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In 1976, Zimmermann first introduced fuzzy set theory into linear programming problems. He considered linear programming problems with a fuzzy goal and fuzzy constraints. Following the fuzzy decision proposed by Bellman and Zadeh (1970) together with linear membership functions, he proved that there exists an equivalent linear programming problem.

The impact of implicit “Fuzziness” is inevitable due to the subjective assessment made by investors. Human judgment of events may be significantly different based on individuals’ subjective perceptions or personality tendencies for judgment, evaluation and decisions; thus human judgment is often fuzzy. So we will use the fuzzy set theory to describe and eliminate the “fuzziness” that is the subjective assessment made by investors. Due to the fluctuation of the financial market from time to time, in reality, the future state of a system might not be known completely due to lack of information, so investment problems are often uncertain or vague in a number of ways. The traditional probability financial model does not take into consideration the fact that investors often face fuzzy factors. Therefore, the fuzzy set theory may be a useful tool for modeling this kind of imprecise problem. This theory permits the gradual assessment of the membership of elements in relation to a set; this is described with the aid of a membership function valued in the real unit interval (0, 1). The fuzzy set theory allows the representation of uncertainty and inexact information in the form of linguistic variables that have been applied to many areas. The application of the fuzzy set theory to finance research is proposed in this article.

Fuzzy logic and genetic algorithms are well-established computational techniques that have been employed to deal with the problem of classification as this is presented in the context of data mining. Based on Fuzzy Miner which is a recently proposed state-of-the-art fuzzy rule based system for numerical data, in this paper we propose GF-Miner which is a genetic fuzzy classifier that improves Fuzzy Miner firstly by adopting a clustering method for succeeding a more natural fuzzy partitioning of the input space, and secondly by optimizing the resulting fuzzy if-then rules with the use of genetic algorithms. More specifically, the membership functions of the fuzzy partitioning are extracted in an unsupervised way by using the fuzzy c- means clustering algorithm, while the extracted rules are optimized in terms of the volume of the rulebase and the size of each rule, using two appropriately designed genetic algorithms. The efficiency of our approach is demonstrated through an extensive experimental evaluation using the IRIS benchmark dataset.

In this chapter we discuss how fuzzy logic extends the envelop of the main data mining tasks: clustering, classification, regression and association rules. We begin by presenting a formulation of the data mining using fuzzy logic attributes. Then, for each task, we provide a survey of the main algorithms and a detailed description (i.e. pseudo-code) of the most popular algorithms. However this chapter will not profoundly discuss neuro-fuzzy techniques, assuming that there will be a dedicated chapter for this issue.

This paper presents an approach for mining fuzzy Association Rules (ARs) relating the properties of composite items, i.e. items that each feature a number of values derived from a common schema. We partition the values associated to properties into fuzzy sets in order to apply fuzzy Association Rule Mining (ARM). This paper describes the process of deriving the fuzzy sets from the properties associated to composite items and a unique Composite Fuzzy Association Rule Mining (CFARM) algorithm founded on the certainty factor interestingness measure to extract fuzzy association rules. The paper demonstrates the potential of composite fuzzy property ARs, and that a more succinct set of property ARs can be produced using the proposed approach than that generated using a non-fuzzy method.

Basic fuzzy controller (BFC) has been successfully adopted to control greenhouse environment temperature. Variable universal adaptive fuzzy controller (VUAFC) which can contracts or expands the universal of the fuzzy variables of the BFC by the value of response error has been developed to get better control performance. By analyzing the result of experiments, VUAFC improved the performance of the fuzzy controller greatly.

This paper put forward the idea of producing fuzzy rules by genetic algorithms based on Takagi-Surgeon Fuzzy Logic System from the dataset of multidimension climate data and crop water requirements, and establishing the fuzzy model to predict crop water requirements. The forecast model was tested and the result showed that it was an effective way to forecast crop water requirements by fuzzy rules model.

Fuzzy set theory and fuzzy logic provide tools for handling uncertainties in data mining tasks. To design a fuzzy rule-based classification system (fuzzy classifier) with good generalization ability in a high dimensional feature space has been an active research topic for a long time. As a powerful machine learning approach for data mining and pattern recognition problems, support vector machine (SVM) is known to have good generalization ability. More importantly, an SVM can work very well on a high (or even infinite) dimensional feature space. This chapter presents a survey of the connection between fuzzy classifiers and kernel machines. A significant portion of the chapter is built upon material from articles we have written, in particular (Chen and Wang, 2003a,Chen and Wang, 2003b).

The aim of this chapter is to study the stability of multi-objective dynamic programming (MODP) problems with fuzzy parameters in the objective functions and in the constraints. These fuzzy parameters are characterized by fuzzy numbers. For such problems, the concept and notion of the stability set of the first kind in parametric nonlinear programming problems are redefined and analyzed qualitatively under the concept of α-Pareto optimality. An interactive fuzzy decision-making algorithm for the determination of any subset of the parametric space that has the same corresponding α-Pareto optimal solution is proposed. A numerical example is given to illustrate the method developed in the chapter.

An offer in a fuzzy negotiation model is rejected or accepted by acceptability based on fuzzy set theory and membership functions. Since different issues have different effect on negotiators, the combined concession in the multi-issue negotiation for negotiators and negotiation agents and genetic learning mechanism are adopted to update their beliefs about incomplete information. The fuzzy negotiation model with genetic algorithms is more practical than the traditional negotiation model.

This paper attempts to present the new approach to design sufficient number of systematic fuzzy linguistics in matrix form and map the Fuzzy Linguistic Variable Matrix, which contains linguistic terms, into numeric domain using Fuzzy Normal Distribution based on the Parabola-based Membership Function. Existing fuzzy set theory is difficult to design the systematic and sufficient fuzzy linguistics. Due to this reason, in most practice, giving insufficient fuzzy linguistics induces inaccurate calculation whilst giving excessive fuzzy linguistics induces the parameter design problems and calculation performance. This paper presents Fuzzy Linguistic Variable Matrix and Parabola-based Fuzzy Normal Distribution (FND) as preferred framework to address the problem.

Classical fuzzy model computes a crisp response for crisp inputs. This paper presents a method for computing fuzzy model response for fuzzy inputs. The method is based on singleton representation of a fuzzy set and it enables to obtain fuzzy response for fuzzy inputs. The presented method is compared with alternative approaches: Zadeh’s possibilistic method and method based on similarity measure. The validity of the proposed method is illustrated with experimental results (in comparison with extension principle results).

This paper presents an application of Fuzzy Logic Systems (FLS) to control a powder deposition process, which is used for depositing coatings on the surface of products. Four input variables: work piece rotation speed, gun to work piece distance, powder volume pressure and gun current and one output variable, the first-pass transfer efficiency was chosen. Fuzzy rules have been introduced in this study. Results showed that FLS can be easily implemented in a company; and has the capability to increase first-pass transfer efficiency, minimize waste, reduce the time involved in selecting the best combination of application variables for coating jobs and improves savings.

Decision trees are commonly employed as data classifiers in various research fields, but also in real-world application domains. In the fuzzy neural framework, decision trees can offer valuable assistance in determining a proper initial system structure, which means not only feature selection, but also rule extraction and organization. This paper proposes a synergistic model that combines the advantages of a subsethood-product neural fuzzy inference system and a CART algorithm, in order to create a novel architecture and generate fuzzy rules of the form “IF - THEN IF”, where the first “IF” concerns the primary attributes and the second “IF” the secondary attributes of the given dataset as defined by our method. The resulted structure eliminates certain drawbacks of both techniques and produces a compact, comprehensible and efficient rulebase. Experiments in benchmark classification tasks prove that this method does not only reduce computational cost, but it also maintains performance at high levels, offering fast and accurate processing during realtime operations.

A fuzzy logic controller with the fuzzy knowledge base: scaling factors of the input/output variables, membership functions and the rules are optimized by the use of the genetic algorithms, is presented in this work, and its application in the highly nonlinear systems. The fuzzy structure is specified by a combination of the mixed Sugeno ’s and Mamdani’s fuzzy reasoning. The mixed, binary-integer, coding is utilized to construct the chromosomes, which define the set of necessary prevailing parameters for the conception of the desired controller. This new controller stands out by a non standard gain (output scaling factor) which varies linearly with the fuzzy inputs. Under certain conditions, it becomes similar to the conventional PID controller with non-linearly variable coefficients. The results of simulation show, well, the efficiency of the proposed controller.

The paper presents a problem of vessel traffic optimisation with the use of fuzzy linear programming. The Swinoujscie-Szczecin fairway has been taken as a practical example. The L-R representation of fuzzy numbers was used for the model description. The interpretation of fuzzy numbers has been presented. As deviations from the fixed values of constraint coefficients have been allowed, the solutions obtained are more realistic. The approach enables a flexible and safer formulation of an optimisation task. The calculations and their results have been presented and interpreted.

This chapter is intended to deal with a special fuzzy variable (FV) including T-FV and trapezoidal FV applied in a programming, build-up LP [Cao96a] and GP [Cao97a] models with T-FV, and GP with trapezoidal FV, respectively. Because this is a new research direction, we propose one method of variable determination in anticipation of many new research achievements constantly being produced. Some ideas introduced in the chapter will give rise to many more valuable ones.

In this chapter we recount the building of four GP models, encompassing models with an interval-valued and a fuzzy-valued coefficient [Cao93c], a type (·, c) one [Cao95a], and an L-R [Cao94a] and a flat fuzzy one [Cao92a][Cao00] by extending a classical GP. Such models play an important part in application of engineering design and budget, mechanical analysis and economical management, etc.

Having been introduced only a short time ago, and with their initial application only in power engineering and postal services, a fuzzy GP and a multi-objective fuzzy GP have great vitality and potential for function applications.

Solution of real world problems often rely on solutions of mathematical models of empirical phenomena. It is well known that the precision and exactness necessary during the construction and solution of such models are not always true in real situations. The major difficulty encountered by a model builder is to express imprecise notions in a seemingly precise form. Conventional mathematics is not equipped to handle vagueness. As researchers and mathematical model builders continue their efforts to construct intelligent systems they are coming to grip with the issue of uncertainty in human knowledge and reasoning. As new fields of study like general system theory, robotics, artificial intelligence and language theory continue to grow, we are forced to specify imprecise notions and analyze them. In 1965, Zadeh [31] introduced a modification of set theory known as fuzzy set theory to study notions with prescribed vagueness.

This chapter gives an overview of distances between fuzzy numbers and the topology that these metrics induce. The metric structure allows the development of fuzzy analysis and various applications to interpolation, approximation and differential equations.

Starting from an input fuzzy set and an IF-THEN rule, implementations of the Generalized Modus Ponens (GMP) in a fuzzy set theoretical framework allow the derivation of an output fuzzy set. If the GMP is implemented by means of the Compositional Rule of Inference (CRI) in general this derivation involves a lot of computational efforts, and the shape of the resulting membership function often seems quite arbitrary compared to that of the input membership function. In this chapter we present a review of techniques, generating as their output a fuzzy set belonging to a predefined class, that are not afflicted with these disadvantages.

In this Chapter, we will describe a large scale application of fuzzy control, which has been realized in the Czech Republic. It concerns the enterprise, which is a part of the Metallurgic Plant Břidlicna, a.s., an important producer of the high quality aluminium in the Czech Republic. The applications of fuzzy control started in 1995. After good experiences with the first furnace fuzzy control, it was decided to apply it on the other four furnaces one by one, too. At present, the system works in the whole enterprise on all five furnaces. We have used the linguistically oriented fuzzy logic controller, for which it is specific to interpret the IF-THEN rules as linguistically characterized logical implications and the inference is logical deduction based on the formal fuzzy logic in broader sense.

This chapter illustrates how fuzzy if-then rules can be used for pattern classification problems. First we describe a heuristic method for automatically generating fuzzy if-then rules for pattern classification problems from training patterns. The heuristic method uses a simple fuzzy grid for partitioning a pattern space into fuzzy subspaces. A fuzzy if-then rule is generated in each fuzzy subspace. Using the heuristic rule generation method, we examine some basic aspects of fuzzy rule-based classification systems such as the shape of membership functions, the definition of the compatibility grade, and the choice of a fuzzy reasoning method. Next we describe a fuzzy rule selection method for designing compact fuzzy rule-based systems with high classification ability. A small number of fuzzy if-then rules are selected from a large number of candidate rules by a genetic algorithm. Finally we describe two genetics-based machine learning algorithms for designing fuzzy rule-based systems for high-dimensional pattern classification problems. In those methods, fuzzy rule-based systems are evolved by genetic operations such as selection, crossover, and mutation. Simulation results on some well-known data sets are shown for illustrating our approaches to the design of fuzzy rule-based systems.

The elements and principles of approximate reasoning were encountered in Chapter 2. We discussed implication, types of implications, fuzzy rule and, at the end, we explained the transition from the conditional to the concluding part of a rule. In Chapter 4 we arrived from one fuzzy rule to the list of fuzzy rules or fuzzy algorithm, which contains several fuzzy rules. So we are familiar with the foundations of approximate reasoning.

Foreword L.A. Zadeh. Preface. Series Foreword. Contributing Authors. General Introduction D. Dubois, H. Prade. Part I: Fuzzy Sets. 1. Fuzzy Sets: History and Basic Notions D. Dubois, et al. 2. Fuzzy Set-Theoretic Operators and Quantifiers J. Fodor, R.R. Yager. 3. Measurement of Membership Functions: Theoretical and Empirical Work T. Bilgic, I.B. Turksen. Part II: Fuzzy Relations. 4. An Introduction to Fuzzy Relations S. Ovchinnikov. 5. Fuzzy Equivalence Relations: Advanced Material D. Boixader, et al. 6. Analytical Solution Methods for Fuzzy Relational Equations B. De Baets. Part III: Uncertainty. 7. Possibility Theory, Probability and Fuzzy Sets: Misunderstandings, Bridges and Gaps D. Dubois, et al. 8. Measures of Uncertainty and Information G.J. Klir. 9. Quantifying Different Facets of Fuzzy Uncertainty N.R. Pal, J.C. Bezdek. Part IV: Fuzzy Sets on the Real Line. 10. Fuzzy Interval Analysis D. Dubois, et al. 11. Metric Topology of Fuzzy Numbers and Fuzzy Analysis P. Diamond, P. Kloeden. Index.

The basic many-sorted fuzzy predicate calculus {tiBL∀} is presented and used to express and prove logical properties of “fuzzy IF-THEN rules”.

In the previous Chapter we got familiar with the basic concepts of a fuzzy variable and a fuzzy set which relate to the fuzzy value, fuzzy number or term. This brings us to the interesting issue of the connectivity or the interaction between the named variables and values. In this Chapter we will focus our attention on the various connections and relations between fuzzy sets; two given fuzzy sets A and B are somehow connected to produce a new fuzzy set C = A ◊ B. Such operations ◊ that create a new fuzzy set out of two are called fuzzy. The nature of fuzzy operations calls for a bit lengthy explanation — even though we are comfortable with the operations in classical algebra, crisp set theory and classical logic.

For a long time it has been recognized that an exact description of many real life physical situations may be virtually impossible. This is due to the high degree of imprecision involved in real world situations. Zadeh, in his seminal papers [Zadeh, 1965; and 1968], proposed fuzzy set theory as the means for quantifying the inherent fuzziness that is present in ill-posed problems (which by many accounts are the majority of the real life problems in decision making). Fuzziness is a type of imprecision which may be associated with sets in which there is no sharp transition from membership to nonmembership [Bellman and Zadeh, 1970]. Examples of fuzzy sets are classes of objects (entities) characterized by such adjectives as large, small, serious, simple, approximate, etc. [Bellman and Zadeh, 1970].

Automatic knowledge generation for Fuzzy Rule Based Systems comprises two main tasks: rules generation and the definition of the semantics of the linguistic variables applied by the rules. The paper focuses on the problem of semantics’ definition by using the Fuzzy c-means algorithm. Fuzzy c-means is a clustering algorithm widely used but with several drawbacks, such as the dependence of the results on the initialization and the need to predefine the number of clusters to be generated. To solve the first problem different authors have used genetic algorithms, defining the genetic fuzzy c-means (GFCM) clustering algorithms. In addition, the paper will present a GFCM clustering algorithm which also finds the suitable number of clusters.

This chapter presents a review of various interpretations of the fuzzy membership function together with ways of obtaining a membership function. We emphasize that different interpretations of the membership function call for different elicitation methods. We try to make this distinction clear using techniques from measurement theory.

The rough sets theory has been proposed by Z. Pawlak in the early 80’s to deal with inconsistency problems following from information granulation. It operates on an information table composed of a set U of objects (actions) described by a set Q of attributes. Its basic notions are: indiscernibility relation on U, lower and upper approximation of a subset or a partition of U, dependence and reduction of attributes from Q, and decision rules derived from lower approximations and boundaries of subsets identified with decision classes. The original rough sets idea has proved to be particularly useful in the analysis of multiattribute classification problems; however, it was failing when preferential ordering of attributes (criteria) had to be taken into account In order to deal with problems of multicriteria decision making (MCDM), like sorting, choice or ranking, a number of methodological changes to the original rough sets theory were necessary. The main change is the substitution of the indiscernibility relation by a dominance relation (crisp or fuzzy), which permits approximation of ordered sets in multicriteria sorting In order to approximate preference relations in multicriteria choice and ranking problems, another change is necessary: substitution of the information table by a pairwise comparison table, where each row corresponds to a pair of objects described by binary relations on particular criteria. In all those MCDM problems, the new rough set approach ends with a set of decision rules, playing the role of a comprehensive preference model. It is more general than the classic functional or relational model and it is more understandable for the users because of its natural syntax. In order to workout a recommendation in one of the MCDM problems, we propose exploitation procedures of the set of decision rules. Finally, some other recently obtained results are given: rough approximations by means of similarity relations (crisp or fuzzy) and the equivalence of a decision rule preference model with a conjoint measurement model which is neither additive nor transitive.

The main purpose of the fuzzy approach of individual preferences is to introduce some new behaviors inside the standard theory. The existing literature is generally divided in two schools, according to the position authors adopt relatively to the axiom of Independence of Irrelevant Alternatives (IIA).1 Actually, most of the works using fuzzy preferences are based on additive measures of satisfaction (Bazu 1984, Butnariu 1987, Orlovsky 1980, Ovchinnikov & Roubens 1992).

This chapter deals with an important issue pertaining to intelligent information processing systems, that of managing information coming from several sources. Possibility theory and the body of aggregation operations from fuzzy set theory provide some tools to address this problem. The fusion of imprecise information is carefully distinguished from the estimation problem. The approach to fusion is set-theoretic and the choice of conjunctive versus disjunctive fusion modes depends on assumptions on whether all sources are reliable or not. Quantified, prioritized and weighted and fusion rules are described. Fuzzy extensions of estimation processes are also discussed. The approach, based on conflict analysis, applies to sensor fusion, aggregation of expert opinions as well as the merging of databases.

The problem of adaptive fuzzy control for a class of nonlinear systems is studied in this paper. Based on the principle of variable structure control and the approximation capability of the second-type fuzzy systems, a design scheme of an adaptive fuzzy controller is proposed. The control architecture employs fuzzy systems in which the unknown parameters are adjusted on-line to adaptively compensate for plant uncertainties. By using the Lyapunov method, the closed-loop fuzzy control system is proved to be globally stable, with tracking errors converging to a neighborhood of zero. The simulation results are given to demonstrate the effectiveness of the approach.

Rather early in the (short) history of fuzzy sets it became clear that there is an intimate relationship between fuzzy set theory and many-valued logic. In the early days of fuzzy sets the main connection was given by fuzzy logic — in the understanding of this notion in those days: and this was as switching logic within a multiple-valued setting.

This paper deals with the approximation of the functions minimum and maximum. Both standard and fuzzy algorithms are discussed and practical applications in trajectory planning and drive control are presented.

An association between qualities or abilities for certain jobs and candidates is often established, and this should come as no surprise, by means of the use of semantic operators such as “and”, “and/or”, “or exclusive”, “no”, etc. Let us take a look at some of them and, in the event of multiple experts participating, we are going to consider the theory of expertons.

While the fuzzy ER/EER model presented in Part II describes the fuzziness inherent in the conceptual aspects of data, the fuzzy relational data model to be discussed in Part III will deal with the fuzziness inherent in the contents and integrity constraints of data. In this chapter (Chapter 6), how, where and to what extent fuzziness could be incorporated into the classical relational model (Codd, 1970) is described, followed by a detailed investigation on the treatments of closeness and redundancy of fuzzy data. In Chapter 7, a kind of integrity constraints that reflect the association between imprecise attribute values, namely, fuzzy functional dependency (FFD), is represented in its general and specific forms. Furthermore, the extended keys and related integrity rules are introduced. Chapter 8 addresses the issue of FFD inference and presents a FFD axiomatic system that is both sound and complete.