Optimal Reliability Demonstration Test Plans for $k$-out-of-$n$ Systems of Gamma Distributed Components

Constrained optimization problems are solved to determine the best reliability sampling plans for
$k$
-out-of-
$n$
:F systems of gamma distributed components. As usual, the constraints are related to the quality levels required by the producer and the consumer, whereas the decision criterion is based on the uniformly most powerful reliability test. Excellent approximate solutions are derived in closed-forms assuming a frequentist perspective. This classical methodology is generalized to those situations in which the prior information on the fraction effective,
$r$
, is appreciable. Because prior knowledge on
$r$
often suggests a restricted parameter interval, a limited beta distribution is used to describe the probabilistic behavior of
$r$
. Optimal sample sizes and acceptance factors are obtained by solving mixed integer nonlinear programming problems. Lower and upper bounds on the smallest sample size, as well as a quite accurate approximation, are also provided. An efficient step-by-step procedure, which only needs a few iterations in most practical cases, is proposed for solving the optimization problem. The incorporation of prior knowledge into the reliability analysis provides better assessment of the true producer's and consumer's risks, and considerable savings in sample size and test duration. The optimal solutions are usually quite robust to small variations in the gamma shape parameter and the prior information. An example concerning a system composed of five water pumps for cooling a reactor is included for illustration. The proposed component test sampling plans can be recommended for use in practical applications, particularly when it is economically infeasible, or even impossible, to test a whole system.