The time average reward for some dynamic fuzzy systems

Applied Mathematics Letters, 2002

In this note, we first define the concept of convex fuzzy processes. Second, we present some basic properties of convex fuzzy processes and important connection between convex fuzzy processes and their graphs.

International Symposium on Algorithms and Computation, 2019

Minimum and maximum operators are two wellknown t-norm and s-norm used frequently in fuzzy systems. In this paper, two different types of fuzzy inequalities are simultaneously studied where the convex combination of minimum and maximum operators is applied as the fuzzy relational composition. Some basic properties and theoretical aspects of the problem are derived and four necessary and sufficient conditions are presented. Moreover, an algorithm is proposed to solve the problem and an example is described to illustrate the algorithm.

This paper investigates fuzzy linear time varying controlled systems with fuzzy initial conditions, also fuzzy optimal control systems where the initial conditions are described by fuzzy numbers. We use a complex number representation of ®-cut sets of fuzzy system and obtain the new approach for solving this systems, and prove theorems for the richness theory.

European Journal of Operational Research, 1996

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.

Fuzzy Sets and Systems, 2011

The aim of this paper is to analyse the solution of a fuzzy system when the classical solution based on standard fuzzy mathematics fails to exist. In particular we analyse the solution of the system Ax=b with A squared matrix with positive fuzzy coefficients and y crisp vector of positive elements. This system is particularly important for financial applications. We propose two different solution methods that are based respectively on the work of and Friedman, . An application to an important financial problem, the derivation of the artificial probabilities in a lattice framework, is provided.

In this article we found the solution of fuzzy linear controlled system with fuzzy initial conditions by using $\alpha$-cuts and presentation of numbers in a more compact form by moving to the field of complex numbers. Next, a fuzzy optimal control problem for a fuzzy system is considered to optimize the expected value of a fuzzy objective function. Based on Pontryagin Maximum Principle, a constructive equation for the problem is presented. In the last section, three examples are used to show that the method in effective to solve fuzzy and fuzzy optimal linear controlled systems.

Kybernetika

The paper concerns Markov decision processes (MDPs) with both the state and the decision spaces being finite and with the total reward as the objective function. For such a kind of MDPs, the authors assume that the reward function is of a fuzzy type. Specifically, this fuzzy reward function is of a suitable trapezoidal shape which is a function of a standard non-fuzzy reward. The fuzzy control problem consists of determining a control policy that maximizes the fuzzy expected total reward, where the maximization is made with respect to the partial order on the α-cuts of fuzzy numbers. The optimal policy and the optimal value function for the fuzzy optimal control problem are characterized by means of the dynamic programming equation of the standard optimal control problem and, as main conclusions, it is obtained that the optimal policy of the standard problem and the fuzzy one coincide and the fuzzy optimal value function is of a convenient trapezoidal form. As illustrations, fuzzy extensions of an optimal stopping problem and of a red-black gambling model are presented.

2003

Based on the more restrictive definition of fuzzy convexity due to Ammar and Metz, some useful extremum properties are developed. We prove that any local maximizer of a convex fuzzy set is also a global maximizer, and that any strictly local maximizer of a quasiconvex fuzzy set is also a global maximizer. We also study the class of strictly convex (resp. strictly quasiconvex) fuzzy sets that is more restrictive than the class of convex (resp. quasiconvex) fuzzy sets. We prove that for both families of strictly convex and strictly quasiconvex fuzzy sets, every local maximizer is also the unique global maximizer. Finally, some composition rules for convex fuzzy sets are given and some applications to fuzzy decision making are discussed.

2021

This paper concerns discounted Markov decision processes with a fuzzy reward function triangular in shape. Starting with a usual and non-fuzzy Markov control model (Hernández-Lerma, 1989) with compact action sets and reward R, a control model is induced only substituting R in the usual model for a suitable triangular fuzzy function R̃ which models, in a fuzzy sense, the fact that the reward R is “approximately” received. This way, for this induced model a discounted optimal control problem is considered, taking into account both a finite and an infinite horizons, and fuzzy objective functions. In order to obtain the optimal solution, the partial order on the α-cuts of fuzzy numbers is used, and the optimal solution for fuzzy Markov decision processes is found from the optimal solution of the corresponding usual Markov decision processes. In the end of the paper, several examples are given to illustrate the theory developed: a model of inventory system, and two others more in an econ...

Fuzzy Sets and Systems, 2002

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.