National Center for Freight and Infrastructure Research and Education (
C
FIRE)
University of Wisconsin – Madison.
Instructor: Bruce X. Wang and Ernie Wittwer.
- 3 -
Economic Order Quantity (EOQ) model
The EOQ model tries to make a balance between ordering cost and inventory holding
cost. If less is ordered each time and orders are placed more frequently, the average
inventory will be lower. One might think this would decrease the average inventory
carrying cost and would be most preferable. However, there is a trade-off here with the
administrative cost, called ordering cost. The ordering cost is a fixed cost associated with
placing an order. It includes direct labor time/cost for paperwork, equipment leasing (e.g.
for shipping), documentation of ordered items, and unloading.
The EOQ model is the most basic model for deciding the optimal order quantity that
balances these costs.
Where D = Demand; K = fixed ordering cost; h = inventory carrying cost per unit; Q =
optimal order quantity.
Example
A distribution center (DC) manages distribution of a product. The unit value of
this product (purchase cost) is $50.00. The annual demand for this product that goes
through the distribution center is 4000 units. A cost of placing the order each time is
$400. If the inventory carrying cost is 20% of the tied inventory value, how many units shall be ordered each time?
4000
*
400
*
2
≈
566 (units).
To conclude, the optimal ordering size is 566 units each time. The order frequency is
determined by the total annual demand and order size.
At first glance, the EOQ model does not appear to capture the effect of transportation. In
fact, transportation does not affect the quantity ordered each time. The EOQ model
introduced might leave readers with an impression that logistics activities are
independent of transportation. Since transportation cost is not considered in the EOQ
model, one might suspect that the shipping behavior has been driven by other factors than
transportation. This is true in that transportation is only one component of a very large
supply chain system. Shipping decisions are often not affected by transportation cost.
However, if the shipping volume dramatically changes the freight rate so that the
h
KD
Q
2
*
=
h
KD
Q
2
*
=

shipping cost could differ significantly, say from 5% to 15% of the total product value,
then EOQ model might not apply and a different model might need to be developed. If it
is only a difference of between 5% to 7% of the total cost, application of the EOQ model
should be fairly accurate.
Transportation cost might not affect the ordering policy in the application of the EOQ
model. However, it affects the constantly carried inventory, the safety stock.
This is
obvious as a longer transit time would increase the risk of running out of stock (stockout)
and necessitate the need for a larger safety stock. Therefore, transportation affects the
total logistical costs in a very fundamental way.
We have an example to demonstrate this point based on the optimal inventory policies.
Example
Suppose that an inventory policy is needed for a consumer product (e.g. TV
set).
Assume that whenever an order is placed for replenishment, the ordering cost is
$4,500, which is independent of the order size. Each unit of product has a cost of $250,
and the annual inventory cost is 18% of the product cost. Lead time (from order placing
to order arrival) is about two weeks. The support data and optimal decision are given
below.
Before we present the optimal policy for the example above, we should introduce more
about the inventory policy with uncertain lead time and demand.
(s, S) Policy for Inventory Management
An effective policy for inventory management is referred to as (
s
,
S
) policy. Whenever
the inventory drops to below s, an order is placed to make the inventory position to
S
.
Here
S
=
s
+
Q
, where
s
is called re-order point,
Q
order quantity and
S
order-up-to level.
Note that
s
is composed of two parts: safety stock to prevent stockout and average
demand during lead time. If everything is ‘normal’, the inventory level shall be equal to
the safety stock at the arrival of an order. If the demand is not as ‘regular’ as expected
during the lead time, the safety stock is used to prevent stockout.
Freight impacts inventory costs primarily through affecting the safety stock. Safety stock
is determined by the uncertainty of demand during lead time. Obviously, the uncertainty
of demand during a longer lead time is larger.
As a result, a longer lead time is
associated with a higher safety stock. Of course, the nature of demand itself (variation of
demand within a unit time, or predictability of demand) affects the overall demand during
the lead time.
In addition, a practical factor in determining the safety stock is the ability
of the vendor to accept a stockout. The vendor may refer the customer to another store
close by, or to satisfy the demand with substitutable products. Measuring all the
outcomes, the stockout cost could be quantified to deciding the inventory policy.
The following paragraphs will illustrate how the lead time, demand, and their associated
uncertainty affect inventory cost through affecting the safety stock.