# Economic Order Quantity (EOQ)

## Definitions

**Economic Order Quantity (EOQ)** is the order quantity that minimizes total inventory costs.

**Order Quantity** is the number of units added to inventory each time an order is placed.

**Total Inventory Costs** is the sum of inventory acquisition cost, ordering cost, and holding cost.

**Ordering Cost** is the cost incurred in ordering inventory from suppliers excluding the cost of purchase such as delivery costs and order processing costs.

**Holding Cost**, also known as carrying cost, is the total cost of holding inventory such as warehousing cost and obsolescence cost.

## Explanation

Total inventory cost is comprised of the following main costs:

- Cost of purchase
- Order Costs
- Holding Costs

If we change the order quantity, it can affect the different types of inventory costs in different ways.

Larger order size results in lower order costs because fewer orders need to be placed to cover the annual demand. This however results in higher holding costs because of the increase in inventory levels.

Conversely, smaller order size results in lower holding costs because of the decline in average inventory level. However, as lower quantity of inventory is ordered each time, the number of orders needed to increase in order to fulfill the annual demand which leads to higher ordering costs. Reducing the order size may also affect the cost of purchase due to the loss of trade discounts that are based on the order quantity.

So the question arises how we can find the optimal order quantity that minimizes the total inventory costs.

EOQ model offers a method of finding the optimal order quantity that minimizes inventory costs by finding a balance between the opposing inventory costs.

## Formula

Economic Order Quantity | = | √ | 2 x C_{o} x D | |

C_{h} |

Where:

*C*is the cost of placing one order_{o}*D*is the annual demand*C*is the annual cost of holding one unit of inventory_{h}

How is the EOQ formula derived? [ - ]

## Relevant Costs

When calculating EOQ, it is important to include only those ordering and holding costs that are relevant. Any costs that are not incremental should be ignored while calculating EOQ. Following examples illustrate the application of relevant costing in the calculation of EOQ.

Order Costs | Relevance to EOQ calculation |

Salary paid to clerk who processes orders. | If increase in number of orders would not result in overtime or hiring an additional clerk, the cost will not be relevant to EOQ. |

Supplier charges $5 delivery cost for each unit of inventory supplied. | The total delivery expense will be the same irrespective of the number of order deliveries so it should be ignored in EOQ calculation. |

Supplier charges $500 fixed delivery charge for each delivery. | Delivery expense will increase with an increase in number of orders so it should be included in EOQ calculation. |

Auto dealer transports cars from the car factory to its showroom using its own trucks. Insurance premium is paid to cover for any accidents during the transportation. $100 premium is paid for each vehicle that is transported. | The annual insurance cost is fixed irrespective of how many cars are transported in one go and should therefore be ignored. |

Holding Costs | Relevance to EOQ calculation |

Company earns a return of 15% on its projects. One unit of inventory costs $100. | The opportunity cost of holding one more unit of inventory for one year is $15(15%x$100) which should be included in EOQ calculation because more the number of units of inventory that are held, higher the opportunity cost of capital tied in inventory purchase. |

Company pays lease rentals of $20,000 for its warehouse. | Warehouse rent is fixed and hence irrelevant to EOQ calculation as the cost does not vary to changes in the number of units of inventory held. |

Insurance premium of $10 per day is paid for each unit of inventory stored in the warehouse to cover the risk of fire and theft. | The insurance cost rises with an increase in number of units held and is therefore relevant to EOQ. |

0.2% of the average number of inventory units stored in the warehouse get damaged or stolen. | The cost of damage and shrinkage (theft) of inventory increases as more units of inventory are held at the warehouse. The cost should therefore be factored into EOQ calculation. |

### Example

**Jason owns a fish shop where he sells an exotic variety of tuna fish which he imports from Japan.**

**Jason refrigerates the fish in a cold storage facility near his shop that charges him a fixed annual fee of $1000 and variable charge of $5 per day for each fish container that is stored.**

**Every morning, Jason brings fish from the cold storage to his shop for sale. Jason estimates that he incurs $10,000 electricity cost each year on refrigerating the fish inside his own shop.**

**Jason incurs the following ordering costs:**

**Delivery charges of $10,000 per delivery****Import duties of $300 per carton****Custom fees of $200 per order****Import license fee of $150 per annum**

**Jason currently imports fish by placing one order of 20 cartons every month. Each carton costs $2,000. **

**Jason is wondering if he can save inventory costs by adopting EOQ model.**

**a) Calculate the current annual total inventory costs****b) Calculate the economic order quantity****c) Calculate the annual total inventory costs if EOQ is used**

### Solution

**a) Current Inventory Cost**

Costs | Working | $ |

Purchase Cost | Annual demand = 20 x 12 = 240 cartons Purchase cost = 240 x $2000 = $480,000 | 480,000 |

Order Cost | ||

Delivery Cost | Number of deliveries = 12 Delivery Cost = 12 x $10,000 = 120,000 | 120,000 |

Import Cost | Import fee = $300 x 240 cartons = $72,000 | 72,000 |

Custom Cost | Custom fee = $200 x 12 orders = $2400 | 2,400 |

Holding Cost | ||

Cold storage | Maximum number of cartons stored = 20 Average number of cartons = 20 ÷ 2 = 10 Variable charge = 10 x $5 x 365 = $18,250 Fixed charge = $1,000 Total = $19,250 | 19,250 |

Electricity | 10,000 | |

Total Inventory Cost (Current) | 703,650 |

**b) Economic Order Quantity**

EOQ | = | √ | 2 x 10,200 (W1) x 240 (W2) |

1,825 (W3) | |||

≈ | 52 cartons |

__Order Cost__ (W1)

Delivery Cost | $10,000 |

Import fees | - |

Custom fees | $200 |

Cost of 1 order | $10,200 |

Note:

Import fees can be ignored in EOQ calculation as they remain the same irrespective of the number of orders.

__Annual Demand__ (W2) = 240 cartons

__Holding Cost__ (W3)

Cold Storage - Variable (365 x $5) | $1,825 |

Cold Storage - Fixed | - |

Electricity | - |

Cost of holding 1 carton for 1 year | $1,825 |

**c) Inventory Cost using EOQ**

Costs | Working | $ |

Purchase Cost | As before | 480,000 |

Order Cost | ||

Delivery Cost | Annual Demand = 240 cartons Number of deliveries = 240/52 ≈ 5 Delivery Cost = 5 x $10,000 = $50,000 | 50,000 |

Import Cost | As before | 72,000 |

Custom Cost | Custom fee = $200 x 5 orders = $1000 | 1,000 |

Holding Cost | ||

Cold storage | Maximum number of cartons stored = 52 Average number of cartons = 52 ÷ 2 = 26 Variable charge = 26 x $5 x 365 = $47,450 Fixed charge (as before) = $1,000 Total = $48,450 | 19,250 |

Electricity (as before) | 10,000 | |

Total Inventory Cost (using EOQ) | 661,450 |

Using EOQ Model will save Jason $42,200 (703,650 - 661,450) annually.

#### Tip

Annual holding cost per unit is sometimes expressed as a percentage of the inventory purchase price.

## Assumptions

- EOQ model assumes a constant demand.
- EOQ calculation assumes that ordering costs and holding costs will remain constant.

## Limitations

- Since no fluctuation in demand is considered in the EOQ calculation, business losses due to potential shortage of inventory are ignored.
- EOQ model does not take into account the seasonal fluctuations in the cost of inventory. In seasonal industries, it would make sense to buy inventory in bulk when it is readily available at a lower price. Inventory may be harder to procure in off season and would usually cost more as well.
- EOQ model does not take into account purchase discounts that could be obtained by buying inventory in bulk. We can however work around this problem as is illustrated in this next lesson.