1. Inventory Control Models
Part II
Applied Management Science for Decision Making, 1e © Pearson Prentice-Hall, Inc Philip A. Vaccaro , PhD

2. The Production Order Quantity Model
USED WHENEVER THE VENDOR
CANNOT DELIVER THE ORDER
( Q* ) ALL IN ONE DAY OR
USED WHENEVER THE FACTORY
CANNOT PRODUCE THE ORDER ( Q* ) ALL IN ONE DAY

3. Variable Interpretations
Service Sector
Manufacturing
P or p
The delivery rate of
purchased items
P or p
The production rate of
manufactured items

4. Variable Interpretations
Service Sector
Manufacturing
Qp*
Qp*
The optimal EOQ
when
purchased items
are
received in partial
shipments
The optimal EOQ
when
manufactured items
cannot
all be produced
in a single day

5. Production Order Quantity Model Cycle Chart
Graphically depicts the relationship between
Maximum Inventory Level ( IMAX )
Replenishment Rate ( P )
Consumption Rate ( D )
Time
Cycle Charts
enhance
understanding
of basic
inventory
concepts

6. Production Order Quantity Model Cycle Chart
SAW TOOTH VERSION
REPLENISHMENT
RATE
MAXIMUM INVENTORY LEVEL
CONSUMPTION
RATE ONLY
UNITS P D P D
CONSUMPTION OCCURS
EVEN AS
REPLENISHMENT
IS TAKING PLACE
AVERAGE INVENTORY LEVEL
ELAPSED TIME

7. Production Order Quantity Model Cycle Chart
SAW TOOTH VERSION
REPLENISHMENT
RATE
MAXIMUM INVENTORY LEVEL
CONSUMPTION
RATE ONLY
INVENTORY
LEVEL PEAKS
AT THIS POINT
UNITS P D P D
An EOQ of 100 units
is delivered at the rate of
20 units per week over
five weeks
AVERAGE INVENTORY LEVEL
INVENTORY
FALLS TO
ZERO OR
REORDER POINT
ELAPSED TIME

8. CYCLE CHART DISCUSSION
The replenishment rate ( P or p ) is diminished
by the consumption rate ( D or d ).
Average inventory will always be less than
[ Q* / 2 ] .
Average inventory is essentially [ IMAX / 2 ] .

9. Production Order Quantity Formula
THE FINITE CORRECTION FACTOR PRODUCES A LARGER
VALUE OF Q* IN ORDER TO COMPENSATE FOR PIECEMEAL
REPLENISHMENT AND CONSUMER DEMAND
2DS
H [ 1 – d / p ] √
Qp* =
THE CONSUMPTION OR
USAGE RATE ( D )
THE REPLENISHMENT OR
PRODUCTION RATE ( P )

10. Production Order Quantity Model EXAMPLE
DA = 1000 units ( annual demand )
S = $10.00 ( cost per order )
H = $.50 ( annual unit carry cost )
P = 8 units ( daily supply rate )
D = 6 units ( daily usage rate )
How many
units should
be ordered
at a time?
How long
to receive
the entire
order?

11. Production Order Quantity Model SOLUTION√
2(1000)(10.00)
.50 [ 1 – 6 / 8 ]
Qp* = √
20,000
.50 [ .25 ] √ = =
160,000 = 400 units

12. The Basic EOQ Model SOLUTION√
2(1000)(10.00)
.50
Q* = √ √
20,000
.50
40,000 = 200 units = =
Q* IS ONLY HALF OF WHAT IT WAS UNDER THE
PRODUCTION ORDER
QUANTITY MODEL

13. Production Order Quantity Model EXAMPLE
ORDER RECEIPT TIME PERIOD ( t )
Qp* P
t = =
= 50 days

14. Production Order Quantity Model EXAMPLE
TOTAL VARIABLE COSTS ( TVC )
Q*p D D
1 -
TVC =
X H +
X S X 2 P
Q*p
400 6
1,000 X
1 -
X .50 + X = 8 2
400
= [ 200 x (.25)(.50) ] + [ 2.5 x ]
= [ ] + [ ] = $50.00

15. Post Solution Comments
The Qp* must be larger than Q* since it is being
drawn down even as it arrives on a piecemeal basis.
The larger Qp* produces no increase in carry costs.
Annual carry costs are actually less than they are
under the basic EOQ model ( Q* ) .
“P” and “D” can be expressed as
daily, weekly, monthly, and annual
figures without changing the value
of the finite correction factor [1-d/p]

16. Production
Order
Quantity
Model

17. WE SELECT THE “INVENTORY”
MODULE

18. WE SELECT THE “PRODUCTION ORDER
QUANTITY MODEL”

19. THE DIALOGUE BOX
ALLOWS US TO
INSERT A TITLE
FOR THIS
PROBLEM

20. THE DATA INPUT TABLE
APPEARS

21. DAILY REPLENISHMENT RATE = 8 UNITS DAILY CONSUMPTION RATE = 6 UNITS
ANNUAL DEMAND = 1,000 UNITS
ORDER COST = $10.00
UNIT CARRY COST = $.50
DAILY REPLENISHMENT RATE = 8 UNITS
DAILY CONSUMPTION RATE = 6 UNITS

22. Optimal Order Quantity
Total Variable Costs

23. Total Variable Cost
Annual Carry Cost
Annual Order Cost

24. Production
Order
Quantity
Model

28. Template
and
Sample Data

30. Sensitivity Analysis
also shows the
optimal solution

31. The Backorder Inventory Model
USED WHENEVER THE
FIRM IS APPEALING
TO ITS CUSTOMERS
TO BUY AND THEN
WAIT FOR THEIR
PURCHASES UNTIL
A NEW SHIPMENT
( Q* ) IS ORDERED
AND RECEIVED

32. Backorder Model Variables
Backorder cost – B
Optimal number of backorders – S*
Optimal order quantity under a backordering
scenario – Qb*
Number of units going into stock after all
backorders have been filled – b or [ Qb* - S* ]

33. Backorder Model Cycle Chart
Optimal Order Quantity,
Optimal Number of Backorders,
Remaining Units Going Into
Stock after Backorders
Have Been Filled
Graphically depicts the relationship between
Cycle charts
enhance
understanding
of basic inventory
concepts

34. Backorder Model Cycle Chart PICKET FENCE VERSION
Q* or EOQ
b = Q*- S*
POSITIVE
INVENTORY
INITIAL
CYCLE
CYCLE 2
CYCLE 3
CYCLE 4
NEGATIVE
INVENTORY S*

35. Backorder Model Cycle Chart PICKET FENCE VERSION
88 UNITS GO
INTO INVENTORY AFTER THE FOUR STOCKOUTS
HAVE BEEN
FILLED
Q* or EOQ = 92 units
b = Q*- S* = 88 units
ASSUME AN
EOQ OF
92 UNITS
POSITIVE
INVENTORY
THIS MODEL
ASSUMES
THAT THE
NEW EOQ IS ORDERED WHEN
BACKORDERS
EQUAL FOUR
UNITS
INITIAL
CYCLE
CYCLE 2
CYCLE 3
CYCLE 4
THE FIRM ALLOWS BACKORDERS
TO REACH
4 UNITS
NEGATIVE
INVENTORY
S* = 4 UNITS

36. Backorder Model Formula
OPTIMAL ORDER QUANTITY
IN BACKORDER-TOLERATED
SITUATIONS
ANNUAL CARRY COST
PER UNIT H
UNIT BACKORDER
COST

37. Backorder Model Formula
THE
OPTIMAL
NUMBER
OF BACKORDERS
OPTIMAL ORDER QUANTITY UNDER
BACKORDERING
UNIT BACKORDER
COST

38. The Backorder Model EXAMPLE
DA = 500 units ( annual demand )
S = $4.00 ( order cost )
H = $ .50 ( annual unit carry cost )
B = $10.00 ( unit backorder cost )
How many
units should
be ordered?
What are the
number of
backorders?

39. The Backorder Model SOLUTION√
2(500)(4.00) ( )
Qb* = X √
= x 1.05 =
≈ 92 units √

40. The Backorder Example SOLUTION
.50
S* = x
= ( x )
≈ 4 units

41. Relationship Between ROP and S*
If leadtime ( L ) = 0 , then ROP = S*
IF LEADTIME IS ZERO, THE REORDER POINT OCCURS WHEN
THE OPTIMAL NUMBER OF BACKORDERS IS REACHED

42. Relationship Between ROP and S*
If leadtime ( L ) > 0, then ROP > S*
IF LEADTIME IS NOT ZERO, THE REORDER POINT OCCURS
BEFORE THE OPTIMAL NUMBER OF
BACKORDERS IS REACHED

43. Backorder Reorder Point EXAMPLE
Given: L = 0 days Qb* = 92 units S* = 4 units
Order 92 units when
the number of backorders
accumulate to 4

44. Backorder Reorder Point EXAMPLE
Given: d = 2 units, L = 6 days, Qb* = 92 units, S* = 4 units
Order 92 units when there are
8 units still left in the
account balance.
THE INITIAL ROP = d x L = [ 2 units x 6 days ] = 12 units
THE FINAL ROP = Initial ROP – S* = [ 12 units – 4 units ] = 8 units

45. ROP under Backordering
EXAMPLE WHERE L IS POSITIVE
Qb*
Qb*
POSITIVE
INVENTORY
REGION
FINAL ROP
FINAL ROP
12 – 4 = +8
12 – 4 = +8
INITIAL ROP = 12 UNITS
INITIAL ROP = 12 UNITS
NEGATIVE
INVENTORY
REGION
S* = 4 units ( - 4 )

46. MAY OR MAY NOT BE AVAILABLE ON YOUR PARTICULAR SOFTWARE
THE BACKORDER MODEL
MAY OR MAY NOT BE AVAILABLE ON YOUR PARTICULAR SOFTWARE

48. Inventory Control Part II
Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc Philip A. Vaccaro , PhD

49. The ABC Classification System
Its purpose is to assist inventory specialists
in establishing policies that focus their
limited resources on the relatively few
critical materials, components, and
products...and not the many trivial ones.
Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc Philip A. Vaccaro , PhD

50. ABC Classification System
RATIONALE
Purchasing personnel are relatively few in number
in the firm.
There are thousands of components, materials,
and finished good inventory accounts in medium
and large firms.
There needs to be a priority system for establishing
and updating inventory control doctrines ( Q* / R ).

51. Class “A” Inventory Items
Comprise only 15% of the
total items in stock yet
represent 70% - 80% of
the total dollar volume.
LARGE APPLIANCES
AUTOMOBILES
FURNITURE
DIAMONDS

52. Class “B” Inventory Items
Comprise 30% of the
total items in stock and
represent 15% - 25% of
the total dollar volume
MID-SIZED APPLIANCES
LAWN MOWERS
MOST SUITS & COATS

53. Class “C” Inventory Items
Comprise 55% of the
total items in stock yet
represent only 5% of
the total dollar volume
TOASTERS / BLENDERS
NUTS, BOLTS, SCREWS
STATIONERY SUPPLIES
MOST ACCESSORIES

54. Policies Based On ABC
“A” items would be inventoried in a more secure area

55. Policies Based On ABC
“A” items would be inventoried in a more secure area
“A” items would warrant more care in forecasting

56. Policies Based On ABC
“A” items would be inventoried in a more secure area
“A” items would warrant more care in forecasting
“A” item records would be verified more frequently

57. Policies Based On ABC
“A” items would be inventoried in a more secure area
“A” items would warrant more care in forecasting
“A” item records would be verified more frequently
“A” items would justify closer attention to customer
service levels

58. Policies Based On ABC
“A” items would be inventoried in a more secure area
“A” items would warrant more care in forecasting
“A” item records would be verified more frequently
“A” items would justify closer attention to customer
service levels
“A” items would qualify for real-time inventory tracking
systems and more sophisticated ordering rules

59. Additional Criteria for ABC
Anticipated engineering changes
Delivery problems
Quality problems
High unit production costs
WE FOCUS ON THE RELATIVELY FEW ITEMS WITH MAJOR PROBLEMS

60. ABC
Analysis

61. WE FIRST SCROLL TO
“INVENTORY”

62. WE THEN SELECT THE SUB MENU
“ABC Analysis”

63. THE DIALOG BOX
APPEARS

64. ITEMS THAT NEED TO BE CLASSIFIED
WE HAVE FIVE ( 5 )
ITEMS THAT NEED TO BE CLASSIFIED

65. THE DATA INPUT TABLE

66. FIVE ITEMS, TOGETHER WITH
WE DESIRE
20% OF ALL
ITEMS BE
“A” CATEGORY
WE DESIRE
30% OF ALL
ITEMS BE
“B” CATEGORY
FIVE ITEMS, TOGETHER WITH
THEIR ANNUAL DEMANDS,
AND UNIT VALUES

67. One “A” Item
One “B” Item
Three “C” Items

68. ABC
Analysis

72. Template
and
Sample Data

76. ABC Classification System
Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc Philip A. Vaccaro , PhD

77. Reorder Point Models Known Stockout Cost Model Service Level Model
Variable Demand / Constant
Lead Time Model
Constant Demand / Variable
Variable Demand / Variable
Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc Philip A. Vaccaro , PhD

78. Reorder Point Models The original reorder point formula ROP = d x L
computes the mean demand
during lead time, that is, the
average demand during the
waiting period for the item.
Where d = daily demand
L = lead time
( in days )

79. Reorder Point Models However, actual demand during
lead time can be much higher
than the mean (average) demand.
For this reason, the reorder point
should contain a built-in safety
stock ( SS or B ) that will meet
unexpectedly higher demands
and consequently reduce
stockout costs.

80. Reorder Point Models The reorder point formula now becomes
ROP = d x L + SS
But larger safety stocks
involve higher carry costs.
We must find a safety
stock that minimizes
both carry costs and
expected stockout
costs annually

81. Reorder Point Cost Tradeoff
KNOWN STOCKOUT COST MODEL
REORDER
POINT TOTAL
COSTS
ANNUAL
SAFETY
( BUFFER )
STOCK
CARRY
COSTS
Cost
THE
SELECTED
SAFETY
( BUFFER ) STOCK
LEVEL
( SS or B )
ANNUAL
EXPECTED
STOCKOUT COSTS ∞
Safety ( Buffer ) Stock in Units

82. Known Stockout Cost Model
ASSUMPTIONS
Stockout cost per unit is known.
Lead time is known and constant.
Lead time demand is variable.
IT IS ALSO ASSUMED THAT THE ANNUAL NUMBER
OF ORDERING PERIODS IS KNOWN ( n )

83. Demand During Lead Time ( DL )
EXAMPLE
Probability
15% % % % %
DEMAND IN UNITS
LEAD TIME DEMAND IS APPROXIMATELY NORMALLY DISTRIBUTED
AND RANGES BETWEEN THIRTY AND SEVENTY UNITS

84. Demand During Lead Time ( DL )
THE HIGHEST
PROBABILITY
Probability
15% % % % %
DEMAND IN UNITS
SINCE THERE IS A 30% CHANCE THAT LEAD TIME DEMAND WILL BE FIFTY ( 50 ) UNITS WE WOULD AT
LEAST SET THE REORDER POINT AT 50 UNITS. OTHERWISE, WE ARE EXTREMELY VULNERABLE TO
LARGE AND RECURRING STOCKOUTS.

85. Demand During Lead Time ( DL )
Probability
15% % % % %
SAFETY
STOCK IS
ZERO
HERE
DEMAND IN UNITS
THE SAFETY OR BUFFER STOCK LEVEL AT THE MINIMUM REORDER POINT IS ZERO BY DEFINITION

86. Reorder Point Carry Cost
ANNUAL UNIT
CARRY COST
SAFETY OR BUFFER
STOCK in units X
ANNUAL SAFETY STOCK
CARRY COSTS =

87. Reorder Point of 50 Units ‘0’ Safety Stock x $5.00 per unit = $0.00
ANNUAL SAFETY STOCK CARRY COSTS
‘0’ Safety Stock x $5.00 per unit = $0.00
per year
carry cost
( BUFFER STOCK )

88. CREATE A STOCKOUT OF 10 UNITS CREATE A STOCKOUT OF 20 UNITS
Reorder Point of 50 Units
EXPECTED STOCKOUTS PER ORDER PERIOD
EXPECTED STOCKOUTS
Probability
15% % % % % 10
Stockouts 20
Stockouts
Reorder
Point
Actual
Demand
A DEMAND OF 60 UNITS WOULD
CREATE A STOCKOUT OF 10 UNITS
WITH A 20% PROBABILITY
A DEMAND OF 70 UNITS WOULD
CREATE A STOCKOUT OF 20 UNITS
WITH A 10% PROBABILITY

89. Reorder Point Stockout Cost
Lead Time Expected
Stockouts in units X
Stockout Cost per unit
Annual Expected
Stockout Costs = X
Annual Number of
Lead Time Periods
( ordering periods )

90. NUMBER OF ORDER PERIODS
Reorder Point of 50 Units
ANNUAL EXPECTED STOCKOUT COSTS
STOCKOUT
COST
( per unit )
NUMBER OF ORDER PERIODS
( per year )
NUMBER OF
STOCKOUTS
EXPECTED
STOCKOUTS
COST
Lead Time
Demand
of 60 10 2
$40.00 6
$480.00
Lead Time
Demand
of 70 20 2
$40.00 6
$480.00
Σ = $960.00

91. Reorder Point Total Cost
ANNUAL SAFETY STOCK
CARRY COSTS +
ANNUAL EXPECTED
STOCKOUT COSTS
THE REORDER POINT THAT HAS THE LOWEST TOTAL COST IS SELECTED

92. Reorder Point Score Board
ANNUAL CARRY COSTS
ANNUAL STOCKOUT COSTS
TOTAL COSTS 50
$0.00
$960.00

93. RAISING THE REORDER POINT TO 60 UNITS AUTOMATICALLY
Reorder Point of 60 Units
ANNUAL SAFETY STOCK CARRY COSTS
‘10’ Safety Stock x $5.00 per unit = $50.00
per year
carry cost
( BUFFER STOCK )
RAISING THE REORDER POINT
TO 60 UNITS AUTOMATICALLY
CREATES A SAFETY STOCK
OF TEN UNITS

94. CREATE A STOCKOUT OF 10 UNITS
Reorder Point of 60 Units
EXPECTED STOCKOUTS PER ORDER PERIOD
EXPECTED
STOCKOUT 1
Probability
15% % % % % 10
Stockouts
Reorder
Point
Actual
Demand
A DEMAND OF 70 UNITS WOULD
CREATE A STOCKOUT OF 10 UNITS
WITH A 10% PROBABILITY

95. NUMBER OF ORDER PERIODS
Reorder Point of 60 Units
ANNUAL EXPECTED STOCKOUT COSTS
STOCKOUT
COST
( per unit )
NUMBER OF ORDER PERIODS
( per year )
NUMBER OF
STOCKOUTS
EXPECTED
STOCKOUTS
COST 10 1
$40.00 6
$240.00
LEAD TIME DEMAND
of 70
Σ = $240.00

96. Reorder Point Score Board
ANNUAL
CARRY COSTS
STOCKOUT COSTS
TOTAL COSTS
50 Units
$0.00
$960.00
60 Units
$50.00
$240.00
$290.00

97. RAISING THE REORDER POINT TO 70 UNITS AUTOMATICALLY
Reorder Point of 70 Units
ANNUAL CARRY COSTS
‘20’ Safety Stock x $5.00 per unit = $100.00
per year
carry cost
( BUFFER STOCK )
RAISING THE REORDER POINT
TO 70 UNITS AUTOMATICALLY
CREATES A SAFETY STOCK
OF 20 UNITS

98. Reorder Point of 70 Units 15% 25% 30% 20% 10% 30 40 50 60 70 EXPECTED
EXPECTED STOCKOUTS PER ORDER PERIOD
EXPECTED
STOCKOUTS
ZERO
Probability
15% % % % %
Stockouts
Reorder
Point
BASED ON HISTORICAL DEMAND DATA,
NO DEMAND SHOULD OCCUR THAT CREATES A STOCKOUT

99. NUMBER OF ORDER PERIODS
Reorder Point of 70 Units
ANNUAL EXPECTED STOCKOUT COSTS
STOCKOUT
COST
( per unit )
NUMBER OF
STOCKOUTS
NUMBER OF ORDER PERIODS
( per year )
EXPECTED
STOCKOUTS
COST
$40.00 6
$0.00
LEAD TIME DEMAND
of 70
Σ = $0.00

100. Reorder Point Score Board
ANNUAL
CARRY COSTS
STOCKOUT COSTS
TOTAL COSTS
50 Units
$0.00
$960.00
60 Units
$50.00
$240.00
$290.00
70 Units
$100.00

101. The Conclusion
The lowest cost option:
ROP = 70 units
SS = 20 units

102. Known
Stockout
Cost
Model

103. TO FIND THE KNOWN STOCKOUT COST
WE SCROLL TO
“INVENTORY”
TO FIND THE KNOWN STOCKOUT COST
REORDER POINT MODEL

104. WE SELECT THE “Reorder Point / Safety Stock” ( Discrete Distribution )
SUB MENU

105. THE DIALOG BOX
APPEARS

106. DISCRETE DEMANDS DURING LEAD TIME
WE HAVE FIVE ( 5 )
DISCRETE DEMANDS DURING LEAD TIME

107. THE DATA INPUT TABLE

108. THE COMPLETED DATA INPUT TABLE

109. THE KNOWN STOCKOUT COST
Reorder Point = 50 UNITS
Reorder Point = 60 UNITS
Reorder Point = 70 UNITS
THE KNOWN STOCKOUT COST
REORDER POINT MODEL
Reorder Point = 70 UNITS
Safety Stock = 20 UNITS

110. Known
Stockout
Cost
Model

114. Template and Sample Data
Insert Selected
Reorder Point, etc.
This is the conditional payoff matrix for this problem. The strategies
are the safety stocks, the events are the five possible demands, and
the conditional payoffs are the cost consequences (expected
stockouts + carry costs) of selecting a particular safety stock
and a certain demand materializing.

115. 5 discrete demand possibilities during any
given lead time ( with probabilities )
Selected Reorder Point
EMV Criterion Maxi-Min Criterion
Conditional Payoffs
For ROP = 50 , total cost = $ , H = $
For ROP = 60 , total cost = $ , H = $
For ROP = 70 , total cost = $ , H = $100.00
Safety stocks associated
with all 5 discrete demands

116. Service Level Model When the stockout cost per unit ( Cs ) is
REORDER POINT DETERMINATION
When the stockout cost per unit ( Cs ) is
difficult or impossible to determine, the
firm may elect to establish a policy of
keeping enough safety stock on hand to
satisfy a prescribed level of customer
service.
FOR EXAMPLE, THE FIRM MAY DESIRE A SERVICE LEVEL
THAT MEETS 95% OF THE DEMAND, OR CONVERSELY,
RESULTS IN STOCKOUTS ONLY 5% OF THE TIME.

117. Service Level Model Assuming that the demand during lead time
REORDER POINT DETERMINATION
Assuming that the demand during lead time
( DL ) follows a normal distribution, only the
mean ( µ ) and standard deviation ( σ ) are
required to define the reorder point as well
as the safety stock ( SS or B ) for any given
service level.
SALES DATA ARE USUALLY ADEQUATE FOR COMPUTING
THE MEAN AND STANDARD DEVIATION

118. Service Level Model
REORDER POINT DETERMINATION EXAMPLE
A firm stocks an item that has a normally-distributed
demand during the lead time period. The average or
mean demand during the lead time period is 400 units
and the standard deviation is 15 units. The firm wants
a reordering policy that limits stockouts to only 5% of
the time.
Requirement:
1. How much safety stock should the firm maintain?
2. What should be the reorder point?

119. Service Level Model Z AREAS UNDER THE STANDARD NORMAL CURVE .03 .04
.05
.06
.07
1.4
.92364
.92507
.92647
.92785
.92922
1.5
.93699
.93822
.93943
.94062
.94179
1.6
.94845
.94950
.95053
.95154
.95254
1.7
.95818
.95907
.95994
.96080
.96164 Z
95% SERVICE LEVEL IS REPRESENTED BY z = 1.65 standard normal deviates

121. Service Level Model μ ROP Z = + 1.65 0 units
REORDER POINT AND SAFETY STOCK FOR 95% SERVICE LEVEL
Z = 5%
STOCKOUT
PROBABILITY
95% STOCKAGE PROBABILITY
SAFETY STOCK
0 units μ
ROP
LEAD TIME MEAN DEMAND = 400 UNITS

122. Service Level Model Reorder Point ( R ) = μ + ( z )( σ )
FORMULAE
Reorder Point ( R ) = μ + ( z )( σ )
Safety Stock ( SS ) = ( z )( σ )
μ = MEAN DEMAND DURING LEAD TIME
z = NUMBER OF STANDARD NORMAL DEVIATES
σ = STANDARD DEVIATION OF LEAD TIME DEMAND

123. Service Level Model Reorder Point ( R ) = 400 + ( 1.65 )( 15 )
EXAMPLE – 95% SERVICE LEVEL
Reorder Point ( R ) = ( 1.65 )( 15 )
= 425 units
Safety Stock ( SS ) = ( 1.65 )( 15 )
= 25 units
μ = 400 units
z = 1.65 ( 95% service level )
σ = 15 units

124. Service Level Model Reorder Point ( R ) = 400 + ( 2.33 )( 15 )
EXAMPLE – 99% SERVICE LEVEL
Reorder Point ( R ) = ( 2.33 )( 15 )
= 435 units
Safety Stock ( SS ) = ( 2.33 )( 15 )
= 35 units
μ = 400 units
z = 2.33 ( 99% service level )
σ = 15 units Z
.03

126. Service
Level
Reorder
Point
Model

127. TO FIND THE SERVICE LEVEL REORDER POINT, SCROLL TO
“ INVENTORY “

128. “ Reorder Point / Safety Stock “ ( Normal Distribution )
SELECT THE SUB MENU
ENTITLED
“ Reorder Point / Safety Stock “
( Normal Distribution )

129. THE DIALOG BOX APPEARS

130. IF THE DAILY DEMAND AND/OR THEN THEIR STANDARD DEVIATION(S) = 0
THE DATA INPUT TABLE REQUIRES
DAILY DEMAND DURING LEAD TIME
THE STANDARD DEVIATION OF DAILY
DEMAND DURING LEAD TIME
THE SERVICE LEVEL DESIRED ( i.e. 95% )
LEAD TIME IN DAYS
THE STANDARD DEVIATION OF LEAD TIME
IF THE DAILY DEMAND AND/OR
LEADTIME ARE CONSTANT,
THEN THEIR STANDARD DEVIATION(S) = 0

131. The mean demand during lead time was given as 400 units.
Since this model requires both daily demand during the
lead time, and the lead time ( in days ), we will assume
here that daily demand = 50 units, and lead time = 8 days.

132. THE 95% “SERVICE LEVEL” REORDER POINT = 423 UNITS
THE 95% “SERVICE LEVEL” SAFETY STOCK = 23 UNITS

133. Service
Level
Reorder
Point
Model

136. Templates for three ( 3 ) different ( we choose the first model )
reorder point models.
( we choose the first model )

137. The Reorder Point = = 425 !

138. Stochastic Reorder Point Models
WHEN DEMAND AND / OR LEAD TIME ARE VARIABLE
These models generally assume that any variability in
either the demand rate or lead time can be adequately
described by a normal distribution. This, however, is
not a strict requirement.
These models will provide approximate reorder points
even in cases where the actual probability distributions
depart substantially from normal.
In all models shown, stockout costs are assumed to be
unknown.

139. Stochastic Reorder Point Models
MODELS CONSIDERED IN THIS PRESENTATION
variable demand rate / constant lead time
constant demand rate / variable lead time
variable demand rate / variable lead time

140. Stochastic Reorder Point Models
THE VARIABLES
d = constant demand rate
d = average demand rate
L = constant lead time
L = average lead time
σD = standard deviation of demand rate
σL = standard deviation of lead time

141. Stochastic Reorder Point Models
LEAD TIME IS CONSTANT
DEMAND RATE IS VARIABLE
Jack’s Pizza Parlor uses 1,000 cans of tomatoes per month at an
average rate of 40 per day for each of 25 days per month. Usage
can be approximated by a normal distribution with a standard de-
viation of 3 cans per day. Lead time is constant at 4 days. Jack
desires a service level of 99% , that is, a stockout risk of only 1%
Requirement:
1. Determine the reorder point ( ROP )
2. Determine the safety or buffer stock ( SS or B )

142. Reorder Point Solution
PIZZA PARLOR _
ROP = ( d x L ) + ( z ) ( L ) ( σD )
= ( 40 x 4 ) ( 4 ) ( 3 )
= ( 6 ) =
= 174 cans
Given:
d = 40 cans daily
σD = 3 cans daily
L = 4 days _

143. Safety Stock Solution SS or B = ( z ) ( L ) ( σD ) = 2.33 ( 4 ) ( 3 )
PIZZA PARLOR
SS or B = ( z ) ( L ) ( σD )
= 2.33 ( 4 ) ( 3 )
= 2.33 ( 6 )
≈ 14 cans

144. Reorder
Point
Where
Lead Time Is
Constant
Demand Rate
Variable

147. Template
for
pizza parlor
reorder point

148. Reorder point = 174
Safety stock = 14

149. Stochastic Reorder Point Models
LEAD TIME IS VARIABLE
DEMAND RATE IS CONSTANT
An oil-driven generator uses 2.1 gallons per day. Lead time is
normally distributed with a mean of 6 days.
The standard deviation of lead time is 2 days. The service level
is 98%, that is, the stockout risk is 2%
Requirement:
1. Determine the reorder point ( ROP )
2. Determine the safety or buffer stock ( SS or B )

150. Reorder Point Solution
THE GENERATOR _
ROP = ( d x L ) + ( z ) ( d ) ( σL )
= ( 2.1 x 6 ) ( 2.1 ) ( 2 )
= ( 4.2 ) =
= gallons
Given:
d = 2.1 gallons daily
σL = 2 days
L = 6 days _

151. Safety Stock Solution SS or B = ( z ) ( d ) ( σL )
THE GENERATOR
SS or B = ( z ) ( d ) ( σL )
= ( 2.1 ) ( 2 )
= ( 4.2 )
= 8.63 gallons

152. Reorder
Point
Where
Lead Time Is
Variable
Demand Rate
Constant

156. Stochastic Reorder Point Models
LEAD TIME IS VARIABLE
DEMAND RATE IS VARIABLE
Beer consumption at a local tavern is known to be normally
distributed with a mean of 150 bottles daily and a standard
deviation of 10 bottles daily. Delivery time is also normally
distributed with a mean of 6 days and a standard deviation
of 1 day. The service level is 90%
Requirement:
1. Determine the reorder point ( ROP )
2. Determine the safety or buffer stock ( SS or B )

157. Reorder Point Solution
TAVERN EXAMPLE _
Given: d = 150 bottles daily
σD = 10 bottles daily
L = 6 days
σL = 1 day _

158. Reorder Point Solution
TAVERN EXAMPLE _ _ _ _ 2 2 2
ROP = ( d x L ) + ( z ) ( L )( σD ) + ( d ) ( σL ) 2 2 2
= ( 150 x 6 ) + ( 1.28 ) (6)(10) + (150) (1)
= ,500
= ( ) ≈ 1,095 bottles

159. Safety Stock Solution SS or B = ( z ) ( L )( σD ) + ( d ) ( σL )
TAVERN EXAMPLE _ _ 2 2 2
SS or B = ( z ) ( L )( σD ) + ( d ) ( σL ) 2 2 2
= (1.28 ) (6)(10) + (150) (1)
= ,500
= 1.28 ( ) ≈ 195 bottles

160. Stochastic Reorder Point Model
Demand
and
Lead Time
Variable

161. TO SOLVE A STOCHASTIC REORDER POINT
AND
SAFETY STOCK
PROBLEM

162. THE DIALOG BOX
APPEARS

163. REQUIRES DAILY DEMAND AND STANDARD DEVIATION AND STANDARD DEVIATION
THE DATA INPUT TABLE
REQUIRES DAILY DEMAND AND STANDARD DEVIATION
( if any )
LEAD TIME ( DAYS )
AND STANDARD DEVIATION

165. STOCHASTIC MODEL REORDER POINT = 1,095 UNITS
SAFETY STOCK = 195 UNITS
THE LOCAL TAVERN

166. Stochastic Reorder Point Model
Demand
Variable
Lead Time

170. Reorder Point Models Inventory Control
Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc Philip A. Vaccaro , PhD

172. Quantity Discount Model
Solved Problem
Quantity Discount Model
Ivonne Callen
Computer-Based
Manual

173. Ivonne Callen Problem
Ivonne Callen sells beauty supplies. Her annual demand
for a particular skin lotion is 1,000 units.
The cost of placing an order is $20.00, while the holding
cost per unit per year is 10 percent of the cost.
The item currently costs $10.00 if the order quantity is
less than 300. For orders of 300 units or more, the cost
falls to $9.80.
To minimize total cost, how many units should Ivonne
order each time she places an order?
What is the minimum total cost?

174. Ivonne Callen Problem D = 1,000 units S = $20.00 I = 10%
P = $10.00 if Q < 300 units
P = $ if Q > 300 units

175. Ivonne Callen Problem To minimize total costs, how many units
should Ivonne order each time she places
an order ?
What is the minimum total cost ?

176. √ √ √ EOQ at $10.00 Item Cost 2 (1,000 ) ( 20 ) 2 x D x S
( .10 ) ( )
2 x D x S
I x P √ =
Q*1 = √
40,000
1.00
= 200 units =

177. Total Cost at $10.00 Item Cost
TC = [ Q* / 2 ] (I)(P) + [ D / Q* ] (S) + [ D x P ]
= [ 200 / 2 ] (.10)(10.00) + [ 1,000 / 200 ] (20.00) + [1,000 x 10.00]
= $ $ $10,000.00
= $10,200.00

178. Total Cost at $9.80 Item Cost
TC = [ Q* / 2 ] (I)(P) + [ D / Q* ] (S) + [ D x P ]
= [ 300 / 2 ] (.10)(9.80) + [ 1,000 / 300 ] (20.00) + [1,000 x 9.80]
= $ $ $9,800.00
= $10,013.67

179. Ivonne Callen Problem Therefore, she should order 300 units.
CONCLUSION
Therefore, she should
order 300 units.

180. Serial Rate Production Model The Handy Manufacturing Company
Solved Problem
Serial Rate Production Model
The Handy Manufacturing Company
Computer-Based
Manual

181. The Handy Mfg. Company
The Handy Manufacturing Company manufactures small
air conditioner compressors. The estimated demand for
the year is 12,000 units. The setup cost for the production
process is $ per run, and the carrying cost is $10.00
per unit per year. The daily production rate is 100 units per
day, and demand has been 50 units per day.
Determine the number of units to produce in each batch,
the number of batches that should be run each year, and
the time interval, in days, between each batch.
( Assume 240 operating days. )

182. The Data D = 12,000 units S = $200.00 H = $10.00 / unit / year
p = 100 units / day
d = 50 units / day
240 working days / year
Use
Production Order
Quantity Model

183. √ √ Production Run Model H ( 1 - d / p ) Qp* = 2 (12,000)(200.00) =
2 (D)(S)
H ( 1 - d / p )
Qp* = √
2 (12,000)(200.00)
(10.00)( / 100 ) =

184. √ √ √ Production Run Model (10.00)(.5) Qp* = 4,800,000 5 = =
960,000 = 979.8
units

185. Batches Run Annually n = D / Q*p n = 12,000 / 980
n ≈ 12 production runs

186. Time Between Runs
t = 240 days / n
= 240 days / 12
= every 20 days

188. Solved Problem
Reorder Point Model
Mr. Beautiful
Computer-Based
Manual

189. Mr. Beautiful Problem
Mr. Beautiful, an organization that sells weight training sets, has
an ordering cost of $40.00 for the BB-1 set. The carry cost for the
BB-1 set is $5.00 per set per year.
To meet demand, Mr. Beautiful orders large quantities of BB-1 (7)
seven times a year. The stockout cost for BB-1 is estimated to be
$50.00 per set. Over the past several years, Mr. Beautiful has ob-
served the following demand during lead time for BB-1:
Lead Time Demand Probability
Starting with a ROP of 60 units, what level of safety stock
should be maintained for BB-1?

190. Mr. Beautiful Problem ROP 40 50 60 70 80 90 10 sets 20 sets 30 sets
10 sets
20 sets
30 sets
Expected Stockouts
[.20 x 10] + [.20 x 20] + [.10 x 30] = 9 sets

191. Mr. Beautiful Problem Carry Cost / Set / Year = $5.00
Stockout Cost / Set = $50.00
Order Periods / Year = 7
At ROP = 60 sets , SS = 0 sets
Stockout Cost:
[.20 x 10 sets x 20 sets x 30 sets] x 7 orders x $50.00
[ ] x 7 x $50.00 = $3,150.00
Carrying Cost:
SS = 0 sets x $5.00 / set / year = $0.00
TOTAL COST = $3,150.00

192. Mr. Beautiful Problem ROP 40 50 60 70 80 90 10 sets 20 sets
10 sets
20 sets
Expected Stockouts
[.20 x 10] + [.10 x 20] = 4 sets

193. Mr. Beautiful Problem Carry Cost / Set / Year = $5.00
Stockout Cost / Set = $50.00
Order Periods / Year = 7
At ROP = 70 sets , SS = 10 sets
Stockout Cost:
[.20 x 10 sets x 20 sets] x 7 orders x $50.00
[ ] x 7 x $50.00 = $1,400.00
Carrying Cost:
SS = 10 sets x $5.00 / set / year = $50.00
TOTAL COST = $1,450.00

194. Mr. Beautiful Problem ROP 40 50 60 70 80 90 10 sets
10 sets
Expected Stockouts
[.10 x 10] = 1 set

195. Mr. Beautiful Problem Carry Cost / Set / Year = $5.00
Stockout Cost / Set = $50.00
Order Periods / Year = 7
At ROP = 80 sets , SS = 20 sets
Stockout Cost:
[.10 x 10 sets] x 7 orders x $50.00
[ 1 ] x 7 x $50.00 = $350.00
Carrying Cost:
SS = 20 sets x $5.00 / set / year = $100.00
TOTAL COST = $450.00

196. Mr. Beautiful Problem ROP 40 50 60 70 80 90 .10 .20 .20 .20 .20 .10
Expected Stockouts
[ .00 x 0 ] = 0 sets

197. Mr. Beautiful Problem Carry Cost / Set / Year = $5.00
Stockout Cost / Set = $50.00
Order Periods / Year = 7
At ROP = 90 sets , SS = 30 sets
Stockout Cost:
[.0 x 0 sets] x 7 orders x $50.00
[ 0 ] x 7 x $50.00 = $0.00
Carrying Cost:
SS = 30 sets x $5.00 / set / year = $150.00
TOTAL COST = $150.00

198. Mr. Beautiful Problem Reorder Point Stockout Cost Carry Total 60 sets
$3,150.00
$0.00
70 sets
$1,400.00
$50.00
$1,450.00
80 sets
$350.00
$100.00
$450.00
90 sets
$150.00
Mr. Beautiful should select a ROP = 90 sets with a SS = 30 sets

199. We Scroll Down To
INVENTORY

200. “Reorder Point / Safety Stock” ( Discrete Distribution )
WE SELECT THE SUB MENU
“Reorder Point / Safety Stock”
( Discrete Distribution )

201. THE DIALOG BOX
APPEARS

202. THERE ARE SIX ( 6 ) DEMANDS THAT CAN OCCUR DURING LEADTIME

203. PROBABILITY DISTRIBUTION OF
THE DATA INPUT TABLE REQUIRES:
STARTING REORDER POINT WHERE
THE SAFETY STOCK IS ZERO ( 0 )
CARRY COST PER UNIT PER YEAR
STOCKOUT COST PER UNIT PER YR
NUMBER OF ORDERS PER YEAR ( 7 )
THE DISCRETE
PROBABILITY DISTRIBUTION OF
LEAD TIME DEMAND

204. Expected Stockout Cost = $0.00 Carry ( Holding ) Costs = $150.00
THE ROP = 90 SETS
THE SS = 30 SETS
Expected Stockout Cost = $0.00
Carry ( Holding ) Costs = $150.00