# Economic order quantity

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Economic order quantity

Economic order quantity is that level of inventory that minimizes the total of inventory holding cost and ordering cost. The framework used to determine this order quantity is also known as Wilson EOQ Model. The model was developed by F. W. Harris in 1913. But still R. H. Wilson is given credit for his early in-depth analysis of the model.

Underlying assumptions

# The ordering cost is constant.
# The rate of demand 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 &times; annual demand quantity. This is P&times;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 &times; D/Q

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

$TC = PD + \left\{frac\left\{CD\right\}\left\{Q + \left\{frac\left\{HQ\right\}\left\{2$.

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

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

The result of this derivation is:

$-\left\{frac\left\{CD\right\}\left\{Q^2 + \left\{frac\left\{H\right\}\left\{2=0$.

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

$\left\{frac\left\{H\right\}\left\{2=\left\{frac\left\{CD\right\}\left\{Q^2$

$Q^2=\left\{frac\left\{2CD\right\}\left\{H$

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

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.

* 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).

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