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 carrying 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.

Underlying assumptions

1. the annual (or monthly or whatever periodicity you desire, here we will use annual) demand for the item is known and constant
2. the lead time is not taken into account
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 ordering cost is constant

Variables

• $Q$ = order quantity
• $Q^*$ = optimal order quantity
• $D$ = annual demand quantity of the product
• $P$ = purchase cost per unit
• $C$ = fixed cost per order (not per unit, in addition to unit cost)
• $H$ = annual holding cost per unit (also known as carrying cost) (warehouse space, refrigeration, insurance, etc. usually not related to the unit cost)

The Total Cost function

The single-item EOQ formula finds the minimum point of the following cost function:

Total Cost = purchase cost + ordering cost + holding cost

- Purchase cost: This is the variable cost of goods: purchase unit price × annual demand quantity. This is P×D

- Ordering cost: This is the cost of placing orders: each order has a fixed cost C, and we need to order D/Q times per year. This is C × D/Q

- Holding cost: the average quantity in stock (between fully replenished and empty) is Q/2, so this cost is H × Q/2

$TC\; =\; PD\; +\; \{\backslash frac\{CD\}\{Q\}\}\; +\; \{\backslash frac\{HQ\}\{2\}\}$.

In order to determine the minimum point of the total cost curve, set its derivative equal to zero:

$\{\backslash frac\{dTC(Q)\}\{dQ\}\}\; =\; \{\backslash frac\{d\}\{dQ\}\}\backslash left(PD\; +\; \{\backslash frac\{CD\}\{Q\}\}\; +\; \{\backslash frac\{HQ\}\{2\}\}\backslash right)=0$.

The result of this derivation is:

$-\{\backslash frac\{CD\}\{Q^2\}\}\; +\; \{\backslash frac\{H\}\{2\}\}=0$.

Solving for Q gives Q* (the optimal order quantity):

$\{\backslash frac\{H\}\{2\}\}=\{\backslash frac\{CD\}\{Q^2\}\}$

$Q^2=\{\backslash frac\{2CD\}\{H\}\}$

Therefore: $Q^*\; =\; \backslash sqrt\{\backslash frac\{2CD\}\{H\}\}$.

Note that interestingly, Q* is independent of P, it is a function of only C, D, H.

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.

See also

• Classical Newsvendor model: Newsvendor

References

• 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).

Links

http://www.inventoryops.com/economic_order_quantity.htm

This article is licensed under the GNU Free Documentation License. It uses material from Wikipedia