Stochastic optimal control of random quality deteriorating hybrid manufacturing/remanufacturing systems

In this paper, we investigate the simultaneous production planning and replacement control problem for a deteriorating hybrid system in closed-loop reverse logistics. The system is composed of unreliable manufacturing and recovery machines in which one part type is produced to satisfy a given demand. In particular, the deterioration of the manufacturing machine, which is caused by the aging process, randomly affects its availability and the quality of the parts it produces. Defective parts produced by the manufacturing machine affect the failure process of the recovery machine during its remediation activity. Due to the deterioration effect, the system is unable to fulfill long-term product demand, and the manufacturing machine can be replaced in order to increase the production capacity of the hybrid system. The main objective of this study is to determine the optimal production plan, in terms of manufacturing and recovery, as well as the replacement strategy, for the manufacturing machine, minimizing the total cost over an infinite planning horizon. The optimality conditions are developed in the form of second-order Hamilton-Jacobi-Bellman (HJB) equations in order to capture the effects of random quality deterioration and of random machine failures and repairs for which first-order HJB equations have been successfully developed in the literature. We adopt numerical methods to solve the optimality equations, and a numerical example is presented to illustrate the proposed approach. Finally, a sensitivity analysis is considered in order to confirm the structure of the joint control policy obtained.