Your Reliability Engineering Professional Development Site
The LinkedIn ASQ RRD group published this question from a reliability manager. Replies included:
Generations of products have similar field reliability functions because they are designed, processed, shipped, sold, and used in similar environments by similar customers. Replacement parts have similar reliability functions depending on replacement number: 1st, 2nd,….
Biostatisticians use David Cox’ proportional hazard (PH) survival function models to quantify effects of treatment or risk factors. Proportional hazard models could describe product’s failure modes, parts’ reliabilities in successive replacements, or products’ reliabilities in successive generations. [Read more…]
The (age-specific or actuarial) force of mortality drives the demand for spares, service parts, and most products. The actuarial demand forecast is Σd(t‑s)*n(s), where d(t-s) is (age-specific) actuarial demand rate and n(s) is the installed base of age s, s=0,1,2,…,t. Ulpian, 220 AD, made actuarial forecasts of pension costs for Roman Legionnaires. (Imagine computing actuarial demand forecasts with Roman numerals.) Actuarial demand rates are functions of reliability. What if reliability changes? We Need Statistical Reliability Control (SRC).
Actuarial demand forecasts require updating as installed base and field reliability data accumulates. Actuarial failure rate function, a(t), is related to reliability function, R(t), by a(t) = (R(t)-R(t-1))/R(t-1), t=1,2,… If products or parts are renewable or repairable, then actuarial demand rate function, d(t), depends on the number of prior renewals or repairs by age t [George, Sept. 2021].
Do you want easy demand forecasts or do you want to learn and use the reliabilities of service parts and make demand forecasts and distribution estimates, without sample uncertainty? Would you like to do something about service parts’ reliability? Would you like demand forecast distributions so you could set inventory policies to meet fill rate or service level requirements? Without sample uncertainty? Without life data? Don’t believe people who write that it can’t be done!
My wife says I am wasting my time trying to change reliability statistics, so I polled the www.linkedin.com Reliability Leadership…, ASQRRD, IEEE Reliability, “Biostatistics, and No MTBF groups. The polls claimed that “Life data, censored or not, is required to estimate MTBF, reliability function, failure rate function, or survivor function. TRUE? FALSE? or DON’T KNOW.” I am grateful for the responses.
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Someone asked, “…if you can give me quick explanation: For Example, EPRD 2014 part, Category: IC, Subcategory: Digital, Subtype1: JK, Failure Rate (FPMH) = 0.083632 per (million) calendar hours! How do you convert that to operational hours?” I.e., time-to-failure T has exponential distribution in calendar (million) hours with MTBF 11.9571 (million) hours.
Did the questioner mean how to convert calendar-hour MTBF into operating-hour MTBF? David Nichols’ article does that for 217Plus MTBF predictions, based on “the percentage of calendar time that the component is in the operating or non-operating (dormant) calendar period, and how many times the component is cycled during that period.” I.e., MTBF/R where R is the proportion of operating hours per calendar hour.
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Dear Larry
Thank you for your data request for breast implant data and apologies for the delay in responding. The data available is:
My colleagues have been copied into this email to show your request has been actioned. I hope this is helpful. [Read more…]
Email from www.smartcorp.com advertised how to forecast inventory requirements using time-series analyses: single and double exponential smoothing, linear and simple moving average, and Winters models. SmartCorp compares alternative times-series forecasts in a “tournament” that picks the best forecast. Charles Smart says forecasting, “…particularly for low-demand items like service and spare parts — is especially difficult to predict with any accuracy.”
Time series forecasts also quantify variance. Excel’s time-series FORECAST() functions do exponential smoothing, account for seasonality and trend, and “pointwise” confidence intervals. Pointwise means only one confidence interval is valid at a time; not a confidence band on several forecasts!
What are the covariances of Kaplan-Meier reliability estimates at different ages? I need them for the variance of actuarial demand forecasts and for confidence bands on reliability. I thought cohort reliability estimate variances and covariances in the previous article were a good idea. How good? Not as good as bootstrap and jackknife resampling alternatives!
The Kaplan-Meier reliability function estimator uses right-censored and grouped time-to-failure counts in periodic cohorts (rows in table 1). The Nelson-Aalen cumulative failure rate function estimators are theoretically independent [Aalen, Nelson], but not for some examples. The Kaplan-Meier reliability and actuarial failure rate function estimates at different ages are dependent, so their covariances matter to actuarial forecasts and confidence bands on reliability.
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The well-known variance of the Kaplan-Meier reliability function estimator [Greenwoood, Wikipedia] can drastically under-or over-estimate variance. The covariances of the Kaplan-Meier reliability pairs at different ages are ignored or neglected. Variance errors and covariance neglect bias the variance of actuarial demand forecasts. Imagine what errors and neglect do to confidence bands on reliability functions.
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My first task at Apple Computer was to recommend the warranty duration for the Apple II computer. Apple didn’t have a warranty! So, I looked at competitors’ warranties and recommended the same, one year. I wish I had known Apple’s computers’ and service parts’ reliabilities before that recommendation; I would have used actuarial forecasts of warranty returns to compare alternative warranties. Apple’s hardware warranty is still one year. Is that equitable to Apple, its customers, and society?
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Imagine observing inputs and outputs of a self-service system, without individual service times. How would you estimate the distribution of service time without following individuals from input to output? The maximum likelihood estimator for an M/G/Infinity self-service-time distribution function from ships and returns counts works for nonstationary arrival process M(t)/G/Infinity self-service systems, under a condition. A constant or linearly increasing arrival (ships) rate satisfies the condition. If you identify outputs by failure mode then you could estimate reliability by failure mode or quantify reliability growth, without life data. [Read more…]
Nonparametric, age-specific field reliability estimates helped deal with a Customer’s bad experience using a Hewlett-Packard part in the Customer’s product: 110 part failures out of 3001 shipped in the first five months. Comparison of HP population vs. Customer reliability estimates showed the Customer’s infant mortality was not typical. Using population ships and failures or returns data eliminated sample uncertainty from the HP population field reliability estimate.
[Read more…]Rupert Miller said, “Surprisingly, no efficiency comparison of the sample distribution function with the mles (maximum likelihood estimators) appears to have been reported in the literature.” (Statistical “efficiency” measures how close an estimator’s sample variance is to its Cramer-Rao lower bound.) In “What Price Kaplan-Meier?” Miller compares the nonparametric Kaplan-Meier reliability estimator with mles for exponential, Weibull, and gamma distributions.
This report compares the bias, efficiency, and robustness of the Kaplan-Meier reliability estimator from grouped failure counts (grouped life data) with the nonparametric maximum likelihood reliability estimator from ships (periodic sales, installed base, cohorts, etc.) and returns (periodic complaints, failures, repairs, replacement, spares sales, etc.) counts, estimator vs. estimator and population vs. sample.
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Dick Mensing said, “Larry, you can’t give an estimate without some measure of its uncertainty!” For seismic risk analysis of nuclear power plants, we had plenty of multivariate earthquake stress data but paltry strength-at-failure data on safety-system components. So we surveyed “experts” for their opinions on strengths-at-failures distribution parameters and for the correlations between pairs of components’ strengths at failures.
If you make estimates from population field reliability data, do the estimates have uncertainty? If all the data were population lifetimes or ages-at-failures, estimates would have no sample uncertainty, perhaps measurement error. Estimates from population field reliability data have uncertainty because typically some population members haven’t failed. If field reliability data are from renewal or replacement processes, some replacements haven’t failed and earlier renewal or replacement counts may be unknown. Regardless, estimates from population data are better than estimates from a sample, even if the population data is ships and returns counts!
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