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An optimal ordering policy of a system subject to shocks

We consider an optimal ordering policy of a system subject to shocks with number-dependent minimal repair and random lead time. By introducing costs due to ordering, repairs and shortage, we derive the expected cost per unit time in the long run as a criterion of optimality and seek the optimum ordering policy by minimizing that cost. We show that, under certain conditions, there exists a finite and unique optimum policy. Various special cases are discussed. A numerical example is given to illustrate the method.