Optimal System Allocations with Penalty Costs

We have N stages to sequentially construct I successful components. At each stage, we allocate a certain amount of money for the construction of a component. If y is the amount allocated, then the component constructed will be a success with probability P(y), where P is a continuous nondecreasing function satisfying P(0) = 0. After each component is constructed, we are informed as to whether or not it is successful. If, at the end of the N stages, we are i components short, then a final penalty cost C(i) is incurred. The problem is to determine at each stage how much money to allocate so as to minimize the total expected cost (construction cost plus penalty cost) incurred. The major result is that if C(i + 1) − C(i) ≤ C(i + 2) − C(i + 1), and if y_n(i) denotes the optimal value to allocate when i components are needed with n stages remaining, then y_n(i) is nondecreasing in i and nonincreasing in n.