Some properties of fuzzy random renewal processes

Fuzzy random variable is a measure function from a probability space to a collection of fuzzy variables. Based on the fuzzy random theory, this paper addresses some properties of fuzzy random renewal processes generated by a sequence of independent and identically distributed (iid) fuzzy random interarrival times. The relationship between the expected value of the fuzzy random renewal variable and the distribution functions of the /spl alpha/-pessimistic values and /spl alpha/-optimistic values of the interarrival times is discussed. Furthermore, the fuzzy random style of renewal equation is provided. Finally, fuzzy random Blackwell's renewal theorem and Smith's key renewal theorem are also given.     

Random fuzzy delayed renewal processes

In renewal processes, fuzziness and randomness often coexist intrinsically. Based on the random fuzzy theory, a delayed renewal process with random fuzzy interarrival times is proposed in this paper. Relations between the renewal number and interarrival times in such a process are investigated. Useful theorems such as the elementary renewal theorem, the Blackwell renewal theorem and the Smith key renewal theorem in a conventional delayed renewal process are extended to their counterparts for random fuzzy delayed renewal processes.     

Renewal process with T-related fuzzy inter-arrival times and fuzzy rewards

In this paper, we consider a renewal process in which the inter-arrival times and rewards are characterized as fuzzy variables under t-norm-based fuzzy operations. A T-related fuzzy renewal theorem and a fuzzy renewal reward theorem are proved using a law of large numbers for fuzzy variables.     

Random fuzzy alternating renewal processes

Random fuzzy theory offers an appropriate mechanism to model random fuzzy phenomena, with a random fuzzy variable defined as a function from a credibility space to a collection of random variables. Based on this theory, this paper presents the results of an investigation into the representation of properties of alternating renewal processes that are described by sequences of positive random fuzzy vectors. It provides a theorem on the limit value of the average chance of a given random fuzzy event in terms of “system being on at time t”. The resultant model coincides with that attainable by stochastic analysis when the random fuzzy vectors degenerate to random vectors.     

Fuzzy random renewal reward process and its applications

This paper studies a renewal reward process with fuzzy random interarrival times and rewards under the ?-independence associated with any continuous Archimedean t-norm ?. The interarrival times and rewards of the renewal reward process are assumed to be positive fuzzy random variables whose fuzzy realizations are ?-independent fuzzy variables. Under these conditions, some limit theorems in mean chance measure are derived for fuzzy random renewal rewards. In the sequel, a fuzzy random renewal reward theorem is proved for the long-run expected reward per unit time of the renewal reward process. The renewal reward theorem obtained in this paper can degenerate to that of stochastic renewal theory. Finally, some application examples are provided to illustrate the utility of the result.     

Modeling Random Fuzzy Renewal Reward Processes

This short paper discusses the modeling of random fuzzy renewal reward processes in which the interarrival times and rewards are represented by nonnegative random fuzzy variables. Based on random fuzzy theory, a random fuzzy variable denotes a measurable function from a credibility space to a collection of random variables. Under this setting, the long-run expected reward per unit time is addressed and the theorem on random fuzzy renewal rewards is established. The utility of this research is demonstrated with a realistic application case.      

The central limit theorems for fuzzy random variables

The new concept of the central limit theorem for fuzzy random variables is discussed in this paper by proposing the convergence in distribution for fuzzy random variables. We first consider the limit properties of fuzzy numbers by invoking the Hausdorff metric and then we extend it to the weak and strong convergence of fuzzy distribution functions. We provide a notion of fuzzy normal distribution. Then the central limit theorem for fuzzy random variables follows naturally.     

Multi-objective decision making model under fuzzy random environment and its application to inventory problems

In this paper, we concentrate on developing a fuzzy random multi-objective model about inventory problems. By giving some definitions and discussing some properties of fuzzy random variable, we design a method of solving solution sets of fuzzy random multi-objective programming problems. These are applied to numerical inventory problems in which all inventory costs, purchasing and selling prices in the objectives and constraints are assumed to be fuzzy random variables in nature, and then the impreciseness of fuzzy random variables in the above objectives and constraints are transformed into fuzzy variables which are similar trapezoidal fuzzy numbers. The exact parameters of fuzzy membership function and probability density function can be obtained through fuzzy random simulating the past dates. By comparing the results with those from the fuzzy multi-objective models, we believe that the proposed fuzzy random multi-objective model and hybrid intelligent algorithm provide significant solutions to construct other inventory models with fuzzy random variables in real life.     

On minimum-risk problems in fuzzy random decision systems

In a decision-making process, we may face a hybrid environment where linguistic and frequent imprecision nature coexists. The problem of frequent imprecision can be solved by probability theory, while the problem of linguistic imprecision can be tackled by possibility theory. Therefore, to solve this hybrid decision-making problem, it is necessary to combine both theories effectively. In this paper, we restrict our attention to this hybrid decision-making problem, where the input data are imprecise and described by fuzzy random variables. Fuzzy random variable is a mapping from a probability space to a collection of fuzzy variables, it is an appropriate tool to deal with twofold uncertainty with fuzziness and randomness in an optimization framework. The purpose of this paper is to present reasonable chances of a fuzzy random event characterized by fuzzy random variables so that they can connect with the expected value operators of a fuzzy random variable via Choquet integrals, just like the relation between the probability of a random event and the mathematical expectation of a random variable, and that between the credibility of a fuzzy event and the expected value operator of a fuzzy variable. Toward that end, we take fuzzy measure and fuzzy integral theory as our research tool, and present three kinds of mean chances of a fuzzy random event via Choquet integrals. After discussing the duality of the mean chances, we use the mean chances to define the expected value operators of a fuzzy random variable via Choquet integrals. To show the reasonableness of the mean chance approach, we prove the expected value operators defined in this paper coincide with those presented in our previous work. Using the mean chances, we present a new class of fuzzy random minimum-risk problems, where the objective and the constraints are all defined by the mean chances. To solve general fuzzy random minimum-risk optimization problems, a hybrid intelligent algorithm, which integrates fuzzy random simulations, genetic algorithm and neural network, is designed, and its feasibility and effectiveness are illustrated by numerical examples.     

Fuzzy set-valued Gaussian processes and Brownian motions

Gaussian processes and Brownian motion are concepts and tools in modelling important uncertain systems in many areas. In view of uncertainty complexity in many real-world problems, we extend these tools to the case where stochastic processes can take on fuzzy sets as values. In this paper, we discuss fuzzy set-valued Gaussian processes based on the results of [S. Li, Y. Ogura, V. Kreinovich, Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables, Kluwer Academic Publishers, Dordrecht, 2002; S. Li, Y. Ogura, H.T. Nguyen, Gaussian processes and martingales for fuzzy valued variables with continuous parameter, Inform. Sci. 133 (2001) 7–21; S. Li, Y. Ogura, F.N. Proske, M.L. Puri, Central limit theorems for generalized set-valued random variables, J. Math. Anal. Appl. 285 (2003) 250–263; N.N. Lyashenko, On limit theorems for sums of independent compact random subsets in the Euclidean space, J. Soviet Math. 20 (1982) 2187–2196] and [M.L. Puri, D.A. Ralescu, The concept of normality for fuzzy random variables, Ann. Probab. 13 (1985) 1373–1379]. We also introduce the concept of fuzzy set-valued Brownian motion, and then prove several properties of such processes.     

Simulation of random fuzzy variables: an empirical approach tostatistical/probabilistic studies with fuzzy experimental data

In this paper, we simulate different types of random fuzzy variables to get some conclusions concerning fuzzy-valued random variables. This simulation was carried out to illustrate certain limit results formalizing the convergence of the arithmetic mean of sampled fuzzy data to the population mean (or an expected value of the random fuzzy variable), like the well-known strong law of large numbers and the law of iterated logarithm. Since the theoretical results have recently been proved, this simulation analysis represents a complementary study in which we can determine sample sizes providing us with suitable approximations. Finally, future directions regarding statistical inference with fuzzy data and other applications are commented     

Some comments on fuzzy variables with different membership functions

Hong and Kim (Fuzzy Sets Syst 93:121–124, 1998) presented a membership function of a finite sum of mutually unrelated fuzzy variables with a common membership function. In this paper, we extend the fuzzy variables with a common membership function to those with different membership functions and then present the membership function of a finite sum of mutually unrelated fuzzy variables with different membership functions.     

Expected value of fuzzy variable and fuzzy expected value models

This paper will present a novel concept of expected values of fuzzy variables, which is essentially a type of Choquet integral and coincides with that of random variables. In order to calculate the expected value of general fuzzy variable, a fuzzy simulation technique is also designed. Finally, we construct a spectrum of fuzzy expected value models, and integrate fuzzy simulation, neural network, and genetic algorithms to produce a hybrid intelligent algorithm for solving general fuzzy expected value models.     

Metric spaces of fuzzy variables

Distance between fuzzy variables has played an important role in fuzzy theory and has been defined in many ways, for example, Hausdorff-like distance, Hamming distance and the distance based on expected value operator of fuzzy variable. This paper proposes a new kind of distances between fuzzy variables, fuzzy random variables and random fuzzy variables and these distances completely satisfy the mathematical axioms of a metric. Furthermore, a metric space of fuzzy variables is defined, the completeness of this space is proved and the properties of new distances are discussed. Finally, the distances between fuzzy vectors, fuzzy random vectors and random fuzzy vectors are also given.     

Blackwell’s Theorem for T-related fuzzy variables

In this paper, we consider Blackwell’s Theorem in which inter-arrival times are characterized as fuzzy variables under t-norm-based fuzzy operations. We first prove that Blackwell’s Theorem for T-related fuzzy variables with respect to necessity measure holds true where T is an Archimedean t-norm. Subsequently, we provide a counter example under which Blackwell’s Theorem does not hold when T = min. Finally, we evaluate the expected value of fuzzy variable with respect to credibility measure and derive fuzzy Blackwell’s Theorem based on the expected value of fuzzy variables.     

The modes of convergence in the approximation of fuzzy random optimization problems

To develop the approximation approach to fuzzy random optimization problems, it is required to introduce the modes of convergence in fuzzy random theory. For this purpose, this paper first presents several novel convergence concepts for sequences of fuzzy random variables, such as convergence in chance, convergence in distribution and convergence in optimistic value; then deals with the convergence criteria and convergence relations among various types of convergence. Finally, we deal with the convergence theorems for sequences of integrable fuzzy random variables, including dominated convergence theorem and bounded convergence theorem.     

A new approach to formulate fuzzy regression models

A fuzzy regression model is developed to construct the relationship between the response and explanatory variables in fuzzy environments. To enhance explanatory power and take into account the uncertainty of the formulated model and parameters, a new operator, called the fuzzy product core (FPC), is proposed for the formulation processes to establish fuzzy regression models with fuzzy parameters using fuzzy observations that include fuzzy response and explanatory variables. In addition, the sign of parameters can be determined in the model-building processes. Compared to existing approaches, the proposed approach reduces the amount of unnecessary or unimportant information arising from fuzzy observations and determines the sign of parameters in the models to increase model performance. This improves the weakness of the relevant approaches in which the parameters in the models are fuzzy and must be predetermined in the formulation processes. The proposed approach outperforms existing models in terms of distance, mean similarity, and credibility measures, even when crisp explanatory variables are used.     

Monte Carlo methods in fuzzy queuing theory

In this paper, we consider an optimization problem in fuzzy queuing theory that was first used in web planning. This fuzzy optimization problem has no solution algorithm and approximate solutions were first produced by computing the fuzzy value of the objective function for only sixteen values of the fuzzy variables. We introduce our fuzzy Monte Carlo method, using a quasi-random number generator, to produce 100,000 random sequences of fuzzy vectors for the fuzzy variables, which will present a much better approximate solution.     

Entailment for intuitionistic fuzzy sets based on generalized belief structures

Entailment for measure-based belief structures can extend the possible probability value range of variables on a space and obtain more information from variables. However, if the variable space comes from intuitionistic fuzzy sets, the classical entailment for measure-based belief structures will not work in this issue. To deal with this situation, we propose the entailment for intuitionistic fuzzy sets based on generalized belief structures in this paper to apply the entailment for measure based belief structures on space, which is made up of non-membership degree, membership degree and hesitancy degree of a given intuitionistic fuzzy sets. Numerical examples are mentioned to prove the effectively and flexibility of this proposed entailment model. The experimental results indicate that the proposed algorithm can extend the possible probability value range of variables of space efficiently and obtain more information from intuitionistic fuzzy sets.     

Hierarchical fuzzy control

In a conventional rule based fuzzy control system, the rules are of the following form: if (condition) then (action), and all rules are essentially in a random order. The number of rules increases exponentially as the number of the system variables, on which the fuzzy rules are based, is increased. In this paper, the rules are structured in a hierarchical way so that the total number of rules will be a linear function of the system variables. The hierarchical fuzzy control algorithm developed in this paper is applied to control the feedwater flow to a steam generator of a power plant. The simulation results show that the hierarchical fuzzy controller yields superior performance over the conventional PID controller.