Yager 2004 PDF

Preview

Summary

Download Yager 2004 PDF

Description

Fuzzy Optimization and Decision Making, 3, 93–107, 2004  2004 Kluwer Academic Publishers. Printed in The Netherlands.

Generalized OWA Aggregation Operators RONALD R. YAGER Machine Intelligence Institute, Iona College, New Rochelle, NY 10801, USA

[email protected]

Abstract. We extend the ordered weighted averaging (OWA) operator to a provide a new class of operators called the generalized OWA (GOWA) operators. These operators add to the OWA operator an additional parameter controlling the power to which the argument values are raised. We look at some special cases of these operators. One important case corresponds to the generalized mean and another special case is the ordered weighted geometric operator. Keywords: aggregation, generalized mean, fuzzy sets, OWA operators

1. Introduction The ordered weighted averaging operator introduced in Yager (1988) provides a parameterized family of aggregation operators which have been used in many applications (Yager and Kacprzyk (1997)). In this work we provide a generalization of this OWA operator by combining it with the generalized mean operator (Dyckhoff and Pedrycz (1984)). This combination leads to a class of operators which we denote as the generalized ordered weighted averaging (GOWA) operators. Here we investigate some properties of these new operators.

2. GOWA Operators The OWA operator is defined by Fða1 ; . . . ; an Þ ¼

n X

wj bj

j¼1

where bj is thePjth largest of the ai and wj are a collection of weights such that wj 2 [0,1] and nj¼1 wj ¼ 1. A convenient vector expression of this aggregation operator can be obtained if we let W be an n-dimension vector whose components are the wj and let B be an n-dimension vector whose components are the bj . We call W the weighting vector and B the ordered argument vector. Using these vectors we can express Fða1 ; . . . ; an Þ ¼ WT B. By selecting different manifestations of W we can implement different aggregations. Particularly notable among the operators that can be obtained are the Max,

YAGER

94

Min and the simple average. These are respectively obtained by the vectors W where w1 ¼ 1 and wj ¼ 0 for j 6¼ 1, W where wn ¼ 1 and wj ¼ 0 for j 6¼ n, and WA where wj ¼ n1. Yager (1993) discuses various different examples of weighting vectors. It has been shown (Yager (1988)) that the OWA operator is a mean operator: it is symmetric, monotonic and bounded, Mini ½ai   Fða1 ; . . . ; an Þ  Maxi ½ai . It is also idempotent, Fða1 ; . . . ; an Þ ¼ a when ai ¼ a for all i. While the OWA operator can take its arguments values from the real line an important special case occurs when the arguments are drawn from the unit interval, I ¼ ½0; 1. In this case F : In ! I. It is this special case we shall focus on. We now introduce a class of aggregation operator which we shall call the generalized OWA operators. We shall denote these as GOWA operators. Definition A mapping M : In ! I is called a generalized ordered weighted aggregation (GOWA) operator of dimension n if

Mða1 ; . . . ; an Þ ¼

n X

wj bkj

j¼1

! 1=k

Pn wj ¼ 1; k is a where, wj are a collection of weights satisfying wj 2 ½0; 1 and j¼1 parameter such that k 2 ½1; 1; bj is the jth largest of the ai . Using vector notation we can express this as Mða1 ; . . . ; an Þ ¼ ðWT Bk Þ1=k where W and B are the vectors introduced earlier. In order to emphasize the parameters W and k at times we shall indicate this operator as MW=k ða1 ; . . . ; an Þ. Two special cases are of great significance. First is the case when k ¼ 1, here we get Mða1 ; . . . ; an Þ ¼

n X

wj bj ¼ WT B

j¼1

which is the usual OWA operator. The other important special case is when wj ¼ n1. In this case Mða1 ; . . . ; an Þ ¼

n X 1 j¼1

n

akj

!1=k

This is the generalized mean operator discussed by Dyckhoff and Pedrycz (1984). We note these are also mean operators: they are symmetric, monotonic and bounded. Before investigating more special cases we look at some properties of the GOWA operators. First we see that the GOWA operator is commutative, if P is any permutation then Mða1 ; . . . ; an Þ ¼ MðaPð1Þ ; . . . ; aPðnÞ Þ: This implies that the initial indexing of the arguments does not matter.

GOWA AGGREGATION OPERATORS

95

We easily see that it is an idempotent operator if aj ¼ a for all j then

Mða1 ; . . . ; an Þ ¼

n X

k

wj a

j¼1

! 1=k

¼ ðak Þ1=k ¼ a

Next we establish the monotonicity of these operators.

THEOREM Mða1 ; . . . ; an ) is monotonic, Mða~1 ; . . . ; a~n Þ  Mða1 ; . . . ; an Þ if a~i  ai for all i. Proof: We prove the monotonicity in two steps. e and B be the associated ordered argument vectors with component b~j and (1) Let B ~ bj respectively. Since a~i  aiit is the case 1=kthat bj  bj for all j. Pn k (2) Consider now the term f ¼ we now show that it is monotonic in j¼1 wj bj P  n k and then we take bj . First we take the natural log of f, log ½f ¼ k1 log j¼1 wj bj the derivative with respect to bj wj b1k d log f 1 kwj b1k ¼ Pn j 1k ¼ Pn j 1k  0 k j¼1 wj bj dbj j¼1 wj bj log f  0 then obofj  0. Furthermore since bj is monotonic with respect to the ai since ddb j then the result follows. The boundness of Mða1 ; . . . ; an Þ can easily be established. Since aj  Maxi ½ai  ¼ a from we get Mða1 ; . . . ; an Þ  Mða ; . . . ; a Þ. Since Mða ; . . . ; a Þ ¼ P the monotonicity 1=k n k ¼ a then Mða1 ; . . . ; an Þ  Maxj ðaj Þ. Similarly we can show that j¼1 wj a

Mða1 ; . . . ; an Þ  Mini ½ai . Thus we see that the GOWA operator is bounded. The satisfaction of these properties, commutativity, boundedness and monotonicity implies that the GOWA operators are mean operators for any choice of k and W. An additional property associated with the GOWA operators is monotonicity with respect to k, if k > k0 then

n X j¼1

wj bkj

!1=k



n X j¼1

wj bkj

0

! 1=k0

YAGER

96

The proof of this is essentially the same as the proof that the generalized mean has this property (Dyckhoff and Pedrycz (1984)). It is also the case that this operator exhibits a monotonicity with respect to the vector W. In particular, as more of the weight is allocated to components higher in W the value of the aggregation increases. Formally we express this as follows. Let k1 e are two weighing vector such and k2 be two indices such that k1 > k2 . If W and W that w~j ¼ wj

for all j 6¼ k1

and

k2 ;

w~k1 ¼ wk1  D;

w~k2 ¼ wk2 þ D

then Mw=k ~ ða1 ; . . . ; an Þ  MW=k ða1 ; . . . ; an Þ Thus in summary the GOWA operator is a mean operator that is monotonic with respect to both its parameters, W and k.

3. Cases of GOWA Operators We now look at some special cases obtained by using different choices of the parameters W and k. First we consider some cases of W. If W ¼ W where w1 ¼ 1 and wj ¼ 0 for all j 6¼ 1 then Mða1 ; . . . ; an Þ ¼

n X j¼1

wj bkj

! 1=k

¼ ðbk1 Þ1=k ¼ b1 ¼ Maxi ½ai 

Thus here with W ¼ W we always get the Max independent of the selection of k. In the case where W ¼ W where wn ¼ 1 and wj ¼ 0 for all j 6¼ n we can show that Mða1 ; . . . ; an Þ ¼ Mini ½aj  independent of the selection of k. More generally if W½k is a focused weighting vector having wk ¼ 1 and wj ¼ 0 for all j 6¼ k then for any k we get Mða1 ; . . . ; an Þ ¼ bk , the kth largest of the arguments. In this case the aggregation is effectively based on only one argument. More generally note that if wk ¼ 0 then Mða1 ; . . . ; an Þ ¼

0

11=k

B C B P C B n C wj bjkC B B C @j¼1 A j 6¼ k

Thus if the kth weight in W is zero then the kth largest

argument is disregarded in the aggregation. Here the kth largest argument plays in the ordering process but not in the actual calculation. We already noted in the special case where wj ¼ n1 for all j we get the generalized mean. Another important special case occurs when w1 ¼ a and wn ¼ 1  a. This corresponds to the Hurwicz weighting vector WH . In this case

GOWA AGGREGATION OPERATORS

97

Mða1 ; . . . ; an Þ ¼ ðabk1 þ ð1  aÞbnkÞ1=k where b1 ¼ Maxi ½ai  and bn ¼ Mini ½ai . An interesting special case is when a ¼ 1=2. Here we get Mða1 ; . . . ; an Þ ¼

  1=k 1 ðbk1 þ bnkÞ1=k 2

Let us now consider the form of the GOWA operator for some particular cases of k. As we have already noted when k ¼ 1 we get the usual OWA operator. When k ¼ 2 we get

Mða1 ; . . . ; an Þ ¼

n X

wj b2j

j¼1

! 1= 2

We get the order weighted square mean aggregation (OWMESA). Consider now the case when k ¼ 1 here we get Qn   wn 1 w1 1 j¼1 bj þþ Mða1 ; . . . ; an Þ ¼ ¼ Pn w j ¼ P n  Q bn b1 n j¼1 bj w b j¼1

j

i

i¼1 i 6¼ 1

This is closely related to the Harmonic average. If we denote Prodj ¼ Qn

Mða1 ; . . . an Þ ¼ Pn

j¼1 bj

j¼1

n Q

bi then

i¼1 i 6¼ j

wj Prodj

Since the bj are indexed in decreasing order we see that for j < i we have Prodj  Prodi . From this we see that as the weights move to the lower elements the value of the resulting aggregation increases since its denominator increases while the numerator remains the same. Consider now the case where k ! 0. In this case we get Mða1 ; . . . ; an Þ ¼

n Y

w

bj j

j¼1

We observe that this is closely related to the geometric mean. This special case with k ! 0 has been studied in Chiclana et al. (2000) and Xu and Da (2002) where it was

YAGER

98

called this the ordered weighted geometric (OWG) operator. In Herrera et al. (in press) the authors have indicated its usefulness in decision making in the case where the criteria are measured on ratio scales. We note the following property for GOWA operators with k ! 0: THEOREM If k ! 0 and if wn 6¼ 0 any aggregation Mða1 ; . . . ; an Þ in which there exists one argument with aj ¼ 0 has Mða1 ; . . . ; an Þ ¼ 0. Proof: We see this since if 9 aj ¼ 0 then bn ¼ 0 and since wn 6¼ 0 then bwnn ¼ 0 thus Qn wj b j¼1 j ¼ 0. This leads us to observe an important property of these GOWA operators. THEOREM For any GOWA operator with k < 0 and having a W for which wn 6¼ 0 any aggregation MW=k ða; . . . ; an Þ in which one argument has value, zero results in Mða1 ; . . . ; an Þ ¼ 0. Proof: We have just show that this holds for k > 0. The monotonicity of the GOWA with respect to k implies the property holds for all k < 0. We now consider the case in which k ! 1. Here then we have

MW=k ða1 ; . . . ; an Þ ¼ Lim

k!1

n X

!1=k

wj bjk

j¼1

¼

Max ½bj :

all j s:t: wj 6¼0

Thus here we get as the aggregated value the largest argument which has a non-zero weight. Since the bj are in descending then MW=k ða1 ; . . . ; an Þ ¼ b1 if w1 6¼ 0. It is interesting to note that if k ! 1 but W is such that wn ¼ 1, wj ¼ 0 for all j 6¼ n, then MW=k ða1 ; . . . ; an Þ ¼ Mini ½ai  ¼ bn : In the case where k ! 1. we get MW=k ða1 ; . . . ; an Þ ¼ Minall j s:t wj 6¼0 ½bj . Here we get the smallest argument which has a non-zero weight. In particular if wn 6¼ 0 then Mða1 ; . . . ; an Þ ¼ bn . However if W is such that w1 ¼ 1 then even though k ! 1 we get Mða1 ; . . . ; an Þ ¼ Maxi ½ai  ¼ b1 . We earlier noted the special case in which our weights are of the Hurwicz type WH (Hurwicz (1951)). In this case w1 ¼ a. and wn ¼ ð1  aÞ and the aggregated value is Mða1 ; . . . ; an Þ ¼ ðabk1 þ ð1  aÞbnkÞ1=k In this Hurwicz case when k ¼ 1 we get a form of the harmonic mean Mða1 ; . . . ; an Þ ¼

b1 bn abn þ ð1  aÞb1

GOWA AGGREGATION OPERATORS

99

If we take the derivatives with respect to b1 and bn we get b2n a oM ¼ ob1 ðabn þ ab1 Þ2

and

b21 a oM ¼ obn ðabn þ ab1 Þ2

We note that since b1 > bn there is a tendency for the smaller value, bn , to have a larger derivative and hence be more influential in the aggregation, this effect is oM ¼ oM when of course modulated by the value of a.As  2 a matter of fact ob1 obn a ¼ bbn1 . b2n a ¼ b21 ð1  aÞbn2a ¼ b2n a hence when 1a bn 1 bn ¼ b2bþb We note we additional assume a ¼ 1=2 then Mða1 ; . . . ; an Þ ¼ 1ðbb11þb . 2Þ 2

1

2

4. Characterization of GOWA Operators Yager (1988) associated with the OWA operator a measure called the attitudinal character of the aggregation. This measure of attitudinal character is a number in the unit interval indicating the ‘‘Andness/Orness’’ or equivalently the Miness/Maxness of the aggregation. Essentially it provides a scalar valued characterization of the type of aggregation being performed. For the ordinary OWA operator the attitudinal character, which is just dependent on W, was defined as

A  CðWÞ ¼

n X j¼1

wj

nj n1

It can be shown that when W ¼ W , w1 ¼ 1 and wj ¼ 0 for all j 6¼ 1, A–CðW Þ ¼ 1. For W ¼ W , wn ¼ 1 and wj ¼ 0 for j 6¼ n, A–CðW Þ ¼ 0. For W ¼ Wn , where wj ¼ 1=n for all j, A–CðWN Þ ¼ 0:5. For the Hurwicz type weighting vector, WH , where w1 ¼ a and wn ¼ ð1  aÞ and all other wj ¼ 0 we get A– CðWH Þ ¼ a. We also note in the case of the weighting vector W½k where wk ¼ 1 we nk. get A–CðW½k Þ ¼ n1 We see the attitudinal character provides an indication of the type of the aggregation being performed: A–C!1 indicates Max type aggregation, A–C!0 indicates Min type aggregation and A–C!0.5 indicates an aggregation that is neutral with respect to this dimension. An important class of weighting vectors are symmetric ones. We say W is symmetric if wj ¼ wnjþ1 . It can be shown that if W is symmetric then A–CðWÞ ¼ 0:5. We note examples of symmetric vectors are WA , the median type vector, W½K with k ¼ 2n, as well as WH when a ¼ 0:5. e be two Another notable situation is that of dual weighting vectors. Let W and W e are duals. We weighting vectors such that w~j ¼ wnjþ1 then we say that W and W

YAGER

100

note that W and W are dual. We also note that a symmetric weighting vector is self ~ e ¼ W. Let us look at the relationship between A–CðWÞ and A–Cð WÞ. dual, W ACðWÞ ¼

e ¼ ACð WÞ

n 1 X wj ðn  jÞ n  1 j¼1

n n 1 X 1 X w~j ðn  jÞ ¼ w ðn  jÞ n  1 j¼1 n  1 j¼1 njþ1

Let i ¼ n  j þ 1, then j ¼ n  i þ 1 using this we get e ¼ ACð WÞ e ¼ ACð WÞ

i n 1 X 1 X wi ðn  ðn  i þ 1ÞÞ ¼ wi ði  1Þ n  1 i¼n n  1 i¼1 n n 1 X 1 X ðn  1Þ wi ði  nÞ þ n  1 i¼1 n  1 i¼1

e Þ ¼ 1  ACðWÞ ACð W

Thus the attitudinal character of dual weighting vectors are complements of each of other. Yager (1988) suggested an interpretation of the attitudinal character of the aggregation that allows us to extend it to GOWA operators having parameters W and k. In Yager (1988) it was noted this attitudinal character is the OWA aggre n1 ; n2 ; . . . ; 1 ; 0 T gation of the argument n1 n1 n1 n1 , A–CðWÞ ¼ W B where B has comnj ponents bj ¼ n1 . Using this we can define the attitudinal character of the GOWA operator

ACðW=kÞ ¼

n X j¼1



nj wj n1

 k! 1=k ¼ ðWT Bk Þ1=k

P  1=k k n 1 We can also express this as A–CðW=kÞ ¼ n1 . It is the GOWA j¼1 wj ðn  jÞ  n1 n2  1 0 aggregation of the linear argument n1 ; n1 ; . . . ; n1 ; n1 : Let us now obtain the attitudinal character for some examples of the GOWA operators. A–CðW=kÞ ¼ Qn  nj First consider the class where k ! 0 here wenget n j¼1 n1 wj . If we additional assume that wn 6¼ 0, since bn ¼ n1 ¼ 0, we get A– CðW=kÞ ¼ 0. As we have previously indicated the GOWA is monotonic with respect

GOWA AGGREGATION OPERATORS

101

to k, and since A–CðW=kÞ is a GOWA aggregation we have that if k~ < k then A– CðW=~kÞ A–CðW=kÞ. This allows us to make the following observation. Observation: If W is such that wn 6¼ 0 then A–CðW=kÞ ¼ 0 for all k < 0. The implication of this is that GOWA operators with k < 0 and having W such that wn 6¼ 0 tend to act like a ‘‘Min/And’’ type aggregations. The smallest valued arguments in the aggregation play an important role. Let us now look at the attitudinal character for the family of operators where W ¼ WA , wj ¼ 1n for all j. In this case we get ACðWA =kÞ ¼

! 1=k  k !1=k  1=k X n n X 1 nj 1 1 k ðn  jÞ ¼ n n1 n1 n j¼1 j¼1

Since wn ¼ n1 from the preceding observation we have that A–CðWA =kÞ ¼ 0 for k  0. We now investigate what happens to A–CðWA =kÞ when k >0 (see Figures1 and 2). 1=k  Pn k 1 1 1=k In Figure 1, we have plotted A–CðWA =kÞ ¼ n1 as a j¼1 ðn  jÞ n

function of n for k ¼ 20, 15, 10, 4, 2. We observe that A–CðWA =kÞ leads to be higher for smaller n, although not significantly, and it asymptotically approaches some limit which depends on k. The bigger k the closer the limit is to one.

In Figure 2, we have Pplotted these 1=kasymptotic limits for n ¼ 100. Here we plotted k 100 1 1 A–CðWA =kÞ ¼ 99 100 for k=1–30. We observe that A–CðWA =kÞ j¼1 ðn  jÞ

increases as k increases going from A–CðWA =kÞ ¼ 0:5 for k ¼ 1 to A–C ðWA =kÞ ¼ 0:9 for k ¼ 30. We now consider the case where W ¼ W½k , here wk ¼ 1. Let us see the effect of k. Here ! 1=k n X nk 1 1 k ½k ððn  kÞk Þk ¼ ACðW =kÞ ¼ ¼ wj ðn  jÞ n1 n  1 j¼1 n1

Figure 1.

102

YAGER

Figure 2.

What is interesting is that A–CðW½k =kÞ is the same for all k, it just depends on k. We now turn to symmetric weighting vectors. We previously noted that for a symmetric weighting vector W we get A–CðWÞ ¼ 0:5. This is not necessarily the case when k 6¼ 1. In the following we consider the special case of symmetric vector where wj ¼ wjnþ1 ¼ 0:5 for some j. For the case where n ¼ 100 and j ¼ 2 in Figure 3 we plot A–CðW=kÞ for k ¼ 20 to 20. We see that as k gets smaller we have A– CðW=kÞ ! 0, we get a kind of Min aggregation. On the other hand when k gets larger we have A–CðW=kÞ ! 1 giving us a more Max like aggregation. In Figure 4, we consider the situation of different values of j in the above symmetric weighting vector. We plot A–CðW=kÞ for k ¼ 30 to 30 and for j ¼ 2, 20, 30 and 49 (in increasing thickness). We see that as the two symmetric weights move closer to the center, j getting larger, this operator acts more like A–CðW=kÞ ¼ 0:5.

Figure 3.

GOWA AGGREGATION OPERATORS

103

Figure 4.

We now look at the Hurwicz weighting vector WH where w1 ¼ a and wn ¼ ð1  aÞ. For this we get ACðWH =kÞ ¼ ðab1k þ ð1  aÞbkn Þ1=k where b1 ¼ 1 and bn ¼ 0. We see that for k < 0 this has value zero. For k > 0 we have ACðWH =kÞ ¼ ðabnkÞ1=k ¼ a1=k Consider now the situation for 0 < k < 1. We see that for k ¼ 1 we get ACðWH =kÞ ¼ a. Increasing k, letting it go to 1, leads us to obtaining ACðWH =kÞ ! 1. On the other hand decreasing k, letting go to zero results in having ACðWH =kÞ ! 0. This situation very clearly displays the effect of k. 5. Functional Defined GOWA Operators Yager (1996) discusses various different methods for obtaining the OWA weighing vectors. One important method for generating the weights is via a function f : ½0; 1 ! ½0; 1 for which fð0Þ ¼ 0, fð1Þ ¼ 1 and fðxÞ ‡ fðyÞ if x > y. These functions are called basic unit interval monotonic(BUM) funct...