Production and replacement policies for a deteriorating manufacturing system under random demand and quality

This work investigates the production planning of an unreliable deteriorating manufacturing system under uncertainties. The effect of the deterioration phenomenon on the machine is mainly observed in its availability and the quality of the parts produced, with the rates of failure and defectives increasing with the age of the machine. The option to replace the machine should be considered to mitigate the effect of deterioration in order to ensure long-term satisfaction of demand. The objective of this paper is to find the production rate and the replacement policy that minimize the total discounted cost, which includes inventory, backlog, production, repair and replacement costs, over an infinite planning horizon. We formulate the stochastic control problem in the framework of a semi-Markov decision process to consider the machine's history. The integration of random demand and quality behaviour led us to propose a new modeling approach by developing optimality conditions in terms of a second-order approximation of Hamilton-Jacobi-Bellman (HJB) equations. Numerical methods are used to obtain the optimal control policies. Finally, a numerical example and a sensitivity analysis are presented in order to illustrate and confirm the structure of the optimal solution obtained.