Economic Order Quantity (EOQ) Calculator

Background.

The economic order quantity answers a question every business holding stock has to answer somehow: how much should you buy at a time? Order in large quantities and you place few orders, but you finance and store a lot of stock. Order in small quantities and stock is cheap to hold, but you pay the fixed cost of ordering over and over. There is a quantity that minimises the sum of the two, and it is the square root of twice annual demand times the cost per order, divided by the annual cost of holding one unit.

That result is not modern. Ford W. Harris published it in Factory, The Magazine of Management in February 1913, in a paper called "How Many Parts to Make at Once", and his own framing has aged remarkably well: "Interest on capital tied up in wages, material and overhead sets a maximum limit to the quantity of parts which can be profitably manufactured at one time; set-up costs on the job fix the minimum." He wrote it as the square root of 240MS/C, where M is monthly movement, S the set-up cost and C the unit cost. The 240 is not a law of nature — it is 24 divided by 0.10, and the 0.10 is an interest and depreciation charge he simply assumed, adding that "it is probable that double this would be fairer in many instances". That rate is an input on this page for exactly that reason.

The model earns its place by being robust rather than precise. MIT's supply chain course puts the sensitivity plainly: a 50% increase in order quantity over the optimum raises total relevant cost by only about 8%. That flatness is the practically useful part of the result. It means you can round the square root up to a pallet, a case pack or a supplier minimum without much loss — and it means the calculation is worth doing mainly to find out whether you are in the right neighbourhood, not to defend the third decimal place. The second mode on this page exists for that: enter the quantity you actually order and it will price the gap for you.

What the model assumes is where it breaks. Demand must be uniform and known. The whole order must arrive at once. There must be no stockouts, no backorders and no quantity discounts, and the cost parameters must not themselves change with order size. Lead time is absent for a reason worth knowing rather than worrying about: it determines when you reorder, not how much, so it has no effect on the answer. Where demand is seasonal or lumpy, or where a supplier's price breaks at 1,000 units, the square root is a starting point and not a policy — with a volume discount the purchase cost stops being constant across order sizes, and this model no longer picks the cheapest option because it never looks at purchase cost at all.

One cost is deliberately left out of every figure below. The purchase price of the goods, annual demand times unit cost, is identical whatever quantity you order in, so including it would swamp a comparison it cannot affect. On the worked example the goods cost $240,000 a year and the ordering and holding costs together come to $2,400 — one per cent of it. That ratio is worth keeping in mind before spending a week optimising the smaller number.

What is economic order quantity calculator?

The economic order quantity is the replenishment lot size that minimises the sum of annual ordering cost and annual inventory holding cost for a single item under constant, known demand. Ford W. Harris derived it in 1913 in "How Many Parts to Make at Once", concluding that "the value for X that will give the minimum value to Y, reduces to the square root of (240MS divided by C)", with M the units used per month, S the set-up cost of an order and C the unit cost. In modern notation the same result is written Q* = √(2DS/H), where D is annual demand, S is the fixed cost per order and H is the annual cost of holding one unit — the two forms are algebraically identical once D is written as 12M and H as the interest rate times C.

The two cost curves the model balances move in opposite directions. Annual ordering cost is S × D/Q, which falls as Q rises. Annual holding cost is H × Q/2, taking average stock as half the order quantity because inventory is drawn down linearly between deliveries, which rises as Q rises. Their sum, the total relevant cost, has a single minimum, and at that minimum the two components are exactly equal. Substituting Q* back gives the minimum attainable relevant cost as √(2SHD).

The model is sometimes called the Harris-Wilson or simply the Wilson formula, after R. H. Wilson's 1934 Harvard Business Review treatment, which popularised it for stock control in the years after Harris's original paper had been forgotten. Harris himself was clear about its limits, writing that "the method given is not rigorously accurate, for many minor factors have purposely been left out", and that no formula "should be depended upon entirely for determining the amount of stock that should be carried".

How to use this calculator.

1. 01Work on one item at a time. The model describes a single stock-keeping unit with its own demand and its own ordering cost — it says nothing useful about a basket of items ordered together on one purchase order.
2. 02Enter annual demand in units. If you only have monthly movement, multiply by twelve; that is exactly what Harris's original formula did internally.
3. 03Enter the cost per order — the part that does not scale with quantity. Purchasing time, approvals, goods-in, inspection and any fixed inbound freight. Harris's warning still applies: most managers underestimate this badly, and in a large factory he found it could exceed a dollar an order in 1913 money.
4. 04Enter the unit cost and the annual holding cost rate. The rate should cover your cost of capital plus storage, insurance, shrinkage and obsolescence. Do not take a number off the internet for this — it is the input the answer is most sensitive to, and it is specific to your business.
5. 05Read the optimum, then switch to the second mode and enter the quantity you actually order. The cost penalty tells you whether the gap is worth an argument with your supplier or not.
6. 06Round the answer to something you can actually buy. The cost curve is flat near the optimum, so a case or pallet quantity close to it costs almost nothing extra — and the penalty figure will tell you exactly how much.
7. 07Check the assumptions before acting. Steady demand, one whole delivery, no stockouts, no volume discount. If a supplier offers a price break at a quantity above your EOQ, this model cannot compare the two — you need to price both options including the purchase cost, which this page excludes by design.

The formula.

Holding cost per unit per year is the unit cost multiplied by the holding rate: at $20 a unit and 20% a year, $4.00. The economic order quantity is then the square root of twice annual demand times cost per order, divided by that figure. With 12,000 units of annual demand and $60 an order, that is the square root of 2 × 12,000 × 60 ÷ 4, which is the square root of 360,000, which is exactly 600 units.

Everything else follows from the quantity. Orders per year is demand divided by the order quantity, here 12,000 ÷ 600 = 20. The cycle length is the year divided by that, 365 ÷ 20 = 18.25 days. Annual ordering cost is $60 × 20 = $1,200. Annual holding cost is $4 × 600 ÷ 2 = $1,200, taking average stock as half the order quantity. Their sum, $2,400, is the total relevant cost — and it matches the closed form √(2 × 60 × 4 × 12,000) = √5,760,000 = $2,400, which is the check that the quantity really is the minimum.

The equality of those two cost figures is not a coincidence of these numbers. At the economic order quantity, annual ordering cost always equals annual holding cost, for every set of inputs. That is the simplest way to sanity-check any EOQ answer by hand.

Rounding stage: nothing is rounded part-way through. The square root, every cost and the penalty are carried at full decimal precision and rounded exactly once, at the point the result is returned, to ten decimal places. The order quantity in particular is not rounded to whole units. That is deliberate — rounding to a practical pack size is your decision, and the cost of it is precisely what the penalty figure measures.

The penalty comes from the sensitivity relation, total relevant cost at any quantity divided by total relevant cost at the optimum, which equals one half of the optimum over your quantity plus your quantity over the optimum. Ordering the 1,000-unit pallet instead of the 600-unit optimum means 12 orders a year at $720 of ordering cost, average stock of 500 units at $2,000 of holding cost, and $2,720 in total — 13.33% above the $2,400 minimum, or $320 a year. Ordering 50% above the optimum, 900 units, would cost only 8.33% more; doubling to 1,200, or halving to 300, both cost exactly 25% more. The penalty can never be negative, because the optimum is a genuine minimum, and a test sweeps hundreds of quantities to prove it.

Invalid states are refused rather than returned. A zero cost per order would make ordering one unit at a time optimal, and a zero holding cost would make ordering a lifetime's supply optimal; both are degenerate rather than useful, so both are rejected with a message naming the field. Zero or negative demand, unit cost, holding rate, days in the year, or current order quantity are refused the same way.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Ford W. Harris, "How Many Parts to Make at Once", Factory, The Magazine of Management, Vol. 10, No. 2, February 1913, pp. 135–136, 152; reprinted in Operations Research, Vol. 38, No. 6, November–December 1990, pp. 947–950 (retrieved 2026-07-29). The original derivation and the source of every Harris quotation and worked example on this page, including "the value for X that will give the minimum value to Y, reduces to the square root of (240MS divided by C)", the assumed ten per cent carrying charge, and the three printed examples (2,190 units; 6,850 pieces; 48.5 pieces). The INFORMS landing page carrying the citation and abstract is open; the reprint body text was read from a course-hosted scan of the same Operations Research reprint at brooklyn.cuny.edu.
2. [2]MITx CTL.SC1x "Supply Chain & Logistics Fundamentals", Key Concepts, Week 5 Lesson 2: Economic Order Quantity, MIT Center for Transportation & Logistics (retrieved 2026-07-29). Independent second authority. Source for the modern form Q* = √(2 ct D / ce), the total relevant cost expression, the closed form TRC(Q*) = √(2 ct ce D), the sensitivity relation TRC(Q)/TRC(Q*) = ½(Q*/Q + Q/Q*), the stated assumptions ("Demand is uniform and deterministic", "Lead time is instantaneous (0) — although this is not restrictive at all since the lead time, L, does not influence the Order Size, Q", "Total amount ordered is received"), and the robustness claim that "a 50% increase in Q over the optimal quantity (Q*) only increase the TRC by ~ 8%". Open access.
3. [3]Donald Erlenkotter, "Ford Whitman Harris's economical lot size model", International Journal of Production Economics, Vol. 155(C), September 2014, DOI 10.1016/j.ijpe.2013.12.008; open-access author copy at eScholarship, University of California (retrieved 2026-07-29). Peer-reviewed source for the provenance of the model: the paper "was published in the A. W. Shaw Company's magazine Factory, The Magazine of Management in February 1913", that "the square-root formula derived by Harris has become one of the most cited and applied results in production and operations management", and that Harris's foundation paper "was lost from sight for many years and was finally rediscovered 75 years later". Open access.
4. [4]R. H. Wilson, "A Scientific Routine for Stock Control", Harvard Business Review, Vol. 13, No. 1, 1934, pp. 116–128. PRINT, BIBLIOGRAPHIC ONLY — deliberately shipped without a URL because none was retrieved. This is the paper that popularised the square-root lot-size rule for purchasing and gave the model its common alternative name, the Wilson formula. No figure, quotation or constant on this page is attributed to it.