Economic Order Quantity (EOQ), also known as Economic Buying Quantity (EPQ), is the order quantity that minimizes the total
holding cost
In marketing, carrying cost, carrying cost of inventory or holding cost refers to the total cost of holding inventory. This includes warehousing costs such as rent, utilities and salaries, financial costs such as opportunity cost, and inventory co ...
s and
ordering costs in
inventory management. It is one of the oldest classical
production scheduling
Scheduling is the process of arranging, controlling and optimizing work and workloads in a Production (economics), production process or manufacturing process. Scheduling is used to allocate plant and machinery resources, plan human resources, plan ...
models. The model was developed by
Ford W. Harris in 1913, but R. H. Wilson, a consultant who applied it extensively, and K. Andler are given credit for their in-depth analysis.
Overview
EOQ applies only when
demand
In economics, demand is the quantity of a good that consumers are willing and able to purchase at various prices during a given time. The relationship between price and quantity demand is also called the demand curve. Demand for a specific item ...
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; an order is assumed to contain only 1 unit. There is also a cost for each unit held in storage, commonly known as
holding cost
In marketing, carrying cost, carrying cost of inventory or holding cost refers to the total cost of holding inventory. This includes warehousing costs such as rent, utilities and salaries, financial costs such as opportunity cost, and inventory co ...
, sometimes expressed as a percentage of the purchase cost of the item. While the EOQ formulation is straightforward there are factors such as transportation rates and quantity discounts to consider in actual application.
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 for a single item 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.
Variables
*
= total annual inventory cost
*
= purchase unit price, unit production cost
*
= order quantity
*
= optimal order quantity
*
= annual demand quantity
*
= fixed cost per order, setup cost (''not'' per unit, typically cost of ordering and shipping and handling. This is not the cost of goods)
*
= annual holding cost per unit, also known as carrying cost or storage cost (capital cost, warehouse space, refrigeration, insurance, opportunity cost (price x interest), etc. usually not related to the unit production cost)
The total cost function and derivation of EOQ formula
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
Where:
* 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 K, and we need to order D/Q times per year. This is K × D/Q
* Holding cost: the average quantity in stock (between fully replenished and empty) is Q/2, so this cost is h × Q/2
:
.
To determine the minimum point of the total cost curve, calculate the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the total cost with respect to Q (assume all other variables are constant) and set it equal to 0:
:
Solving for Q gives Q* (the optimal order quantity):
:
Therefore:
Q* is independent of P; it is a function of only K, D, h.
The optimal value Q* may also be found by recognizing that
:
where the non-negative quadratic term disappears for
which provides the cost minimum
Example
*annual requirement quantity (D) = 10000 units
*Cost per order (K) = 40
*Cost per unit (P)= 50
*Yearly carrying cost per unit = 4
*Market interest = 2%
Economic order quantity =
= 400 units
Number of orders per year (based on EOQ)
Total cost
Total cost
If we check the total cost for any order quantity other than 400(=EOQ), we will see that the cost is higher. For instance, supposing 500 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 firm.
Extensions of the EOQ model
Quantity discounts
An important extension to the EOQ model is to accommodate quantity discounts. There are two main types of quantity discounts: (1) all-units and (2) 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 when 150 units are ordered, the total cost is $30*100 + $28*50.
* All units discount: an order of 1–1000 units costs $50 each; an order of 1001–5000 units costs $45 each; an order of more than 5000 units costs $40 each. So when 1500 units are ordered, the total cost is $45*1500.
In order to find the optimal order quantity under different quantity discount schemes, one should use algorithms; these algorithms are developed under the assumption that the EOQ policy is still optimal with quantity discounts. Perera et al. (2017) establish this optimality and fully characterize the (s,S) optimality within the EOQ setting under general cost structures.
Design of optimal quantity discount schedules
In presence of a strategic customer, who responds optimally to discount schedules, the design of an 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.
Backordering costs and multiple items
Several extensions can be made to the EOQ model, including backordering costs and multiple items. In the case backorders are permitted, the inventory carrying costs per cycle are:
:
where s is the number of backorders when order quantity Q is delivered and
is the rate of demand. The backorder cost per cycle is:
:
where
and
are backorder costs,
, T being the cycle length and
. The average annual variable cost is the sum of order costs, holding inventory costs and backorder costs:
: