Consider an item with the following properties:

A = $20 v = $2/unit r = 0.24 $/$/year At time 0, the inventory has dropped to zero and a replenishment (with negligible lead time) must be made. The demand pattern for the next 12 months is Month j 1 2 3 4 5 6 Demand (units) D(j) 50 70 100 120 110 100 Month j 7 8 9 10 11 12 Demand (units) D(j) 100 80 120 70 60 40 All the requirements of each month must be available at the beginning of the month.

Replenishments are restricted to the beginnings of the months. No shortages are allowed.

Using each of the following methods, develop the pattern of replenishments to cover the 12 months and the associated total costs of each pattern (do not bother to count the costs of carrying D(j) during its period of consumption, namely, period j). In each case, the size of the last replenishment should be selected to end month 12 with no inventory.

a. Fixed EOQ (rounded to the nearest integer number of months of supply; i.e., each time the EOQ, based on the average demand through the entire 12 months, is adjusted so that it will last for exactly an integer number of months).

b. A fixed time supply (an integer number of periods) based on the EOQ expressed as a time supply, using the average demand rate for the 12 months.

c. On each replenishment, the selection of Q (or, equivalently, the integer T), which minimizes the costs per unit of quantity ordered to cover demand through T.

d. The Silver–Meal heuristic.

e. One replenishment at the start of month 1 to cover all the requirements to the end of month 12.

f. A replenishment at the beginning of every month.

Hint: For each case, it would be helpful to develop a table with at least the following columns: (1) Month, (2) Replenishment Quantity, (3) Starting Inventory, (4) Demand, and

(5) Ending Inventory.