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In this article, we investigate a dynamic control problem of aproduction-inventory system. Here, demands arrive at the production unitaccording to a Poisson process and are processed in an FCFS manner. Theprocessing time of the customers' demand is the exponential distribution. Theproduction manufacturers produce the items on a make-to-order basis to meetcustomer demands. The production is run until the inventory level becomessufficiently large. We assume that an item's production time followsexponential distribution and the amount of time for the produced item to reachthe retail shop is negligible. Also, we assume that no new customer joins thequeue when there is a void inventory. This yields an explicit product-formsolution for the steady-state probability vector of the system. The optimalpolicy that minimizes the discounted/average/pathwise average total cost perproduction is derived using a Markov decision process approach. We find optimalpolicy using value/policy iteration algorithms. Numerical examples arediscussed to verify the proposed algorithms.

Optimal control for production inventory system with various cost criterion