Owing to the complexity of the problems, the objective functions have so fartreated using the information about preferences or aspirations of decision makers in mathematical programming problems with fuzzy coefficients. Namely, it intends to satisfy rather than to optimize. Recently, extending the concepts of optimality and efficiency, a few researchers attempt to optimize the objective functions of fuzzy mathematical programming problems. In this paper, the concepts of efficient solutions or nondominated solutions for usual multiple objective programming probelms are extended for fuzzy multiple objective programming problems in two different manners, which are direct extension manner and the manner based on constructing dominance relation in decision space from the relation in objective space. Using modality measures, i.e., possibility measure and necessity measure, two kinds of efficient solutions are derived from the former manner and six kinds of nondominated solutions from the latter. As the result, we have eight kinds of solutions. In order to arrange these eight solutions, the relationships are investigated. Two of these eight solutions are equivalent to the other two and eight kinds are reduced to six kinds. Moreover, the six kinds of solutions are reduced to four kinds when a usual relation ≧ is adopted for a dominance relation in fuzzy multiple objective linear programming problems with non-interactive fuzzy coefficients. Since these six kinds of solutions are constructed using modality measures, they have linguistic representations, and may play a significant role when we discuss the rationality of the solutions.

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We consider a generalization of the transportation problem. That is, we formulate the problem by introducing membership functions which show a satisfaction degree for flow values from source nodes and those to sink nodes. The membership function is decided by a decision maker, and each source and sink has the corresponding membership function. The objective is to determine the flow which attains an equitable distribution such that all membership functions have the same values as possible.

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Investigations on fuzzy control of a mobile robot for obstacle avoidance in house based on finding permissible passageways using edges between the floor and the wall or obstacles obtained by processing image from CCD camera in front of the robot have been already conducted by T.Takeuchi et al. One objective of guidance control for autonomous mobile robot is to enable the robot to maneuver over an appropriate path to a destination point through an environment that is already known. Here we discuss guidance over paths that are specified in terms of maps, in which the passageways consist of only 90°turns and straight segments ; typical coditions in indoor workspaces. In order for the robot to advance along the designated path autonomously, the processing treats wrong path as virtual obstacles on the screen and the robot to advance in the designated direction. The robot guidance control based on the above method is simulated on personel computer using C-language and experimented, and its usefulness is demonstrated. The control program developed with simulation is transplanted into the robot and a path to a destination point from one point is specified due to order of passing intersections and directions to advance to next intersection in the map.

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This note points out that conditional probabilities can be considered as fuzzy sets. Knopfmacher's fuzzy entropy (J. Math. Anal. Appl. 49,529-534. 1975) shows that, the coarser the σ-subalgebra is, the fuzzier the conditional probability is.

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In order to let a computer laugh, this note deals with popularity and inferiority as the relevant factors then forms the scaling of a laugh in the fuzzy method. And we discuss popularity/inferiority.

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