Product Maturity, Volume 2
Description
While there have been theoretical and practical advances in reliability from the 1960s to the end of the 1990s, to take into account the effect of maintenance, the maturity of a product is often only partially addressed. Product Maturity 2 fills this gap as much as possible; a difficult exercise given that maturity is a transverse activity in the engineering sciences; it must be present throughout the lifecycle of a product.
Franck Bayle is an electronic engineer by training. He has practiced for almost 15 years, working at Crouzet and then at Thalès in Valence, France. He has also worked in reliability and maturity.
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Content
Foreword by Laurent Denis ix
Foreword by Serge Zaninotti xiii
Acknowledgements xv
Introduction xvii
Chapter 1 Sampling in Manufacturing 1
1.1 Cost aspects 2
1.2 Considering the distribution of defects 7
1.3 Considering the test coverage 10
Chapter 2 Compliance Test 13
Chapter 3 Non-Regression Tests 17
3.1 Non-regression on a physical quantity 17
3.2 Non-regression depending on time 20
Chapter 4 Zero-Failure Reliability Demonstration 23
4.1 Purpose of zero-failure tests 23
4.2 Theoretical principle 23
4.2.1 Non-maintained products 24
4.2.2 Maintained products 29
4.2.3 Estimation of parameter ß 32
4.2.4 Physical laws of failure 35
4.3 Optimization of test costs 42
4.4 Specific cases 48
4.4.1 Imposed number of parts 48
4.4.2 Imposed testing time 48
4.4.3 Imposed testing time and number of parts 49
4.4.4 A test was already conducted and the demonstrated reliability should be estimated 50
4.4.5 One test was already conducted and failure to demonstrate reliability must be known 51
4.4.6 Two tests were conducted 51
4.4.7 A second test is conducted 60
4.4.8 Reliability objective is a failure rate 69
4.4.9 Reliability data are available from the manufacturer 71
4.4.10 Demonstration of reliability at the product level 74
4.4.11 Taking into account a complex life profile 76
Chapter 5 Reliability Management 79
5.1 Context 79
5.2 Physical architecture division 80
5.3 Classification of subsets 81
5.4 Allocation of initial reliability 81
5.5 Estimation of the reliability of subsets 82
5.5.1 Consistency with the experience feedback 85
5.5.2 Estimation of the power of the test 85
5.5.3 Simulation algorithm 85
5.6 Optimal allocation of the reliability of subsets 90
5.7 Illustration 90
5.8 Definition of design rules 103
5.9 Construction of a global predicted reliability model with several manufacturers 107
Chapter 6 Confirmation of Maturity 115
6.1 Internal data from equipment manufacturer 115
6.2 System manufacturer data 117
6.2.1 Original fit removal rate or "zero hour returns" 117
6.3 End-customer data 121
6.3.1 Burn-in effectiveness 121
6.3.2 First failure analysis 121
6.3.3 Method based on failure analysis 124
6.3.4 Observed reliability 124
6.3.5 Estimation of the forecasting number of catastrophic failures 128
6.4 Burn-in optimization 134
6.4.1 Distribution of failures observed during HASS cycles 134
6.4.2 Verification of the degradation of the manufacturing process 136
List of Notations 139
List of Definitions 141
List of Acronyms 147
References 151
Index 155
1
Sampling in Manufacturing
Chapters 1 to 6 of Volume 1 described various methods for building maturity. However, from a manufacturing perspective, these methods must be cost-effective. One of the solutions that can be considered to reduce costs is to test less than 100% of the products before delivery to the system manufacturer. This is called sampling.
As expected, there are various standards dealing with this subject, such as ISO 28590. These standards clarify the sampling rules to be applied, and the interested reader is invited to read them for further details.
However, the standards do not cover several aspects that are very important for the manufacturer:
- - The cost aspect, which leads to the following questions:
- - What is the benefit of applying a sampling rule?
- - Is a sampling rule adapted for my application?
- - What rule should I use to minimize costs?
- - What is the impact of test coverage rates if they are not 100%?
- - What is the impact of considering a distribution of potential defects?
This chapter aims to suggest a solution for each of these cases in order to formulate optimum sampling in terms of quality and cost.
Theoretically speaking, sampling techniques rely on discrete probability distributions (the random variable can only take certain values), unlike the probability distributions for estimating the reliability of a failure mechanism, which are continuous (e.g. exponential, Weibull, etc.). The Bernoulli distribution is used for the result of a test (failure or success). When this test is repeated several times, two cases are possible.
- - The "draw" is unrestricted, and in this case the binomial distribution is applicable.
- - The "draw" is restricted, and in this case the hypergeometric distribution is applicable.
Sampling is obviously a "restricted" draw. Therefore, the theoretical basis of the sampling norms is the hypergeometric distribution.
1.1. Cost aspects
The function of "cost associated with a size n sampling" is a random variable (the batch is accepted or rejected). Therefore, its average value (mathematical expectation) is considered here.
Given:
- - C is the cost per unit of the non-compliance test;
- - K is the cost per unit of accepting a non-compliant product in a batch;
- - N is the number of products to be tested;
- - X is the number of "non-compliant" products in the batch;
- - n is the size of the sample.
The average cost is equal to the cost of compliant products plus the cost of non-compliant products. The cost of compliant products (if the batch was accepted) is given by:
[1.1]P1 is the probability of having "no non-compliant parts" in a size "n" sample, randomly drawn from a batch of size "N". Therefore it is:
[1.2]where is the number of combinations of x out of y elements.
It is worth considering the details of this result. The probability of an event can be estimated by the ratio of the number of possible cases to the total number of cases. It is clear that the total number of cases is equal to the number of combinations of "n" taken out of "N", or
The number of possible cases is equal to the number of combinations of compliant parts, or N-X, and is therefore equal to hence the result of equation [1.2].
The cost of non-compliant products (if the batch was rejected) is given by:
[1.3]Based on equations [1.1], [1.2] and [1.3], the total cost is:
[1.4]As an illustration, assume that:
- - there are 100 products to be delivered N = 100;
- - the cost of a compliant part is C = 30?;
- - the cost of a non-compliant part is K = 1,500?.
According to this data, the cost of a non-compliant part is very high compared to the cost of a compliant part. Let us now consider the evolution of the total average cost depending on the size (n) and the number of defective products (X) (see Figure 1.1).
It can be noted that starting with X > 1, the sampling rate of 100% is optimal. In this example, the sampling is not interesting in terms of cost.
Now assume that:
- - there are 100 products to be delivered N = 100;
- - the cost of a compliant part is C = 1,000?;
- - the cost of a non-compliant part is K = 1,500?.
Figure 1.1. Evolution of the total average cost depending on the size of the sample and the number of defective products
Although low, the cost of a compliant part is of the order of the cost of non-compliant parts. Let us now consider the evolution of the total average cost depending on the sample size (n) and the number of defective products (X) (see Figure 1.2).
Figure 1.2. Evolution of the total average cost depending on the size of the sample and the number of defective products
In contrast to the previous example, in this case the sampling is better than the test at 100%.
Now assume that:
- - there are 100 products to be delivered N = 100;
- - the cost of a compliant part is C = 100?;
- - the cost of a non-compliant part is K = 1,500?.
Let us now consider the evolution of the total average cost depending on the sample size (n) and the number of defective products (X) (see Figure 1.3).
Figure 1.3. Evolution of the total average cost depending on the size of the sample and the number of defective products
This situation is an intermediate one between the two previously mentioned examples, as the sampling is optimal when the number of defects is equal to or greater than 7.
1.2. Considering the distribution of defects
Given p(X), the probability density of the defects observed during this phase of the test, the mathematical expectation of the cost is:
[1.5]If this distribution is not known, the uniform probability distribution can be used:
Given that the number of non-compliant products ranges between 0 and N, the probability density is then:
[1.6]Using equations [10.5] and [10.6], the mathematical expectation of the cost is given by:
[1.7]Once again, as an illustration, let us resume the following example and assume that:
- - there are 100 products to be delivered N = 100;
- - the cost of a compliant part is C = 30?;
- - the cost of a non-compliant part is K = 1,500?.
Let us now consider the evolution of the total average cost depending on the sample size (n) (see Figure 1.4).
It can be noted that for a small sample size, the "average" cost can be very high.
Figure 1.4. Evolution of the total average cost depending on the sample size - Example 1
Assume that:
- - there are 100 products to be delivered N = 100;
- - the cost of a compliant part is C = 100?;
- - the cost of a non-compliant part is K = 1,500?.
Let us now examine the evolution of the total average cost depending on the sample size (n) (see Figure 1.5).
It can be noted that for a small sample size, the "average" cost can be very high. However, a slight optimum is obtained for n = 20, or a sampling rate of 20%, in this example.
Assume that:
- - there are 100 products to be delivered N = 10;
- - the cost of a compliant part is C = 1,000?;
- - the cost of a non-compliant part is K = 1,500?.
Let us now examine the evolution of the total average cost depending on the sample size (n) (see Figure 1.6).
Figure 1.5. Evolution of the total average cost depending on the sample size - Example 2
Figure 1.6. Evolution of the total average cost depending on the sample size - Example 3
It can be noted that for a small sample size, the "average" cost is the lowest.
1.3. Considering the test coverage
Given Pt, the probability of detecting a non-compliant product for a test, knowing that it will be detected by the client, let us denote by pnMNXt the probability of having M "non-compliant" parts in a sample of size n, drawn from a batch of size N, in which X parts are non-compliant.
Figure 1.7. Illustration
Here, the number of possible cases is the number of possibilities of drawing M defective parts from X and drawing "n-M" compliant parts from the "N-X" compliant parts of the batch. The probability pnMNXt is then written as (hypergeometric...
