Economic Order Quantity (EOQ) is the level of inventory that minimizes the total cost of holding and ordering inventory over a period of time. Usually the time period is one year.

The total cost of inventory is the sum of the purchase, ordering and holding costs. As a formula:

**TC** = **PC** + **OC** + **HC**, where **TC** is the Total Cost; **PC** is Purchase Cost; **OC** is Ordering Cost; and **HC** is Holding Cost.

The risk when using the EOQ is that ordering costs and lead times may be regarded as constant.

Within lean, the goal is to reduce lead times and setup times using methods such as SMED and Kanban.

At any time, optimal order size can be calculated, but when the optimal order size is 1, we have reached one-piece production, a final goal in lean manufacturing.

To determine the Economic Order Quantity, these costs must be analyzed further. Some assumptions are required:

- Purchase Cost is a straight-forward "unit cost X number of units" calculation. In other words, volume discounts do not apply. As well, the unit cost remains constant over the year.
- Order Cost is a fixed overhead cost, and remains constant over the year. It represents, for example, the time value for employees to write up an order, mail it, follow up, inspect the received goods, and make the payment.
- Holding Cost is fixed over the year. It represents warehouse space, with services such as refrigeration or insurance or security. It also includes the interest cost, which should be set to the "risk-free opportunity-cost" rate.
- The rate of demand is constant over the year.
- The total quantity ordered is delivered in one batch.
- The lead time (between placing an order and receiving it) is constant and does not depend on the order quantity.
- The quantities are large enough that calculus may be used to determine a minimum point. Calculus requires smooth functions on the real number system. Order quantities assume integral units.

We will use the following variables:

**Q**=**Q**uantity being ordered**Q***= the optimal order**Q**uantity: the result being sought**D**= annual**D**emand for the item, over the year**P**= unit**P**urchase cost**O**= cost of one**O**rder, regardless of the number of units in the order- Sometimes shown as
**S**, meaning the cost to Set-Up the next production run **H**= annual cost to**H**old one unit

It is important to note which variables are annualized, which are per-order and which are per-unit.

Using the variables, here are the components of the first equation (**TC** = **PC** + **OC** + **HC**):

**PC** = **P** x** D** : Purchase Cost = unit Purchase cost times the annual Demand

**OC** = (**D** x **O**) / **Q** : Order Cost = annual Demand times cost per Order,

divided by the order Quantity (number of units)

**HC** = (**H** x **Q**) / 2: Holding Cost = annual unit Holding cost times order Quantity (number of units),

divided by 2 (because throughout the year, on average the warehouse is half full).

So **TC** = **PC** + **OC** + **HC** = (**P** x** D**) + ( (**D** x **O**) / **Q**) + ( (**H** x **Q**) / 2).

To minimize **TC** for **Q**, determine the first derivative of this formula and solve for zero.

d**TC**(**Q**)/d**Q** = d ( (**P** x** D**) + ( (**D** x **O**) / **Q**) + ( (**H** x **Q**) / 2) )/d**Q**

= (**H** / 2) – (**D** x **O**) / ( **Q****2** ) )

= zero

To solve for **Q***: (the optimal order Quantity):

(**H** / 2) = (**D** x **O**) / ( **Q*****2** ) )

Therefore **Q*****2** = 2 x (**D** x **O**) / **H**.

Thus **Q*** = the square root of 2 x (**D** x **O**) / **H**, and does not depend on the unit purchase cost.

In English: the optimal order Quantity is the square root of 2 times the annual **D**emand times the cost of one **O**rder divided by the annual cost to **H**old one unit.

**References:**

`1 `Matching Supply with Demand: An Introduction to Operations Management

*By Oskar Olofsson*

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