4 Economic Order Quantity: Determining How Much to Order
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200
CHAPTER 6 • INVENTORY CONTROL MODELS
FIGURE 6.2
Inventory Usage
over Time
Inventory
Level
Order Quantity = Q =
Maximum Inventory Level
Minimum
Inventory
0
Time
Inventory Costs in the EOQ Situation
The objective of the simple EOQ
model is to minimize total
inventory cost. The relevant costs
are the ordering and holding
costs.
The average inventory level is
one-half the maximum level.
The objective of most inventory models is to minimize the total costs. With the assumptions just
given, the relevant costs are the ordering cost and the carrying, or holding cost. All other costs,
such as the cost of the inventory itself (the purchase cost), are constant. Thus, if we minimize
the sum of the ordering and carrying costs, we are also minimizing the total costs.
The annual ordering cost is simply the number of orders per year times the cost of placing
each order. Since the inventory level changes daily, it is appropriate to use the average inventory
level to determine annual holding or carrying cost. The annual carrying cost will equal the average inventory times the inventory carrying cost per unit per year. Again looking at Figure 6.2,
we see that the maximum inventory is the order quantity (Q), and the average inventory will be
one-half of that. Table 6.2 provides a numerical example to illustrate this. Notice that for this
situation, if the order quantity is 10, the average inventory will be 5, or one-half of Q. Thus:
Average inventory level =
Q
2
(6-1)
Using the following variables, we can develop mathematical expressions for the annual ordering
and carrying costs:
Q = number of pieces of order
EOQ = Q* = optimal number of pieces to order
D = annual demand in units for the inventory item
Co = ordering cost of each order
Ch = holding or carrying cost per unit per year
IN ACTION
F
Global Fashion Firm Fashions Inventory
Management System
ounded in 1975, the Spanish retailer Zara currently has more
than 1,600 stores worldwide, launches more than 10,000 new
designs each year, and is recognized as one of the world’s principal fashion retailers. Goods were shipped from two central warehouses to each of the stores, based on requests from individual
store managers. These local decisions inevitably led to inefficient
warehouse, shipping, and logistics operations when assessed on
a global scale. Recent production overruns, inefficient supply
chains, and an ever-changing marketplace (to say the least)
caused Zara to tackle this problem.
A variety of operations research models were used in
redesigning and implementing an entirely new inventory
management system. The new centralized decision-making
system replaced all store-level inventory decisions, thus providing results that were more globally optimal. Having the
right products in the right places at the right time for customers has increased sales from 3% to 4% since implementation. This translated into an increase in revenue of over $230
million in 2007 and over $350 million in 2008. Talk about
fashionistas!
Source: Based on F. Caro, J. Gallien, M. Díaz, J. García, J. M. Corredoira,
M. Montes, J.A. Ramos, and J. Correa. “Zara Uses Operations Research
to Reengineer Its Global Distribution Process,” Interfaces 40, 1 (January–
February 2010): 71–84.
6.4
TABLE 6.2
Computing Average
Inventory
ECONOMIC ORDER QUANTITY: DETERMINING HOW MUCH TO ORDER
201
INVENTORY LEVEL
DAY
BEGINNING
ENDING
AVERAGE
April 1 (order received)
10
8
9
April 2
8
6
7
April 3
6
4
5
April 4
4
2
3
April 5
2
0
1
Maximum level April 1 = 10 units
Total of daily averages = 9 + 7 + 5 + 3 + 1 = 25
Number of days = 5
Average inventory level = 25/5 = 5 units
Annual ordering cost = 1Number of orders placed per year2 * 1Ordering cost per order2
Annual demand
=
* 1Ordering cost per order2
Number of units in each order
=
D
C
Q o
Annual holding or carrying cost = 1Average inventory2 * 1Carrying cost per unit per year2
Order quantity
* 1Carrying cost per unit per year2
2
Q
= Ch
2
=
A graph of the holding cost, the ordering cost, and the total of these two is shown in Figure 6.3.
The lowest point on the total cost curves occurs where the ordering cost is equal to the carrying
cost. Thus, to minimize total costs given this situation, the order quantity should occur where
these two costs are equal.
FIGURE 6.3
Total Cost as a Function
of Order Quantity
Cost
Curve for Total Cost
of Carrying
and Ordering
Minimum
Total
Cost
Carrying Cost Curve
Ordering Cost Curve
Optimal
Order
Quantity
Order Quantity
202
CHAPTER 6 • INVENTORY CONTROL MODELS
Finding the EOQ
We derive the EOQ equation by
setting ordering cost equal to
carrying cost.
When the EOQ assumptions are met, total cost is minimized when:
Annual holding cost = Annual ordering cost
Q
D
Ch =
Co
2
Q
Solving this for Q gives the optimal order quantity:
Q 2 Ch = 2DCo
Q2 =
Q =
2DCo
Ch
2DCo
A Ch
This optimal order quantity is often denoted by Q*. Thus, the economic order quantity is given
by the following formula:
2DCo
A Ch
EOQ = Q* =
This EOQ is the basis for many more advanced models, and some of these are discussed later in
this chapter.
Economic Order Quantity (EOQ) Model
Annual ordering cost =
D
C
Q o
(6-2)
Annual holding cost =
Q
C
2 h
(6-3)
2DCo
A Ch
(6-4)
EOQ = Q* =
Sumco Pump Company Example
Sumco, a company that sells pump housings to other manufacturers, would like to reduce its
inventory cost by determining the optimal number of pump housings to obtain per order. The
annual demand is 1,000 units, the ordering cost is $10 per order, and the average carrying cost
per unit per year is $0.50. Using these figures, if the EOQ assumptions are met, we can calculate
the optimal number of units per order:
Q* =
=
2DCo
A Ch
211,00021102
B
0.50
= 140,000
= 200 units
The relevant total annual inventory cost is the sum of the ordering costs and the carrying costs:
Total annual cost = Order cost + Holding cost
The total annual inventory cost
is equal to ordering plus holding
costs for the simple EOQ model.
In terms of the variables in the model, the total cost (TC) can now be expressed as
TC =
Q
D
Co + Ch
Q
2
(6-5)
6.4
ECONOMIC ORDER QUANTITY: DETERMINING HOW MUCH TO ORDER
203
The total annual inventory cost for Sumco is computed as follows:
TC =
=
Q
D
Co + Ch
Q
2
1,000
200
1102 +
10.52
200
2
= $50 + $50 = $100
The number of orders per year 1D>Q2 is 5, and the average inventory 1Q>22 is 100.
As you might expect, the ordering cost is equal to the carrying cost. You may wish to try
different values for Q, such as 100 or 300 pumps. You will find that the minimum total cost
occurs when Q is 200 units. The EOQ, Q*, is 200 pumps.
USING EXCEL QM FOR BASIC EOQ INVENTORY PROBLEMS The Sumco Pump Company exam-
ple, and a variety of other inventory problems we address in this chapter, can be easily solved
using Excel QM. Program 6.1A shows the input data for Sumco and the Excel formulas
needed for the EOQ model. Program 6.1B contains the solution for this example, including
the optimal order quantity, maximum inventory level, average inventory level, and the number
of setups or orders.
Purchase Cost of Inventory Items
Sometimes the total inventory cost expression is written to include the actual cost of the material purchased. With the EOQ assumptions, the purchase cost does not depend on the particular
order policy found to be optimal, because regardless of how many orders are placed each year,
we still incur the same annual purchase cost of D * C, where C is the purchase cost per unit
and D is the annual demand in units.*
PROGRAM 6.1A
Input Data and Excel QM
Formulas for the
Sumco Pump
Company Example
Enter demand rate, setup/ordering
cost, holding cost, and unit price.
If unit price is available, it is entered here.
On input screen, you may specify
whether the holding cost is fixed
amount or a percentage of the
unit (purchase) cost.
Total unit (purchase) cost is given here.
*Later
in this chapter, we discuss the case in which price can affect order policy, that is, when quantity discounts are offered.
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CHAPTER 6 • INVENTORY CONTROL MODELS
PROGRAM 6.1B
Excel QM Solution for the Sumco Pump Company Example
Total cost includes holding cost,
ordering/setup cost, and unit/purchase
cost if the unit cost is input.
It is useful to know how to calculate the average inventory level in dollar terms when the
price per unit is given. This can be done as follows. With the variable Q representing the quantity of units ordered, and assuming a unit cost of C, we can determine the average dollar value
of inventory:
Average dollar level =
I is the annual carrying cost as a
percentage of the cost per unit.
1CQ2
2
(6-6)
This formula is analogous to Equation 6-1.
Inventory carrying costs for many businesses and industries are also often expressed as an
annual percentage of the unit cost or price. When this is the case, a new variable is introduced.
Let I be the annual inventory holding charge as a percent of unit price or cost. Then the cost of
storing one unit of inventory for the year, Ch, is given by Ch = IC, where C is the unit price or
cost of an inventory item. Q* can be expressed, in this case, as
Q* =
2DCo
B IC
(6-7)
Sensitivity Analysis with the EOQ Model
The EOQ model assumes that all input values are fixed and known with certainty. However,
since these values are often estimated or may change over time, it is important to understand
6.5
REORDER POINT: DETERMINING WHEN TO ORDER
205
how the order quantity might change if different input values are used. Determining the effects
of these changes is called sensitivity analysis.
The EOQ formula is given as follows:
EOQ =
2DCo
B Ch
Because of the square root in the formula, any changes in the inputs 1D, Co, Ch2 will result in
relatively minor changes in the optimal order quantity. For example, if Co were to increase by a
factor of 4, the EOQ would only increase by a factor of 2. Consider the Sumco example just presented. The EOQ for this company is as follows:
EOQ =
211,00021102
B
0.50
= 200
If we increased Co from $10 to $40,
EOQ =
211,00021402
B
0.50
= 400
In general, the EOQ changes by the square root of a change in any of the inputs.
6.5
Reorder Point: Determining When to Order
The reorder point (ROP)
determines when to order
inventory. It is found by
multiplying the daily demand
times the lead time in days.
Now that we have decided how much to order, we look at the second inventory question: when
to order. The time between the placing and receipt of an order, called the lead time or delivery
time, is often a few days or even a few weeks. Inventory must be available to meet the demand
during this time, and this inventory can either be on hand now or on order but not yet received.
The total of these is called the inventory position. Thus, the when to order decision is usually
expressed in terms of a reorder point (ROP), the inventory position at which an order should
be placed. The ROP is given as
ROP = 1Demand per day2 * 1Lead time for a new order in days2
= d * L
(6-8)
Figure 6.4 has two graphs showing the ROP. One of these has a relatively small reorder point,
while the other has a relatively large reorder point. When the inventory position reaches the
ROP, a new order should be placed. While waiting for that order to arrive, the demand will be
met with either inventory currently on hand or with inventory that already has been ordered but
will arrive when the on-hand inventory falls to zero. Let’s look at an example.
PROCOMP’S COMPUTER CHIP EXAMPLE Procomp’s demand for computer chips is 8,000 per
year. The firm has a daily demand of 40 units, and the order quantity is 400 units. Delivery of an
order takes three working days. The reorder point for chips is calculated as follows:
ROP = d * L = 40 units per day * 3 days
= 120 units
Hence, when the inventory stock of chips drops to 120, an order should be placed. The order
will arrive three days later, just as the firm’s stock is depleted to 0. Since the order quantity is
400 units, the ROP is simply the on-hand inventory. This is the situation in the first graph in
Figure 6.4.
Suppose the lead time for Procomp Computer Chips was 12 days instead of 3 days. The reorder point would be:
ROP = 40 units per day * 12 days
= 480 units
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CHAPTER 6 • INVENTORY CONTROL MODELS
FIGURE 6.4
Reorder Point Graphs
Inventory
Level
Q
ROP
0
Lead time = L
ROP < Q
Time
Inventory
Level
On Order
Q
On hand
0
Lead time = L
ROP > Q
Time
Since the maximum on-hand inventory level is the order quantity of 400, an inventory position
of 480 would be:
Inventory position = 1Inventory on hand2 + 1Inventory on order2
480 = 80 + 400
Thus, a new order would have to be placed when the on-hand inventory fell to 80 while there
was one other order in-transit. The second graph in Figure 6.4 illustrates this type of situation.
6.6
EOQ Without the Instantaneous Receipt Assumption
The production run model
eliminates the instantaneous
receipt assumption.
When a firm receives its inventory over a period of time, a new model is needed that does not
require the instantaneous inventory receipt assumption. This new model is applicable when
inventory continuously flows or builds up over a period of time after an order has been placed or
when units are produced and sold simultaneously. Under these circumstances, the daily demand
rate must be taken into account. Figure 6.5 shows inventory levels as a function of time. Because
this model is especially suited to the production environment, it is commonly called the
production run model.
In the production process, instead of having an ordering cost, there will be a setup cost. This
is the cost of setting up the production facility to manufacture the desired product. It normally
includes the salaries and wages of employees who are responsible for setting up the equipment,
engineering and design costs of making the setup, paperwork, supplies, utilities, and so on. The
carrying cost per unit is composed of the same factors as the traditional EOQ model, although
the annual carrying cost equation changes due to a change in average inventory.