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http://ssrn.com/abstract=3475239
ECONOMIC ORDER QUANTITY (EOQ)
Samithambe Senthilnathan
PhD (Business/Finance), MSc (Mgmt.), CMA (Aus.)
Independent Academic Consultant, New Zealand
Abstract: In stock management, Economic Order Quantity (EOQ) is an
important inventory management system that demonstrates the quantity of an
item to reduce the total cost of both handling of inventory (Handling Cost) and
order processing (Ordering Cost). The purpose of determining the EOQ is to
minimise the Total Incremental Cost (TIC), beyond the cost of purchasing of a
product/material, in consideration of two main total costs: Total Ordering Cost
(TOC) and Total Handling Cost (THC). This paper contextually highlights two
basic methods of determining the EOQ: Trial and error method and
Mathematical approach and emphasises the mathematical model as highly
useful to enhance the inventory management of a product.
Keywords: economic order quantity, incremental cost, ordering cost,
handling cost, lead-time, safety stock
JEL code: C00, C02, C6, C60, C61
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ECONOMIC ORDER QUANTITY (EOQ)
Samithambe Senthilnathan
Abstract: In stock management, Economic Order Quantity (EOQ) is an
important inventory management system that demonstrates the quantity of an
item to reduce the total cost of both handling of inventory (Handling Cost) and
order processing (Ordering Cost). The purpose of determining the EOQ is to
minimise the Total Incremental Cost (TIC), beyond the cost of purchasing of a
product/material, in consideration of two main total costs: Total Ordering Cost
(TOC) and Total Handling Cost (THC). This paper contextually highlights two
basic methods of determining the EOQ: Trial and error method and
Mathematical approach and emphasises the mathematical model as highly
useful to enhance the inventory management of a product.
1. INTRODUCTION
Various aspects are very important in warehouse management system, such as inventory
management, warehouse maintenance, overhead management, pricing systems, etc. However,
determining optimal ordering quantity is one of the main aspects in inventory management that can
facilitate the inventory management to run with optimal cost. In this context, this paper is devised to
illustrate the basic model of Economic Order Quantity (EOQ) from a learner’s point of view.
The purpose of determining the EOQ is to minimise the Total Incremental Cost (TIC), beyond the cost
of purchasing, in consideration of two main total costs: Total Ordering Cost (TOC) and Total Handling
Cost (THC).
1
In this context, this paper highlights two basic methods of determining the EOQ: Trial
and error method and Mathematical approach. However, in this illustration, mathematical model is
highly emphasised to enhance the inventory management applications.
As further explanations, EOQ related other measures also illustrated supports to the inventory
management system, mainly the relationships of EOQ to Economic Number of Orders (ENO), length of
inventory cycle, and reorder point of quantity stored. As a contextual explanation of EOQ, this paper
has been following with: Definition and determination of EOQ, Extensions of EOQ with other related
concerns and Concluding remarks.
1
An incremental cost is the difference in total costs as the result of a change in some activity. Incremental costs are also referred to
as the differential costs and they may be the relevant costs (source: https://www.accountingcoach.com/blog/what-is-an-
incremental-cost).
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2. DEFINITION AND DETERMINATION OF ECONOMIC ORDER QUANTITY (EOQ)
Economic Order Quantity (EOQ) is an inventory management system that demonstrates the quantity
of an item to reduce the total cost of both handling of inventory (Handling Cost) and order processing
(Ordering Cost). EOQ as a model has been introduced in 1913 by Ford W. Harris; and R. H. Wilson and
K. Andler are given credit for their in-depth analysis and application of the EOQ model (Hax and
Candea, 1984).
With respect to an item to be ordered, from a business point of view, the EOQ model establishes the
amount of quantity to be placed in an order in consideration of minimising the annual total cost of
inventory handling and order processing. In this context, these specific two types of costs are the main
categories of determining the EOQ in its basic explanation. However, the model has been presented
with certain assumptions for the initial understanding; and from that point onward, its extensions are
used widely in businesses, especially in inventory management.
2.1 Assumptions of the Basic Model of Economic Order Quantity (EOQ)
The determination of EOQ consists of the following assumptions:
a) The EOQ will be determined for every product individually in a business.
b) Annual requirement (Demand) for product in units is known with certainty.
c) Ordering cost is known and constant throughout the year.
d) Inventory handling cost is known and constant throughout the year. Notably, if the handling
cost of an item is given as the percentage of price of the item, the unit price of the item
remains same throughout the year.
e) No cash or quantity discount is allowed.
f) The ordered quantity of the product is delivered at once as a single batch.
g) Immediate replenishment of ordered quantity on time (No delay and stock shortage).
h) Constant lead time is only allowed (no fluctuation is permitted).
2.2 Annual Demand of the Product
The annual requirement of a product is constant and known in its measurement units and this known
as annual demand of the product (D). Using various forecasting technique, it is important for a
business to predict the annual demand for the specific item, for which the business need to know the
EOQ. As the demand produces the primary purchasing cost of the item, the total purchasing cost is
irrelevant in determining the EOQ.
2.3 Order Processing Cost
The Ordering Cost refers to the cost of orders to be placed for the product in consideration of order
communication, allowances to purchase officers, order printing and stationery, costs of inspection,
receiving the product, and transport cost, etc. Notably, these costs remain constant and unchanged for
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the period, irrespective the number of orders to be placed. As the ordering cost per order is constant,
the relationship between the quantity ordered and number of orders to be placed is negative, i.e.,
higher the quantity ordered (Q) per order, lower the number of orders to be placed; and lower the
quantity ordered (Q) per order, higher the number of orders to be placed. This implies the negative
relationship between the quantity ordered (Q) and total cost of order processing (TOC) as in Figure 1.
2.4 Inventory Handling Cost
This cost refers to the handling and maintaining the product inventory in workplace in consideration
of warehousing costs, shrinkage loss, evaporation, deterioration and spoilage costs, insurance,
warehouse rent, obsolescence, and other related overhead cost of warehouse. As these costs are
constant to maintain the total demand (annual requirement) of the product, the handling cost per unit
remains same. Therefore, total cost of handing has positive relationship with the number of products
handled in the workplace (warehouse), i.e., higher the number of products (Q) in store, higher the total
cost of handling (THC); and lower the number of products in store, lower the total handling cost (THC)
as in Figure 2. It is notable that handling of stock would be a half the number of quantity to be ordered
throughout the year; and therefore, the average stock to handle would be Q/2.
Figure 1:
Figure 2:
Negative Relationship between Total Ordering Cost
(TOC) and Quantity to be ordered (Q)
Positive Relationship between Total Handling Cost
(THC) and Quantity to be ordered (Q) for storing
2.5 The Model: Economic Order Quantity (EOQ)
The basic EOQ model, with all assumptions in consideration, deal with two types of costs: Total
Ordering Cost (TOC) and Total Handling (THL). It is obvious from the above explanation that these
costs are moving in opposite directions, when certain number of quantities are ordered for storage
purpose. Therefore, it is important for a business to find a trade-off point of ordering quantity in order
to minimise the total cost of both: TOC plus THC. In this context, the quantity to be ordered to minimise
the total cost of both TOC and THC is known as the Economic Order Quantity (EOQ).
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As TOC has negative relationship to quantity to be ordered and THC has positive relationship to
quantity to be ordered, the total minimum cost of both TOC and THC is the intersection point of both
cost lines that can produce: (a) the total cost of both TOC and THC as minimum as possible; and (b)
the number of quantities to be ordered (known as EOQ) to meet the minimised cost (see Figure 3).
Figure 3: Graphical Determination of Economic Order Quantity
Notably, from purchasing point of view, TOC and THC are the additional costs, which incur above cost
of a material purchased. Therefore, the aggregation of both costs (TOC and THC) are known as Total
Incremental Cost, i.e., TIC = TOC + THC. In the context of EOQ, TOC and THC are the additional costs
incurring beyond the original purchasing cost of an item.
2
Generally, there are two basic methods to determine the EOQ.: (a) Trial and Error method in
combination of graphical representation; and (b) Mathematical Approach – this is widely used popular
method. These methods can be explained with an exhibit for easy understanding.
Exhibit 1
Consider a small production process, which need sawdust as raw material. The production process
requires 20,000 cubic meters (m3) annually. If an order is to be placed, every order can cost $ 50.00
and the cost of handling one unit of cubic meter is $ 2.
From the above, the following information is available.
Annual Demand (D) = 20000 m3 for the sawdust,
2
An incremental cost is the difference in total costs as the result of a change in some activity. Incremental costs are also referred to
as the differential costs and they may be the relevant costs (source: https://www.accountingcoach.com/blog/what-is-an-
incremental-cost).
Q
Economic Order
Quantity (EOQ): Q*
Combined cost of
TOC and THC
TOC
THC
Cost in $
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Cost per order (CO) = $ 50.00 and Handling cost per unit (CH) = $ 2.00
Following table can produce various possible number of cubic meters of sawdust to be ordered (this
should be assumed independently) and the related cost calculations.
3
Table 1: Various number of cubic meters (quantity) and related cost calculations
Ordering
Quantity
(Q)
No. of
Orders
(N = D/Q)
Average
Stock to
handle
(Q/2)
Total Ordering Cost
(TOC) @ $50
Total Handling Cost
(THC) @ $2
Total Incremental Cost
(TIC = TOC + THC)
250
80
125
4,000.00
250.00
4,250.00
400
50
200
2,500.00
400.00
2,900.00
500
40
250
2,000.00
500.00
2,500.00
1000
20
500
1,000.00
1,000.00
2,000.00
2000
10
1000
500.00
2,000.00
2,500.00
2500
8
1250
400.00
2,500.00
2,900.00
4000
5
2000
250.00
4,000.00
4,250.00
From the table, it is possible to observe that the (last) column TIC has a minimum of $ 2000.00, where
TOC = THC = $ 1000.00 and the quantity Q = 1000. The information available in the table can be shown
in a diagram with all three costs: TIC, TOC and THC (see Figure 4).
Figure 4: Economic Order Quantity (EOQ) with TOC, THC and TIC
3
As various quantities are assumed independently for table formulation, this method is called trial and error method.
-
500.00
1,000.00
1,500.00
2,000.00
2,500.00
3,000.00
3,500.00
4,000.00
4,500.00
0500 1000 1500 2000 2500 3000 3500 4000 4500
Cost in $
Order Quantity in number of cubes (m3)
Total Ordering Cost (TOC) @ $50 Total Handling Cost (THC) @ $2
Total Incremental Cost (TIC = TOC + THC)
EOQ
TIC
TOC = THC
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Using the same information of Exhibit 1, the following deals with the mathematical approach, which
is widely used in determining the EOQ in inventory control system.
The basics of mathematical model therefore need to be illustrated to determine the EOQ to have
minimum of TIC. Accordingly, it is important to determine total individual cost of both (TOC and THC),
in terms of quantities to be ordered and handled.
Therefore, TOC = Cost per order . (Demand/Quantity Ordered per year)
TOC = CO . (D/Q) =
and
THC = Cost per unit for handling . (Average quantity maintained in store for a year)
THC = CH . (Q/2) =
This results in TIC = TOC + THC =
+
As this function TIC depends on the quantity ordered (Q) to minimise the total cost, the function needs
to be differentiated with respect to Q.
This can result in:
and for minimisation/maximisation
.
Therefore,
and
; and solving for Q can result in
and this must be confirmed for minimising the TIC.
To confirm the minimisation of TIC for a value of Q, the second derivative of TIC should be greater
than zero for the value of Q.
As such, from the first derivative, the second derivative of TIC must result in
and for a value of Q,
.
Therefore, TIC optimally produce a minimum cost for the value of Q* (where Q* = EOQ).
Now, we can substitute the values available in Exhibit 1, where Annual Demand (D) = 20000 m3 for
the sawdust, Cost per order (CO) = $ 50.00 and Handling cost per unit (CH) = $ 2.00.
Therefore, TOC =
=
and THC =
=
= Q
Then, TIC =
+ Q
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Differentiating TIC with respect to Q can result in
For an optimum of TIC with respect to Q,
.
i.e.,
and solving for Q can result in
Q2 = (20000) . (50) and Q* = m3.
It is now possible to note that the number of orders to be place for the year is
(D/Q*) = (20000/1000) = 20, to have the optimal TIC.
Q* = EOQ =1000 can be applied to determine the TOC, THC and TIC.
TOC =
= $ 1000, THC =
=
= $ 1000
And TIC = TOC + THC = $(1000 + 1000) = $ 2000.
Explicitly, the results are obvious about the trade off point of Q (as EOQ) between the costs of TOC and
THC as equal; and the TIC results in at the minimum of the sum of costs (TOC + THC).
3. EXTENSION OF EOQ WITH OTHER RELATED CONCERNS
As EOQ is explained to determine in both: Trial and Error Method and Mathematical Approach, it is
notable how the EOQ determination can be useful to determine other related concerns. In this context,
this section particularly deals with the following.
3.1 Determining TIC with Demand (D), per Order Cost (CO) and per unit product Handling Cost (CH)
As TIC = TOC + THC =
+
and EOQ = Q* =
, now Quantity (Q) in the TIC function
can be represented with EOQ = Q*. Therefore, Q can be substituted with
. This can result in
TIC =
+
=
=
=
TIC =
=
=
TIC =
As in Exhibit 1, Annual Demand (D) = 20000 m3 for the sawdust, Cost per order (CO) = $ 50.00 and
Handling cost per unit (CH) = $ 2.00,
TIC = = = $ 2000.
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3.2 Transforming EOQ into Economic (Optimum) Number of Orders (ENO)
When EOQ (Q*) is determined and termed as optimal quantity to minimise the TIC, the number of
orders for the year can be determined as Demand (D) divided by EOQ (i.e., N = D/Q*). This in other
term can be interpreted that the Economic (optimal) Number of Orders (ENO = N*) can provide the
minimum TIC. In this context, Q can be substituted in terms of N and the mathematical approach can
be extended to determine the Economic Number of Orders (ENO) and TIC relatively, as shown below.
Therefore, TOC = Cost per order . (Demand/Quantity Ordered per year)
TOC = CO . (D/Q) =
and as N = (D/Q),
TOC = N . CO
THC = Cost per unit for handling . (Average quantity maintained in store for a year)
THC = CH . (Q/2) =
and as N = (D/Q),
THC = CH . (Q/2) =
This results in TIC = TOC + THC =
As this function TIC depends on the quantity ordered (Q) to minimise the total cost, the function needs
to be differentiated with respect to N.
This can result in:
and for minimisation/maximisation
.
Therefore,
and
; and solving for Q can result in
and this must be confirmed for minimising the TIC.
To confirm the minimisation of TIC for a value of N, the second derivative of TIC should be greater
than zero for the value of N.
As such, from the first derivative, the second derivative of TIC must result in
and for a value of Q,
.
Therefore, TIC optimally produce a minimum cost for the value of N* (where N* = ENO).
As in Exhibit 1, Annual Demand (D) = 20000 m3 for the sawdust, Cost per order (CO) = $ 50.00 and
Handling cost per unit (CH) = $ 2.00, and substituting these values in
N*
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With respect to the number of orders to be placed, ENO can be found can be represented in a diagram
with TOC, THC and TIC, respectively (see Table 2 and Figure 5).
Table 1: Various number of orders and related cost calculations
No. of
Orders
(N)
Ordering
Quantity
(Q = D/N)
Average
Stock to
handle
(Q/2)
Total Ordering Cost
(TOC) @ $50
Total Handling Cost
(THC) @ $2
Total Incremental Cost
(TIC = TOC + THC)
80
250
125
4,000.00
250.00
4,250.00
50
400
200
2,500.00
400.00
2,900.00
40
500
250
2,000.00
500.00
2,500.00
20
1000
500
1,000.00
1,000.00
2,000.00
10
2000
1000
500.00
2,000.00
2,500.00
8
2500
1250
400.00
2,500.00
2,900.00
5
4000
2000
250.00
4,000.00
4,250.00
Figure 5: Economic Number of Orders (ENO) with TOC, THC and TIC
It is now possible to note that the number of quantities (known as EOQ in other term) to be placed in
an order is (D/N*) = (20000/20) = 1000, to have the optimal TIC.
3.3 Length of inventory cycle (provided with daily usage ‘d’ of the product)
Length of inventory cycle is a measure that gives a time period how long a batch of EOQ can last in the
storage. As production/supply of the items takes place, the daily usage (d) of the item becomes
-
500.00
1,000.00
1,500.00
2,000.00
2,500.00
3,000.00
3,500.00
4,000.00
4,500.00
010 20 30 40 50 60 70 80 90
Cost in $
Number of Orders to be placed
Total Ordering Cost (TOC) @ $50 Total Handling Cost (THC) @ $2
Total Incremental Cost (TIC = TOC + THC)
ENO
TIC
TOC = THC
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reduction in the EOQ stored. Therefore, length of inventory cycle indicated how many times (T) of
daily usage of the item in total equates the EOQ.
Therefore, EOQ = T . d where T = length of inventory cycle and d = daily usage of the item.
Note that the daily usage should have been provided with certainty (consider it as an assumption).
Referring to the information above, EOQ = 1000 units and assume daily use (d) of the item is 100 units.
Therefore, length of inventory cycle
days.
3.4 Determining number of annual working days
It is also notable that every batch of EOQ last for certain period (10 days in the example above) and
this will happen for every order of EOQ in a year. Therefore, the T (=10 days) time cycle is applicable
for every order. As EOQ can be transformed into Economic Number of Orders (ENO = N*) in a year and
every order (a batch of EOQ) last for T (=10 days) time as a cycle, the annual number of working days
in a year on this product can be determined as:
Annual number of working days = N* . T* = (20) . (10) = 200 days
Note in the example, N* = (D/EOQ) = (20000/1000) = 20 and
T* = (EOQ/d) = (1000/100) = 10.
3.5 Reorder point quantity (provided with lead-time for stock replenishment)
The reorder point quantity/stock level of an item is measure at which the product needs an order
placement for the replenishment of the stock, as for not to interrupt the trade operations. In other
term, it is the stock level to use during the lead-time of stock replenishment. After the immediate
replenishment of EOQ, daily usage of the product is taken from the stock in the store. In this context,
reorder point can be determined in consideration of daily usage (d), lead-time of replenishing the EOQ
(TL) and safety stock (GS)of the product/item.
• If a firm has no policy of maintaining a safety stock level of the item,
Reorder Level (ROL) = (Daily Usage) . (Lead-time in days)
ROL = (d).(TL)
As previously stated, consider again the daily usage d = 100 units and the lead time TL = 6 days.
Accordingly, the reorder level ROL = (d).(TL) = (100).(6) = 600 units. This implies that when the stock
level is at 600 units a new order of EOQ need to be placed to get it after 6 days as immediate
replenishment, and available 600 units can meet the 6 days requirements at the daily usage of 100
units (see Figure 6).
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Figure 6: Daily stock level and Reorder Level (ROL) without safety stock
• If a firm maintains a safety stock level of the item,
Reorder Level (ROL) = (Daily Usage) . (Lead-time in days) + (Safety Stock)
ROL = (d).(TL) + (GS)
When we consider a safety stock level GS = 200 units of the product as an additional information, the
reorder level ROL = (d).(TL) + (GS) = (100).(6) + 200 = 800 units. This is to safeguard the daily
operations of the firm in case of expected delay of two (2) days for replenishing the stock ordered
(EOQ). This implies that a deviation of additional two days to the lead time cannot have impacts on
continuing operations of the firm (see Figure 7).
Figure 7: Daily stock level and Reorder Level (ROL) with safety stock
0
200
400
600
800
1000
1200
Quantity of the product
Dates of Operations
Stock Level ROL
0
200
400
600
800
1000
1200
1400
Quantity of the product
Dates of Operations
Safety Stock Stock Level ROL
ROL
EOQ
EOQ
ROL
Safety Stock Level
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Further, there is a possibility of having irregular daily usage of the product/item. In this case, it is wise
to have the maximum daily usage of the product to determine its ROL. This always happen when there
is uncertainty of daily usage with varying demands, considering maximum usage of the item can
facilitate the continuing operations. In case of ‘No Safety Stock’, ROL = (Maximum of d).(TL); and if
‘With Safety Stock’, ROL = (Maximum of d).(TL) + (GS).
4. CONCLUDING REMARKS
In inventory management, determination of EOQ is an important measure to regulate other concerns
in warehouse management. Main objective of determining the EOQ is to minimise the total
incremental cost (TOC and THC) that incur beyond the cost of purchasing the product. In this context,
this paper attempts to highlight two basic methods of determining the EOQ: Trial and Error Method
and Mathematical Approach. However, it is advisable to apply mathematical approach to make
decisions objectively.
This paper has more explanations on mathematical approach of EOQ and further explanations are also
provided with the relationships of EOQ to Economic Number of Orders (ENO), length of inventory
cycle, and reorder point of quantity stored. This paper is contextually presented for the learners of
EOQ within its assumptions. However, the assumptions themselves become the limitations of the
model. This explanation can be extended in particular by analysing: (a) How an EOQ measure can be
determined where TOC ≠ THC, (b) What will happen, if per order cost and/or per unit handling cost is
dependent on EOQ/ENO, (c) What will happen to EOQ, if discount is allowed, and (d) Sensitivity
Analysis, relatively. Considering them, this paper can provide a base those extensions.
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