0:5796i; A e 2 ¼ h0:450:20 ; 1 ð1 0:2Þ50:20 i ¼ h0:4000; 0:2000i; A e 4 ¼ h0:350:18 ; 1 ð1 0:3Þ50:18 i ¼ h0:287; 0:275i; e 3 ¼ h0:550:15 ; 1 ð1 0:4Þ50:15 i ¼ h0:504; 0:318i; A A 50:22 50:22 e 5 ¼ h1 ð1 0:1Þ ; 0:7 i ¼ h0:7203; 0:1094i: A According to Definition 3, we obtain that
e 1 Þ ¼ 0:5796; Sð A
e 2 Þ ¼ 0:2000; Sð A
e 3 Þ ¼ 0:2763; Sð A
e 4 Þ ¼ 0:0638; Sð A
e 5 Þ ¼ 0:5660: Sð A
Obviously,
e 5 Þ > Sð A e 3 Þ > Sð A e 2 Þ > Sð A e 4 Þ > Sð A e 1 Þ: Sð A Therefore,
e rð1Þ ¼ h0:6755; 0:1094i; A e rð5Þ ¼ h0; 0:5796i: A
e rð2Þ ¼ h0:5946; 0:3183i; A
e rð3Þ ¼ h0:4000; 0:2000i; A
Supposing that w = (w1, w2, . . . , w5)T is determined by w = (0.112, 0.236, 0.304, 0.236, 0.112). By Definition 10, we get
A ¼ IFHGAw;x ðA1 ; A2 ; A3; A4 ; A5 Þ ¼
* 5 Y j¼1
D
wj
ðue Þ ; 1 A rðjÞ
5 Y j¼1
the
normal
e rð4Þ ¼ h0:3384; 0:2746i; A
distribution
based
method
[31],
+ ð1 v e Þ
wj
A rðjÞ
¼ 0:67550:112 0:59460:236 0:40:304 0:33840:236 0:00000:112 ; 1 ð1 0:1094Þ0:112 ð1 0:3183Þ0:236 ð1 0:2Þ0:304 ð1 0:2746Þ0:236 ð1 0:5796Þ0:112
E
¼ h0; 0:2911i: While, by Definition 11 and Theorem 11, we have
D E e 1 ¼ ð1 0:5Þ50:25 ð1 0:5Þ50:25 ; 1 ð1 0:5Þ50:25 ¼ h0; 0:5796i; A D E e 2 ¼ ð1 0:2Þ50:20 ð1 0:6Þ50:20 ; 1 ð1 0:2Þ50:20 ¼ h0:4000; 0:2000i; A D E e 3 ¼ ð1 0:4Þ50:15 ð1 0:9Þ50:15 ; 1 ð1 0:4Þ50:15 ¼ h0:5039; 0:3183i; A D E e 4 ¼ ð1 0:3Þ50:18 ð1 0:6Þ50:18 ; 1 ð1 0:3Þ50:18 ¼ h0:2870; 0:2746i; A D E e 5 ¼ ð1 0:1Þ50:22 ð1 0:8Þ50:22 1 ð1 0:1Þ50:22 ¼ h0:7203; 0:1094i: A According to Definition 5, we have
e 1 Þ ¼ 0:5796; Sð A
e 2 Þ ¼ 0:2000; Sð A
e 3 Þ ¼ 0:1856; Sð A e 4 Þ ¼ 0:0125; Sð A e 5 Þ ¼ 0:6109: Sð A
Obviously,
e 5 Þ > Sð A e 2 Þ > Sð A e 3 Þ > Sð A e 4 Þ > Sð A e 1 Þ: Sð A As a result, we have
e rð1Þ ¼ h0:7203; 0:1094i; A e rð5Þ ¼ h0; 0:5796i: A
e rð2Þ ¼ h0:4000; 0:2000i; A
e rð3Þ ¼ h0:5039; 0:3183i; A
e rð4Þ ¼ h0:2870; 0:2746i; A
By Theorem 11, it follows that
A ¼ IFHGIAw;x ðA1 ; A2 ; A3; A4 ; A5 Þ * + 5 5 5 Y Y Y wj wj wj ¼ h0:4093; 0:2919i: ½1 v e ½1 ðue þ v e Þ ; 1 ð1 v e Þ ¼ A rð1jÞ A rð1jÞ A rð1jÞ A rð1jÞ j¼1
j¼1
j¼1
It is evident that uIFHGIAx;w ðA1 ;...;A5 Þ ¼ 0:4093 – 0. Therefore, uA1 does not play a decisive role. Practically, the IFHGIA operator is a generalization of the IFWGIA operator or the IFOWGIA operator. Remark 7. If w ¼ ðw1 ; w2 ; . . . ; wn ÞT ¼
Remark 8. If x ¼ ðx1 ; x2 ; . . . ; xn ÞT ¼
1
1 T n;...n ,
1 n
the IFHGIA operator reduces to the IFWGIA operator.
T ; . . . 1n , the IFHGIA operator reduces to the IFOWGIA operator.
then
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Y. He et al. / Information Sciences 259 (2014) 142–159
5. Multi-criteria decision making with the IFHGIA operator For a multi-criteria decision making problem, it is assumed that X = {x1, x2, . . . , xn} is a set of alternatives, G = {G1, G2, P . . . , Gm} is a set of attributes with the associated weighting vector w = (w1, w2, . . . , wm)T, satisfying wi 2 [0, 1] and m i¼1 wi ¼ 1. Suppose that the characteristics of the alternatives xi (i = 1, 2, . . . , n) are represented by intuitionistic fuzzy sets Aij ¼ huAij ; v Aij iði ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; mÞ in Table 1, where uAij indicates the degree that the alternative xi satisfies the attribute Gj, v Aij denotes the degree that the alternative xi does not satisfy the attribute Gj, i = 1, 2, . . . , n; j = 1, 2, . . . , m. Then we have the following decision making method. Step 1: Assume that x = (x1, x2, . . . , xm)T is the weighting vector of Ai1, Ai2, . . . , Aim, i = 1, 2, . . . , n, satisfying xj 2 [0, 1] and Pm mxj e ði ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; mÞ in Table 2. j¼1 xj ¼ 1, m is the balancing coefficient. By Definition 5, we get A ij ¼ Aij e e e e e i1 ; A e i2 ; . . . A e im , Step 2: By Definition 4, we get A rði1Þ ; A rði2Þ ; . . . A rðimÞ , i = 1, 2, . . . , n, where A rðijÞ is the jth largest of A i = 1, 2, . . . , n. Step 3: According to the aggregation operators in Definition 11, we get the final intuitionistic fuzzy sets Ai (i = 1, 2, . . . , n) by using the normal distribution based method [31] to determine weights of the IFOWGIA operator. Step 4: According to Definition 4, we get the order of the final intuitionistic fuzzy sets Ai(i = 1, 2, . . . , n). Step 5: Rank all the alternatives xi(i = 1, 2, . . . , n) and select the best one (s).
6. Numerical example and systematic comparison 6.1. Numerical example Assume that an investment company wants to invest a sum of money in the best option. Three possible alternatives are to be considered by analyzing the market. x1 is a car company. x2 is a food company. x3 a computer company. In order to assess these alternatives, the investors have brought panel data. After careful review of the information, they summarize the ability of companies with five attributes G = {G1, G2, G3, G4, G5}.
G1: G2: G3: G4: G5:
The The The The The
risk analysis. growth analysis. social-political impact analysis. environmental impact analysis. development of the society.
The three possible alternatives are to be evaluated by the intuitionistic fuzzy information in Table 3 under above five attributes.
Table 1 Intuitionistic fuzzy matrix (Aij)nm. G1
G2
...
Gm1
Gm
A11 A21 ... An1
A12 A22 ... An2
... ... ... ...
A1,m1 A2,m1 ... An,m1
A1,m A2,m ... An,m
G1
G2
...
Gm1
Gm
e 12 A e 22 A
...
x2
e 11 A e 21 A
...
e 1;m1 A e 2;m1 A
e 1;m A e 2;m A
... xn
... e n;1 A
... e n;2 A
... ...
... e n;m1 A
... e n;m A
x1 x2 ... xn
Table 2 e ij Þ Intuitionistic fuzzy matrix ð A . nm
x1
155
Y. He et al. / Information Sciences 259 (2014) 142–159 Table 3 Intuitionistic fuzzy matrix (Aij)35.
x1 x2 x3
G1
G2
G3
G4
G5
h0.2, 0.5i h0.2, 0.7i h0.2, 0.7i
h0.4, 0.2i h0.6, 0.3i h0.5, 0.3i
h0.5, 0.4i h0.4, 0.3i h0.4, 0.5i
h0.3, 0.3i h0.4, 0.4i h0.3, 0.4i
h0.7, 0.1i h0.6, 0.1i h0.6, 0.2i
Table 4 e ij ÞÞ . Score matrix ðSð A 35
i=1 i=2 i=3
j=1
j=2
j=3
j=4
j=5
0.3811 0.6122 0.6122
0.2000 0.3000 0.2000
0.1856 0.1252 0.0114
0.0125 0.0280 0.0755
0.6109 0.5152 0.3944
Table 5 e rðijÞ Þ . Intuitionistic fuzzy ordered matrix ð A 35
i=1 i=2 i=3
j=1
j=2
j=3
j=4
j=5
h0.7203, 0.1094i h0.6246, 0.1094i h0.6120, 0.2176i
h0.4000, 0.2000i h0.6000, 0.3000i h0.5000, 0.3000i
h0.5039, 0.3183i h0.3599, 0.2347i h0.4168, 0.4054i
h0.2870, 0.2746i h0.3965, 0.3686i h0.2930, 0.3686i
h0.1984, 0.5796i h0.1658, 0.7780i h0.1658, 0.7780i
e ij ¼ Anxj , where the balancing coefficient n = 5, Step 1: According to operational law in Definition 5, we obtain A ij T x = {0.25, 0.20, 0.15, 0.18, 0.22} is the weighting vector of Ai1, Ai2, . . . , Ai5, i = 1, 2, 3; j = 1, 2, . . . , 5. e ij ÞÞ Step 2: By Definition 4, we get the score matrix ðSð A 35 in Table 4. Obviously,
e 15 Þ > Sð A e 12 Þ > Sð A e 13 Þ > Sð A e 14 Þ > Sð A e 11 Þ; Sð A e 25 Þ > Sð A e 22 Þ > Sð A e 23 Þ > Sð A e 24 Þ > Sð A e 21 Þ Sð A and
e 35 Þ > Sð A e 32 Þ > Sð A e 33 Þ > Sð A e 34 Þ > Sð A e 31 Þ: Sð A e rðijÞ Þ e rðijÞ ði ¼ 1; 2; 3; j ¼ 1; 2; . . . ; 5Þ is the jth largThen, we get the intuitionistic fuzzy ordered matrix ð A in Table 5, where A 35 e i1 ; A e i2 ; A e i3 ; A e i4 ; A e i5 ; i ¼ 1; 2; 3. est of A Step 3: The weights w = (0.112, 0.236, 0.304, 0.236, 0.112) are determined by the normal distribution based method [31]. According to aggregation operators in Definition 9, we obtain the final intuitionistic fuzzy sets Ai(i = 1, 2, 3):
A1 ¼ IFHGIAw;x ðA11 ; A12 ; A13; A14 ; A15 Þ ¼
* 5 Y j¼1
1 ve
wj
A rð1jÞ
5 Y 1 ðue
þ ve
wj 5 Y Þ ; 1 ð1 v e
5 Y 1 ðue
5 Y 1 ðue
j¼1
A rð1jÞ
+
Þ
wj
þ ve
wj 5 Y Þ ; 1 ð1 v e
Þ
wj
þ ve
wj 5 Y Þ ; 1 ð1 v e
Þwj
A rð1jÞ
j¼1
A rð1jÞ
¼ h0:4294; 0:2988i; A2 ¼ IFHGIAw;x ðA21 ; A22 ; A23; A24 ; A25 Þ ¼
* 5 Y j¼1
1 ve
wj
A rð2jÞ
j¼1
A rð2jÞ
A rð2jÞ
j¼1
A rð2jÞ
+
¼ h0:4383; 0:3659i; A3 ¼ IFHGIAw;x ðA31 ; A32 ; A33; A34 ; A35 Þ ¼
* 5 Y j¼1
1 ve
¼ h0:3927; 0:4212i: Step 4: According to Definition 4, we have
SðA1 Þ ¼ 0:1306; SðA2 Þ ¼ 0:0724; SðA3 Þ ¼ 0:0285: Thus, S(A1) > S(A2) > S(A3). Step 5: According to the scores in Step 4, we have
A rð3jÞ
wj
j¼1
A rð3jÞ
A rð3jÞ
j¼1
A rð3jÞ
+
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Y. He et al. / Information Sciences 259 (2014) 142–159
Table 6 e ij ÞÞ . Score matrix ðSð A 35 j=1
j=2
j=3
j=4
j=5
0.5796 0.6442 0.6442
0.2000 0.3000 0.2000
0.2763 0.2683 0.0976
0.0638 0.0698 0.0302
0.5660 0.4607 0.3525
j=1
j=2
j=3
j=4
j=5
h0.6755, 0.1094i h0.5701, 0.1094i h0.5701, 0.2176i
h0.5946, 0.3183i h0.6000, 0.3000i h0.5000, 0.3000i
h0.4000, 0.2000i h0.5030, 0.2347i h0.5030, 0.4054i
h0.3384, 0.2746i h0.4384, 0.3686i h0.3384, 0.3686i
h0, 0.5796i h0.1337, 0.7780i h0.1337, 0.7780i
i=1 i=2 i=3
Table 7 e rðijÞ Þ . Intuitionistic fuzzy ordered matrix ð A 35
i=1 i=2 i=3
Table 8 e ij ÞÞ . Score matrix ðSð A 35
i=1 i=2 i=3
j=1
j=2
j=3
j=4
j=5
0.5796 0.6122 0.6122
0.2000 0.3000 0.2000
0.1856 0.1252 0.0144
0.0125 0.0280 0.0755
0.6109 0.5152 0.3944
x1 x2 x3 : Therefore, the optimal alternative is x1. If we use the method with the operator proposed in [37], we also have x1 x2 x3. i.e., the best alternative is x1. Thus, the results of the above two methods are same. Therefore, the method proposed in this paper is effective and valid. 6.2. Systematic comparison with corresponding work of other papers However, the method developed in [37] has some differences from the proposed method in this paper. For example, if A11 ¼ A011 ¼ h0; 0:5i, we have the following steps using the method in [37]. e ij ¼ A5xj i ¼ 1; 2; 3; j ¼ 1; 2; . . . ; 5, where x = {0.25, 0.20, 0.15, 0.18, 0.22}T is Step 1: By the operational law of Definition 5, A ij the weighting vector of Ai1, Ai2, . . . , Ai5, i = 1, 2, 3. e ij ÞÞ Step 2: By Definition 4 we obtain the score matrix ðSð A in Table 6. 35 Obviously,
e 15 Þ > Sð A e 13 Þ > Sð A e 12 Þ > Sð A e 14 Þ > Sð A e 11 Þ; Sð A e 25 Þ > Sð A e 22 Þ > Sð A e 24 Þ > Sð A e 23 Þ > Sð A e 21 Þ Sð A and
e 35 Þ > Sð A e 32 Þ > Sð A e 34 Þ > Sð A e 33 Þ > Sð A e 31 Þ: Sð A e rðijÞ Þ e rðijÞ is the jth largest of Then, we get the intuitionistic fuzzy ordered matrix ð A in Table 7, where A 35 e i1 ; A e i2 ; . . . A e i5 ; i ¼ 1; 2; 3. A Step 3: By the aggregation operator in Definition 8, and w = (0.112, 0.236, 0.304, 0.236, 0.112). It follows that
A1 ¼ IFHGAw;x ðA11 ; A12 ; A13; A14 ; A15 Þ ¼
* 5 Y j¼1
wj
ðue Þ ; 1 A rð1jÞ
5 Y j¼1
+ ð1 v e
A rð1jÞ
Similarly, A2 = h0.4438, 0.3659i, A3 = h0.3999, 0.4211i. Step 4: By Definition 4, S(A1) = 0.2911, S(A2) = 0.0779, S(A3) = 0.0212. Thus, S(A2) > S(A3) > S(A1). Step 5: Therefore, x2 x3 x1 Therefore, the optimal alternative is x2.
wj
Þ
¼ h0; 0:2911i;
157
Y. He et al. / Information Sciences 259 (2014) 142–159 Table 9 e rðijÞ Þ . Intuitionistic fuzzy ordered matrix ð A 35
i=1 i=2 i=3
j=1
j=2
j=3
j=4
j=5
h0.7203, 0.1094i h0.6246, 0.1094i h0.6120, 0.2176i
h0.4000, 0.2000i h0.6000, 0.3000i h0.5000, 0.3000i
h0.5039, 0.3183i h0.3599, 0.2347i h0.4168, 0.4054i
h0.2870, 0.2746i h0.3965, 0.3686i h0.2930, 0.3686i
h0, 0.5796i h0.1658, 0.7780i h0.1658, 0.7780i
e 1j Þ > Sð A e 3j Þðj ¼ 1; 2; . . . ; 5Þ. By Table 7 and Definition 4, we have that By Table 6, we know that Sð A e e Sð A rð1jÞ Þ P Sð A rð3jÞ Þ; j ¼ 1; . . . ; 5. Thus, it is more acceptable for us to get the result that x1 x3 from monotonicity point of view. Therefore, the method proposed in [37] is not very stable. However, if we use the method developed by this paper, we have following steps. e ij ¼ A5xj i ¼ 1; 2; 3; j ¼ 1; 2; . . . ; 5, Step 1: According to operational law in Definition 5, we get A ij T x = {0.25, 0.20, 0.15, 0.18, 0.22} is the weighting vector of Ai1, Ai2, . . . , Ai5, i = 1, 2, 3. e ij ÞÞ Step 2: By Definition 4, we get the score matrix ðSð A in Table 8. 35 Obviously,
where
e 15 Þ > Sð A e 12 Þ > Sð A e 13 Þ > Sð A e 14 Þ > Sð A e 11 Þ; Sð A e 25 Þ > Sð A e 22 Þ > Sð A e 23 Þ > Sð A e 24 Þ > Sð A e 21 Þ; Sð A and
e 35 Þ > Sð A e 32 Þ > Sð A e 33 Þ > Sð A e 34 Þ > Sð A e 31 Þ: Sð A e rðijÞ Þ e rðijÞ is the jth largest of Then, we get the intuitionistic fuzzy ordered matrix ð A in Table 9, where A 35 e e e A i1 ; A i2 ; . . . A i5 ; i ¼ 1; 2; 3. Step 3: By the aggregation operator in Definition 11 and Theorem 11, and w = (0.112, 0.236, 0.304, 0.236, 0.112), we get that
A1 ¼ IFHGIAw;x ðA11 ; A12 ; A13; A14 ; A15 Þ ¼
* 5 Y j¼1
1 ve
wi
A rð1jÞ
5 Y j¼1
1 ðue
A rð1jÞ
þ ve
A rð1jÞ
wi Þ
;1
5 Y ð1 v e j¼1
A rð1jÞ
+ Þwj
¼ h0:4093; 0:2919i; Similarly, A2 = h0.4383, 0.3659i, A3 = h 0.3927, 0.4212i. Step 4: By Definition 4, we have
SðA1 Þ ¼ 0:1105;
SðA2 Þ ¼ 0:0724 and SðA3 Þ ¼ 0:0285:
Therefore, S(A1) > S(A2) > S(A3). Step 5: According to the scores S(Ai)(i = 1, 2, 3), we rank all the alternatives xi(i = 1, 2, 3) and obtain that
x1 x2 x3 : Therefore, the optimal alternative is x1. By Table 8, we have
e 1j Þ P Sð A e 3j Þ; j ¼ 1; . . . ; 5: Sð A By Table 9 and Definition 4, we have
e rð1jÞ Þ P Sð A e rð3jÞ Þ; j ¼ 1; . . . ; 5: Sð A Thus, it is acceptable for us to get the result that x1 x3 from monotonicity point of view. That is to say, the result of this paper is acceptable, which means the proposed method is a good complement to the existing works on IFSs in the case that one of the membership degree is zero. We have summarized the systematic comparison in four aspects: (1) Atanassov [4] originally defined the multiplication operation on IFSs, De et al. [12] defined the power operation by reasoning from the multiplication operation. In this paper, we define the new geometric operations of IFSs from the probability point of view and give the geometric interpretations of these geometric operations. We propose the new multiplication operation on IFSs based on probability non-membership (PN) function operator, probability membership (PM) function operator and probability hetergeneous (PH) operator. And the geometric meanings of these new operations are interpreted in Figs. 1–5. (2) The existing geometric operations just consider the effects of membership degree or non-membership degree of different IFSs, while the new geometric operations also take the interactions of non-membership degree and membership de-
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gree of different IFSs into consideration, which could be used and explained reasonably in more case, especially when one of the membership degree is zero. It can be seen clearly in Example 1. (3) By the aggregation operators in [37], when there exists only one membership degree of IFS equals to zero, the membership degree of aggregation result of n IFSs is zero even if the membership degrees of n 1 IFSs are not zero, which is the weakness of the aggregation operators in [37]. In the above cases, if we use the operators presented in this paper, the aggregation result can be explained reasonably, which could be seen clearly by Example 2. (4) In the process of intuitionistic fuzzy multi-criteria decision making, the evaluated values of candidate alternatives under attributes are IFSs. In Section 6.1, we know that if none of the membership degree of IFSs is zero, we can get the same optimal alternative by the approaches developed in [37], and the ranking result is same. But when only one membership degree of IFSs is zero, the result gotten by the proposed approach in this paper is acceptable from monotonicity point of view. 7. Conclusions Intuitionistic fuzzy information aggregation is an interesting research field of the IFS theory. The rational operational laws on IFS are very important to aggregate the intuitionistic fuzzy information. In this paper, we propose the new operational laws on IFSs and develop the intuitionistic fuzzy geometric interaction averaging (IFGIA) operator, which provide a good complement to the existing works on IFSs. In the succeeding work, we plan to extend our research to the interval-valued intuitionistic fuzzy environment, we will develop some new operational laws on interval-valued intuitionistic fuzzy sets and propose the corresponding interval-valued intuitionistic fuzzy geometric interaction averaging (IVIFGIA) operators. We intend to apply these approaches to multiattribute decision making, pattern recognition, data mining, clustering and medical diagnosis. In decision-making, the generalization capability of extracting fuzzy rules is the key index. Recently, some new refinement techniques [28,44] related to maximize the uncertainty or to combine multiple classifiers have been proposed to improve the generalization of the decision rules. The fuzzy information systems is related to extracting rules because it contains conditional attributes and decision attributes. We will conduct some numerical experiments on comparing with some existing fuzzy rule extraction methods [17,43] in the future research under the intuitionistic fuzzy environment. Acknowledgments This work was supported by National Natural Science Foundation of China (71071002, 71371011, 71301001), Higher School Specialized Research Fund for the Doctoral Program (20123401110001), The Scientific Research Foundation of the Returned Overseas Chinese Scholars, Anhui Provincial Natural Science Foundation (1308085QG127), Provincial Natural Science Research Project of Anhui Colleges (KJ2012A026), Humanity and Social Science Youth foundation of Ministry of Education (13YJC630092), Humanities and social science Research Project of Department of Education of Anhui Province (SK2013B041), Natural Science Foundation of Anhui Provincial Higher School and Foundation for the Young Scholar of Anhui University (2009QN022B). The authors are thankful to the anonymous reviewers and the editor for their valuable comments and constructive suggestions that have led to an improved version of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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