EOQ

EOQ – how much to order at once, and why the answer is flat

How much should you order at once? The EOQ formula explained with a worked example, the elegant property at the optimum, why the cost curve is so flat that precision barely matters – and what lean does to the whole calculation.

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What this covers

The two costs pulling opposite ways
The 1913 formula, and the man who was forgotten
A worked example
The elegant property at the optimum
Why the curve is flat – and what that means
What the model assumes
The lean attack on the formula itself

What this covers (cont.)

The link to the bullwhip effect

The two costs

The whole problem is that two costs pull in opposite directions.

Order often, or order a lot

Small, frequent orders

Cheap stock, expensive ordering

Little capital tied up in goods
Little space in use
But: many orders a year
Each order costs admin, freight and changeover

Large, infrequent orders

Cheap ordering, expensive stock

Few orders a year
Discounts and a full lorry
But: a lot of capital tied up
Space, shrinkage, insurance, obsolescence

Note that neither column is wrong. Both are sensible, and they are sensible for opposite reasons. When two costs behave like that – one falling as the other rises – there is a point where the sum is lowest. Finding that point is all EOQ is.

The 1913 formula

EOQ is the square root of twice demand times order cost, divided by holding cost.

D is annual demand, in units.
S is what it costs to place one order – admin, freight, machine changeover. Note: per order, not per unit.
H is what it costs to hold one unit for one year – cost of capital, space, shrinkage, insurance.
And so: EOQ = √(2DS / H)

A worked example

Take the screws from the bookcase factory.

From three numbers to one answer

Demand: 100,000 screws a yearD. Taken from the production schedule.
Order cost: 500 kronerS. It costs the same whether you order 100 or 100,000.
Holding cost: 1 krone per screw per yearH. Capital, space and risk.
EOQ = √(2 × 100,000 × 500 / 1) = 10,000Order 10,000 screws at a time, that is ten times a year.

Note how little is needed. Three numbers, one square root. The hard part of EOQ is not the mathematics – it is working out what the order cost and holding cost actually are in your own business, and those figures are rarely sitting ready in any system.

The elegant part

Work out what the two costs come to at 10,000, and something neat happens.

At the optimum, ordering cost and holding cost are exactly equal.

Ordering cost: 100,000 / 10,000 = 10 orders, times 500 kroner = 5,000 kroner.
Holding cost: average stock is half the batch, that is 5,000 screws, times 1 krone = 5,000 kroner.
Total: 10,000 kroner a year.
And this is no coincidence: it always holds. If the two costs differ, you are not at the optimum – which gives you a quick way to check an answer without running the formula again.

The curve is flat

Here comes the most useful insight in the episode, and it is nearly always left out when EOQ is taught.

What it costs to be wrong

Batch	Ordering	Holding	Total	vs optimum
2,500	20,000	1,250	21,250	+112%
5,000	10,000	2,500	12,500	+25%
7,500	6,667	3,750	10,417	+4.2%
10,000	5,000	5,000	10,000	optimum
12,500	4,000	6,250	10,250	+2.5%
15,000	3,333	7,500	10,833	+8.3%
20,000	2,500	10,000	12,500	+25%

Look at the three middle rows. Being 25 per cent out in either direction costs you between 2.5 and 4.2 per cent. The curve is flat around the bottom, and that has a practical consequence people rarely draw: you do not need precise figures for holding cost and order cost. A rough estimate gives you a decision that is good enough. It only really hurts when you are out by a factor of four – 2,500 against 10,000.

What the model assumes

The formula is this simple because it assumes a lot away. The assumptions are worth knowing, because they tell you where the model does not fit.

Four assumptions

Assumption	When it breaks
Steady, known demand	Seasonal goods, promotions, trends. Anything that swings.
Instant delivery	Lead times exist. They are handled with a reorder point, not by EOQ.
Fixed unit price	Quantity discounts make the curve jagged, and the optimum can jump.
No limit on space or capital	The warehouse has a wall, and the budget has an edge.

It is easy to dismiss the model because of these. But remember the flat curve: even when the assumptions are badly broken, EOQ usually gives a batch closer to right than the gut feeling of whoever is placing the order. The model is a direction, not an answer.

The lean attack

Here it gets interesting, and this is where EOQ meets Toyota.

What happens to EOQ when the order cost is cut

S = 500 kr → EOQ = 10,000Ten orders a year. Total cost 10,000 kroner.
S = 125 kr → EOQ = 5,000Twenty orders. Total cost 5,000 kroner.
S = 50 kr → EOQ = 3,162Over thirty orders. Total cost 3,162 kroner.
S = 5 kr → EOQ = 1,000A hundred orders a year, and the cost down to a tenth.

Cutting from 500 to 50 kroner makes EOQ fall by 68 per cent. This is the whole point of changeover work in lean: spend a day getting a machine changeover from four hours down to ten minutes, and you have not merely saved those hours – you have moved the optimum, and made small batches economic. Lean and EOQ do not disagree. Lean changes one of the inputs, and lets the formula give a new answer.

And the link to bullwhip

Finally, a reminder that every optimisation has a price somewhere else.

What is worth taking with you

That EOQ balances order cost against holding cost, and that both are real.

Key points

EOQ balances the cost of ordering often against the cost of holding stock.
The formula is the square root of twice demand times order cost, divided by holding cost.
At the optimum the two costs are exactly equal.
The curve is flat – being 25 per cent out costs only a few per cent.

Key points (cont.)

That means you do not need precise figures to make a good enough decision.
The model assumes steady demand and instant delivery. Neither is true.
Lean does not dispute the answer, it attacks the input: cut the order cost and EOQ falls.

Questions this answers

What is EOQ?
Which two costs are being balanced?
What is the formula, and how is it used?
What happens if I get the order quantity wrong?
What does the model assume?
Why do lean practitioners want small batches when EOQ says large?

Questions this answers (cont.)

What does EOQ have to do with the bullwhip effect?

Sources

Ford W. Harris – How Many Parts to Make at Once, Factory, The Magazine of Management 10 (1913)
Donald Erlenkotter – Ford Whitman Harris and the Economic Order Quantity Model, Operations Research 38(6), 1990
Economic order quantity – overview article

Spørretimen

Read the whole thing at

www.sporretimen.no/en/episodes/eoq-optimal-bestillingsmengde/