The Economic Order Quantity formula is Q* = √(2DS/H), where D is annual demand in units, S is the fixed cost per order, and H is the annual holding cost per unit. When a problem asks you to "calculate EOQ from the following," it gives you those three values and expects you to substitute them into the formula, compute the square root, and report the order quantity that minimizes annual relevant cost. The result is the number of units to order each time stock is replenished, under the assumption that demand, ordering cost, and holding cost are all constant and known. Plug the same numbers into the EOQ Calculator to verify the formula step, including the per-year cost components and the order frequency that follows.
Reading the Variables in an EOQ Problem
Every "calculate EOQ from the following" question hides the same three inputs inside a word problem. Spotting them from the wording is half the work, because the closed-form formula only accepts three numbers and they must all share one annual basis. Demand is units per year, ordering cost is currency per order, and holding cost is currency per unit per year. If the problem gives monthly demand, multiply by 12 to get D. If it gives a monthly cost, divide by 12 to keep the basis consistent.
| Symbol | Name | Units | Where it appears in the problem |
|---|---|---|---|
| D | Annual demand | units per year | "sold 1,200 units last year," "uses 60 per month," "annual usage" |
| S | Fixed cost per order | currency per order | "setup cost," "administrative cost per order," "freight + paperwork per PO" |
| H | Holding cost per unit per year | currency per unit per year | "carrying cost of $1.20 per unit," "warehouse cost per unit per year" |
If holding cost is supplied as a percentage of unit price, convert it to a currency amount per unit per year before entry. The calculator does not infer currencies or perform that conversion. Output cost uses whatever consistent currency was entered, so the same currency symbol must apply to both S and H.
Solve the EOQ From the Following Data
Once D, S, and H are on the same annual basis, the calculation is the same whether you do it by hand or in a tool. The square-root form is the closed-form answer to minimizing the annual relevant cost function TC(Q) = DS/Q + HQ/2. The first term falls as Q grows because fewer orders are placed; the second rises because average cycle stock is Q/2. Setting the derivative to zero and rearranging produces the square-root formula Q* = √(2DS/H). At this optimum, the two variable cost terms balance.
- List D, S, and H with units. Example: D = 1,200 units per year, S = $5 per order, H = $1.20 per unit per year.
- Compute the numerator 2 × D × S. With the example values: 2 × 1,200 × 5 = 12,000.
- Divide by H to get the value under the square root: 12,000 / 1.2 = 10,000.
- Take the square root to reach Q*: √10,000 = 100 units per order.
- Compute supporting metrics: orders per year = D / Q* = 1,200 / 100 = 12 orders per year; cycle length = 365 × Q* / D ≈ 30.4 days between orders.
- Compute the cost components: annual ordering cost = DS / Q* = 1,200 × 5 / 100 = $60; annual holding cost = HQ* / 2 = 1.2 × 100 / 2 = $60; total relevant cost = $60 + $60 = $120.
- Confirm the cost balance: at the EOQ, annual ordering cost equals annual holding cost. If the two numbers do not match, the quantity is not the EOQ.
The same six steps run inside the EOQ Calculator in your browser, so you can plug the values in once and copy the supporting metrics if you need to show the cost components or order frequency in your answer. The calculator validates positive annual D, S, and H and rejects zero, negative, non-finite, or excessively large inputs, so a clean output is a sign that the inputs were internally consistent.
Reading the Output: Orders, Cycle Time, and Cost Components
EOQ is rarely the only number a textbook problem asks for. The cost-balance invariant at the optimum gives you six related values from the same three inputs, and each one is useful for a different check on the answer.
| Metric | Formula | What it tells you |
|---|---|---|
| Orders per year | D / Q* | How many purchase orders you would place annually |
| Cycle length | 365 × Q* / D | Approximate days between orders (assumes 365 calendar days) |
| Annual ordering cost | DS / Q* | Ordering overhead spent per year at the EOQ |
| Annual holding cost | HQ* / 2 | Carrying cost spent per year at the EOQ |
| Total relevant cost | DS/Q* + HQ*/2 | Variable inventory cost (excluding purchase price) |
The cost-balance invariant is the easiest sanity check. At the continuous optimum the two variable cost terms are mathematically equal, so if your calculator reports them as different by more than rounding, the quantity is not the EOQ. This equality is also why the answer is stable: the relevant-cost curve is relatively flat near its minimum, so a quantity one or two steps away from the formula answer is usually only marginally more expensive.
When the Basic EOQ Doesn't Fit the Problem
The square-root formula Q* = √(2DS/H) is the closed-form solution to a deliberately narrow model. Before you trust the number, confirm the problem actually fits the model. Basic EOQ assumes constant and known demand, constant and known ordering cost, constant and known holding cost, instantaneous complete replenishment, no shortages, no safety stock, no quantity discounts, no capacity or cash constraint, and a unit purchase price that does not change with Q. If any of those breaks, the best quantity can move materially.
Lead time is not an EOQ input. The formula answers "how much should I order," not "when should I reorder." Reorder timing is a separate calculation based on demand during lead time plus any justified safety stock. Seasonal demand, uncertain lead time, perishability, obsolescence, production replenishment, quantity discounts, or service-level targets all require a different model or an extension — for example, a quantity-discount model that compares total cost including purchase price across price breaks.
Common Limits and Sensitivity Checks
The three inputs D, S, and H are estimates, not constants. Forecasts change, warehouse and insurance costs move, capital costs shift, and supplier terms change. A practical EOQ is a baseline, not a purchasing commitment. Recalculate whenever those inputs change, and compare realistic scenarios rather than preserving one precise EOQ from stale numbers. Sensitivity matters because D, S, and H are estimates rather than timeless facts.
The continuous EOQ from the formula can be fractional because it is a mathematical optimum, not a real order. Real orders may require whole units, cases, pallets, minimum order quantities, or supplier pack multiples. When those constraints are large, blind rounding can be inappropriate. Compare feasible whole or pack-multiple quantities immediately below and above the formula result using the total-cost equation, and pick the one that minimizes total cost under the constraint. The cost curve is often relatively flat near its minimum, so the discrete optimum is usually close to the continuous one.
For a guided walk-through that ties the formula to the cost components and extends the result into total inventory cost, see the related guide on calculating EOQ and total cost of inventory step by step. The underlying quadratic formula and the cost-balance property used above are standard in operations-management references, including the SUNY New Paltz Operations Management EOQ page.