In this short paper, we give a simple proof for a quantization theorem for Kleene algebras. Moreover we establish the theorem for universal algebras and show that it is a special case of the homomorphism theorem for universal algebras.

View full abstract

The aim of the present paper is to show significance of toll sets proposed by Dubois and Prade by considering complementation operations, t-norms, t-conorms, and two applications in the following way. 1. A complementation operation N (T) of a toll set T that satisfies N (N(T)) = T is considered. Note that the complement proposed by Dubois and Prade does not satisfy this property. 2. Based on this complementation, t-norms and t-conorms on toll sets are defined and in particular a t-norm of the type of Minkowski's norm is discussed. 3. Correspondence between a t-norm (or conorm) on toll sets and a t-norm (or conorm) on fuzzy sets is shown using a 1-1 mapping between tool sets and fuzzy sets. 4. A decision making problem is formulated as the minimization of the t-norm of the Minkowski type and the t-norm is proved to be convex. 5. Hierarchical clustering based on toll graphs is discussed and it is shown that a toll graph plays a complementary role to the fuzzy graph in clustering.

View full abstract

In recent papers, a new threshold function is defined on a continuous logic. The function is denominated the fuzzy threshold function and it provides a way of expressing reasoning rules in fuzzy reasoning into a mathematical function. It has been shown in the previous works that the method is fast and stable, but there remains a problem of devising an effective scheme for expressing non-linear fuzzy rules into mathematical formulas. In this paper, firstly, a fuzzy multi-threshold threshold function is newly defined, and its theoretical aspects are considered. Next, a general mehtod is proposed in order to represent a reasoning rule table which involves non-linearlity with a mathematical formula employing the fuzzy multi-threshold threshold function. The formulas obtained by the proposed method are not necessarily minimum but the method is available for any reasoning rule table. Together with this method, other means are discussed on their feasibility.

View full abstract

Equilibrium solutions in terms of an attainment degree of a fuzzy goal for games in fuzzy and multiobjective environments are examined. We introduce a fuzzy goal for a payoff in order to incorporate ambiguity of human judgements and assume that a player tries to maximize his attainment degree of the fuzzy goal. A fuzzy goal for a payoff and the equilibrium solution with respect to an attainment degree of a fuzzy goal are defined. Two basic methods, one by weighting coefficients and the other by a minimum component, are employed to aggregate multiple fuzzy goals. When membership functions are linear functions, the computational methods for the equilibrium solutions are developed. It is shown that hte equilibrium solutions are equal to optimal solutions to mathematical programming problems in both cases. The relations between equilibrium solutions for multiobjective bimatrix games incorporating fuzzy goals and the Pareto optimal equilibrium solutions are considered.

View full abstract

Fuzzy mathematical programming problems are convenient for decision making under uncertainty. Therefore, a lot of formulations for them have been proposed in the recent decade. We get optimal solutions easily, because many problems are transformed into linear programming problems. But, they can hardly deal with some of relationship between uncertain coefficients. In this paper, in order to improve these defects, we suggest a new method using scenarios which are used in stochastic linear programming. In usual methods, scenarios are composed of crisp numbers. However, in this case, there are some defects that the size of problems become very large and uncertainty of problems are not described well in a small number of scenarios. So, we suggest new scenarios composed of fuzzy numbers and a new method for solving multiple objective linear programming problems which can get the solutions considering the balance of all scenarios, all goals, optimistic viewpoints and pessimistic ones.

View full abstract

Recently, Genetic Algorithms, a new learning paradigm that models a natural evolution mechanism, have received a great deal of attention regarding their potential as optimization techniques for solving combinatorial optimization problems. In this paper, we focus on multiobjective combinatorial optimization problems as a generalization of the traditional single objective combinatorial optimization ones. By considering the imprecise nature of human judgements, we assume that the decision maker may have fuzzy goals for each of the objective functions. After eliciting the linear membership functions through the interaction with the decision maker, we adopt the minimum-operator for combining them. In order to investigate the applicability of the conventional simple Genetic Algorithms for the solution of the formulated problems, a lot of numerical simulations are performed by assuming several genetic operators. Then, instead of using the penalty function for treating the constraints, we propose three types of revised Genetic Algorithms which generate only feasible solutions. Illustrative numerical examples demonstrate the both feasibility and efficiency of the proposed methods.

View full abstract

In the recent decade, a lot of formulations for fuzzy mathematical programming problems have been proposed. In order to deal with general cases, it is desirable to formulate problems, where the aspiration levels as well as the coefficients in the objective functions may be fuzzy. In this case, we suppose that an ill-defined problem ought have an ill-defined solution like a well-defined problem has a well-defined solution. So we proposed a method for solving fuzzy multi-goal linear programming problems involving both fuzzy decision variables and fuzzy parameters. The fuzzy solution is a fuzzy theorem's unique idea, and its merit is that decision maker understands possibility of the solution visually. Applying an interactive method to this approach is effective way of finding the satisfying solution of the decision maker. However proposed interactive methods need a considerable knowledge of mathematical programming, so it is difficult for decision maker to handle the methods. In this paper, we present a new interactive method using fuzzy reasoning. Finally, we apply this approach to the diet problem and an illustrative numerical example is also demonstrated.

View full abstract

For allocating room-units of an office on a floor, such complex factors as relative distances from windows/entrances, correlationships among room-units should be considered. It is necessary to acquire experts' knowledge for realizing an automatic zoning. The experts, however, often have knowledge unconciously and without clear linguistic expressions. Knowledge acquisition is a difficult work in constructing expert systems. The authors have proposed a knowledge acquisition support system with a presentation of suggestions for modifying pre-input knowledge. This paper describes a basic study on knowledge acquisition for zoning of an office on a floor. New notions of degree of belonging and degree of fixation for each room-unit are introduced. This paper also presents a curve of belonging which expresses the above two notions. the knowledge acquisition supprot system proposed by the authors is used to acquire the degree of correlationships among room units and the two notions : degree of belonging and degree fixation. Simulations are carried out to show the feasibility of the proposed method.

View full abstract

In this paper, we consider to grasp ceiterion-referenced evaluation structure of a practical subject by a function on the basis of teacher's subjectivity. A membership function in the fuzzy theory is used as its function. Reached levels are guessed from freely written criticism for some evaluation items. The membership function corresponding to the reached level is found by using fuzzy reasoning. The center of gravity in area of the membership function is computed for one of evaluation items. The work's evaluation is the value divided the sum of the values in the centers of gravity by the number of the evaluation items written in the criticism. Evaluation values are computed by changing the membership functions. We compute an average of absolute subtraction between the ten-point ratings and the fuzzy evaluation for works. Teacher's evaluation structure is presumed by the membership function with the minimal average. Work's evaluation is obtained using the membership function. This is called fuzzy evaluation method. We asked teacher's questionnaires about the criticism and ten-point ratings for real calligraphy works of four pupils. The fuzzy evaluation obtained using the questionnaires shows to be exactly similar to the ten-point rating.

View full abstract