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```An analytic approach for obtaining maximal
entropy OWA operator weights ∗
Robert Fullér
rfuller@abo.fi
Péter Majlender
peter.majlender@abo.fi
Abstract
One important issue in the theory of Ordered Weighted Averaging (OWA)
operators is the determination of the associated weights. One of the first approaches, suggested by O’Hagan, determines a special class of OWA operators having maximal entropy of the OWA weights for a given level of orness;
algorithmically it is based on the solution of a constrained optimization problem. In this paper, using the method of Lagrange multipliers, we shall solve
this constrained optimization problem analytically and derive a polinomial
equation which is then solved to determine the optimal weighting vector.
Keywords: OWA operator, dispersion, degree of orness
1
Introduction
An OWA operator of dimension n is a mapping F : Rn → R that has an associated
weighting vector W = (w1 , . . . , wn )T of having the properties
w1 + · · · + wn = 1, 0 ≤ wi ≤ 1, i = 1, . . . , n,
and such that
F (a1 , . . . , an ) =
n
X
wi bi ,
i=1
where bj is the jth largest element of the collection of the aggregated objects
{a1 , . . . , an }.
∗
The final version of this paper appeared in: Fuzzy Sets and Systems, 124(2001) 53-57.
1
In [3], Yager introduced two characterizing measures associated with the weighting vector W of an OWA operator. The first one, the measure of orness of the
aggregation, is defined as
n
1 X
orness(W ) =
(n − i)wi .
n−1
i=1
and it characterizes the degree to which the aggregation is like an or operation. It
is clear that orness(W ) ∈ [0, 1] holds for any weighting vector.
The second one, the measure of dispersion of the aggregation, is defined as
disp(W ) = −
n
X
wi ln wi
i=1
and it measures the degree to which W takes into account all information in the
aggregation.
It is clear that the actual type of aggregation performed by an OWA operator
depends upon the form of the weighting vector. A number of approaches have been
suggested for obtaining the associated weights, i.e., quantifier guided aggregation
[3, 4], exponential smoothing [6] and learning [5].
Another approach, suggested by O’Hagan [2], determines a special class of
OWA operators having maximal entropy of the OWA weights for a given level
of orness. This approach is based on the solution of he following mathematical
programming problem:
n
X
−
wi ln wi
maximize
i=1
subject to
1
n−1
n
X
n
X
(n − i)wi = α, 0 ≤ α ≤ 1
(1)
i=1
wi = 1, 0 ≤ wi ≤ 1, i = 1, . . . , n.
i=1
Using the method of Lagrange multipliers we shall transfer problem (1) to
a polinomial equation which is then solved to determine the optimal weighting
vector.
2
2
Obtaining maximal entropy weights
First we note that disp(W ) is meaningful if wi > 0 and by letting wi ln wi to zero
if wi = 0, problem (1) turns into
disp(W ) → max; subject to {orness(W ) = α, w1 + · · · + wn = 1, 0 ≤ α ≤ 1}.
If n = 2 then from orness(w1 , w2 ) = α we get w1 = α and w2 = 1 − α. Furthemore, if α = 0 or α = 1 then the associated weighting vectors are uniquely
defined as (0, 0, . . . , 0, 1)T and (1, 0, . . . , 0, 0)T , respectively, with value of dispersion zero.
Suppose now that n ≥ 3 and 0 < α < 1. Let us
L(W, λ1 , λ2 ) = −
n
X
wi ln wi + λ1
i=1
X
n
i=1
X
n
n−i
wi − α + λ2
wi − 1 .
n−1
i=1
denote the Lagrange function of constrained optimization problem (1), where λ1
and λ2 are real numbers. Then the partial derivatives of L are computed as
∂L
n−j
= − ln wj − 1 + λ1 +
λ2 = 0, ∀j
∂wj
n−1
n
X
∂L
=
wi − 1 = 0
∂λ1
(2)
i=1
n
X
∂L
n−i
=
wi − α = 0.
∂λ2
n−1
i=1
For j = n equation (2) turns into
− ln wn − 1 + λ1 = 0 ⇐⇒ λ1 = ln wn + 1,
and for j = 1 we get
− ln w1 − 1 + λ1 + λ2 = 0,
and, therefore,
λ2 = ln w1 + 1 − λ1 = ln w1 + 1 − ln wn − 1 = ln w1 − ln wn .
For 1 ≤ j ≤ n we find
j−1
n−j
ln wj =
ln wn +
ln w1 ⇒ wj =
n−1
n−1
3
q
w1n−j wnj−1 .
n−1
(3)
If w1 = wn then (3) gives
w1 = w2 = · · · = wn =
1
⇒ disp(W ) = ln n,
n
which is the optimal solution to (1) for α = 0.5 (actually, this is the global optimal
value for the dispersion of all OWA operators of dimension n). Suppose now that
w1 6= wn . Let us introduce the notations
1
1
u1 = w1n−1 , un = wnn−1 .
Then we may rewrite (3) as wj = u1n−j uj−1
n , for 1 ≤ j ≤ n.
From the first condition, orness(W ) = α, we find
n
n
X
X
n−i
wi = α ⇐⇒
(n − i)u1n−i uni−1 = (n − 1)α,
n−1
i=1
i=1
and from
n−1
X
1
n
i n−i
(n − 1)u1 −
u1 un
u1 − un
i=1
1
− unn−1
un−1
1
n
=
(n − 1)u1 − u1 un
u1 − un
u1 − un
1
n
n
n
(n − 1)u1 (u1 − un ) − u1 un + u1 un
=
(u1 − un )2
1
n+1
n
n
=
(n − 1)u1 − nu1 un + u1 un ,
(u1 − un )2
n
X
i−1
=
(n − i)un−i
1 un
i=1
we get
(n − 1)un+1
− nun1 un + u1 unn = (n − 1)α(u1 − un )2
1
nun1 − u1 = (n − 1)α(u1 − un )
1
n
un =
((n − 1)α + 1)u1 − nu1
(n − 1)α
un (n − 1)α + 1 − nw1
=
.
(4)
u1
(n − 1)α
4
From the second condition, w1 + · · · + wn = 1, we get
n
X
j−1
un−j
= 1 ⇐⇒
1 un
j=1
un1 − unn
=1
u1 − un
⇐⇒ un1 − unn = u1 − un
⇐⇒ u1n−1 −
un
un
× un−1
=1−
n
u1
u1
(5)
(6)
Comparing equations (4) and (6) we find
w1 −
(n − 1)α + 1 − nw1
nw1 − 1
× wn =
(n − 1)α
(n − 1)α
and, therefore,
wn =
((n − 1)α − n)w1 + 1
.
(n − 1)α + 1 − nw1
(7)
Let us rewrite equation (5) as
un1 − unn = u1 − un
u1 (w1 − 1) = un (wn − 1)
w1 (w1 − 1)n−1 = wn (wn − 1)n−1
n−1
w1 (w1 − 1)
(n − 1)α(w1 − 1 n−1
((n − 1)α − n)w1 + 1
=
×
(n − 1)α + 1 − nw1
(n − 1)α + 1 − nw1
w1 [(n − 1)α + 1 − nw1 ]n = ((n − 1)α)n−1 [((n − 1)α − n)w1 + 1].
(8)
So the optimal value of w1 should satisfy equation (8). Once w1 is computed
then wn can be determined from equation (7) and the other weights are obtained
from equation (3).
Remark 2.1 If n = 3 then from (3) we get
w2 =
√
w1 w3
independently of the value of α, which means that the optimal value of w2 is always
the geometric mean of w1 and w3 .
5
3
Computing the optimal weights
Let us introduce the notations
f (w1 ) = w1 [(n − 1)α + 1 − nw1 ]n ,
g(w1 ) = ((n − 1)α)n−1 [((n − 1)α − n)w1 + 1].
Then to find the optimal value for the first weight we have to solve the following
equation
f (w1 ) = g(w1 ),
where g is a line and f is a polinom of w1 of dimension n + 1.
Without loss of generality we can assume that α < 0.5, because if a weighting
vector W is optimal for problem (1) under some given degree of orness, α < 0.5,
then its reverse, denoted by W R , and defined as
wiR = wn−i+1
is also optimal for problem (1) under degree of orness (1 − α). Really, as was
shown by Yager [4], we find that
disp(W R ) = disp(W ) and orness(W R ) = 1 − orness(W ).
Therefore, for any α > 0.5, we can solve problem (1) by solving it with level of
orness (1 − α) and then taking the reverse of that solution.
From the equations
1
1
0 1
0 1
f
=g
and f
=g
n
n
n
n
we get that g is always a tangency line to f at the point w1 = 1/n. But if w1 = 1/n
then w1 = · · · = wn = 1/n also holds, and that is the optimal solution for α = 0.5.
Consider the the graph of f . It is clear that f (0) = 0 and by solving the
equation
f 0 (w1 ) = [(n − 1)α + 1 − nw1 ]n − n2 w1 [(n − 1)α + 1 − nw1 ]n−1 = 0
we find that its unique solution is
ŵ1 =
(n − 1)α + 1 1
< ,
n(n + 1)
n
and its second derivative, f 00 (ŵ1 ) is negative, which means that ŵ1 is the only
maximizing point of f on the segment [0, 1/n].
6
Figure 1: Graph of f withr n = 4 and α = 0.2.
We prove now that g can intersect f only once in the open interval (0, 1/n). It
will guarantee the uniqueness of the optimal solution of problem (1). Really, from
the equation
f 00 (w1 ) = −2n2 [(n−1)α+1−nw1 ]n−1 +n3 (n−1)w1 [(n−1)α+1−nw1 ]n−2 = 0
we find that its unique solution is
w̄1 = 2
(n − 1)α + 1
1
= 2ŵ1 < , (since α < 0.5).
n(n + 1)
n
with the meaning that f is strictly concave on (0, w̄1 ), has an inflextion point at
w̄1 , and f is strictly convex on (w̄1 , 1/n). Therefore, the graph of g should lie
below the graph of g if ŵ1 < w1 < 1/n and g can cross f only once in the interval
(0, ŵ1 ).
4
Illustrations
Let us suppose that n = 5 and α = 0.6. Then from the equation
w1 [4 × 0.6 + 1 − 5w1 ]5 = (4 × 0.6)4 [1 − (5 − 4 × 0.6)w1 ].
we find
w1∗ = 0.2884
(4 × 0.6) − 5)w1∗ + 1
= 0.1278
4 × 0.6 + 1 − 5w1∗
q
w2∗ = 4 (w1∗ )3 w5∗ = 0.2353,
q
w3∗ = 4 (w1∗ )2 (w5∗ )2 = 0.1920,
q
w4∗ = 4 (w1∗ )(w5∗ )3 = 0.1566.
w5∗ =
and, disp(W ∗ ) = 1.5692.
Using exponential smoothing [1], Filev and Yager [6] obtained the following
weighting vector
W 0 = (0.41, 0.10, 0.13, 0.16, 0.20),
7
with disp(W 0 ) = 1.48 and orness(W 0 ) = 0.5904.
We first note that the weights computed from the constrained optimization
problem have better dispersion than those ones obtained by Filev and Yager in
[6], however the (heuristic) technology suggested in [6] needs less computational
efforts.
Other interesting property here is that small changes in the required level of
orness, α, can cause a big variation in weighting vectors of near optimal dispersity,
(for example, compare the weighting vectors W ∗ and W 0 ).
References
[1] R.G. Brown, Smoothing, Forecasting and Prediction of Discrete Time Series
(Prentice-Hall, Englewood Cliffs, 1963).
[2] M. O’Hagan, Aggregating template or rule antecedents in real-time expert
systems with fuzzy set logic, in: Proc. 22nd Annual IEEE Asilomar Conf.
Signals, Systems, Computers, Pacific Grove, CA, 1988 81-689.
[3] R.R.Yager, Ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. on Systems, Man and Cybernetics,
18(1988) 183-190.
[4] R.R.Yager, Families of OWA operators, Fuzzy Sets and Systems, 59(1993)
125-148.
[5] R. R. Yager and D. Filev, Induced ordered weighted averaging operators,
IEEE Trans. on Systems, Man and Cybernetics – Part B: Cybernetics,
29(1999) 141-150.
[6] D. Filev and R. R. Yager, On the issue of obtaining OWA operator weights,
Fuzzy Sets and Systems, 94(1998) 157-169
8
5
Citations
[A13] Robert Fullér and Péter Majlender, An analytic approach for obtaining maximal entropy OWA operator weights, FUZZY SETS AND SYSTEMS, 124(2001) 53-57. [Zbl.0989.03057]
in journals
A13-c134 J.-W. Liu; C.-H. Cheng; Y.H. Chen; S.-F. Huang, OWA based
PCA information fusion method for classification problem, INTERNATIONAL JOURNAL OF INFORMATION AND MANAGEMENT
SCIENCES, 21(2010), Issue 2, pp. 209-225. 2010
A13-c133 Chang, Liang; Shi, Zhong-Zhi; Chen, Li-Min; Niu, Wen-Jia,
Family of extended dynamic description logics, Ruan Jian Xue Bao
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A13-c130 Victor M Vergara, Shan Xia, Minimization of uncertainty for ordered weighted average, INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS 25(2010), pp. 581-595. 2010
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There are several methods that find the OWA weights. [3,912] Yager [3] proposed finding weights that maximizes their
entropy (also known as weights dispersion). Later, Fuller and
Majlender [A13] found an analytical solution to Yager’s theory based on Lagrange multipliers. In addition, these two authors proposed minimizing the variance of weights [A10] instead of maximizing their entropy. The goal of these methods
is to maximize similarity. Maximizing the entropy or minimizing the variance of the weights {w1 , w2 , . . . , wn } makes
each weight wi more similar in magnitude tot he others. (page
582)
9
A13-c129 Yao-Hsien Chen, Ching-Hsue Cheng, Jing-Wei Liu, Intelligent
preference selection model based on NRE for evaluating student learning achievement, COMPUTERS & EDUCATION, 54(2010), pp. 916926. 2010
http://dx.doi.org/10.1016/j.compedu.2009.09.020
This step computes the aggregate values of the test-score examples and typical score examples, for different orders of
evaluation items by using the OWA operator. First, according
to the Fullér and Majlenders’ equation introduced in Section
3, we can obtain a setP
of OWA weights Wα {w1 , w2 , . . . , wn },
where 0 ≤ w1 ≤ 1, ni=1 wi = 1 and α ∈ [0, 1]. Second, to
compute the aggregate values, we multiply the values of the
evaluation items, which are permuted by all possible orders,
by the corresponding OWA weights, and then sum up these
multiplication values. (page 920)
A13-c128 A. Emrouznejad, G.R. Amin, Improving minimax disparity model
to determine the OWA operator weights, INFORMATION SCIENCES,
180(2010) 1477-1485. 2010
http://dx.doi.org/10.1016/j.ins.2009.11.043
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pp. 2362-2370. 2010
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the electronic industry, EXPERT SYSTEMS WITH APPLICATIONS,
37(2010), pp. 610-617. 2010
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Fullér and Majlender (2001) proposed a new method by using
Lagrange multipliers to improve the problem in Eq. (5) and
got the following: (page 611)
10
A13-c124 Pan Yuhou; Yao Shuang; Guo Yajun, Comprehensive Multiindex Group Evaluation Method for Scheme of Oilfield Development
Adjustment and Its Application, TECHNOLOGY ECONOMICS, 29(2010),
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averaging method, COMPUTERS & INDUSTRIAL ENGINEERING,
57(2009), Issue 4, pp. 1274-1281. 2009
http://dx.doi.org/10.1016/j.cie.2009.06.007
Additionally, Fuller and Majlender (2001) used Lagrange multipliers on Yager’s OWA equation to derive a polynomial equation, which determines the optimal weighting vector under
maximal entropy (ME-OWA operator). The proposed approach
thus determines the optimal weighting vector under maximal
entropy, and the OWA operator ascertains the optimal reliability allocation rating after an aggregation process. This
method is both a simple and effective approach that can efficiently resolve the shortcomings of the FOO technique and
average weighting allocation. (page 1275)
A13-c122 S. Zadrozny, J. Kacprzyk, Issues in the practical use of the OWA
operators in fuzzy querying, JOURNAL OF INTELLIGENT INFORMATION SYSTEMS, 33(2009), No. 3, pp. 307-325. 2009
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for Modeling Rating Functions, INTERNATIONAL JOURNAL OF
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11
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2009
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2009
http://dx.doi.org/10.1002/int.20384
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http://dx.doi.org/10.1016/j.ins.2009.07.015
A13-c115 Badredine Arfi, Probing the Democratic Peace Argument Using
Linguistic Fuzzy Logic, INTERNATIONAL INTERACTIONS: EMPIRICAL AND THEORETICAL RESEARCH IN INTERNATIONAL
RELATIONS, 35(2009), pp. 30-57. 2009
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SYSTEMS WITH APPLICATIONS, 36(2009), pp. 4988-4995. 2009
http://dx.doi.org/10.1016/j.eswa.2008.06.013
2.1.2. Fullér and Majlender’s OWA
Fullér and Majlender (2001) transform Yager’s OWA equation to a polynomial equation by using Lagrange multipliers.
12
According to their approach, the associated weighting vector
can be obtained by (5)-(7).
A13-c110 YAO Shuang; GUO Ya-jun; YI Ping-tao, Multi-variable Induced
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Maps Based on Ordered Weighted Averaging Operators COMPUTER
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13
ideal and anti-ideal points for location site selection, EXPERT SYSTEMS WITH APPLICATIONS, 35 (2008) pp. 2041-2048. 2008
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In this paper we use a special class of OWA operators which
have maximum entropy for a given level of orness (O’Hagan,
1988). The weighing vector of an OWA operator with maximum entropy is calculated applying the results of Fullér and
Majlender (2001). (page 2044)
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KNOWLEDGE-BASED SYSTEMS, 21(2008), Issue 8, pp. 847-851.
2008
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However to apply the OWA operator for decision making, a
very crucial issue is to determine its weights. O’Hagan [16]
suggested a maximum entropy method as the rst approach to
determine OWA operator weights in which he formulated the
OWA operator weight problem to a constrained nonlinear optimization model with a predened degree of orness. Fullér and
Majlender [A13] transformed the maximum entropy method
into a polynomial equation that can be solved analytically.
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entropy in decision analysis, CONTROL AND DECISION, Volume
23, Issue 4, April 2008, Pages 361-366. 2008
A13-c99 Xinwang Liu, A general model of parameterized OWA aggregation with given orness level INTERNATIONAL JOURNAL OF APPROXIMATE REASONING, vol. 48, pp. 598-627. 2008
http://dx.doi.org/10.1016/j.ijar.2007.11.003
Filev and Yager [12] further proposed a method to generate
MEOWA weight vector by an immediate parameter. Fullér
and Majlender [A13] transformed the maximum entropy model
into a polynomial equation, which can be solved analytically.
(page 599)
14
A13-c98 Xinwang Liu and Da Qingli, On the properties of regular increasing monotone (RIM) quantifiers with maximum entropy, INTERNATIONAL JOURNAL OF GENERAL SYSTEMS, Volume 37, Issue 2,
pp. 167-179. 2008
http://dx.doi.org/10.1080/03081070701192675
A13-c97 Byeong Seok Ahn, Some Quantier Functions From Weighting
Functions With Constant Value of Orness, IEEE TRANSACTIONS
ON SYSTEMS, MAN, AND CYBERNETICSPART B: CYBERNETICS, 38: (2) 540-546. 2008
http://dx.doi.org/10.1109/TSMCB.2007.912743
A13-c96 Xinwang Liu and Shilian Han, Orness and parameterized RIM
quantifier aggregation with OWA operators: A summary, INTERNATIONAL JOURNAL OF APPROXIMATE REASONING, Volume 48,
Issue 1, Pages 77-97. 2008
http://dx.doi.org/10.1016/j.ijar.2007.05.006
A13-c95 B.S. Ahn, Preference relation approach for obtaining OWA operators weights, INTERNATIONAL JOURNAL OF APPROXIMATE
REASONING, 47 (2), pp. 166-178. 2008
http://dx.doi.org/10.1016/j.ijar.2007.04.001
The resulting weights are called maximum entropy OWA (MEOWA) weights for a given degree of orness and analytic forms
and property for these weights are further investigated by several researchers [25,A13]. (page 167)
A13-c94 B.S. Ahn, H. Park, Least-squared ordered weighted averaging
operator weights, INTERNATIONAL JOURNAL OF INTELLIGENT
SYSTEMS, 23(1), pp. 33-49. 2008
http://dx.doi.org/10.1002/int.20257
A13-c93 D.H. Hong and K.T. Kim, A note on the maximum entropy weighting function problem, JOURNAL OF APPLIED MATHEMATICS AND
COMPUTING, 23(2007), No. 1-2, pp. 547-552. 2007
http://www.mathnet.or.kr/mathnet/thesis_file/DHHong0613F.pdf
A13-c90 Zhenbang Lv and Lihua Zhou, Advanced Fuzzy Cognitive Maps
Based on OWA Aggregation, INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION, 5(2007) pp. 31-34. 2007
http://www.yangsky.com/ijcc/pdf/ijcc524.pdf
15
A13-c89 Xinwang Liu, Hongwei Lou, On the equivalence of some approaches to the OWA operator and RIM quantier determination, FUZZY
SETS AND SYSTEMS, vol. 159, pp. 1673-1688. 2007
http://dx.doi.org/10.1016/j.fss.2007.12.024
Filev and Yager [2] further analyzed the properties of MEOWA operators, and proposed a method for obtaining weights
as a function of one parameter. Recently, Fullér and Majlender [A13] proposed an analytical solution by transforming the
problem into a polynomial equation. (page 1677)
A13-c88 X. Liu, Some OWA operator weights determination methods with
RIM quantifier, JOURNAL OF SOUTHEAST UNIVERSITY (ENGLISH EDITION), vol. 23 (SUPPL.), pp. 76-82. 2007
A13-c61 Z.-B. Lu, L.-H. Zhou, Hybrid fuzzy cognitive maps, JOURNAL
OF XIDIAN UNIVERSITY (NATURAL SCIENCE), 34 (5), pp. 779783 (in Chinese). 2007
A13-c60 B.S. Ahn, The OWA aggregation with uncertain descriptions on
weights and input arguments, IEEE TRANSACTIONS ON FUZZY
SYSTEMS, 15 (6), pp. 1130-1134. 2007
http://dx.doi.org/10.1109/TFUZZ.2007.895945
A13-c87 L. Zhenbang, Z. Lihua, A hybrid fuzzy cognitive model based
on weighted OWA operators and single-antecedent rules, INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 22 (11), pp. 11891196. 2007
http://dx.doi.org/10.1002/int.20243
A13-c86 Sadiq, R., Tesfamariam, S., Probability density functions based
weights for ordered weighted averaging (OWA) operators: An example of water quality indices, EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 182 (3), pp. 1350-1368. 2007
http://dx.doi.org/10.1016/j.ejor.2006.09.041
A13-c85 Llamazares, B., Choosing OWA operator weights in the field of
Social Choice, INFORMATION SCIENCES, 177 (21), pp. 4745-4756.
2007
http://dx.doi.org/10.1016/j.ins.2007.05.015
In the field of OWA operators, one of the first approaches,
suggested by O’Hagan [18], lies in selecting the vector that
maximizes the entropy of the OWA weights for a given level
of orness. This methodology has also been used by Fullér and
Majlender [A13]. (page 4745)
16
A13-c84 Wang YM, Luo Y, Liu XW, Two new models for determining
OWA operator weights COMPUTERS & INDUSTRIAL ENGINEERING 52 (2): 203-209 MAR 2007
http://dx.doi.org/10.1016/j.cie.2006.12.002
Fullér and Majlender (2001) showed that the maximum entropy model could be transformed into a polynomial equation
that can be solved analytically. (page 203)
A13-c83 Yeh DY, Cheng CH, Yio HW, Empirical research of the principal
component analysis and ordered weighted averaging integrated evaluation model on software projects CYBERNETICS AND SYSTEMS, 38
(3): 289-303 2007
http://dx.doi.org/10.1080/01969720601187347
A13-c82 Wang YM, Parkan C, A preemptive goal programming method
for aggregating OWA operator weights in group decision making INFORMATION SCIENCES, 177 (8): 1867-1877 APR 15 2007
http://dx.doi.org/10.1016/j.ins.2006.07.023
Fullér and Majlender [A13] showed that the maximum entropy model could be converted into a polynomial equation
that can be solved analytically. (page 1867)
A13-c81 Wu J, Liang CY, Huang YQ, An argument-dependent approach to
determining OWA operator weights based on the rule of maximum entropy INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS,
22 (2): 209-221 FEB 2007
http://dx.doi.org/10.1002/int.20201
Fullér and Majlender [A13] transformed the maximum entropy model into a polynomial equation that can be solved
analytically. (page 209)
A13-c80 Xu ZS, Chen J, An interactive method for fuzzy multiple attribute
group decision making INFORMATION SCIENCES, 177 (1): 248263 JAN 1 2007
http://dx.doi.org/10.1016/j.ins.2006.03.001
Fullér and Majlender [A13] used the method of Lagrange
multipliers to solve O’Hagan’s procedure analytically. (page
251)
A13-c79 Sadiq, R., Tesfamariam, S. Probability density functions based
weights for ordered weighted averaging (OWA) operators: An example
17
of water quality indices EUROPEAN JOURNAL OF OPERATIONAL
RESEARCH, 182 (3), pp. 1350-1368. 2007
http://dx.doi.org/10.1016/j.ejor.2006.09.041
Yager and Filev (1999) suggested an algorithm to obtain the
OWA weights from a collection of samples with the relevant aggregated data. Fullér and Majlender (2001) used the
method of Lagrange multipliers to solve O’Hagan’s procedure analytically. (page 1356)
A13-c78 Liu, X., The solution equivalence of minimax disparity and minimum variance problems for OWA operators, INTERNATIONAL JOURNAL OF APPROXIMATE REASONING, 45 (1), pp. 68-81. 2007
http://dx.doi.org/10.1016/j.ijar.2006.06.004
Recently, Fullér [A13] transformed the maximum entropy model
into a polynomial equation, which can be solved in an analytical way. (page 69)
A13-c77 Cheng CH, Chang JR, Ho TH Dynamic fuzzy OWA model for
evaluating the risks of software development CYBERNETICS AND
SYSTEMS, 37 (8): 791-813 DEC 2006
http://dx.doi.org/10.1080/01969720600939797
A13-c76 Xu ZS, A note on linguistic hybrid arithmetic averaging operator in multiple attribute group decision making with linguistic information GROUP DECISION AND NEGOTIATION, 15 (6): 593-604
NOV 2006
http://dx.doi.org/10.1007/s10726-005-9008-4
Fullér and Majlender (2001) used the method of Lagrange
multipliers to solve O’Hagan’s procedure analytically. (page
595)
A13-c75 Liu XW, Lou HW Parameterized additive neat OWA operators
with different orness levels INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 21(10): 1045-1072 OCT 2006
http://dx.doi.org/10.1002/int.20176
The maximum entropy OWA operator was first suggested by
O’Hagan [40] and later was discussed by Filev and Yager [21]
and Fullér and Majlender [A13]. (page 1055)
A13-c74 Liu XW, On the properties of equidifferent OWA operator INTERNATIONAL JOURNAL OF APPROXIMATE REASONING, 43
(1): 90-107 SEP 2006
18
http://dx.doi.org/10.1016/j.ijar.2005.11.003
The consistent condition of geometric (maximum entropy)
OWA operator was proved, some properties associated with
the orness level are discussed, which extended the results of
O’Hagan [14], Filev and Yager [6,7], Fullér and Majlender
[A13]. (page 91)
A13-c73 Cheng CH, Chang JR, MCDM aggregation model using situational ME-OWA and ME-OWGA operators, INTERNATIONAL JOURNAL OF UNCERTAINTY FUZZINESS AND KNOWLEDGE-BASED
SYSTEMS, 14 (4): 421-443 AUG 2006
http://dx.doi.org/ 10.1142/S0218488506004102
A13-c72 Ahn BS, On the properties of OWA operator weights functions
with constant level of orness, IEEE TRANSACTIONS ON FUZZY
SYSTEMS, 14(4): 511-515 AUG 2006
http://dx.doi.org/10.1109/TFUZZ.2006.876741
A13-c71 Xu ZH, Induced uncertain linguistic OWA operators applied to
group decision making, INFORMATION FUSION, 7(2): 231-238 JUN
2006
http://dx.doi.org/10.1016/j.inffus.2004.06.005
A13-c70 Marchant T, Maximal orness weights with a fixed variability for
owa operators, INTERNATIONAL JOURNAL OF UNCERTAINTY
FUZZINESS AND KNOWLEDGE-BASED SYSTEMS, 14(3): 271276 JUN 2006
http://dx.doi.org/10.1142/S021848850600400X
A13-c69 Wang JW, Chang JR, Cheng CH, Flexible fuzzy OWA querying method for hemodialysis database, SOFT COMPUTING, 10 (11):
1031-1042 SEP 2006
http://dx.doi.org/10.1007/s00500-005-0030-x
Fullér and Majlender [A13] use the method of Lagrange multipliers to transfer Eq. (12) to a polynomial equation, which
can determine the optimal weighting vector. By their method,
the associated weighting vector is easily obtained by Eqs.
(13)-(18). (page 1033)
A13-c68 Xinwang Liu, On the maximum entropy parameterized interval
approximation of fuzzy numbers, FUZZY SETS AND SYSTEMS, 157,
pp. 869-878. 2006
http://dx.doi.org/10.1016/j.fss.2005.09.010
19
A13-c67 J.-R. Chang, T.-H. Ho, C.-H. Cheng, A.-P. Chen, Dynamic fuzzy
OWA model for group multiple criteria decision making, SOFT COMPUTING, 10 543-554. 2006
http://dx.doi.org/10.1007/s00500-005-0484-x
To resolve this problem, this study proposes a dynamic OWA
aggregation model based on the faster OWA operator, which
has been introduced by Fullér and Majlender [A13] and can
work like a magnifying lens and adjust its focus based on the
sparest information to change the dynamic attribute weights
to revise the weight of each attribute based on aggregation
situation, and then to provide suggestions to decision maker
(DM). (page 544)
A13-c66 Liu XW, Some properties of the weighted OWA operator, IEEE
TRANSACTIONS ON SYSTEMS MAN AND CYBERNETICS PART
B-CYBERNETICS, 36(1): 118-127 FEB 2006
http://dx.doi.org/10.1109/TSMCA.2005.854496
Comparing the researches on the weights obtaining methods
in OWA operator, such as the quantifier guided aggregation
[2], [37], exponential smoothing [14], learning [25], especially the maximum entropy method [16], [28], [38], [A13],
the WOWA aggregation methods are relatively rare [30], [40].
(pages 118-119)
A13-c65 Nasibov, E.N., Aggregation of fuzzy information on the basis
of decompositional representation, CYBERNETICS AND SYSTEMS
ANALYSIS, 41 (2), pp. 309-318. 2005
http://dx.doi.org/10.1007/s10559-005-0065-0
A13-c64 Xu ZS, An overview of methods for determining OWA weights,
INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 20 (8):
843-865 AUG 2005
http://dx.doi.org/10.1002/int.20097
Fullér and Majlender [A13] used the method of Lagrange
multipliers to solve problem 12 analytically and got the following:
1. If n = 2 then w1 = α and w2 = 1 − α.
2. If α = 0 or α = 1 then the associated weighting vectors
are uniquely defined as w = (0, 0, . . . , , 1)T and w =
(1, 0, . . . , 0)T respectively, with value of dispersion zero.
20
3. If n ≥ 3 and 0 < α < 1 then
q
n−1
wj =
w1n−j wnj−1
((n − 1)α − n)w1 + 1
(n − 1)α + 1 − nw1
w1 [(n − 1)α + 1 − nw1 ]n = ((n − 1)α)n−1 [((n − 1)α − n)w1 + 1]
wn =
(15)
(16)
(17)
Solving Equations 15-17, the optimal OWA weights can be
determined. (page 847)
A13-c63 Lan H, Ding Y, Hong J, Decision support system for rapid prototyping process selection through integration of fuzzy synthetic evaluation and an expert system INTERNATIONAL JOURNAL OF PRODUCTION RESEARCH, 43 (1): 169-194 JAN 1 2005
http://dx.doi.org/10.1080/00207540410001733922
A13-c60 Arfi B, Fuzzy decision making in politics: A linguistic fuzzy-set
approach (LFSA), POLITICAL ANALYSIS, 13 (1): 23-56 WIN 2005
http://dx.doi.org/10.1093/pan/mpi002
A13-c59 Ying-Ming Wang, Celik Parkan, A minimax disparity approach
for obtaining OWA operator weights, INFORMATION SCIENCES,
175(2005) 20-29. 2005
http://dx.doi.org/10.1016/j.ins.2004.09.003
Fullér and Majlender [A13] showed that the maximum entropy model could be transformed into a polynomial equation
that can be solved analytically. (page 21)
A13-c58 Liu Xinwang, Preference Representation with Geometric OWA
Operator, SYSTEMS ENGINEERING, 22(2004), number 9, pp. 8286 (in Chinese). 2004
http://d.wanfangdata.com.cn/Periodical_xtgc200409019.aspx
A13-c57 Liu Xinwang, Three methods for generating monotonic OWA operator weights with given orness level, JOURNAL OF SOUTHEAST
UNIVERSITY (ENGLISH EDITION), Vol. 20 No. 3, pp. 369-373.
2004
http://www.wanfangdata.com.cn/qikan/periodical.articles/dndxxb-e/dndx2004/0403/040321.htm
A13-c56 Liu XW, On the methods of decision making under uncertainty
with probability information, INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 19(12): 1217-1238 DEC 2004
http://dx.doi.org/10.1002/int.20045
21
The maximum entropy OWA operator was first suggested by
O’Hagan [11] and later was discussed by Filev and Yager [10]
and Fullér and Majlender [A13]. (page 1225)
A13-c55 Xinwang Liu and Lianghua Chen, On the properties of parametric geometric OWA operator, INTERNATIONAL JOURNAL OF APPROXIMATE REASONING, 35 pp. 163-178. 2004
http://dx.doi.org/10.1016/j.ijar.2003.09.001
Recently, Fullér and Majlender [A13] proposed another method
to generate MEOWA weights, the method get the weights by
solving a polynomial equation. (page 164)
A13-c54 Chiclana F, Herrera-Viedma E, Herrera F, et al. Induced ordered
weighted geometric operators and their use in the aggregation of multiplicative preference relations INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 19(3): 233-255 MAR 2004
http://dx.doi.org/10.1002/int.10172
A13-c53 Beliakov G., How to build aggregation operators from data INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 18(8):
903-923 AUG 2003
http://dx.doi.org/10.1002/int.10120
A13-c52 Xu, Z., Da, Q. Approaches to obtaining the weights of the ordered
weighted aggregation operators Dongnan Daxue Xuebao (Ziran Kexue
Ban)/Journal of Southeast University (Natural Science Edition), 33 (1),
pp. 94-96. 2003
in proceedings and in
edited volumes
A13-c32 Gleb Beliakov, Optimization and Aggregation Functions, in: Weldon A Lodwick, Janusz Kacprzyk eds., Fuzzy Optimization: Recent
Advances and Applications, Studies in Fuzziness and Soft Computing
vol. 254/2010, Springer [ISBN 978-3-642-13934-5], pp. 77-108. 2010
http://dx.doi.org/10.1007/978-3-642-13935-2_4
A13-c31 Jia-Wen Wang and Jing-Wen Chang, A Fusion Approach for Multicriteria Evaluation, in: Ngoc Thanh Nguyen, Radosław Katarzyniak,
and Shyi-Ming Chen eds., Advances in Intelligent Information and
Database Systems, Studies in Computational Intelligence, vol. 283/2010,
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A13-c30 X. Liu, On the methods of OWA operator determination with
different dimensional instantiations, 6th International Conference on
Fuzzy Systems and Knowledge Discovery, FSKD 2009, 14-16 August 2009, Tianjin, China, Volume 7, [ISBN 978-076953735-1], Article number 5359982, pp. 200-204. 2009
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A13-c29 Mostafa Keikha and Fabio Crestani, Effectiveness of Aggregation
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Henrik Bulskov, Henning Christiansen, Henrik Legind Larsen eds.,
Flexible Query Answering Systems, Lecture Notes in Computer Science, vol. 5822/2009, Springer, [ISBN 978-3-642-04956-9], pp. 157167. 2009
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A13-c28 Matteo Brunelli, Michele Fedrizzi, A Fuzzy Approach to Social Network Analysis, Social Network Analysis and Mining, International Conference on Advances in Social Network Analysis and Mining, Athens, Greece, July 20-July 22, [ISBN 978-0-7695-3689-7] pp.
225-230. 2009
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A13-c27 B. Fonooni; S. Moghadam, Applying induced aggregation operator in designing intelligent monitoring system for financial market,
IEEE Symposium on Computational Intelligence for Financial Engineering, March 30, 2009 - April 2, 2009, Nashville, TN, [ISBN 978-14244-2774-1], pp. 80-84. 2009
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2009
A13-c25 Victor M. Vergara, Shan Xia, and Thomas P. Caudell, Information fusion across expert groups with dependent and independent components, Multisensor, Multisource Information Fusion: Architectures,
Algorithms, and Applications 2009, Proceedings of SPIE - The International Society for Optical Engineering, 7345, art. no. 73450C.
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23
A13-c24 Ming Li, Yan-Tao Zheng, Shou-Xun Lin, Yong-Dong Zhang and
Tat-Seng Chua, Multimedia evidence fusion for video concept detection via OWA operator, Lecture Notes in Computer Science, vol. 5371/2008,
pp. 208-216. 2008
http://dx.doi.org/10.1007/978-3-540-92892-8_21
We therefore formulate the multi-modal fusion as an information aggregation task in the framework of group decision
making (GMD) problem. Specifically, we employ the Ordered Weighted Average (OWA) operator to aggregate the
group of decisions by uni-modal detectors, as it has been reported to be an effective solution for GMD problem [A13].
(page 209)
In Dispersion Maxspace represented by Eqn (8), the weights
of different Orness are given with maximal dispersion which
means most individual criteria are being used in the aggregation that gives more robustness [A13]. (page 212)
A13-c23 Yao-Hsien Chen, Jing-Wei Liu, Ching-Huse Cheng, Intelligent
Preference Selection for Evaluating Studentsapos; Learning Achievement Third International Conference on Convergence and Hybrid Information Technology, Volume 2, Article number 4682412, 11-13 November, 2008, pp. 1214 -1219. 2008
http://dx.doi.org/10.1109/ICCIT.2008.410
A13-c22 K.-M. Björk, Obtaining minimum variability OWA operators under a fuzzy level of orness, ICINCO 2008 - Proceedings of the 5th
International Conference on Informatics in Control, Automation and
Robotics ICSO, Volume ICSO, pp. 114-119. 2008
A13-c21 Xinwang Liu; Xiaoguang Yang; Yong Fang, The relationships between two kinds of OWA operator determination methods, IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2008), pp. 264270, 1-6 June 2008
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A13-c20 B. Fonooni; S. J. Moghadam, Designing financial market intelligent monitoring system based on OWA, in: Proceedings of the WSEAS
international Conference on Applied Computing, (Istanbul, Turkey, May
27 - 30, 2008). M. Demiralp, W. B. Mikhael, A. A. Caballero, N.
Abatzoglou, M. N. Tabrizi, R. Leandre, M. I. Garcia-Planas, and R.
24
S. Choras, Eds. Mathematics And Computers In Science And Engineering. World Scientific and Engineering Academy and Society
(WSEAS), Stevens Point, Wisconsin, pp. 35-39. 2008
A13-c19 Calvo, T., Beliakov, G., Identification of weights in aggregation
operators, in: Bustince, Humberto; Herrera, Francisco; Montero, Javier
(Eds.) Fuzzy Sets and Their Extensions: Representation, Aggregation
and Models Intelligent Systems from Decision Making to Data Mining,
Web Intelligence and Computer Vision Series: Studies in Fuzziness
and Soft Computing , Vol. 220 Springer, [ISBN: 978-3-540-73722-3]
2008, pp. 145-162. 2008
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A13-c18 B. Llamazares, J.L. Garcia-Lapresta, Extension of some voting
systems to the field of gradual preferences, in: Bustince, Humberto;
Herrera, Francisco; Montero, Javier (Eds.) Fuzzy Sets and Their Extensions: Representation, Aggregation and Models Intelligent Systems
from Decision Making to Data Mining, Web Intelligence and Computer Vision Series: Studies in Fuzziness and Soft Computing , Vol.
220 Springer, [ISBN: 978-3-540-73722-3] 2008, pp. 297-315. 2008
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A13-c17 Benjamin Fonooni, Rational-Emotional Agent Decision Making
Algorithm Design with OWA, 19th IEEE International Conference on
Tools with Artificial Intelligence, October 29-31, 2007, Paris, France,
pp. 63-66. 2007
http://doi.ieeecomputersociety.org/10.1109/ICTAI.2007.123
A13-c16 Cheng, Ching-Huse; Liu, Jing-Wei; Wu, Ming-Chang, OWA Based
Information Fusion Techniques for Classification Problem, 2007 International Conference on Machine Learning and Cybernetics, 19-22
Aug. 2007, vol.3, [ISBN 978-1-4244-0973-0 ], pp.1383-1388. 2007
http://dx.doi.org/10.1109/ICMLC.2007.4370360
Fullér and Majlender [A13] transform Yager’s OWA equation
to a polynomial equation by using Lagrange multipliers. According to their approach, the associated weighting vector can
be obtained by (2) - (4). (page 1384)
A13-c15 Wang, Jia-Wen; Cheng, Ching-Hsue, Information Fusion Technique for Weighted Time Series Model, International Conference on
25
Machine Learning and Cybernetics, 19-22 Aug. 2007, [ISBN 978-14244-0973-0 ], vol.4, pp.1860-1865. 2007
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Fullér and Majlender use the method of Lagrange multipliers to transfer equation (7) to a polynomial equation, which
can determine the optimal weighting vector. By their method,
the associated weighting vector is easily obtained by (8)-(9)
[A13].
(page 1861)
A13-c14 Na Cai; Ming Li; Shouxun Lin; Yongdong Zhang; Sheng Tang;
AP-Based Adaboost in High Level Feature Extraction at TRECVID, in:
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A13-c13 Chang, Jing-Rong; Liao, Shu-Ying; Cheng, Ching-Hsue, Situational ME-LOWA Aggregation Model for Evaluating the Best Main
Battle Tank, 2007 International Conference on Machine Learning and
Cybernetics, 19-22 Aug. 2007, vol.4, pp.1866-1870. 2007
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A13-c12 Ching-Huse Cheng, Jing-Wei Liu, OWA Rough Set to Forecast
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A13-c11 Benjamin Fonooni, Behzad Moshiri and Caro Lucas, Applying
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50th Anniversary of Artificial Intelligence, Lecture Notes in Computer
Science, Volume 4850/2007, Springer, 2007, pp. 320-331. 2007
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A13-c10 Beliakov, G., Pradera, A., Calvo, T., Other types of aggregation and additional properties, in: Aggregation Functions: A Guide
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[ISBN 978-3-540-73720-9], Springer, pp. 297-304. 2007
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A13-c9 Eric Levrat, Jean Renaud, Christian Fonteix, Decision compromise
modelling based on OWA operators, Ninth IFAC Symposium on Automated Systems Based on Human Skill and Knowledge, Automated
Systems Based on Human Skill and Knowledge, May 22-24, 2006,
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A13-c8 Yeh, Duen-Yian Cheng, Ching-Hsue Yio, Hwei-Wun, OWA and
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Automation Congress, WAC’06, 24-26 July 2006, Budapest, Hungary,
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A13-c7 Zadrozny S, Kacprzyk J, On tuning OWA operators in a flexible
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SCIENCE 4027: 97-108 2006
http://dx.doi.org/10.1007/11766254_9
Filev and Yager [11] simplified this optimization problem using the Lagrange multipliers method. Then the problem boils
down to finding the root of a polynomial of degree m − 1.
Fullér and Majlender [A13], assuming the same approach,
proposed a simpler formulae for the weight vector W . (page
101)
A13-c6 Xu ZS, Dependent OWA operators, in: Modeling Decisions for
Artificial Intelligence, Third International Conference, MDAI 2006,
LECTURE NOTES IN ARTIFICIAL INTELLIGENCE 3885: 172178 2006
http://dx.doi.org/10.1007/11681960_18
A13-c5 Troiano L, Yager RR On the relationship between the quantifier
threshold and OWA operators, LECTURE NOTES IN ARTIFICIAL
INTELLIGENCE 3885: 215-226 2006
http://dx.doi.org/10.1007/11681960_22
A13-c4 Ching-Hsue Cheng, Jing-Rong Chang, Tien-Hwa Ho, and An-Pin
Chen, Evaluating the Airline Service Quality by Fuzzy OWA Operators
in: Vincent Torra, Yasuo Narukawa, Sadaaki Miyamoto (Eds.): Proceedings of the Modeling Decisions for Artificial Intelligence: Second
27
International Conference, MDAI 2005, Tsukuba, Japan, July 25-27,
2005, LNAI 3558, Springer, pp. 77-88. 2005
http://dx.doi.org/10.1007/11526018_9
Fullér and Majlender [A13] used the method of Lagrange
multipliers to transfer Yager’s OWA equation to a polynomial
equation, which can determine the optimal weighting vector.
By their method, the associated weighting vector is easily obtained by (5)-(7).
q
j−1
n−j
ln wj =
ln wn +
ln w1 ⇒ wj = n−1 w1n−j wnj−1
n−1
n−1
((n − 1)α − n)w1 + 1
and wn =
(n − 1)α + 1 − nw1
then w1 [(n − 1)α + 1 − nw1 ]n = ((n − 1)α)n−1 [((n − 1)α − n)w1 + 1]
(5)
(6)
(7)
(pages 79-80)
A13-c3 L. Troiano and R.R. Yager, A meaure of dispersion for OWA operators, in: Y. Liu, G. Chen and M. Ying eds., Proceedings of the Eleventh
International Fuzzy systems Association World Congress, July 28-31,
2005, Beijing, China, 2005 Tsinghua University Press and Springer,
[ISBN 7-302-11377-7] pp. 82-87. 2005
Entropy has been generally adopted as a measure of weight
dispersion of the OWA operators. O’Hagan [2], in his ground
braking work, suggests to select the vector that maximizes the
entropy of OWA weights (ME-OWA). Analytical solutions to
this problem have been proposed by Filev and Yager [3], and
Fullér and Majlender [A13]. (page 82)
A13-c2 Liu, X.-W., Chen, L.-H. The equivalence of maximum entropy
OWA operator and geometric OWA operator, International Conference
on Machine Learning and Cybernetics, 5, pp. 2673-2676. 2003
http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1259989
A13-c1 H.B. Mitchell, Data Mining Using a Probabilistic Weighted Ordered Weighted Average (PWOWA) Operator, in: Vicenc Torra ed.,
Information Fusion in Data Mining Series: Studies in Fuzziness and
Soft Computing , Vol. 123, Springer, [ISBN: 978-3-540-00676-3] pp.
41-58. 2003
28
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