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The formula of the “Economic batch” of Wilson or “formula of Wilson“, created in 1934, allows to calculate the quantity of optimal order and the time between two orders of a product for a given entity (factory, Logistic Center…). Introduced for the first time in 1913 by Harris, it is sometimes called the Harrys-Wilson formula.

## Introduction

Wilson’s formula is based on observing different phenomena related to stock management. Place an order generates a cost that increases depending on the number of orders issued over a period of time. Similarly, owning the stock has a cost (surface, electricity…). And the latter also increases, depending on the quantities stored in the stores.

Wilson’s method is developed through a simple logic:

• By reducing the number of orders placed over a period, the overall cost of the orders is reduced.
• On the other hand, it is possible to limit and maintain the quantities of the managed stock at an acceptable level simply by renewing it more regularly.

We are faced with two contradictory objectives. On one side we want to reduce the number of orders, on the other hand, the maintenance and the regular renewal of the stock cause an increase in the number of orders.

Wilson offers a mathematical and graphical formula that allows you to obtain the lowest overall cost of inventory management for a given number of orders and average stock: the optimal cost.

## The assumptions

• The process is considered to be consistent over time.
• The demand is continuous, known and homogeneous in time.
• The supply time is constant and well-known.
• Stock failures are not accepted.
• The purchase price is constant.
• The batch will always have the same size, to keep the parameters of the model constant.

The most economical, under these conditions, is that a batch goes into the system at the time the inventory level is zero. This assumes that the order must be made at a time when the stock level is sufficient to provide the demand during the supply period.

## Economic batch size

The economic batch of each order will be:

Q = √ (2 * CL * X/CP)

• Q: quantity per batch
• CL: Cost of launching an order in \$/Commande
• X: Consumption per unit of time
• CP: Cost of possession of the stock in €/unit

### The cost of launching

Each order generates costs for the company. They are determined according to:

• Cost of supply: It represents the costs associated with the supply of parts. You can determine them by calculating the transport costs, the costs of the employees in charge of the orders, the costs of electricity… In general, an estimate is done by dividing the costs of the overall purchase service by the number of orders placed.
• launch Cost: It represents the costs associated with the launch of the production. These are the costs of machine adjustments, testing, preseries… Generally, the cost of scheduling service is divided by the number of launches.

### The cost of Ownership

Owning a stock has a cost. It consists of loads related to storage (surface, Handling…) but also by the costs of fixed assets (non-remuneration, loans…).

The ownership rate is calculated by dividing the total cost of the possession costs by the average stock. A percentage is generally in the range of 15 to 35%.

## The Wilson graph

To visualize the economic quantity, Wilson offers a chart that easily identifies the batchsize. We find 3 curves:

• The cost of launching: declining according to the quantities
• The cost of ownership: theoretically proportional to the quantities
• The cumulative cost curve

## Example

The accounting department of the company provided the following information for article A:

• Annual consumption: 500 pieces
• Ordering Unit Cost: €200
• Purchase price of item A: \$250/unit
• Cost of ownership of stock: 10% of average stock
• At the beginning of the period, we have a zero initial stock. (SI = 0);
• At the end of the period, as the consumptions are regular, the present stock is the equivalent of the last order entered. (SF = Order qty);
• The average stock is the mean of the initial stock and the final stock. (SM = (SI + SF)/2)
• Average stock value = Unit Cost of item X average stock
• Cost of orders = number of orders in period x cost of ordering;
• Economic batch per order = Total consumption of the period divided by the number of orders;
• Total Cost = Order Cost + inventory ownership cost
 Number of orders Initial Stock Final Stock Average Stock Cost of ownership Cost of Orders Total Cost Economic batch 1 0 500 250 6250 200 6450 500 2 0 250 125 3125 400 3525 250 3 0 166,7 83,3 2083,3 600 2683,3 167 4 0 125 62,5 1562,5 800 2362,5 125 5 0 100 50 1250 1000 2250 100 6 0 83,3 41,7 1041,7 1200 2241,7 83 7 0 71,4 35,7 892,9 1400 2292,9 71 8 0 62,5 31,3 781,3 1600 2381,3 63 9 0 55,6 27,8 694,4 1800 2494,4 56 10 0 50 25 625 2000 2625 50 11 0 45,5 22,7 568,2 2200 2768,2 45 12 0 41,7 20,8 520,8 2400 2920,8 42

According to Wilson’s method, the most economical solution would be to pass for this item, 6 Annual orders of 83 units each, for an optimized total cost of €2241.7. It is noticed that for a different Q quantity, the total cost varies on the rise.

## The Limits Wilson’s method

The method of Wilson is in reality difficult to apply with such accuracy, because it only takes into account the hypothesis defining a certain future.

It is a formula that resides only on two parameters: the cost of possession of the stocks and the cost of ordering. But at this level too, the reality is quite different. Apart from the cost of transport which actually varies according to the number of orders, the other cost elements (rents, wages, electricity…) taken into account in the assessment of the costs of possession and of the transfer are not necessarily variable in Depending on the quantity or number of orders.

In addition, the calculation of the batch size is independent of the actual demand. Which is the exact inverse of the Takt Time.

## Source

Y. Pimor, M. Fender (2008)-Logistics

S. Larryi, A. Thomas (2000)-Inventory management in a context of independent applications

B. Doriath, M. Lozato, p. Mendes-Miniatura, p. Nicolle (2012)-Accounting and management of organizations