Sampling for assurance of future reliability

Abstract

Ensuring measurement trueness, compliance with regulations and conformity with standards are key tasks in metrology which are often considered at the time of an inspection. Current practice does not always verify quality after or between inspections, calibrations, laboratory comparisons, conformity assessments, etc. Statistical models describing behavior over time may ensure reliability, i.e. they may give the probability of functioning, compliance or survival until some future point in time.It may not always be possible or economic to inspect a whole population of measuring devices or other units. Selecting a subset of the population according to statistical sampling plans and inspecting only these, allows conclusions about the quality of the whole population with a certain confidence.Combining these issues of sampling and aging, raises questions such as: How many devices need to be inspected, and at least how many of them must conform, so that one can be sure, that more than p 100 % of the population will comply until the next inspection? This research is to raise awareness and offer a simple answer to such time-and sample-based quality statements in metrology and beyond.Reliability demonstration methods, such as the prevailing Weibull binomial model, quantify the confidence in future reliability on the basis of a sample. We adapt the binomial model to be applicable to sampling without replacement and simplify the Weibull model so that sampling plans may be determined on the basis of existing ISO standards. Provided the model is suitable, no additional information and no software are needed; and yet, the consumer is protected against future failure.We establish new sampling plans for utility meter surveillance, which are required by a recent modification of German law. These sampling plans are given in similar tables to the previous ones, which demonstrates their suitability for everyday use.

Reducing sample size by tightening test conditions

“…Instead of requiring a large sample to prove the high reliability , it is sufficient to apply a smaller sample to prove the smaller reliability at tighter specifications, while achieving the same statistical type 1 error (ie, the same probability of falsely accepting an unreliable population). For example, when ( n , c ) denotes the sampling plan requiring n devices to be verified and c of these devices to comply, then the sampling plan (80,3) is sufficient instead of (141,1) for a population of size 1201 to 3200, for p =0.973, q =0.92 (table D1 in ISO 2859‐2, and Klauenberg and Elster) and thus γ N =1.263 for Normally (and γ t =1.557 for t 3 ‐) distributed deviations (see Section 4.2 and Equation for the calculation of these factors). That means that, for this particular example, the sample size almost halves and in addition the acceptance number relaxes considerably when the maximum permissible error is decreased to 80% (or 64%, respectively).…”

“…In particular, let us assume a population of size N = 3182 and a required reliability of 100 q =97.3% at the confidence level of 90% and at the specification Δ. Then sampling plan ( n , c )=(141,1) is to be applied when sampling without replacement (cf ISO 2859‐2, and Klauenberg and Elster and the hypergeometric distribution). That is, if 2.7% of the devices in the population fail—due to any reason—then the probability of falsely accepting this population is 9.8%.…”

“…The new German regulation for the monitoring of utility meters (section 35 in the regulation) implies that inspections need to provide evidence for high reliability levels in meter populations (cf Section 1). The portion of complying meters which are required at inspections typically ranges from 96% to 98%, depending on the age of the meters, the proposed extension period, and the assumed behavior of the reliability over time . These high reliability levels entail stringent sampling plans with large sample sizes n and low acceptance numbers c , especially for small population sizes (see Table , third column).…”

“…For example, the most common gas meters as well as their installation together cost about €100. The replacement of cold water meters in flats or houses costs about €57 or €75 on average, and a third of these costs are attributed to the meter itself . It is estimated that in Germany the replacement of water meters currently adds up to €800 million every year…”

“…By exploiting additional information on measurement characteristics, relation (R) helps to alleviate the higher legal requirements about the reliability of utility meters. Without violating section 35 in the regulation Theorem enables us to implement more efficient sampling plans than in Klauenberg and Elster . In particular, the sampling plans applied under the old regulation may be resumed when, at the same time, sampled meters are tested for tighter specifications.…”