Fuzzy Sets and Systems 47 (1992) 395-400 395 North-Holland

Bulletin

Editorial Fuzzy Relations in Taiwan

This Bulletin contains important conference an- nouncements and research reports along with some book reviews. The review of Zimmermann's book is the second we have printed (see FSS 42(3)) and that of Coad's is of a sequel to an earlier book reviewed in these pages (see FSS 40(3)). Please continue to send in your book reviews and write to me with your interests if you wish to become a reviewer.

lan Graham February 1992

The Study of Water Pollution

Under contract No. 3888-89-12EDISPI with Joint Re- search Center of Euratom Ispra (Varese, Italy), the situation of the water of two torrents in the north of Italy was studied in order to calculate the biological situation measured by the Extended Biotic Index (EBI) by some chemical data. The two torrents are near to each other and are similar for the kind of possible pollution and geographical~morphological situation. The case study was also interesting for the very different flows that the two torrents have during the year.

The situation was modelled using part of the available data by a Fuzzy Reasoning model. This model was tested on the used data and on new ones giving very good results. For a short description of the method see: N. Prati, Per un modello matematico di biologia ambien- tale, Biologia Ambientale, 5 (6) (1991) 15-20 (in italian).

By this we are now capable of: (1) filling in biological indices in the case they have

not been measured; (2) predicting the biological indices (and situation) in

future cases; (3) looking for other possible pollution factors

different from the ones taken into account by the model in the case the measured situation is different from the calculated one.

A first attempt was made to model the situation using generalized statistical methods coming from t-norms and t-conorms.

We are trying to enlarge the study to other rivers and waterways.

For information write to:

A. G. Colombo Engineering Division, J.R.C. Ispra 21020 Varese, Italy

N. Prati Via Gabbi 6 42100 Reggio Emilia, Italy

Consider a fuzzy relation equation in the form of Po Q = R, where relation matrices P and Q are composed with max-min operator o to obtain a relation matrix R. We are interested in both theoretical and application aspects of this equation.

For theoretical development, we have focussed on (1) the structures and properties of P if Q and R are

given with different forms; (2) the solution procedures with sensitivity analysis

when there exist solutions; and (3) the calibration methods when there is no solution. These structures and properties are considered from

investigating the necessary and sufficient conditions of matrix P when P is either existent with multiple or unique values or nonexistent. Therefore, when R and Q are given as interval-valued, the properties and the solution procedure were proposed in the paper "Resolu- tion of interval-valued fuzzy relation equations" [FSS 44, 1991], which was improved on its efficiency by an alternative method [CIIE, 1991]. For constant-valued R and Q, we have defined a Characteristic Matrix of Q to identify the solution properties of P, which facilitates the finding of solutions [FSS 45, 1992; IEEE, 1990] as well as the sensitivity analysis and calibrations [Gen. Systs., 1991]. In addition, since finding exact solutions is an NP-hard problem, we thus also studied the properties and solution procedures of approximate solutions with different operators [FSS 45, 1992].

For applications, we have investigated the structures and properties of efficient solutions when a multiobjec- tive mathematical programming problem is considered with fuzzy relation constraints [Comp. & O.R., 1992]. This model with the developed decision procedure has been applied to determine the optimal treatment and follow- up care decisions for gastric cancer patients, with prom- ising results.

Apart from the relation equations, we have carried out a comparative study on fuzzy t-norm operators from six different aspects [Systs. & Contl. Lett., 1992]. Besides, to develop a prototype of a decision support system for diagnosis, treatment and fol low-up care of gastric cancer patients is one of our ongoing projects, which is spon- sored by a joint program of Tsing-Hua University and Veterans General Hospital. In this project, apart from the applications of fuzzy relation equations to the treatment and follow-up care models as mentioned above, a fuzzy KNN method is applied to the diagnosis subsystem [Systs. Eng., 1991]. This subsystem has been put in practice and has shown its accuracy and potential in decision aid.

While developing this subsystem, we have found some clustering problems that inspired further studies

0165-0114/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved

396 Bulletin

on this subject. Therefore, another ongoing project is focused on theoretical development and applications of fuzzy clustering methods, which is sponsored by the National Science Council of Taiwan, Republic of China.

Hsiao-Fan Wang Department of Industrial Engineering National Tsing Hua University Hsinchu, Taiwan, ROC

Book Reviews

Vagueness: An Investigation into Natural Languages and the Sorites Paradox

This book by Linda Claire Burns was published by Kluwer Academic Publishers (1991 ; ISBN 0-7923-1489-1 ).

The Sorites paradox arises when there is a continuous range of cases where a description applies at one end and not at the other but it is not clear exactly where it ceases to apply. Thus, when we throw stones into a pond, we start with a pond which becomes gradually shallower but at some point we wil l agree that there is now no pond but a pile of stones.. It appears as a paradox when it is claimed that the borderline case is simultaneously a heap and not a heap. Similarly, if there are three shades of red A, B and C with A indistinguish- able from B and B from C though A and C are distinguishable, does this mean that B is of two distinct shades at the same time? Obviously, this is an important problem for fuzzy set theorists and for all artificial intelligence workers who wish to capture this aspect of reasoning in computer systems.

Burns' book is a monograph steeped in the conven- tions of analytical and linguistic philosophy. It starts with Frege's characterization of vagueness and surveys the work of several modern philosophers such as Dummett who have discussed the issue and their various solu- tions in terms of tolerance rules for predication, rejection of the induction hypothesis for vague predicates and even the retreat into idealism (where there are no heaps). The survey is both penetrating and comprehen- sive though no pre-analytic contributions are dealt with. This is slightly disappointing since this has been a topic for Western philosophers from Herakleitos to Hegel. This neglect of history shows most clearly in the part of the discussion which suggests that it is necessary to decide cleanly whether the Sorites is an epistemological or semantic (i.e. objective) phenomenon. Some non- analytic thinkers would argue that this distinction cannot be made since knowledge is related dialectically to its content and object. Also the assumption throughout is that negation applies to predicates and that a mathe- maticization is possible. The result is that fuzzy set theory is squarely demolished as an approach to vague- ness and if you agree with the assumptions then you wil l give up fuzzy sets and indeed all multivalent logics forthwith, for the arguments themselves can't be faulted.

The chief reason is that assigning a truth value of 0.5 is equally as problematical as assigning a value of 1 or 0 when it comes to borderline cases. The arguments and assumptions remind me of Haack's criticism of deviant logics. Burns' programme is to rescue the classical laws of logic such as the excluded middle, and she does this using several reductio arguments (which of course depend on this very law). At least three reactions are possible: one can say that fuzzy sets is mere engineer ing-so it doesn't matter; find no reason to believe the classical laws of logic in the first place; or suggest that the way the problem is posed is wrong.

Most of the book discusses other people's solutions. In fact it is slightly annoying that the author makes the reader wait to the end before her solution is revealed. This makes a second reading almost imperative. Let me give the game away by telling you that the solution is found in the context of utterances or the pragmatics of language. What constitutes a heap is determined only in context. This is something of an over-simplification of course and Burns argues the mathematical consistency of her solution over the weaknesses she finds in her predecessors' formulations very well and clearly.

This is an important book that should not be ignored by theorists interested in the foundations of fuzzy sets. It is a serious contribution to our understanding of vague- ness and needs to be answered more thoroughly than is either appropriate or possible in a review. One further point; even though it assumes a good deal of back- ground in modern Philosophy, it is extraordinarily well written.

lan Graham

Fuzzy Set Theory and its Applications

This book by Hans-J~irgen Zimmermann was pub- lished in a second edition by Kluwer Academic Pub- lishers (Dordrecht, 1991; 399 pages).

From the purely mathematic view point, the evolution of fuzzy logic (soft logic) has been very exciting, but complex. Many scientific theories start by borrowing notions from the already developed areas of mathe- matics, but in this case, Professor Lotfi A. Zadeh intro- duced the basic notion of vagueness which has no sharp morphology and which is so common in human thought processes. This notion of vagueness is modelled using the notion of graded membership in fuzzy logic. Since the introduction of the theory, the applications of it can be found in diverse fields such as artificial intelligence, expert systems, medical diagnosis, control systems, operations research, and management science.

This book on "Fuzzy Set Theory and its Application" by Hans Zimmermann is the revised second edition (1991). It introduces the basic notion of fuzzy set theory using simple to complex examples and easy to under- stand mathematical steps, and then leads the readers to