**Reliability is a concept that uses performance, costs but also the environment.**

## Introduction

Reliability is a key indicator of maintenance. It gives a state of the situation with regard to the level of confidence that we carry to equipment. By definition:

” *Reliability is the ability of a system or component to properly perform the functions provided under specific conditions and for a specified period of time*“^{1}

In other words, reliability is the probability that there will be no failure in a given time interval. Three terms must be defined precisely :

**precise conditions:**One must be able to define clearly and unambiguously what are the conditions of use of the system and therefore its limits.**functions correctly:**You have to be able to define what the correct functioning of the system is.**Period of time :**The period of time is the most significant unit of use for the system. This can be days, seconds, minutes, years, but also kilometers, number of cycles…

The reliability will be expressed in terms of two indicators :

**the failure rate****λ :**It represents the ratio between a number of failures and a duration of use.**The function R :**It is representative of the proportion of*Survivors*at the moment T. Clearly, at the end of X usage time, we have for example a 30% chance of having a breakdown.

**Programmed obsolescence**

According to the definition given by ADEME^{2}, * “The concept of” programmed obsolescence “denounces a ploy whereby a property would see its normative duration knowingly reduced from its conception, thus limiting its Duration of use for reasons of economic model. *»

This notion refers to the fact that manufacturers would be able to identify the right level of reliability to ensure a level of sales that would allow them to increase their profitability. Even though we could have very good reliability, we would design the product to make it break down and the consumer would have to buy back.

Here is a report on the lamp that shines since 1901 while the official lifespan is 1000 hrs.

## The failure rate

The failure rate, also called breakdown rate, gives a number of average failures per unit of use. In other words, this rate gives us the number of failures per km, time… that we have on average. It is calculated using the following formula:

Most often, if the failure rate is traced as a function of time, the so-called ” *bath* ” curve is obtained.

**Area A**

** **

This is the time of *Youth* “or” *Debugging *“. The system is still new and in lapping. We learn to use it little by little and we eliminate the early failures.

**Zone B**

** **

This is the time of *Normal operation “* . The quality of the product has reached a plateau for which the rate of failures becomes constant over time. At this point, the causes of failures can be undetectable defects, too small safety margins, unexpected random stresses, human factors and natural failures.

**Zone C**

** **

This is the time of obsolescence. The system is slowly ageing and more and more failures of wear and tear and old age outages are encountered.

## The Reliability function

Modeling the probability of survival at a time t allows to identify for a certain level of wear (1000km, 500 hours…) What is the probability of failure.

This function is modelled via the law of Weibull. This one has the mathematical peculiarity of adapting to all forms of distributions: normal, exponential…

In its most general Shape , the distribution of Weibull depends on 3 parameters:

**β:**the Shape parameter. It defines the type of degradation phenomenon. At the most it is large, at the most the average reliability is great.**η:**the scale parameter. It defines the operating times of the components studied. This can be in hour, minute, or also, in turn, km, number of cycle… It all depends on the element being studied. It represents the mode and can therefore be taken as a reference or ”*average*” value.**υ:**the position parameter. It takes into account the level of wear of the components.- υ = 0: All components are new
- Υ > 0: The mechanism can only have a failure beyond a certain pre-operating time.
- Υ < 0: There is either a mixture of population or the fact that the element has already been used

## Example

We are a reliability engineer in charge of setting up preventative maintenance. We know that our most frequently lost Part is a joint in the reducer. It crashes quite regularly and especially it is very long to change. So we want to know how often we have to change it in a preventative way to better organize itself.

### 1. Make the data statement

The first step is to make a record of the working times and the downtimes according to the elements causing the failure.

We note the following data:

The Times are in time and we read the table in this way :

My equipment broke down after 559 hours due to a joint problem.

The time to replace it and restart in normal manner was 8 hours.

Then we left for 992 before we met a new joint problem…

**Excerpt from the statement sheet**

### 2. Calculate the first indicators

By parts, we calculate the MTBF (average of the good working times) and the TMP (average time of failure), which allows us to deduce the average reliability.

Taking our example, we get the following results:

- MTBF = 611.5 Hours
- TMP = 6.6 hours
- Average reliability = MTBF/(TMP + MTBF) = 98.94%

In other words, on average our equipment runs 98.94% of the time without joint problem.

### 3. Identify the parameters of Weibull’s Law

It is recommended with regard to the complexity of the Weibull function to go through a statistical software. In Minitab:

- Put your working time on a Minitab worksheet
- Go to
*graph*, then*histogram* - Select
*with Curve* - In view, select Distribution
*Weibull-3 parameters* - Finally, you get the distribution graph with the Weibull curve and the 3 parameters you are looking for.

### 4. Identify your likelihood of failure

At this stage, we have the distribution of the good working times and the associated Weibull curve. The whole issue is how often you need to change the joint in a preventative way to avoid significant losses.

Imagine, that one wishes to change it at a frequency corresponding to a probability of 50% chance that it is still alive (value totally arbitrary). To do this, the inverse function of the Weibull distribution is used. In Minitab:

- Go to
*graph*, then*Distribution of probabilities* - Select
*View Probabilities* - Finally, enter your Weibull settings and then choose the level of probability you want. For 50%, you get the graph below.

Thus, in our example, we have 50% chance that our joint is still in good condition for a running time of 671.4 hours.

### 5. Optimize your Maintenance

By successive iteration and confronting cost of the breakdown/cost of maintenance, we will optimize our process.

## Reliability calculation for a system as a whole

In case, for example, we are forced to change a complete subset while only one Part is down.

To study this type of system, we are going to be in 2 cases:

**Series Reliability**

**Parallel reliability**

With:

**fi:**Individual reliability**I:**number of Part of the system

To identify the type of modeling, simply answer the following question for each of the system components:

*Does the failure of this component cause the system to fail?*

**The answer is yes:**the components are in series**the answer is no:**the components are in parallel

## Source

1 – Institute of Electrical and Electronics Engineers (1990) – IEEE Standard Computer Dictionary: A compilation of IEEE standards glossaries

2 – ADEME (2012) – Study on the lifespan of electrical and electronic equipment

H. Procaccia, E. Ferton, M. Procaccia (2011) – Reliability and maintenance of repairable and non-repairable industrial equipment.

W. Torell, V. Avelar (2004) – Mean Time Between Failure or average working time: definitions and standards

C. Lasse (2009)-English-French dictionary for the Maintenance Professional