Jun-ichi Nakagami

A limit theorem in some dynamic fuzzy systems

Abstract In this paper, using a fuzzy relation we define a dynamic fuzzy system with a compact state space. By introducing a contraction property, we study the limit of a sequence of fuzzy states and obtain a theorem for the existence and uniqueness of the solution of a fuzzy relational equation. As an application, we deal with a dynamic fuzzy system with a terminal fuzzy gain and give some characterizations for the fuzzy expected gain. A numerical example is given to comprehend our idea in this paper.

A new evaluation of mean value for fuzzy numbers and its application to American put option under uncertainty

This paper discusses two topics on fuzzy random variables in decision making. One is a new evaluation method of fuzzy random variables, and the other is to present a mathematical model in financial engineering by fuzzy random variables. The evaluation method is introduced as mean values defined by fuzzy measures, and it is also applicable to fuzzy numbers and fuzzy stochastic process defined by fuzzy random variables. The other is to apply the method to an American put option with uncertainty formulated as an optimal stopping problem for fuzzy random variables, and the randomness and fuzziness are estimated by the probabilistic expectation and the mean values. The optimal expected price of the American put option is given by the mean values with decision makers subjective parameters. Numerical examples are given to illustrate our idea.

Markov-type fuzzy decision processes with a discounted reward on a closed interval

Abstract We formulate a new multi-stage decision process with Markov-type fuzzy transition, which is termed Markov-type fuzzy decision process. In the general framework of the decision process, both of state and action are assumed to be fuzzy itself. The transition of states is defined using the fuzzy relation with Markov property and the discounted total reward is described as a fuzzy number on a closed bounded interval. To discuss the optimization problem, a partial order of convex fuzzy numbers is introduced. In this paper the discounted total reward associated with an admissible stationary policy is characterized by a unique fixed point of the contractive mapping. Moreover, the optimality equation for the fuzzy decision model is derived under some continuity conditions. Also, an illustrated example is given to explain the theoretical results and the computation in the paper.

A fuzzy relational equation in dynamic fuzzy systems

For a dynamic fuzzy system, the fundamental method is to analyze its recursive relation of the fuzzy states. It is similar in that the Bellman equation is an important tool in the dynamic programming. Here we will consider the existence and the uniqueness of solution of a fuzzy relational equation. Two examples, which satisfies our conditions, are given to illustrate the results.

A potential of fuzzy relations with a linear structure: the contractive case

Abstract In this paper we develop a potential theory of fuzzy relations on the positive orthant in a Euclidean space. By introducing a linear structure for fuzzy relations, the existence of a potential and its characterization by fuzzy relational equation are derived under the assumption of contraction and compactness. In the one-dimensional unimodal case, a potential is given explicity. Also, a numerical example is shown to illustrate our approaches.

A potential of fuzzy relations with a linear structure: the unbounded case

This paper is a sequel to Yoshida et al. (1993), in which the potential theory for linear fuzzy relations on the positive orthant R+n is considered in the class of fuzzy sets with a compact support under the contractive assumption. In this paper, potential treatment for unbounded fuzzy sets is developed without the assumption of contraction and compactness. The objective of this paper is to give the existence and the characterization of potentials for linear fuzzy relations under some reasonable conditions. Also, introducing a partial order in fuzzy sets, we prove Riesz decomposition theorem in the fuzzy potential theory. The proofs are shown by using only the linear structure and the monotonicity of fuzzy relations. In the one-dimensional case, the potential and its α-cuts are explicitly calculated. Numerical examples are given to comprehend further discussions.

A limit theorem in dynamic fuzzy systems with a monotone property

Abstract A dynamic fuzzy system defined on a compact state space is considered under a monotone property of the fuzzy relation. We study the limit of a sequence of fuzzy states and obtain a convergence theorem. The limit is characterized as the solution of a fuzzy relational equation. A numerical example is given to comprehend our idea in this paper.

A fuzzy approach to Markov decision processes with uncertain transition probabilities

In this paper, Markov decision models with uncertain transition matrices, which allow a matrix to fluctuate at each step in time, is described by the use of fuzzy sets. We find a Pareto optimal policy maximizing the infinite horizon fuzzy expected discounted reward (FEDR) over all stationary policies under some partial order. The Pareto optimal policies are characterized by maximal solutions of an optimal inclusion including efficient set-functions. As a numerical example, a machine maintenance problem is considered.

An approach to stopping problems of a dynamic fuzzy system

We formulate a stopping problem for dynamic fuzzy systems concerning with fuzzy decision environment. It could be regarded as a natural fuzzification of non-fuzzy stopping problem with a deterministic dynamic system. The validity of the approach by α-cuts of fuzzy sets will be discussed in constructing One-step Look Ahead policy of an optimal fuzzy stopping time. A numerical example is given to illustrate the theoretical results and to show a merit to use fuzzy stopping times in dynamic fuzzy systems.

Constrained markov decision processes with compact state and action spaces: the average case

Constrained Markov decision processes with compact state and action spaces are studied under long-run average reward or cost criteria. By introducing a corresponding Lagrange function, a saddle-point theorem is given, by which the existence of a constrained optimal pair of initial state distribution and policy is shown. Also, under the hypothesis of Doeblin, the functional characterization of a constrained optimal policy is obtained