This paper presents interactive fuzzy programming for multi-level 0-1 programming problems through genetic algorithms. In fuzzy programming for multi-level linear programming problems, recently developed by Lai et al., since the fuzzy goals are determined for both an objective function and decision variables at the upper level, undesirable solutions are produced when these fuzzy goals are inconsistent. In order to overcome such problems, after eliminating the fuzzy goals for decision variables, interactive fuzzy programming for multi-level 0-1 programming problems through genetic algorithms is presented. In our interactive method, after determining the fuzzy goals of the decision makers at all levels, a satisfactory solution is derived efficiently by updating the satisfactory degrees of decision makers at the upper level with considerations of overall satisfactory balance among all levels. Illustrative numerical examples for two-level and three-level 0-1 programming problems are provided to demonstrate the feasibility of the proposed method.

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A proportional learning vector quantization (PLVQ) algorithm has been developed. The algorithm employs a fuzzy learning law to solve the normalization and initialization problems that are encountered in traditional learning vector quantization (LVQ). The performance of the new algorithm has been compared to that of the learning vector quantization (LVQ) and generalized learning vector quantization (GLVQ) methods by two special examples. The results show that the presented method does not only avoid the initialization problem but also solve the normalization problem.

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Previous works on fuzzy regression models have been developed as an extension of the interval regression model. On account of this line of approach, in previous works, observed data of dependent variables are not contained in minimizing objectives but in constraint conditions only. As a result, the estimated fuzzy numbers that are derived from those models can be said that they have little reflection of the dependent fuzzy numbers used for identifying models. In this paper, the author proposes a method for identifying fuzzy regression models. My method is based on the concept of least square estimate as well as interval linear regression of the previous type, and include two types of problems to solve. The first type problem is given as a single-stage problem, and the second is given as a two-stage one. Since observed data are explicitly taken into objective functions in both problems, fuzzy regression models identified by our method are more sensitive, with respect to observed data, than those of previous method. And the author can compare models by defining the function that judges models. With this function, the author can quantify the difference between previous model and new one.

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Minimum Spanning Tree (MST) is one of the network design problems. The problem is to find a subgraph spanning all vertices with the minimal total weights on a finite graph. The cost, time or distance defined on arcs in a network are usually determined according to the experience of experts, but can not be determined exactly in practice. Compared with the defination of fuzzy membership on arcs, the interval coefficient of arcs is a more adequate representation of them on this case. On the other hand, usually there is such case that the MST problem has two objectives because cost and time may exist at the same time in real world. However, such kind of problem is difficult to deal with because of its NP-hard complexity. Recently, more attention has been paid to the MST problems by using genetic algorithm. Several tree encodings have been suggested to code the solution of the problem, such as Prufer number and so on. But all of them are focused on dealing with the MST problem on a complete graph. Actually, it is not always the case in practice because most practical network structures are not a complete graph. In this paper, we formulate this problem as bicriteria minimum spanning tree problem with interval coefficients shortly for i-BMST and present a new approach based on genetic algorithm with matrix-based tree encoding to cope with the MST problem on a sparse graph. Numerical example is given out to illustrate the effectiveness of the proposed method.

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In this paper, a systematic method to design rating scale is proposed. The rating scale used by respondents in various subjective evaluation ought to be easy to rate for them. This study aims (1) to clarify existence of individual preference towards scale design, i.e., easy / difficult to answer, and (2) to propose a system that treat differences of the preference and individual meaning of categories. To obtain fundamental findings for constructing system, two psychological experiments are executed; the experiments consist of (1) tasks of choosing and assigning rating categories, and (2) ranking of the variously designed scales in terms of easiness of rating and assignment of the rating categories. From the results obtained, existence of individual preference towards various scales is exemplified, first. Next, some factors that affiliate evaluation of easiness of rating are extracted to clarify the factors required to treat in the system. Then, relationship between easiness of rating and assignment of categories is examined through comparison among subjective rankings of easiness of rating scales, subjective rankings of assignment of categories and 14 mathematical indices for evaluating the assignment to compose a strategy for treating these factors. In the next stage, methodology of estimating easiness of rating from those index values will be discussed to show scheme of the system. Lastly, combining methodologies and findings obtained in this study, a system that supports to design a customized rating scale for each respondent through interaction is proposed.

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Since unconstrained moving objects are sensitive to circumstance change and their moving velocity can not be handled directly, their trajectory tracking control is considered to be a difficult problem. In this paper, a trajectory tracking experiment system taking an official table-tennis ball as its control object is constructed, and a fuzzy controller based on the recently proposed SIRMs dynamically connected fuzzy inference model is presented. The estimated tracking error, velocity and acceleration of the ball are selected as the input items of the fuzzy controller. For each input item, a SIRM (Single Input Rule Module) is designed and an importance degree is assigned. Especially for the input item corresponding to the velocity, its importance degree is allowed to change with the moving situation. The summation of the products of the importance degree and the fuzzy inference result of the SIRMs is outputted to control the angle of a table with level surface, making the ball on the table move along a desired trajectory. In order for the ball in any position to smoothly approach and stably track a trajectory, asymptotic trajectory from its current position to the real trajectory is introduced. Experiments are done for desired trajectories of three kinds of circles and one kind of ellipses. In more than 80% of the experiments, the biggest tracking error is less than 0.05m despite of different sizes and different kinds of desired trajectories, and the unevenness of sampling steps necessary for one cycle of the trajectories is very small. All the results show that the unconstrained ball can stably and accurately track the desired trajectories under the proposed control scheme.

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Although the SIRMs dynamically connected fuzzy inference model can improve drastically the control performance of first-order delay plants and second-order delay plants compared with the SIRMs connected fuzzy inference model, no systematic method of determining the parameters of its dynamic importance degrees is established. In this paper the model is applied to first-order delay plants, and a fuzzy controller is constructed taking the output error and the change in the output error as the input items and the change in the manipulated variable as the output item. It is proved that the fuzzy controller completely corresponds to the conventional PI controller in each control action, and is essentially a nonlinear PI controller. For first-order delay plants with a time constant from 0.20 to 30.00, the setting method of the parameters, i.e., the base values and the changing breadths, of the dynamic importance degrees are given based on the data collected by the random optimization search method. Simulation results show that by using the proposed method, the plant output can rise to a desired value quickly, and the overshoot or undershoot amount is suppressed to about 1.0% without vibration or steady-state error.

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