Most studies on tuning of fuzzy inference are concerned with numerical inputs and outputs only, and very few research has been done on tuning of fuzzy inference with fuzzy inputs and outputs. Moreover, in many cases the objects of tuning are fuzzy predicates only, apart from the other factors intervening in fuzzy inference. In this paper we propose a method to tune the fuzzy inference when teaching signals are given as fuzzy sets. The objects of tuning are the parameters of aggregation operators, implication functions and combination functions, which are important factors of the fuzzy inference method, as well as the parameters of fuzzy predicates. In the proposed method, we adjust the value of the parameters by the gradient descent method, using the network representing the calculation process of the fuzzy inference.

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We have fuzzy mathematical programming which is used in making a plan under uncertainty.A lot of methods for solving fuzzy multiple objective programming problems have been proposed in the recent decade. But almost all methods can not deal with some of relationshipbetween uncertain coefficients.So, Ohta et all proposed a method for fuzzy multiple objective linear programming problems with relationship between coefficients. This method focuses on common point betweentwo-stage programming problem and goal programming problem. This method expresses uncertainty as scenarios composed of fuzzy numbers in order to deal with some of relationship between uncertain coefficients. And regrets of objective functions are measuredby indexes which are caught optimistic and pessimistic viewpoints. This method can get the ・solutions considering the balance of all scenarios, all goals, optimistic and pessimistic viewpoints. But this method can not apply in making a plan which includes ratio goals such ・as return on sales, because this method deeals with only linear objective functions. In this paper, we extend the method which is proposed by Ohta et al, propose a method for fuzzy multiple objective linear fractional programming problems with relationship between coefficients. And we propose new method for standardizing indexes that measure regrets in order to consider the balance between objective functions based on aspiration revel which is set by decision maker.

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A method to get Propagation Diagram for friendship is proposed using the sociomatrix obtained from the sociometric test based on interval scale measurement. All names of pupils in the class are written in the sheet. The diagram is a kind of sociogram and is gotten by using the fuzzy structural modeling. The diagram graphically represents group relationships and propagation of friendhips among pupils. A teacher can easily find which pupil may require his/her guidance. The grades of membership of five characteristic values of the diagram in the fuzzy set "small numbers" are defined. It is defined that the grade of unity of a class is equal to the minimum value in their grades of membership. The classes are classified into the two sets, one of which is well united classes and the other lacks unity, by the grade of unity. It is shown that the characteristic values can clearly discriminate between the two sets. A teacher can grasp the changing of friendship by comparing the diagrams and the grades of unity at two points of time. Many teachers favorably estimate that it is very easy and useful to grasp real state in their classes from the diagrams.

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Information retrieval of a conventional fuzzy detabase is executed on the base of equality of attribute values of stored data and the one of a retrieval condition. But, when the attribute values are very fuzzy, the information retrieval on the base of "equality" often gives us unreasonable results in comparison with our ordinary common sense. In order to overcome the difficulty of the information retrieval for very large fuzzy data, the authors express the fuzzy attribute value of stored data and a retrieval condition in a label and its membership function, and formulate the new method of fuzzy information retrieval by defining the satisfaction grade, that estimates how the attributes of stored data satisfy the one of a fuzzy retrieval condition. By example of paintings, the authors discuss usability and effectiveness of the information retrieval formulated here.

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The information entropy introduced by Shannon is a measure which represents the value of information in numerical value. Generally, when the result is obvious before receiving related information, the value of information is low. On the contrary, the more ambiguous the result is, the higher the value of information becomes. Following Shannon's information entropy, we can assume the ambiguous object as the fuzziness of fuzzy sets. For example, if some of whose factors are "very high dollar value" and "high dollar value" is included in the whole sets, "the rate of exchange", the fuzziness of "very high dollar value" is smaller than that of "high dollar value". In this paper we report a consideration of methods for approximate reasoning employing a fuzzy entropy theory as the parameter of the defuzzification operation after composition of "THEN" conclusive part membership function. A desuzzifier methods called fuzzy entropy method are realized as robust natures as any the area method, hight method, and the center of gravity method used usually in the approximate reasoning.

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In this paper, we consider the least core, its related solution, and the nucleolus in n-person cooperative games with fuzzy coalitions. Since there are an infinite number of inequality constraints for fuzzy coalitions in the mathematical programming problem for obtaining the least core, it is difficult to directly solve the problem. We employ a computational method based on the relaxation method, which repeatedly solves mathematical programming problems with a part of the inequality constraints. The nucleolus is considered in the case where there are a finite number of fuzzy coalitions, and the nucleolus based on aggregated excesses is also examined.

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Some methods of fuzzy structural modeling are proposed as applications of fuzzy theory to structural modeling.FISM/Fuzzy is such a modeling among them. In this modeling a trainsitive coupling problem yields in embedding process. For the problem, however, there are merely some algorithms to find particular solutions to the transitive coupling problem. An algorithm to generate general solutions has not been jet found. In this paper, generalizing the transitive coupling problem to a problem to solve simultaneous equations with two fuzzy eigen relation equations and a fuzzy relation inequality we present an algorithm to generate solutions to the generalized problem, and also it is clear that the presented algorithm consequently generates all solutions to the transitive coupling problem.

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There are a lot of implication functions in the field of fuzzy logic, and the nature of the fuzzy inference changes variously depending on the implication function to be used. However, it is very difficult to select a suitable implication function for actual applications. Then we intend to define a parameterized implication function whose nature changes variously depending on the values of the parameters, and adjust the parameters so that the implication function has a suitable nature for actual applications using a parameter tuning method. We think this work can reduce the efforts to select a suitable implication function. Moreover, it can clarify the nature of the rule to select a suitable implication function by parameter tuning. Therefore, the tuning of the parameters can be regarded as a kind of quantification of the meaning of the natural language sentences expressed in IF-THEN rules.In this paper, we propose a parameterized implication function, defined to satisfy some desirable relations between a premise and the conclusion of a rule used in fuzzy inference. The desirable relations are defined using some parameters indicating the nature of the implication. We also demonstrate some tuning simulations of the parameters used in the implication function using the tuning mechanism of a fuzzy inference tool named FINEST.

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In this paper, we introduce the concept of fuzzy interflow into facility layout design problem and address fuzzy facility layout problem, where the uncertainty of material flows among facilities is represented as trapezoidal fuzzy numbers. Genetic algorithms are applied to such hard fuzzy combinatorial problem. Polish expression is adopted as the coding scheme of chromosome. The condition of legality for Polish expression coding and the condition for searching cut point in a chromosome are given. Based on these conditions, effective initialization procedure and layout construction procedure are built. Fuzzy ranking method is used to select the best layout in fuzzy context. A penalty to the violation of aspect ratio for each facilities is used to guide genetic search effectively towards to the promising part of solution space. The possibility theory and fuzzy integral are used to meaningfully interpret the fuzzy results. The simulation results demonstrate that genetic algorithm and fuzziness approach can be the efficient tools to solve large-scale layout problem.

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On an automatic rule extraction from set of input-output data examples, decision tree generating methods such as ID3 proposed by Quinlan (1986) are major. ID3 is, however, applicable only to the case that both input and output data are discrete or symbolic. As a method extended to case that input data are numerical and extracting fuzzy rules, Fuzzy ID3 has been proposed by Umano (1993) and Sakurai (1993). These crisp or fuzzy decision tree generating methods, however, need to recreate the trees from the beginning as often as a tendency of learning examples and are hard to be applied in such case that a tendency of examples changes changes dynamically during the inference process is in progress. As a method to overcome above two shortcomings and extract fuzzy rules adaptively, clustering input-output data space with neural networks, especially, Kohonen's Self-Organizing Map (SOM) is considerable.In this paper, we propose a neural network that has the architecture of SOM and the function of fuzzy clustering, called "Fuzzy Self-Organizing Map (FSOM)", and the learning methods based on Competitive Learning.And we propose a neural network that learns tendency of examplese, represents results of learning as fuzzy rules, and does fuzzy inference with the architecture of FSOM, called "Fuzzy Inference Network (FIN)", and a method of an automatic and adaptive rule extraction with the neural network. Furthermore, we present results of simulations for the comparison with other methods of automatic and adaptive rule extraction.

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A framework for qualitative modeling from linguistic data is proposed. Basic elements of the model are qualitative propositions, gradual and similarity rules, arithmetical rules and knowledge partition and execution strategy. These elements are introduced and discussed. The framework is applied to develop the extensive behavioral model of the Mulder's theory of power capable of prediction of power distribution in social groups. Knowledge is separated into loosely coupled modules called knowledge sources, and the backward search with forward evaluation is applied to evaluate the qualitative descriptions of power domains. The simulation results are presented and discussed.

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Genetic algorithms have been quite successfully applied to various optimization problems such as scheduling, transportation problems, traveling salesman problems, optimal control problems, and so on.In this paper, we formulate an optimal design problem of system reliability as a nonlinear integer programming problem with interval coefficients, tranform it into a bicriteria nonlinear integer problem without interval coefficients, and solve it directly with keeping nonlinearity of the objective function by using the improved genetic algorithm. We also demonstrated the efficiency of this method with a numerical example.

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In fuzzy modeling, the most important problem is how to divide the input space. Several methods have been proposed. For example, methods to incorporate variables sequentially, methods based on clusters, or stochastic searching methods using genetic algorithm are the notable ones. In this paper, we propose a genetic algorithm-based machine learning method for emergent generation of necessary rules. The action part is the linear model, and with small number of rules, the nonlinearity is to be achieved by the mixture of fuzzy rules.Moreover, the actual data of a plant factory was used for the modeling experiment. A plant factory is made of translucent materials and equipped with shading and supplemental light. The objective of the plant is to realize an optimal environment for vegetables to grow. The outside solar radiation, the status of the control equipments and optimal environment for vegetables to grow. The outside solar radiation, the status of the control equipments and the information of the sun locations are used as the input of the model, and the illuminance inside the house are taken as the output. A good model was derived using our method.

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This paper deals with a stability analysis using the describing function method to control systems which have a fuzzy controller designed using indirect fuzzy inference. First, the describing function of fuzzy controller designed using indirect fuzzy inference is found out analytically. Next, asymptotically stable conditions of this system are investigated. Using these conditions, stability bounds of loop gain for the first order lag system with dead time and the second order linear system are shown analytically. The result was that stability bounds of loop gain using describing function method were nearly equal to those of simulation studies. It was found that the describing function method was available as stability criterion for fuzzy control systems with indirect fuzzy inference.It was seen from the stability analysis of extended circle criterion and the simulation analysis that stability bounds of loop gain are in inverse proportion to √<c_1^2+c_2^2>, where c_1 and c_2 are input parameters for fuzzy logic controllers. This is also proved through analysis using the describing function method.

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