Journal of Information, Control and Management Systems, Vol. 1(2003), No.2
43
REVIEW ON THE HARRIS´S EOQ MODEL IN
INVENTORY MANAGEMENT
Jaroslav KRÁL
Faculty of Management and Informatics, University of Zilina
e-mail: [email protected]
Abstract
Inventory is a stock of items kept by an organization to meet internal and
external customer demand. Inventory is the subject of interest of inventory
management. The main objective of inventory management has been to keep
enough inventory to meet customer demand and also be cost-effective. The
paper reviews the oldest inventory model introduced by F. W. Harris. Harris´s
formula, computing EOQ and discussed here, is usually a part of decision
support systems or advanced planning modules of an enterprise resource
planning. Inventory plays a key role in the logistics behavior of manufacturing
systems, but inventory modeling is a very poor field of enterprise practices.
Keywords: Inventor: Economic Order Quantity; Setup Cost; Lot Size; Holding
Cost; Economic Order Quantity; Sensitivity Analysis; Continuous Inventory
System.
1
INTRODUCTION
One of the earliest applications of mathematics to manufacturing management
was the work of F. W. Harris on the problem of setting manufacturing lot sizes (How
Many Parts to Make at Once, 1913). Harris´s Economic Order Quantity (EOQ) model
has been widely studied and is staple of virtually every management science or
inventory management textbook. The problem Harris had on mind was that of a plant
producing various products and switching between products entails a costly setup. He
puts it thus: “Interest on capital tied up in wages, material and overhead sets a
maximum limit to the quantity of parts which can be profitably manufactured at one
time; ´set-up´ costs on the job fix the minimum. Experience has shown one manager a
way to determine the economical size of lots.” As an example he described a
metalworking shop that produced cooper connectors.
Harris defined the sum of the labor and material costs to ready the shop to
produce a product to be the setup cost. If the connectors had been purchased, instead
of manufactured, then the problem would remain similar, but setup cost would
44
Review on The Harris´s EOQ Model in Inventory Management
correspond to the cost of placing a purchase order, usually known as ordering cost.
Large lots reduce setup costs by requiring less frequent changeovers. But small lots
reduce inventory by bringing in product closer to the time it is used. The EOQ model
was Harris´s systematic approach to striking a balance between these two concerns.
THE EOQ MODEL
Despite his note in the above quote that the EOQ is based on experience, Harris
was consistent with the scientific emphasis of his day on precise mathematical
approaches to manufacturing management. To derive a lot size formula, he made the
following assumptions about manufacturing systems:
1. Production is instantaneous. There is no capacity constraint, and the entire lot
is produced simultaneously.
2. Delivery is immediate. There is no time lag between production and
availability to satisfy demand.
3. Demand is deterministic. There is no uncertainty about quantity or timing of
demand.
4. Demand is constant over time. In fact, it can be represented as a straight line,
so that if annual demand is 365 units, this translates to a daily demand of one
unit.
5. A production run incurs a fixed setup cost. Regardless of the size of the lot or
the status of the plant, the setup cost is the same.
6. Products can be analyzed individually. Either there is only a single product or
there are no interactions, e.g. shared machines, between products.
With these assumptions, we can use Harris´s notation, with slight modifications
for easy to presentation, to develop the EOQ model for computing optimal production
lot sizes. The notation can be follows:
D … demand rate [units per year]
C… unit production cost, not counting setup or inventory costs [SKK per unit]
co ... fixed setup or ordering cost to produce or purchase a lot [SKK]
ch ... holding or carrying cost [SKK]
Q …lot size [units]; this is the decision variable.
For modeling purposes, Harris represented both time and product as continuous
quantities. Since he assumed constant, deterministic demand, ordering Q units each
time the inventory reaches zero results in an average inventory level of Q/2 – see
Figure 1. The holding cost associated with this inventory is therefore ch Q/2 per year.
The setup cost is co per order, or co D/Q per year, since it must be placed D/Q orders to
satisfy demand. The production or purchase cost c is per unit, or c D per year. Thus,
the total (inventory, setup, or production) cost per year can be expressed as
Q
D
Y (Q ) = c h + c o + cD .
(1)
2
Q
2
Inventory [units]
Journal of Information, Control and Management Systems, Vol. 1(2003), No.2
45
Q
Q/2
Q/D
2Q/D
3Q/D
Time
Figure 1 Inventory and time in the EOQ model
If we practically use the cost function Y(Q) it would indicate the following
results:
1. The ch Q/2 holding cost increases linearly in the lot size Q and eventually
becomes the dominant component of total annual cost for large Q.
2. The co D/Q setup cost diminishes quickly in Q, indicating that while increasing
lot size initially generates substantial savings in setup cost, the returns from
increased lot sizes decrease rapidly.
3. The cD unit cost does not affect the relative cost for different lot sizes, since it
does not include the decision variable, Q.
4. The Y(Q) total annual cost is minimized by some lot size Q. Interestingly, this
minimum turns out to occur precisely at the value of Q for which the holding
cost and setup cost are exactly balanced.
Harris wrote that finding the value of Q that minimizes Y(Q) “involves higher
mathematics” and simply gives the solution without further derivation. That means:
Q
D
ch = co .
2
Q
Then, the lot sizes that minimizes Y(Q) in cost function (1) is:
c
Q * = 2D o .
ch
(2)
This square root formula is the well-known economic order quantity (EOQ),
also referred to as the economic lot size.
3
TWO KEY INSIGHTS OF EOQ
The obvious implication of the above formula (2) is that the optimal order
quantity increases with the square root of the setup cost or the demand rate and
decreases with the square root of the holding cost. However, a more fundamental
insight from the Harris´s work is the realization that “There is a trade-off between lot
size and inventory.” Increasing the lot size increases the average amount of inventory
46
Review on The Harris´s EOQ Model in Inventory Management
on hand, but reduces the frequent of ordering. By using a setup cost to penalize
frequent replenishments, Harris articulated this tradeoff in clear economic terms.
The basic insight above is incontrovertible. However, the specific mathematical
result – the EOQ formula – depends on the modeling assumptions, some of which we
could certainly question, e.g., how realistic is instantaneous production. Moreover, the
usefulness of the EOQ formula for computational purposes depends on the realism of
the input data. Although Harris claimed that “The setup cost proper is generally
understood” and “may, in a large factory, exceed one dollar per order” (a number
larger than 1), estimating setup costs may be a huge difficult task. Setups in a
manufacturing system have a variety of other impacts, e.g. on capacity, variability and
quality, and therefore not easily reduced to a single invariant cost. In purchasing
systems, however, where some of these other effects are not an issue and the setup cost
can be clearly interpreted as the cost of placing a purchase order, the model can be very
useful.
It is essential noting that the insight here there is a tradeoff between lot size and
inventory without even resorting to Hariss´s EOQ formula. As the average number of
lots per year N is
N=
D
,
Q
the total inventory investment I is
Q
D
=c
.
2
2N
Now, we could simply plot inventory investment I as a function of replenishment
frequency N in lots per year. The analysis can always show that there are decreasing
returns to additional replenishments. If we can attach a value to these production runs
or purchase orders, e.g. the setup cost co, then we can compute the optimal lot size
using the EOQ formula. However, if this cost is unknown, as it may well be, then the
function of replenishment frequency at least gives us an idea of the impact on total
inventory of an additional annual replenishment. Armed with this tradeoff information,
a decision maker can select a reasonable number of changeovers or purchase orders per
year and thereby specify a reasonable lot size.
A second insight that follows from the EOQ model is that holding and setup costs
are fairly intensive to lot size. This implies that if, for any reason, we use a lot size that
is slightly different than Q*, the increase of holding plus setup costs will not be large.
To examine the sensitivity of the cost to lot size it begins by substituting Q* for Q into
expression (1), but omitting c, since this is not affected by lot size, and we can find that
the minimum holding plus setup cost per unit is given by
I =c
Journal of Information, Control and Management Systems, Vol. 1(2003), No.2
Y * = Y (Q *) = c h
Q*
D
+ co
2
Q*
2c o D / c h
= ch
47
2
+ co
D
2c o D / c h
(3)
= 2c o c h D
Now, suppose that instead of using Q*, we use some other arbitrary lot size Q´,
which might be larger or smaller than Q*. From expression (1) for Y(Q), we see that
the annual holding plus setup cost under Q´ can be written
Y (Q´) = ch
Q´
D
.
+ co
Q´
2
Hence, the ratio of the annual cost using lot size Q´ to the optimal annual cost (using
Q*) is given
Y (Q ) c h Q´/ 2 + c o D / Q´
=
Y*
2c h c o D
=
2 2
ch 2
Q´
1 co D
+
2 2c o c h D Q´ 2c o c h D
(4)
Q´
Q*
=
+
2Q * 2Q´
=
1  Q´ Q * 


+
2  Q * Q´ 
To review (4), suppose that Q´=2Q*, which implies that we use a lot size twice as
large as optimal. Than, the ratio of the resulting holding plus setup cost to the optimum
is
1
1
 2 +  = 1,25 ; that is, a 100 % error in lot size results in a 25 % in the cost
2
2
function. If Q´=Q*/2, we also get an error of 25 % in the cost function.
Further sensitivity insight from the EOQ formula we can get by noting that
because demand is deterministic, the order interval is completely determined by the
order quantity. We need to express the time between orders T:
Q
(5)
T= .
D
Dividing (2) by D, we can get the following expression for the optimal order interval
2c o
(6)
T* =
ch D
48
Review on The Harris´s EOQ Model in Inventory Management
and substituting (5) into (4), we get the following expression for the ratio of the cost
resulting from an arbitrary order interval T´ and the optimum cost:
Annual cost under T ´ 1  T ´ T * 
= 
+

Annual cost under T * 2  T * T ´ 
(7)
Expression (7) is useful in multi-product settings, where it is desirable to order
such different products those are frequently replenished at the same time (mainly, in
order to facilitate sharing of delivery trucks). A method for facilitating this is to order
items at intervals given by powers of 2. That is, make the order interval one week, two
weeks, four weeks, eight weeks, etc. To be complete, we need also to consider negative
powers of 2, i.e. one-half week, one-fourth week, one-eighth week. (But if we are used
a smaller time unit such as days instead of weeks, setting of the negative powers of 2
will not be necessary). The result is that items ordered at 2n week intervals will be
placed at the same time as orders for items with 2k intervals for all k smaller than n –
see Figure 2. This will facilitate, e.g., sharing of trucks, simplification of shipping
schedules.
Moreover, the sensitivity results we derived above for the EOQ model imply that
the error introduced by restricting order intervals to powers of 2 will not be excessive.
To see this, suppose that the optimal interval for an item T* lies between 2m and 2m+1
[
]
for some m – see Figure 3. Then T* lies ether in the interval 2 m ,2 m 2 or in the
[
interval 2
m
2 ,2
m +1
]. All points in [2
m
,2
[
m
]
2 are no more than
]
2 times as large
as 2m. Likewise, all points in the interval 2m 2 ,2m +1 are no less than 2m+1 divided by
m
2 . In Figure 3, 2 is within a multiplicative factor of
2 of T1* , and 2m+1 is within a
multiplicative factor of 1 / 2 of T2* . Hence, the power of 2 order interval T´ must lie
[
]
in the interval T * / 2 , 2T * around the optimal order interval T*. Thus, the
maximum error in cost will occur when T ´= 2T * , or T ´= T * / 2 . From (7), the
error from using T ´= 2T * is
1
1 
 2 +
 = 1,06
2
2
and is the same when T ´= T * / 2 . Hence, the error in the holding plus setup costs
resulting from using the optimal power of 2 order interval instead of the optimal order
interval is guaranteed to be no more than 6 %. A lot of management science sources
offer algorithms for computing the optimal power of 2 policy and extend the above
results to more general multipart settings.
49
Journal of Information, Control and Management Systems, Vol. 1(2003), No.2
Time
Order Interval
0
1
2
3
4
5
6
7
8
1=20
2=21
4=22
8=23
Figure 2 Powers of 2 order intervals
2m
T1*
2m 2
T2*
2m+1
Figure 3 The root-2 interval
CONCLUSION
The objective of inventory management is to employ an inventory control system
that will indicate how much should be ordered and when orders should take place so
that the sum of the inventory costs will be minimized. An inventory management
system controls the level of inventory by determining how much to order – the level of
replenishment – and, when to order. There are two basic types of inventory systems: a
continuous (or fixed-order-quantity) system and a periodic (or fixed-time-period)
system. In continuous system, an order is placed for the same constant amount
whenever the inventory on hand decreases to a certain level, whereas in a periodic
system, an order is placed for a variable amount after specific regular intervals. In a
continuous inventory system, a continual record of the inventory level for every item is
maintained. Whenever the inventory on hand decreases to a predetermined level – the
reorder point – a new order is placed to replenish the stock of inventory. This amount
is calculated by the economic order quantity, discussed above or some its applications.
Harris´s original EOQ formula has been extended in a variety of ways over the
years. One of the earliest variation was a model called the economic production lot
(EPL). It is the case in which replenishment is not instantaneous; instead, there is a
finite, but constant and deterministic, production rate. The other applications of the
EOQ are used in statistical inventory modeling. A very useful model with a wide
application is called the (r, Q) model. The acronym means: When inventory of the item
falls to the reorder point r, order the replenish quantity Q.
4
50
Review on The Harris´s EOQ Model in Inventory Management
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[1]
[2]
[3]
[4]
[5]
[6]
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Referee: Jozef Majerčák