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The problem of production control for a hybrid manufacturing/remanufacturing system under uncertainty is analyzed. Two sources of uncertainty are considered: machines are subject to random breakdowns and repairs, and demand level is modeled as a diffusion type stochastic process. Contrary to most of studies where the demand level is considered constant and fewer results where the demand is modeled as a Poisson process with few discrete levels and exponentially distributed switching time, the demand is modeled here as a diffusion type process. In particular Wiener and OrnsteinUhlenbeck processes for cumulative demands are analyzed. We formulate the stochastic control problem and develop optimality conditions for it in the form of Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs). We demonstrate that HJB equations are of the second order contrary to the case of constant demand rate (corresponding to the average demand in our case), where HJB equations are linear PDEs. We apply the Kushner-type finite difference scheme and the policy improvement procedure to solve HJB equations numerically and show that the optimal production policy is of hedging-point type for both demand models we have introduced, similarly to the known case of a constant demand. Obtained results allow to compute numerically the optimal production policy in hybrid manufacturing/ remanufacturing systems taking into account the demand variability, and also show that Kushner-type discrete scheme can be successfully applied for solving underlying second order HJB equations.

IntroductionIn the manufacturing environment, the availability of the machine often decreases due to its age and also to imperfect maintenance activities. In general, corrective or preventive maintenance brings the state of the machine to a level which is not new, and it may not be able to meet the demand rate for the commodity produced. For this reason, the machine has to be replaced. We consider a machine that is subject to random breakdowns and repairs. It undergoes deterioration while in operation, and the failure rate increases with its age. The aging of the machine is an increasing function of its production rate. The corrective maintenance activities performed are imperfect and restore the machine to as-bad-as-old conditions. Replacement activities for their part renew the machine, which is similar to restoring the machine to as-good-as-new conditions (resetting its age to zero).The first objective of this paper is to simultaneously determine production and replacement policies in a manufacturing environment under deterioration and imperfect repairs. We enhance existing mathematical models by including the production cost in the objective function. Given that the dynamics of the machine aging process depends on the production rate, penalizing the latter helps to amplify the impact of aging and push the system (optimal control policies) towards an appropriate solution. The proposed model appears to be better at addressing industrial reality, and is yet to be used in the literature in analyzing age-dependent production and replacement strategies. The obtained solution provides the simultaneous optimal control of production and replacement of the machine (assuming that one replacement is performed). The decision variables are the production rate and the replacement policy. The dynamics of the machine is described by a semi-Markov decision model due to the machine's deterioration and imperfect repairs (as-bad-as-old). The optimal control policies are determined in order to satisfy a deterministic customer demand and minimize inventory, backlog, production, repair and replacement costs over an infinite planning horizon. AbstractA failure-prone manufacturing system that consists of one machine producing one type of product is studied. The random phenomena examined are machine breakdowns and repairs. We assume that the machine undergoes a progressive deterioration while in operation and that the machine failure rate is a function of its age. The aging of the machine (the dynamics of the machine age) is assumed to be an increasing function of its production rate. Corrective maintenance activities are imperfect and restore the age of the machine to as-bad-as-old conditions (ABAO). When a failure occurs, the machine can be repaired, and during production, the machine can be replaced, depending on its age. When the replacement action is selected, the machine is replaced by a new and identical one. The decision variables are the production rate and the replacement policy. The objective of this paper is to address th...

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