Question

For the simple EOQ model, show that the average annual inventory carrying charges are equal to the average annual ordering costs when Q = Q*

Answer 1

The assumptions made while deriving the EOQ formula includes that each order is received in a single delivery, lead times does not vary and annual demands are known. There are no quantity discounts and the demand is constant and continuous over time. Only holding cost and ordering costs are relevant and purchase cost is not considered as it is a constant since the quantity discounts are zero.

As the demand is considered constant over time, the order Q is received each time at the instant the inventory on hand falls to zero.

Hence, Average cycle inventory = (0 + Q)/2 =Q/2

The inventory holding cost can be calculated by multiplying the average inventory with the cost of holding one item for the stated period.

If H is the carrying cost per unit, the annual carrying cost can be calculated as

Annual carrying cost = average inventory x annual carrying cost per unit

                                       = Q/2 x H

If D is the annual demand for the product,

The number of order placed per year = D/Q

If S denotes the cost of placing one order,

Annual ordering cost = D/Q x S

Hence the total cost can be calculated as TC = Annual carrying cost + annual ordering cost

                                                                                 = (Q/2) H + (D/Q) S

Economical order quantity is the quantity that minimizes the total cost. Let us denote EOQ as Q*.

We can find the order quantity that minimizes the total cost by taking derivative of the cost function with respect to Q on both sides.

d(TC)/dQ = d/dQ(QH/2) + d/dQ(DS/Q)

We can equate the derivative of total cost to zero while minimizing cost,

Hence d (TC)/dQ = 0

0 = H/2 – DS/Q²

H/2 = DS/Q²

H (Q/2) = (D/Q) S

Hence at Q = Q*, average annual inventory carrying charges are equal to the average annual ordering costs.