The Economic Order Quantity

A shop sells a steady 1,000 boxes of tea a year. It could order all 1,000 at once — one big delivery, almost no paperwork, but a stockroom groaning under a year's supply. Or it could order a handful at a time — a nearly empty stockroom, but a courier at the door every other day and an ordering clerk driven to despair. Somewhere between "one huge order" and "a thousand tiny ones" sits the sweet spot. Finding it is the oldest optimisation in operations management: the Economic Order Quantity, or EOQ, worked out by Ford Harris in 1913 and still humming inside inventory software today.

Two costs pulling in opposite directions

The whole problem is a tug-of-war between two costs, and the order size Q is the rope.

Ordering cost — a fixed charge K every time you place an order (the paperwork, delivery fee, set-up). With annual demand D you place D/Q orders a year, so this cost is K D/Q. Bigger orders → fewer orders → less ordering cost.
Holding cost — a charge h per unit per year to keep stock on the shelf (capital tied up, storage, spoilage). Stock sawtooths between Q and 0, averaging Q/2, so this cost is h Q/2. Bigger orders → more stock sitting around → more holding cost.

Ordering cost wants Q large; holding cost wants it small. The total annual cost is their sum:

C(Q) = \underbrace{\frac{KD}{Q}}_{\text{ordering}} \;+\; \underbrace{\frac{hQ}{2}}_{\text{holding}}.

The U-shaped cost curve

Plot those two pieces and their sum. The ordering cost slides down like a slippery dip; the holding cost climbs a straight line; their sum is a gentle U. Drag the order size and hunt for the bottom:

The minimum sits exactly where the two costs cross — where ordering cost equals holding cost. That is not a coincidence, and a line of calculus shows why.

Minimising by calculus

To find the bottom of the U, differentiate C(Q) and set it to zero:

C'(Q) = -\frac{KD}{Q^2} + \frac{h}{2} = 0 \quad\Longrightarrow\quad \frac{h}{2} = \frac{KD}{Q^2}.

Notice the two terms being equal is precisely "holding cost rate = ordering cost rate" — the crossing point. Solving for Q gives the celebrated square-root formula:

\displaystyle Q^{*} = \sqrt{\frac{2KD}{h}} — the order size that minimises total cost.
The minimum cost itself is C(Q^{*}) = \sqrt{2KDh}, split evenly between ordering and holding.

The square root is the signature of EOQ: to serve four times the demand you order only twice as much each time. Economies of scale, quantified.

Worked example

Our tea shop: annual demand D = 1000 boxes, ordering cost K = \pounds 40 per order, holding cost h = \pounds 2 per box per year. Then

Q^{*} = \sqrt{\frac{2 \times 40 \times 1000}{2}} = \sqrt{40000} = 200 \text{ boxes.}

So order 200 boxes at a time — that is 1000/200 = 5 orders a year. Check the costs: ordering = 40 \times 1000/200 = \pounds 200; holding = 2 \times 200/2 = \pounds 200 — equal, as promised, for a total of \pounds 400 a year. Compare that with ordering monthly (Q ≈ 83): total cost climbs to about \pounds 565. The EOQ saves a third, from one square root.

Look again at the U — near its base it is remarkably flat. Order 25% more or less than Q^{*} and the total cost rises by only a couple of percent. This robustness is EOQ's secret superpower: you do not need to know K, D and h precisely. A rough estimate lands you in the flat valley, and being a little off costs almost nothing. It also means you can round Q^{*} to a convenient case-quantity or pallet size with a clear conscience.

The classic formula is built on strong idealisations: demand is constant and known, stock is replenished instantly the moment it hits zero, and the costs K and h never change with order size. Real demand is bumpy, deliveries take time, and suppliers offer quantity discounts that bend the curve. So EOQ is a starting point, not the last word — you bolt on reorder points and safety stock to cope with the messiness. Happily, that flat bottom means EOQ stays a useful guide even when its assumptions are only roughly true.