Using Weibull Analysis for Cases With an Unknown Susceptible Population

Abstract

Weibull analysis is a powerful predictive tool for studying failure trends of engineering systems. [1] One noted shortcoming is that traditional techniques require the size of the susceptible population to be known. The method described in this paper allows for estimation of the size of the susceptible population using only failure data and no assumptions about total population size or susceptible portion. In the analysis of failures of mass-produced products, a large amount of failure data may be available, but all the conditions that define the susceptible population may never be known. For example, units with a particular usage condition may be expected to fail over time following a Weibull model, but the number of units subjected to that usage condition may never be known. To assume that the entire population is susceptible to the failure mode would greatly over-predict future failures, and the model could not be used to guide decision-making. By doing a least squares fit to the trend of failures versus time, a Weibull model can be fit to the data and then used to estimate the total number of susceptible units expected in the population. The ability to accurately estimate the size of the susceptible sub-population from failure data will be explored as a function of the size of the data set used, for known sets of failure data. For example, for a failure distribution that has increased, peaked, and then decreased to zero, almost the entire population has failed, so an estimate of the size of the susceptible population from this data is likely to be accurate. On the contrary, for only a few data points that show an increasing failure rate over time, little can be determined. Monte Carlo simulations will be used in order to estimate the error associated with this technique. Our analysis will show that predictions of total susceptible populations become similar to the actual susceptible populations when the predicted mean time to failure (MTTF) from the observed data is shorter than the observation time. In effect, predictions become accurate when it is clear to the observer that the number of failures per unit time has peaked.