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The ordered multiplicative modular geometric operator
Knowledge-Based Systems 39 (2013) 144–150
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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
The ordered multiplicative modular geometric operator Liuhao Chen a, Zeshui Xu b,⇑, Xiaohan Yu a a b
Institute of Communications Engineering, PLA University of Science and Technology, Nanjing, Jiangsu 210007, China Institute of Sciences, PLA University of Science and Technology, Nanjing, Jiangsu 210007, China
a r t i c l e
i n f o
Article history: Received 19 March 2012 Received in revised form 6 September 2012 Accepted 21 October 2012 Available online 10 November 2012 Keywords: Modularity Symmetric capacity Ordered multiplicative modular geometric Aggregation function Operator
a b s t r a c t The aim of this paper is to investigate an ordered multiplicative modular geometric operator and its relevant properties. The ordered multiplicative modular geometric operator is a generalized form of the ordered weighted geometric operator which has been designed incorporating the advantages of the geometric mean to deal with ratio judgments and the advantages of the ordered weighted averaging (OWA) operator to represent the concept of fuzzy majority in the process of information aggregation. Besides, the ordered multiplicative modular geometric operator can be seen as a symmetrized multiplicative modular aggregation function, characterized by comonotone multiplicative modularity. It is worth pointing that lots of the existing operators (such as the ordered weighted geometric operator, the weighted geometric operator, the ordered weighted maximum, and the Max and Min operators) can be regarded as the special cases of the ordered multiplicative modular geometric operator, which is of value in developing the theory of geometric operators. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Aggregation operators are a useful tool which can be widely utilized to multi-criteria decision making, neural networks, image process, pattern recognition, and machine learning, etc. From a general point of view, the aggregation operators are used to integrate different pieces of information so as to come to conclusions or make decisions. The aggregation operators have the common structural feature that integrates finite arguments into a single value so as to carry out the sequent operations. There are various aggregation operators with different forms. Due to respective different properties and application fields, we introduce some typical aggregation operators as follows. A useful operator, the ordered weighted averaging (OWA) operator, was introduced by Yager [1] in 1988. Since then, the OWA operator has been used in a wide range of application fields including neural networks [2,3], multi-criteria decision making [4,5], database systems [6], fuzzy logic controllers [7,8], and group decision making [9,10], etc. A fundamental aspect of the OWA operator is a reordering step in which the input arguments are rearranged according to their values. Therefore, the OWA operator is a nonlinear aggregation operator. The traditional Yager’s OWA operator focuses exclusively on the aggregation of crisp numbers. In real decision making situations, the information is not always given by the form of crisp numbers, but intuitionistic fuzzy numbers or ⇑ Corresponding author. E-mail addresses: [email protected] (L. Chen), [email protected] (Z. Xu), [email protected] (X. Yu). 0950-7051/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.knosys.2012.10.014
uncertain information. So a lot of scholars have extended the OWA operator to its generalized forms under intuitionistic fuzzy or other uncertain environments [11–19]. Similarly, the ordered weighted geometric (OWG) operator [20,21], is a useful tool for aggregating a finite collection of arguments. Since it was proposed, the OWG operator has been mentioned repeatedly in many theoretical papers and applied in lots of practical problems. Sequentially, Xu and Da [22] proposed an induced ordered weighted geometric (IOWG) operator, which is an extension of the OWG operator and the weighted geometric mean. Chiclana et al. [23] gave its use in the aggregation of multiplicative preference relations. Besides, harmonic mean [24] and Bonferroni mean [25] are the aggregation operators that can capture the interrelationships among arguments. As the natural development of the above aggregation operators, recently, Mesiar and Mesiarová-Zemánková [26] introduced the ordered modular averages (OMAs) to generalize the ordered weighted averages operators, with the replacement of the additivity in the latter by the modularity. Both the OWA and OWG operators have been widely used in the multi-criteria decision making problems. The decision makers can use each of them for aggregating information in the multi-criteria decision making problems according to their personal preferences. For some special problems, the OWG operator may be more suitable to be used than the OWA operator. For example, there are several multiplicative preference relations with the same sizes. We want to get a total preference relation through aggregating the corresponding elements of each multiplicative preference relation. The OWG operator will be suitable for aggregating decision information taking the form of multiplicative preference
L. Chen et al. / Knowledge-Based Systems 39 (2013) 144–150
relations, but the OWA operator is unsuitable in this case. Therefore, the study on the OWG operator is significant. Inspired by the idea of the OMAs, in this paper, we develop a new aggregation operator called the ordered multiplicative modular geometric (OMMG) operator, which is based on the OWG operator. Some special cases of the OMMG operator are discussed in this paper. It is worth noting that a class of aggregation operators based on Sugeno integral (such as the ordered weighted maximum (OWMax) operator [27]) is proven to be the special case of the OMMG operator. Therefore, the OMMG operator is a very general aggregation operator which generalizes lots of the existing ones. The rest of the paper is organized as follows. In Section 2, we introduce some basic concepts and terminologies. Section 3 develops the OMMG operator, and illustrates it with some examples. In Section 4, we study how to get the OMMG operator and then investigate some of its desirable properties. Section 5 ends the paper with concluding remarks. 2. Preliminaries
Aðx _ yÞ þ Aðx ^ yÞ ¼ AðxÞ þ AðyÞ
Definition 1 (Aggregation function [27–29]). An aggregation function f is a function of n(>1) arguments that maps the (n-dimensional) unit cube onto the unit interval, i.e., f: [0, 1]n ? [0, 1], satisfying the following properties: 1) f(0,0, . . . ,0) = 0 and f(1, 1, . . . , 1) = 1. 2) x 6 y implies f(x) 6 f(y), for all x, y 2 [0, 1]n. Definition 2. (Weighted geometric mean [30]). Let w = (w1, w2, . . . , w ) be a weighted vector satisfying (w1, w2, . . . , wn) 2 [0, 1]n and Pnn i¼1 wi ¼ 1, then, for any fixed vector x = (x1, x2, . . . , xn), the corresponding weighted geometric mean is defined as:
WGMw ðxÞ ¼
n Y w xi i
ð1Þ
i¼1
Based on the weighted geometric mean, Chiclana et al. [20], and Xu and Da [21] introduced the ordered weighted geometric operator, which was defined as follows: Definition 3 (Ordered weighted geometric operator [20,21]). Let w = (w1, w2, . . . , wn) be a weighting vector satisfying (w1, w2, Pn . . . , wn) 2 [0, 1]n and i¼1 wi ¼ 1, then, for any fixed vector x = (x1, x2, . . . , xn), the corresponding ordered weighted geometric operator can be given as:
OWGw ðxÞ ¼
n Y wi xrðiÞ
3. The ordered multiplicative modular geometric (OMMG) operator In order to introduce the OMMG operator, we first develop two special aggregation functions (i.e., the multiplicative aggregation function and the multiplicative modular aggregation function): Definition 5. Let A: [0, 1]n ? [0, 1] be an aggregation function, x = (x1, x2, . . . , xn) 2 [0, 1]n and y = (y1, y2, . . . , yn) 2 [0, 1]n. Then A is multiplicative, if
Aðx _ yÞ Aðx ^ yÞ ¼ AðxÞ AðyÞ
Definition 4 (27,28). Let A: [0, 1]n ? [0, 1] be an aggregation function, x = (x1, x2, . . . , xn) 2 [0, 1]n and y = (y1, y2, . . . , yn) 2 [0, 1]n. Then A is additive whenever
Aðx þ yÞ ¼ AðxÞ þ AðyÞ n
for all x, y, x + y 2 [0, 1] . A is modular whenever
ð3Þ
ð6Þ
where x _ y = (max (x1, y1), . . . , max (xn, yn)), and x ^ y = (min (x1, y1), . . . , min (xn, yn)). Evidently, due to (x _ y) (x ^ y) = x y, the multiplicative aggregation functions are also multiplicative modular. In Definition 5, the formula (5) is the well-known Cauchy’s equation, more details can be found in Ref. [31]. As we all know, the formulas (5) and (6) are the functional equations. The multiplicative and multiplicative modular aggregation functions are given by the form of implicit functions. Thus, it is not convenient to use the multiplicative and multiplicative modular aggregation functions. In order to understand the multiplicative and multiplicative modular aggregation functions sufficiently and use them efficiently, we give a proposition as below: Proposition 1. Let A: [0, 1]n ? [0, 1] be an aggregation function. Then, we have 1) A is multiplicative if and only if
AðxÞ ¼
n Y w xi i
ð7Þ
i¼1
with the corresponding weights (w1,.w2, . . . , wn) 2 [0, + 1)n and Pn i¼1 wi > 0, which is a generalization form of the weighted geometric mean; 2) A is multiplicative modular if and only if
AðxÞ ¼
where r is a permutation of (1, 2, . . . , n) such that xr(1) P xr(2) P . . . P xr(n). Recently, Mesiar and Mesiarová-Zemánková [26] replaced the additive property by the modularity in their paper so as to get the OMA operator, which is a general form of the ordered weighted average operator. The concepts of additive and modular functions can be given as follows:
ð5Þ
where x y = (x1y1, x2y2, . . . , xnyn); A is multiplicative modular, if
ð2Þ
i¼1
ð4Þ
where x _ y = (max (x1, y1), . . . , max (xn, yn)), and x ^ y = (min (x1, y1), . . . , min (xn, yn)). Similar to the concepts of additive and modular aggregation functions, we will introduce the concepts of multiplicative and multiplicative modular aggregation functions in the next section.
Aðx yÞ ¼ AðxÞ AðyÞ In what follows, we first introduce some basic concepts and terminologies related to the aggregation functions and operators:
145
n Y fi ðxi Þ
ð8Þ
i¼1
where fi: [0, 1] ? [0, 1] (i = 1, 2, . . . , n) are the non-decreasing functions satisfying fi(1) = 1, for i = 1, 2, . . . , n, and fi(0) = 0 for at least one i in {1,2, . . . , n}. Proof. 1) The proof of (7): (Sufficiency). Let the aggregation function A(x) satisfy (7), i.e., Q i AðxÞ ¼ ni¼1 xw for any x = (x1, x2, . . . , xn) 2 [0, 1]n, then A(x y) i = A(x) A(y), i.e., A(x) satisfies (5), where y = (y1, y2, . . . , yn) 2 [0, 1]n. Thus A(x) is multiplicative. (Necessity). Let A be a multiplicative aggregation function and A(x y) = A(x) A(y), where x = (x1, x2, . . . , xn) 2 [0, 1]n and y = (y1, y2, . . . , yn) 2 [0, 1]n, then we have
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L. Chen et al. / Knowledge-Based Systems 39 (2013) 144–150
Aðx1 ; x2 ; . . . ; xn Þ ¼ Aðx1 ; 1; . . . ; 1Þ Að1; x2 ; 1; . . . ; 1Þ . . . Að1; . . . ; 1; xn Þ
ð9Þ
Let F i ðxi Þ ¼ Að1; . . . ; 1; xi ; 1; . . . ; 1Þ, for any i = 1, 2, . . ., n, then |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
F i ðxi Þ F i ðyi Þ ¼ F i ðxi yi Þ
ð10Þ
a
b
In this case, there exist a, b 2 (1, 0] such that e = xi and e = yi, and we can transform (10) into
F i ðea ÞF i ðeb Þ ¼ F i ðea eb Þ ¼ F i ðeaþb Þ
ð11Þ
If there is a function G: (1, 0] ? [0, 1] such that Gi(a) = Fi(ea) (a 2 (1, 0]), then (11) can be rewritten as:
Gi ðaÞ Gi ðbÞ ¼ Gi ða þ bÞ
ð12Þ
which is equivalent to
Gi ðaÞ Gi ðbÞ Gi ðaÞ ¼ Gi ða þ bÞ Gi ðaÞ
ð13Þ
Pn where ðw1 ; w2 ; ; wn Þ 2 ½0; 1n , i¼1 wi ¼ 1, and gi: [0, 1] ? [0, 1] (i = 1, 2, . . . , n) are the non- decreasing functions satisfying gi(1) = 1 Q (i = 1, 2, . . . , n) and ni¼1 g i ð0Þ ¼ 0. Thus, the multiplicative modular aggregation function A can also be called a multiplicative modular geometric operator. Definition 6 26. Let A: [0, 1]n ? [0, 1] be an aggregation function, then the corresponding symmetrized aggregation function SA: [0, 1]n ? [0, 1] can be
SA ðxÞ ¼ Aðxrð1Þ ; . . . ; xrðnÞ Þ
where r is a permutation of (1, 2, . . . , n) such that xr(1) P xr (2) P . . . P xr(n). In particular, if A is multiplicative with the weights w = (w1, Pn w2, . . . , wn) and i¼1 wi ¼ 1, then SA (x) is an ordered weighted geometric (OWG) operator (see Definition 3), i.e.,
and then,
Gi ðbÞ Gi ð0Þ Gi ða þ bÞ Gi ðaÞ Gi ðaÞ lim ¼ lim b!0 b!0 b b
SA ðxÞ ¼ OWGw ðxÞ ¼ ð14Þ
where Gi(0) = Fi(1) = A(1, . . . ,1) = 1. In this case,
Gi ðaÞ G0i ð0Þ ¼ G0i ðaÞ
ð15Þ
G0i ðaÞ
where denotes the derivative of Gi(a). We solve the differential 0 Eq. (15) and get Gi ðaÞ ¼ eaGi ð0Þ . Since Gi(a) = Fi(ea) and ea = xi, then we have F 0 ð1Þ
F i ðxi Þ ¼ xi i
ð16Þ
Let wi ¼ F 0i ð1Þ, then
ð17Þ
i¼1
Aðx1 ; x2 ; . . . ; xn Þ ¼ Aðx1 ; x2 ; . . . ; xn Þ Að1; 1; . . . ; 1Þ ð18Þ
Aðx1 ; x2 ; . . . ; xn Þ ¼ Aðx1 ; 1; . . . ; 1Þ Að1; x2 ; 1; . . . ; 1Þ . . . ð19Þ
Similarly, suppose that fi ðxi Þ ¼ Að1; . . . ; 1; xi ; 1; . . . ; 1Þ; i ¼ 1; 2; . . . ; n, |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} then fi(xi)(i = 1, 2, . . . , n) are the non-decreasing functions satisfying Qn for i = 1, 2, . . . , n, and Therefore, fi(1) = 1, i¼1 fi ð0Þ ¼ 0. Qn AðxÞ ¼ i¼1 fi ðxi Þ. h 3.1. The formulation of the OMMG operator In addition, if wi 2 (0, 1], then g i ðxi Þ ¼ fi ðxi Þ1=wi for i 2 {1,2, . . . , n}; otherwise if wi = 0, let gi(xi) = 1. Then the corresponding multiplicative modular aggregation function A can be rewritten equivalently as:
AðxÞ ¼
n Y i¼1
g i ðxi Þwi
ð22Þ
Similarly, a new geometric operator, called the OMMG operator, can be defined as follows: Definition 7. Let A: [0, 1]n ? [0, 1] be a multiplicative modular aggregation function (or a multiplicative modular geometric Q operator), i.e., AðxÞ ¼ ni¼1 fi ðxi Þ. Then, its symmetrized form SA is called an ordered multiplicative modular geometric (OMMG) operator, i.e., n Y
fi ðxrðiÞ Þ
ð23Þ
where the permutation r satisfies xr(1) P xr(2) P . . . P xr(n). i Through observation, we find that if fi ðxrðiÞ Þ ¼ xw rðiÞ in (23), then the corresponding OMMG operator reduces to an OWG operator. Thus, the OWG operator is a special case of the OMMG operator. 3.2. Illustrative examples Below, we give two examples to illustrate the OMMG operator: Example 1. Let f1(x) = x and f2(x) = x2, then the corresponding OMMG operator is given by
OMMGðx; yÞ ¼ maxðx; yÞ ðminðx; yÞÞ2 If we reverse the order of the function above, i.e., f1(x) = x2 and f2(x) = x, then
From (18), and by induction we can obtain
Að1; . . . ; 1; xn Þ
w
i xrðiÞ
i¼1
where (w1, w2, . . . , wn) 2 [0, + 1)n. 2) The proof of (8): Q (Sufficiency). Suppose that AðxÞ ¼ ni¼1 fi ðxi Þ, where fi: [0, 1] ? [0, 1] (i = 1, 2, . . . , n) are the non-decreasing functions satisfying Q fi(1) = 1, for i = 1,2, . . . , n, and ni¼1 fi ð0Þ ¼ 0, then A(x) satisfies the Eq. (6), i.e., A is multiplicative modular. (Necessity). Assume that A is multiplicative modular, then
¼ Aðx1 ; 1; . . . ; 1Þ Að1; x2 ; . . . ; xn Þ
n Y i¼1
SA ðxÞ ¼ OMMGðxÞ ¼
Aðx1 ; x2 ; . . . ; xn Þ ¼ Aðx1 ; 1; . . . ; 1Þ Að1; x2 ; 1; . . . ; 1Þ . . . n Y w xi i Að1; . . . ; 1; xn Þ ¼
ð21Þ
ð20Þ
OMMGðx; yÞ ¼ ðmaxðx; yÞÞ2 minðx; yÞ where x, y 2 [0, 1]2. Example 2. An enterprise produces a kind of products, which is composed of four sections: c1, c2, c3, c4. The relationships of four sections are not linear, but multiplicative, i.e., if any section is faulty, the product will be rejected. For instance, the bad quality of a car’s tyre or engine leads to the bad quality of the car, although other parts are of high quality. There are four products, denoted by a set, Y = {y1, y2, y3, y4}. The satisfaction degrees of each product for each section are shown in Table 1: To obtain the ordering of products, the following steps are given: Step 1. Rearrange the satisfaction degrees in descending order for each product according to their values (see Table 2):
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L. Chen et al. / Knowledge-Based Systems 39 (2013) 144–150 Table 1 Satisfaction degree matrix.
y1 y2 y3 y4
Table 3 Transformed matrix.
c1
c2
c3
c4
0.60 0.41 0.62 0.58
0.64 0.49 0.28 0.22
0.10 0.28 0.42 0.48
0.45 0.52 0.67 0.53
y1 y2 y3 y4
0.52 0.40 0.56 0.46
0.60 0.49 0.62 0.53
0.45 0.41 0.42 0.48
0.32 0.53 0.53 0.47
OWMaxðxÞ ¼ _ni¼1 ðv i ^ xrðniþ1Þ Þ Table 2 Rearranged matrix. y1 y2 y3 y4
ð25Þ n
0.64 0.52 0.67 0.58
0.60 0.49 0.62 0.53
0.45 0.41 0.42 0.48
0.10 0.28 0.28 0.22
Step 2. Let f1(x) = (x + x2)/2, f2(x) = x, f3(x) = x, f4 (x) = x1/2, calculate the values of fi(cr(i)(yj)) for i, j 2 {1, 2, 3, 4} where ci(yj) represents the satisfaction degree of the section ci for the alternative yj and r is a permutation of (1, 2, 3, 4) such that cr(1) (yj) P cr(2)(yj) P cr(3)(yj) P cr(4)(yj), for example,
f1 ðc2 ðy1 ÞÞ ¼ f1 ð0:64Þ ¼
0:64 þ 0:642 ¼ 0:52 2
sðy2 Þ ¼ 0:0420;
Proposition 2. Let the OWMax function be given by (25), then, it satisfies (23), i.e., OWMax = SA, where A: [0, 1]n ? [0, 1] is a Q multiplicative modular aggregation function AðxÞ ¼ ni¼1 fi ðxi Þ with the condition:
fi ðxÞ ¼ min 1; max
x
v niþ1
;
v niþ2 v niþ1
sðy3 Þ ¼ 0:0771;
i ¼ 1; 2; . . . ; n;
v nþ1 ¼ 0
Case 1. Suppose that OWMax = vj for some j, then vj 6 xr(nj+1). Due to that vi and xr(i) are non-increasing, if i 6 n j + 1, then we have vni+1 6 vj 6 xr(nj+1) 6 xr(i) ) vni+1 6 xr(i), and thus
xrðiÞ v niþ2 xrðiÞ ¼ min 1; ¼1 fi ðxrðiÞ Þ ¼ min 1; max ;
v niþ1 v niþ1
sðy4 Þ ¼ 0:0547
but if i > n j + 2, then we have
Since s(y3) > s(y4) > s(y1) > s(y2), then we can obtain the priority of the products yi(i = 1, 2, . . . , 4): y3 y4 y1 y2. In Step 2, if we let f1(x) = x2, f2(x) = x, f3(x) = x, f4 (x) = x0.5, then we find that the OMMG operator reduces into a simple operator associated with an OWG operator:
OMMGða1 ; a2 ; a3 ; a4 Þ ¼ ðarð1Þ Þ2 arð2Þ arð3Þ ðarð4Þ Þ1=2 ðarð4Þ Þ1=9 Þ9=2 ð24Þ
where w = (4/9, 2/9, 2/9, 1/9) is the weight vector of the OWG operator. In this example, consider that the relationships of four sections are multiplicative, the OWA operator is not suitable for aggregating the satisfaction degrees of four sections. Additionally, through analysis, we have found that the OMMG operator is more flexible than the OWG operator. The OMMG operator can use varieties of distinct functions according to the attitude of the decision maker, while the OWG operator can only adjust the associated weights. 4. Some properties of the ordered multiplicative modular geometric operator The ordered weighted maximum (OWMax) [32] operator was introduced by Dubois [32]. It is symmetric (neutral), non-decreasing, and idempotent [26,27,33]. Definition 8 (26,27,32). Let 1 = v1 P v2 P . . . P vn P 0 be the given weights. Then, an OWMax function OWMax: [0, 1]n ? [0, 1] can be formularized as:
v niþ1
vni+2 P xr(i1) P xr(i), and thus
v niþ2 xrðiÞ ¼ min 1; fi ðxrðiÞ Þ ¼ min 1; max ;
v niþ1 v niþ1
v niþ2 ¼ v niþ1 otherwise, we have
v niþ2 v niþ1
vni+2 P xr(i), and then
v niþ2 xrðiÞ ¼ min 1; fi ðxrðiÞ Þ ¼ min 1; max ;
v niþ1 v niþ1
v niþ2 ¼ v niþ1
¼ ððarð1Þ Þ4=9 ðarð2Þ Þ2=9 ðarð3Þ Þ2=9 ¼ ðOWGw ðarð1Þ ; arð2Þ ; arð3Þ ; arð4Þ ÞÞ9=2
;
Proof. Evidently, each function fi: [0, 1] ? [0, 1] is non-decreasing.
In the same way, we can get all the rest values, listed in Table 3: Step 3. Calculate the scores s(yi) of yi by the formula (23):
sðy1 Þ ¼ 0:0448;
where x = (x1, x2, . . . , xn) 2 [0, 1] and r is a permutation of (1, 2, . . . , n) such that xr(1) P xr(2) P . . . P xr(n). In the following, we will verify that the OWMax operator is an OMMG operator:
v niþ2 v niþ1
Summarizing the above analysis, we can get
( fi ðxrðiÞ Þ ¼
1;
if i 6 n j þ 1
v niþ2 v niþ1 ;
otherwise
ð26Þ
Therefore,
OWMaxðxÞ ¼ SA ðxÞ ¼
n Y
fi ðxrðiÞ Þ ¼ v j
i¼1
Case 2. Assume that OWMax = xr(j) for some j, then xr(j) 6 vnj+1, vnj+2 6 xr(j1) and vnj+2 6 xr(j). In this case, if i < j, then we have xr(i) P xr(j1) P vnj+2 P vni+1, and thus
xrðiÞ v niþ2 xrðiÞ ¼ min 1; ¼1 fi ðxrðiÞ Þ ¼ min 1; max ;
v niþ1 v niþ1
v niþ1
but if i = j, then
xrðjÞ v niþ2 xrðiÞ ¼ min 1; fi ðxrðiÞ Þ ¼ min 1; max ;
v niþ1 v niþ1
¼
xrðjÞ
v njþ1
v njþ1
148
L. Chen et al. / Knowledge-Based Systems 39 (2013) 144–150
otherwise, we have
vni+1 P vni+2 P xr(i1) P xr(i), and then
The comonotonicity implies that there exists a permutation r such that xr(1) P xr(2) P . . . P xr(n) and yr(1) P yr(2) P . . . P yr(n). Thus we have (xr(1) _ yr(1)) P . . . P (xr(n) _ yr(n)) and (xr(1) ^ yr(1)) P (xr(2) ^ yr(2)) P . . . P (xr(n) ^ yr(n)). Therefore
v niþ2 xrðiÞ v niþ2 fi ðxrðiÞ Þ ¼ min 1; max ; ¼ min 1;
v niþ1 v niþ1
v niþ2 ¼ v niþ1
v niþ1
OMMGðx _ yÞ ¼
Based on which, we can get
fi ðxrðiÞ Þ ¼
8 1; > > < xrðjÞ v
OMMGðx ^ yÞ ¼
if i < j ; if i ¼ j
n Y i¼1 n Y
fi ðxrðiÞ _ yrðiÞ Þ fi ðxrðiÞ ^ yrðiÞ Þ
i¼1
ð27Þ
njþ1 > > : v niþ2 ; otherwise v
OMMGðxÞ ¼
niþ1
n Y fi ðxrðiÞ Þ i¼1
Therefore,
OMMGðyÞ ¼
n Y
fi ðyrðiÞ Þ
i¼1
n Y OWMaxðxÞ ¼ SA ðxÞ ¼ fi ðxrðiÞ Þ ¼ xrðjÞ
Specially, for each i = 1, 2, . . ., n, we have
i¼1
It follows from the above result that OWMax(x) = SA(x) always holds. h Example 3. Let
x ; 0 ; f 2 ðxÞ f1 ðxÞ ¼ min 1; max 0:3 x 0:3 ; f 3 ðxÞ ¼ minð1; maxðx; 0:5ÞÞ ¼ min 1; max ; 0:5 0:5
fi ðxrðiÞ _ yrðiÞ Þ fi ðxrðiÞ ^ yrðiÞ Þ ¼ fi ðxrðiÞ Þ fi ðyrðiÞ Þ (Sufficiency). Let A: [0, 1]n ? [0, 1] be a symmetric and comonotone multiplicative modular aggregation function. Because of the symmetry property, we can assume that x = (x1, x2, . . . , xn) with x1 P . . . P xn. In this case, there exist the non-decreasing functions fi: [0, 1] ? [0, 1] (i = 1, 2, . . ., n), satisfying fi(1) = 1, for i = 1, 2, . . ., n, Q Q and ni¼1 fi ð0Þ ¼ 0, such that AðxÞ ¼ ni¼1 fi ðxi Þ. Let
g i ðxÞ ¼ Að1; 1; . . . ; 1; x; . . . ; xÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} i1times
then OMMG(x, y, z) = OWMax(x, y, z), where the weights of the OWMax operator are v1 = 1, v2 = 0.5, and v3 = 0.3 respectively. Let x = 0.2, y = 0.5, and z = 0.3, then
for all x 2 [0, 1] and i = 1, 2, . . ., n. Then, according to the comonotone multiplicative modularity of A, we have
OMMGðx; y; zÞ ¼ OMMGð0:3; 0:5; 0:2Þ ¼ 1 0:6 0:5 ¼ 0:3
Að1; . . . ; 1; xn Þ Að1; . . . ; 1; xn1 ; xn1 Þ ¼ Að1; . . . ; 1; xn1 Þ Að1; . . . ; 1; xn1 ; xn Þ
and
OWMaxð0:3; 0:5; 0:2Þ ¼ maxðminðv 1 ; xrð3Þ Þ; minðv 2 ; xrð2Þ Þ; minðv 3 ; xrð1Þ ÞÞ ¼ maxð0:2; 0:3; 0:3Þ ¼ 0:3
and thus,
Að1; . . . ; 1; xn1 ; xn Þ ¼ g n ðxn Þ
In this case, OMMG(x, y, z) = OWMax(x, y, z).
g n1 ðxn1 Þ g n ðxn1 Þ
Similarly, for two comonotone vectors: Definition 9. Two vectors x, y 2 Rn (R denotes the set of all real numbers) are called comonotone, if there exists at least a mutual permutation r of (1, 2, . . . , n), such that xr(1) P xr(2) P . . . P xr(n) and yr(1) P yr(2) P . . . P yr(n). Generally speaking, this condition is frequently formularized as:
ð1; . . . ; 1; xniþ1 ; . . . ; xn Þ and
ð1; . . . ; 1; xni ; . . . ; xni Þ |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} iþ1
ðxi xj Þðyi yj Þ P 0; for i; j 2 f1; 2; . . . ; ng The comonotone additivity of the ordered weighted averaging (OWA) operator means that OWA(x + y) = OWA(x) + OWA(y), whenever x, y, x + y 2 [0, 1]n, and x and y are comonotone. Similarly, we can define an n-ary aggregation function A to be comonotone multiplicative modular, whenever it holds that A(x _ y) A(x ^ y) = A(x) A(y) for the comonotone x and y (x, y 2 [0, 1]n). For any OMMG operator, we have the following theorem:
we have
Að1; . . . ; 1; xniþ1 ; . . . ; xn Þ Að1; . . . ; 1; xni ; . . . ; xni Þ |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} iþ1
¼ Að1; . . . ; 1; xni ; . . . ; xni Þ Að1; . . . ; 1; xni ; . . . ; xn Þ |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} i
In this case, n
Theorem 1. An aggregation function A: [0, 1] ? [0, 1] is an OMMG operator if and only if it is symmetric and comonotone multiplicative modular. Proof (Necessity). Let A: [0, 1]n ? [0, 1] be an OMMG operator, then Q AðxÞ ¼ OMMGðxÞ ¼ ni¼1 fi ðxrðiÞ Þ, where xr(1) P xr(2) P . . . P xr(n) 2 n [0, 1] and fi: [0, 1] ? [0, 1] (i = 1, 2, . . ., n) are the non-decreasing Q functions satisfying fi (1) = 1, for i = 1, 2, . . ., n and ni¼1 fi ð0Þ ¼ 0. Thus, the OMMG operator is symmetric. Suppose that x and y are comonotone, then according to Definition 9, we have
OMMGðx _ yÞ OMMGðx ^ yÞ ¼ OMMGðxÞ OMMGðyÞ
Að1; . . . ; 1; xni ; . . . ; xn Þ ¼ g n ðxn Þ
g n1 ðxn1 Þ g ðxni Þ ni g n ðxn1 Þ g niþ1 ðxni Þ
If fn(x) = gn(x) and fi ðxÞ ¼ ggiþ1i ðxÞðxÞ ði ¼ 1; 2; . . . ; n 1Þ, for all x 2 [0, 1], then
Að1; . . . ; 1; xni ; . . . ; xn Þ ¼ fn ðxn Þ fn1 ðxn1 Þ . . . fni ðxni Þ and especially, we get
Aðx1 ; . . . ; xn Þ ¼ fn ðxn Þ fn1 ðxn1 Þ . . . f1 ðx1 Þ To conclude the proof, we have to show that each fi is nonQ decreasing, fi(1) = 1(i = 1, 2, . . . , n) and ni¼1 fi ð0Þ ¼ 0. Since
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L. Chen et al. / Knowledge-Based Systems 39 (2013) 144–150
Að1; 1; . . . ; 1; x; . . . ; xÞ ¼ g i ðxÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
Property 3 (Monotonicity). The OMMG operator is monotone, i.e.,
i1times
then we have gi(1) = 1, and thus fi(1) = 1. Furthermore, based on the property of the aggregation function A, we have
g ð0Þ g ð0Þ fn ð0Þ fn1 ð0Þ . . . f1 ð0Þ ¼ g n ð0Þ n1 ... 1 ¼ g 1 ð0Þ g n ð0Þ g 2 ð0Þ
ð30Þ
where xi 6 yi and xi, yi 2 [0, 1](i = 1, 2, . . . , n). Property 4. The OMMG operator reduces to the OWG operator (see P w (22)), if fi ðxi Þ ¼ xi i with wi 2 [0, 1] and ni¼1 wi ¼ 1. In this case,
¼ Að0; 0; . . . ; 0Þ ¼ 0 i.e., n Y
OMMGðx1 ; x2 ; . . . ; xn Þ 6 OMMGðy1 ; y2 ; . . . ; yn Þ
OMMGðx1 ; x2 ; . . . ; xn Þ ¼
n n Y Y xi fi ðxrðiÞ Þ ¼ xrðiÞ i¼1
fi ð0Þ ¼ g 1 ð0Þ ¼ Að0; 0; ; 0Þ ¼ 0
ð31Þ
i¼1
i¼1
Since fi(i = 1, . . . , n) is not non-decreasing, then there exist x and y satisfying x < y and fi(x) > fi(y), which means that
Property 5. The OMMG operator reduces to the maximum, if f1(x) = x and fi (x) = 1(i = 2,3, . . . , n), i.e.,
g i ðxÞ g ðyÞ > i g iþ1 ðxÞ g iþ1 ðyÞ
OMMGðx1 ; x2 ; . . . ; xn Þ ¼
n Y fi ðxrðiÞ Þ ¼ maxðxi Þ
ð32Þ
i¼1
i.e.,
Að1; 1; . . . ; 1; x; . . . ; xÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} i1times
Að1; 1; . . . ; 1; x; . . . ; xÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} itimes
Að1; 1; . . . ; 1; y; . . . ; yÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} >
Property 6. The OMMG operator reduces to the minimum, if fn(x) = x and fi (x) = 1 (i = 1, 2, . . . , n 1), i.e.,
i1times
Að1; 1; . . . ; 1; y; . . . ; yÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} itimes
OMMGðx1 ; x2 ; . . . ; xn Þ ¼
Thus,
n Y fi ðxrðiÞ Þ ¼ minðxi Þ
ð33Þ
i¼1
Að1; 1; . . . ; 1; x; . . . ; xÞ Að1; 1; . . . ; 1; y; . . . ; yÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} i1times
itimes
Property 7. If fj(x) = x and fi(x) = 1(i – j), then
> Að1; 1; . . . ; 1; x; . . . ; xÞ Að1; 1; . . . ; 1; y; . . . ; yÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} itimes
ð28Þ
i1times
OMMGðx1 ; x2 ; . . . ; xn Þ ¼
According to the comonotone multiplicative modularity:
i¼1
j ¼ 1; 2; . . . ; n
Að1; 1; . . . ; 1; x; . . . ; xÞ Að1; 1; . . . ; 1; y; . . . ; yÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} itimes
itimes
5. Concluding remarks
i1times
together with (28), we have
Að1; 1; . . . ; 1; x; . . . ; xÞ > Að1; 1; . . . ; 1; y; x; . . . ; xÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} i1times
which is in contradiction to the monotonicity of the aggregation function A. h Similar to [21,34,35], other desirable properties of the OMMG operator can be given as follows: Property 1 (Boundedness). The OMMG operator is bounded, i.e.,
0 6 OMMGðx1 ; x2 ; . . . ; xn Þ 6 1;
for ðx1 ; x2 ; . . . ; xn Þ 2 ½0; 1n
Proof. Let A be an OMMG operator, then it can be given by Q AðxÞ ¼ ni¼1 fi ðxrðiÞ Þ, where fi: [0, 1] ? [0, 1] (i = 1, 2, . . ., n) are the non-decreasing functions satisfying fi(1) = 1, for i = 1, 2, . . ., n, and Qn i¼1 fi ð0Þ ¼ 0. Thus 0 6 fi(xr(i)) 6 1, for i = 1, 2, . . ., n, and we can conclude that
06
ð34Þ
i1times
¼ Að1; 1; . . . ; 1; y; . . . ; yÞ Að1; 1; . . . ; 1; y; x; . . . ; xÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
i1times
n Y fi ðxrðiÞ Þ ¼ xj ; for any
n Y fi ðxrðiÞ Þ ¼ OMMGðx1 ; x2 ; . . . ; xn Þ 6 1
Acknowledgements The work was partly supported by the National Natural Science Foundation of China (Nos. 71071161 and 61273209). References
i¼1
Property 2 (Commutativity). Let d be any permutation of (1, 2, . . . , n), then
OMMGðxdð1Þ ; xdð2Þ ; . . . ; xdðnÞ Þ ¼ OMMGðx1 ; x2 ; . . . ; xn Þ
In this paper, we have introduced the concepts of the multiplicativity and multiplicative modularity of an aggregation function, and first given the concrete forms of the two kinds of aggregation functions. Furthermore, on the basis of the multiplicative modularity, we have developed the ordered multiplicative modular geometric (OMMG) operator by extending the ordered weighted geometric (OWG) operator. Meanwhile, how to judge whether an aggregation function is the OMMG operator has been given. In particular, we have discussed some special cases of the OMMG operator, such as the OWMax operator, the OWG operator, the Max function and the Min function, etc. At length, some good properties of the OMMG operator have been analyzed in detail. Through in-depth research of the OMMG operator in this paper, we have enriched the theory of ordered aggregations.
ð29Þ
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Contents lists available at SciVerse ScienceDirect
Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
The ordered multiplicative modular geometric operator Liuhao Chen a, Zeshui Xu b,⇑, Xiaohan Yu a a b
Institute of Communications Engineering, PLA University of Science and Technology, Nanjing, Jiangsu 210007, China Institute of Sciences, PLA University of Science and Technology, Nanjing, Jiangsu 210007, China
a r t i c l e
i n f o
Article history: Received 19 March 2012 Received in revised form 6 September 2012 Accepted 21 October 2012 Available online 10 November 2012 Keywords: Modularity Symmetric capacity Ordered multiplicative modular geometric Aggregation function Operator
a b s t r a c t The aim of this paper is to investigate an ordered multiplicative modular geometric operator and its relevant properties. The ordered multiplicative modular geometric operator is a generalized form of the ordered weighted geometric operator which has been designed incorporating the advantages of the geometric mean to deal with ratio judgments and the advantages of the ordered weighted averaging (OWA) operator to represent the concept of fuzzy majority in the process of information aggregation. Besides, the ordered multiplicative modular geometric operator can be seen as a symmetrized multiplicative modular aggregation function, characterized by comonotone multiplicative modularity. It is worth pointing that lots of the existing operators (such as the ordered weighted geometric operator, the weighted geometric operator, the ordered weighted maximum, and the Max and Min operators) can be regarded as the special cases of the ordered multiplicative modular geometric operator, which is of value in developing the theory of geometric operators. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Aggregation operators are a useful tool which can be widely utilized to multi-criteria decision making, neural networks, image process, pattern recognition, and machine learning, etc. From a general point of view, the aggregation operators are used to integrate different pieces of information so as to come to conclusions or make decisions. The aggregation operators have the common structural feature that integrates finite arguments into a single value so as to carry out the sequent operations. There are various aggregation operators with different forms. Due to respective different properties and application fields, we introduce some typical aggregation operators as follows. A useful operator, the ordered weighted averaging (OWA) operator, was introduced by Yager [1] in 1988. Since then, the OWA operator has been used in a wide range of application fields including neural networks [2,3], multi-criteria decision making [4,5], database systems [6], fuzzy logic controllers [7,8], and group decision making [9,10], etc. A fundamental aspect of the OWA operator is a reordering step in which the input arguments are rearranged according to their values. Therefore, the OWA operator is a nonlinear aggregation operator. The traditional Yager’s OWA operator focuses exclusively on the aggregation of crisp numbers. In real decision making situations, the information is not always given by the form of crisp numbers, but intuitionistic fuzzy numbers or ⇑ Corresponding author. E-mail addresses: [email protected] (L. Chen), [email protected] (Z. Xu), [email protected] (X. Yu). 0950-7051/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.knosys.2012.10.014
uncertain information. So a lot of scholars have extended the OWA operator to its generalized forms under intuitionistic fuzzy or other uncertain environments [11–19]. Similarly, the ordered weighted geometric (OWG) operator [20,21], is a useful tool for aggregating a finite collection of arguments. Since it was proposed, the OWG operator has been mentioned repeatedly in many theoretical papers and applied in lots of practical problems. Sequentially, Xu and Da [22] proposed an induced ordered weighted geometric (IOWG) operator, which is an extension of the OWG operator and the weighted geometric mean. Chiclana et al. [23] gave its use in the aggregation of multiplicative preference relations. Besides, harmonic mean [24] and Bonferroni mean [25] are the aggregation operators that can capture the interrelationships among arguments. As the natural development of the above aggregation operators, recently, Mesiar and Mesiarová-Zemánková [26] introduced the ordered modular averages (OMAs) to generalize the ordered weighted averages operators, with the replacement of the additivity in the latter by the modularity. Both the OWA and OWG operators have been widely used in the multi-criteria decision making problems. The decision makers can use each of them for aggregating information in the multi-criteria decision making problems according to their personal preferences. For some special problems, the OWG operator may be more suitable to be used than the OWA operator. For example, there are several multiplicative preference relations with the same sizes. We want to get a total preference relation through aggregating the corresponding elements of each multiplicative preference relation. The OWG operator will be suitable for aggregating decision information taking the form of multiplicative preference
L. Chen et al. / Knowledge-Based Systems 39 (2013) 144–150
relations, but the OWA operator is unsuitable in this case. Therefore, the study on the OWG operator is significant. Inspired by the idea of the OMAs, in this paper, we develop a new aggregation operator called the ordered multiplicative modular geometric (OMMG) operator, which is based on the OWG operator. Some special cases of the OMMG operator are discussed in this paper. It is worth noting that a class of aggregation operators based on Sugeno integral (such as the ordered weighted maximum (OWMax) operator [27]) is proven to be the special case of the OMMG operator. Therefore, the OMMG operator is a very general aggregation operator which generalizes lots of the existing ones. The rest of the paper is organized as follows. In Section 2, we introduce some basic concepts and terminologies. Section 3 develops the OMMG operator, and illustrates it with some examples. In Section 4, we study how to get the OMMG operator and then investigate some of its desirable properties. Section 5 ends the paper with concluding remarks. 2. Preliminaries
Aðx _ yÞ þ Aðx ^ yÞ ¼ AðxÞ þ AðyÞ
Definition 1 (Aggregation function [27–29]). An aggregation function f is a function of n(>1) arguments that maps the (n-dimensional) unit cube onto the unit interval, i.e., f: [0, 1]n ? [0, 1], satisfying the following properties: 1) f(0,0, . . . ,0) = 0 and f(1, 1, . . . , 1) = 1. 2) x 6 y implies f(x) 6 f(y), for all x, y 2 [0, 1]n. Definition 2. (Weighted geometric mean [30]). Let w = (w1, w2, . . . , w ) be a weighted vector satisfying (w1, w2, . . . , wn) 2 [0, 1]n and Pnn i¼1 wi ¼ 1, then, for any fixed vector x = (x1, x2, . . . , xn), the corresponding weighted geometric mean is defined as:
WGMw ðxÞ ¼
n Y w xi i
ð1Þ
i¼1
Based on the weighted geometric mean, Chiclana et al. [20], and Xu and Da [21] introduced the ordered weighted geometric operator, which was defined as follows: Definition 3 (Ordered weighted geometric operator [20,21]). Let w = (w1, w2, . . . , wn) be a weighting vector satisfying (w1, w2, Pn . . . , wn) 2 [0, 1]n and i¼1 wi ¼ 1, then, for any fixed vector x = (x1, x2, . . . , xn), the corresponding ordered weighted geometric operator can be given as:
OWGw ðxÞ ¼
n Y wi xrðiÞ
3. The ordered multiplicative modular geometric (OMMG) operator In order to introduce the OMMG operator, we first develop two special aggregation functions (i.e., the multiplicative aggregation function and the multiplicative modular aggregation function): Definition 5. Let A: [0, 1]n ? [0, 1] be an aggregation function, x = (x1, x2, . . . , xn) 2 [0, 1]n and y = (y1, y2, . . . , yn) 2 [0, 1]n. Then A is multiplicative, if
Aðx _ yÞ Aðx ^ yÞ ¼ AðxÞ AðyÞ
Definition 4 (27,28). Let A: [0, 1]n ? [0, 1] be an aggregation function, x = (x1, x2, . . . , xn) 2 [0, 1]n and y = (y1, y2, . . . , yn) 2 [0, 1]n. Then A is additive whenever
Aðx þ yÞ ¼ AðxÞ þ AðyÞ n
for all x, y, x + y 2 [0, 1] . A is modular whenever
ð3Þ
ð6Þ
where x _ y = (max (x1, y1), . . . , max (xn, yn)), and x ^ y = (min (x1, y1), . . . , min (xn, yn)). Evidently, due to (x _ y) (x ^ y) = x y, the multiplicative aggregation functions are also multiplicative modular. In Definition 5, the formula (5) is the well-known Cauchy’s equation, more details can be found in Ref. [31]. As we all know, the formulas (5) and (6) are the functional equations. The multiplicative and multiplicative modular aggregation functions are given by the form of implicit functions. Thus, it is not convenient to use the multiplicative and multiplicative modular aggregation functions. In order to understand the multiplicative and multiplicative modular aggregation functions sufficiently and use them efficiently, we give a proposition as below: Proposition 1. Let A: [0, 1]n ? [0, 1] be an aggregation function. Then, we have 1) A is multiplicative if and only if
AðxÞ ¼
n Y w xi i
ð7Þ
i¼1
with the corresponding weights (w1,.w2, . . . , wn) 2 [0, + 1)n and Pn i¼1 wi > 0, which is a generalization form of the weighted geometric mean; 2) A is multiplicative modular if and only if
AðxÞ ¼
where r is a permutation of (1, 2, . . . , n) such that xr(1) P xr(2) P . . . P xr(n). Recently, Mesiar and Mesiarová-Zemánková [26] replaced the additive property by the modularity in their paper so as to get the OMA operator, which is a general form of the ordered weighted average operator. The concepts of additive and modular functions can be given as follows:
ð5Þ
where x y = (x1y1, x2y2, . . . , xnyn); A is multiplicative modular, if
ð2Þ
i¼1
ð4Þ
where x _ y = (max (x1, y1), . . . , max (xn, yn)), and x ^ y = (min (x1, y1), . . . , min (xn, yn)). Similar to the concepts of additive and modular aggregation functions, we will introduce the concepts of multiplicative and multiplicative modular aggregation functions in the next section.
Aðx yÞ ¼ AðxÞ AðyÞ In what follows, we first introduce some basic concepts and terminologies related to the aggregation functions and operators:
145
n Y fi ðxi Þ
ð8Þ
i¼1
where fi: [0, 1] ? [0, 1] (i = 1, 2, . . . , n) are the non-decreasing functions satisfying fi(1) = 1, for i = 1, 2, . . . , n, and fi(0) = 0 for at least one i in {1,2, . . . , n}. Proof. 1) The proof of (7): (Sufficiency). Let the aggregation function A(x) satisfy (7), i.e., Q i AðxÞ ¼ ni¼1 xw for any x = (x1, x2, . . . , xn) 2 [0, 1]n, then A(x y) i = A(x) A(y), i.e., A(x) satisfies (5), where y = (y1, y2, . . . , yn) 2 [0, 1]n. Thus A(x) is multiplicative. (Necessity). Let A be a multiplicative aggregation function and A(x y) = A(x) A(y), where x = (x1, x2, . . . , xn) 2 [0, 1]n and y = (y1, y2, . . . , yn) 2 [0, 1]n, then we have
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L. Chen et al. / Knowledge-Based Systems 39 (2013) 144–150
Aðx1 ; x2 ; . . . ; xn Þ ¼ Aðx1 ; 1; . . . ; 1Þ Að1; x2 ; 1; . . . ; 1Þ . . . Að1; . . . ; 1; xn Þ
ð9Þ
Let F i ðxi Þ ¼ Að1; . . . ; 1; xi ; 1; . . . ; 1Þ, for any i = 1, 2, . . ., n, then |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
F i ðxi Þ F i ðyi Þ ¼ F i ðxi yi Þ
ð10Þ
a
b
In this case, there exist a, b 2 (1, 0] such that e = xi and e = yi, and we can transform (10) into
F i ðea ÞF i ðeb Þ ¼ F i ðea eb Þ ¼ F i ðeaþb Þ
ð11Þ
If there is a function G: (1, 0] ? [0, 1] such that Gi(a) = Fi(ea) (a 2 (1, 0]), then (11) can be rewritten as:
Gi ðaÞ Gi ðbÞ ¼ Gi ða þ bÞ
ð12Þ
which is equivalent to
Gi ðaÞ Gi ðbÞ Gi ðaÞ ¼ Gi ða þ bÞ Gi ðaÞ
ð13Þ
Pn where ðw1 ; w2 ; ; wn Þ 2 ½0; 1n , i¼1 wi ¼ 1, and gi: [0, 1] ? [0, 1] (i = 1, 2, . . . , n) are the non- decreasing functions satisfying gi(1) = 1 Q (i = 1, 2, . . . , n) and ni¼1 g i ð0Þ ¼ 0. Thus, the multiplicative modular aggregation function A can also be called a multiplicative modular geometric operator. Definition 6 26. Let A: [0, 1]n ? [0, 1] be an aggregation function, then the corresponding symmetrized aggregation function SA: [0, 1]n ? [0, 1] can be
SA ðxÞ ¼ Aðxrð1Þ ; . . . ; xrðnÞ Þ
where r is a permutation of (1, 2, . . . , n) such that xr(1) P xr (2) P . . . P xr(n). In particular, if A is multiplicative with the weights w = (w1, Pn w2, . . . , wn) and i¼1 wi ¼ 1, then SA (x) is an ordered weighted geometric (OWG) operator (see Definition 3), i.e.,
and then,
Gi ðbÞ Gi ð0Þ Gi ða þ bÞ Gi ðaÞ Gi ðaÞ lim ¼ lim b!0 b!0 b b
SA ðxÞ ¼ OWGw ðxÞ ¼ ð14Þ
where Gi(0) = Fi(1) = A(1, . . . ,1) = 1. In this case,
Gi ðaÞ G0i ð0Þ ¼ G0i ðaÞ
ð15Þ
G0i ðaÞ
where denotes the derivative of Gi(a). We solve the differential 0 Eq. (15) and get Gi ðaÞ ¼ eaGi ð0Þ . Since Gi(a) = Fi(ea) and ea = xi, then we have F 0 ð1Þ
F i ðxi Þ ¼ xi i
ð16Þ
Let wi ¼ F 0i ð1Þ, then
ð17Þ
i¼1
Aðx1 ; x2 ; . . . ; xn Þ ¼ Aðx1 ; x2 ; . . . ; xn Þ Að1; 1; . . . ; 1Þ ð18Þ
Aðx1 ; x2 ; . . . ; xn Þ ¼ Aðx1 ; 1; . . . ; 1Þ Að1; x2 ; 1; . . . ; 1Þ . . . ð19Þ
Similarly, suppose that fi ðxi Þ ¼ Að1; . . . ; 1; xi ; 1; . . . ; 1Þ; i ¼ 1; 2; . . . ; n, |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} then fi(xi)(i = 1, 2, . . . , n) are the non-decreasing functions satisfying Qn for i = 1, 2, . . . , n, and Therefore, fi(1) = 1, i¼1 fi ð0Þ ¼ 0. Qn AðxÞ ¼ i¼1 fi ðxi Þ. h 3.1. The formulation of the OMMG operator In addition, if wi 2 (0, 1], then g i ðxi Þ ¼ fi ðxi Þ1=wi for i 2 {1,2, . . . , n}; otherwise if wi = 0, let gi(xi) = 1. Then the corresponding multiplicative modular aggregation function A can be rewritten equivalently as:
AðxÞ ¼
n Y i¼1
g i ðxi Þwi
ð22Þ
Similarly, a new geometric operator, called the OMMG operator, can be defined as follows: Definition 7. Let A: [0, 1]n ? [0, 1] be a multiplicative modular aggregation function (or a multiplicative modular geometric Q operator), i.e., AðxÞ ¼ ni¼1 fi ðxi Þ. Then, its symmetrized form SA is called an ordered multiplicative modular geometric (OMMG) operator, i.e., n Y
fi ðxrðiÞ Þ
ð23Þ
where the permutation r satisfies xr(1) P xr(2) P . . . P xr(n). i Through observation, we find that if fi ðxrðiÞ Þ ¼ xw rðiÞ in (23), then the corresponding OMMG operator reduces to an OWG operator. Thus, the OWG operator is a special case of the OMMG operator. 3.2. Illustrative examples Below, we give two examples to illustrate the OMMG operator: Example 1. Let f1(x) = x and f2(x) = x2, then the corresponding OMMG operator is given by
OMMGðx; yÞ ¼ maxðx; yÞ ðminðx; yÞÞ2 If we reverse the order of the function above, i.e., f1(x) = x2 and f2(x) = x, then
From (18), and by induction we can obtain
Að1; . . . ; 1; xn Þ
w
i xrðiÞ
i¼1
where (w1, w2, . . . , wn) 2 [0, + 1)n. 2) The proof of (8): Q (Sufficiency). Suppose that AðxÞ ¼ ni¼1 fi ðxi Þ, where fi: [0, 1] ? [0, 1] (i = 1, 2, . . . , n) are the non-decreasing functions satisfying Q fi(1) = 1, for i = 1,2, . . . , n, and ni¼1 fi ð0Þ ¼ 0, then A(x) satisfies the Eq. (6), i.e., A is multiplicative modular. (Necessity). Assume that A is multiplicative modular, then
¼ Aðx1 ; 1; . . . ; 1Þ Að1; x2 ; . . . ; xn Þ
n Y i¼1
SA ðxÞ ¼ OMMGðxÞ ¼
Aðx1 ; x2 ; . . . ; xn Þ ¼ Aðx1 ; 1; . . . ; 1Þ Að1; x2 ; 1; . . . ; 1Þ . . . n Y w xi i Að1; . . . ; 1; xn Þ ¼
ð21Þ
ð20Þ
OMMGðx; yÞ ¼ ðmaxðx; yÞÞ2 minðx; yÞ where x, y 2 [0, 1]2. Example 2. An enterprise produces a kind of products, which is composed of four sections: c1, c2, c3, c4. The relationships of four sections are not linear, but multiplicative, i.e., if any section is faulty, the product will be rejected. For instance, the bad quality of a car’s tyre or engine leads to the bad quality of the car, although other parts are of high quality. There are four products, denoted by a set, Y = {y1, y2, y3, y4}. The satisfaction degrees of each product for each section are shown in Table 1: To obtain the ordering of products, the following steps are given: Step 1. Rearrange the satisfaction degrees in descending order for each product according to their values (see Table 2):
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L. Chen et al. / Knowledge-Based Systems 39 (2013) 144–150 Table 1 Satisfaction degree matrix.
y1 y2 y3 y4
Table 3 Transformed matrix.
c1
c2
c3
c4
0.60 0.41 0.62 0.58
0.64 0.49 0.28 0.22
0.10 0.28 0.42 0.48
0.45 0.52 0.67 0.53
y1 y2 y3 y4
0.52 0.40 0.56 0.46
0.60 0.49 0.62 0.53
0.45 0.41 0.42 0.48
0.32 0.53 0.53 0.47
OWMaxðxÞ ¼ _ni¼1 ðv i ^ xrðniþ1Þ Þ Table 2 Rearranged matrix. y1 y2 y3 y4
ð25Þ n
0.64 0.52 0.67 0.58
0.60 0.49 0.62 0.53
0.45 0.41 0.42 0.48
0.10 0.28 0.28 0.22
Step 2. Let f1(x) = (x + x2)/2, f2(x) = x, f3(x) = x, f4 (x) = x1/2, calculate the values of fi(cr(i)(yj)) for i, j 2 {1, 2, 3, 4} where ci(yj) represents the satisfaction degree of the section ci for the alternative yj and r is a permutation of (1, 2, 3, 4) such that cr(1) (yj) P cr(2)(yj) P cr(3)(yj) P cr(4)(yj), for example,
f1 ðc2 ðy1 ÞÞ ¼ f1 ð0:64Þ ¼
0:64 þ 0:642 ¼ 0:52 2
sðy2 Þ ¼ 0:0420;
Proposition 2. Let the OWMax function be given by (25), then, it satisfies (23), i.e., OWMax = SA, where A: [0, 1]n ? [0, 1] is a Q multiplicative modular aggregation function AðxÞ ¼ ni¼1 fi ðxi Þ with the condition:
fi ðxÞ ¼ min 1; max
x
v niþ1
;
v niþ2 v niþ1
sðy3 Þ ¼ 0:0771;
i ¼ 1; 2; . . . ; n;
v nþ1 ¼ 0
Case 1. Suppose that OWMax = vj for some j, then vj 6 xr(nj+1). Due to that vi and xr(i) are non-increasing, if i 6 n j + 1, then we have vni+1 6 vj 6 xr(nj+1) 6 xr(i) ) vni+1 6 xr(i), and thus
xrðiÞ v niþ2 xrðiÞ ¼ min 1; ¼1 fi ðxrðiÞ Þ ¼ min 1; max ;
v niþ1 v niþ1
sðy4 Þ ¼ 0:0547
but if i > n j + 2, then we have
Since s(y3) > s(y4) > s(y1) > s(y2), then we can obtain the priority of the products yi(i = 1, 2, . . . , 4): y3 y4 y1 y2. In Step 2, if we let f1(x) = x2, f2(x) = x, f3(x) = x, f4 (x) = x0.5, then we find that the OMMG operator reduces into a simple operator associated with an OWG operator:
OMMGða1 ; a2 ; a3 ; a4 Þ ¼ ðarð1Þ Þ2 arð2Þ arð3Þ ðarð4Þ Þ1=2 ðarð4Þ Þ1=9 Þ9=2 ð24Þ
where w = (4/9, 2/9, 2/9, 1/9) is the weight vector of the OWG operator. In this example, consider that the relationships of four sections are multiplicative, the OWA operator is not suitable for aggregating the satisfaction degrees of four sections. Additionally, through analysis, we have found that the OMMG operator is more flexible than the OWG operator. The OMMG operator can use varieties of distinct functions according to the attitude of the decision maker, while the OWG operator can only adjust the associated weights. 4. Some properties of the ordered multiplicative modular geometric operator The ordered weighted maximum (OWMax) [32] operator was introduced by Dubois [32]. It is symmetric (neutral), non-decreasing, and idempotent [26,27,33]. Definition 8 (26,27,32). Let 1 = v1 P v2 P . . . P vn P 0 be the given weights. Then, an OWMax function OWMax: [0, 1]n ? [0, 1] can be formularized as:
v niþ1
vni+2 P xr(i1) P xr(i), and thus
v niþ2 xrðiÞ ¼ min 1; fi ðxrðiÞ Þ ¼ min 1; max ;
v niþ1 v niþ1
v niþ2 ¼ v niþ1 otherwise, we have
v niþ2 v niþ1
vni+2 P xr(i), and then
v niþ2 xrðiÞ ¼ min 1; fi ðxrðiÞ Þ ¼ min 1; max ;
v niþ1 v niþ1
v niþ2 ¼ v niþ1
¼ ððarð1Þ Þ4=9 ðarð2Þ Þ2=9 ðarð3Þ Þ2=9 ¼ ðOWGw ðarð1Þ ; arð2Þ ; arð3Þ ; arð4Þ ÞÞ9=2
;
Proof. Evidently, each function fi: [0, 1] ? [0, 1] is non-decreasing.
In the same way, we can get all the rest values, listed in Table 3: Step 3. Calculate the scores s(yi) of yi by the formula (23):
sðy1 Þ ¼ 0:0448;
where x = (x1, x2, . . . , xn) 2 [0, 1] and r is a permutation of (1, 2, . . . , n) such that xr(1) P xr(2) P . . . P xr(n). In the following, we will verify that the OWMax operator is an OMMG operator:
v niþ2 v niþ1
Summarizing the above analysis, we can get
( fi ðxrðiÞ Þ ¼
1;
if i 6 n j þ 1
v niþ2 v niþ1 ;
otherwise
ð26Þ
Therefore,
OWMaxðxÞ ¼ SA ðxÞ ¼
n Y
fi ðxrðiÞ Þ ¼ v j
i¼1
Case 2. Assume that OWMax = xr(j) for some j, then xr(j) 6 vnj+1, vnj+2 6 xr(j1) and vnj+2 6 xr(j). In this case, if i < j, then we have xr(i) P xr(j1) P vnj+2 P vni+1, and thus
xrðiÞ v niþ2 xrðiÞ ¼ min 1; ¼1 fi ðxrðiÞ Þ ¼ min 1; max ;
v niþ1 v niþ1
v niþ1
but if i = j, then
xrðjÞ v niþ2 xrðiÞ ¼ min 1; fi ðxrðiÞ Þ ¼ min 1; max ;
v niþ1 v niþ1
¼
xrðjÞ
v njþ1
v njþ1
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L. Chen et al. / Knowledge-Based Systems 39 (2013) 144–150
otherwise, we have
vni+1 P vni+2 P xr(i1) P xr(i), and then
The comonotonicity implies that there exists a permutation r such that xr(1) P xr(2) P . . . P xr(n) and yr(1) P yr(2) P . . . P yr(n). Thus we have (xr(1) _ yr(1)) P . . . P (xr(n) _ yr(n)) and (xr(1) ^ yr(1)) P (xr(2) ^ yr(2)) P . . . P (xr(n) ^ yr(n)). Therefore
v niþ2 xrðiÞ v niþ2 fi ðxrðiÞ Þ ¼ min 1; max ; ¼ min 1;
v niþ1 v niþ1
v niþ2 ¼ v niþ1
v niþ1
OMMGðx _ yÞ ¼
Based on which, we can get
fi ðxrðiÞ Þ ¼
8 1; > > < xrðjÞ v
OMMGðx ^ yÞ ¼
if i < j ; if i ¼ j
n Y i¼1 n Y
fi ðxrðiÞ _ yrðiÞ Þ fi ðxrðiÞ ^ yrðiÞ Þ
i¼1
ð27Þ
njþ1 > > : v niþ2 ; otherwise v
OMMGðxÞ ¼
niþ1
n Y fi ðxrðiÞ Þ i¼1
Therefore,
OMMGðyÞ ¼
n Y
fi ðyrðiÞ Þ
i¼1
n Y OWMaxðxÞ ¼ SA ðxÞ ¼ fi ðxrðiÞ Þ ¼ xrðjÞ
Specially, for each i = 1, 2, . . ., n, we have
i¼1
It follows from the above result that OWMax(x) = SA(x) always holds. h Example 3. Let
x ; 0 ; f 2 ðxÞ f1 ðxÞ ¼ min 1; max 0:3 x 0:3 ; f 3 ðxÞ ¼ minð1; maxðx; 0:5ÞÞ ¼ min 1; max ; 0:5 0:5
fi ðxrðiÞ _ yrðiÞ Þ fi ðxrðiÞ ^ yrðiÞ Þ ¼ fi ðxrðiÞ Þ fi ðyrðiÞ Þ (Sufficiency). Let A: [0, 1]n ? [0, 1] be a symmetric and comonotone multiplicative modular aggregation function. Because of the symmetry property, we can assume that x = (x1, x2, . . . , xn) with x1 P . . . P xn. In this case, there exist the non-decreasing functions fi: [0, 1] ? [0, 1] (i = 1, 2, . . ., n), satisfying fi(1) = 1, for i = 1, 2, . . ., n, Q Q and ni¼1 fi ð0Þ ¼ 0, such that AðxÞ ¼ ni¼1 fi ðxi Þ. Let
g i ðxÞ ¼ Að1; 1; . . . ; 1; x; . . . ; xÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} i1times
then OMMG(x, y, z) = OWMax(x, y, z), where the weights of the OWMax operator are v1 = 1, v2 = 0.5, and v3 = 0.3 respectively. Let x = 0.2, y = 0.5, and z = 0.3, then
for all x 2 [0, 1] and i = 1, 2, . . ., n. Then, according to the comonotone multiplicative modularity of A, we have
OMMGðx; y; zÞ ¼ OMMGð0:3; 0:5; 0:2Þ ¼ 1 0:6 0:5 ¼ 0:3
Að1; . . . ; 1; xn Þ Að1; . . . ; 1; xn1 ; xn1 Þ ¼ Að1; . . . ; 1; xn1 Þ Að1; . . . ; 1; xn1 ; xn Þ
and
OWMaxð0:3; 0:5; 0:2Þ ¼ maxðminðv 1 ; xrð3Þ Þ; minðv 2 ; xrð2Þ Þ; minðv 3 ; xrð1Þ ÞÞ ¼ maxð0:2; 0:3; 0:3Þ ¼ 0:3
and thus,
Að1; . . . ; 1; xn1 ; xn Þ ¼ g n ðxn Þ
In this case, OMMG(x, y, z) = OWMax(x, y, z).
g n1 ðxn1 Þ g n ðxn1 Þ
Similarly, for two comonotone vectors: Definition 9. Two vectors x, y 2 Rn (R denotes the set of all real numbers) are called comonotone, if there exists at least a mutual permutation r of (1, 2, . . . , n), such that xr(1) P xr(2) P . . . P xr(n) and yr(1) P yr(2) P . . . P yr(n). Generally speaking, this condition is frequently formularized as:
ð1; . . . ; 1; xniþ1 ; . . . ; xn Þ and
ð1; . . . ; 1; xni ; . . . ; xni Þ |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} iþ1
ðxi xj Þðyi yj Þ P 0; for i; j 2 f1; 2; . . . ; ng The comonotone additivity of the ordered weighted averaging (OWA) operator means that OWA(x + y) = OWA(x) + OWA(y), whenever x, y, x + y 2 [0, 1]n, and x and y are comonotone. Similarly, we can define an n-ary aggregation function A to be comonotone multiplicative modular, whenever it holds that A(x _ y) A(x ^ y) = A(x) A(y) for the comonotone x and y (x, y 2 [0, 1]n). For any OMMG operator, we have the following theorem:
we have
Að1; . . . ; 1; xniþ1 ; . . . ; xn Þ Að1; . . . ; 1; xni ; . . . ; xni Þ |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} iþ1
¼ Að1; . . . ; 1; xni ; . . . ; xni Þ Að1; . . . ; 1; xni ; . . . ; xn Þ |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} i
In this case, n
Theorem 1. An aggregation function A: [0, 1] ? [0, 1] is an OMMG operator if and only if it is symmetric and comonotone multiplicative modular. Proof (Necessity). Let A: [0, 1]n ? [0, 1] be an OMMG operator, then Q AðxÞ ¼ OMMGðxÞ ¼ ni¼1 fi ðxrðiÞ Þ, where xr(1) P xr(2) P . . . P xr(n) 2 n [0, 1] and fi: [0, 1] ? [0, 1] (i = 1, 2, . . ., n) are the non-decreasing Q functions satisfying fi (1) = 1, for i = 1, 2, . . ., n and ni¼1 fi ð0Þ ¼ 0. Thus, the OMMG operator is symmetric. Suppose that x and y are comonotone, then according to Definition 9, we have
OMMGðx _ yÞ OMMGðx ^ yÞ ¼ OMMGðxÞ OMMGðyÞ
Að1; . . . ; 1; xni ; . . . ; xn Þ ¼ g n ðxn Þ
g n1 ðxn1 Þ g ðxni Þ ni g n ðxn1 Þ g niþ1 ðxni Þ
If fn(x) = gn(x) and fi ðxÞ ¼ ggiþ1i ðxÞðxÞ ði ¼ 1; 2; . . . ; n 1Þ, for all x 2 [0, 1], then
Að1; . . . ; 1; xni ; . . . ; xn Þ ¼ fn ðxn Þ fn1 ðxn1 Þ . . . fni ðxni Þ and especially, we get
Aðx1 ; . . . ; xn Þ ¼ fn ðxn Þ fn1 ðxn1 Þ . . . f1 ðx1 Þ To conclude the proof, we have to show that each fi is nonQ decreasing, fi(1) = 1(i = 1, 2, . . . , n) and ni¼1 fi ð0Þ ¼ 0. Since
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L. Chen et al. / Knowledge-Based Systems 39 (2013) 144–150
Að1; 1; . . . ; 1; x; . . . ; xÞ ¼ g i ðxÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
Property 3 (Monotonicity). The OMMG operator is monotone, i.e.,
i1times
then we have gi(1) = 1, and thus fi(1) = 1. Furthermore, based on the property of the aggregation function A, we have
g ð0Þ g ð0Þ fn ð0Þ fn1 ð0Þ . . . f1 ð0Þ ¼ g n ð0Þ n1 ... 1 ¼ g 1 ð0Þ g n ð0Þ g 2 ð0Þ
ð30Þ
where xi 6 yi and xi, yi 2 [0, 1](i = 1, 2, . . . , n). Property 4. The OMMG operator reduces to the OWG operator (see P w (22)), if fi ðxi Þ ¼ xi i with wi 2 [0, 1] and ni¼1 wi ¼ 1. In this case,
¼ Að0; 0; . . . ; 0Þ ¼ 0 i.e., n Y
OMMGðx1 ; x2 ; . . . ; xn Þ 6 OMMGðy1 ; y2 ; . . . ; yn Þ
OMMGðx1 ; x2 ; . . . ; xn Þ ¼
n n Y Y xi fi ðxrðiÞ Þ ¼ xrðiÞ i¼1
fi ð0Þ ¼ g 1 ð0Þ ¼ Að0; 0; ; 0Þ ¼ 0
ð31Þ
i¼1
i¼1
Since fi(i = 1, . . . , n) is not non-decreasing, then there exist x and y satisfying x < y and fi(x) > fi(y), which means that
Property 5. The OMMG operator reduces to the maximum, if f1(x) = x and fi (x) = 1(i = 2,3, . . . , n), i.e.,
g i ðxÞ g ðyÞ > i g iþ1 ðxÞ g iþ1 ðyÞ
OMMGðx1 ; x2 ; . . . ; xn Þ ¼
n Y fi ðxrðiÞ Þ ¼ maxðxi Þ
ð32Þ
i¼1
i.e.,
Að1; 1; . . . ; 1; x; . . . ; xÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} i1times
Að1; 1; . . . ; 1; x; . . . ; xÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} itimes
Að1; 1; . . . ; 1; y; . . . ; yÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} >
Property 6. The OMMG operator reduces to the minimum, if fn(x) = x and fi (x) = 1 (i = 1, 2, . . . , n 1), i.e.,
i1times
Að1; 1; . . . ; 1; y; . . . ; yÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} itimes
OMMGðx1 ; x2 ; . . . ; xn Þ ¼
Thus,
n Y fi ðxrðiÞ Þ ¼ minðxi Þ
ð33Þ
i¼1
Að1; 1; . . . ; 1; x; . . . ; xÞ Að1; 1; . . . ; 1; y; . . . ; yÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} i1times
itimes
Property 7. If fj(x) = x and fi(x) = 1(i – j), then
> Að1; 1; . . . ; 1; x; . . . ; xÞ Að1; 1; . . . ; 1; y; . . . ; yÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} itimes
ð28Þ
i1times
OMMGðx1 ; x2 ; . . . ; xn Þ ¼
According to the comonotone multiplicative modularity:
i¼1
j ¼ 1; 2; . . . ; n
Að1; 1; . . . ; 1; x; . . . ; xÞ Að1; 1; . . . ; 1; y; . . . ; yÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} itimes
itimes
5. Concluding remarks
i1times
together with (28), we have
Að1; 1; . . . ; 1; x; . . . ; xÞ > Að1; 1; . . . ; 1; y; x; . . . ; xÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} i1times
which is in contradiction to the monotonicity of the aggregation function A. h Similar to [21,34,35], other desirable properties of the OMMG operator can be given as follows: Property 1 (Boundedness). The OMMG operator is bounded, i.e.,
0 6 OMMGðx1 ; x2 ; . . . ; xn Þ 6 1;
for ðx1 ; x2 ; . . . ; xn Þ 2 ½0; 1n
Proof. Let A be an OMMG operator, then it can be given by Q AðxÞ ¼ ni¼1 fi ðxrðiÞ Þ, where fi: [0, 1] ? [0, 1] (i = 1, 2, . . ., n) are the non-decreasing functions satisfying fi(1) = 1, for i = 1, 2, . . ., n, and Qn i¼1 fi ð0Þ ¼ 0. Thus 0 6 fi(xr(i)) 6 1, for i = 1, 2, . . ., n, and we can conclude that
06
ð34Þ
i1times
¼ Að1; 1; . . . ; 1; y; . . . ; yÞ Að1; 1; . . . ; 1; y; x; . . . ; xÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
i1times
n Y fi ðxrðiÞ Þ ¼ xj ; for any
n Y fi ðxrðiÞ Þ ¼ OMMGðx1 ; x2 ; . . . ; xn Þ 6 1
Acknowledgements The work was partly supported by the National Natural Science Foundation of China (Nos. 71071161 and 61273209). References
i¼1
Property 2 (Commutativity). Let d be any permutation of (1, 2, . . . , n), then
OMMGðxdð1Þ ; xdð2Þ ; . . . ; xdðnÞ Þ ¼ OMMGðx1 ; x2 ; . . . ; xn Þ
In this paper, we have introduced the concepts of the multiplicativity and multiplicative modularity of an aggregation function, and first given the concrete forms of the two kinds of aggregation functions. Furthermore, on the basis of the multiplicative modularity, we have developed the ordered multiplicative modular geometric (OMMG) operator by extending the ordered weighted geometric (OWG) operator. Meanwhile, how to judge whether an aggregation function is the OMMG operator has been given. In particular, we have discussed some special cases of the OMMG operator, such as the OWMax operator, the OWG operator, the Max function and the Min function, etc. At length, some good properties of the OMMG operator have been analyzed in detail. Through in-depth research of the OMMG operator in this paper, we have enriched the theory of ordered aggregations.
ð29Þ
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