Optimal control for a closed-loop supply chain with stochastic demand and return depending return payment price

For a closed-loop supply chain system with one serviceable inventory and used inventory, a finite period optimal control problem is considered. The demand and return are stochastic, and the return quantity of used product is influnced by the return patment price in order to control thr return process of used product. Because the return payment is a function of price discount for customer, we assume that the demand is also influnced by the return payment price. Under each of demand and return is a linear function of return payment price, the joint concave of the period profit function and optimal value function are proved, and the optimal decision rule is given. And then, the algorithm for the parameters in decisoin rule is proposed.