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Management controls the resources for corrective actions. Consequently, the effectiveness factors are part of the management strategy. For the BD mode failure intensity that has been seen during development testing, 100d percent will be removed and 100 (1 – d) percent will remain in the system. Therefore, after the corrective actions have been made, the current system instantaneous failure intensity consists of the failure intensity due to the A modes plus the failure intensity for the unseen BC modes, plus the failure intensity for the unseen BD modes, plus the failure intensity for the BD modes that have been seen. Figure 9.5 shows how the system's instantaneous failure intensity can be broken down into its individual pieces based on the current failure mode strategy.
|
Figure 9.5: System failure intensity after the corrective actions |
Keep in mind that the individual components of the system's instantaneous failure intensity will depend on the classifications defined in the data. For example, if BC modes are not present within the data then the BC mode MTBF will not be a part of the overall system MTBF. The individual pieces of the pie, as shown Figure 9.5, are calculated using the following equations.
Let:

where T is the test time and
and
are the maximum likelihood estimates of the Crow-AMSAA
model for all of the data.
is the biased estimate of
β. Therefore:


where N is the total number of failures and Xi is the ith time-to-failure. Let the successive failures 0 < X1 < X2 < ... < X3 < XN be partitioned into the A mode failures (NA), BC first occurrence failures (NBCF), BC remaining failures (NBCR), BD first occurrence failure (NBDF) and the BD remaining failures (NBDR). For continuous data, each portion of the pie chart due to each of the modes is calculated as follows:







For grouped data, from the Crow-AMSAA (NHPP) chapter the maximum likelihood estimates of β and λ are calculated such that the following equations are satisfied:


where K is the number of groups
and
.



BD modes unseen




Consider the data in Table 9.3. There were 56 total failures and T = 400. The effectiveness factors of the BD modes are given in Table 9.4. Determine the following:
Table 9.3 - Test-fix-find-test data
|
i |
Xi |
Mode |
|
i |
Xi |
Mode |
|
1 |
0.7 |
BC17 |
|
29 |
192.7 |
BD11 |
|
2 |
3.7 |
BC17 |
|
30 |
213 |
A |
|
3 |
13.2 |
BC17 |
|
31 |
244.8 |
A |
|
4 |
15 |
BD1 |
|
32 |
249 |
BD12 |
|
5 |
17.6 |
BC18 |
|
33 |
250.8 |
A |
|
6 |
25.3 |
BD2 |
|
34 |
260.1 |
BD1 |
|
7 |
47.5 |
BD3 |
|
35 |
263.5 |
BD8 |
|
8 |
54 |
BD4 |
|
36 |
273.1 |
A |
|
9 |
54.5 |
BC19 |
|
37 |
274.7 |
BD6 |
|
10 |
56.4 |
BD5 |
|
38 |
282.8 |
BC27 |
|
11 |
63.6 |
A |
|
39 |
285 |
BD13 |
|
12 |
72.2 |
BD5 |
|
40 |
304 |
BD9 |
|
13 |
99.2 |
BC20 |
|
41 |
315.4 |
BD4 |
|
14 |
99.6 |
BD6 |
|
42 |
317.1 |
A |
|
15 |
100.3 |
BD7 |
|
43 |
320.6 |
A |
|
16 |
102.5 |
A |
|
44 |
324.5 |
BD12 |
|
17 |
112 |
BD8 |
|
45 |
324.9 |
BD10 |
|
18 |
112.2 |
BC21 |
|
46 |
342 |
BD5 |
|
19 |
120.9 |
BD2 |
|
47 |
350.2 |
BD3 |
|
20 |
121.9 |
BC22 |
|
48 |
355.2 |
BC28 |
|
21 |
125.5 |
BD9 |
|
49 |
364.6 |
BD10 |
|
22 |
133.4 |
BD10 |
|
50 |
364.9 |
A |
|
23 |
151 |
BC23 |
|
51 |
366.3 |
BD2 |
|
24 |
163 |
BC24 |
|
52 |
373 |
BD8 |
|
25 |
164.7 |
BD9 |
|
53 |
379.4 |
BD14 |
|
26 |
174.5 |
BC25 |
|
54 |
389 |
BD15 |
|
27 |
177.4 |
BD10 |
|
55 |
394.9 |
A |
|
28 |
191.6 |
BC26 |
|
56 |
395.2 |
BD16 |
Table 9.4 - Effectiveness factor for each unique BD mode
|
BD Mode |
EF di |
|
1 |
.67 |
|
2 |
.72 |
|
3 |
.77 |
|
4 |
.77 |
|
5 |
.87 |
|
6 |
.92 |
|
7 |
.50 |
|
8 |
.85 |
|
9 |
.89 |
|
10 |
.74 |
|
11 |
.70 |
|
12 |
.63 |
|
13 |
.64 |
|
14 |
.72 |
|
15 |
.69 |
|
16 |
.46 |
, use the traditional
Crow-AMSAA model for test-fix-test to fit all 56 data points, regardless
of the failure mode classification to get:
Thus the achieved or demonstrated
failure intensity is estimated by:
The achieved or demonstrated MTBF, MCA , is the system reliability attained at the end of test,
T = 400, and is estimated by:



Calculate
and
of the BD modes using
Eqns. (3) and (4):
Then:
Therefore:
The Crow Extended model projected
MTBF is:
Consequently, based on the Crow Extended
model and the data in Tables 9.3 and 9.4, the MTBF grew to 7.85 as a result
of the corrective actions for the BC failure modes during the test. The
MTBF then jumped to 11.29 after the test as a result of the delayed corrective
actions for the BD failure modes. The management strategy can be summarized
by the Failure Mode Strategy plot shown in Figure 9.6.
|
Figure 9.6: Failure Mode Strategy plot |
Figure 9.6 shows that 9.48% of the system's failure intensity has been left in (A modes), 31.81% of the failure intensity due to the BC modes has not been seen yet and 13.40% was removed during the test (BC modes - seen). In addition, 33.23% of the failure intensity due to the BD modes has not been seen yet, 3.37% will remain in the system since the corrective actions will not be completely effective at eliminating the identified failure modes and 8.72% will be removed after the delayed corrective actions.

As shown in Figure 9.7, the MTBF of each individual failure mode can be plotted and the failure modes with the lowest MTBF can be identified. These are the failure modes that cause the majority of the system failures.

|
Figure 9.7: Individual Mode MTBF chart |