The square-root formula worked on one part number, the cost curve around it, and why the inputs matter more than the formula.
The economic order quantity is EOQ = √(2 × D × S ÷ H): twice the annual demand times the cost of placing one order, divided by the cost of holding one unit for a year, square-rooted. For part AF-7, with 95,680 units a year, 385 EUR an order and 3.8324 EUR to hold a unit for a year, the EOQ is 4,384.5 units, about 22 orders a year, at a minimum annual cost of 16,803.18 EUR.
Worked in full in Supply Chain Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The formula balances two costs that pull in opposite directions. Order often in small batches and ordering cost climbs; order rarely in large batches and the stock carried, and its cost, climbs instead. The EOQ is the batch where the sum of the two is lowest. The formula is a century old and still right. What goes wrong in practice is the inputs.
| Input | Value |
|---|---|
| Mean weekly demand | 1,840 units |
| Annual demand D (52 weeks) | 95,680 units |
| Cost of placing one order S | 385 |
| Purchase cost delivered | 28.60 |
| Cost of capital | 7.4% |
| Storage | 3.1% |
| Obsolescence | 2.25% |
| Insurance | 0.65% |
| Holding rate, total | 13.4% |
| Holding cost per unit per year H | 3.8324 |
=SQRT(2*B2*B3/(B4*B9)) with D in B2, S in B3, unit cost in B4 and the holding rate in B9At that batch the company places 95,680 ÷ 4,384.5 = 21.82 orders a year, one every 2.4 weeks. Ordering costs 21.82 × 385 = 8,401.59 EUR. The average cycle stock is half a batch, 2,192 units worth 62,698 EUR, and holding it costs 2,192 × 3.8324 = 8,401.59 EUR. Total: 16,803.18 EUR a year.
The two costs are equal at the EOQ, and that is not a coincidence of these numbers: it is where the minimum of the curve always lies. It gives a two-second test of any existing batch. If the annual ordering cost is well above the holding cost of the cycle stock, the batch is too small; if well below, too large.
| Batch (units) | Orders a year | Ordering | Holding | Total | Above minimum |
|---|---|---|---|---|---|
| 2,000 | 47.8 | 18,418 | 3,832 | 22,251 | 32.4% |
| 3,000 | 31.9 | 12,279 | 5,749 | 18,028 | 7.3% |
| 4,385 | 21.8 | 8,402 | 8,402 | 16,803 | 0.0% |
| 6,000 | 15.9 | 6,139 | 11,497 | 17,637 | 5.0% |
| 8,000 | 12.0 | 4,605 | 15,330 | 19,934 | 18.6% |
| 10,000 | 9.6 | 3,684 | 19,162 | 22,846 | 36.0% |
The curve is flat around its minimum. The penalty of ordering x times the EOQ is ½(x + 1/x) − 1, whatever the part: 2.5 per cent at 1.25 times, 8.3 per cent at 1.5 times, 25.0 per cent at double or half. The curve is not symmetric, though: 0.75 times costs 4.2 per cent, so an error on the low side hurts more than the same error on the high side. That flatness is a licence to round. AF-7 ships on pallets of 600, and seven pallets, 4,200 units, cost 16,819 EUR a year: 16 EUR, or 0.09 per cent, above the theoretical minimum. Nobody should order 4,384.5 units.
Holding cost as the cost of capital alone. Finance often supplies the 7.4 per cent cost of capital and nothing else. H falls to 2.1164, the EOQ rises to 5,900 units, 1.35 times the right answer, and the true annual cost rises by 746 EUR, 4.4 per cent. Storage, obsolescence and insurance all grow with the stock; leaving them out tells the model that stock is cheaper than it is.
Order cost loaded with fixed overhead. The cost of an order is what changes when one more order is placed: the receiving, inspection, invoice handling and transport that scale with orders. If the buyer's salary and the warehouse rent are divided by the number of orders, S can easily reach 960 instead of 385. The EOQ then becomes 6,924, 1.58 times too large, and costs 1,784 EUR a year more, 10.6 per cent. Because the formula takes a square root, an S that is two and a half times too large still only moves the batch by 1.58 times, which is why this error survives for years without being noticed.
Setting the batch on its own. The EOQ ignores the shortage cost. A larger batch means fewer cycles a year and therefore fewer exposures to a stock-out, so strictly the batch and the safety stock should be set together. The companion files for AF-7 do this: set jointly with the 98 per cent service level, the batch rises to 5,840.71 units and the year costs 683.37 EUR less than setting the two separately.
The free workbook for this case at the Supply Chain Management companion page computes AF-7's order quantity, its service level table and the cost curve, with every input in a blue cell. The buffer that sits under the cycle stock is worked in how to calculate safety stock when lead time varies.
Because at the minimum of D/Q times S plus Q/2 times H, the slopes of the two lines cancel, which happens exactly where the two costs are equal. For part AF-7 both are 8,401.59 EUR a year at 4,384.5 units. A quick check of any order quantity: if ordering cost is well above holding cost, the batch is too small, and the reverse.
Not very. The cost penalty of ordering x times the EOQ is half of (x plus 1/x) minus one. Ordering 25 per cent too much costs 2.5 per cent more, 50 per cent too much 8.3 per cent. Rounding AF-7's 4,384.5 units to seven pallets of 600 costs 0.09 per cent, 16 EUR a year.
Every cost that rises with the stock held: cost of capital, storage, obsolescence and insurance. For AF-7 these are 7.4, 3.1, 2.25 and 0.65 per cent, 13.4 per cent in all, or 3.8324 EUR per unit per year. Using the cost of capital alone gives an EOQ of 5,900 and a true annual cost 746 EUR higher.
This article is one calculation from Supply Chain Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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