1. Introduction

We know that human thinking and reasoning frequently involve fuzzy information which was originating from inherently in exact human concepts, since our systems are unable to answer many questions, and then fuzzy sets have been able to provide solutions to many real world problems. We know that the fuzzy set theory is an extension of classical set theory where the elements have degree of membership. The world “fuzzy” means “vagueness”, this is occurs when the boundary of a piece of information is not clear – cut, Zadeh (1965, 1975) give an extension study of fuzzy sets and he introduce the concepts of fuzzy set and fuzzy theory. While (Zadeh & Sugeno [1970]) developed the probability measures of fuzzy event, Kwaker Naak, Puri and Relescu [1978, 1986] introduced fuzzy random variable, after them Thomas [1979, 1995], elaborates the likelihood semantics for fuzzy sets. In 1988 military handbook electronic reliability design handbook in this field of study. In 1993 Cai etal developed fuzzy system reliability based on fuzzy state assumption and probability assumption [21], but (Cai etal, Chen and Cai [1991, 1994, 1995]) gives the relationship between fuzzy and reliability, many other researchers worked on finding the membership of any given system reliability. Nozer D. [2004] and Sigh Puewalla & June [2006] introduced the members function and probability measures of fuzzy set, and work on developing fuzzy reliability concept for Weibull fuzzy probability distribution and other fuzzy distribution. Here we complete the work in this field to explain the fuzzy reliability models with exponential fuzzy distribution.

2. Some Basic Definitions

v The fuzzy set theory represent the extension of classical set theory where elements have degree of membership.

v The sample space , is a collection of all possible outcomes of random experiments, but the fuzzy subset is defined by the membership function denoted by which produce [0,1] for all , so is mapping function into [0,1], and [4];

And the fuzzy set has the properties;

Ÿ The height of a fuzzy set denoted by is the maximum of the membership grades of A, i.e [8]

Ÿ A fuzzy set A is normal if

Ÿ The support of a set A is the crisp subs of A with non – zero membership grades [3];

Ÿ Many other notations related to fuzzy sets are [3];

Since out research deals with fuzzy probability distribution and reliability, so we need to explain some probability measures of fuzzy events, then of reliability function [16].

v The reliability of a device of a system represent the probability that it will give satisfactory performance for a specified period under specific operating conditions which is denoted by [R(t)] where;

(1)

v Let (T) be continuous random variable then the reliability function is;

(2)

Also the probability that the failure time (T) occurs in an interval is;

(3)

From the last definition we can say that the reliability function or the probability of a device not failing prior to sometime (t), is given by;

(4)

The probability of failure in a given time interval between can be expressed by reliability function as;

(5)

Now we can say that the rate at which the failure occur in the interval is denoted by , which is given by;

(6)

According to above, we can say that the fuzzy reliability represent the probability of a device performing its purpose in varying degrees of success for the period of time intended under operating conditional encountered and it can denoted by , which is a function of a fuzzy set [18, 19].

Let represent the degree of membership (R) in , then;

(7)

Where

(8)

3. Fuzzy Probability Function and its Reliability

This section introduce the probability density function of exponential distribution which is used commonly in reliability engineering and is used to model the behavior of units that have a constant failure rate (or units that do not degrade with time or wear out).

The of two parameters exponential is defined by [10];

(9)

is scale parameter, is location parameter, and its cumulative distribution is [10];

(10)

While the reliability function;

(11)

Also the mean and variance;

The failure rate function (hazard function);

(12)

We assume the values of a fuzzy random variable are fuzzy number [7];

Then the vagueness is a real triangular fuzzy number, ;

(13)

When the random variable (T) have a crisp exponential probability function , then the corresponding fuzzy random variable with fuzzy exponential probability distribution , with the following characteristics;

For all , the cumulative fuzzy distribution function [10];

(14)

Then the fuzzy distribution function where , we have

(15)

While the fuzzy reliability function is defined by;

Then of fuzzy reliability function can be written as;

(16)

4. Estimation Methods

First of all we explain the estimator of by;

4.1. Moment Estimator

Since

(18)

(19)

4.2. Maximum Likelihood Estimator

Let be random sample from (9), then;

Then according to different values of and to specified value of , we can estimate fuzzy reliability function;

(20)

5. Simulation Procedures

To find the estimator's for MOM, MLE, and fuzzy reliability function we perform simulation experiments using Monte Carlo assuming that;

Table (1). Two parameters estimators      , when

Table (2). MSE values for

Table (3). Two parameters estimators      , when

Table (4). MSE values for

Table (5). Comparison between real and estimated fuzzy reliability by two methods

Table (6). Comparison between real and estimated fuzzy reliability by two methods

Table (7). Comparison between real and estimated fuzzy reliability by two methods

Table (8). Comparison between real and estimated fuzzy reliability by two methods

6. Conclusions

The best estimator for is found to be MLE, also for fuzzy reliability when . We conclude that the fuzzy estimator of (R) is best than ordinary estimator since it takes all the variation under consideration.