Optimum attributes component test plans for k-out-of-n:F Weibull systems using prior information

Assuming that the proportion defective in a production process p is variable and the component lifetimes are Weibull distributed, integer nonlinear programming problems are formulated and solved in order to determine the optimum component inspection scheme by attributes for k-out-of-n:F system reliability demonstration using available prior knowledge. A limited beta prior model is adopted to reflect the random fluctuations on p. The required quantity of components to test and the permissible number of component failures up to a specified censoring time are found by solving a minimisation problem with nonlinear constraints which are related to the tolerable average producer and consumer risks. First-order Taylor polynomials of the operating characteristic function are used to derive a quite accurate approximate solution. Lower and upper bounds are also deduced. Optimal solutions are usually robust to small variations in the Weibull parameters and prior information. Existing technical knowledge and experience are shown to be of great practical value in designing optimal sampling plans for lot acceptance purposes. The proposed approach generalises the classical viewpoint to those cases in which appreciable prior information on the fraction of defective systems exists, and also allows the analyst to determine the acceptability of a k-out-of-n:F system before assembly and to limit the range of p. Moreover, the practitioners may attain substantial savings in sample size and improved assessments of the true producer and consumer risks. A 4-out-of-5:F system of independent water pumps for cooling a reactor is considered to illustrate the suggested component test plans.