Stationary and time-dependent policies for the substitute teacher pool problem

This work provides multi-period generalizations of some of the analytical and simulation models that were originally developed to find the optimal size of a substitute teacher pool in a large city school district. Our objective is to minimize the expected weekly cost given that the daily demand for substitute teachers is random with possibly different distributions for each day of the week. We first develop a model that finds the stationary pool size during a five-day school week. We then discuss a dynamic programming formulation of the problem where the pool sizes can fluctuate, at a cost. Finally, we transform this dynamic program into a non-linear programming model whose cost minimizing solution is computed easily as the unique solution of a system of non-linear equations. Several examples illustrate the results.