# Optimal order quantity

The optimum order size referred to in the supply logistics and materials management that order size , wherein the sum of the ordering and storage costs for a given level of service in the period at a minimum. A known annual requirement can be covered by many orders of small quantities; however, the high order costs lead to a low average stock level and thus low storage costs and an acceptable storage risk . If you order a few large quantities, the reverse is true.

The order quantity is only optimal if the stated criteria are fully met. In reality, these are only rarely met.

## Order costs

The order costs include the entire processing costs of an order, from order preparation to order completion and order processing. Often the order costs per unit depend on the order quantity, for example due to discounts when purchasing larger quantities.

${\ displaystyle {\ text {Order costs per order}} = {\ frac {\ text {Sum of order costs per period}} {\ text {Number of orders per period}}}}$

## Storage costs

The storage costs include the costs for the staff, the storage rooms and the tied-up capital, including costs for depreciation through shrinkage, obsolescence, etc., as well as the insurance of stocks and rooms.

${\ displaystyle {\ text {Storage cost rate}} = {\ text {Interest rate of the tied capital}} + {\ text {Storage cost rate}}}$

## Static order quantity calculation (classic order quantity formula)

Static order quantities mean that an "optimal" order quantity is determined, which is then always ordered in this volume. It is the earliest thought to solve the order quantity size problem.

## Graph cost history

optimal order quantity = minimum of the total cost curve.
There is a trade-off between order costs (blue) and storage costs (red)

further formula from financial mathematics:

The Andlersche formula

${\ displaystyle Q = {\ sqrt {{200 \ times F \ times S} \ over {C \ times P}}}}$

With:

 ${\ displaystyle Q:}$ optimal order quantity in pieces. for the respective order ${\ displaystyle F:}$ Annual requirement in unit of measure (e.g. 5000 pieces) ${\ displaystyle S:}$ Order costs * per order (e.g. 50 €) ${\ displaystyle C:}$ Storage cost rate ** (e.g. 20 ) ${\ displaystyle P:}$ Purchase price per unit of measure (e.g. € 20 / piece)

Legend:

• * Procurement costs are the sum of fixed procurement costs (order costs) and variable procurement costs (purchase costs).
• ** The storage cost rate is always a percentage (e.g. 20%). In the context of Andler's calculation formula, however, only whole numbers are used (e.g. 20).

### example

A company has an annual requirement of 5,000 items for a warehouse item. The following additional data are known:

• Order costs per order = 120.00 EUR
• Storage rate = 20%
• Purchase price per piece = 100.00 EUR

The optimal order quantity, i.e. the quantity at which the total costs are lowest, is calculated as 244.95 pieces or 245 pieces (pieces are only available as a whole).

## Dynamic lot sizes

Methods of dynamic lot size determination are used when future demand is not constant but is subject to fluctuations. With these methods, the optimum is not calculated analytically, but rather through a step-by-step approximation, which is usually sufficient in practice.

## problem

Conflicting objectives between procurement costs and storage costs :

• If small quantities are purchased at short intervals, this results in increasing procurement costs (fixed order costs, no discounts)
• If large quantities are purchased at long intervals, this results in increasing storage costs (large inventory, high capital commitment )

## Defects in the model

The model of the optimal order quantity is a theoretical approach and does not represent reality. This model is therefore kept too concise for the needs in practice. In order for the model to work in theory, the following premises apply:

• the demand per period is known and remains the same over time
• the warehouse leaving rate is constant
• the filling time for the warehouse is = 0, so the filling speed is infinitely high
• there are no shortfalls
• only a warehouse with unlimited capacity is considered
• the amount of the storage cost is known and remains constant
• Ordering processes only cause costs that are fixed to the order
• the cost price of the goods is constant
• Changes in quality are excluded

Furthermore, in reality, only whole-number solutions are possible within a planning period. Among other things, possible loss and deterioration of goods are neglected. Volume discounts and partial deliveries as well as the formation of safety stocks are also not taken into account in the model.

## Excluded goods

Another problem is that some goods cannot be procured using the optimal order quantity methods, including: a. Z-goods , the consumption of which cannot be predicted, such as B. road salt , or goods with a use-by date (dairy products, etc.).