On optimal assembly of systems

Abstract

AbstractWe are concerned with the following reliability problem: A system has k different types of components. Associated with each component is a numerical value. Let {aj}, (j= 1,…,k), denote the set of numerical values of the k components. Let R(a1,…, ak) denote the probability that the system will perform satisfactorily (i. e., R(a1,…. ak) is the reliability of the system) and assume R(a1,…,ak) has the properties of a joint cumulative distribution function.Now suppose aj1 ≤… ≤ ajn are n components of type j (j= 1,…, k). Then n systems can be assembled from these components. Let N denote the number of systems that perform satisfactorily. N is a random variable whose distribution will depend on the way the n systems are assembled. Of all different ways in which the n systems can be assembled, the paper shows that EN is maximized if these n systems have reliability R(a1j,…, akj) (i = 1,…,n). The method used here is an extension of a well known result of Hardy, Littlewood, and Polya on sums of products. Furthermore, under certain conditions, the same assembly that maximizes EN minimizes the variance of N.Finally, for a similar problem in reliability, it is shown that for a series system a construction can be found that not only maximizes the expected number of functioning modules, but also possesses the stronger property of maximizing the probability that the number of functioning modules is at least r, for each 0 ≤ r ≤ n.