This paper concerns a CPM network in which individual job times are random variables. Specifically the time for each job consists of a component which is a linear function of the investment (up to some maximum) in that job and a random variable that is independent of the investment. It is desired to find the minimum investment required as a function of expected project completion time. The problem is solved by a cutting plane technique in which the investment allocations yield feasibility cuts. Because of the special structure of this problem, these cuts can be generated by solving a sequence of longest path problems in an acyclic network.