Economic order quantity calculator

What is the EOQ formula?

EOQ = √(2DS ÷ H), where D is annual demand in units, S the cost of placing one order and H the cost of holding one unit for a year.

Example

An MIT logistics course uses demand of 2,000 units a year, $500 per order and a holding cost of $12.50 per unit a year (25% of a $50 unit cost): √(2 × 2,000 × 500 ÷ 12.50) = 400 units per order — 5 orders a year.

Why is that the best order size?

Bigger orders mean fewer ordering costs but more stock sitting on the shelf. EOQ is where total ordering cost plus holding cost is lowest — and at that point the two are equal, $2,500 each in the example.

What does the model assume?

Steady, known demand; a fixed cost per order; a constant holding cost; and no quantity discounts or stockouts.

What goes into holding cost?

Everything it costs to keep a unit in stock for a year — storage, insurance, spoilage and the money tied up in inventory.

What if I order a different quantity than the EOQ?

The total of ordering and holding cost rises, but slowly near the optimum. On the example the best total is $5,000 a year at 400 units. Ordering 500 units costs $5,125, only 2.5% more; ordering 200 costs $6,250, 25% more; ordering all 2,000 units at once costs $13,000. The MIT lecture draws the same conclusion: total cost is relatively insensitive to rounding the quantity and to forecast errors.

How often should I place an order?

Every 73 days on the example: 5 orders a year, one every 365 ÷ 5 days, which the lecture writes as a cycle of 0.2 years.

Where the formula comes from

If you order Q units each time, the average stock is Q ÷ 2, so holding cost for the year is H × Q ÷ 2. You place D ÷ Q orders a year, so ordering cost is S × D ÷ Q. The total, S × D ÷ Q + H × Q ÷ 2, falls as Q grows from small values and rises again once holding dominates; setting its slope to zero gives Q = √(2DS ÷ H). At that point the two parts are equal. The purchase price of the goods is left out because, with no discounts, it does not depend on the order size.

Worked example: 2,000 units, $500 per order, $12.50 holding

2 × 2,000 × $500 = 2,000,000, divided by $12.50 gives 160,000, and the square root is 400 units per order. That means 5 orders a year, $2,500 of ordering cost, $2,500 of holding cost (average stock 200 × $12.50) and $5,000 in all. The $12.50 is the MIT lecture’s holding charge of 25% on a $50 item, so a percentage must be turned into dollars per unit per year before you enter it.

Common mistakes

Typing the holding rate instead of the holding cost is the most frequent error: entering 25 instead of 12.50 gives an order of 283 units instead of 400. Mixing time units is next, since demand and holding cost must both be per year. Using the formula where the supplier offers quantity discounts, where demand swings, or where stockouts are acceptable ignores the model’s own assumptions; the MIT lecture lists no discounts, uniform known demand, zero lead time and no shortages as the basic model.

Sources: Chris Caplice, MIT OpenCourseWare ESD.260J Logistics Systems (Fall 2006), Lecture 7, Inventory Management: Fundamental Concepts & EOQ (assumptions of the basic model, the 2,000-unit example and the sensitivity table); F. W. Harris, “How Many Parts to Make at Once,” Factory, The Magazine of Management 10(2), 1913, reprinted in Operations Research 38(6), 947–950 (1990), where the formula first appeared. The break-even and ordering-cost arithmetic is the same algebra. See also the break-even calculator, the margin calculator and the price elasticity calculator.