Module 4:
Topic 1: Basic Economic Order Quantity (EOQ) Model
1. Inventory System
1.1. Inventory
Inventory is the stock of any item or resource used in an organization and can include:
 raw materials,
 work-in-process,
 maintenance/repair/operating (MRO) parts, and
 finished goods.
1.2. Inventory System
An inventory system is the set of policies and controls that monitor levels of inventory and
determine what inventory levels should be maintained, when stock should be replenished,
and how large orders should be.
2. Inventory Costs
2.1 Elements of Inventory Costs
(1) Holding (Carrying) costs
- Costs of holding or “carrying” inventory over time. They include cost of storage, opportunity
cost of capital, and risks (obsolescence, theft, spoilage etc).
(2) Ordering Cost (outside procurement) or Setup Cost (internal production)
- Costs of placing orders. They include transaction costs, labor, materials, and transportation
costs.
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2.2 Optimal Order Quantity
The figure above illustrates how the inventory setup (or order) and holding (or carrying) costs
change with different sizes of order quantity (that is the amounts included in each order). As
the size of order quantity increases, the carrying costs increase since you need to keep the high
level of inventory; however, the ordering costs decrease since you place smaller number of
orders.
For example, suppose that you have 300 demands for the next three month periods.
 If you set the order quantity that equals to 100, you may place three separate orders
every month to meet this demand (i.e., relatively high setup costs), keeping relatively
small amounts of inventory in the warehouse (i.e., relatively low holding costs).
 Alternatively, you may place only one order with the order quantity that equals to 300
(i.e., relatively low setup costs), keeping relatively large amounts of inventory in the
warehouse (i.e., relatively high holding costs).
Which option will be the better strategy in terms of effective inventory management? The
objective of inventory management is to minimize the total costs of inventory by setting up the
optimal order quantity.
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3. Basic Economic Order Quantity Model (EOQ)
We answer the above question (i.e., how to minimize the inventory costs) using the Economic
Order Quantity (EOQ) model. We start with the simple base model by assuming that most
parameters are constant and then, further extend this base model by relaxing the constant
assumption and combining a statistical approach in the model.
3.1 Assumption
(1) All parameters are constant
 Demand for the product is constant and uniform throughout the period
: Annual demand is constant (D)
 Lead time (time from ordering to replenish) is constant (L)
 Ordering or setup costs are constant (S)
 Holding costs per unit is constant (H)
(2) All demands for the product will be satisfied (No back order or shortages is allowed)
NOTE: Definitions about “H” and “S”
 H = cost to hold 1 unit for the planning horizon
: 1 year planning horizon is most commonly used. For example, H=$3 means that it costs
$3 to store one unit of product in the warehouse for one year.
 S = cost of placing one order or doing one setup
3.2. Decision Rule
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(1) Q units will be ordered every time an order is placed (fixed order quantity)
(2) The optimal Q will minimize total inventory costs over the planning horizon (Q*, EOQ)
(3) Reorder point (ROP) that is a predetermined number such that when inventory reaches ROP
units, an order will be placed. During the given lead time (L), ROP units will be used to fulfill the
demand.
3.3 Decision of Economic Order Quantity, Q*
We find the optimal Q (i.e., Q* or EOQ) that minimizes the total inventory costs.
(1) Total Annual Ordering (Setup) Costs.
Total annual ordering (Setup) Costs =
where D is annual demand and S is the cost to place one order.
NOTE:
 You always order the same quantity Q each time.
 (D/Q) is the actual number of orders placed per year.
(2) Total Annual Holding (Carrying) Costs.
Total annual holding (carrying) Costs =
where H is the cost to hold one unit for one year
Note:
 (Q/2) is the “average inventory level.”
 When you receive a new order, the inventory level goes up to Q and you may use them
up at a constant speed throughout the period (See the figure above). Therefore, you
may not always keep Q units in the warehouse, but keep on average (Q/2) units in the
warehouse.
(3) Economic Order Quantity: Q*
Total annual cost function (TC) for a given Q
TC(Q) =
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Optimal order quantity is found under the condition that annual setup cost equals annual
holding costs (See the above figure).
(D/Q*) x S = (Q*/2) x H
Solving for Q*
2DS = Q*2H
Q*2 = 2DS/H
EOQ  Q* 
2 DS
H
Reference:
We can derive the same result by setting up the first derivative of TC(Q) = 0.
TC(Q) = [(D/Q) x S] + [(Q/2) x H]
d (TC )  DS H

 0
dQ
2
Q2
DS H

2
Q2
2 DS
Q2 
H
2 DS
EOQ  Q* 
H
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(4) Reorder Point : ROP
 L = lead time (e.g., 3 days).
 d = demand per unit of time (expressed in the same unit of time for L, e.g., 5units/day).
During the given lead time (L) ROP units will be used to fulfill the demand. Therefore,
ROP =
Note: d (e.g., daily demand) is different from D (e.g., annual demand)
Example 1: EOQ
Annual demand for widgets is 10,000 per year. The cost to hold one item for the year is $5 and
the cost to place an order is $9.25. The store opens 250 days per year and the lead time for an
order is 3 working days.
a. What is EOQ?
b. Compare the total annual cost of the current policy (of ordering 1000 units at a time) with
the total annual cost using the EOQ.
c. what is ROP to implement the EOQ policy?
Answer:
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Module 4:
Topic 2: EOQ with Variability
1. EOQ with Variability Model
1.1 Assumption
As an extension of the basic EOQ model we learned in the previous topic, we will relax the
constant demand assumption.
As a result,
(1) Demands (d or D) are not constant. Instead, the daily demand, d is assumed to be normally
distributed
d ~ N (d , d2 )
where d is the average daily demand and  d is the standard deviation of d.
(2) All other assumptions same as basic EOQ model
1.2 Decision of Economic Order Quantity, Q*
EOQ  Q* 
2 DS
H
where D is the "average" annual demand (not a constant).
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1.3. Decision of Safety Stock using Service Level
(1) Distribution of demand during lead time (DLT)
We assumed that the daily demand (d) is not constant but follows a normal distribution. Under
this assumption, what is the distribution of demand during lead time (DLT),
DLT = d L
where
 d is daily demand that is assumed to be a random variable and
 L is lead time that is assumed to be a constant.
To get the distribution of DLT we can use the following theorem:
Theorem:
It is known that when a random variable X follows a normal distribution
X ~ N (  , 2 )
then, Y = kX (where k is a constant) follows a normal distribution that is
Y ~ N (k , Y2 )
where  Y   k
Example:
If X ~ N (2, 0.12 ) then,
Y= 4X ~ N (8, 0.22).
Using the above theorem and the assumption about the daily demand, DLT (= dL) follows a
normal distribution such that
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DLT ~ N (d L, dlt
)
where  dlt   d L
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(2) Service Level
 The stock out happens when the demand during lead time (DLT) is greater than ROP.
Recall the definition of ROP that is a predetermined number such that when inventory
reaches ROP units, an order will be placed.
 Service level is the probability of no stock out during the lead time.
Therefore, in terms of DLT and ROP we can define
Service Level = Prob (DLT ≤ ROP)
For example, Service Level = 95% means that
Prob (DLT ≤ ROP) = 0.95
(3) Safety Stock (ss)
 Amount of inventory carried in addition to the expected demand since demand is not
constant any more.
 Safety stock can protect against stock-outs.
 The amounts of safety stock depends on the service level
We can formulate the safety stock as
ss  z dlt  z d L
where z depends on the service level.
The value of z can be found using the standard normal distribution table for a given service
level. For example, for a given service Level = 95% we can find z=1.645 as shown below.
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(4) ROP with safety stock
As shown in the above figure, the ROP should be the sum of the expected demand during lead
time and additional safety stock that will prevent possible stock outs.
ROP  d L  ss
where
ss  z dlt  z d L .
NOTE:
 ROP = dL for the basic EOQ model
where d is a fixed constant and no safety stock is required.
 Recall that Q/2 is average inventory level for the basic EOQ model. By carrying the
safety stock, in this extended EOQ model
Average inventory level = Q/2 + ss
as a result,
Average annual holding cost = (Q/2+ss) * H
Example 1: EOQ with Variability
Suppose daily demand for a product is normally distributed with a mean of 75 units and a
variance of 49. The cost of placing an order is $5. Each item costs $2. The annual carrying cost
is 20% of the cost of the item. The lead time is 6 days. At least a 95% service level is desired.
(Assume demand occurs over 360 days/yr) Find:
a. The amount to order (EOQ)
b. Average demand during lead time
c. Standard deviation of demand during lead time
d. Safety stock (ss)
e. Reorder point (ROP)
Answer:
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