Eleven systems from the field were chosen for the purposes of a fleet
analysis. Each system had at least one failure. All of the systems
had a start time equal to zero and the last failure for each system corresponds
to the end time. Group the data based on a fixed interval of 3000 hours
and assume a fixed effectiveness factor equal to 0.4. Do the following:
Based on the analysis
does it appear that the systems were randomly ordered?
After the implementation
of the delayed fixes, how many failures would you expect within the next
4000 hours of fleet operation.
Table 10.9 - Fleet data
for Example 5
System
Times-to-Failure
1
1137 BD1, 1268 BD2
2
682 BD3, 744 A, 1336 BD1
3
95 BD1, 1593 BD3
4
1421 A
5
1091 A, 1574 BD2
6
1415 BD4
7
598 BD4, 1290 BD1
8
1556 BD5
9
55 BD4
10
730 BD1, 1124 BD3
11
1400 BD4, 1568 A
Solution to Fielded Systems Example 5
The estimated Crow Extended
parameters are shown in Figure 10.8.
Upon observing the estimated
parameter it does appear that the systems
were randomly ordered since . This value
is close to one. You can also verify that the confidence bounds on include one by going to the QCP and calculating
the parameter bounds or by viewing the Beta Bounds plot. However, you
can also determine if the systems were randomly ordered graphically using
the System Operation plot as shown in Figure 10.9. Looking at the Cum.
Time Line it does not appear that the failures have a trend associated
with them. Therefore, the systems can be assumed to be randomly ordered.
Figure 10.8: Estimated
Crow Extended parameters
Figure 10.9: System Operation
plot
Fielded Systems Example 6
This case study is based on the data given in the article "Graphical Analysis of Repair Data" by Dr. Wayne
Nelson. The data in Table 10.10 represents repair data on an automatic
transmission from a sample of 34 cars. For each car, the data set shows
mileage at the time of each transmission repair, and the latest mileage.
Car 1, for example, had a repair at 7068 miles and was observed until
26,744 miles. The + indicates the latest mileage observed without failure.
Do the following:
Estimate the parameters
of the Power Law model.
Estimate the number
of warranty claims for a 36,000 mile warranty policy for an estimated
fleet of 35,000 vehicles.
Table 10.10 - Automatic
transmission data
Car
Mileage
Car
Mileage
1
7068, 26744+
18
17955+
2
28, 13809+
19
19507+
3
48, 1440, 29834+
20
24177+
4
530, 25660+
21
22854+
5
21762+
22
17844+
6
14235+
23
22637+
7
1388, 18228+
24
375, 19607+
8
21401+
25
19403+
9
21876+
26
20997+
10
5094
27
19175+
11
21691+
28
20425+
12
20890+
29
22149+
13
22486+
30
21144+
14
19321+
31
21237+
15
21585+
32
14281+
16
18676+
33
8250, 21974+
17
23520+
34
19250, 21888+
Figure 10.10: Entered
transmission data and the estimated Power Law parameters
Figure 10.11: Cumulative
number of failures at 36,000 miles
Solution to Fielded Systems Example 6
The estimated Power
Law parameters are shown in Figure 10.10.
The expected number
of failures at 36,000 miles can be estimated using the QCP as shown in
Figure 10.11. The model predicts that 0.3553 failures per system will
occur by 36,000 miles. This means that for a fleet of 35,000 vehicles,
the expected warranty claims are 0.3553 * 35,000 = 12,436.
Fielded Systems Example 7
Field data has been collected for a system which begins its wearout
phase at time zero. The start time for each system is equal to zero and
the end time for each system is 10,000 miles. Each system is scheduled
to undergo an overhaul after a certain number of miles. It has been determined
that the cost of an overhaul is four times more expensive than a repair.
The data is presented in Table 10.11. Do the following:
Estimate the parameters
of the Power Law model.
Determine the optimum
overhaul interval.
If , would it be cost efficient to implement an overhaul policy?
Table 10.11 - Field data
System 1
System 2
System 3
1006.3
722.7
619.1
2261.2
1950.9
1519.1
2367
3259.6
2956.6
2615.5
4733.9
3114.8
2848.1
5105.1
3657.9
4073
5624.1
4268.9
5708.1
5806.3
6690.2
6464.1
5855.6
6803.1
6519.7
6325.2
7323.9
6799.1
6999.4
7501.4
7342.9
7084.4
7641.2
7736
7105.9
7851.6
8246.1
7290.9
8147.6
7614.2
8221.9
8332.1
9560.5
8368.5
9575.4
8947.9
9012.3
9135.9
9147.5
9601
Solution to Fielded Systems Example 7
The estimated Power
Law parameters are shown in Figure 10.12.
The QCP can be used
to calculate the optimum overhaul interval as shown in Figure 10.13.
If then it would not be cost efficient to implement an overhaul
policy. Since , the systems are not wearing
out. An overhaul policy only makes sense if the systems are wearing out.
Otherwise, an overhauled unit would have the same probability of failing
as a unit that was not overhauled.
Figure 10.12: Entered
data and the estimated Power Law parameters
Figure 10.13: Calculate
the optimum overhaul interval
it does appear that the systems
were randomly ordered since 
. This value
is close to one. You can also verify that the confidence bounds on 
include one by going to the QCP and calculating
the parameter bounds or by viewing the Beta Bounds plot. However, you
can also determine if the systems were randomly ordered graphically using
the System Operation plot as shown in Figure 10.9. Looking at the Cum.
Time Line it does not appear that the failures have a trend associated
with them. Therefore, the systems can be assumed to be randomly ordered.
, would it be cost efficient to implement an overhaul policy?
then it would not be cost efficient to implement an overhaul
policy. Since 
, the systems are not wearing
out. An overhaul policy only makes sense if the systems are wearing out.
Otherwise, an overhauled unit would have the same probability of failing
as a unit that was not overhauled.
