The EOQ Square-Root Formula, and Where It Stops Being True

Why the answer is a square root

Two costs move in opposite directions when you change the size of an order. Ordering cost is the fixed work of raising a purchase order, receiving it and paying the invoice, and you incur it D/Q times a year — so it falls as the order gets bigger. Holding cost is capital, space, insurance and obsolescence on the average stock you carry, which is Q/2 — so it rises as the order gets bigger. Write the sum, D/Q × S + Q/2 × H, differentiate, set it to zero, and the two terms turn out to be equal at the optimum. Solve that equality and the square root falls out: Q* = √(2DS/H).

That equality is worth remembering as a check. At 25,000 units a year, $80 an order and $4 per unit-year of holding, the formula gives 1,000 units. Twenty-five orders a year at $80 is $2,000 of ordering cost; an average stock of 500 units at $4 is $2,000 of holding cost. They match, as they must. The total, $4,000, can also be read straight off as √(2DSH) — the same numbers rearranged. If your spreadsheet produces an EOQ where the two cost lines are not equal, the arithmetic is wrong, not the theory.

The flat bottom is the honest headline

Divide the cost at any quantity by the cost at the optimum and almost everything cancels. What is left is ½(Q/Q* + Q*/Q) — a ratio that depends only on how far off you are, not on demand, not on S, not on H. Order 20% too many and you pay 1.7% more. Order 20% too few and you pay 2.5% more. Order 50% too many and you pay 8.3% more. You have to go all the way to double or half the optimum before the bill rises by 25%. Near the bottom, the curve is almost a horizontal line.

This is the single most practical thing EOQ tells you, and it is usually buried. The formula rarely produces a number a warehouse can use: 1,000 units may be 41 cases or two-thirds of a pallet. The flatness says round it. Going to a full pallet at 1,200 units costs $67 a year on a $4,000 base. Going to a truckload at 1,500 costs $333. Set against the handling, damage and admin savings of shipping whole units of packaging, both are usually a good trade. EOQ is not an instruction to order exactly 1,000; it is a statement that anything between roughly 800 and 1,250 is within 2.5% of the best you can do.

Which is lucky, because S and H are the hardest numbers here

Annual demand you can pull from the sales system. The other two you have to construct. S is supposed to be the incremental cost of placing one more order — but most of what a purchasing department costs is salary that does not change whether it raises 25 orders or 30. Divide the department's budget by the number of orders and you get an average, not a marginal cost, and the average is almost always too high. H is worse: capital cost, storage, insurance, shrinkage, obsolescence. The capital component alone forces you to pick a discount rate, and the obsolescence component is a guess about the future.

The flatness rescues you. Because Q* is proportional to √(S/H), only the ratio matters, and an error in the ratio is halved by the square root before it reaches the quantity. Get S/H wrong by a factor of two in either direction and the recommended quantity moves by 41% — which, from the table above, costs 6.1%. Get it wrong by a factor of four and the quantity doubles or halves, costing 25%. Anything short of a fourfold error in your cost assumptions is a rounding difference. Spend the effort on the demand figure and on the constraints instead; those are what actually move the answer.

Break one: quantity discounts

The formula never sees the price of the goods, because at a constant unit price the purchase cost is D × C no matter how you split the orders — it is the same in every scenario and drops out of the comparison. A quantity discount destroys that. Suppose the unit cost is $20, holding cost is set at 20% of value (which gives exactly the $4 used above), and the supplier offers 2% off for orders of 2,500 units or more. The discount is worth 2% of $500,000, or $10,000 a year. That is not a small number next to a $4,000 total of ordering and holding cost.

Work it properly. Buying 2,500 at a time means 10 orders a year, $800 of ordering cost. The discounted unit cost is $19.60, so holding at 20% of value becomes $3.92 per unit-year, and an average stock of 1,250 units costs $4,900. Ordering plus holding rises from $4,000 to $5,700 — an increase of $1,700 — while the purchase bill falls from $500,000 to $490,000. Everything in: $504,000 against $495,700, a net saving of $8,300 a year. The EOQ answer is comprehensively beaten by a supplier's price list, which is why any real discount evaluation computes the total cost at each price break rather than at the unconstrained optimum.

Break two: lumpy demand and a finite replenishment rate

EOQ assumes demand arrives at a smooth, constant rate. Take the same 25,000 units a year but concentrate 15,000 of them into two months. During the peak the annualised rate is 90,000 units, and the correct order quantity there is √(2 × 90,000 × 80 / 4) = 1,897. For the remaining 10,000 spread over ten months the annualised rate is 12,000 and the correct quantity is 693. Run the annual EOQ of 1,000 through the peak and the ratio formula charges you 21.2% over what a seasonal quantity would cost. Any product with a real season needs its EOQ recomputed per season, or a dynamic lot-sizing method — Wagner-Whitin is the exact one, Silver-Meal the practical approximation.

The second structural assumption is that the whole order lands at once. When you make the goods yourself rather than buying them, stock builds while production runs and drains while it does not, so the average inventory is lower than Q/2 and you can afford bigger runs. The economic production quantity corrects for it: √(2DS / (H(1 − D/P))). At a production rate of 50,000 units a year against demand of 25,000, the factor (1 − D/P) is 0.5, the optimal run rises to 1,414 units, peak stock is only 707, and the true minimum cost falls to $2,828. Apply the purchasing EOQ of 1,000 in that setting and you pay $3,000 — 6.1% too much, and you also under-size every production run.

What EOQ is still good for

Treat it as a scale check, not an instruction. It tells you whether the right order is roughly a hundred units or roughly ten thousand, and that is the question most people actually get wrong — usually by ordering far too often on cheap, bulky items and far too rarely on expensive, compact ones. Once EOQ has fixed the order of magnitude, the constraints take over: pallet quantities, minimum order values, container fill, shelf life, the supplier's price breaks and the warehouse's slot capacity. Because the curve is flat, letting those constraints win costs almost nothing.

And keep it in its lane. EOQ answers how much to order; it says nothing about when. That is the reorder point, which depends on lead time and on the variability of demand during that lead time, and it is where safety stock lives. The two questions are independent in the classical model and are usually best kept independent in practice: size the order with EOQ and the constraints, trigger it with a reorder point that reflects the service level you have promised.

Frequently asked questions

How far off can my order quantity be before it costs real money?

Use ½(Q/Q* + Q*/Q). At 20% off the optimum in either direction the penalty is under 2.5%; at 50% too many it is 8.3%; you have to double or halve the quantity to reach 25%. On the worked example that means anything from about 800 to 1,250 units costs no more than $100 a year above the best possible $4,000. Round to whatever your packaging, pallet or truck wants.

What should I actually put in for the ordering cost S?

Only costs that genuinely change when you place one more order: inbound freight that is charged per shipment, goods-in labour and inspection, and any per-order supplier or customs fee. Leave out the buyer's salary, the ERP licence and the department overhead — those are there whether you order 25 times or 30. Dividing total purchasing cost by the number of orders gives an average that overstates S, which biases orders larger. Because the curve is flat, even a twofold error costs only 6.1%, so do not agonise; just be consistent between S and H.

Does EOQ tell me when to reorder?

No. EOQ answers how much; the reorder point answers when. The reorder point is average demand during the lead time plus safety stock, and safety stock depends on how variable demand and lead time are and on the service level you have chosen. In the classical model the two decisions do not interact, so you can size the order with EOQ and set the trigger separately without either one spoiling the other.

A supplier is offering a discount above my EOQ. How do I decide?

Compute the full annual cost — purchases plus ordering plus holding — at your EOQ and again at each price break, and compare the totals. Remember to recompute holding cost on the discounted value if you express it as a percentage. In the example above, a 2% discount at 2,500 units raises ordering plus holding by $1,700 but cuts the purchase bill by $10,000, so the discount wins by $8,300 a year. Discounts usually win, because the purchase bill is an order of magnitude larger than the costs EOQ optimises.

Is EOQ still relevant with just-in-time supply?

Yes, but it points the other way than people expect. Just-in-time is not a rejection of EOQ; it is an attack on S. Halve the cost of placing an order — through electronic ordering, blanket agreements, faster changeovers — and the optimal quantity falls by 29%, and the total relevant cost falls with it, because the optimum is √(2DSH). Small, frequent deliveries are economic only once the fixed cost of a delivery has been driven down. EOQ is the formula that tells you how much that reduction is worth.