In fuzzy theory, possibility valuation is defined as an extention of possibility measure for a fuzzy set, which is used as the valuation for the fuzzy set under possibilistic uncertainty. This definition is also regarded as a fuzzy integral of a membership function on possibility measure. However, various kinds of fuzzy integrals are poposed. Therefore, another possibility valuations can be defined. By the way, membership functions and possibility distribution functions are identified subjectively by decision makers. Therefore, the valur membership functions and possibility distribution functions should not be considered as cardinal value but ordinal value. In this paper, new definition of possibility valuation under uncertainty are proposed based on oridinal utility theory. And also, necessity valuation, which is a dual concept of possibility valuation, is proposed. Moreover, relationship between these definirions and identification of membership functions are discussed.

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Logical models for document retrieval are theoretically reformulated as possible-worlds models from the point of view of restriction over a set of possibilities using evidence. The restriction can be regarded as crisp evidence and so the models can be further extended to belief models by means of logic of belief based on evidence. Then, nonmonotonic reasoning is shown to be implicit in indexing in logicl retrieval.

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There are several methods in Fuzzy reasoning, proposed Zadeh, Baldwin, Tsukamoto, Mizumoto, and Yager. In this note, we investigate relations between these methos. For the purpose, we study implicfation functions in section 2,and obtain our main results in sectin 3,which are as follows : B'_<Zad>(y)=B'_<Bal>(y)=B'_<tsu>(y) (Theorem 3.1〜3.2), B'_<Miz>(y)≦B'_<Bal>(y) (Theorem 3.3), B'_<Yag>(y)≦B'_<Bal>(y) (Theorem 3.4), B'_<Miz>(y)≦B'_<Yag>(y) (Theorem 3.5), where B'_<Zad>, B'_<Bal>, B'_<Tsu>, B'_<Miz>, and B'_<Yag> are consequents by Zadeh's method, Baldwin's, Tsukamoto's, Mizumoto's and Yager's, respectively.Also we give a lisy showing the results using several kinds of "implication functions".

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In this paper, we will propose new relations of trapezoidal fuzzy numbers(TrFNs) each of which is defined using three parameters. And we will show that the proposed relations include the order relations produced by using both any cutting level α∈[0,1] and anyone of the indices by Dubois and Prade. Moreover, we will propose a method for solving linear programming(LP) problems with TrFN coefficients, using the proposed relations. We will demonstrate that in this methof decision maker(DM) can treat the constraint with more reflection of DM's intention than in the method using the order relations by indices proposed by Dubois and Prade.

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Various types of formulations of solving methods have been proposed with fuzzy multigoal programming, but most of them exclusively take linear functions as objectives. However among the elements exhibited as goals, some goals that can be expressed with a ratio equation are available. Therefore, thus models that can handle such a fractional goal are preferable. On the other hand, most of the methods are of the type with which solutions can be obtained by crisp values notwithstanding that vague numerical values are considered. However since the elements composing problems are given by vague numerical values, the solutions should depend on the degree of the vagueness. Accordingly it is natural that the solutions themselves are calculated as vague numerical values. As one of the solutions based upon such an idea, Kono et al.'s method is recommendable. However, even the said method cannot handle the fractional goal.In this study, a method that expands Kono et al.'s method and allows also the fraction-al goal to be complied with si proposed in consideration of such a situation mentioned above.

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A probabilistic model for the determination of membership function based on fuzzy data is proposed. First, an experiment on the perception of tiangle area by human is carried out in order to see the relationship between subjective value , y, and objective value , x. It is found that the subjective value can be expressed by a probability density function fγ(y), in which mean of fγ(y) has a distance from x and standard deviation of fγ(y) is a function of x. By introducing subjective classification boundaries, the process of subjective classification is expressed by a probabilistic model. In the model, membership grade, which is same as the probability that an objective data x is classified into small, medium of large, can be calculated by the integration of fγ(y) in the range between respective classification boundaries.Actual calculation of membership grade is carried out on objective coordinate. The fγ(y) and the subjective classification boundaries are all mapped onto objective coordinate and a probability density function fx(x) and objective classification boundaries are newly introduced. Normal, beta and uniform distributions are assumed for fx(x). The mean of fx(x) is approximated by the objective vaue x. The standard deviation of fx(x) and the objective classification boundaried are determined from the fuzzy data by the likelihood analysis.The proposed method is applied to the fuzzy data obtained by the experiments on the perception of triangle area. It is concluded tha membership function can be determined for every type of fx(x). Further, it is found that the proposed method can generate triangle and trapezoid membership functions too.

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Projection Pursuit is a computer-intensive method for multivariate statistical analysis which intends to clarify properties of multivariate data by projecting them onto lower-dimensional subspace.In practice, the projection pursuit is performed based on an algorithm to find a lower-dimensional subspace by maximizing an objective function called projection index which is a measure of interestingness of the projected data.Adapting this method for fuzzy data, it was proposed to use measures of fuzziness as projection index, and to find the subspace by minimizing the measures of fuzziness of projected data. So the subspace in which they are as crisp as possible is searched.But from the viewpoint of statistical analysis, important properties such as cluster or functional relationship are not considered.In this paper, we propose a method that includes both interestingness of data and the measures of fuzziness called hybrid index. And we confirm the efficiency of projection pursuit using this index by numerical experiments.

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Up to now, the psychological impression to acoustic noise by use of fuzzy set theory has been quantitatively studied from the viewpoint that the psychological impression as judged by human beings is a fuzzy event. The membership functions of categorized impression on the noise level universe in the studies played an important role in the discussion of the quantitative relationship between the noise as the objective input and impression as the subjective output. However, this being the beginning stage of the studies the method used to estimate the membership function was very primitive. At the next stage, the system for measuring impression to the noise easily and effectively, and for estimating the membership function, is developed for the purpose of investigating individuality of response to the noise, changes of the function by various kinds of factor and so on. The validity and usefulness of the system are confirmed by applying it to the actual measurement for the functions of four subjects.Finally, several methods for estimating the function proposed before by other researchers are summarized and the differences between the present method and previous ones, and the features unique to the present method, are discussed. It is shown from the discussion that the proposed method is a new one based on the decomposition theorem, and a practical one with a wide application based on a small number of assumptions.Noisiness, Calmness, Individuality, Decomposition theorem

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The estimation of consolidtion yield stress of soils in significant in the design process of civil engineering structures. The consolidation settlements occur gradually with time when structures are build on the soil foundation. The total amount of settlements are related to the stress of soil foundations. To estimate the stress appropriately is generally very hard. Because, limited lab data would available on wide and deep area of site where structures are buid. To inspect the whole characteristic data on soil foundation is very expensive and therefore it is impossible in economic reasons. Civil engineering is dependent upon the public economies and investments. It is, therefore, very useful to use the fuzzy possiblistic regression estimating the consolidation yield stress properly in civil engineering design.

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On decision-making for person perception, choice of excellent persons and so on, there are several problems in reliability and validity of the merit-rating, since scale of the existing rating depends on crisp values. It is easily dominanted by human subjective judgement to acknowledge and examine a person. Vagueness in the rating appears to reflect basic festures in cognitive process of human social information. Therefore, it is desired to develop a new kind of scales of the person perception.In this research, the decision-making problem based on Inverse Problem in Fuzzy Relational Equations with Fuzzy elements is formulated by fuzzy measure which can reflect human subjective judgement. Farther, a solution of the Inverse Problem on Fuzzy Relation with Fuzzy Element is proposed. By this method, some quantities which can not be observed but important in decision-making can be infered from bjectively observed date. Especially vangueness which in merit-rating is sufficently expressed, and inprovement of reliability and validity can be expected.

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Generalization of conventional methods of free-form surface design results in simplified fuzzy logic. This yields a flexible free-form surface construction method, and relates free-form surface with nonparametric regression. Reconstruction of a corn leaf using this algorithm illustrates its superior performance.

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Roles of the fuzzy group in the fuzzy quantification method 1 are studied using the linear regression model. First, relationship between the fuzzy group and linear approximation of a nonlinear model is discussed. Secondly, it is pointed out that the fuzzy group should be related to error varianves of the linear regression model, and effect of fuzzy group data on regression analysis is studied by computer simulation.

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