MOGUL: A methodology to obtain genetic fuzzy rule-based systems under the iterative rule learning approach

The main aim of this paper is to present MOGUL, a Methodology to Obtain Genetic fuzzy rule-based systems Under the iterative rule Learning approach. MOGUL will consist of some design guidelines that allow us to obtain different genetic fuzzy rule-based systems, i.e., evolutionary algorithm-based processes to automatically design fuzzy rule-based systems by learning and/or tuning the fuzzy rule base, following the same generic structure and able to cope with problems of a different nature. A specific evolutionary learning process obtained from the paradigm proposed to design unconstrained approximate Mamdani-type fuzzy rule-based systems will be introduced, and its accuracy in the solving of a real-world electrical engineering problem will be analyzed. ©1999 John Wiley & Sons, Inc.     

Fuzzy classifier system architectures for mobile robotics: An experimental comparison

We present an experimental comparison between two approaches to optimization of the rules for a fuzzy controller. More specifically, the problem is autonomous acquisition of an “investigative” obstacle avoidance competency for a mobile robot. We report on results from investigating two alternative approaches to the use of a Learning Classifier System (LCS) to optimize the fuzzy rule base. One approach operates at the level of whole rule bases, the “Pittsburgh” LCS. The other approach operates at the level of individual rules, the “Michigan” LCS. In this work, both of these Fuzzy Classifier Systems were designed to operate only on the rules of fuzzy controllers, with predefined fuzzy membership functions. There are two main results from this work. First, both approaches were capable of producing fuzzy controllers with subtle interactions between rules leading to competencies exceeding that of the hand‐coded fuzzy controller presented in this article. Second, the Michigan approach suffered more seriously than the Pittsburgh approach from the well‐known LCS “cooperation/competition” problem, which is accentuated here by the structural combination of Evolutionary Computation and a fuzzy system. This problem was alleviated a little by the combination of a clustered subpopulation niche system and a fitness‐sharing scheme applied to the Michigan approach, but still remains. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 993–1019, 2007.     

Relevance in systems having a fuzzy‐set‐based semantics

Future automated question answering systems will typically involve the use of local knowledge available on the users' systems as well as knowledge retrieved from the Web. The determination of what information we should seek out on the Web must be directed by its potential value or relevance to our objective in the light of what knowledge is already available. Here we begin to provide a formal quantification of the concept of relevance and related ideas for systems that use fuzzy‐set‐based representations to provide the underlying semantics. We also introduce the idea of ease of extraction to quantify the ability of extracting relevant information from complex relationships. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 385–396, 2007.     

A two-stage evolutionary process for designing TSK fuzzy rule-basedsystems

Nowadays, fuzzy rule-based systems are successfully applied to many different real-world problems. Unfortunately, relatively few well-structured methodologies exist for designing and, in many cases, human experts are not able to express the knowledge needed to solve the problem in the form of fuzzy rules. Takagi-Sugeno-Kang (TSK) fuzzy rule-based systems were enunciated in order to solve this design problem because they are usually identified using numerical data. In this paper we present a two-stage evolutionary process for designing TSK fuzzy rule-based systems from examples combining a generation stage based on a (mu, lambda)-evolution strategy, in which the fuzzy rules with different consequents compete among themselves to form part of a preliminary knowledge base, and a refinement stage in which both the antecedent and consequent parts of the fuzzy rules in this previous knowledge base are adapted by a hybrid evolutionary process composed of a genetic algorithm and an evolution strategy to obtain the final Knowledge base whose rules cooperate in the best possible way. Some aspects make this process different from others proposed until now: the design problem is addressed in two different stages, the use of an angular coding of the consequent parameters that allows us to search across the whole space of possible solutions, and the use of the available knowledge about the system under identification to generate the initial populations of the Evolutionary Algorithms that causes the search process to obtain good solutions more quickly. The performance of the method proposed is shown by solving two different problems: the fuzzy modeling of some three-dimensional surfaces and the computing of the maintenance costs of electrical medium line in Spanish towns. Results obtained are compared with other kind of techniques, evolutionary learning processes to design TSK and Mamdani-type fuzzy rule-based systems in the first case, and classical regression and neural modeling in the second.     

Increasing fuzzy rules cooperation based on evolutionary adaptive inference systems

This article presents a study on the use of parametrized operators in the Inference System of linguistic fuzzy systems adapted by evolutionary algorithms, for achieving better cooperation among fuzzy rules. This approach produces a kind of rule cooperation by means of the inference system, increasing the accuracy of the fuzzy system without losing its interpretability. We study the different alternatives for introducing parameters in the Inference System and analyze their interpretation and how they affect the rest of the components of the fuzzy system. We take into account three applications in order to analyze their accuracy in practice. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 1035–1064, 2007.     

Differences between t‐norms in fuzzy control

The application of conjunctive aggregation functions in fuzzy control systems with n inputs is discussed, and the effect of the choice of a continuous t‐norm in the inference phase for Takagi–Sugeno–Kang (TSK) systems is computed. A continuous t‐norm modeling AND connective in antecedent part of fuzzy rules can be reduced just to strict or nilpotent t‐norm. The isomorphism of strict (nilpotent) t‐norms enables simpler fitting of TSK fuzzy system parameters and reduces the computational complexity. Similar principle can also be used in the case of some noncommutative conjunctive aggregation functions modeling AND connective. The effect of the choice of a continuous t‐norm is then evaluated on well‐known case studies in fuzzy control, the Sinc function, and the urban traffic noise control system.     

Genetic fuzzy systems to evolve interaction strategies in multiagent systems

This article suggests an evolutionary approach to designing interaction strategies for multiagent systems, focusing on strategies modeled as fuzzy rule‐based systems. The aim is to learn models evolving database and rule bases to improve agent performance when playing in a competitive environment. In competitive situations, data for learning and tuning are rare, and rule bases must jointly evolve with the databases. We introduce an evolutionary algorithm whose operators use variable length chromosomes, a hierarchical relationship among individuals through fitness, and a scheme that successively explores and exploits the search space along generations. Evolution of interaction strategies uncovers unknown and unexpected agent behaviors and allows a richer analysis of negotiation mechanisms and their role as a coordination protocol. An application concerning an electricity market illustrates the effectiveness of the approach. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 971–991, 2007.     

Intelligent evolutionary design: A new approach to optimizing complex engineering systems and its application to designing heat exchangers

A new method for optimizing complex engineering designs is presented that is based on the Learnable Evolution Model (LEM), a recently developed form of non‐Darwinian evolutionary computation. Unlike conventional Darwinian‐type methods that execute an unguided evolutionary process, the proposed method, called LEMd, guides the evolutionary design process using a combination of two methods, one involving computational intelligence and the other involving encoded expert knowledge. Specifically, LEMd integrates two modes of operation, Learning Mode and Probing Mode. Learning Mode applies a machine learning program to create new designs through hypothesis generation and instantiation, whereas Probing Mode creates them by applying expert‐suggested design modification operators tailored to the specific design problem. The LEMd method has been used to implement two initial systems, ISHED1 and ISCOD1, specialized for the optimization of evaporators and condensers in cooling systems, respectively. The designs produced by these systems matched or exceeded in performance the best designs developed by human experts. These promising results and the generality of the presented method suggest that LEMd offers a powerful new tool for optimizing complex engineering systems. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 1217–1248, 2006.     

Learning cooperative linguistic fuzzy rules using the best–worst ant system algorithm

Within the field of linguistic fuzzy modeling with fuzzy rule‐based systems, the automatic derivation of the linguistic fuzzy rules from numerical data is an important task. In the last few years, a large number of contributions based on techniques such as neural networks and genetic algorithms have been proposed to face this problem. In this article, we introduce a novel approach to the fuzzy rule learning problem with ant colony optimization (ACO) algorithms. To do so, this learning task is formulated as a combinatorial optimization problem. Our learning process is based on the COR methodology proposed in previous works, which provides a search space that allows us to obtain fuzzy models with a good interpretability–accuracy trade‐off. A specific ACO‐based algorithm, the Best–Worst Ant System, is used for this purpose due to the good performance shown when solving other optimization problems. We analyze the behavior of the proposed method and compare it to other learning methods and search techniques when solving two real‐world applications. The obtained results lead us to remark the good performance of our proposal in terms of interpretability, accuracy, and efficiency. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 433–452, 2005.     

Solution of fuzzy matrix games: An application of the extension principle

Conventional game theory is concerned with how rational individuals make decisions when they are faced with known payoffs. This article develops a solution method for the two‐person zero‐sum game where the payoffs are only approximately known and can be represented by fuzzy numbers. Because the payoffs are fuzzy, the value of the game is fuzzy as well. Based on the extension principle, a pair of two‐level mathematical programs is formulated to obtain the upper bound and lower bound of the value of the game at possibility level α. By applying a dual formulation and a variable substitution technique, the pair of two‐level mathematical programs is transformed to a pair of ordinary one‐level linear programs so they can be manipulated. From different values of α, the membership function of the fuzzy value of the game is constructed. It is shown that the two players have the same fuzzy value of the game. An example illustrates the whole idea of a fuzzy matrix game. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 891–903, 2007.     

Incremental Evolutionary Design of TSK Fuzzy Controllers

This paper proposes a novel method for the incremental design and optimization of first order Tagaki-Sugeno-Kang (TSK) fuzzy controllers by means of an evolutionary algorithm. Starting with a single linear control law, the controller structure is gradually refined during the evolution. Structural augmentation is intertwined with evolutionary adaptation of the additional parameters with the objective not only to improve the control performance but also to maximize the stability region of the nonlinear system. From the viewpoint of optimization the proposed method follows a divide-and-conquer approach. Additional rules and their parameters are introduced into the controller structure in a neutral fashion, such that the adaptations of the less complex controller in the previous stage are initially preserved. The proposed scheme is evaluated at the task of TSK fuzzy controller design for the upswing and stabilization of a rotational inverted pendulum. In the first case, the objective is a time optimal controller that upswings the pendulum in to the upper equilibrium point in shortest time. The stabilizing controller is designed as a state optimal controller. In a second application the optimization method is applied to the design of a fuzzy controller for vision-based mobile robot navigation. The results demonstrate that the incremental scheme generates solutions that are similar in control performance to pure parameter optimization of only the gains of a TSK system. Even more important, whereas direct optimization of control systems with more than 35 rules fails to identify a stabilizing control law, the incremental scheme optimizes fuzzy state-space partitions and gains for hundreds of rules.     

Fuzzy multi‐attribute cost–benefit analysis of e‐services

E‐service evaluation is a complex problem in which many qualitative attributes must be considered. These kinds of attributes make the evaluation process hard and vague. Cost–benefit analyses applied to various areas are usually based on the data under certainty or risk. In case of uncertain, vague, and/or linguistic data, the fuzzy set theory can be used to handle the analysis. In this article, after the evaluation attributes of e‐services and the fuzzy multi‐attribute decision‐making methods are introduced, a fuzzy hierarchical TOPSIS model is developed and applied to an e‐service provider selection problem with some sensitivity analyses. The developed model is a useful tool for the companies that prefer outsourcing for e‐activities. It is shown that service systems can be effectively evaluated by the proposed method. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 547–565, 2007.     

A fuzzy modeling method via Enhanced Objective Cluster Analysis for designing TSK model

This paper proposes a fuzzy modeling method via Enhanced Objective Cluster Analysis to obtain the compact and robust approximate TSK fuzzy model. In our approach, the Objective Cluster Analysis algorithm is introduced. In order to obtain more compact and more robust fuzzy rule prototypes, this algorithm is enhanced by introducing the Relative Dissimilarity Measure and the new consistency criterion to represent the similarity degree between the clusters. By these additional criteria, the redundant clusters caused by iterations are avoided; the subjective influence from human judgment for clustering is weakened. Moreover the clustering results including the number of clusters and the cluster centers are considered as the initial condition of the premise parameters identification. Thus the traditional iteration modeling procedure for determining the number of rules and identifying parameters is changed into one-off modeling, which significantly reduces the burden of computation. Furthermore the decomposition errors and the approximation errors resulted from premise parameters identification by Fuzzy c-Means clustering are decreased. For the consequence parameters identification, the Stable Kalman Filter algorithm is adopted. The performance of the proposed modeling method is evaluated by the example of Box–Jenkins gas furnace. The simulation results demonstrate the power of our model.     

Robust TSK fuzzy modeling for function approximation with outliers

The Takagi-Sugeno-Kang (TSK) type of fuzzy models has attracted a great attention of the fuzzy modeling community due to their good performance in various applications. Most approaches for modeling TSK fuzzy rules define their fuzzy subspaces based on the idea of training data being close enough instead of having similar functions. Besides, training data sets algorithms often contain outliers, which seriously affect least-square error minimization clustering and learning algorithms. A robust TSK fuzzy modeling approach is presented. In the approach, a clustering algorithm termed as robust fuzzy regression agglomeration (RFRA) is proposed to define fuzzy subspaces in a fuzzy regression manner with robust capability against outliers. To obtain a more precision model, a robust fine-tuning algorithm is then employed. Various examples are used to verify the effectiveness of the proposed approach. From the simulation results, the proposed robust TSK fuzzy modeling indeed showed superior performance over other approaches     

Integration on finite sets

Various types of integrals with respect to signed fuzzy measures on finite sets with cardinality n can be presented as corresponding rules for partitioning the integrand. The partition can be expressed as an n‐dimensional vector, whereas the signed fuzzy measure is also an n‐dimensional vector. Thus, the integration value is the inner product of these two vectors. Two pairs of extremes, the Lebesgue‐like integral versus the Choquet integral and the upper integral versus the lower integral, are discussed in detail. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 1073–1092, 2006.     

Online identification of evolving Takagi–Sugeno–Kang fuzzy models for crane systems

This paper suggests new evolving Takagi–Sugeno–Kang (TSK) fuzzy models dedicated to crane systems. A set of evolving TSK fuzzy models with different numbers of inputs are derived by the novel relatively simple and transparent implementation of an online identification algorithm. An input selection algorithm to guide modeling is proposed on the basis of ranking the inputs according to their important factors after the first step of the online identification algorithm. The online identification algorithm offers rule bases and parameters which continuously evolve by adding new rules with more summarization power and by modifying existing rules and parameters. The potentials of new data points are used with this regard. The algorithm is applied in the framework of the pendulum–crane system laboratory equipment. The evolving TSK fuzzy models are tested against the experimental data and a comparison with other TSK fuzzy models and modeling approaches is carried out. The comparison points out that the proposed evolving TSK fuzzy models are simple and consistent with both training data and testing data and that these models outperform other TSK fuzzy models.     

A hybrid fuzzy cognitive model based on weighted OWA operators and single‐antecedent rules

Conventional fuzzy cognitive maps (FCMs) can only represent monotonic or symmetric causal relationships and cannot simulate the AND/OR combinations of the antecedent nodes. The rule‐based fuzzy cognitive maps (RBFCMs) usually suffer from the well‐known combinatorial rule explosion problem. A hybrid fuzzy cognitive model based on weighted OWA operators and single‐antecedent rules is proposed to eliminate the drawbacks of the existing FCM models. Hybrid fuzzy cognitive maps (HFCMs) represent the causal relationships with single‐antecedent fuzzy rules and handle the various AND/OR relationships among the antecedent nodes with weighted OWA aggregation operators. Compared with conventional FCMs, HFCMs have more powerful cognitive capability. Compared with RBFCMs, HFCMs reduce the scale and complexity of the rule bases significantly and have better representation and inference performance. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 1189–1196, 2007.     

A fuzzy logic approach to experience‐based reasoning

Experience‐based reasoning (EBR) is a reasoning paradigm that has been used in almost every human activity such as business, military missions, and teaching activities since early human history. However, EBR has not been seriously studied from either a logical or mathematical viewpoint, although case‐based reasoning (CBR) researchers have paid attention to EBR to some extent. This article will attempt to fill this gap by providing a unified fuzzy logic‐based treatment of EBR. More specifically, this article first reviews the logical approach to EBR, in which eight different rules of inference for EBR are discussed. Then the article proposes fuzzy logic‐based models to these eight different rules of inference that constitute the fundamentals for all EBR paradigms from a fuzzy logic viewpoint, and therefore will form a theoretical foundation for EBR. The proposed approach will facilitate research and development of EBR, fuzzy systems, intelligent systems, knowledge management, and experience management. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 867–889, 2007.     

Relevancy transformation operators: Construction methods

This article deals with the construction of relevancy transformation (RET) operators for fuzzy systems. The notion of pseudo‐duality is introduced to obtain new RET operators, and t‐norms, t‐conorms, nullnorms, and uninorms are used in different ways for the same purpose. Finally, several other methods to construct new RET operators from old ones are pointed out. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 155–171, 2006.