LESSON 12
INVENTORY MODELS (DETERMINISTIC)
FINITE PRODUCTION RATE MODEL
Outline
• Finite Production Rate Model
– The model
– Cost curves
– Characteristics
– Example
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Finite Production Rate
• Finite production rate model is an extension of the EOQ
model
• The following assumption of the EOQ model is not used in
the finite production rate model
– All the quantities ordered are received at the same time
(the assumption is suitable when the quantities are
purchased)
• An assumption in the finite production rate model
– The quantity ordered is produced at an uniform rate,P
(the assumption is suitable when the quantities are
produced)
• The finite production run model is also called the
Economic Production Quantity (EPQ) model
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Finite Production Rate
• As in the EOQ model, assume that the inventory at the
beginning is zero.
• The production facility is set up to produce Q units. If it’s
optimal to produce Q units in the beginning, it’s also
optimal to produce Q units next time when the inventory
level reaches zero, because no cost parameter changes in
between two production runs.
• Therefore, like the EOQ model, the finite production rate
model also has a constant order size and an inventory
cycle.
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Finite Production Rate
• The length of the inventory cycle is the length of time over
which the demand is Q . Thus, the length of cycle in years
Q
T

• In order to meet the demand (feasibility), the production
rate, P  . Thus, for some time after the start of a new
production run, the inventory level increases. The rate of
increase is P   . The length of each production run is
called the uptime and is denoted by T1 . The uptime is the
time required to produced order quantity, Q . So,
Q
T1 
P
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Finite Production Rate
• If P   the production must be stopped before the end of
the cycle in order to avoid producing more than the order
quantity, Q units. Thus, there is a idle time or downtime in
each cycle. The downtime is denoted by T2 .
Q Q
T2  T  T1  
 P
• Unlike the EOQ model, the maximum inventory in the finite
production rate model is not Q because some items are
used to meet the demand before the receipt of all of the
order quantity. The maximum inventory is the inventory
accumulated over the uptime. Since the inventory level
increases at the rate of P  , the maximum inventory,
Q
 
H  T1 P     P     Q1  
P
 P
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Inventory
Finite Production Rate
Slope = P - 
H
Slope = - 
T1
T2
Time
T
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Finite Production Rate
• Note that the downtime demand is met entirely from the
maximum inventory. Hence, downtime can also be
obtained as follows:
H Q  Q Q
T2 
 1    
   P  P
• The formula for the optimal order quantity is shown in the
next slide.
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Finite Production Rate
 = annual demand
P = the production rate in units/year
Q = size of each production run
K = cost of setting up production
h = annual holding cost per unit
h´ = h(1- /P)
Optimal solution, Q*, Economic Production Quantity (EPQ)
2 K

h'
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Finite Production Rate
At EPQ
h ' Q * h ' 2 K
K h '
Annual holding cost 


2
2
h'
2
K
Annual setup cost  * 
Q
K
K h '

2
2 K
h'
h ' Q * K
Total annual cost 
 *  2 K h '
2
Q
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The Finite Production Rate Model Cost Curves
Slope = 0
Annual
cost ($)
Total Cost
h' Q K

2
Q
h'Q
Holding Cost =
2
Minimum
total cost
K
Ordering Cost =
Q
Optimal solution, Q*, Economic Production Quantity (EPQ)
Order Quantity, Q
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Some Important Characteristics of the EPQ
Cost Function
• As in the EOQ model
– The annual holding cost is the same as the annual
setup cost at the EPQ
– The total cost curve is flat near the EPQ
• So, the total cost does not change much with a
slight change in the order quantity
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Example 2: Vision Optics makes microscope lens
housings. Annual demand is 100,000 units per year.
Assume that the product can be produced at the rate of
200,000 units per year. Each production run costs $5,000
to set up, and the variable production cost of each item is
$10. The annual cost per dollar value of holding items of
inventory is $0.20. Compute
a. Economic production quantity
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b. Cycle time
c. The length of each production run (uptime per cycle)
d. The length of downtime per cycle
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e. The maximum inventory
f. Total annual cost
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READING AND EXERCISES
Lesson 12
Reading:
Section 4.6 , pp. 211-213 (4th Ed.) pp. 202-204 (5th
Ed.)
Exercise:
17 and 19, pp. 213-214 (4th Ed.), p. 204 (5th Ed.)
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