## Figure 215

Inventory Holdings for the Eyssell Corporation

process produces a sawtooth pattern for inventory holdings; this pattern is illustrated in Figure 21.5. As the figure shows, Eyssell always starts with 3,600 units in inventory and ends up at zero. On average, then, inventory is half of 3,600, or 1,800 units.

The Carrying Costs As Figure 21.4 illustrates, carrying costs are normally assumed to be directly proportional to inventory levels. Suppose we let Q be the quantity of inventory that Eyssell orders each time (3,600 units); we will call this the restocking quantity. Average inventory would then just be Q/2, or 1,800 units. If we let CC be the carrying cost per unit per year, Eyssell's total carrying costs will be:

Total carrying costs = Average inventory X Carrying costs per unit

In Eyssell's case, if carrying costs were $.75 per unit per year, then total carrying costs would be the average inventory of 1,800 multiplied by $.75, or $1,350 per year.

The Shortage Costs For now, we will focus only on the restocking costs. In essence, we will assume that the firm never actually runs short on inventory, so that costs relating to safety reserves are not important. We will return to this issue later.

Restocking costs are normally assumed to be fixed. In other words, every time we place an order, there are fixed costs associated with that order (remember that the cost of the inventory itself is not considered here). Suppose we let T be the firm's total unit sales per year. If the firm orders Q units each time, then it will need to place a total of T/Q orders. For Eyssell, annual sales are 46,800, and the order size is 3,600. Eyssell thus places a total of 46,800/3,600 = 13 orders per year. If the fixed cost per order is F, the total restocking cost for the year would be:

Total restocking cost = Fixed cost per order X Number of orders = F X (T/Q)

For Eyssell, order costs might be $50 per order, so the total restocking cost for 13 orders would be $50 X 13 = $650 per year.

The Total Costs The total costs associated with holding inventory are the sum of the carrying costs and the restocking costs:

Total costs = Carrying costs + Restocking costs = (Q/2) X CC + F X (T/Q)

Our goal is to find the value of Q, the restocking quantity, that minimizes this cost. To see how we might go about this, we can calculate total costs for some different values of Q. For the Eyssell Corporation, we had carrying costs (CC) of $.75 per unit per year, fixed costs (F) of $50 per order, and total unit sales (T) of 46,800 units. With these numbers, some possible total costs are (check some of these for practice):

Restocking Quantity (Q) |
Carrying Costs (Q/2 x CC) |
+ |
Restocking Costs (F x T/Q) |
= |
Total Costs |

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