Progress in Field Reliability?

COVID-19 Case Fatality Rate (CFR) is easy to estimate: CFR=deaths/cases. Regression forecasts of COVID-19 cases and deaths are easy but complicated by variants and nonlinearity. Epidemiologists use SIR models (Susceptible, Infectious, and “Removed”) to estimate Ro. These are baseball statistics. Reliability people need age-specific reliability and failure-rate function estimates, by failure mode, to diagnose problems, recommend spares, plan maintenance, do risk analyses, etc. Markov models use actuarial transition rates.

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Age-specific reliability of a standby system depends on components’ failure rates. Reliability computation is interesting when part failure rates depend on age, which is what motivates having a standby system. A Markov chain, approximates the age-specific reliability and availability, which are complicated to compute exactly, unless you assume constant failure rates. Why not use age-specific (actuarial) rates? They are Markov chain transition rates.

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Suppose installed base or cohorts in successive periods have different reliabilities due to nonstationarity? What does that do to forecasts, estimates, reliability predictions, diagnostics, spares stock levels, maintenance plans, etc.? Assuming stationarity is equivalent to assuming all installed base, cohorts, or ships have the same reliability functions. At what cost? Assuming a constant failure rate is equivalent to assuming everything has exponentially distributed time to failure or constant failure rate. At what cost?

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In the 1960s, my ex-wife’s father set safety stock levels and order quantities for Pep Boys. He used part sales rates and the Wilson square-root formula to set order quantities. 

Why not use the ages of the cars into which those parts go, to forecast part sales and recommend stock levels? Imagine you had vehicle counts (year, make, model, and engine) in the neighborhoods of parts stores, catalogs of which parts and how many go into which cars, and store sales by part number.

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As I rode, I thought, how could I use reliability statistics to optimize a solar-tube production line? Then I noticed a brass glint in the scrub brush. It didn’t look like trash, so I stopped and found an old brass oil lamp like Aladdin’s. Naturally, I rubbed it. There was a flash and a puff of smoke, and out popped the genie who said, “Yes master, by the powers vested in me, I grant you three wishes.” 

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Would you like age-specific field reliability of your products and their service parts? Age-specific field reliability is useful for reliability prediction, diagnoses, forecasting, warranty reserves, spares stock levels, warranty extensions, and recalls. Nonparametric estimation of age-specific field reliability is easy, if you track parts or products by name and serial number for life data. What if there are life limits? What if there’s no life data?

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I needed multivariate fragility functions for seismic risk analysis of nuclear power plants. I didn’t have any test data, so Lawrence Livermore Lab paid “experts” for their opinions! I set up the questionnaires, asked for percentiles, salted the sample to check for bias, asked for percentiles of conditional fragility functions to estimate correlations, and fixed pairwise correlations to make legitimate multivariate correlation matrixes. Subjective percentiles provide more distribution information than parameter or distribution assumptions, RPNs, ABCD, high-medium-low, or RCM risk classifications.

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How to allocate subsystems’ MTBF requirements with testing? Name-withheld-to-protect-the- guilty proposed “Top-Down” reduction in subsystem MTBF requirements; the more subsystems (in series) that you test, the lower the subsystem required MTBF! “The correct formula is

1/MTBF(subsystem requirement) = 1/MTBF(system requirement) –
((# of subsystems in series – # of subsystems tested)/MTBF(subsystem).”

This “Top-Down…” method is uncited and not found in Internet search. 

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“The effects of chance are the most accurately calculable, and the least doubtful of all factors in the evolutionary situation.”

R. A. Fisher, ca. 1953

COVID-19 vaccination claims have changed from “prevention” to “reduced severity.” FDA approved Pfizer’s vaccine for 95% efficacy, compared with the placebo control sample. Pfizer’s placebo sample had 86% efficacy, compared with the US population case rate! Sample subjects resembled each other but not the US population! 

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Reliability-based forecasts can be made from field data on complaints, failures, repairs, age-replacements (life limits), NTFs (no trouble found), WEAP (warranty expiration anticipation phenomenon), spares, warranty claims, or deaths. Some spares inventory forecasting software says… “Please enter forecast______” No kidding. 1800 years ago Roman Jurist Ulpian made actuarial pension cost forecasts for retiring Roman Legionnaires. Would you like actuarial forecasts? Their distributions? Stock recommendations?

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The title was inspired by Rupert Miller’s report “What Price Kaplan Meier?” That report compares nonparametric vs. parametric reliability estimators from censored age-at-failure data. This article compares alternative, nonparametric estimators from different data: grouped, censored age-at-failure data vs. population ships and returns data required by generally accepted accounting principles. This article compares data storage and collection requirements and costs, and bias, precision, and information of nonparametric reliability estimators. 

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The title is a Statistician’s Lament. “Design of Experiments (DoE) is the design of any task that aims to describe or explain the variation of information under conditions that are hypothesized to reflect the variation.” [Wikipedia] Are you using DoE to design reliability tests? What do PH, GMDH, and |D|-optimality have to do with design of DoE of reliability tests?

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Bob Butler nuclear engineer, musician (www.pleasantonband.org), former city councilman and Mayor of Pleasanton, California  died October fifth https://www.pleasantonweekly.com/news/2021/10/14/what-a-week-remembering-bob-butler-former-pleasanton-mayor-and-councilman. He helped me get traffic counts data from the Pleasanton Traffic Department.

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IEC 60601-1 says… Estimate the probability per time pe of an electrical failure and of an oxygen leak po. Determine the accepted probability of dangerous failures [fire] per time r. Calculate the inspection time interval tc = r/(0.5*pe*po).

A friend asked, “What’s the 0.5 for? It doesn’t account for the fire event sequence: leak before spark.” I posted correction tc = r/((po/(po+pe))*pe*po) and notified the IEC committee which acknowledged, “We’ll consider your suggestion for edition 4.”

[An earlier, shorter version of this article on www.LinkedIn.com, July 5, 2018. This version describes an inspection-time and risk-analysis template.]

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When nuclear power plants were built, companies had quality assurance programs and US Nuclear Regulatory Commission risk standards. Now the nuclear industry faces obsolescence. Qualifying replacement parts and replacing analog instrumentation and controls with digital systems generates some reliability testing work. NASA solicits unmanned nuclear power plants on the moon and Mars. Nevertheless, the demand for nuclear engineers is decreasing. Fortunately, the nuclear industry spawned risk analyses useful in other industries. 

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