Tuesday, June 3, 2008

Economic order quantity

Economic order quantity is that level of inventory that minimizes the total of inventory holding cost and ordering cost.

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 + CxD/2 + HxQ/2

In order to determine the minimum point of the total cost curve, set its derivative equal to zero:
dTC(Q)/dQ = (d/dQ)(PD + CxD/Q + HxQ/2) = 0

= 0 + (-C*D/Q*Q) + H/2) = 0

The result is
Q = root of ((2*C*D)/H)

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