Consider a replacement problem where it is possible to keep one spare item in stock in case the operating item fails

Consider a replacement problem where it is possible to keep one spare item in stock in case the operating item fails. After a fixed time T1Ā 0 of operation, an identical spare is ordered. It is delivered after a constant lead time, L. From time T1 + L onwards, we have a spare unit and at some time T2 ( T L T) the operating item is replaced with the spare item. If the operating item fails before time T1 , an emergency order is placed and the failed item is replaced as soon as the ordered item arrives. If the operating item fails between T1 and T1 L, it is replaced at time T1 L when the ordered spare arrives, otherwise it is replaced at time T2 . Assume that the inventory cost of holding a spare per unit time is h, the regular ordering and emergency ordering costs are Ar and Ae (Ae r A ), respectively. There is also a penalty cost Cā„“ per unit time when the item is waiting for a spare to arrive. Develop a model for deciding the optimal time of ordering T1 and the optimal time of replacement T2 that will minimize the asymptotic expected total cost per unit time.

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