Analysis of Recurrence Data on Repairs

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
Age	Sample MCF	Standard Error	Lower	Upper	Unit ID
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.

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