The RELIABILITY Procedure |
Analysis of Recurrence Data on Repairs
This example illustrates analysis of recurrence data from
repairable systems.
Repair data analysis differs from life data analysis, where units
fail only once. As
a repairable system ages, it accumulates repairs and costs of repairs.
The RELIABILITY procedure provides a nonparametric
estimate and plot of the mean cumulative function (MCF) for the
number or cost of repairs for a population of repairable systems.
The nonparametric estimate of the MCF, the variance of the MCF
estimate, and confidence limits for the MCF estimate are based
on the work of Nelson (1995).
The MCF, also written as M(t), is defined by Nelson (1995) to be the
population mean of the
distribution of the cumulative number or cost of repairs at age t.
The method does not assume any underlying structure for the repair process.
Figure 30.17 is a listing of the SAS data set VALVE,
which contains repair histories
of 41 diesel engines in a fleet (Nelson 1995).
The valve seats in these engines wear out and must be
replaced. The variable ID is a unique identifier for individual
engines. The variable DAYS provides the engine age in days.
The value of the variable VALUE is
1 if the age is a valve seat replacement age
or -1 if the age
is the end of history, or censoring age, for the engine.
Obs |
id |
days |
value |
1 |
251 |
761 |
-1 |
2 |
252 |
759 |
-1 |
3 |
327 |
98 |
1 |
4 |
327 |
667 |
-1 |
5 |
328 |
326 |
1 |
6 |
328 |
653 |
1 |
7 |
328 |
653 |
1 |
8 |
328 |
667 |
-1 |
9 |
329 |
665 |
-1 |
10 |
330 |
84 |
1 |
11 |
330 |
667 |
-1 |
12 |
331 |
87 |
1 |
13 |
331 |
663 |
-1 |
14 |
389 |
646 |
1 |
15 |
389 |
653 |
-1 |
16 |
390 |
92 |
1 |
17 |
390 |
653 |
-1 |
18 |
391 |
651 |
-1 |
19 |
392 |
258 |
1 |
20 |
392 |
328 |
1 |
21 |
392 |
377 |
1 |
22 |
392 |
621 |
1 |
23 |
392 |
650 |
-1 |
24 |
393 |
61 |
1 |
25 |
393 |
539 |
1 |
26 |
393 |
648 |
-1 |
27 |
394 |
254 |
1 |
28 |
394 |
276 |
1 |
29 |
394 |
298 |
1 |
30 |
394 |
640 |
1 |
31 |
394 |
644 |
-1 |
32 |
395 |
76 |
1 |
33 |
395 |
538 |
1 |
34 |
395 |
642 |
-1 |
35 |
396 |
635 |
1 |
36 |
396 |
641 |
-1 |
37 |
397 |
349 |
1 |
38 |
397 |
404 |
1 |
39 |
397 |
561 |
1 |
40 |
397 |
649 |
-1 |
41 |
398 |
631 |
-1 |
42 |
399 |
596 |
-1 |
43 |
400 |
120 |
1 |
44 |
400 |
479 |
1 |
45 |
400 |
614 |
-1 |
46 |
401 |
323 |
1 |
47 |
401 |
449 |
1 |
48 |
401 |
582 |
-1 |
49 |
402 |
139 |
1 |
50 |
402 |
139 |
1 |
51 |
402 |
589 |
-1 |
52 |
403 |
593 |
-1 |
53 |
404 |
573 |
1 |
54 |
404 |
589 |
-1 |
55 |
405 |
165 |
1 |
56 |
405 |
408 |
1 |
57 |
405 |
604 |
1 |
58 |
405 |
606 |
-1 |
59 |
406 |
249 |
1 |
60 |
406 |
594 |
-1 |
61 |
407 |
344 |
1 |
62 |
407 |
497 |
1 |
63 |
407 |
613 |
-1 |
64 |
408 |
265 |
1 |
65 |
408 |
586 |
1 |
66 |
408 |
595 |
-1 |
67 |
409 |
166 |
1 |
68 |
409 |
206 |
1 |
69 |
409 |
348 |
1 |
70 |
409 |
389 |
-1 |
71 |
410 |
601 |
-1 |
72 |
411 |
410 |
1 |
73 |
411 |
581 |
1 |
74 |
411 |
601 |
-1 |
75 |
412 |
611 |
-1 |
76 |
413 |
608 |
-1 |
77 |
414 |
587 |
-1 |
78 |
415 |
367 |
1 |
79 |
415 |
603 |
-1 |
80 |
416 |
202 |
1 |
81 |
416 |
563 |
1 |
82 |
416 |
570 |
1 |
83 |
416 |
585 |
-1 |
84 |
417 |
587 |
-1 |
85 |
418 |
578 |
-1 |
86 |
419 |
578 |
-1 |
87 |
420 |
586 |
-1 |
88 |
421 |
585 |
-1 |
89 |
422 |
582 |
-1 |
|
Figure 30.17: Listing of the Valve Seat Data
The following statements produce the graphical
display in Figure 30.18.
symbol c=blue v=plus;
proc reliability data=valve;
unitid id;
mcfplot days*value(-1) / cframe = ligr
ccensor = megr;
inset / cfill = ywh ;
run;
The UNITID statement specifies that the variable ID uniquely identifies
each system. The MCFPLOT statement requests a plot of the MCF estimates
as a function of the age variable DAYS, and it specifies -1 as the value
of the variable
VALUE, which identifies the end of history for each engine (system).
In Figure 30.18,
the MCF estimates and confidence limits are plotted versus
system age in days. The end-of-history ages are plotted
in an area at the top of the plot.
Except for the last few points, the plot is essentially
a straight line, suggesting
a constant replacement rate.
Consequently, the prediction of future replacements
of valve seats can be based on a fitted line in this case.
Figure 30.18: Mean Cumulative Function for the Number of Repairs
A partial listing of the tabular output is shown in Figure 30.19.
It contains a summary of the repair data, estimates of the MCF,
the Nelson (1995) standard errors,
and confidence intervals for the MCF. The MCF estimates,
standard errors, and confidence limits are shown as missing values (.)
at the end of history points, since they are not computed at
these points.
The RELIABILITY Procedure |
Repair Data Summary |
Input Data Set |
WORK.VALVE |
Observations Used |
89 |
Number of Units |
41 |
Number of Events |
48 |
Repair Data Analysis |
Age |
Sample MCF |
Standard Error |
95% Confidence Limits |
Unit ID |
Lower |
Upper |
61.00 |
0.024 |
0.024 |
-0.023 |
0.072 |
393 |
76.00 |
0.049 |
0.034 |
-0.018 |
0.116 |
395 |
84.00 |
0.073 |
0.041 |
-0.008 |
0.154 |
330 |
87.00 |
0.098 |
0.047 |
0.006 |
0.190 |
331 |
92.00 |
0.122 |
0.052 |
0.021 |
0.223 |
390 |
98.00 |
0.146 |
0.056 |
0.037 |
0.256 |
327 |
120.00 |
0.171 |
0.059 |
0.054 |
0.287 |
400 |
139.00 |
0.195 |
0.063 |
0.072 |
0.318 |
402 |
139.00 |
0.220 |
0.074 |
0.074 |
0.365 |
402 |
165.00 |
0.244 |
0.076 |
0.094 |
0.394 |
405 |
166.00 |
0.268 |
0.078 |
0.115 |
0.422 |
409 |
202.00 |
0.293 |
0.080 |
0.136 |
0.449 |
416 |
206.00 |
0.317 |
0.089 |
0.143 |
0.491 |
409 |
249.00 |
0.341 |
0.090 |
0.165 |
0.517 |
406 |
254.00 |
0.366 |
0.091 |
0.188 |
0.544 |
394 |
258.00 |
0.390 |
0.092 |
0.211 |
0.570 |
392 |
265.00 |
0.415 |
0.092 |
0.234 |
0.595 |
408 |
276.00 |
0.439 |
0.099 |
0.245 |
0.633 |
394 |
298.00 |
0.463 |
0.111 |
0.246 |
0.681 |
394 |
323.00 |
0.488 |
0.111 |
0.270 |
0.706 |
401 |
326.00 |
0.512 |
0.111 |
0.294 |
0.730 |
328 |
328.00 |
0.537 |
0.116 |
0.309 |
0.765 |
392 |
344.00 |
0.561 |
0.116 |
0.333 |
0.788 |
407 |
348.00 |
0.585 |
0.126 |
0.339 |
0.832 |
409 |
349.00 |
0.610 |
0.125 |
0.364 |
0.855 |
397 |
367.00 |
0.634 |
0.125 |
0.390 |
0.879 |
415 |
377.00 |
0.659 |
0.133 |
0.397 |
0.920 |
392 |
389.00 |
. |
. |
. |
. |
409 |
404.00 |
0.684 |
0.138 |
0.414 |
0.953 |
397 |
408.00 |
0.709 |
0.142 |
0.431 |
0.986 |
405 |
410.00 |
0.734 |
0.141 |
0.457 |
1.010 |
411 |
449.00 |
0.759 |
0.144 |
0.475 |
1.042 |
401 |
479.00 |
0.784 |
0.148 |
0.494 |
1.073 |
400 |
497.00 |
0.809 |
0.151 |
0.512 |
1.105 |
407 |
538.00 |
0.834 |
0.154 |
0.531 |
1.136 |
395 |
539.00 |
0.859 |
0.157 |
0.551 |
1.166 |
393 |
561.00 |
0.884 |
0.164 |
0.563 |
1.205 |
397 |
563.00 |
0.909 |
0.166 |
0.583 |
1.234 |
416 |
570.00 |
0.934 |
0.172 |
0.596 |
1.272 |
416 |
573.00 |
0.959 |
0.171 |
0.623 |
1.294 |
404 |
578.00 |
. |
. |
. |
. |
419 |
578.00 |
. |
. |
. |
. |
418 |
581.00 |
0.985 |
0.173 |
0.645 |
1.325 |
411 |
582.00 |
. |
. |
. |
. |
422 |
582.00 |
. |
. |
. |
. |
401 |
585.00 |
. |
. |
. |
. |
421 |
585.00 |
. |
. |
. |
. |
416 |
586.00 |
1.014 |
0.176 |
0.669 |
1.359 |
408 |
586.00 |
. |
. |
. |
. |
420 |
587.00 |
. |
. |
. |
. |
417 |
587.00 |
. |
. |
. |
. |
414 |
589.00 |
. |
. |
. |
. |
404 |
589.00 |
. |
. |
. |
. |
402 |
593.00 |
. |
. |
. |
. |
403 |
594.00 |
. |
. |
. |
. |
406 |
595.00 |
. |
. |
. |
. |
408 |
596.00 |
. |
. |
. |
. |
399 |
601.00 |
. |
. |
. |
. |
411 |
601.00 |
. |
. |
. |
. |
410 |
603.00 |
. |
. |
. |
. |
415 |
604.00 |
1.060 |
0.188 |
0.692 |
1.427 |
405 |
606.00 |
. |
. |
. |
. |
405 |
608.00 |
. |
. |
. |
. |
413 |
611.00 |
. |
. |
. |
. |
412 |
613.00 |
. |
. |
. |
. |
407 |
614.00 |
. |
. |
. |
. |
400 |
621.00 |
1.119 |
0.211 |
0.705 |
1.532 |
392 |
631.00 |
. |
. |
. |
. |
398 |
635.00 |
1.181 |
0.210 |
0.768 |
1.594 |
396 |
640.00 |
1.244 |
0.231 |
0.791 |
1.696 |
394 |
641.00 |
. |
. |
. |
. |
396 |
642.00 |
. |
. |
. |
. |
395 |
644.00 |
. |
. |
. |
. |
394 |
646.00 |
1.320 |
0.233 |
0.864 |
1.777 |
389 |
648.00 |
. |
. |
. |
. |
393 |
649.00 |
. |
. |
. |
. |
397 |
650.00 |
. |
. |
. |
. |
392 |
651.00 |
. |
. |
. |
. |
391 |
653.00 |
1.432 |
0.259 |
0.923 |
1.940 |
328 |
653.00 |
1.543 |
0.324 |
0.908 |
2.177 |
328 |
653.00 |
. |
. |
. |
. |
390 |
653.00 |
. |
. |
. |
. |
389 |
663.00 |
. |
. |
. |
. |
331 |
665.00 |
. |
. |
. |
. |
329 |
667.00 |
. |
. |
. |
. |
330 |
667.00 |
. |
. |
. |
. |
328 |
667.00 |
. |
. |
. |
. |
327 |
759.00 |
. |
. |
. |
. |
252 |
761.00 |
. |
. |
. |
. |
251 |
|
Figure 30.19: Partial Listing of the Output for the Valve Seat Data
Parametric modeling of the repair
process requires more assumptions than nonparametric modeling,
and considerable work has been done in this area.
Ascher and Feingold (1984)
describe parametric models for repair processes.
For example, repairs are sometimes modeled as a nonhomogeneous
Poisson process. The current release of the RELIABILITY procedure does not
include this type of parametric modeling, although it is planned
for future releases.
The MCF plot might be a first step in
modeling a repair process, but, in many cases, it provides the required
answers without further analysis.
An estimate of the MCF for a sample of systems aids engineers in
determining the repair rate at any age and the
increase or decrease of repair rate with population age.
The estimate is also useful for predicting the number of
future repairs.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.