Failure Discounting

During a reliability growth test, once a failure has been analyzed and corrective actions for that specific failure mode have been implemented, the probability of its recurrence is diminished, as given in Lloyd [4]. Then for the success/failure data that follow, the value of the failure for which corrective actions have already been implemented should be subtracted from the total number of failures. However, certain questions arise, such as to what extent should the failure value be diminished or discounted, and how should the failure value be defined? One answer would be to use engineering judgment (e.g., a panel of specialists would agree that the probability of failure has been reduced by 50% or 90% and therefore, that failure should be given a value of 0.5 or 0.9). The obvious disadvantage of this approach is its arbitrariness and the difficulty of reaching an agreement. Therefore, a statistical basis needs to be selected, one that is repeatable and less arbitrary. Failure discounting is applied when using the Lloyd-Lipow, logistic, and the standard and modified Gompertz models.

The value of the failure, [math]\displaystyle{ f\,\! }[/math], is chosen to be the upper confidence limit on the probability of failure based on the number of successful tests following implementation of the corrective action. The failure value is given by the following equation:

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

where:

• [math]\displaystyle{ CL\,\! }[/math] is the confidence level.
• [math]\displaystyle{ {{S}_{n}}\,\! }[/math] is the number of successful tests after the first success following the corrective action.

For example, after one successful test following a corrective action, [math]\displaystyle{ {{S}_{n}}=1\,\! }[/math], the failure is given a value of 0.9 based on a 90% confidence level. After two successful tests, [math]\displaystyle{ {{S}_{n}}=2\,\! }[/math], the failure is given a value of 0.684, and so on. The procedure for applying this method is illustrated in the next example.

Example

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	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			2.436
6	F3	0.438	1.000	1.000			2.438
7	S	0.369	0.900	1.000			2.269
8	S	0.319	0.684	0.900			1.902
9	S	0.280	0.536	0.684			1.500
10	S	0.250	0.438	0.536			1.224
11	S	0.226	0.369	0.438			1.032
12	S	0.206	0.319	0.369			0.894
13	S	0.189	0.280	0.319			0.788
14	S	0.175	0.250	0.280			0.705
15	S	0.162	0.226	0.250			0.638
16	S	0.152	0.206	0.226			0.584
17	F4	0.142	0.189	0.206	1.000		1.537
18	S	0.134	0.175	0.189	1.000		1.498
19	F5	0.127	0.162	0.175	0.900	1.000	2.364
20	S	0.120	0.152	0.162	0.684	1.000	2.118
21	S	0.114	0.142	0.152	0.536	0.900	1.844
22	S	0.109	0.134	0.142	0.438	0.684	1.507

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	28.617
5	51.283	3.937	45.883
6	59.372	4.084	60.841
7	67.585	4.213	72.017
8	76.219	4.334	79.654
9	83.334	4.423	84.600
		[math]\displaystyle{ {{S}_{1}}\,\! }[/math] = 24.558
10	87.764	4.475	87.701
11	90.614	4.507	89.609
12	92.555	4.528	90.769
13	93.939	4.543	91.469
14	94.964	4.553	91.891
15	95.746	4.562	92.143
		[math]\displaystyle{ {{S}_{2}}\,\! }[/math] = 27.167
16	96.356	4.568	92.295
17	90.960	4.510	92.385
18	91.681	4.518	92.439
19	87.560	4.472	92.472
20	89.411	4.493	92.491
21	91.219	4.513	92.503
		[math]\displaystyle{ {{S}_{3}}\,\! }[/math] = 27.076
22	93.152	4.534	92.510

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_{2}-S_{3}}{S_{2}-S_{1}} \right )^\frac{1}{n\cdot I} \\ &= \left [ \frac{27.167-27.076}{27.167-24.558} \right ]^\frac{1}{6} \\ &= 0.572 \\ \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 (24.558 + \frac{27.167 - 24.558}{1-0.572^{6}}\right ) \right ] \\ &= 94.024% \\ \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{(27.167-24.558)(0.572-1)}{(1-0.572^{6})^{2}} \right ] \\ &= 0.301 \\ \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)}=94.024,\,\! }[/math] [math]\displaystyle{ g_{2}^{(0)}=0.301,\,\! }[/math] [math]\displaystyle{ g_{3}^{(0)}=0.572\,\! }[/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}&= 0.9252 \\ \widehat{b} &= 0.1404 \\ \widehat{c} &= 0.5977 \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 &= 0.9252{{(0.1404)}^{{{0.5977}^{19}}}} \\ &= 0.9251 \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.