The question EOQ actually answers
EOQ answers one narrow question well: given steady demand for an item, how many units should be on each purchase order? It does not tell you when to order — that is the reorder point — and it does not tell you how much buffer to keep against variability, which is safety stock. It answers batch size only.
The reason batch size has an optimum is that two costs pull in opposite directions. Every order carries a fixed cost that does not care how many units are on it: the buyer's time raising and chasing the PO, the freight minimum, the inbound inspection, the receiving and put-away labour, the accounts-payable transaction. Order twelve times a year instead of four and you pay that three times as often.
Against that sits holding cost. Stock you own is capital you cannot use elsewhere, plus rack space, insurance, cycle counting, damage, theft and the risk that the item is obsolete before it sells. Holding cost is proportional to the average inventory you carry, and with steady demand average inventory is half the order quantity. Double the order quantity and you double the holding bill.
Ordering cost per year falls as 1/Q; holding cost rises linearly in Q. A falling curve crossing a rising line has a minimum, and the EOQ is where it sits. Ford W. Harris published the derivation in 1913 for a printing company; R. H. Wilson popularised it in the 1930s, which is why practitioners still call it the Wilson formula.
Reading the formula: why a square root, and why the two costs are equal at the optimum
Total relevant cost for a year is the sum of the two curves:
TC(Q) = (D/Q)·S + (Q/2)·H
The first term counts orders — annual demand divided by lot size, times the fixed cost each. The second term prices average inventory — half the lot, times the annual cost of holding one unit. Purchase cost D·C is deliberately absent, because with a flat unit price it is the same whatever Q you choose, so it cannot influence the optimum.
Differentiate and set to zero: −DS/Q² + H/2 = 0, so Q² = 2DS/H and Q* = √(2DS/H). The square root is not a quirk. It is why the answer is so insensitive: to double the EOQ you must quadruple either demand or the order cost, or quarter the holding cost.
Two consequences are worth memorising. First, at Q* the two cost terms are exactly equal — substitute Q* back in and each side comes to √(DSH/2). If your ordering and holding costs are not equal, you are not at the EOQ, and the larger one tells you which way to move. Second, the minimum total is TC(Q*) = √(2DSH), a clean closed form you can check in your head.
Getting H right is the hard part
D and S are usually knowable. H is where estimates go wrong, because most people count only warehouse rent. A defensible carrying rate stacks: cost of capital (your real borrowing rate or hurdle rate), storage and handling, insurance and taxes, shrinkage and damage, and obsolescence or markdown risk. Add them and most product businesses land between 15% and 30% a year — fashion and electronics higher because obsolescence dominates, staple commodities lower. Apply that rate to landed cost, not the supplier's invoice price, because landed cost is what you have tied up.
Worked example: 12,000 units a year, $250 per order, 22% carrying cost
You sell 12,000 units a year of an item that lands at $12.50. Raising and receiving a purchase order costs you $250 all-in. Your finance team puts carrying cost at 22% a year.
- Convert the carrying rate to dollars. H = 22% × $12.50 = $2.75 per unit per year.
- Build the numerator. 2 × D × S = 2 × 12,000 × $250 = 6,000,000.
- Divide by H. 6,000,000 ÷ 2.75 = 2,181,818.
- Take the square root. √2,181,818 = 1,477 units. That is the EOQ.
- Orders per year. 12,000 ÷ 1,477.1 = 8.12 orders.
- Days between orders. 365 ÷ 8.12 = 44.9 days, so roughly every six and a half weeks.
- Ordering cost. 8.12 × $250 = $2,031.
- Holding cost. (1,477.1 ÷ 2) × $2.75 = 738.5 × $2.75 = $2,031. Equal, as the algebra promised.
- Total relevant cost. $2,031 + $2,031 = $4,062. Check against the closed form: √(2 × 12,000 × 250 × 2.75) = √16,500,000 = $4,062. ✓
- Total including purchases. $4,062 + 12,000 × $12.50 = $154,062.
Now test the answer against a practical round number. Suppose the supplier ships in pallets of 500 and you order 1,500. TC(1,500) = (12,000 ÷ 1,500) × 250 + (1,500 ÷ 2) × 2.75 = $2,000 + $2,062.50 = $4,062.50 — fifty cents worse than the theoretical optimum on a $154,000 spend. That flatness is the most useful property of the model, and the reason you should never argue about the last twenty units.
How to read the result: the cost curve is flatter than you think
Treat the EOQ as the centre of a comfortable range, not a target to hit. The cost penalty for missing it follows an exact formula: TC(Q)/TC(Q*) = ½(Q/Q* + Q*/Q). Order 20% more than the EOQ and you pay 1.7% more in total relevant cost. Order half as much, or twice as much, and you pay exactly 25% more. Nothing in between is expensive.
That has three practical consequences. Round to the case pack, the pallet or the container without guilt. Prefer the round number that also gives you a clean reorder cycle — a flat 1,500 units, three pallets of 500, beats 1,477 every 44.9 days for a planner who has to live with it. And spend your effort on the inputs instead: because the EOQ varies with 1÷√H, overstating your holding cost by 30% pulls the EOQ down by only 12%, but a wrong order cost — for example, forgetting that a below-minimum freight order carries a surcharge — can be off by a factor of two.
Look next at orders per year and sanity-check it against how you actually buy. More than one order a week says you should be negotiating a blanket order with scheduled releases rather than raising individual POs; the fixed cost per release is much lower than a full PO cycle, which changes S and therefore the EOQ. Fewer than two orders a year says the order cost is dominating and it is worth asking whether it is genuinely fixed.
Finally, convert the answer into a turn rate and see whether it is one your business can live with. With an EOQ of 1,477 on 12,000 units of annual demand and no safety stock, average inventory is 739 units, which is about 16 inventory turns a year. If your category norm is four turns, the EOQ is telling you your competitors are carrying too much — or that their order costs are much higher than yours.
Cost penalty for ordering off the EOQ
| Order quantity as a multiple of EOQ | Total relevant cost vs minimum | Extra cost |
|---|---|---|
| 0.50 × | 1.250 | +25.0% |
| 0.60 × | 1.133 | +13.3% |
| 0.70 × | 1.064 | +6.4% |
| 0.80 × | 1.025 | +2.5% |
| 0.90 × | 1.006 | +0.6% |
| 1.00 × (EOQ) | 1.000 | 0.0% |
| 1.10 × | 1.005 | +0.5% |
| 1.25 × | 1.025 | +2.5% |
| 1.50 × | 1.083 | +8.3% |
| 2.00 × | 1.250 | +25.0% |
| 3.00 × | 1.667 | +66.7% |
The penalty is symmetric in the ratio: ordering half the EOQ costs exactly as much extra as ordering double it. Anything between 0.7 and 1.4 times the EOQ is within 7% of optimal.
If the supplier offers a quantity discount, EOQ alone is the wrong tool
The plain EOQ assumes unit price does not depend on order size. When a supplier offers a price break, the purchase cost term D·C stops being constant and must be brought back into the comparison. The standard procedure is short: for each price tier, compute the EOQ using that tier's unit cost; if the EOQ is not feasible in that tier, use the tier's minimum quantity instead; then evaluate full total cost — purchases plus ordering plus holding — at each candidate and take the cheapest.
Put numbers on it with the worked example below. A 2% break on the $150,000 annual purchase saves $3,000 a year. Total relevant cost at the EOQ is $4,062, and ordering double the EOQ raises that by only 25%, or $1,015. The discount is worth nearly three times the penalty, so you take it. That is the shape of the comparison: the discount is a percentage of a large number, the penalty a percentage of a small one, so the EOQ's job becomes checking that the extra holding cost stays below the discount rather than finding the exact minimum.
Where EOQ estimates go wrong
- Counting only warehouse rent as holding cost. Capital, insurance, shrinkage and obsolescence usually exceed rent. Understate H and the EOQ comes out too high.
- Treating variable freight as a fixed order cost. Freight that scales with volume belongs in unit cost, not in S. Only the truly per-order part — the minimum charge, the paperwork, the inspection — is S.
- Using purchase price instead of landed cost. Carrying cost applies to the money tied up, which includes freight-in and duty.
- Applying it to lumpy or seasonal demand. EOQ assumes a flat demand rate. For a product with a Christmas peak, use a period order quantity or a lot-sizing heuristic instead.
- Applying it to perishables or short-life goods. If the item expires, shelf life caps the order quantity regardless of what the formula says.
- Ignoring the case pack and container fill. An EOQ of 1,477 is useless if the master carton is 240 and a full container is 12,000. Round to what the supply chain can actually ship.
- Recalculating monthly and chasing the number. The curve is flat; changing lot size every month adds transaction cost and destabilises the supplier. Review quarterly or when an input moves by a third.
The model's assumptions, and what to use when they break
The Wilson EOQ rests on five assumptions: demand is known and constant; the whole order arrives at once; lead time is known and does not vary; unit cost is independent of order size; and stockouts do not happen. Real supply chains violate all five to some degree, which is why the model is a planning aid rather than a control rule.
When an assumption breaks badly, there is a named extension:
- Replenishment is not instantaneous — you make the item, or the supplier delivers over time. Use the economic production quantity, which multiplies the EOQ by √(1/(1 − d/p)) where d is demand rate and p production rate.
- Unit cost falls at volume — use the quantity-discount procedure above.
- Demand is lumpy or time-varying — use the Wagner-Whitin dynamic lot-sizing algorithm, or a period order quantity that covers a fixed number of weeks.
- Demand and lead time are uncertain — EOQ still sets the lot size, but you must add safety stock and a reorder point on top. The two decisions are separable in the standard continuous-review (Q,R) policy, which is exactly why this calculator and the reorder-point calculator are separate tools.
- Shelf life or space is binding — treat the constraint as a hard cap and take the lower of the cap and the EOQ.
One more limit worth naming: EOQ optimises cost, not service. It says nothing about fill rate, and following it with no buffer stock guarantees you run out during any demand spike. Cost minimisation and availability are separate objectives, and the second one usually matters more to a customer.
How EOQ fits with the rest of inventory planning
A working replenishment policy needs three numbers, and EOQ is one of them. How much comes from EOQ, adjusted to case pack and container fill. When comes from the reorder point: average daily demand times lead time, plus safety stock. How much buffer comes from a service-level decision against demand and lead-time variability.
Downstream, the lot size you choose drives the financial statements. Average inventory of half the order quantity plus safety stock is what shows up on the balance sheet, feeds days inventory outstanding, and consumes working capital. A planner who doubles every lot size to reduce PO admin has just moved cash out of the bank and into the racks, and the EOQ is the number that tells you whether the trade was worth it.
Lean and just-in-time practice attacks the problem from the other end. Rather than accept S as given and optimise Q, it drives S down — standardised ordering, EDI, kanban pulls, supplier-managed replenishment — so that small lots become economic. The formula agrees: cut the order cost by a factor of nine and the EOQ falls by a factor of three. That is the mathematical case for setup reduction, and it is why EOQ and lean are complements rather than rivals.
Key terms
- Ordering cost (S)
- The cost of placing and receiving one replenishment order, independent of its size: buyer time, PO administration, freight minimums, inbound inspection, receiving labour and invoice processing.
- Holding cost (H)
- The annual cost of keeping one unit in stock: cost of capital, storage, insurance and taxes, shrinkage and damage, and obsolescence risk. Often expressed as a percentage of unit value.
- Total relevant cost
- Ordering cost plus holding cost — the only two costs the lot-size decision changes. Purchase cost is excluded because a flat unit price makes it identical at every quantity.
- Order cycle
- The time between two consecutive replenishments: order quantity divided by the demand rate. Equivalently, the planning year divided by orders per year.
- Economic production quantity
- The EOQ variant for items produced rather than purchased, where stock builds up gradually during a production run instead of arriving all at once.
- Period order quantity
- A lot-sizing rule that orders a fixed number of weeks of forecast demand rather than a fixed number of units. Handles seasonality that EOQ cannot.