Intuitionistic Fuzzy Sets: Theory and Applications

335 Pages·1999·6.902 MB·English

by  Krassimir T. Atanassov

Intuitionistic Fuzzy Sets
Studies in Fuzziness and Soft Computing
Editor-in-chief
Prof. Janusz Kacprzyk
Systems Research Institute
Polish Academy of Sciences
ul. Newelska 6
01-447 Warsaw, Poland
E-mail: [email protected]
Vol. 3. A. Geyer-Schulz Vol. 14. E. Hisdal
Fuzzy Rule-Based Expert Systems and Genetic Logical Structures for Representation
Machine Learning. 2nd ed. 1996 of Knowledge and Uncertainty, 1998
ISBN 3-7908-0964-0 ISBN 3-7908-1056-8
Vol. 4. T. Onisawa and J. Kacprzyk (Eds.) Vol. 15. G.J. K1ir and M.J. Wiennan
Reliability and Safety Analyses under Uncertainty-Based Information, 1998
Fuzziness. 1995 ISBN 3-7908-1073-8
ISBN 3-7908-0837-7
Vol. 16. D. Driankov and R. Palm (Eds.)
Vol. 5. P. Bosc and J. Kacprzyk (Eds.)
Advances in Fuzzy Control. 1998
Fuzziness in Database Management
ISBN 3-7908-1090-8
Systems. 1995
ISBN 3-7908-0858-X
Vol. 17. L. Reznik, V. Dimitrov and
Vol. 6. E. S. Lee and Q. Zhu J. Kacprzyk (Eds.)
Fuzzy Systems Design. 1998
Fuzzy and Evidence Reasoning. 1995
ISBN 3-7908-1118-1
ISBN 3-7908-0880-6
Vol. 7. B.A. Juliano and W. Bandler Vol. 18. L. Polkowski and A. Skowron (Eds.)
Tracing Chains-of-Thought. 1996 Rough Sets in Knowledge Discovery 1. 1998,
ISBN 3-7908-1119-X
ISBN 3-7908-0922-5
Vol. S. F. Herrera and J. L. Verdegay (Eds.) Vol. 19. L. Polkowski and A. Skowron (Eds.)
Genetic Algorithms and Soft Computing. 1996. Rough Sets in Knowledge Discovery 2. 1998.
ISBN 3-790S-0956-X ISBN 3-7908-1120-3
Vol. 9. M. Sato et aI. Vol. 20. J. N. Mordeson and P. S. Nair
Fuzzy Clustering Models and Applications. 1997,
Fuzzy Mathematics. 1998
ISBN 3-7908-1026-6 ISBN 3-7908-1121-1
Vol. 10. L.C. Jain (Ed.)
Soft Computing Techniques in Knowledge-based In Vol. 21. L. C. Jain and T. Fukuda (Eds.)
telligent Engineering Systems. 1997. Soft Computing for Intelligent Robotic Systems.
ISBN 3-790S-1035-5 1998
ISBN 3-7908-1147-5
Vol. II. W. Mielczarski (Ed.)
Fuzzy Logic Techniques in Power Systems. 1998, Vol. 22. J. Cardoso and H. Camargo (Eds.)
ISBN 3-7908-1044-4 Fuzziness in Petri Nets. 1999
ISBN 3-7908-1158-0
Vol. 12. B. Bouchon-Meunier (Ed.)
Aggregation and Fusion of Imperfect
Vol. 23. P. S. Szczepaniak (Ed.)
Information. 1998
Computational Intelligence and Applications, 1999
ISBN 3-7908-1048-7
ISBN 3-7908-1161-0
Vol. 13. E. Ortowska (Ed.)
Incomplete Information: Vol. 24. E. Ortowska (Ed.)
Rough Set Analysis. 1998 Logic at Work. 1999
ISBN 3-7908-1049-5 ISBN 3-7908-1164-5
continued on page 324
Krassimir T. Atanassov
Intuitionistic
Fuzzy Sets
Theory and Applications
With 121 Figures
Springer-Verlag Berlin Heidelberg GmbH
Prof. Krassimir T. Atanassov
Centre of Biomedical Engineering
Bulgarian Academy of Sciences
ul. Acad. G. Bonchev, Bl. 105
Sofia-lI13
Bulgaria
Email: [email protected]
ISBN 978-3-7908-2463-6
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnabme
Atanasov, Krassimir T.: Intuitionistic fuzzy sets: theory and applications I Krassimir
T. Atanassov.
(Studies in fuzziness and soft computing; Vol. 35)
ISBN 978-3-7908-2463-6 ISBN 978-3-7908-1870-3 (eBook)
DOI 10.1007/978-3-7908-1870-3
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To the three ladies:
my mother,
my wife, and
my daughter.
Foreword
In the beginning of 1983, I came across A. Kaufmann's book "Introduction
to the theory of fuzzy sets" (Academic Press, New York, 1975). This was my
first acquaintance with the fuzzy set theory.
Then I tried to introduce a new component (which determines the degree
of non-membership) in the definition of these sets and to study the properties
of the new objects so defined. I defined ordinary operations as "n", "U", "+"
and "." over the new sets, but I had began to look more seriously at them
since April 1983, when I defined operators analogous to the modal operators
of "necessity" and "possibility".
The late George Gargov (7 April 1947 - 9 November 1996) is the "god
father" of the sets I introduced - in fact, he has invented the name "intu
itionistic fuzzy", motivated by the fact that the law of the excluded middle
does not hold for them.
Presently, intuitionistic fuzzy sets are an object of intensive research by
scholars and scientists from over ten countries.
This book is the first attempt for a more comprehensive and complete
report on the intuitionistic fuzzy set theory and its more relevant applications
in a variety of diverse fields. In this sense, it has also a referential character.
Naturally, the theory is far from being complete, there are some gaps
and still unsolved, open questions and problems - some of them are formu
lated at the end of the book. A solution to any of them would stimulate the
development of the intuitionistic fuzzy set theory, and contribute to fruitful
applications.
April, 1999 Krassimir T. Atanassov
Sofia, Bulgaria
Acknowledgements
First of all, I wish to convey my most sincere gratitude to all of my Bul
garian co-authors and collaborators, especially to Hristo Aladjov, Plamen
Angelov, Ognian Asparoukhov, Lilija Atanassova, Stefan Danchev, Dinko
Dimitrov, Peter Georgiev, Stefan Hadjitodorov, Lubomir Hadjyisky, Ludmila
Kuncheva, Dimiter Lakov, Nikolai Nikolov, Valentina Radeva, Dimiter Sas
selov, Joseph Sorsich, Stefka Stoeva and Darinka Stoyanova, Peter Vassilev,
for their active participation and collaboration in developing the intuitionistic
fuzzy set theory.
Anthony Shannon (Australia), Christer Carlsson (Sweden), Didier Dubois
and Henri Prade (France), Siegfried Gottwald and Hans-Jiirgen Zimmer
mann (Germany), Humberto Bustince Sola, Pedro Burillo Lopez, Victoria
Mohedano, Ana Burusco and R. Fuentes-Gonzalez (Spain), Eulalia Szmidt,
Tadeusz Gerstenkorn and Jacek Manko (Poland), Toader Buhaescu (Ro
mania), Dogan Coker, Es Haydar, Sadik Bayhan, Ibrahim Ibrahimoglu and
Mustafa Demirci (Turkey), Hung T. Nguyen, Misha Koshelev, Vladik Krei
novich, Bhuvan Rachamreddy, Cecilia Temponi, Haris Yasemis (USA), Berlin
Wu (Taiwan), Soon-Ki Kim and Young Hyun Kim (S. Korea), Tuyun Chen,
Li Zou, Zhang Cheng, YauaXue-hai and Yin Guo-min (P.R. of China), Ranjit
Biswas, Srijit Biswas, Kankana Chakrabarty, K. C. Chattopadhyay, Supriya
Kumar De, Tapas Kumar Mondal, U. K. Mukherjee, Sudarsan Nanda, Akhil
Ranjan Roy and S. K. Samanta (India), and T. E. Kaminsky (Russia) deserve
my thanks for their very important research on intuitionistic fuzzy sets and
for a moral support I received from them.
Special thanks are due to Janusz Kacprzyk (Poland), the editor-in-chief
of this book series, who urged me to finalise my several-year-Iong work on
the theory and applications of intuitionistic fuzzy sets, and to prepare this
book.
My friends and post-graduate students Nikolai Nikolov, Peter Georgiev
and Valentina Radeva, whose knowledge and dedication have helped attain
a professional camera-ready form of the book, deserve my deep thanks and
appreciation.
x
I wish to thank Dr. Martina Bihn and Mrs. Gabriele Keidel from Physica
Verlag (A Springer-Verlag Company), Heidelberg and New York, for their
kind consideration and help in arranging and running this book publication
project.
And last but not least, I would like to thank the three most important
ladies in my life: my mother, my wife, and my daughter, who have been
encouraging and stimulating all my scientific research.
April, 1999 Krassimir T. Atanassov
Sofia, Bulgaria
Intuitionistic Fuzzy Sets: Past and Present
Present-day science and technology is featured with complex processes and
phenomena for which complete information is not always available. For such
cases, mathematical models are developed to handle various types of systems
containing elements of uncertainty. A large part of these models are based
on a recent extension of the ordinary set theory, namely, the so-called fuzzy
sets.
Fuzzy sets (FSs, for short) were introduced by L.A. Zadeh [507] in 1965.
The interest in this theory is constantly rising, which is proved by an in
creasing number of publications devoted to the field. Twenty years ago, for
example, according to some estimations, there have been about 500 publi
cations, while 10 years after their number was as large as 5000. Today it is
hard to estimate even the approximate number of research works related to
the fuzzy sets theory.
Two years after the emergence of the concept of a fuzzy set, it was
generalized by J. Goguen in [297] who proposed L-fuzzy sets. Currently there
are also some other extensions of FSs, and the aim of this book is to present
and discuss one of them.
One generalisation of the notion of FS was proposed by the author of this
book in the beginning of 1983. To some extent, it began as a mathematical
game, inspired by the Russian translation of "Introduction to the theory of
fuzzy sets" by A. Kaufmann [324].
In May 1983 it turned out that the new sets allow the definition of
operators which are, in a sense, analogous to the modal ones (in the case
of ordinary fuzzy sets such operators are meaningless, since they reduce to
identity). It was then that the author realized that he had found a promising
direction of research and published the results in [20].
George Gargov not only gave the new sets their name, intuitionistic fuzzy
sets (IFSs, for short), as their fuzzification denies the law of the excluded
middle, one of the main ideas of intuitionism, but also encouraged the author
to continue his work on them.
XII
Later, with Stefka Stoeva we further generalized that concept to an intu
itionistic L-fuzzy set [117] (see Section 3.1), where L stands for some lattice
coupled with a special negation.
Together with Lilija Atanassova we gave an example [123] of a genuine
IFS (an IFS which is not a fuzzy set) (see Section 1.1). An analogous example
for intuitionistic L-fuzzy sets is given in [26] (see Section 3.1).
Some basic results on the IFSs were published in [21, 24, 25, 28, 40, 46,
50, 52, 56, 57, 58, 59, 60, 116, 119].
Various operators were defined on the IFSs, which are analogous to the
modal logic operators [28, 29, 32, 36, 50, 51, 53, 55, 74, 75] (see Chapter 1).
A further generalisation of the IFSs, the interval-valued IFSs (IVIFSs, for
short) is introduced in [28,61,106] and George Gargov and the author showed
that interval valued fuzzy sets are equivalent to IFSs [106] (see Chapter 2).
A geometric interpretation of an IFS was given [31] (see Chapter 1).
Temporal IFSs were introduced in [42] (see Section 3.3). A second type of
IFSs (see Section 3.4) is introduced in [47], IFSs over an universe which in
turn is an IFS over another universe are described in [38] (see Section 1.15).
Elements Qf intuitionistic fuzzy logic (IFL, for short): two versions of
intuitionistic fuzzy propositional calculus, intuitionistic fuzzy predicate calcu
lus, two versions of intuitionistic fuzzy modal logic, and temporal intuitionistic
fuzzy logic (TIFL, for short) were defined and investigated in a series of com
munications [30, 34, 107, 275] (the last two together with George Gargov).
These results will be published in the present book.
Versions of FORTRAN, C, and PASCAL software packages implement
ing operations, relations and operators over the IFSs are introduced.
In the area of applications we have investigated the so-called V -fuzzy
Petri nets, reduced V -fuzzy generalized nets [442], intuitionistic fuzzy gener
alized nets of type I [23] and II [41, 90] (see Section 4.1) and intuitionistic
fuzzy programs (with Stefka Stoeva) [118]. A gravity field of many bodies
(with Dimitar Sasselov) is announced in [428]. Intuitionistic fuzzy models of
neural networks (with Lubomir Hadjyisky [306, 307]) were developed on the
basis of IFSs. Intuitionistic fuzzy expert systems [54, 62, 63, 411, 412] (see
Section 4.3), intuitionistic fuzzy systems [68] (see Section 4.5), intuitionistic
fuzzy PROLOG [110, 111, 283] and intuitionistic fuzzy constraint logic pro
gramming [45, 49] were developed, too. The last two topics will be included
in a next book.
Some new results on the IFS theory and its applications were reported
to the "Mathematical Foundation of Artificial Intelligence" Seminars held
in Sofia in October, 1989 [33, 274, 283, 308], March, 1990 [37, 38, 109, 282,
445,449]' June, 1990 [446,447]' November, 1990 [39,448] and October, 1994
[67,77,160,162,176].
In the beginning of the 1990s, research on IFSs was reported outside
Bulgaria: in Romania - by Toader Buhaescu [150, 151, 152, 153], in Poland -
by Tadeusz Gerstenkorn and Jacek Manko [290, 291, 292, 293, 294, 295, 367,