Table Of ContentTHE ORDERED WEIGHTED
AVERAGING OPERATORS
Theory and Applications
TIIE ORDERED WEIGHTED
AV ERAGING OPERATORS
Theory and Applications
edited by
Ronald R. Yager
Iona College
New Rochelle, New York USA
and
Janusz Kacprzyk
Polish Academy of Sciences
Warsaw, Poland
~.
"
Springer-Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data
A C.I.P. Catalogue record for this book is available
from the Library of Congress.
ISBN 978-1-4613-7806-8 ISBN 978-1-4615-6123-1 (eBook)
DOI 10.1007/978-1-4615-6123-1
Copyright © 1997 Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1997
Softcover reprint of the hardcover 1s t edition 1997
AlI rights reserved. No part of this publication may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, mechanical, photo
copying, recording, or otherwise, without the prior written permission of the
publisher, Springer-Science+Business Media, LLC.
Printed on acid-free paper.
Table of Contents
Preface ix
1. BASIC ISSUES IN AGGREGATION
Kolmogorov's theorem and its impact on soft computing 3
H.T. Nguyen and V. Kreinovich
Possibility and necessity in weighted aggregation 18
C. Carlsson, R Fuller and S. Fuller
OWA operators and an extension oft he contrast model 29
B. Bouchon-Meunier and M. Rifqi
Equivalence ofc hanges in proportions at crossroads of 36
mathematical theories
J. Aczel, G. Rote and J. Schwaiger
2. FUNDAMENTAL ASPECTS OF OW A OPERATORS
On the inclusion ofi mportances in OWA aggregation 41
RR Yager
On the linguistic OWA operator and extensions 60
F. Herrera and E. Herrera-Viedma
Alternative representations ofo wA operators 73
M. Grabisch
3. MATHEMATICAL ISSUES AND OWA OPERATORS
Useful tools for aggregation procedures: some consequences 89
and applications ofStrassen 's measurable Hahn-Banach theorem
H,J. Skala
vi
OWA specificity 98
A. Ramer
Ordered continuous means and in/ormation 107
A. Ramer
4. OW A OPERATORS IN DECISION ANALYSIS
OWA operators in decision making with uncertainty and nonnumeric 123
payoffs
RR Yager and M.T. Lamata
On the role o/immediate probability in various decision making models 139
KJ. Engemann and RR Yager
Risk management using/uzzy logic and genetic algorithms 155
T.C. Rubinson and G. Geotsi
OWA operators/or doctoral student selection problem 167
C. Carlsson, R. Fuller and Sz. Fuller
5. OWA OPERATORS IN MULTICRITERIA AND MULTIPERSON
DECISION MAKING
Beyond min aggregation in multicriteria decision: (ordered) weighted 181
mean, discri-min, leximin
D. Dubois, H. Fargier and H. Prade
OWA operators in group decision making and consensus reaching 193
under fuzzy preferences and/uzzy majority
J. Kacprzyk, M. Fedrizzi and H. Nurmi
Applications o/the linguistic OWA operators in group decision making 207
F. Herrera, E. Herrera-Viedma and 1.L. Verde gay
Aggregation rules in committee procedures 219
1. Montero and V. Cutello
6. OW A OPERATORS IN QUERYING AND INFORMA TION RETRIEVAL
Quantified statements and some interpretations/or the OWA operators 241
P. Bose and L. Lietard
vii
Using OWA operators inflexible query processing 258
M.-A. Vila, J.-C. Cubero, J.-M. Medina and O. Pons
Application ofOWA operatrors to soften information retrieval systems 275
G. Bordogna and G. Pasi
Implementation ofOWA operators in fuzzy querying for Microsoft Access 293
J. Kacprzyk and S. Zadrozny
7. OWA OPERATORS IN LEARNING AND CLASSIFICATION
OWA-based computing: learning algorithms 309
W.Pedrycz
OWA operators in machine learning from imperfect examples 321
J. Kacprzyk
An application ofOWA operators to the aggregation ofm ultip ie 330
classification decisions
L.I. Kuncheva
INDEX 345
PREFACE
Aggregation plays a central role in many of the technological tasks we are
faced with. The importance of this process will become even greater as we
move more and more toward becoming an information-cent.ered society, us is
happening with the rapid growth of the Internet and the World Wirle Weh.
Here we shall be faced with many issues related to the fusion of information.
One very pressing issue here is the development of mechanisms to help
search for information, a problem that clearly has a strong aggregation-related
component. More generally, in order to model the sophisticated ways in which
human beings process information, as well as going beyond the human capa
bilities, we need provide a basket of aggregation tools.
The centrality of aggregation in human thought can be be very clearly seen
by looking at neural networks, a technology motivated by modeling the human
brain. One can see that the basic operations involved in these networks are
learning and aggregation.
The Ordered Weighted Averaging (OWA) operators provide a parameter
ized family of aggregation operators which include many of the well-known
operators such as the maximum, minimum and the simple average.
While the original insights leading to the creation of these operators owe
a large debt to the rich framework provided by fuzzy sets theory, the use of
these operators are not in any way restricted to fuzzy sets. For example, the
simple process of taking the average of a collection of numbers In which we
give preferential treatment to the higher numbers, as may be the case when
we are averaging multiple estimates of future interest rates anrl want to be
conservative, is an example of the use of OWA operators.
In this book we see papers from a number of domains where these opera
tors have been applied: decision analysis under uncertainty, multi-crit.eria de
cision making, multi-person decision making and consensus formation, flexihle
database querying and information retrieval, learning and classification.
The editors would like to thank all the contributors without whose effort.s
this volume would be impossible.
Ronald R. Yager, New York
Janusz Kacprzyk, Warsaw
1
BASIC ISSUES IN AGGREGATION
KOLMOGOROV.S THEOREM
AND ITS IMPACT ON SOFT COMPUTING
Hung T. Nguyenl and Vladik Kreinovich2
1 Department of Mathematical Sciences,
New Mexico State University,
Las Cruces, NM 88003, USA,
e-mail [email protected]
2Department of Computer Science,
University of Texas at EI Paso,
EI Paso, TX 79968, USA,
e-mail [email protected]
Abstract. In this chapter, we describe various applications of the Kol
mogorov's theorem on representing continuous functions of several variables
(as superpositions of functions of one and two variables) to soft computing.
Kolmogorov's theorem leads to a theoretical justification, as well as to de
sign methodologies, for neural networks. In the design of intelligent systems,
Kolmogorov's theorem is used to show that general logical operators can be
expressed in terms of basic fuzzy logic operations.
In the area of reliable computing (i.e., computing that takes into consid
eration the accuracy of the input data), an extended version of Kolmogorov's
theorem justifies the need to use operations with three or more operands in soft
computing. Such operations have already been actively used in soft computing;
the simplest (and, so far, most used) of such operations are ordered weighted
averaging (OWA) operators proposed by R. R. Yager.
Keywords: Neural networks, soft computing, Kolmogorov's theorem, reliable
(and interval) computing, fuzzy logic, logical operations
1. INTRODUCTION
The area of Soft Computing, as defined by L. A. Zadeh, deals with methodolo
gies and techniques for analyzing complex systems. These methodologies and
R. R. Yager et al. (eds.), The Ordered Weighted Averaging Operators
© Springer Science+Business Media New York 1997
4
techniques include probability and statistics, neural networks, fuzzy logic, etc.
Normally in scientific investigations, heuristics precede formal methods. The
path from heuristics to formal methods is paved, in general, by some math
ematical results, which may belong to another domain of pure mathematics.
The story we are going to report in this chapter is no exception.
Among tremendous contributions to mathematics, ranging from Probability
Theory to Analysis, A. Kolmogorov has left us with many fundamental th&
orems. His investigations into Hilbert's famous conjecture (known as 13th
problem) has led to Kolmogorov's famous 1957 theorem on the possibility of
representing continuous functions of several variables as superpositions of con
tinuous functions of one and two variables. This theorem is the one we refer
to in this chapter. While Kolmogorov's theorem is a purely mathematical,
non-algorithmic result (disproving Hilbert's conjecture) that has applications
in fields related to approximations of functions, its impact on Artificial Intelli
gence has not been noticed until recently.
In the next section, we will first describe Hilbert's 13th problem and Kol
mogorov's theorem. In Sections 3-5, we will describe how Kolmogorov's the
orem impacts soft computing. In general, complex systems are difficult to
describe directly, so, a natural way to describe them is:
• to describe simple systems, and then
• to combine the descriptions of simple systems into a description of a complex
system.
The simplest possible combination of systems is when the output of one sub
system becomes an input to another subsystem. In mathematical terms, such a
combination corresponds to superposition of the corresponding functions. For
example:
• In the neural network methodology, we describe an arbitrary complex sys
tem as a combination (superposition) of simple elements called neuron.9, ill
which:
• first, input signals Xl, ••. ,:en undergo linear combination :el •...• Xu ->
+ ...
:e = WIX! WnXn - WO;
• second, a non-linear function g(x) (called activation function) is applied
to the result X of the linear combination, yielding the value y = g(x).
• In fuzzy systems methodology, a general description of complex systems
comes from combining such simpler descriptions as:
• membership functions (that describe fuzzy properties);
• hedges like "very" (that describe a degree to which a property is tme),
and
• t-norms and t-conorms (fuzzy analogues of "and" and "or" logical op-
erations), that combine the simpler descriptions.
In both cases, Kolmogorov's theorem shows that these methodologies can actu
ally describe an arbitrary complex system. A similar result is proven for logical
operators in fuzzy logic. Here, the impact of Kolmogorov's theorem is this:
All (continuous) logical operations in fuzzy logic can be represented by basic
