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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 1913, though R. H. Wilson is credited for his early in-depth analysis of the model.

1. the monthly demand for the item is known, deterministic and constant
2. the lead time is known and constant
3. the receipt of the order occurs in a single instant and immediately after ordering it
4. quantity discounts are not calculated as part of the model
5. the setup cost is constant

Note that deterministic does not imply the constancy of the demand. For instance, the sine function is deterministic, but not constant.

• Q ^* = optimal order quantity
• C = cost per order event (not per unit)
• R = monthly demand of the product
• P = purchase cost per unit
• F = 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)
• H = holding cost per unit per month (H = PF)

A graph illustrating the relationship amongst the Ordering Costs curve, the Holding Costs curve, the Total Costs curve and the Economic ordering quantity

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:

.

In order to determine the minimum value of the total cost curve, take the derivative of both sides of the equation and set the result equal to zero to obtain

.

The result of this differentiation is:

.

Solving for QQQQQ:

.

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

EOQ models provide several key insights. The most important one is that the optimal order quantity Q* varies with the square root of annual demand — NOT directly with annual demand.

This provides an important economy of scale; if demand doubles, for example, the optimal inventory does NOT double — it goes up by the square root of 2, or approximately 1.4.

This also means that inventory rules based on time-supply are not optimal. For example: many inventory managers maintain a "month's worth" of inventory. If demand doubles, then a "month's worth" of inventory is twice as large. As noted above, this is more than is optimal; double demand should increase inventory by the square root of 2, or about 1.4.

Remember that this model makes some important assumptions. If these assumptions are not accurate, then results may differ.

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.

• Classical Newsvendor model: Newsvendor

• Harris, F.W. "How Many Parts To Make At Once" Factory, The Magazine of Management, 10(2), 135-136, 152 (1913).
• 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).