Economic Order Quantity Inventory Models
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 (Wikipedia 2011).
The EOQ model is a continuous replenishment system, which means that inventory is checked upon every withdrawal to see if that withdrawal will cause the inventory level to fall below the restocking level. If so, an order is immediately place for the EOQ number of units. This is also known as a fixed order quantity model because the quantity ordered (EOQ) does not change unless the parameters of the model change.
Basic EOQ Model
Total Inventory Cost Formula
Total annual cost = annual purchase cost + annual ordering cost + annual holding cost
TC = DC + (D/Q)*S + (Q/2)*H, where:
 TC = Total annual cost
 D = Demand (annual)
 C = Cost per unit
 Q = Quantity to be ordered
 S = Setup cost or cost of placing an order
 R = Reorder point
 L = Lead time (in days)
 H = Annual holding and storage cost per unit of average inventory (often holding cost is taken as a percentage of the cost of the item, such as H=iC, where I is the percent carrying cost).
Basic EOQ and Reorder Point Formula
Quantity to Order = EOQ = Square Root of ((2*D*S)/H)
Reorder Point = R = d * L
where,
 D = Demand (annual)
 S = Setup cost or cost of placing an order
 H = Annual holding and storage cost per unit of average inventory
 d = is the average daily demand (constant)
 L = Lead time (in days)
EOQ Model with Demand Uncertainty
When demand varies and is not constant, which is the usual case in business, you must protect against stockouts by holding additional inventory. This additional inventory held to protect against variation is called safety stock. To protect against variation given demand uncertainty, the Reorder Point is increased for safety stock with the following formula:
Reorder Point = R = (d * L) + (z * σL), where
 z = the zscore of the normal distribution given a desired service level
 σL = the standard deviation of demand during the lead time
 σd = the standard deviation of daily demand (demand variation)
 σd = STDEV(daily demand times) when using Excel
 p = the probability of not stocking out (desired service level)
Using Microsoft Excel, the zscore of a given desired service level, p and the standard deviation of demand during the lead time (σL) is:
 z = NORMSINV(p)
 σL = SqRoot(L * σd^2)
Reorder Point = R = (d * L) + (NORMSINV(p) * [SqRoot(L * σd^2)])
Quantity to Order = EOQ = Square Root of ((2*D*S)/H)
EOQ Model with Demand & Delivery Uncertainty
If you have both demand and delivery (lead time) uncertainty, you must use a convolution formula (Bowersox 2010) to calculate the safety stock level.
Standard Deviation of Combined Probabilities
σc = Square Root of [(L * σd^2) + (d^2 * σl^2)], where
 L = Lead time (in days)
 d = the average daily demand
 σd = the standard deviation of daily demand (demand variation)
 σd = STDEV(daily demand times) when using Excel
 σl = Standard Deviation of lead time = STDEV(lead times)
Reorder Point = R = (d * L) + (NORMSINV(p) * σc)
Reorder Point = R = (d * L) + (NORMSINV(p) * Square Root of [(L * σd^2) + (d^2 * σl^2)])
Total Inventory Cost with Safety Stock
When you include safety stock in a model to accomodate variation, the amount of the average inventory on hand (Q/2) is incresaed by the amount of the safety stock. This new total average inventory is multiplied by the annual holding cost to obtain the total cost.
TC = DC + (D/Q)*S + [(Q/2)+SS]*H, where:
 SS = Amount of Safety Stock
Echelon Inventory EOQ Models
If a supply chain is either owned by one firm or the supplier partnerships in the supply chain are such that information is immediately shared and all members of the chain operate as one, you can use an Echelon inventory model.
The echelon inventory at any stage of the system is equal to the inventory on hand at that echelon, plus all downstream inventory (SimechiLevy 2008).
In the case of an echelon inventory system, the Reorder Point (R) uses the Echelon Lead Time (Le) instead of the standard lead time in its calculation. The Echelon Lead Time (Le) is the lead time between the entity doing its inventory calculation PLUS the lead times between all downstream supply chain partners.
For example, for a distributor to a retailer, the echelon lead time (Le) is the lead time between the retailer and distributor PLUS the lead time between the distributor and their supplier.
The average demand and standard deviations of demand are those of all the retail customers in this model, even when calculating reorder point for the distributor.
For systems with stable, constant lead times, use the following formula:
Reorder Point = R = (d * Le) + (NORMSINV(p) * [SqRoot(Le * σd^2)])
References
Bowersox, Closs & Cooper, 2010. Supply Chain Logistics Management, 3rd Edition. McGrawHill, New York
SimchiLevi, Kaminsky, SimchiLevi, 2008. Designing and Managing the Supply Chain: Concepts, Strategies and Case Studies, 3rd Edition. McGrawHill, New York
