Age Replacement Model. An item (e.g., battery, vehicle, tool, or electronic component) whose use…

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Age Replacement Model. An item (e.g., battery, vehicle, tool, or electronic component) whose use is needed continuously is replaced whenever it fails or reaches age a, whichever comes first. The successive items are independent and have the same lifetime distribution G. The cost of a failure is cf dollars and the cost of a replacement at age a is cr. Show that the average cost per unit time is

Find the optimal age a that minimizes this average cost.