M.Sc Student Polukarov Mariya A Two Stage Supply Problem with Stochastic Demands Department of Industrial Engineering and Management Professor Michal Penn Professor Moshe Kress

Abstract

The problem of determining the deployment and content of logistic units in the battlefield is an important problem in military logistics. Here we model it in the form of a ``supply problem with stochastic demands''. We define the supply problem as the problem of distributing goods to a number of depots and customers in such a way that at each stage all customers’ demands are met with prescribed response probabilities, and the objective is to minimize the total operation cost. We formulate this chance constrained optimization problem as a binary linear programming problem. Setting the problem in the form of a mixed-integer programming problem enables us to incorporate simultaneously chance constraints and stochastic recourse in our model.

In this study, we consider four different versions of the supply problem. The problems differ with respect to the decision-making policy (a priori (no recourse) vs. a posteriori (with recourse)) and the way the chance constraints are incorporated in the model (separate vs. joint chance constraints). Assuming the cost for the consumer is higher than the cost for the depot (which is usually the real situation), we find combinatorial algorithms for solving two of the versions of the two-stage model: a priori decision under joint chance constraints and a posteriori decision under separate chance constraints. From these solutions we easily derive a solution for the former case for any cost function. For the remaining two versions we construct a method to obtain feasible solutions.

Finally, we analyze a special model in which the supply at the first stage is distributed among the customers in equal amounts. Using a simple geometric interpretation of the problem we characterize the set of optimal solutions of the problem for two of its versions, and present combinatorial algorithm to solve the problem in two special cases. The first one is if the cost for the consumer is higher than the cost for the depot and the second if the cost for the depot is higher than the cumulative cost for the consumer.