Optimal allocation of cost to detectors in a two-unit series system

A two-unit series system, in which each unit is equipped with a separate detector to detect failure, is considered. The probability that a detector operates successfully at the time of need (i.e. when system fails) is a function of cost spent on the detector. The problem is that of allocation of total resources (cost) to the two detectors such that the overall expected profit is maximized. The optimization problem has been formulated. Two examples are included to show the uses of results.