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

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