Test-Find-Test Data Example: Difference between revisions

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<noinclude>{{Banner RGA Examples}}
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''This example appears in the [[Crow_Extended|Reliability Growth and Repairable System Analysis Reference book]]''.
''This example appears in the [[Crow_Extended|Reliability Growth and Repairable System Analysis Reference]]''.
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This example appears in the Reliability Growth and Repairable System Analysis Reference.


Consider the data in the first table below. A system was tested for [math]\displaystyle{ T=400\,\! }[/math] hours. There were a total of [math]\displaystyle{ N=42\,\! }[/math] failures and all corrective actions will be delayed until after the end of the 400 hour test. Each failure has been designated as either an A failure mode (the cause will not receive a corrective action) or a BD mode (the cause will receive a corrective action). There are [math]\displaystyle{ {{N}_{A}}=10\,\! }[/math] A mode failures and [math]\displaystyle{ {{N}_{BD}}=32\,\! }[/math] BD mode failures. In addition, there are [math]\displaystyle{ M=16\,\! }[/math] distinct BD failure modes, which means 16 distinct corrective actions will be incorporated into the system at the end of test. The total number of failures for the [math]\displaystyle{ {{j}^{th}}\,\! }[/math] observed distinct BD mode is denoted by [math]\displaystyle{ {{N}_{j}}\,\! }[/math], and the total number of BD failures during the test is [math]\displaystyle{ {{N}_{BD}}=\underset{j=1}{\overset{M}{\mathop{\sum }}}\,{{N}_{j}}\,\! }[/math]. These values and effectiveness factors are given in the second table.

Do the following:

  1. Determine the projected MTBF and failure intensity.
  2. Determine the growth potential MTBF and failure intensity.
  3. Determine the demonstrated MTBF and failure intensity.
Test-Find-Test Data
[math]\displaystyle{ i\,\! }[/math] [math]\displaystyle{ {{X}_{i}}\,\! }[/math] Mode [math]\displaystyle{ i\,\! }[/math] [math]\displaystyle{ {{X}_{i}}\,\! }[/math] Mode
1 15 BD1 22 260.1 BD1
2 25.3 BD2 23 263.5 BD8
3 47.5 BD3 24 273.1 A
4 54 BD4 25 274.7 BD6
5 56.4 BD5 26 285 BD13
6 63.6 A 27 304 BD9
7 72.2 BD5 28 315.4 BD4
8 99.6 BD6 29 317.1 A
9 100.3 BD7 30 320.6 A
10 102.5 A 31 324.5 BD12
11 112 BD8 32 324.9 BD10
12 120.9 BD2 33 342 BD5
13 125.5 BD9 34 350.2 BD3
14 133.4 BD10 35 364.6 BD10
15 164.7 BD9 36 364.9 A
16 177.4 BD10 37 366.3 BD2
17 192.7 BD11 38 373 BD8
18 213 A 39 379.4 BD14
19 244.8 A 40 389 BD15
20 249 BD12 41 394.9 A
21 250.8 A 42 395.2 BD16

Effectiveness Factors for the Unique BD Modes
BD Mode Number [math]\displaystyle{ {{N}_{j}}\,\! }[/math] First Occurrence EF [math]\displaystyle{ {{d}_{i}}\,\! }[/math]
1 2 15.0 .67
2 3 25.3 .72
3 2 47.5 .77
4 2 54.0 .77
5 3 54.0 .87
6 2 99.6 .92
7 1 100.3 .50
8 3 112.0 .85
9 3 125.5 .89
10 4 133.4 .74
11 1 192.7 .70
12 2 249.0 .63
13 1 285.0 .64
14 1 379.4 .72
15 1 389.0 .69
16 1 395.2 .46

Solution

  1. The maximum likelihood estimates of [math]\displaystyle{ {{\beta }_{BD}}\,\! }[/math] and [math]\displaystyle{ {{\lambda }_{BD}}\,\! }[/math] are determined to be:
    [math]\displaystyle{ \begin{align} {{{\hat{\beta }}}_{BD}} = & \frac{M}{\underset{i=1}{\overset{M}{\mathop{\sum }}}\,\ln (\tfrac{T}{{{X}_{i}}})} \\ = & 0.7970 \\ {{{\hat{\lambda }}}_{BD}} = & 0.1350 \end{align}\,\! }[/math]
    The unbiased estimate of [math]\displaystyle{ \beta \,\! }[/math] is:
    [math]\displaystyle{ \begin{align} {{\overline{\beta }}_{BD}} = & \frac{M-1}{M}{{{\hat{\beta }}}_{BD}} \\ = & 0.7472 \end{align}\,\! }[/math]
    Based on the test data, [math]\displaystyle{ \overline{d}=\tfrac{1}{M}\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{d}_{i}}= 0.72125\,\! }[/math]. Therefore, [math]\displaystyle{ B(T)=\overline{d}\tfrac{M{{\overline{\beta }}_{BD}}}{T}=0.0215\,\! }[/math]. The projected failure intensity due to incorporating the 16 corrective actions is:
    [math]\displaystyle{ \begin{align} r(T) = & \left( \frac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{{{N}_{i}}}{T} \right)+\overline{d}\left( \frac{M}{T}{{\overline{\beta }}_{BD}} \right) \\ = & 0.0661 \end{align}\,\! }[/math]
    The projected MTBF is:
    [math]\displaystyle{ M\widehat{T}B{{F}_{P}}={{[r(T)]}^{-1}}=15.127\,\! }[/math]
  2. To estimate the maximum reliability that can be attained with this management strategy, use the following calculations.
    [math]\displaystyle{ \begin{align} {{N}_{A}}/T=0.0250 \end{align}\,\! }[/math]
    [math]\displaystyle{ \frac{1}{T}\underset{i=1}{\overset{16}{\mathop \sum }}\,(1-{{d}_{i}}){{N}_{i}}=0.0196\,\! }[/math]
    The growth potential failure intensity is estimated by:
    [math]\displaystyle{ \begin{align} {{\widehat{r}}_{GP}}(T) = & \left( \frac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{{{N}_{i}}}{T} \right) \\ = & 0.0250+0.0196 \\ = & 0.0446 \end{align}\,\! }[/math]
    The growth potential MTBF is:
    [math]\displaystyle{ M\widehat{T}B{{F}_{GP}}={{[{{\widehat{r}}_{GP}}]}^{-1}}=22.4467\,\! }[/math]
  3. The demonstrated failure intensity and MTBF are estimated by:
    [math]\displaystyle{ \begin{align} {{\widehat{\lambda }}_{D}}(T) = & \frac{{{N}_{A}}+{{N}_{BD}}}{T} \\ = & \frac{42}{400} \\ = & 0.1050 \end{align}\,\! }[/math]
    [math]\displaystyle{ \begin{align} M\widehat{T}B{{F}_{D}} = & {{[{{\widehat{\lambda }}_{D}}(T)]}^{-1}} \\ = & 9.5238 \end{align}\,\! }[/math]
    The first chart below shows the demonstrated, projected and growth potential MTBF. The second shows the demonstrated, projected and growth potential failure intensity.
    Demonstrated, projected and growth potential MTBF.
    Demonstrated, projected and growth potential failure intensity.