Economic distribution of budget among producers for fulfilling orders under delivery chance constraints

At a common due date, k orders with known quantities of the same product should be delivered. The product can be manufactured by n producers with stochastic production capabilities (random yields). The expected capability and its standard deviation increase linearly with the allocated budget, according to the pre-given plant's capability-cost trade-off functions. Our problem is to minimize the total budget and to determine its distribution among the producers, in order to ensure the complete fulfillment of all orders in accordance with their pre-given confidence levels (probabilities). Necessary and sufficient conditions for the existence of an optimal solution of the considered problem, as well as the proof that if the problem has a feasible solution then the optimal solution is unique, are provided. We propose a method for evaluating the optimal solution and demonstrate it through a numerical example.