Fundamenta l Fuzzy Re la t ion Concepts of a D . S . S . for the E s t i m a t i o n of Natural Disasters ' Risk ( The Case of a Trapezoidal

K e y w o r d s S o f t c o m p u t i n g , Fuzzy sets, Membership functions, Natural disaster risk. 1. I N T R O D U C T I O N Recently, the world has witnessed scary natural disasters with fatal results. Forest fires and floods are major global issues with a huge environmental impact on a global scale. All of the Mediterranean countries, Australia, and U.S.A. face a serious forest fire problem. Data reveal that Greece has the most severe forest fire problem among the E.U. countries, not only according to the number of fires that break out every year, but according to the average burnt area per fire as well. It has been estimated that almost 39.4 Ha are burned per fire in Greece, while in 0895-7177/05/$ see front matter @ 2005 Elsevier Ltd. All rights reserved. Typeset by .Ajt4S-TEX doi:10.1016/j.mcm.2005.09.004 748 L . S . ILIADIS AND S. I. SPARTALIS Spain this number is 28.47, in I taly 19.74, and in Portugal 15.29. It is not only a problem of climatic conditions and vegetation, but a problem of political will and a mat te r of the existing forest policy and forest laws [1]. Floods also are a very dangerous type of natural disaster tha t should be studied and a proper protection policy should be made. It is obvious tha t the proper calculation of the degree of natural disaster risk (D.N.D.R.) for each area is very crucial. The case of the floods of the summer of 2002 in central Europe revealed a totally unexpected situation [2]. It is obvious tha t a flexible and reliable way of measuring natural disaster risk is required. The problem of the existing approaches (for the D.N.D.R. estimation) is tha t they use crisp sets. A crisp set is based in the concept that something either belongs to it or it does not. Based on this logic, an area either belongs to the highest (or lowest) risk group or not. In this way, specific boundaries are drawn between the areas in order to cluster them. For example, an area with 20 to 30 annual forest fire breakouts is considered to be high risky. At the same time, an area with 19 forest fires is considered not risky and another with 31 is considered as maximum risk area. However, it does not seem rational to differentiate two areas with one forest fire breakout difference. The purpose of this s tudy is the extension of the decision support system tha t has been developed by Iliadis et al. in 2002 [3,4]. This is done in such a way tha t it can perform D.N.D.R. estimation, by applying trapezoidal membership functions and successful forecasting of the extent of the natural disaster problem for the following year. A parallel target is the comparison of the results obtained by this effort, to the results taken by the application of the older version of the system [3,4]. 2. M A T E R I A L S A N D M E T H O D S There are two main types of sets. The crisp (or classic) sets and the fuzzy sets. For example, a crisp set can be defined by a membership function in the following way, 1, if X E S, ~ s ( X ) = 0, i f X ¢ S . In crisp sets, a function of this type is also called characteristic function. Fuzzy sets can be used to produce the rational and sensible clustering. For fuzzy sets, there exists a degree of membership #s(X) tha t is mapped on [0,1] and every area belongs to all clusters at the same t ime (from lowest risk to highest) with a different degree of membership [5]. In the 1960s, Zadeh developed a linguistic approach to deal with linguistic vague information based on fuzzy sets and fuzzy logic [6]. Since then there have been a number of applications of the approach to a large variety of fields including meteorology, engineering, medicine, management , computer science, expert systems, and systems science [7]. Of course, the characteristic cluster for each prefecture is the one with the highest value of #s(X). A trapezoidal membership function can be applied to produce five different cases of degrees of membership. However, in many cases the problem is the estimation of a joint D.N.D.R. For example, it is very impor tant to define the degree of flood risk for an area based on the height of rain and on the est imated economic damage at the same time. The production of a unique risk index would be very important in such a case. The decision support system (D.S.S.) would be asked to cluster the areas and characterize them as "Areas with the most floods AND areas with the largest financial damage". To achieve this kind of characterization or others of the same nature, fuzzy mathemat ica l operations like T-norms or S-norms can be applied on the #,(X)~,j where i = 1, 2, 3, 4 . . . n (n is the number of the areas under examination) and j = 1, 2, 3 . . . m (where rn is the criterion for which the risk is calculated). Various types of joint D.N.D.R. can be produced Fundamental Fuzzy Relation Concepts 749 for all of the prefectures of Greece by the application of fuzzy relations and by using matrix multiplications. The developed D.S.S. is able to provide ranking of the areas of Greece based on a specific type of D.N.D.R. Authorities will be able to use the risk groups in order to distribute their forces rationally and to plan appropriate prevention and recovery policies. 3. E X I S T I N G A P P R O A C H E S I N G R E E C E F O R F O R E S T F I R E R I S K E S T I M A T I O N A N D O U R S Y S T E M This s tudy focuses on the application of the developed DSS on the problem of forest fires. Various approaches have been proposed to estimate the annual forest fire risk in Greece. The oldest crisp approaches were developed 30 years ago. More specifically, Kailidhs et al. introduced the fire ignition index (F.I.I.) [8] by dividing the burnt area to the number of forest fires, for each type of vegetation. Katsanos classified the forest departments of Greece [9], into seven classes of forest fire risk, according to the burnt area, for every 10,000 Ha. As mentioned before both approaches use crisp sets and the cases that are dose to the boundaries of the clusters are classified irrationally. Today, fuzzy systems can be used for different kinds of purposes such as modeling, prediction, classification, and control in the field of systems science [7]. In particular the possible use of fuzzy systems in modeling and control has generated great attention [7]. Recently, the laboratory of Forest Informatics of the Democritus University of Thrace, has developed two D.S.S.: a heuristic and a fuzzy one which can be considered as innovative [3,4]. These systems perform forest fire risk estimation of the forest fire departments of Greece and forecasting of the problem extent for the following year. They are using the concept of fuzzy expected intervals, triangular membership flmctions and various sophisticated heuristics. The approach proposed in this paper is a major extension of the existing ones [3,4]. It is based on supervised machine learning algorithms and it uses a trapezoidal membership function to estimate the forest fire risk for each area of interest and for the first time fuzzy expected intervals (F.E.I.) are produced for the burned areas (Ha) of the forest departments. The use of F.E.I. has been proposed by Iliadis [4] but it is the first time that it is materialized and applied here. 4. T H E S E M I T R A P E Z O I D A L M E M B E R S H I P F U N C T I O N Usually, the human reasoning is very approximate. Our statements depend on the contents and we describe our physical and spiritual world in rather vague terms. Imprecisely defined "classes" are an important part of human thinking [10]. It would be essential to understand that the term "forest fire risky area" is both imprecise and subjective and it is determined by a membership function. Membership functions may have different shapes. The choice of a shape for each particular linguistic variable is both subjective and problem-dependent [10]. Any function #(x)-+ [0,1] describes a membership function associated with some fuzzy set. A trapezoidal membership function is a special case of the following expression [10],