Analysis and control of a single-item production–inventory system with shortages and back orders

An inventory model of a single-item production system incorporating deterioration and incorporating shortages and back orders is presented. The linear quadratic theory is used to develop an optimal policy which minimizes a quadratic cost criterion of production, inventory holding and shortages. The cases of both deterministic (not necessarily constant) and stochastic demands are considered. In the stochastic case, the demand is modeled as a Gaussian process with a certain mean and variance. Results of simulations are also presented and show very good agreement with the theory developed.