Optimal control for a collection system of used products

An optimal control problem for collecting used products was considered in a reverse logistics system when a remanufacturing firm faced random arrivals and deterministic demands for used products. The remanufacturing firm purchased used products to remanufacture, which made firm control the returns of used products. Firstly, a multi-period inventory system with random input which was controlled by the purchase price was proposed. Secondly, the Markov decision process model was formulated, and moreover, a state-dependent policy with two critical states was proved to be the optimal control policy of the problem. Finally, a numerical example of optimal control problem was given.