A mission-critical production system has n stages that have to be performed sequentially; stage i is performed by machine Mi . Each machine Mi has a probability ri of functioning reliably and a probability 1 − ri of failing (and the failures are independent). Therefore, if we implement each stage with a single machine, the probability that the whole system works is r1 · r2 · · · rn. To improve this probability we add redundancy by having mi copies of the machine Mi that performs stage i. The probability that all mi copies fail simultaneously is only (1−ri) mi , so the probability that stage i is completed correctly is 1−(1−ri) mi and the probability that the whole system works is Πn i=1[1 − (1 − ri) mi ]. Each machine Mi has a cost ci , and there is a total budget B to buy machines. (Assume that B and the ci are positive integers.) Given the probabilities r1,...,rn, the costs c1,...,cn, and the budget B, find the redundancies m1,...,mn that are within the available budget and that maximize the probability that the system works correctly.