Abstract
We formulate a stochastic optimal production control problem of a manufacturing system with deteriorating items in the presence of random disturbances for a single machine multi-product. The dynamics of the inventory can be governed by an Itô stochastic differential equation. It is found in real market that the deteriorating inventory systems appears in several industrial sectors, such as food production, chemical and radioactive material, and pharmaceutical manufacturing. In the considered production system, the machine’s times to failure and times to repair are modeled as a semi-Markov process. The goal is to find the production rates in real-time to meet the random demands, through a minimization of the expected discounted cost of inventory holdings and shortages over a finite horizon. The optimality conditions are obtained in the form of time-dependent second-order Hamilton-Jacobi-Bellman (HJB) equations by using the dynamic programming and the stochastic calculus introduced by Itô. The model is formulated for n-items, and in particular, is illustrated with two items for some numerical data. The proposed model in the case of non-exponential distributions enables us to independently improve the coefficient of variation (CVup/down). Numerical methods are used to solve the optimality conditions and the structure of the control policies is confirmed by a sensitivity analysis.
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