# Economic Order Quantity : key formula for optimizing inventory management

2021年12月14日

In 1934, R.H. Wilson analyzed then developed a production scheduling model : the Economic Order Quantity, which determines the quantity and, therefore, the optimal period of replenishment of a site (factory, store, warehouse).

The model exists under several names :

• Economic Order Quantity (EOQ),
• Economic Purchase Quantity (EPQ),

In accordance with these variables :

• Q = The optimal order quantity. This is what we are looking for,
• D = The annual demand quantity. It is generally evaluated on a fixed and annual period (12 months). The question is about defining how many units are to be used for sales or production,
• The fixed cost per order. It must include costs of transport or reception but also costs of management of the goods which are called costs of placing (administrative process, accounting, etc.),
• h = The annual holding cost per unit, also known as carrying cost or storage cost. This includes labor costs and everything related to the building that is storing the goods (electricity, heating, insurance, inventory, etc.). This cost is also calculated over a fixed and precise period.

We wish to find the optimal quantity to order, thus Q. The formula to find Q is then written :

Q = √2D x K / h

It is possible to construct the formula by accepting a state of shortage for the company. This is to get as close as possible to reality and respond to the case of a company which assumes a future stock shortage by having calculated its demand first. The company will not respond to all the demand and we must therefore include in the formula the element = proportion of the cost of shortage on the overall holding cost of the goods (which can be called service rate).

We then obtain this formula :

Q= √ 2D x K / h x 1 / Z

The optimal quantity of material, merchandise or product to order has been defined. To know how many orders to make during the chosen period, here is how to proceed:

• N = The number of orders to be made.

N = D / Q

Finally, to know the frequency of your orders, you just need to know the number of orders and the number of days that contains the period you have chosen.

• F = The frequency of orders

F = number of days / N

The model established by Wilson is a key formula for optimized inventory management. This method minimizes holding costs by avoiding over-stocking. The production unit knows how much to buy and when.

But it remains a formula which, obviously, has certain limits when introduced in reality, in particular because of the economic situation and instability of events.

## The demand

The main problem that can be highlighted from this model is that the formula only considers constant parameters

Even if an evolution of the formula makes it possible to integrate a future period of shortage, the demand in general is not stable and drawn with peaks of demand according to conjunctural elements or simply because of the seasonality, for example.

The blocking element is the frequency since according to the peaks of demand, the need for replenishment will not be the same. As the quantity remains relevant, it is sufficient to set aside the optimal frequency (F) given by the model to order more in periods of high demand and less when demand is low.

## The price

The model does not take into account the sliding scale prices. Indeed, the purchase price is not always constant because prices can vary according to the quantity ordered. The optimal quantity to order that the formula will give according to the demand may not be the optimal quantity according to the purchase prices

It may therefore be interesting to order more during one period and less during another.

## Storage costs

The Economic Order Quantity model also considers all warehouse costs to be fixed. However, there are many variable costs in addition to fixed costs

• Labor,
• Electricity,
• Transportation,
• Etc.

These are costs that, for example, needs to be added the depreciation cost of the machines in the factory which are fixed costs.

## Delivery times

The model considers order and delivery times as stable and constant. However, delivery times can vary due to supplier delays or shortages

Today, global pandemic has forced many Asian factories to close their doors. Many productions have been stopped and therefore lead times have been extended.

Another example: in March 2021, the Ever-Given ship crossing the Suez Canal with nearly 20,000 containers ran aground. Billions of dollars worth of goods were lost and many ships were blocked and therefore unable to deliver on time.

Therefore, the solution is either to build two Wilson formulas according to the two supply delivery times, or to take into account an average delivery time.

## Safety stocks

Finally and still because the parameters of the formula remain constant, safety stock is not considered. However, many companies hold a safety stock in order to deal with unforeseen events: delivery delays, changes in demand, etc. This stock is all the more essential as today's markets are more and more volatile.

It is then possible to combine the Wilson formula and the safety stock to calculate what is called the Order Point P. If we follow the diagram below, we understand that it is necessary to restock when the stock level reaches the safety stock. The order will be placed X days before reaching the safety stock.

It should be noted that even safety stock reassure the production unit, it is also an excess stock that can hide other issues: stock management, anticipation, relations with the supplier, etc. It is necessary to be vigilantwhen integrating the safety stock into the Wilson formula.

Wilson's model helps with stock management by determining essential points in the production process. Since the formula is built from constants, it is necessary to work with a warehouse management software to consider more complex and unstable situations

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