In this article, mathematical modeling for the evaluation of reliability is studied using two methods. One of the methods, is developed based on possibility theory. The performance of the reliability of the system is of prime concern. In view of this, the outcomes for the failure are required to evaluate with utmost care. In possibility theory, the reliability information data determined from decision-making experts are subjective. The same method is also related to the survival possibilities as against the survival probabilities. The other method is the one that is developed using the concept of approximation of closed interval including the piecewise quadratic fuzzy numbers. In this method, a decision-making expert is not sure of his/her estimates of the reliability parameters. Numerical experiments are performed to illustrate the efficiency of the suggested methods in this research. In the end, the paper is concluded with some future research directions to be explored for the proposed approach.

In the past three decades, one of the challenging endeavors has been the design and manufacturing of large-scale reliable systems. These large-scale reliable systems are used in various application areas. Some of them are military applications, space exploration, commercial operations, and others deal in power distribution. Unlike the manufacturing of components, the manufacturing of such reliable systems includes the broader aspects of organizing composite manufacturing tools, operating organization and costs, as well as maintenance schedules, plus the skills required to make the system performance as an integrated entity too.

The study field of reliability engineering contains various activities, of which reliability modeling happens to be one area of the utmost significant. The traditional solution techniques are applied extensively based on probabilistic techniques, while the system survival probability is formulated by considering the statistical information (reliabilities or survival probabilities) of its sub-systems or components. In 1982, Martz et al. [

A fuzzy set is adopted to cope with the uncertainty in mathematical models. In 1965, Zadeh [

In 2007, Hryniewicz [

Several researchers used the fuzzy interval to deal with uncertainty in reliability models (Washio et al. [

The above-reported literature review reveals the scope still remaining to innovate and study regarding both survival possibilities and the closed interval approximation of fuzzy numbers. Thus, a research gap is realized. In this paper, the reliability methods for modeling and evaluation based on survival possibilities and closed interval approximation of fuzzy numbers are introduced.

The outlay of the paper is constructed as follows:

In this section, some basics and concepts based on fuzzy numbers, piecewise quadratic fuzzy numbers, and their closed approximation intervals are recalled.

The interval of confidence at a level

Consider a fuzzy number

Also, consider the fuzzy number represents the possibility of failure at time

Now, consider

The cumulative failure possibility distribution of

For

Therefore,

Assume that

In the probability theory of reliability, the logarithmic derivative of the reliability function

Similarly, in the possible theory of reliability, we characterize the failure rate by

The below demonstrated

In the case of

If the failure rate

For

The failure rate

Once can get the initial failure rate

The solution of the differential

For

···

Here

Now, we examine some examples of reliability function as follows:

The corresponding survival function is written by

There is another indicator of reliability called cumulative failure rate

It is noted in the

Proceeding in the same way, we obtain

Now, let us consider the case where a system is introduced into a new service at

Here, it is observed that

The corresponding failure rate is computed by

In addition, the cumulative failure rate is given by

This section is divided into two sub-sections as follows.

Consider the situation where an expert is uncertain about the estimation of model parameters. Instead of describing the reliability function using one survival law, he/she gives two subjective functions forming an interval of confidence. For example, he/she gives

Through the closed interval approximation, the subjective survival law can be represented by

From the Definition of the failure rate and the survival function, we have

This completes the proof.

In this sub-section, we present some examples as follows:

The corresponding closed interval approximation for the survival law is presented by

The data of failure rate and survival law for the Example 1 is illustrated in

Now, let the failure rate be

Then, the corresponding survival law is presented by

The failure rate is not necessary monotonic, but the cumulative failure rate is always monotonic as in the definition, therefore we obtain

The closed interval approximation of the cumulative failure rate is expressed by

In addition, it is possible to define in another way the closed interval approximation for the survival law

The corresponding closed intervals approximation for time to survival

The divergence of these two cases is evaluated and are expressed in

It is quite evident that for

In terms of failure rate,

By applying the survival law, one can write

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||
---|---|---|---|---|---|---|---|---|---|---|

0 | 0.1 | 0.12 | 0.25 | 0.3 | 0.63 | 0.79 | 0.93 | 0.98 | 1 | |

1 | 1 | 0.85 | 0.646 | 0.381 | 0.182 | 0.045 | 0.004 | 0 | 0 |

Using

Also, the mean failure rate using

Thus, the closed interval approximation for the mean failure time

In industrial problems, the performance of the system reliability is of utmost concern. Thus, the outcomes for the failure of one or more machines are required to compute with extreme care. In this paper, we have demonstrated that when uncertainty is associated with the system, the reliability function, as well as some other similar criteria, are formulated as a mathematical model using the closed interval approximations of a piecewise quadratic fuzzy number. Apart from this, the intervals of confidence of fuzzy numbers, instead of the probabilistic, are also used in suggested modeling to handle the associated uncertainty. Extensions of the possibilistic criteria to several scenarios of reliability evaluation may lead to some interesting studies. There are several future possibilities to extend the proposed work in many application areas. One can consider the fuzzy type-2 sets (Tang et al. [

Researchers would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.