On Optimum Stochastic Allocation

Analytic results involving the optimum return and strategy functions of an inventory allocation problem are presented. The problem requires that a number of randomly functioning units be apportioned among randomly many points to maximize the probability that at least one unit functions at each point. The maximum probability function has been shown often, but not always, to be a monotonic non-increasing function of the number of units. When it is monotonic nonincreasing, it is proven here that the ratio improvement of maximum probability is a decreasing function of the number of units. When the ratio is only nonincreasing, it is proven¹ here that the strategy is a monotonic nonincreasing function of the number of units. The theoretical results relate to the management use of increment and ratio measures of effectiveness and to properties of the optimum allocation strategy.