Fuzzy Set Theory

Abstract: The purpose of this paper is to introduce fuzzy set theory which has a number of properties that makes it suitable for formalizing the uncertain information in dealing with systems comprising a very large number of interacting elements or involving a large number of variables in their decision trees. Basically Fuzzy set theory is affecting every aspect of life; social life is one of them. Fuzzy set theory has proved as mile stone in its development. Selection of one of the solutions for social life’s problems and selection of possible way for the development of social life is become possible only due to ranking of fuzzy set theory.

Application of Fuzzy Theory Theory for Diagnosing DiseasesThe development of the application of the fuzzy set theory is widely used in the health sector to diagnose various diseases. Harmful diseases usually require a quick and accurate diagnosis in treatment. But often the diagnosis of the disease is constrained because expert doctors have weaknesses in services such as limited working hours and have to wait in line to do the service. These obstacles can be fatal for sufferers of dangerous diseases. To overcome this, an expert system has been used as a result of the development of the fuzzy set theory. Expert systems are formed through the web, so they can be more easily accessed. Expert systems are built to facilitate the process of diagnosing diseases, so that diagnostic results can be obtained more quickly and precisely. Here are some application results from the use of Fuzzy set theory to diagnose diseases. The development of this expert system can use certain methods in the fuzzy set theory. Some diseases that can be diagnosed using an expert system include liver disease, diabetes mellitus, dengue fever (DHF) and typhoid fever, cardiovascular disease, umbilical cord blood analysis, heart disease, thyroid disease, dental and oral diseases [10][11][12][13][14][15][16][17][18]. Following are the results of studies on the use of expert systems as a diagnostic tool for disease.

Implication functions in interval-valued fuzzy set theory Glad Deschrijver Abstract Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed in- terval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results per- taining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a par- ticular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory.

Keywords: Fuzzy Sets; Prediction; Rainfall; Water Resources 1. Introduction Application of fuzzy set theory has rapidly increased with establishing its utility in numerous areas of the sci- entific world. Any system consisting of vague and am- biguous input variables may contribute to an ultimate effect. In fuzzy logic approach, it is possible to express crisp intervals in terms of linguistic subsets by fuzzy ex- pressions like low, medium, high, good, moderate, poor etc. Each of these expressions represents the sub-range of the entire variability of the variables concerned [1-7].

There is need for a way or means to transform vague ordinal ratings to more appreciable and precise numerical estimates. The paper transforms the ordinal performance ratings of some LAN performance strategies to numerical ratings using Fuzzy Set Theory. Keywords: Fuzzy Set Theory, L A N Performance, P erformance Strategies, Transformation.

KEYWORDS: risk management; risk assessment; fuzzy set theory; reliability. Abstract: Safety and reliability are essential issues in modern sciences. Modern technical systems should meet technical, safety and environmental protection requirements. The risk can appear as personal injury or death, mission degradation, property technical damage or destruction. The risk is a measure of harm or loss associated with the human activity. It is the combination of the probability and the consequence of a specified hazard being realized. For decision making we need the opinions of other sciences, sometimes we have to consider moral questions. To make reliable decision, the risk of the given system or process should be known correctly. Fuzzy set theory is a new mathematical tool to model inaccuracy and uncertainty of the real world and human thinking.

One of the earliest works on medical diagnosis allowing fuzziness was proposed by Sanchez [4]. Thereafter some researchers have contributed in this area successfully. In this concern, Esogbue and Elder proposed fuzzy mathematical models [5,6,7]. An application of fuzzy set theory was given by Adlassing [8]. Chen [9] presented weighted fuzzy algorithms and Belacel proposed PROCFTN methodology to handle uncertainties [10]. In the same direction Yao and Yao used the concept of fuzzy number and compositional rule of inference to make decisions [11]. Roychowdhury et al. gave diagnostic decision model using a GA- fuzzy approach [12]. Roy and Biswas [13] defined compositions for interval valued fuzzy sets and used them for the same. Type-2 fuzzy set is a generalization of type-1 fuzzy set as well as of interval valued fuzzy sets. Quite recently, Own [14] proposed a switching function and type-2 fuzzy similarity and presented applications of these in medical diagnosis and pattern recognition. Pandey et al. [15] proposed diagnostic decision model using vague sets. Celik and Yamak [16] applied fuzzy soft set theory to medical diagnosis. They used the concept of fuzzy arithmetic operations through Sanchez’s approach to make the decisions. Elizabeth and Sujatha [17] used interval valued fuzzy number matrices in medical diagnosis present paper extends the Sanchez’s approach of medical diagnosis in the type-2 fuzzy atmosphere.This work uses the concept of type-2 fuzzy relations and is different from the work given by Own.

Sa`etak Planiranje rada skidera (SMA) pomo}u neizrazite teorije U~inkovito planiranje rada jedno je od glavnih zadataka u gospodarenju {umama, posebice kada postoje}a sredstva za rad nisu jednako dostupna u svim radnim jedinicama. Smanjenje ukupnih tro{kova rada svakako je bitna sastavnica gospodarenja {umama posebice u radno nedostupnim ili te`e dostupnim podru~jima. S obzirom na visoke tro{kove rada specijaliziranih {umskih vozila, u ovom slu~aju skidera, potrebno je smanjiti jedini~ne tro{kove pridobivanja drva odgovaraju}im planiranjem rada skidera te njihovim smje{tanjem u prostoru (Skidding Machines Allocation – SMA). Neizrazita teorija (fuzzy set theory) i sustav za potporu odlu~ivanja primijenjeni su za obradu nestalnih varijabli te raspona podataka kao {to su obujam obloga drva i terenski uvjeti radili{ta, tj.

project network schedule (Lorterapong and Moselhi, 1996); reliability assessment (Booker and Singpurwalla, 2002); and range cost estimating (Shaheen et al., 2007). As to schedule compression models, fuzzy set theory is used along with simulation and/or genetic algorithm (e.g., Zheng and Thomas Ng, 2005; Eshtehardian, 2008a; Eshtehardian, 2008 b). The later methods consider only cost in the schedule compression process. This paper presents a multi-objective method to circumvent the above stated limitations in schedule compression of construction projects. The developed method accounts for cost, contractors’ judgment and for the uncertainties associated with the direct cost of crashing activity durations. The use of FST, as presented in this paper, is particularly suited for the problem at hand due to two main reasons. Firstly, crashing durations of project activities is frequently carried out during construction subjected to the unique conditions of each project environment. As such, there is no historical data for each and every activity being considered for crashing. This is contrary to cost planning of the original projects’ scope of work prior to construction; where historical data may exist to support the use of probabilistic methods. Secondly, FST facilitates the direct utilization of expert knowledge that applies to the unique conditions of each project at hand through the use of membership functions that best suit these unique conditions.

Keywords: - Linguistic Values, Fuzzy logic. 1. INTRODUCTION The Fuzzy set theory has been studying nearly the past 55 years. Most of an early interest in fuzzy set theory pertained to representing uncertainty in human cognitive processes according Zadeh (1965)). Fuzzy set theory is now used to minimize problems in engineering, business, medical and related health sciences, and the natural sciences. In an effort to gain a better understanding of the use of fuzzy set theory in construction industry management research and to provide a basis for future research, a literature review of fuzzy set theory in construction industry management has been conducted.

Abstract. Fault tree analysis (FTA) has been modified in different ways to make it capable of performing quantitative and qualitative safety analysis with temporal gates, thereby overcoming its limitation in capturing sequential failure behaviour. However, for many systems, it is often very difficult to have exact failure rates of components due to increased complexity of systems, scarcity of necessary statistical data etc. To overcome this problem, this paper presents a methodology based on fuzzy set theory to quantify temporal fault trees. This makes the imprecision in available failure data more explicit and helps to obtain a range of most probable values for the top event probability.

PG Scholar, Dept. of EEE, Mar Athanasius College of Engineering, Kothamangalam, Kerala, India 1 Professor, Dept. of EEE, Mar Athanasius College of Engineering, Kothamangalam, Kerala, India 2,3 ABSTRACT: Fuzzy logic controller (FLC) is the most widely used applications of fuzzy set theory. A Fuzzy Logic Controlled (FLC) buck-boost converter for solar energy-battery systems is analyzed. Solar Photovoltaic system has been increasingly playing a significant role in new energy systems. PV system has some disadvantages including low conversion efficiency and inconsistent values of output voltage due to irregular solar power which is caused by weather changes and shading effects. To address these disadvantages, DC-DC converters have been used to control PV output voltage and output power. Fuzzy logic controllers are non linear controllers which do not require exact information of mathematical model. Hybridization of the two controllers can be done to use the positive sides of both controllers to obtain better performance. The comparison of result obtained from fuzzy logic controller, fuzzy - PI hybrid controller, fuzzy - PID hybrid controller for DC-DC converter shows the benefit of the hybrid algorithm in terms of transient response. General design of a fuzzy logic controller based on MATLAB/Simulink is performed. The control system has been developed, analyzed, and validated by simulation study.

Copyright © 2013 Mahbub Hasan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT The paper presents occurrence of rainfall using principles of fuzzy set theory and principles of reliability analysis. Both the abstract and the rest of the paper are discussed from these two points of view. First, a fuzzy inference model for pre- dicting rainfall using scan data from the USDA Soil Climate Analysis Network Station at Alabama Agricultural and Mechanical University (AAMU) campus for the year 2004 is presented. The model further reflects how an expert would perceive weather conditions and apply this knowledge before inferring a rainfall. Fuzzy variables were selected based on judging patterns in individual monthly graphs for 2003 and 2004 and the influence of different variables that caused rainfall. A decrease in temperature (TP) and an increase in wind speed (WS) when compared between the ith and (i − 1)th day were found to have a positive relation with a rainfall (RF) occurrence in most cases. Therefore, TP and WS were used in the antecedent part of the production rules to predict rainfall (RF). Results of the model showed better perform- ance when threshold values for 1) Relative Humidity (RH) of ith day; 2) Humidity Increase (HI) between the ith and (i

“slow car” or the actions of defining the weather, choosing clothes, preferring a car have nonrandom type of uncertainties and they cannot be clarified by occurrence or tests. Since the decision of complex daily issues are generally related to human decisions and the general probabilistic theories are not satisfactorily explain the uncertainty resulting from human subjectivity, a new powerful tool was proposed by Lutfi Asker Zadeh (1965) called Fuzzy Set Theory. Fuzzy set theory is a mathematical theory which is used for modeling the imprecise, ambiguous, vagueness nature of complex systems when there is not enough of information about the problem. The idea behind the fuzzy sets is related to fuzzy logic. In classical logic, the world is defined by binary extremes such as zero or one, black or white, good or bad, true or false, big or small, short or tall, guilty not guilty etc.

Lee et al. (1990) introduce the application of fuzzy set theory to lot-sizing in material requirements planning. A modified version of the part-period balancing algorithm is presented. Uncertainty in demand is modeled using triangular fuzzy numbers. A numerical example for an eight period horizon and four sets of demand data is used to demonstrate the algorithm. The authors identify two advantages in using fuzzy numbers and membership functions to model demand. First, fuzzy set theory allows both the uncertain demand and the subjective judgment of the decision maker to be incorporated into the lot-sizing decision. Second, fuzzy part-period balancing provides a richer source of data for the decision maker to use in terms of the membership values associated with the lot-sizes and costs. Lee et al. (1991) extend their 1990 treatment of the MRP lotsizing to include fuzzy modifications to the Silver-Meal, Wagner-Whitin, and part-period balancing algorithms. The authors argue that when demands of the master schedule are truly fuzzy, demand should be modeled using membership functions. The performance of the three fuzzy lot-sizing algorithms is compared based on nine sample problems.

to poverty study is required involving a number of non-monetary indicators also of living conditions. Thus poverty, which is a manifestation of insufficient well-being, and hence development, which is a manifestation of sufficient well-being, are multidimensional phenomena and should therefore involve both monetary and non-monetary dimensions. It is true one can argue that with a higher income he may be able to improve some of his non-monetary dimensions of well-being. But there are things that his money cannot buy, just because they are simply not available; for example, a public good like flood control or malaria prevention program or a school or a hospital in an underdeveloped country. Hence it is essential that the monetary factor be supplemented with other non-monetary attributes, such as housing, literacy, life expectancy at birth, nutritional status, provision of public goods etc. This requirement for multi-dimensionality in turn also implies that information on a single attribute is not sufficient in the face of ambiguity in the concept of poverty/development and more information from diverse dimension is essential. Now, according to Amartya Sen, “If a concept has some basic ambiguity (as ideas of what constitutes ‘inequality’ tend to have), then a precise representation of that ambiguous concept must preserve that ambiguity, rather than try to remove it through some arbitrarily completed ordering. This issue is quite central to the need for descriptive accuracy in inequality assessment, which has to be distinguished from fully ranked, unambiguous assertions (irrespective of the ambiguities in the underlying concept).” (Sen 1997: 121). It is here that the fuzzy set theory of Zadeh (1965) plays a fundamental role in tackling problems arising from ambiguity. Thus, it goes without saying that the poverty status of a person or a nation is intrinsically fuzzy, as the concept of poverty itself is vague. This in turn justifies a fuzzy set approach to poverty measurement sufficiently.

Keywords: Fuzzy set theory, Fuzzy similarity relation, Fuzzy equivalence relation, Performance variation. Abstract. Based on the fuzzy set theory, a method for bearing vibration performance variation analysis is proposed. Firstly, the original vibration data of the bearing is grouped and the fuzzy similarity relation matrix is constructed by the linear mapping formula. The fuzzy similarity relation matrix in the engineering practice is transformed into the fuzzy equivalence matrix of the spatial vector by the transfer closure method. Then, The equivalence coefficient of the fuzzy equivalence relation matrix is compared with the set threshold to diagnose the variation of bearing vibration performance.

consiguiente, es imposible desarraigarlo mediante le intelección .Detec.- ter esto exige una metodología rieurosa, coherente y persistente,ca- paz de perseguir al dualis[r]

z Based on discrete shape representation, we estimate features of a real continuous imaged shape z Estimation of area, perimeter, compactness, moments of higher order, signature of a shape all exhibit higher precision if estimated from the fuzzy, instead of the crisp representation of a shape

6. Conclusion Both rough sets and IF sets capture facets of imprecision, a natural extension is to combine the two set theories into a new hybrid one. In this paper, we have introduced two classes of IF approximation operators and investigated their properties. We have defined rough IF sets and IF rough sets which, respectively, resulted from the approximations of AIF sets w.r.t. a crisp approximation space and a fuzzy approximation space. Properties of rough IF approximation operators and IF rough approximation operators corresponding to special approximation spaces have been discussed. An axiomatic approach has been introduced to characterize the intuitionistic fuzzy approximation operators. By this way, we have solved the problem of finding assumptions permitting given AIF set-theoretic operators to represent upper and lower approximations derived from special crisp or fuzzy relations, that is, we have proved that axiom sets of rough IF approximation operators (and IF rough approximation operators, respectively) guarantee the existence of certain types of crisp relations (and fuzzy relations, respectively) producing the same operators.