Economic order quantity

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Economic Order Quantity (also known as the Wilson EOQ Model or simply the EOQ Model) is a model that defines the optimal quantity to order that minimizes total variable costs required to order and hold inventory.

The model was originally developed by F. W. Harris in 1915, though R. H. Wilson is credited for his early in-depth analysis of the model.

Contents 1 Underlying assumptions 2 Variables 3 Formula 4 Extensions 5 References

Underlying assumptions

1. the demand for the item is known
2. the lead time is known and fixed
3. the receipt of the order occurs in a single instant
4. quantity discounts are not calculated as part of the model
5. stockouts or shortage do not occur

Variables

• <math>Q^*</math> = optimal order quantity
• <math>C</math> = cost per order event (not per unit)
• <math>R</math> = monthly demand of the product
• <math>P</math> = purchase cost per unit
• <math>F</math> = holding cost factor; the factor of the purchase cost that is used as the holding cost (this is usually set at 10-15%, though circumstances can require any setting from 0 to 1)
• <math>H</math> = holding cost per unit per month (<math>H=PF</math>)

Formula

Image:EconomicOrderingQuantity.png The single item EOQ formula can be seen as the minimum point of the following cost function:

Total cost = purchase cost + order cost + holding cost, which corresponds to:

<math>TC(Q) = PR + {\frac{CR}{Q}} + {\frac{PFQ}{2}}</math>.

Taking the derivative of both sides of the equation and setting equal to zero, one obtains

<math>{\frac{dTC(Q)}{dQ}} = {\frac{d}{dQ}}\left(PR + {\frac{CR}{Q}} + {\frac{PFQ}{2}}\right)=0</math>.

The result of this differentiation is:

<math>{\frac{PF}{2}}-{\frac{CR}{Q^2}}=0</math>.

Solving for Q:

<math>{\frac{PF}{2}}={\frac{CR}{Q^2}}</math>

<math>Q^2={\frac{2CR}{PF}}</math>

<math>Q^* = \sqrt{\frac{2CR}{PF}} = \sqrt{\frac{2CR}{H}}</math>.

The superscript asterisk (*) indicates the optimal order quantity.

Extensions

Several extensions can be made to the EOQ model, including backordering costs and multiple items. Additionally, the economic order interval can be determined from the EOQ and the economic production quantity model (which determines the optimal production quantity) can be determined in a similar fashion.

References

• Harris, F. W. Operations Cost (Factory Management Series), Chicago: Shaw (1915).
• Wilson, R. H. "A Scientific Routine for Stock Control" Harvard Business Review, 13, 116-128 (1934).de:Bestellmenge

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