Optimal decisions of closed-loop supply chain considering substitution in new and old product

In a closed-loop supply chain system with new and old product, remanufacturing and substitution was considered in the problem. Before random demand and random return are realized, the manufacturing quantities of new and old product, and remanufacturing quantity of used product need to be decided. Firstly, the closed loop system model with substitution and random return was proposed, and the optimal decision problem was given. Secondly, the problem was translated into other optimal problem, the existence and uniqueness of optimal solution were proved, and the optimal decision structure was given.