# Economic Batch Quantity (EBQ)

A business which manufactures products in batches of identical units needs to establish the quantity of units in a batch which will minimize production costs. This batch size is referred to as the economic batch quantity or EBQ.

## Production Costs

For a given demand the total production costs accumulated in a batch costing system comprise the following three items.

1. Product costs including the cost of materials, labor and expenses to manufacture the product itself.
2. Setup costs incurred in setting up machinery etc. ready for a batch production run.
3. Inventory holding costs including for example interest on finance used to fund inventory working capital, warehousing costs, insurance, and obsolescence.

### Product Cost

The total product cost of materials, labor and expenses is determined solely by the total demand for the product and the unit product cost.

Total product cost = Unit product cost x Demand

Assuming the demand is fixed the total product cost remains the same irrespective of the batch size. These costs are therefore not relevant for determining the economic batch size.

It should be noted however that the cost of the product itself can be a factor in determining the finance cost and therefore the inventory holding cost when considering batch sizes.

### Setup Costs

The total setup costs will be determined by the setup cost per batch (s), which is assumed to be constant, and the number of batches used in production of a given demand quantity.

Total setup costs = Setup cost per batch x Number of batches

Assuming demand (D) is known and constant within a certain period of time then the number of batches will be given by the demand divided by the batch size (q). The total setup costs are as follows.

```Total setup cost = Setup cost per batch x Demand / Batch size
Total setup cost = s x D / q
```

As the batch size (q) increases the total setup cost decreases.

### Inventory Holding Costs

The total inventory holding costs are determined by the average number of units held in the warehouse and the holding cost per unit (h) for the period of demand.

Total holding cost = Holding cost per unit x Average inventory level

#### Average Inventory Level

Assuming demand is known and constant within a certain period of time then the inventory level will start at the batch quantity, fall to zero, and then be placed by the next batch; as shown in the diagram below. The average inventory level is therefore the batch quantity divided by two, and the total holding cost is given as follows.

```Total holding cost = Holding cost per unit x Average inventory level
Total holding cost = h x q / 2
```

As the batch size (q) increases the total holding cost increases.

## Economic Batch Quantity Formula

It can be seen from the two formulas above that the total setup cost and total holding cost move in opposite directions as the batch size (q) changes. This is demonstrated in the diagram below. From the graph above it can be seen that there is a batch size at which the total costs are at a minimum. At this point the total setup cost and the total holding cost are equal.

The batch size when the total cost is at a minimum is referred to as the economic batch quantity (EBQ) or economic batch size and is given by the following formula.

EBQ = √ (2 x Demand x Unit setup cost / Unit holding cost)

The derivation of this formula is set out at the bottom of this post.

## Economic Batch Quantity Example

A business uses batch production for one of its products. The annual demand for the product is 4,000 units. The business estimates that the cost of holding one unit of inventory for a year is 3.25 and that the cost of setting up each batch production run is 65.00.

Using the economic batch size formula the EBQ is calculated as follows.

```EBQ = √(2 x Demand x Setup cost/Holding cost)
EBQ = √(2 x D x s/h)
EBQ = √(2 x 4,000 x 65.00/3.25) = 400 units
```

The graph above shows that at the economic batch quantity the total setup cost and the total holding cost are equal. This can be shown by calculation as follows.

```Total setup cost = Setup cost per batch x Demand / Batch size
Total setup cost = s x D / q
Total setup cost = 65.00 x 4,000 / 400 = 650
```

and

```Total holding cost = Holding cost per unit x Batch quantity / 2
Total holding cost = h x q / 2
Total holding cost = 3.25 x 400 / 2 = 650
```

At the economic batch quantity of 400 units the total setup cost and the total holding cost are both 650.

### Derivation of Economic Batch Quantity Formula

The formula can also be derived using differential calculus as follows.

Demand quantity = D
Unit product cost = p
Unit setup cost = s
Unit holding cost = h
Batch size = q

```Total cost = Product cost + Setup cost + Holding Cost
T = p x D + s x D/q + h x q/2

Using differential calculus the total cost is at a minimum when dT/dq = 0
dT/dq = s x D/q2 + h/2 = 0
q = EBQ = √(2 x D x s/h)
```
Last modified July 16th, 2019

## About the Author

Chartered accountant Michael Brown is the founder and CEO of Double Entry Bookkeeping. He has worked as an accountant and consultant for more than 25 years and has built financial models for all types of industries. He has been the CFO or controller of both small and medium sized companies and has run small businesses of his own. He has been a manager and an auditor with Deloitte, a big 4 accountancy firm, and holds a degree from Loughborough University.