Economic order quantity

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Economic order quantity is the level of inventory that minimizes the total inventory holding costs and ordering costs. It is one of the oldest classical production scheduling models. The framework used to determine this order quantity is also known as Wilson EOQ Model or Wilson Formula. The model was developed by F. W. Harris in 1913. But still R. H. Wilson, a consultant who applied it extensively, is given credit for his early in-depth analysis of the model.[1]

Contents

[edit] Overview

Assume that the demand for a product is constant over the year and that each new order is delivered in full when the inventory reaches zero. There is a fixed cost charged for each order placed, regardless of the number of units ordered. There is also a holding or storage cost for each unit held in storage (sometimes expressed as a percentage of the purchase cost of the item).

We want to determine the optimal number of units of the product to order so that we minimize the total cost associated with the purchase, delivery and storage of the product

The required parameters to the solution are the total demand for the year, the purchase cost for each item, the fixed cost to place the order and the storage cost for each item per year. Note that the number of times an order is placed will also affect the total cost, however, this number can be determined from the other parameters

=

  1. The ordering cost is constant.
  2. The rate of demand is constant
  3. The lead time is fixed
  4. The purchase price of the item is constant i.e no discount is available
  5. The replenishment is made instantaneously, the whole batch is delivered at once.

EOQ is the quantity to order, so that ordering cost + carrying cost finds its minimum. (A common misunderstanding is that formula tries to find when these are equal.)

[edit] 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 or storage cost) (warehouse space, refrigeration, insurance, etc. usually not related to the unit cost)

At EOQ Ordering Cost And Carrying Cost Are Same.....

[edit] 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 + {\frac{CD}{Q}} + {\frac{HQ}{2}}.

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

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

The result of this derivation is:

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

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

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

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

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

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

[edit] 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.

[edit] See also

[edit] References

  1. ^ Hax, AC and Candea, D. (1984), Production and Operations Management, Prentice-Hall, Englewood Cliffs, NJ, pp. 135, http://catalogue.nla.gov.au/Record/772207 

See also:

  • 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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[edit] External links

Retrieved from "http://en.wikipedia.org/wiki/Economic_order_quantity"