Optimal system-profit design of series-parallel systems with multiple failure modes

We treat the problem of achieving optimal subsystem size (m) for series-parallel systems assuming that failure may take either of two forms. We assume that components are independent and identically distributed (i.i.d.) and that the two kinds of system failures can have different costs. In this paper, we determine the optimal m that maximizes the average system-profit. We study how optimal subsystem size m depends on the system parameters. We also determine the optimal subsystem size, m, which maximizes the average system-profit subject to a restricted type I design error. Numerical examples are given to illustrate the results.