MB Matheblog # 11 |
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2021-07-01 | deutsche Version |

We address a problem occurring with functionality checks of multi-component assemblies. A mass-produced technical assembly consists of \(n\) components which may interact and can be activated independently by \(n\) switches. In this case Quality Control (QC) is assumed to imply a functionality check for varying positions of the switches. We ask for the probability that after \(m\) runs of these checks all components have been in ON position at least once. This provides useful information within the prerequisites of Statistical Process Control (SPC). A comprehensive SPC as a part of QC would consist of \(2^n-1\) functionality checks (for each member of a sample) covering all possible ON/OFF-positions of the switches, skipping simultaneous OFF for all switches.

A once-at-a-time check of each singular switch would not be appropriate in terms of QC. In technical assemblies an internal interaction of the components should normally be assumed. But the switches will be independently wired to the components. This leads to the requirement to test various combinations of ON/OFF positions of the switches.

This reminds us of one of W.E. Deming's famous 14 rules (cited e.g. in [6; 1.6]):

While this rule is quite commonplace in modern quality management the cited

\(2^n-1\) checks would indeed represent mass inspection within the sample and could in practice turn out to be a too big number of checks to deal with efficiently, so a different (statistical) approach should be considered: For each test of a produced assembly a random string of \(n\) \(1\)s and \(0\)s is generated (excluding an all \(0-\)string) which defines the ON/OFF positions of the switches during the functionality check.

We will present two operational scenarios based on this random procedure. The first will deal with \(m\) checks for each produced assembly which receives quality inspection. As we have mass production in mind a second scenario will be taken into consideration. If we look at (sufficiently many) \(m\) consecutively produced assemblies which are each tested only once it should be highly unlikely that a component which turns out to be defective repeatedly in consecutive assemblies would remain undetected for a long time. Our task will be to provide a corresponding exact probability as a function of the number \(m\) of checks.

We present an example. A small machine has a lot of different parts but \(n=6\) components have an independent power supply controlled by \(6\) switches. Because the parts may interact it is not sufficient to check each component individually. In QC one of the following scenarios is chosen:

In both scenarios a simple binary matrix representation of the testing procedure suggests itself. Each matrix row stands for a switch and each column stands for a check. The \(1\)s and \(0\)s in a column will then show the ON/OFF status of the switches during a certain check.

With \(n=6\) and \(m=7\) the resulting \((6\times 7)-\)matrix \(B\) would show a random example (as in table 1 below) for the ON/OFF-positions of the \(6\) switches during each of the \(7\) functionality checks.

\(B=\left(\begin{array}{lllllll} 1 & 1 & 0 & 1 & 0 & 1 & 1\\ 1 & 0 & 0 & 1 & 1 & 1 & 1\\ 0 & 1 & 0 & 1 & 1 & 1 & 0\\ 1 & 0 & 0 & 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 1 & 1 & 1 & 1\\ 1 & 0 & 1 & 0 & 0 & 1 & 0\end{array}\right)\)

Which probability \(p(n,m)\) can we assign to the event that each component has been activated at least once after \(m\) checks - on the same machine in scenario 1, or on \(m\) consecutive machines in scenario 2 ?

Section 4 will provide an easily evaluable formula for this probability \(p(n,m)\) . For \(n=6\) and \(m=7\) we will get \(p(6,7)\approx 0.9587\). So \(7\) checks will nearly always suffice to reach the goal of each switch having been ON at least once.

The needed random binary numbers can easily be automatically produced; any standard calculation software tool provides a random number program.

The problem we addressed above concerns

\(M = \{m_1,~...~,m_n\}\) will stand for a numbered set of \(n\) components (e.g. \(m_3=\)

\((A_1,~...~,A_m)\) is called a

In our QC application we define \(1=\) ON and \(0=\) OFF for the switch positions. Then

The number \(u(n,m)\) of all coverings of a \(n-\)element set \(M\) by \(m\) non-empty subsets is the number of \(m-\)tuples \((A_1,... ,A_m)\) with \(\bigcup A_k = M\) , i.e. \(M\) being the union of the non-empty sets \(A_k\) . Note that in this definition the \(A_k\) are ordered. \(u(n,m)\) was derived

The theorem can be expressed in stochastic terms. We pick \(m\) non-empty subsets from \(M\) randomly and independently. In the QC context these are \(m\) random choices of positions of the switches. We ask for the probability that these subsets cover \(M\) . As \(M\) has \(2^n-1\) non-empty subsets there are \((2^n-1)^m\) possible choices of ordered subsets. So the probability in question is \[p(n,m) = \frac{u(n,m)}{(2^n-1)^m}\] Now the theorem gives the final result. We will express it for QC purposes:

The probability that after \(m\) runs of functionality checks with random choices of ON/OFF-positions for \(n\) components all components will have been activated at least once is \[p(n,m) = \frac{1}{(2^n-1)^m}\cdot \sum_{j=0}^m (-1)^{m-j} \binom{m}{j} (2^j-1)^n\] This is our main result. It applies to both scenarios in section 1. The complement probability \(1-p(n,m)\) is the risk of missing at least one component in the testing procedure.

Statistical process control features a wide range of powerful instruments for gaining statistical evidence concerning quality criteria. One of these instruments is the

We revisit the example in section 1, scenario 1 and take now \(m\) as a variable. Each machine is checked \(m\) times with randomly generated settings of the activation of the \(n=6\) components. We want to choose a sufficiently small \(m\) . \(p(6,m)\) gives the probability of each component having been activated at least once. This is shown in table 2.

1 0.0159 6 0.9175

2 0.1832 7 0.9587

3 0.4618 8 0.9795

4 0.6935 9 0.9899

5 0.8381 10 0.9950

Which \(m\) is "sufficiently small"? We can read table 2 as a list of confidence values: \(m = 7\) gives a \(95\%\) confidence for each component having been activated at least once, \(m = 10\) gives a \(99\%\) confidence.

Quality engineers may be interested in a condensed information in order to avoid the evaluation of \(p(n,m)\) . Table 3 provides the minimum \(m\) for two confidence levels as a function of the number \(n\) of components. As a rule-of-thumb it seems safe to perform \(m=10\) checks on not too complex assemblies. Table 3 also shows that for assemblies with few components smaller values of \(m\) will suffice. - On the other hand, checking every component in ON position at least once in a test cycle may not be the only goal of SPC. We should keep in mind that the procedure described in this article tests only a small sample of potential switch positions. If there are many interactions between the components it would be advisable to increase \(m\) .

2 4 5

3 5 7

4 6 8

5 7 9

6 7 10

7 - 10 8 10

11 - 13 8 11

14 - 20 9 11

21 - 26 9 12

27 - 41 10 12

42 - 52 10 13

53 - 82 11 13

83 - 105 11 14

106 - 150 12 14

[1]

[2] Allen, T.:

[3] Dietrich, E., Schulze, A.:

[4] Joglekar, A.:

[5] Pyzdek, T.:

[6] Thompson, J., Koronacki, J.:

Comments are welcome.

Last update 2020-10-21

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