A polynomial time algorithm for solving a quality control station configuration problem

We study unreliable serial production lines with known failure probabilities for each operation. Such a production line consists of a series of stations; existing machines and optional quality control stations (QCS). Our aim is to simultaneously decide where and if to install the QCSs along the line and to determine the production rate, so as to maximize the steady state expected net profit per time unit from the system.We use dynamic programming to solve the cost minimization auxiliary problem where the aim is to minimize the time unit production cost for a given production rate. Using the above developed O(N2)O(N2) dynamic programming algorithm as a subroutine, where N   stands for the number of machines in the line, we present an O(N4)O(N4) algorithm to solve the Profit Maximization QCS Configuration Problem.