128 Cell
i
i
N
i
p
i
Np
i i
Np N
i i
i
Np Np
N
2
1 31
0.1 49.2
-18.2 6.73252
2 61
0.1 49.2
11.8 2.830081
3 37
0.1 49.2
-12.2 3.025203
4 46
0.1 49.2
-3.2 0.20813
5 56
0.1 49.2
6.8 0.939837
6 48
0.1 49.2
-1.2 0.029268
7 52
0.1 49.2
2.8 0.15935
8 58
0.1 49.2
8.8 1.573984
9 52
0.1 49.2
2.8 0.15935
10 51
0.1 49.2
1.8 0.065854
Total 492
1 15.72358
Table 6-12: Calculating the goodness-of-fit
The computation of the test is shown in
Table 6-12
and the value of the test statistics is 7235
. 15
2
. Referring to the chi-square table, we see that
2 95
. ,
9
16.919, which is
not exceeded by
2
, so we cannot reject the null hypothesis when
05 .
. Thus, this
test gives us a reason to conclude that the distribution of sample data follows its population distribution.
6.11 The Decision Model
The essential decision to make at different monitoring points is whether we should replace the engine or not, given all monitored information available. If the answer is
‗yes‘, then what is the best time for such replacement, and should we wait for a suitable production window, such as a scheduled shutdown? Suppose that we need to develop a
decision model: the decision variables themselves might depend on several criteria such as cost, reliability, safety or any other criterion of interest. All of them require input,
that is, the residual time probability distribution. As an example, we may consider the decision model developed by Wang et al 2000 as
129
i
t T
i i
i i
i i
i i
i i
i i
p i
i i
i f
dz z
zp t
T X
P t
T t
t T
X P
C t
T X
P C
t C
| |
1 |
1 |
6-40
where t
C is the total expected cost per unit time
i
t
is the current monitoring point measured since new
T
is the planned replacement time
|
i i
i i
t T
X P
is the probability that
i i
t T
X
time
i
t
f
C
is a failure cost
p
C
is a preventive cost.
The decision model above assumes that the cost of monitoring is negligible. Suppose that the mean cost of a failure is 10 times higher than for preventive maintenance; we
can use the estimated parameters and equation 6-40 above to compute the decision model based on engines life data. By plotting the result of the calculation of equation
6-40 at every time
T
and assuming it is discrete, we could identify the best time to carry out preventive replacement,
T
. Two examples are shown from our case studies. In the first case, the engine is monitored for 102 data points, the interval between
monitoring points is 9.2 days, and the engine failed after 944.45 days of operation. We see that from
Figure 6-18
the cost of preventive replacement decreases at the early stage, which indicates that any effort to replace the engines is unnecessary until it starts to
increase at the
th 80
checking point. This suggests that a preventive replacement should be undertaken to reduce the cost.
130
Figure 6-18: Case 1 – Expected cost per day in terms of planned replacement at time T
given that the current monitoring check is time
i
t
A similar plot is obtained in Figure 6-19, where in this case the engine was monitored
for 72 data points from new, the interval between monitoring points was 9.2 days and it failed at 658.2 days. The expected cost per day is decreasing at an early stage, until it
starts to increase at the
th 69
monitoring point. This suggests that if preventive replacement were carried out, the cost would be minimized. Both cases show that our
model works well in recommending preventive replacement before failure occurs.
Figure 6-19: Case 2 – Expected cost per day in terms of planned replacement at time T
given that the current monitoring check is time
i
t
131
6.12 Summary
