The EOQ formula
Ordering a little at a time means many orders a year, and every order has a cost. Ordering a lot at a time means stock on the shelf, which costs money too. The economic order quantity (EOQ) is the order size where the two add up to the least:
EOQ = √(2 × D × S ÷ H)
Yearly cost = (D ÷ Q) × S + (Q ÷ 2) × H
| Term | Meaning |
|---|---|
| D | annual demand for the item, in units |
| S | cost per order: quoting, raising, receiving, inspecting, paying |
| H | cost of holding one unit for a year, or carrying rate × unit cost |
| Q | lot size |
The first part of the yearly cost is ordering cost (it falls as the lot grows) and the second is holding cost (it rises with the lot, because average stock is half a lot). At the EOQ, the two are equal. Purchase cost (D × unit cost) does not depend on the lot, so it stays out of the formula unless there is a quantity discount.
Worked example
The values already in the calculator are for a purchased part:
- Annual demand: 12,000 units
- Cost per order: $150
- Unit cost: $40, with a carrying rate of 25% a year
- A year of 250 working days
The math:
- Holding cost H = 25% × $40 = $10 per unit per year
- EOQ = √(2 × 12,000 × 150 ÷ 10) = √360,000 = 600 units
- Orders per year = 12,000 ÷ 600 = 20, one every 250 ÷ 20 = 12.5 working days
- Ordering cost = 20 × $150 = $3,000. Holding cost = 600 ÷ 2 × $10 = $3,000
- Yearly ordering and holding cost = $6,000, on top of $480,000 of purchases
What if the supplier only sells in lots of 1,000 units? That is 12 orders ($1,800) and an average stock of 500 units ($5,000): $6,800 a year, 13.3% above the EOQ, or $800.
The flat bottom of the curve
The chart shows why the EOQ is forgiving: near the minimum, the total cost curve is almost flat. The cost of any lot Q compared with the EOQ is:
Cost(Q) ÷ Cost(EOQ) = ½ × (Q ÷ EOQ + EOQ ÷ Q)
Lots from 73% to 137% of the EOQ cost at most 5% more. In the example, any lot from 438 to 822 units will do. That lets you round to a full case, a pallet or the supplier’s multiple with a clear conscience. Getting the costs wrong hurts little too: if the ordering cost is off by 50%, the lot moves 22% and total cost rises only 2%.
Economic production quantity (EPQ)
When the lot is made in your own plant, stock does not arrive all at once: it builds up while the machine runs and usage carries on. The formula gains a factor:
EPQ = √(2 × D × S ÷ (H × (1 − d ÷ p)))
Here S is the setup cost, d the usage rate and p the production rate (in the same unit, for example parts per day). If the machine makes twice what the line uses (d ÷ p = 0.5), the example lot goes from 600 to 849 units. And since the lot follows the square root of the setup cost, halving the setup cuts the lot by about 30%: that is the math behind quick changeover.
How to read the result
- The EOQ is a starting point, not an order. Round it to the supplier’s pack and check what that costs in the comparison.
- Many orders a year at short intervals mean the ordering cost deserves a look: blanket orders and scheduled deliveries cut S.
- Average stock for the item is half a lot plus safety stock. For the safety stock and when to order, use the safety stock and reorder point calculator.
- On the line, parts pulled by cards follow the kanban math, and a production lot has to fit the production capacity left after changeovers.
Common mistakes
- Monthly holding cost with annual demand. D and H must cover the same period, usually the year.
- Using the unit cost as the holding cost. H is the cost of keeping the part for a year, a fraction of its price.
- Loading the ordering cost with costs that do not change with the number of orders, such as the whole purchasing team’s fixed salaries. Only what grows with more orders belongs there.
- Applying the formula to seasonal or very lumpy demand. EOQ assumes steady usage through the year. When demand shifts a lot, recalculate by period.
- Ignoring quantity discounts. If the supplier discounts above a certain lot, compare total cost including purchases at both sizes.
Sources
- Harris, Ford W. “How Many Parts to Make at Once”. Factory, The Magazine of Management, vol. 10, no. 2, pp. 135–136, 1913.
- Wilson, R. H. “A Scientific Routine for Stock Control”. Harvard Business Review, vol. 13, 1934.
- Silver, Edward A.; Pyke, David F.; Thomas, Douglas J. Inventory and Production Management in Supply Chains. 4th ed. CRC Press, 2017.