Economic Order Quantity and Reorder Point

Ordering in larger batches cuts the cost of placing orders but raises the cost of holding stock. This finds the batch size where the two together are cheapest, as √(2 × annual demand × order cost ÷ holding cost), along with the stock level at which to reorder.

Ordering is a balancing act. Order in large batches and you place fewer orders, so the paperwork and delivery charges per year go down. But the stock then sits in the warehouse, tying up space and cash. The order size where those two costs together are lowest is the economic order quantity, also known as the Wilson formula.

Q = \sqrt{\dfrac{2DS}{H}}

$Q$ is the economic order quantity
$D$ is annual demand
$S$ is the cost of placing one order
$H$ is the cost of holding one unit for a year

Annual ordering cost is $(D \div Q) \times S$ , which falls as $Q$ grows. Annual holding cost is $(Q \div 2) \times H$ , which rises as $Q$ grows. The half appears because stock runs down steadily from $Q$ to zero, so on average $Q \div 2$ units are on the shelf.

Example

Take annual demand of 12000, an ordering cost of 5000 and a holding cost of 100 per unit per year.

$2 \times 12000 \times 5000 \div 100$ is 1200000, whose square root is about 1095. Ordering 1095 units at a time is the cheapest policy. That means $12000 \div 1095$ , about 10.95 orders a year, or one roughly every 33 days.

Split the cost at that point. Ordering costs $10.95 \times 5000$ , about 54772. Holding costs $547.7 \times 100$ , about 54772. Exactly the same figure. That is no coincidence: when one cost falls as the other rises, the total bottoms out precisely where they balance. The total is about 109545.

Reorder point

How much to order is only half the question; when to order is the other half. Deliveries take time, so waiting until the shelf is empty is too late.

The order goes in while there is still enough stock to cover sales during the wait. With a seven-day lead time that is $12000 \div 365 \times 7$ , about 230 units. When stock falls through that level, it is time to reorder.

Points to watch

The formula assumes demand is steady through the year and that you never run out. Real demand fluctuates and deliveries slip, which is why practitioners add safety stock on top of the reorder point calculated here.

The total cost curve is very flat near its minimum. Rounding 1095 to either 1000 or 1200 gives an annual cost of 110000, just 455 above the true minimum of 109545. Feel free to round to whatever case, pallet or carton size is convenient.