The economic order quantity (EOQ) from a given set of information equals √(2DS/H), where D is annual demand in units, S is the fixed cost per order in currency, and H is the annual holding cost per unit in currency. To use this formula on a textbook-style problem, identify which given value plays each role, ensure all three inputs share one annual basis, then take the square root of 2DS/H. If the problem states carrying cost as a percentage of unit price rather than a per-unit amount, multiply that percentage by unit price first to get H in currency per unit per year. The resulting Q is a continuous mathematical optimum, so comparing feasible whole-unit or pack-multiple quantities just above and below the formula value is often the realistic next step. All arithmetic for the basic EOQ model — order frequency, cycle length, annual ordering cost, annual holding cost, and their combined relevant cost — follows from D, S, H, and Q without any further information. Once those four numbers are pinned down, the answer to the original problem is just substitution into the formula.
Map Each Given Value to D, S, or H
Most "calculate EOQ from the following information" problems present a short paragraph of inventory facts and expect you to sort them into the right slots of the formula. The slot names themselves are the main hurdle.
Three inputs map cleanly:- D — annual demand: the number of units used per year, whether stated directly ("annual requirement 1,600 units") or as a rate ("200 units per month" multiplied by 12).
- S — fixed cost per order: the currency cost of placing and receiving one purchase order. Setup cost, ordering cost, and "cost of placing and receiving one order" all name the same slot.
- H — annual holding cost per unit: the currency cost of carrying one unit in stock for one full year. Sometimes given directly, sometimes as a percentage of unit price.
If carrying cost is supplied as a percentage, convert it before entry: H = (carrying-cost rate) × (unit purchase price). The calculator does not perform this conversion itself, so the conversion is on you.
| Textbook Label | Maps To | What to Check |
|---|---|---|
| Annual requirement / annual demand | D (units per year) | Already annual; no conversion |
| Cost of placing and receiving one order | S (currency per order) | Currency basis must match H |
| Annual carrying cost of inventory | H (currency per unit per year) | If given as %, multiply by unit price |
| Material cost per unit | Used to build H when carrying cost is a % | Not an EOQ input on its own |
All three inputs must share one currency basis. The calculator does not infer currencies or perform the percentage-to-currency conversion; output cost uses whatever consistent currency was entered.
Compute the EOQ Using Q = √(2DS/H)
Once D, S, and H are correctly identified, the calculation itself is direct substitution. The how-to for a textbook problem is short.
- Write down D in units, S in currency per order, and H in currency per unit per year, all on one currency basis.
- Compute the numerator 2 × D × S, then divide by H to get the radicand inside the square root.
- Take the square root. The result is Q*, the basic EOQ.
- Compute orders per year as D/Q, days between orders as 365 × Q/D, annual ordering cost as DS/Q, and annual holding cost as HQ/2.
- Add DS/Q and HQ/2 to get the combined annual relevant cost. The two halves should match — that is the cost-balance invariant.
The verified university example anchors the method: with D = 1,200 units per year, S = 5 currency per order, and H = 1.2 currency per unit per year, Q* = √(2 × 1,200 × 5 / 1.2) = √10,000 = 100 units. Orders per year = 1,200 / 100 = 12. Annual ordering cost = (1,200 × 5) / 100 = 60. Annual holding cost = (1.2 × 100) / 2 = 60. Combined relevant cost = 60 + 60 = 120 per year. Every value falls out of the same four numbers.
For a textbook problem with carrying cost expressed as a percentage, the conversion comes before substitution: with annual demand 1,600 units, ordering cost 50 per order, unit price 40, and carrying cost 10% of inventory value, H = 0.10 × 40 = 4 currency per unit per year. Plug those three inputs into the formula and the radicand works out cleanly. Hand arithmetic is fine for textbook values. Use the EOQ Calculator when the radicand is awkward, the unit basis needs verifying, or you also want orders per year, cycle length, and the cost split without writing them out by hand.
Check the Cost-Balance Invariant
At the continuous optimum produced by Q* = √(2DS/H), the two variable cost components are mathematically equal. Annual ordering cost DS/Q equals annual holding cost HQ/2 at Q*. That equality is a useful sanity check on the arithmetic and on the inputs.
Looking back at the worked example above, DS/Q = 60 and HQ/2 = 60. The match is not a coincidence — the square-root formula is derived precisely from setting the derivative of TC(Q) = DS/Q + HQ/2 to zero, which forces those two terms to balance. If the two halves do not match at the EOQ you compute, either the square root was taken incorrectly or one of D, S, H is on the wrong basis.
This invariant is verified in the calculator's implementation and matches standard operations-management material from sources such as SUNY New Paltz's EOQ module. The cost balance is not a rough approximation — it is the defining property of the continuous optimum.
Compare Feasible Quantities Near the Formula Result
The formula returns a continuous number. Real purchases must be in whole units, and often in cases, pallets, supplier packs, or minimum-order quantities. Round without checking, and the textbook answer may not be the cheapest practical one.
The pragmatic move is to evaluate the total relevant cost TC(Q) = DS/Q + HQ/2 at two or three feasible quantities immediately below and above the formula value, then pick the cheapest. The cost curve is often relatively flat near its minimum, so moderate pack-size swings usually have small cost impact — but not always, particularly when the pack is large compared with Q.
For example, if the formula gives Q* ≈ 200 but a supplier ships in cases of 24, the candidates are 192 (8 cases), 216 (9 cases), and 240 (10 cases). Plug each into TC(Q) and choose the cheapest. The math is straightforward: plug each candidate into the total-cost equation and pick the cheapest.
When Basic EOQ Assumptions Break
The basic EOQ model assumes a specific, simplified world. When the problem statement — or the real situation behind it — leaves those assumptions behind, a different model or an extension is appropriate.
| Assumption | When It Breaks | Alternative |
|---|---|---|
| Constant known demand | Seasonal sales, trends, lumpy demand | Lot-sizing heuristics, period-order quantities, MRP |
| Constant known ordering and holding costs | Variable transport, fuel surcharges, shifting warehouse rates | Recompute or use stochastic cost models |
| Instant complete replenishment | Production runs, gradual build-up | EPQ (Economic Production Quantity) model |
| No shortages | Backorders are economically justified | EOQ with planned shortages |
| No safety stock | Variable lead time, demand uncertainty | Pair EOQ with reorder-point and safety stock |
| No quantity discounts | All-units or incremental price breaks | Quantity-discount EOQ with cost comparison at each break |
| No capacity or cash constraint | Warehouse caps, working-capital limits | Constrained lot-sizing models |
| Unit price independent of Q | Volume discounts, freight breaks | Discount-aware model |
Lead time is not an EOQ input. EOQ answers how much to order; a separate reorder-point model answers when, using demand during lead time plus any justified safety stock. Mixing the two confuses the math.
Use the basic model for coursework, first-baseline inventory sizing, and sensitivity exploration. Treat it as a starting number, not an automatic purchasing commitment. Operations staff still need to validate forecasts, carrying-cost composition, supplier constraints, cash, shelf life, lead-time risk, service levels, and feasible pack sizes before any reorder is released. For the cost-curve side of the problem — particularly the combined relevant cost that follows from a chosen Q — a worked total-cost treatment is laid out in this step-by-step total-cost walkthrough.
Recalculate When Inputs Change
D, S, and H are estimates, not timeless constants. Demand forecasts move with the market, ordering cost shifts when administrative effort changes, and holding cost moves with warehouse rates, insurance, shrinkage, capital cost, and obsolescence. Any of those changes is a reason to recompute Q* — and often to run a small sensitivity grid rather than one precise number.
A useful pattern is to plug in low, base, and high values for each of D, S, and H, and watch how Q* moves. Because the formula is a square root of a ratio, a 10% change in D moves Q* by roughly 5%, a 10% change in S moves Q* by roughly 5%, and a 10% change in H moves Q* by roughly 5% in the opposite direction. Comparing realistic scenarios is more informative than preserving one precise EOQ from stale inputs.
For a homework problem this rarely comes up. For an operations decision it is the difference between a defensible order quantity and a confidently wrong one.