The Bathtub Curve Doesn’t Always Hold Water

by Larry George

Nonparametric estimates of failure rate functions may show something you didn’t expect, retirement. If you don’t recognize retirement, you might do unnecessary maintenance and get stuck with obsolete spares.

I helped a company buy back $18M of obsolete spare parts, at 50¢ per dollar of original purchase price. The spares went to a landfill. The company had required its dealers to stock those spares, and nobody recognized the onset of retirement and its effect on spares requirements.

First estimate a nonparametric cumulative distribution function, F(t), where t_0 represents calendar time-to-failure on some convenient time scale. (Substitute time-to-repair or time-to-return if appropriate.) Use the nonparametric estimator appropriate for your data:

These estimators yield estimates that change only at failure times. Then compute the failure rate function,

a(t) = f(t)/[1-F(t)].

Finally graph the failure rate function vs. time-to-failure. Your results probably won’t look as smooth as the figures.

Figure 1. The bathtub curve with premature wearout and retirement

Figure 2. The bathtub doesn’t hold water because retirement occurred before wearout

THE BATHTUB CURVE

The bathtub curves in the left and middle of the figures show:

Unfortunately, no family of distribution functions fits these failure rate functions, so you should look at nonparametric estimates.

Both failure rate functions in the figures show retirement. Figure 1 shows retirement following onset of wearout, and figure 2 shows retirement before wearout. Retirement means fewer operating hours per calendar time unit. This causes failure rate functions to decrease.

Unreported failures could also cause failure rate functions to decrease:

Both these causes could occur late in product life.

Regardless of the cause, use nonparametric failure rate functions to guide maintenance policies and to divest spares before they’re obsolete!

One would prefer the product with the failure rate function in figure 2, because he wouldn’t have to deal with wearout. However, the product may cost too much, so either figure may describe products.

If the failure rate function isn’t constant, maintenance should be age specific. If the failure rate function is decreasing, leave the product alone, unless it breaks. Maintenance moves the product to the left on the failure rate function. Don’t move to a higher failure rate unless you have reason. If the failure rate function is increasing, consider inspection, preventive maintenance or replacement. Maintenance depends on the costs of replacement and the cost of failure as well as the shape of the failure rate function.

You need spares to meet expected demand plus extras to maintain a desired service level. When you anticipate end of production or when you no longer need to maintain a desired service level, reduce spares inventory to expected demand. If production hasn’t stopped yet, use excess spares in final production. Otherwise, let attrition reduce your spares to expected demand and restock only to expected demand (assuming you can restock after end of production).

If the failure rate function is constant, expected demand is a(t)*[installed base]*T, where T is the remaining time you plan to support the product. Modify that formula if you plan to ship more products.

If the failure rate function isn’t constant, take into account ages of installed product base and extrapolate a(t) somehow. (I discretize the failure rate function to

This is the actuarial failure rate function. I use various methods to extrapolate failure rate functions. The best is to use the shape of failure rate functions of comparable products that have gone through their complete life cycles.) The expected demands in future periods i = 1, 2,...T, are

where n(i;t) is the number of products of age t in period i. Integer k(i) is the age of the oldest product in period i. Decrease n(i;t) as i increases, to account for retirement. The total expected demand is

This is essentially the same as "future risk" computed in [7] except that, if products aren’t new, you should use actuarial failure rates instead of unconditional failure probabilities.)

If you would like help, send your installed base and failure data and I’ll send back the nonparametric failure rate function estimate and expected demand. Send data on disk, by internet or fax.

[1] A. H. Bowker and G. J. Lieberman, Engineering Statistics, second edition, Prentice-Hall, Englewood Cliffs, NJ, 1972, pp2-6.

[2] C. L. Chiang, Introduction to Stochastic Processes in Biostatistics, Wiley, New York, 1968.

[3] E. L. Kaplan and P. Meier, "Nonparametric Estimator From Incomplete Observations," J. Amer. Statist. Assn., Vol. 53, 1958, pp. 457-481.

[4] B. Efron and V. Petrosian, "Survival Analysis of the Gamma-Ray Burst Data," J. Amer. Statist. Assn., Vol. 89, No. 426, 1994, pp. 452-462.

[5] M. Woodroofe, "Estimating a Distribution Function With Truncated Data," The Annals of Statistics, Vol. 13, 1985, pp. 163-177.

[6] L. L. George and A. Agrawal, "Estimation of a Hidden Service Distribution of an M/G/_ Service System," Naval Research Logistics Quarterly, Vol. 20, No. 3., 1973, pp. 549-555.

[7] James McLinn, "The Weibull Function part III," Reliability Review, Vol. 14, No. 2, 1994, pp. 5-9.

About the Author

Larry has a Ph. D. in industrial engineering and operations research. He’s past Division Director of the IIE QC&RE Division, editor of the Software Entrepreneurs’ Forum newsletter and Reliability Review reviewer. He taught for eleven years. He worked for Lawrence Livermore National Laboratory for eleven years, Apple Computer for two years, EPRI for one year, and Abbott Laboratories for two years. He now works for Triad Systems Corporation estimating reliability, forecasting demands for spares and recommending auto parts stock levels. Contact Larry at 1573 Roselli Drive, Livermore, CA 94550-5852, 510-447-4969 (call first to fax), or pstlarry@holonet.net.