Economic order quantity is the order quantity that minimizes 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 Ford W. Harris in 1913, but R. H. Wilson, a consultant
who applied it extensively, is given credit for his in-depth analysis.
Overview EOQ applies only when demand for a product
is constant over the year and each new order is delivered in full when inventory reaches
zero. There is a fixed cost for each order placed, regardless of the number of units
ordered. There is also a cost for each unit held in storage, commonly known as holding
cost, sometimes expressed as a percentage of the purchase cost of the item.
We want to determine the optimal number of units 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, though this number can be determined from the other parameters.
Underlying assumptions The ordering cost is constant.
The rate of demand is known, and spread evenly throughout the year.
The lead time is fixed. The purchase price of the item is constant
i.e. no discount is available The replenishment is made instantaneously,
the whole batch is delivered at once. Only one product is involved.
EOQ is the quantity to order, so that the sum of ordering cost and holding cost is at
its minimum. These costs will be equal to one another at the minimized cost point.
Variables = purchase price, unit production cost
= order quantity = optimal order quantity
= annual demand quantity = fixed cost per order, setup cost
= annual holding cost per unit, also known as carrying cost or storage cost
The Total Cost function
The single-item EOQ formula finds the minimum point of the following cost function:
Total Cost = purchase cost or production cost + ordering cost + holding cost
- Purchase cost: This is the variable cost of goods: purchase unit price × annual demand
quantity. This is c × D - Ordering cost: This is the cost of placing
orders: each order has a fixed cost K, and we need to order D/Q times per year. This
is K × D/Q - Holding cost: the average quantity in stock
is Q/2, so this cost is h × Q/2 .
To determine the minimum point of the total cost curve, partially differentiate the total
cost with respect to Q and set to 0:
Solving for Q gives Q*:
Therefore:
Q* is independent of c; it is a function of only K, D, h.
The optimal value Q* may also be found by recognising that
where the non-negative quadratic term disappears for which provides the cost minimum
Quantity Discounts An important extension to the EOQ model of
Wilson is to accommodate quantity discounts. There are two main types of quantity discounts:
all-units and incremental. Here is a numerical example.
Incremental unit discount: Units 1-100 cost $30 each; Units 101-199 cost $28 each; Units
200 and up cost $26 each. So 150 units are ordered, the total cost is $30*100+$28*50.
All units discount: An order of 1-1000 units costs $30 each; an order of 1001-5000 units
costs $45 each; an order of more than 5000 units costs $40 each. So if 1500 units are
ordered, the total cost is $45*1500. Design of Optimal Quantity Discount Schedules
In presence of a strategic customer, who responds optimally to discount schedule, the design
of optimal quantity discount scheme by the supplier is complex and has to be done carefully.
This is particularly so when the demand at the customer is itself uncertain. An interesting
effect called the “reverse bullwhip” takes place where an increase in consumer demand
uncertainty actually reduces order quantity uncertainty at the supplier.
Other Extensions Several extensions can be made to the EOQ
model developed by Mr. Pankaj Mane, including backordering costs and multiple items. Additionally,
the economic order interval can be determined from the EOQ and the economic production quantity
model can be determined in a similar fashion. A version of the model, the Baumol-Tobin model,
has also been used to determine the money demand function, where a person's holdings
of money balances can be seen in a way parallel to a firm's holdings of inventory.
Example Suppose annual requirement quantity = 10000
units Cost per order = $2
Cost per unit= $8 Carrying cost percentage(percentage of c)
= 0.02 Annual carrying cost per unit = $0.16
Economic order quantity = = 500 units Number of orders per year
Total cost Total cost
If we check the total cost for any order quantity other than 500(=EOQ), we will see that the
cost is higher. For instance, supposing 600 units per order, then
Total cost Similarly, if we choose 300 for the order
quantity then Total cost
This illustrates that the Economic Order Quantity is always in the best interests of the entity.
Multi-Criteria EOQ Malakooti has introduced the multi-criteria
EOQ models where the criteria could be minimizing the total cost, Order quantity, and Shortages.
See also Costant fill rate for the part being produced:
Economic Production Quantity Demand is random: classical Newsvendor model
Demand varies over time: Dynamic lot size model
Several products produced on the same machine: Economic Lot Scheduling Problem
Reorder point References
^ Harris, Ford W. [Reprint from 1913]. "How Many Parts to Make at Once". Operations Research
38: 947–950. JSTOR 170962. Retrieved Nov 21, 2012.
^ Hax, AC and Candea, D., Production and Operations Management, Prentice-Hall, Englewood Cliffs,
NJ, p. 135 ^ Grubbström, Robert W.. "Modelling production
opportunities — an historical overview". International Journal of Production Economics
41: 1–14. doi:10.1016/0925-5273(95)00109-3. ^ Nahmias, Steven. Production and operations
analysis. McGraw Hill Higher Education. ^ Altintas, N.; Erhun, F.; Tayur, S.. "Quantity
Discounts Under Demand Uncertainty". Management Science 54: 777–792. doi:10.1287/mnsc.1070.0829.
^ Andrew Caplin and John Leahy, "Economic Theory and the World of Practice: A Celebration
of the Model", Journal of Economic Perspectives, Winter 2010, V 24, N 1
^ Malakooti, B. Operations and Production Systems with Multiple Objectives. John Wiley
& Sons. ISBN 978-1-118-58537-5. Further reading
Harris, Ford W. Operations Cost, Chicago: Shaw.
Wilson, R. H. "A Scientific Routine for Stock Control" Harvard Business Review, 13, 116-128.
External links The EOQ Model
http:www.inventoryops.com/economic_order_quantity.htm Simple EOQ Calculator for Android devices
http:www.scmfocus.com201410 Plossel, George. Orlicky's Material Requirement's
Planning. Second Edition. McGraw Hill. 1984.