Advances in Multicriteria Analysis
As its title implies, Advances in Multicriteria Analysis presents the most recent developments in multicriteria analysis and in some of its principal areas of application, including marketing, research and development evaluation, financial planning, and m
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Nonconvex Optimization and Its Applications Volume 5
Managing Editors:
Panos Pardalos University of Florida, U.S.A.
Reiner Horst University of Trier, Germany
Advisory Board:
Ding-Zhu Du University of Minnesota, U.S.A.
C. A. Floudas Princeton University, U.S.A.
G. lnfanger Stanford University, U.S.A.
J.Mockus Lithuanian Academy of Sciences, Lithuania
H. D. Sherali Virginia Polytechnic Institute and State University, U.S.A.
The titles published in this series are listed at the end of this volume.
Advances in
Multicriteria Analysis Edited by
Panos M. Pardalos University of Florida
Y annis Siskos Technical University of Crete
and
Constantin Zopounidis Technical University of Crete
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data Advances 1n mult1criter1a analysis I ed1ted by Panos M. Pardalos, Yann1s Siskos, Constantln Zopounldis. p. em.-- (S(a, b)) for some strictly increasing and one-to-one transformation on [0, 1]. It should be noticed that no relation in F(A)\U(A) can be "ordinally-equivalent" to a relation in U(A) since only one-to-one transformations are invoked by ordinality. Thus, these two axioms impose very few constraints on the desirable behavior of a choice procedure when applied to fuzzy relations outside U(A). In particular, they leave room for "discontinuities", which seem rather paradoxical. Let us illustrate the possibility of discontinuities on a simple example involving a crisp relation and an "almost crisp" one. Consider the relations Rand R' on A= {a, b, c} defined by the following tables (to be read from row to column): R
a
b
c
R'
a
b
c
a
1
1
1
a
1
1
A.
b
0
1 0
b
0
1
0
1
c
0
0
1
c
0 0
where 0 < A. < 1. It is easy to see that R is crisp and that G(R) = {a}. Let C be a faithful choice procedure. We have C(R) = {a}. Even if Cis ordinal, it may happen that a ~ C(R') whatever the value of A.. As a result C(R)nC(R') will be empty even when R' is arbitrarily "close" to R. Our final axiom is designed to prevent such situations. Consider a sequence of valued relations (Ri e F(A), i = 1, 2, ... ). We say that this sequence converges to converges to R e F(A) if, for all E e IR with E > 0, there is an integer k such that, for all j ~ k and all a, b e A, IRj (a, b) - R( a, b )I < E. A choice procedure C is said to be continuous if, for all R e F(A) and all sequences (Ri E F(A), i = 1, 2, ... ) converging toR, [a e C(Ri) for all Ri in the sequence]=> [a e C(R)]. Our definition of continuity implies that an alternative that is always chosen with fuzzy relations arbitrarily close to a given relation should remain chosen with this relation. It is not difficult to see that CmF is continuous. The result presented in the next section combines ordinality and continuity. This may appear awkward since ordinality implies that the cardinal properties of the numbers R(a, b) should not be used whereas continuity involves a measure of distance between fuzzy relations using these properties. In presence of ordinality, it would be more
