This paper discusses a perception-based theory for a multi-variate stopping problem with a monoto... more This paper discusses a perception-based theory for a multi-variate stopping problem with a monotone rule by . The problem is developed by using the perceptive analysis of the previous work ). We will show a recursive equation which determines the perceptive value of its equilibrium point for the model.
This paper is concerned with a fuzzy stopping time for a dynamic fuzzy system. A new class of fuz... more This paper is concerned with a fuzzy stopping time for a dynamic fuzzy system. A new class of fuzzy stopping times which is called as a monotone fuzzy stopping time is introduced. The notion of monotonicity is well-known and important in the stochastic optimization theory. Here we will try to define a monotone property and discuss a stopping problem which is corresponding a dynamic fuzzy system. Since the fuzzy stopping time can be constructed using by a-cuts of fuzzy states, the explicit derivation of an optimal one is derived under appropriate assumptions. The key point of our discussion for the optimization of a stopping problem is to induce an additive weighting function for the fuzzy reward.
Journal of the Operations Research Society of Japan, 2000
AbstTuct Coneerning with the topics of a fuzzy max order, a briefsurvey on orderi-g of fuzzy numb... more AbstTuct Coneerning with the topics of a fuzzy max order, a briefsurvey on orderi-g of fuzzy numbers is presented im this article, and we wil] consider an extensien to that of fuzzy sets. An extension of the fuzzy max order as a pseudo order is investigated and defined on a class of fuzzy sets on R" (n ) 1). This order is developed by using a non-empLy closed convex cone and characterized by the projection into its dual cone. Especially a structure of the lattice can be illustrated with the class of rectang]e-type fuzzy sets. = Ai (a J g).+ A2(b ･ 1]). = (Ai(a ･ 3) + A2(b ･ g))..
Our study is carried out by restricting the class of fuzzy sets into the subclass in which $\neg\... more Our study is carried out by restricting the class of fuzzy sets into the subclass in which $\neg\prec K$ becomes a partial order and a monotone convergence theorem is proved. This restricted subclass of fuzzy sets is created and characterized in the concept of a determining class. These results are applied to obtain the limit theorem for a sequence of fuzzy sets defined by the dynamic fuzzy system with a monotone fuzzy relation.
In a stochastic and fuzzy environment, a multi-objective fuzzy stopping problem is discussed. The... more In a stochastic and fuzzy environment, a multi-objective fuzzy stopping problem is discussed. The randomness and fuzziness are evaluated by probabilistic expectations and scalarization functions respectively. Pareto optimal fuzzy stopping times are given under the assumption of regularity for stopping rules, by using λ-optimal stopping times.
In this paper, we consider the model that the information on the rewards in vector-valued Markov ... more In this paper, we consider the model that the information on the rewards in vector-valued Markov decision processes includes imprecision or ambiguity. The fuzzy reward model is analyzed as follows: The fuzzy reward is represented by the fuzzy set on the multi-dimensional Euclidian space R and the infinite horizon fuzzy expected discounted reward(FEDR) from any stationary policy is characterized as a unique fixed point of the corresponding contractive operator. Also, we fined a Pareto optimal policy which maximizes the infinite horizon FEDR over all stationary policies under the pseudo order induced by a convex cone R . As a numerical example, the machine maintenance problem is considered.
We formulate a fuzzy perceptive model for Markov decision processes with discounted payoff in whi... more We formulate a fuzzy perceptive model for Markov decision processes with discounted payoff in which the perception for transition probabilities is described by fuzzy sets. Our aim is to evaluate the optimal expected reward, which is called a fuzzy perceptive value, based on the perceptive analysis. It is characterized and calculated by a certain fuzzy relation. A machine maintenance problem is discussed as a numerical example.
As the same framework of Fuzzy decision processes with the discounted case we will specify an ave... more As the same framework of Fuzzy decision processes with the discounted case we will specify an average fuzzy criterion model and develop its optimization by "fuzzy max order" under appropriate conditions. The average reward is characterized, by introducing a relative value function, as a unique solution of the associated equation. Also we derive the optimality equation using the "vanishing discount factor" approach.
Journal of Mathematical Analysis and Applications, 2000
In a stochastic and fuzzy environment, two kinds of stopping models are discussed and compared. T... more In a stochastic and fuzzy environment, two kinds of stopping models are discussed and compared. The optimal fuzzy stopping times are given under the assumptions of monotonicity and regularity for stopping rules. Also, we find that fuzzy stopping times are favored in a comparison between fuzzy and classical stopping models.
In this paper, we formulate the fuzzy perceptive model for discounted Markov decision processes i... more In this paper, we formulate the fuzzy perceptive model for discounted Markov decision processes in which the perception for transition probabilities is described by fuzzy sets. The optimal expected reward, called a fuzzy perceptive value, is characterized and calculated by a new fuzzy relation. As a
This paper is a sequel to Kurano et al , , in which the fuzzy perceptive models for optimal stopp... more This paper is a sequel to Kurano et al , , in which the fuzzy perceptive models for optimal stopping or discounted Markov decision process is given. We proposed a method of computing the corresponding fuzzy perceptive values. Here, we deal with the average case for Markov decision processes with fuzzy perceptive transition matrices and characterize the optimal average expected reward, called the average perceptive value, by a fuzzy optimality relation. Also, we give a numerical example.
In this paper, Markov decision models with uncertain transition matrices, which allow a matrix to... more 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.
This paper discusses two topics on fuzzy random variables in decision making. One is a new evalua... more 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 maker's subjective parameters. Numerical examples are given to illustrate our idea.
In a continuous-time fuzzy stochastic system, a stopping model with fuzzy stopping times is prese... more In a continuous-time fuzzy stochastic system, a stopping model with fuzzy stopping times is presented. The optimal fuzzy stopping times are given under an assumption of regularity for stopping rules. Also, the optimal fuzzy reward is characterized as a unique solution of an optimality equation under a differentiability condition. An example in the Markov models is discussed.
we will try to consider a perceptive analysis of the optimal stopping problem. In this paper, the... more we will try to consider a perceptive analysis of the optimal stopping problem. In this paper, the fuzzy perception value of the expectation of the optimal stopped reward is characterized and calculated by a new recursive equation. Also, a numerical example described by triangular fuzzy numbers is given.
We will define a new multi-stage decision process, which is termed Markov-type fuzzy decision pro... more We will define a new multi-stage decision process, which is termed Markov-type fuzzy decision process. By the general framework in the decision process, the optimization problem of the discounted reward is discussed under a partial order of convex fuzzy numbers.
ln this paper, by using a fuzzy relation, we define a dynamic fuzzy system with a bounded convex ... more ln this paper, by using a fuzzy relation, we define a dynamic fuzzy system with a bounded convex fuzzy reward on the positive orthant R~_ of an n-dimensional Euclidean space. As a measure of the system's performance, we introduce the time average fuzzy reward, which is characterized by the limiting fuzzy state under the contractive properties of the fuzzy relation. In the one-dimensional case, the average fuzzy reward is expressed explicitly by the functional equations concerning the extreme points of its a-cuts. Also, a numerical example is given to illustrate the theoretical results.
This paper discusses mean values and variance defined by fuzzy measures as evaluation methods of ... more This paper discusses mean values and variance defined by fuzzy measures as evaluation methods of fuzzy numbers/fuzzy random variables, and the methods are applicable to decision making with both randomness and fuzziness. We find the method with λ-mean functions has proper properties. The variance and the corresponding co-variance and correlation are introduced and their fundamental properties are discussed. The measurement of fuzziness regarding fuzzy numbers is also presented, where fuzziness is another uncertainty different from randomness and comes from the imprecise of data. An example is given as an application in financial engineering of portfolio.
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Papers by Yuji Yoshida