Failure Discounting Example

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This example appears in the article Failure Discounting.

Assume that during the 22 launches given in the first table below, the first failure was caused by Mode 1, the second and fourth failures were caused by Mode 2, the third and fifth failures were caused by Mode 3, the sixth failure was caused by Mode 4 and the seventh failure was caused by Mode 5.

1. Find the standard Gompertz reliability growth curve using the results of the first 15 launches.
2. Find the predicted reliability after launch 22.
3. Calculate the reliability after launch 22 based on the full data set from the second table, and compare it with the estimate obtained for question 2.

Launch sequence with failure modes and failure values
Launch Number	Result/Mode	Failure 1	Failure 2	Failure 3	Failure 4	Failure 5	Failure 6	Failure 7	Sum of Failures
1	F1	1.000							1.000
2	F2	1.000	1.000						2.000
3	F3	0.900	1.000	1.000					2.900
4	S	0.684	0.900	1.000					2.584
5	F2	0.536	1.000	0.900	1.000				3.436
6	F3	0.438	1.000	1.000	1.000	1.000			4.438
7	S	0.369	0.900	1.000	0.900	1.000			4.169
8	S	0.319	0.684	0.900	0.684	0.900			3.486
9	S	0.280	0.536	0.684	0.536	0.684			2.720
10	S	0.250	0.438	0.536	0.438	0.536			2.197
11	S	0.226	0.369	0.438	0.369	0.438			1.839
12	S	0.206	0.319	0.369	0.319	0.369			1.581
13	S	0.189	0.280	0.319	0.280	0.319			1.387
14	S	0.175	0.250	0.280	0.250	0.280			1.235
15	S	0.162	0.226	0.250	0.226	0.250			1.114
16	S	0.152	0.206	0.226	0.206	0.226			1.014
17	F4	0.142	0.189	0.206	0.189	0.206	1.000		1.931
18	S	0.134	0.175	0.189	0.175	0.189	1.000		1.861
19	F5	0.127	0.162	0.175	0.162	0.175	0.900	1.000	2.701
20	S	0.120	0.152	0.162	0.152	0.162	0.684	1.000	2.432
21	S	0.114	0.142	0.152	0.142	0.152	0.536	0.900	2.138
22	S	0.109	0.134	0.142	0.134	0.142	0.438	0.684	1.783

Comparison of the predicted reliability with the actual data
Launch Number	Calculated Reliability (%)	ln(R)	Gompertz Reliability (%)
1	0.000
2	0.000
3	3.333	1.204
4	35.406	3.567	16.426
5	31.283	3.443	26.691
6	26.039	3.260	37.858
7	40.442	3.670	48.691
8	56.422	4.033	58.363
9	69.783	4.245	66.496
		[math]\displaystyle{ {{S}_{1}}\,\! }[/math] = 22.218
10	78.029	4.357	73.044
11	83.281	4.422	78.155
12	86.824	4.464	82.055
13	89.331	4.492	84.983
14	91.175	4.513	87.155
15	92.573	4.528	88.754
		[math]\displaystyle{ {{S}_{2}}\,\! }[/math] = 26.776
16	93.660	4.540	89.923
17	88.639	4.484	90.774
18	89.661	4.496	91.392
19	85.787	4.452	91.839
20	87.841	4.476	92.163
21	89.820	4.498	92.396
		[math]\displaystyle{ {{S}_{3}}\,\! }[/math] = 26.946
22	91.896 4.521 92.565

Solution

1. In the table above, the failures are represented by columns "Failure 1", "Failure 2", etc. The "Result/Mode" column shows whether each launch is a failure (indicated by the failure modes F1, F2, etc.) or a success (S). The values of failure are based on [math]\displaystyle{ CL=0.90\,\! }[/math] and are calculated from:

   [math]\displaystyle{ f=1-{{(1-CL)}^{\tfrac{1}{{{S}_{n}}}}}\,\! }[/math]

   These values are summed and the reliability is calculated from:

   [math]\displaystyle{ R=\left[ 1-\left( \frac{\mathop{}_{i=1}^{N}{{f}_{i}}}{n} \right) \right]\cdot 100\text{ }%\,\! }[/math]

   where [math]\displaystyle{ N\,\! }[/math] is the number of failures and [math]\displaystyle{ n\,\! }[/math] is the number of events, tests, runs or launches.
   • Failure 1 is Mode 1; it occurs at launch 1 and it does not recur throughout the process. So at launch 3, [math]\displaystyle{ {{S}_{n}}=1\,\! }[/math], and so on.
   • Failure 2 is Mode 2; it occurs at launch 2 and it recurs at launch 5. Therefore, [math]\displaystyle{ {{S}_{n}}=1\,\! }[/math] at launch 4 and at launch 7, and so on.
   • Failure 3 is Mode 3; it occurs at launch 3 and it recurs at launch 6. Therefore, [math]\displaystyle{ {{S}_{n}}=1\,\! }[/math] at launch 5 and at launch 8, and so on.
   • Failure 6 is Mode 4; it occurs at launch 17 and it does not recur throughout the process. So at launch 19, [math]\displaystyle{ {{S}_{n}}=1\,\! }[/math], and so on.
   • Failure 7 is Mode 5; it occurs at launch 19 and it does not recur throughout the process. So at launch 21, [math]\displaystyle{ {{S}_{n}}=1\,\! }[/math], and so on.
   For launch 3 and failure 1, [math]\displaystyle{ {{S}_{n}}=1\,\! }[/math].

   [math]\displaystyle{ \begin{align} {{f}_{1/3}}=1-{{(1-0.90)}^{1/1}}=0.900 \end{align}\,\! }[/math]

   For launch 4 and failure 1, [math]\displaystyle{ {{S}_{n}}=2\,\! }[/math].

   [math]\displaystyle{ \begin{align} {{f}_{1/4}}=1-{{(1-0.90)}^{1/2}}=0.684 \end{align}\,\! }[/math]

   And so on. Calculate the initial values of the Gompertz parameters using the second table above. Based on the equations from the Gompertz Models chapter, the initial values are:

   [math]\displaystyle{ \begin{align} c &= \left ( \frac{S_{3}-S_{2}}{S_{2}-S_{1}} \right )^\frac{1}{n\cdot I} \\ &= \left [ \frac{26.946-26.776}{26.776-22.218} \right ]^\frac{1}{6} \\ &= 0.578 \\ \end{align}\,\! }[/math]

   [math]\displaystyle{ \begin{align} a &= e^\left [\frac{1}{n}\left (S_{1} + \frac {S_{2}-S_{1}}{1-e^{n\cdot I}} \right )\right ] \\ &= e^\left [\frac{1}{6}\left (22.218 + \frac{26.776 - 22.218}{1-0.578^{6}}\right ) \right ] \\ &= 89.31% \\ \end{align}\,\! }[/math]

   [math]\displaystyle{ \begin{align} b &= e^\left [\frac{(S_{2}-S_{1})(c-1)}{(1-c^{n})^{2}} \right ] \\ &= e^\left [\frac{(26.776-22.218)(0.578-1)}{(1-0.578^{6})^{2}} \right ] \\ &= 0.127 \\ \end{align}\,\! }[/math]

   Now, since the initial values have been determined, the Gauss-Newton method can be used. Substituting [math]\displaystyle{ {{Y}_{i}}={{R}_{i}},\,\! }[/math] [math]\displaystyle{ g_{1}^{(0)}=89.31,\,\! }[/math] [math]\displaystyle{ g_{2}^{(0)}=0.127,\,\! }[/math] [math]\displaystyle{ g_{3}^{(0)}=0.578\,\! }[/math]. The iterations are continued to solve for the parameters. Using the RGA software, the estimators of the parameters for the given example are:

   [math]\displaystyle{ \begin{align} \widehat{a}&= 0.9299 \\ \widehat{b} &= 0.0943 \\ \widehat{c} &= 0.7170 \end{align}\,\! }[/math]

   The next figure shows the entered data and the estimated parameters.
   

   The Gompertz reliability growth curve may now be written as follows where [math]\displaystyle{ {{L}_{G}}\,\! }[/math] is the number of launches, with the first successful launch being counted as [math]\displaystyle{ {{L}_{G}}=1\,\! }[/math]. Therefore, [math]\displaystyle{ {{L}_{G}}\,\! }[/math] is equal to 19, since reliability growth starts with launch 4.

   [math]\displaystyle{ R=0.9299{{(0.0943)}^{{{0.7170}^{{{L}_{G}}}}}}\,\! }[/math]

2. The predicted reliability after launch 22 is therefore:

   [math]\displaystyle{ \begin{align} R &= 0.9299{{(0.0943)}^{{{0.7170}^{19}}}} \\ &= 0.9260 \end{align}\,\! }[/math]

   The predicted reliability after launch 22 is calculated using the Quick Calculation Pad (QCP), as shown next.
   
3. In the second table, the predicted reliability values are compared with the reliabilities that are calculated from the raw data using failure discounting. It can be seen in the table, and in the following figure, that the Gompertz curve appears to provide a good fit to the actual data.