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The impact of customer impatience on production control

Most analyses of make-to-stock production control assume that either all orders are eventually met (complete backordering) or that no customers are willing to wait (lost sales). We consider a more nuanced model of customer behavior, where the fraction of potential customers who place orders depends on the current backlog, and hence the lead time. A continuous one-part-type, single machine model with Markov modulated demand and deterministic production is considered. We show that the impact of customer impatience is captured by one quantity, the mean sojourn time in the backlog states. A simple procedure finds this quantity and the optimal policy, which has hedging point form. In applications, observing the durations of stockouts gives a practical method of incorporating the effect of customer impatience.