qwrew
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 Qni=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 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