Optimal control of a three-stages supply chain model with shortages

This paper considers an optimal control of a three-stage supply chain model that consists of a manufacturer, a vendor and a retailer. The model is developed under a continuous-time review policy where the deterioration rate is fixed, shortages are allowed and the demand is assumed to be a linear function of price. Optimal control theory for non-smooth dynamic systems is applied in order to maximize the supply chain profit by optimizing both of the production rate and the dynamic price. To illustrate the result of the proposed model, a numerical example is presented.