Inventory Policy
Decisions
“Every management mistake ends up in inventory.”
Michael C. Bergerac
Former Chief Executive
Revlon, Inc.
Chapter 9
CR (2004) Prentice Hall, Inc.
9-1
CONTROLLING
Customer
service goals
• The product
• Logistics service
• Ord. proc. & info. sys.
Transport Strategy
• Transport fundamentals
• Transport decisions
PLANNING
Inventory Strategy
• Forecasting
• Inventory decisions
• Purchasing and supply
scheduling decisions
• Storage fundamentals
• Storage decisions
ORGANIZING
Inventory Decisions in
Strategy
Location Strategy
• Location decisions
• The network planning process
CR (2004) Prentice Hall, Inc.
9-2
What are Inventories?
•Finished product held for sale
•Goods in warehouses
•Work in process
•Goods in transit
•Staff hired to meet service needs
•Any owned or financially controlled
raw material, work in process, and/or
finished good or service held in
anticipation of a sale but not yet sold
CR (2004) Prentice Hall, Inc.
9-3
Where are Inventories?
Inbound
transportation
Production
Outbound
transportation
Finished goods
warehousing
Customers
Receiving
Material
sources
Production
materials
Finished goods
Shipping
Inventories
in-process
Inventory
locations
CR (2004) Prentice Hall, Inc.
9-4
Reasons for Inventories
•Improve customer service
-Provides immediacy in product availability
•Encourage production, purchase, and transportation
economies
-Allows for long production runs
-Takes advantage of price-quantity discounts
-Allows for transport economies from larger shipment sizes
•Act as a hedge against price changes
-Allows purchasing to take place under most favorable price
terms
•Protect against uncertainties in demand and lead times
-Provides a measure of safety to keep operations
running when demand levels and lead times cannot be known
for sure
•Act as a hedge against contingencies
-Buffers against such events as strikes, fires, and
disruptions in supply
CR (2004) Prentice Hall, Inc.
9-5
Reasons Against Inventories
•They consume capital resources that might be put to
better use elsewhere in the firm
•They too often mask quality problems that would more
immediately be solved without their presence
•They divert management’s attention away from careful
planning and control of the supply and distribution
channels by promoting an insular attitude about
channel management
CR (2004) Prentice Hall, Inc.
9-6
Types of Inventories
•Pipeline
-Inventories in transit
•Speculative
-Goods purchased in anticipation of price increases
•Regular/Cyclical/Seasonal
-Inventories held to meet normal operating needs
•Safety
-Extra stocks held in anticipation of demand and
lead time uncertainties
•Obsolete/Dead Stock
-Inventories that are of little or no value due to being
out of date, spoiled, damaged, etc.
9-7
Nature of Demand
•Perpetual demand
-Continues well into the foreseeable future
•Seasonal demand
-Varies with regular peaks and valleys throughout
the year
Accurately forecasting
demand is singly the
•Lumpy demand
most important factor
-Highly variable (3  Mean)
in good inventory
•Regular demand
management
-Not highly variable (3 < Mean)
•Terminating demand
-Demand goes to 0 in foreseeable future
•Derived demand
-Demand is determined from the demand of another
item of which it is a part
9-8
Inventory Management
Philosophies
•Pull
-Draws inventory into the stocking location
-Each stocking location is considered independent
-Maximizes local control of inventories
•Push
-Allocates production to stocking locations based on
overall demand
-Encourages economies of scale in production
•Just-in-time
-Attempts to synchronize stock flows so as to just
meet demand as it occurs
-Minimizes the need for inventory
CR (2004) Prentice Hall, Inc.
9-9
Inventory Management
Philosophies (Cont’d)
•Supply-Driven
-Supply quantities and timing are unknown
-All supply must be accepted and processed
-Inventories are controlled through demand
•Aggregate Control
-Classification of items:
›Groups items according to their sales level
based on the 80-20 principle
›Allows different control policies for 3 or more
broad product groups
CR (2004) Prentice Hall, Inc.
9-10
Pull vs. Push Inventory Philosophies
PUSH - Allocate supply to each
warehouse based on the forecast
for each warehouse
PULL - Replenish inventory with
order sizes based on specific needs
of each warehouse
Demand
forecast
Warehouse #1
Q1
A1
A2
Q2
Plant
Warehouse #2
A3
Demand
forecast
Q3
A = Allocation quantity to each warehouse
Q = Requested replenishment quantity
by each warehouse
CR (2004) Prentice Hall, Inc.
Warehouse #3
Demand
forecast
9-11
Costs Relevant to Inventory
Management
•Carrying costs
-Cost for holding the inventory over time
-The primary cost is the cost of money tied up in
inventory, but also includes obsolescence,
insurance, personal property taxes, and storage
costs
-Typically, costs range from the cost of short term
capital to about 40%/year. The average is about
25%/year of the item value in inventory.
CR (2004) Prentice Hall, Inc.
9-12
Relevant Costs (Cont’d)
•Procurement costs
-Cost of preparing the order
-Cost of order transmission
-Cost of production setup if appropriate
-Cost of materials handling or processing at the
receiving dock
-Price of the goods
CR (2004) Prentice Hall, Inc.
9-13
Relevant Costs (Cont’d)
•Out-of-stock costs
-Lost sales cost
›Profit immediately foregone
›Future profits foregone through loss of goodwill
-Backorder cost
›Costs of extra order handling
›Additional transportation and handling costs
›Possibly additional setup costs
CR (2004) Prentice Hall, Inc.
9-14
Inventory Management Objectives
Good inventory management is a careful balancing act
between stock availability and the cost of holding
inventory.
Customer Service,
i.e., Stock Availability
Inventory Holding costs
•Service objectives
-Setting stocking levels so that there is only a
specified probability of running out of stock
•Cost objectives
-Balancing conflicting costs to find the most
economical replenishment quantities and timing
CR (2004) Prentice Hall, Inc.
9-15
Inventory’s Conflicting Cost Patterns
Minimum cost
reorder quantity
Cost
Total cost
Procurement cost
Stockout cost
CR (2004) Prentice Hall, Inc.
Replenishment quantity
9-16
Glossary of Terms
D = average annual demand, units
d = average period demand, units
S = procurement cost per order, $/order
I = carrying costs as a percent of product value, % per year
C = product value, $ per unit
sd = standard deviation of demand (d), units
k = out - of - stock cost, $ per unit
p = purchase price
s ' = standard deviation of compound demand distribution
E( z ) = partial expectation or unit normal loss integral
P = probability of being in - stock during lead time (Q - system)
or during lead time plus order cycle time (P - system
Q = order quantity
ROP = reorder point quantity, units
T = order interval, e.g., days
MAX = target inventory level, units
z = normal deviate or number of standard deviations from mean
on compound demand distribution
r = safety stock, or z x s' , units
TC = total relevant cost, $
SL = service level as a percent of total annual demand
LT , sLT = average and standard deviation of lead time
CPn = probability of n units being sold
9-17
Single Order Purchasing
Make a one-time purchase of an item. How much to order?
Procedure: Balance incremental profit against incremental loss.
Profit = Price per unit  Cost per unit
Loss = Cost per unit  Salvage value per unit
If CPn is probability of n units being sold, then
CPn x Loss = (1  CPn) x Profit
or
CPn = Profit/(Profit + Loss)
Daily stocking of
newspapers in
vending
machines is a
good example
Now, increase order quantity until CPn just matches cumulative
probability of selling additional units.
CR (2004) Prentice Hall, Inc.
9-18
Single Order Purchasing (Cont’d)
Example A clothing item is purchased for a seasonal sale. It
costs $35, but it has a sale price of $50. After the season is
over, it is marked down by 50% to clear the merchandise.
The estimated quantities to be sold are:
Probability of
Number of selling exactly n
items, n
items
10
0.15
15
0.20
20
0.30
25
0.20
30
0.10
35
0.05
1.00
CR (2004) Prentice Hall, Inc.
Cumulative
probability
0.15
0.35
0.65
0.85
0.95
1.00
9-19
Single Order Purchasing (Cont’d)
Solution
Profit = $50 35 = $15
Loss = $35  (0.5)(50) = $10
CPn = 15/(15 + 10) = 0.60
CPn is between 15 and 20 items, round up and order 20
items.
CR (2004) Prentice Hall, Inc.
9-20
Simple Two-Bin Pull Method
Given:
d = 50 units/week
I = 10%/year
S = $10/order
C = $5/unit
LT = 3 weeks
Note: No uncertainty
in demand or lead
time—manage
regular (cycle) stock
only
Develop a simple control system by finding the
replenishment quantity (Q) and the reorder point
(ROP).
The relevant total cost is:
TC = ordering cost  carrying costs
= DS  ICQ
2
Q
9-21
Reorder Point Method Under Certainty
for a Single Item
Quantity on-hand
plus on-order
Q
Reorder
point, R
0
CR (2004) Prentice Hall, Inc.
Lead
time
Order
Order
Placed Received
Lead
Time
time
Order
Order
Placed
Received
9-22
Two-Bin Method (Cont’d)
Using differential calculus, the optimal value for Q will
be:
Q* = 2DS/IC = 2(50x52)(10)/(0.10x5) = 322 units
The reorder point is:
Famous EOQ
formula
ROP = d(LT) = 3(50) = 150 units
Rule When the inventory level drops to 150 units
(ROP) then reorder 322 units (Q*).
CR (2004) Prentice Hall, Inc.
9-23
Reorder Point Control with
Demand Uncertainty
Given:
d = 50 units/week
sd = 10 units/week
I = 10%/year
S = $10/order
C = $5/unit
LT = 3 weeks
P = 99% during lead time
Find Q* and ROP
From the EOQ formula
Q* = 2(50x52)(10) = 322 units
0.10(5)
CR (2004) Prentice Hall, Inc.
Good method for products:
1. Of high value
2. That are purchased from
one vendor or plant
3. Having few economies of
scale in production,
purchasing, or
transportation
9-24
Quantity on hand
Reorder Point Control for a Single Item
Q
Place
order
Q
DDLT
ROP
Receive
order
0
P
Stockout
LT
LT
Time
CR (2004) Prentice Hall, Inc.
9-25
Reorder Point Control for a Single Item
Quantity on hand
+on order
backorders
Inventory level
Quantity for
control
Actual
on hand
Q
ROP
Safety stock
0
LT
CR (2004) Prentice Hall, Inc.
Time
LT
9-26
Reorder Point Control (Cont’d)
Finding the reorder point requires an understanding of
the demand-during-lead-time distribution
DDLT
P
Week 1
Week 2
+
Week 3
=
+
sd=10
sd=10
sd=10
d =100
d =100
d =100
Weekly demand is normally distributed
with a mean of d = 100 and a standard
deviation of sd = 10
Lead time is 3 weeks
S’=17.3
z
X = 300 ROP
X = d  LT = 100(3) = 300
s ' = sd LT = 10 3 = 17.3
9-27
Reorder Point Control (Cont’d)
Now,
X = d(LT ) = 50(3) = 150 units
s' = sd LT = 10 3 = 17.32 units
Hence,
ROP = X  zs' = X  r
= 150  2.33(17.32) = 190 units
where 2.33 is the normal deviate at a probability of
0.01 taken from a normal distribution table.
CR (2004) Prentice Hall, Inc.
9-28
Reorder Point Control (Cont’d)
Total relevant cost
The total relevant cost equation is now extended to
include the costs of safety stock as well as out-of-stock.
The out-of-stock cost (k) is $2/unit. The price term is
dropped. Hence,
TC = DS  IC Q  ICr  k D s'E(z)
2
Q
Q
= 2,600(10)  (0.1)(5) 322  (0.1)(5)(40)
322
2
 2 2,600(17.32)(0.0034)
322
= $182.20
where E(z) = 0.0034 from a unit normal loss table at a
z value of 2.33
9-29
Reorder Point Control (Cont’d)
With known stockout costs k
Setting Q involves balancing both costs and service
level at optimum. Since P and Q are interrelated,
an iterative approach is required.
1 Solve initially for Q
Q = 2DS
IC
2 Using Q, find
P =1  QIC If backordering is allowed
Dk
or
P =1  QIC
Dk  QIC If sales are lost
CR (2004) Prentice Hall, Inc.
9-30
Reorder Point Control (Cont’d)
3 Using P, find revised Q
2D[S  ks' E ]
d (z)
Q=
IC
4 Repeat steps 2 and 3 until no further change
5 Compute ROP and other statistics
CR (2004) Prentice Hall, Inc.
9-31
Reorder Point Control (Cont’d)
Example Given:
Monthly demand forecast, d
Std. error of forecast., sd
Replenishment lead-time, LT
Item value, C
Cost for processing
vendor order, S
Carrying cost, I
Stockout cost, k
Backordering is allowed
11,107 units
3,099 units
1.5 months
$0.11/unit
$10/order
20%/year
$0.01/unit
Find optimal Q and P
CR (2004) Prentice Hall, Inc.
9-32
Reorder Point Control (Cont’d)
Solution
Estimate Q
Q = 2DS = 2(11,107)(12)(10) =11,008 units
0.20(0.11)
IC
Estimate P
P =1  QIC = 1 11,008(0.20)(0.11) = 0.82
Dk
11,107(12)(0.01)
Revise Q
Find App A, z@0.82=0.92 and from App B,
E(0.92)=0.0968
For these data, s'd was previously calculated as
3,795 units
CR (2004) Prentice Hall, Inc.
9-33
Reorder Point Control (Cont’d)

Q=


(z) 
2DS  ks' E
d

IC
= 2(11,107)(12)[(10 0.01(3,795)(0.068)]
0.20(0.11)
= 12,872 units
Revise P
P =1 12,872(0.20)(0.11) = 0.79
11,107(12)(0.01)
Now, z@0.81=0.81 and E(0.81)=0.1181
CR (2004) Prentice Hall, Inc.
9-34
Reorder Point Control (Cont’d)
Revise Q
Q = 2(11,107)(12)[10 0.01(3,795)(0.1181)] =13,246 units
0.20(0.11)
Continue to revise Q and P until no further
change occurs. P=78% and Q=13,395 units.
Note Although the in-stock probability during the
lead time is 78%, the actual service level is
SL=96%
CR (2004) Prentice Hall, Inc.
9-35
Pull Methods (Cont’d)
Noninstantaneous resupply
At times, production or supply continues while demand
is depleting inventories. This requires a slight
modification of the EOQ formula. That is,
p
Qp* = 2DS
IC p  d
Just add
this term
where
p = output or supply rate
d = demand rate
and p > d. ROP remains unchanged.
CR (2004) Prentice Hall, Inc.
9-36
Pull Methods (Cont’d)
Reorder point control with demand and lead time uncertainties
The combined effect of these two uncertainties is particularly
hard to estimate accurately. It is the standard deviation of the
demand-during-lead-time distribution that is the problem,
especially if the level of demand and the length of the lead time
are related to each other. Ideally, we would simply observe the
actual demand occurring over each lead time period. If the
demand and lead time are independent of each other and each
are represented by separate distributions, we may estimate the
standard deviation (s′) from
2 )
s' = LT (sd2)  d 2(sLT
Caution: Can result in very
high safety stock levels when
lead-time variability is high
After computing s’, calculation of the ordering policy would
9-37
be identical to that presented previously.
Pull Methods (Cont’d)
Periodic review control with demand uncertainty
The inventory is reviewed at the time interval (T) to
determine the quantity on hand. The replenishment
quantity (Q) to be ordered is the difference between a
target level called MAX and the quantity on hand. We
need to find MAX and T*.
Good method for
Given:
d = 50 units/week
sd = 10 units/week
I = 10%/year
S = $10/order
CR (2004) Prentice Hall, Inc.
C = $5/unit
LT = 3 weeks
P = 0.99
k = $2/unit
products:
1. Of low value
2. That are purchased
from the same vendor
3. Having economies of
scale in production,
purchasing, and
transportation
9-38
Periodic Control for a Single Item
Quantity on hand
M
Q2
~
Q1
q
Stock
level
reviewed
Order
received
0
LT
T
M = maximum level
M - q = replenishment quantity
LT = lead time
Time
LT
T
T = review interval
q = quantity on hand
Qi = order quantity
9-39
Periodic Review (Cont’d)
Estimate Q* from the EOQ formula as if under demand
certainty conditions. Recall that this is Q* = 322 units.
Now,
T* = Q*/d = 322/50 = 6.4 weeks
Construct the demand-during-lead-time-plus-ordercycle-time distribution.
T is order
review time
CR (2004) Prentice Hall, Inc.
9-40
Periodic Review (Cont’d)
DD(T* + LT)
P
s′
s = sd T  LT
'
*
CR (2004) Prentice Hall, Inc.
Z(s′)
X
= d(T* + LT)
MAX
9-41
Periodic Review (Cont’d)
where
X = d(T *  LT ) = 50(6.4  3) = 470
s' = s T *  LT = 10 6.4  3 = 30.66
d
Find MAX
MAX = d(T* + LT) + z(s’)
= 50(6.4 + 3) + 2.33(30.66)
= 470 + 71.44 = 541 units
Rule Review the inventory every 6.4 weeks and place
an order for the difference between the MAX level of 541
units and the quantity on hand + quantity on order –
backorders.
CR (2004) Prentice Hall, Inc.
9-42
Periodic Review (Cont’d)
The total relevant cost for this design is:
TC = DS/Q + ICQ/2 + ICr + ks’(D/Q)E(z)
= 2600(10)/322 + (.10)(5)(322/2)
+ (.10)(5)(71) + 2 (30.66)(2600/322)(.0034)
= $198
Note Compare this cost with that of the reorder point
method to see that periodic review control carries a
slight premium in cost due to more safety stock.
CR (2004) Prentice Hall, Inc.
9-43
Pull Methods (Cont’d)
In/
Date
Customer
10/26
Bal Fwd
10/26
100M
10/30
Progression
10/30
Ogleby
11/2
Mid Ross
11/9
Unt Sply
11/29
Berea Lit
12/1
Dol Fed
12/13
Card Fed
12/14
Belmont
12/15
Shkr Sav
1/8
BFK
1/8
100M
1/8
Card Fed
1/9
Pt of View
1/17
Am Safety
1/23
Foster
1/24
Gib Prtg
1/26
Bel-Gar
1/26
Copies
1/29
Slvr Lake
1/29
100M
2/2
Sagamore
Size
M/Wgt
8½x14
12.72
*
Sales
20000
25000
15000
50000
25000
10000
20000
15000
5000
500
30000
10000
5000
15000
5000
5000
20000
5000
20000
Basis
20
On
hand
Date
80500 2/2
180500 2/5
160500 2/6
135500 2/6
120500 2/6
70500 2/6
45500 2/8
35500 2/14
15500 2/15
500 2/16
500* 2/21
0 2/26
100000 2/27
70000 2/28
60000 2/28
55000 3/1
40000 3/2
35000 3/8
30000 3/8
10000 3/12
5000 3/12
105000 3/12
85000 3/20
Grain
Color
L
White
In/
Customer
Copies
Bel-Gar
Bel-Gar
Superior
Unt Sply
Berea Prtg
Sagamore
100M
50M
Bel-Gar
Bel-Gar
Inkspot
Lcl 25UAW
Ptrs Dvl
Shkr Sav
Copies
Untd Tor
Sagamore
Sagamore
150M
Untd Tor
Preston
Midland
Finish
RmSeal
No stock or insufficient stock to meet demand
On
Sales
hand
50000
35000
5000
30000
15000
15000
25000
0*
15000
0*
15000
0*
5000
0*
100000
150000
5000
145000
15000
130000
5000
125000
50000
75000
2500
72500
25000
47500
35000
12500
10000
2500
2500
0
12500
0*
150000
40000
110000
50000
60000
15000
45000
Grade
Advantage Bond
In/
Date
Customer
Sales
3/30
Sup Meats
25000
3/30
Copies
50
3/30
Ptrs Dvl
5000
3/30
Belmont
10000
4/2
Berea Prtg
4950
4/2
Berea Prtg
15050
4/9
REM
500
4/12
Mid Ross
5000
5/7
Ohio Ost
5000
5/8
Inkspots
5000
5/8
Prts Dvl
2500
5/11
100M
5/14
BVR
5000
5/15
Guswold
10000
5/16
ESB
15000
5/16
Superior
50000
5/16
J Stephen
5000
5/16
Am Aster
15000
5/16
Am Aster
10000
5/22
Sagamore
15000
21200
Coding
M. Base Cost
Date Min
2.64
4/2
Max
Location
Ctn. Skid Cont.
F 14
5M
On
Hand
20000
19950
14950
4950
0
0*
0*
0*
0*
0*
0*
100000
95000
85000
70000
20000
15000
0
0*
0*
125M
250M
Att.
9-44
Pull Methods (Cont’d)
Supply chain example
Suppose that inventory is to be maintained on a
distributor’s shelf for an item whose demand is
forecasted to be d = 100 units per day and sd = 10 units
per day. A reorder point is the method of inventory
control. The supply channel is shown in the diagram.
Determine the average inventory to be held at the
distributor where we have:
I = 10%/year
S = $10/order
CR (2004) Prentice Hall, Inc.
C = $5/unit
P = 0.99 during lead time
9-45
Supply Chain Example (Cont’d)
Supplier
Processing time
X
p
= 1, s 2p = 0 .1
Transport time
Inbound transport
2
X i = 4 , si = 1.0
Outbound transport
Pool point
Transport time
X
CR (2004) Prentice Hall, Inc.
o
= 2 , s o2 = 0 .25
Distributor
9-46
Supply Chain Example (Cont’d)
Solution The reorder point inventory theory applies.
However, determining the statistics of the demandduring-lead-time distribution requires taking the leadtime for the entire channel into account.
Recall,
2
s' = LT(sd2 )  d 2(sLT
)
where
2
sLT
= sp2  si2  so2
= 0.1  1.0  0.25
= 1.35 days
CR (2004) Prentice Hall, Inc.
9-47
Supply Chain Example (Cont’d)
Average lead time
LT = X p  X i  X o =1 4  2 = 7 days
Now
s' = 7x102 1002 x1.35 = 14,200 = 119.16 days
and
Q* = 2(100)(10) = 63 units
0.1(5)
*
Q
AIL =  z(s' ) = 63  2.33(199.16) = 309 units
2
2
CR (2004) Prentice Hall, Inc.
9-48
Pull Methods (Cont’d)
Joint ordering
Perpetual inventory control for most firms is the problem of
managing items jointly rather than singly. This occurs since
more than one item is typically purchased from the same
vendor. The approach to joint ordering is to find a common
order review interval (T) and then to set separate target levels
(MAX) based on specific item costs and service levels.
A common review time may be specified, or it may be
computed based on appropriate economics.
T* =
2(O  S )
i
(I C D )
i
i
CR (2004) Prentice Hall, Inc.
where
O = common procurement cost, $/order
Note: Q* = T*xd
9-49
Joint Ordering Example
Given
Item
A
30
8
14
25
30
Average daily demand (d)
Demand std. dev. ( sd)
Average lead time (LT)
Annual carrying cost (I)
Procurement cost (S)
with common cost (O)
In-stock probability (P)
80
Product value (C)
170
Out-of-stock cost (k)
25
Selling days per year
365
CR (2004) Prentice Hall, Inc.
B
75 units
10 units
14 days
25 %
20 $/order
80 $/order
92 %
200 $/unit
45 $/unit
365 days
9-50
Joint Ordering Example (Cont’d)
Find common review time
T* =
2[80  (30  20)]
= 4.35 days
[0.25/365][170(30) 200(75)]
Find target quantity (MAX) for item A
s'A = sd
T *  LT = 8 4.35 14 = 34.3 units
A
A
then z@80%=0.84
MAX A = X  z(sA' ) = 30(4.35 14)  0.84(34.3) = 579 units
CR (2004) Prentice Hall, Inc.
9-51
Joint Ordering Example (Cont’d)
which has an average inventory of
Avg. Inventoryi =T *(di / 2)  zi (si' )
Avg. Inventory A = 4.35(30 / 2)  0.84(34.3) = 94.1units
Find target quantity (MAX) for item B
sB' = sd
T *  LT = 10 4.35 14 = 42.8 units
B
B
then for z@90%=1.41
MAX B = 75(4.35 14) 1.41(42.8) = 1437 units
which has an average inventory of
Avg. inventoryB = 4.35(75 / 2) 1.41(42.8) = 223 units
CR (2004) Prentice Hall, Inc.
9-52
Pull Methods (Cont’d)
The Min-Max variant
This is basically a reorder point system, but the order
quantity is incremented by the amount of the difference
between the reorder point quantity and the quantity on
hand + quantity on order  backorders. This takes into
account that demand does not decrement inventory
levels evenly. Therefore, inventory levels may fall
below the reorder point at the time that it is reached.
CR (2004) Prentice Hall, Inc.
9-53
Min-Max Inventory Control
Add increment ROPq to order size
Quantity on hand
M
Q1
~
Q2
Q*
ROP
q
LT
CR (2004) Prentice Hall, Inc.
LT
Time
9-54
Pull Methods (Cont’d)
The T, R, M variant
This is a combination of the min-max and the periodic
review systems. The stock levels are reviewed
periodically, but control the release of the replenishment
order by whether the reorder point is reached. This
method is useful where demand is low, such that small
quantities might be released under a periodic review
method.
CR (2004) Prentice Hall, Inc.
9-55
Pull Methods (Cont’d)
Inventory not
below R, so don’t
place an order
Inventory level
T,R,M variant
Q1
Q2
R
q
LT
LT
T
Time
T
T = review time
R = reorder point M – Q = replenishment quantity
CR (2004) Prentice Hall, Inc.
9-56
Pull Methods (Cont’d)
Stock to demand (a periodic review method)
This is an important periodic review method, not so much
because of its accuracy but because of its popularity in
practice. The method is synchronized with the period of
the forecast. The target quantity (MAX) is developed as
follows.
An example
•Set the period of the forecast, say 4 weeks
•Add time for lead time, say 1 week
•Add an increment of time for safety stock, say 1 week
CR (2004) Prentice Hall, Inc.
9-57
Stock to Demand (Cont’d)
Therefore, MAX is 6/4 times the monthly forecast. The
replenishment quantity is determined as follows.
At the time (T) of the monthly stock-level review, make a
forecast and determine the MAX level.
MAX = Forecast x 6/4
Plus: Backorders
Less: Quantity on hand*
Less: Quantity on order
Order quantity (Q)
Units
12,500
0
-5,342
-4,000
3,158
*Quantity on hand = actual quantity on hand +
quantity on order – backorders
CR (2004) Prentice Hall, Inc.
9-58
Pull Methods (Cont’d)
Multiple item, multiple-location control
The theory that has been discussed previously is
useful when designing inventory control systems for
the practical problem of controlling many items at
many locations. Consider how a specialty chemical
company designed such a practical system. TASO is
the time to accumulate a stock order (truckload) for all
items in warehouse.
CR (2004) Prentice Hall, Inc.
9-59
Q3
Q2
~
Q1
Stock
order
Order
received
~
0
LT
TASO
TASO
M = maximum level
TASO = time to accumulate stock order
CR (2004) Prentice Hall, Inc.
LT
TASO
Time
Qi = order quantity
LT = lead time
Multiple-Item, Multiple-Location Control
Quantity on hand
M
9-60
Customer Service Level
For individual items
The service level (stock availability) actually achieved by
inventory control methods is not best represented by the
probability (P) of a stockout during the lead time. It is
more accurate to compute it as follows.
SL = 1
s 'E D /Q
(z)
D
= 1
s 'E
(z)
Q
Using data from the reorder point under uncertainty
example, the service level would be:
SL =1 17.32(.0034)(2,600 / 322) = 0.999
2,600
CR (2004) Prentice Hall, Inc.
Note: Higher
than P
9-61
Customer Service Level (Cont’d)
This actual level is higher than P = 0.99 that was
used to set the inventory level. The reason is that
there are periods of time when the stock level is
above the reorder point and there is no risk of being
out of stock.
Methods for defining stock availability include:
•Probability of filling all item demand
•Probability of filling an order completely
•Probability of filling a percent of all item demand
•Weighted average of items filled on an order (fill
rate)
CR (2004) Prentice Hall, Inc.
9-62
Customer Service Level (Cont’d)
For multiple items on the same order
If all items on an order have the same service level,
what is the probability of filling the order complete?
The service level for multiple items is the combination
of the individual item service levels as follows:
SL = SL1 x SL2 x SL3 …x SLn
Suppose 3 items have the following service levels—
0.95, 0.89, and 0.92. The probability of filling the order
complete is:
SL = 0.95 x 0.89 x 0.92 = 0.78
CR (2004) Prentice Hall, Inc.
9-63
Push Inventory Control
Example
Three warehouses are used to supply 900 retail
drugstores. Each warehouse serves approximately 300
stores. A large purchase of clock radios is made, where
radios were to be a promotional item in the next forecast
period. The special buy will result in more stock than
needed, but the company expects to sell all stock
eventually. Warehouses are to have a 92% in-stock
probability. All of the purchased radios are to be allocated
to the warehouses based on the anticipated demand
levels at each warehouse. Account is taken of the
inventory already on hand. A total of 5000 radios is
purchased. The next purchase will be made in one
month. Further information is given below.
CR (2004) Prentice Hall, Inc.
9-64
Push Inventory Control (Cont’d)
Warehouse
1
2
3
Current
stock
level,
units
400
350
0
Forecasted
demand,
units
2,300
1,400
900
4,600
Forecast
error (std.
dev.), units
100
55
20
How should the allocation to the warehouses be
made?
CR (2004) Prentice Hall, Inc.
9-65
Push Inventory Control (Cont’d)
Solution
Warehouse
1
2
3
a2,428
Total
requirements
a
2,428
1,470
926
4,824
= 2,300 + 1.28(100)
Total requirements = Forecast + z(Forecast error)
where z@90% = 1.28. Therefore,
CR (2004) Prentice Hall, Inc.
9-66
Push Inventory Control (Cont’d)
(1)
(2)
(3)=
(4)
(5)=
Pro(4)+(3)
(1)(2)
Total
Onration
Net
require- hand requireof
AlloWare- ments, stock, ments, excess, cation,
house units
units
units
units
units
a
b
1
2,428
400 2,028
463
2,491
2
1,470
350 1,120
282
1,402
3
926
0
926
181
1,107
c
Total 4,824
4,074
926
5,000
requirements less (quantity on hand + quantity on order – backorders)
bExcess purchase quantity times forecast for warehouse divided by total
forecast quantity. For example, (5,000 – 4,074) x 2,300/4,600 = 463
c5,000 – 4,074 = 926
aTotal
CR (2004) Prentice Hall, Inc.
9-67
Multi-Echelon Inventories
Control the entire channel inventory levels, not just a
single echelon.
Warehouse
echelon
Warehouse
lead-time, LTw
S
Supplier
ade
l
tail LT R
e
R e,
tim
W
R1
R2
d 1 , s d1
d 2 , sd 2
Warehouse
R3
End customer demand
How much stock here when
retailers also carry stock?
d 3 , sd3
Retailer
CR (2004) Prentice Hall, Inc.
9-68
Multi-Echelon Inventories (Cont’d)
Example
An item has the following cost characteristics. Item
values are CR=$10/unit and CW=$5/unit. Carrying cost
is I = 20%/year. Ordering costs are SR=$40/order and
SW=$75/order. Lead times are LTR=0.25 month and
LTW=0.5 months. In-stock probability for retailers and
warehouses is 90%. Monthly demand statistics are:
Monthly
avg., units
Std. dev.,
units
Retailer 1
202.5
16.8
Retailer 2
100.5
15.6
Retailer 3
302.5
18.0
Combined
605.5
32.4
CR (2004) Prentice Hall, Inc.
9-69
Multi-Echelon Inventories (Cont’d)
Solution
Based on reorder point inventory control, the retailers’
inventory statistics are
Retailer 1
Retailer 2
Retailer 3
Reorder qty, Q
312
220
381
Reorder point, ROP
61
35
87
Avg. inv., AIL
167
120
202
The warehouse echelon order quantity is
QW =
2DW SW
= 2(605.5x12)(75) =1,043.98, or 1,044 units
0.20(5)
ICW
ROPW = dW xLTW  zsW LTW = 605.5(.5) 1.28(32.4) .5
= 332 units
CR (2004) Prentice Hall, Inc.
9-70
Multi-Echelon Inventories (Cont’d)
The warehouse echelon inventory is
QW
AILW =
 zsW LTW
2
= 1,043.98 1.28(32.4) 0.5
2
= 551.32. or 551units
The average warehouse inventory is the warehouse
echelon inventory less the retailers’ inventory, or 551
– 167 –120 – 202 = 62 units.
Rule When the total warehouse inventory (sum of
retailers’ inventory, inventory at the warehouse and on
order, and retailers’ orders less any inventory committed
to customers drops below 332 units, order 1,044 units.
CR (2004) Prentice Hall, Inc.
9-71
100
90
Total sales (%)
80
70
60
50
40
30
A items
B items
C items
20
10
0
0
CR (2004) Prentice Hall, Inc.
20
40
60
Total items (%)
80
100
Aggregate Inventory Control
Product items can be grouped according to 80-20
curve, each with different stocking policies
9-72
Inventory Consolidation
(“Risk Pooling”)
Illustration of risk pooling
Suppose there is a product stocked in two warehouses.
The replenishment quantities are determined by the
economic order quantity formula. The replenishment
lead-time is 0.5 months, the cost for a replenishment
order is $50, the inventory carrying cost is 2% per
month, and the item value is $75 per unit. The
probability of an out of stock during the lead-time period
is 5%. The demand is normally distributed with typical
demand over six months as follows.
CR (2004) Prentice Hall, Inc.
9-73
Risk Pooling (Cont’d)
6
Month
1
2
3
4
5
6
Avg. (D)
Std. Dev. (sd)
Demand
in Whse
A
35
62
46
25
37
43
41.33
11.38
Combined
Demand Demand in a
in Whse
Central
B
Whse
67
102
83
145
71
117
62
87
55
92
66
109
67.33
108.66
8.58
19.07
Estimate the average inventory levels for twowarehouse and one-warehouse supply channels.
CR (2004) Prentice Hall, Inc.
9-74
Risk Pooling (Cont’d)
Regular stock
2DS
RS = Q = IC
2
2
2(41.33)(50)
0.02(75)
RSA =
= 52.49 = 26.25 units
2
2
2(67.33)(50)
0.02(75)
RSB =
= 67.00 = 33.50 units
2
2
Regular stock in system is
RSs = RSA  RSB = 26.25  33.50 = 59.75 units
CR (2004) Prentice Hall, Inc.
9-75
Risk Pooling (Cont’d)
Regular stock if item is entirely in one warehouse
2(108.66)(50)
0.02(75) = 85.11= 42.56 units
RSC =
2
2
Safety stock
SS = z(sd ) LT
SSA = 1.96(11.38) 0.5 = 15.77 units
SSB = 1.96(8.58) 0.5 = 11.89 units
System safety stock in 2 warehouses
SSA  SSB = 15.77 11.89 = 27.66 units
CR (2004) Prentice Hall, Inc.
9-76
Risk Pooling (Cont’d)
Safety stock in 1 warehouse
SSc =1.96(19.07) 0.5 = 26.43 units
Total inventory
AIL = Regular stock + Safety stock
Two
warehouses
AIL = 59.75 + 27.66 = 87.41 units
In a one-warehouse channel
AIL = 42.56 + 26.43 = 68.99 units
Conclusion There is a reduction in the average
inventory level of an item as the number of stocking
points in the supply channel is decreased. In this
example, both regular stock and safety stock decline.
CR (2004) Prentice Hall, Inc.
9-77
Risk Pooling (Cont’d)
System-wide Inventory as a Fraction of the
Demand Divided Between Two Warehouses
Percent of Peak
System-Wide Inventory
100
95
90
85
80
75
70
0
0.2
0.4
0.6
0.8
1
One warehouse's demand as a fraction
of the total
CR (2004) Prentice Hall, Inc.
9-78
Virtual Inventories
•Stockouts are filled from other stocking locations in
the distribution network
• Customers assigned to a primary stocking location
• Backup locations are usually determined by
“zoning” rules
• Expectation is that lower system-wide inventories
can be achieved while maintaining or improving
stock availability levels
• Total distribution costs should be lower to support
the cross filling of customer demand
CR (2004) Prentice Hall, Inc.
9-79
Cross Filling Among 2
Stocking Locations
Demand 1
Virtual Inventories
Stock
location B
Secondary
assignment
Primary
assignment
Primary
assignment
Stock
location A
Demand 2
9-80
Potential Benefit
of Cross Filling
Suppose that an item is stocked at a fill rate of 80% in
4 stocking locations. If cross filling is used, what is
the effective fill rate for the customer?
Fill rate = [1 – (.20)(.20)(.20)(.20)] x 100 = 99.8%
Customer service levels can be quite high
even if the item fill rate is low!
But are inventory costs lower?
Virtual Inventories
CR (2004) Prentice Hall, Inc.
9-81
Regular Stock in 2 Locations
Demand 1
Method of stock control
Stock
location B
Secondary
assignment
Primary
assignment
Demand dispersion
Stock
location A
Primary
assignment
•Meaning of regular stock
•How it varies with:
Demand 2
Fill rate
Virtual Inventories
CR (2004) Prentice Hall, Inc.
9-82
Stock Control Methods
and Regular Stock
If control is EOQ-based, average inventory level (AIL) is
EOQ
formula





2S D
AIL = Q = IC
2
2
0.5





= kD0.5
If stock-to-demand control
AIL is a function of
demand with
exponents ranging
from 0.5 to 1.0
AIL = kD1.0
Virtual Inventories
CR (2004) Prentice Hall, Inc.
9-83
Percent of peak sytem-w ide
inventory
Regular Stock as a Percent of Demand Divided
Stock-toBetween Two Warehouses
D0.9
D1.0
demand
control
100
95
D0.5
90
D0.7
85
80
75
EOQ-based
control
70
0
20
40
60
80
100
O n e w a re h o u s e 's d e m a n d
a s a p e rc e n t o f to ta l d e m a n d
Virtual Inventories
CR (2004) Prentice Hall, Inc.
9-84
Observation about
Regular Stock
A system of multiple stocking locations
will carry its maximum regular stock
when demand is balanced among them
Virtual Inventories
CR (2004) Prentice Hall, Inc.
9-85
Fill Rate and Regular Stock
Cross filling increases regular stock as lower fill rates
are specified
Example
•2 locations
•Demand is dispersed 50 and 150
•Fill rate is 90%
•Stocking policy is D0.5 with k=1
Virtual Inventories
CR (2004) Prentice Hall, Inc.
9-86
Example (Cont’d)
No cross filling
Location A
Cross filling
Location B
Location A
Location B
45a
5b
Demand 1
50
0
Demand 2
0
150
15
135
Total
50
150
60
140
Regular stock
7.1
12.2
7.7
11.8
System inv.
a50x.90=45
b[50x(1-0.90)]x0.905
19.3
19.5
Regular stock increases
with cross filling
Virtual Inventories
CR (2004) Prentice Hall, Inc.
9-87
% of inventory compared
w ith no cross-filling
Regular Stock Penalty for Cross Filling Under
Various Stocking Policies
EOQ-based
control
150
140
D0.5
130
D0.7
120
D0.9
110
D1.0
100
50
Stock-to-demand
control
CR (2004) Prentice Hall, Inc.
60
70
80
90
100
A v e ra g e fill ra te o n w a re h o u s e s
Virtual Inventories
9-88
Safety Stock in 2 Locations
•Meaning of safety stock
•Safety stock depends on
Demand dispersion (variance is proportional
Primary
assignment
Stock
locationB
locationB
Primary
assignment
Stock
locationA
locationA
Secondary
assignment
Demand 1
Demand 2
to (demand)
Fill rate
Observation
A system of multiple stocking
locations will carry its minimum
safety stock when demand is
balanced among them
Virtual Inventories
CR (2004) Prentice Hall, Inc.
9-89
Safety Stock Estimation
Safety stock is estimated by
ss = zs * LT
where s* is the demand standard deviation at
location N
When cross filling,
s * = FR 2sd2
where sd is the demand standard deviation at the
primary location
At any location N
sN* = [FR(1  FR )N 1]2 sd2
CR (2004) Prentice Hall, Inc.
Virtual Inventories
9-90
Safety Stock in 2 Locations
Example
•2 locations
•Weekly demand and std. dev. are (50,5) and
(150,15)
•Lead time is 1 week
•Fill rate (FR) is 95%
•z is 1.65 for 95% stocking level (demand normally
distributed)
•Inventory control is EOQ based
Virtual Inventories
CR (2004) Prentice Hall, Inc.
9-91
Safety Stock for 2 Locations
No cross filling
Location A
Location B
Cross filling
Location A
Location B
Std. Dev. 1
5
0
4.7500
0.2375
Std. Dev. 2
0
15
0.7125
14.2500
Combined
5
15
4.8
14.3
8.3
24.8
7.9
23.5
Safety stock
System inv.
33.1
31.4
Safety stock decreases
with cross filling
Virtual Inventories
CR (2004) Prentice Hall, Inc.
9-92
Safety Stock Reduction Due to Cross Filling as a
Percent of Demand Divided Between Two Warehouses
Percent reduction
25
FR=70%
20
Lower safety
stocks from
lower fill rates
15
10
FR=90%
5
No crossfilling
0
0
5
15 25 35 45 55 65 75 85 95 100
One warehouse's demand as a percent of
the system-wide demand
Virtual Inventories
CR (2004) Prentice Hall, Inc.
9-93
Simplifying the Decision
Problem
An item potentially can be cross filled from 1 backup
warehouse. The item has a value of $200/unit, a carrying
cost of 25%/per year, a stocking level of 6-weeks demand, a
replenishment lead-time of 8 weeks, and a target fill rate of
95%. To cross haul the item from the secondary warehouse
incurs an extra $10/ unit in transportation. The stock control
policy is not known. Demand statistics are as follows:
Location
Mean demand,
units
Std. Dev.,
units
1
300
138
2
100
80
System
400
160
Should the item be cross filled?
CR (2004) Prentice Hall, Inc.
Virtual Inventories
9-94
Simplifying the Decision
Answer
Solve for K in AIL=KD. K=D1-/TO, where TO is 52
weeks/6 weeks of demand or 8.67. This assumes a
control policy based on  of 0.7. Hence,
K=(400x52)1-.7/8.67=2.28. The demand ratio r between
the 2 locations is 100/400=0.25. Now solve for two
parameters of the cross-filling curve.
1.7
1
tD
X=
= 10([400x52]) =1.73
ICK 0.25(200)(2.28)
and
1.96(160) 8 = 0.4
Y = zs LT
=
KD
2.28(400x52)0.7
Now, from the decision curve, don’t cross fill this item.
Virtual Inventories 9-95
CR (2004) Prentice Hall, Inc.
LEGEND
D = annual system demand
s = system demand std.
dev.
C = item unit cost
I = annual carrying cost
rate (%)

K, = inventory
control parameters
t = transportation rate
FR = item fill rate
r = ratio of minimum
demand to system
demand
LT = lead-time in demand
std. dev. time units
z = normal deviate at FR %

tD 1
X =
ICK
Y =
Ratio of minimum demand to total
demand, r
Decision Curve for FR=.95 and =0.7
zs LT

KD
Below
decision
curve—don’t
cross fill
X
Virtual Inventories
CR (2004) Prentice Hall, Inc.
9-96
Square Root Law
of Inventory Consolidation
The amount of inventory (regular stock) at multiple stocking
points can be estimated by the square root law when
•Inventory control at each point is based on EOQ
principles
•There is an equal amount on inventory at each point
The square root law is:
IT = Ii n
where
IT = amount of inventory at one location
Ii = amount of inventory at each of n locations
n = number of stocking points
CR (2004) Prentice Hall, Inc.
9-97
Square Root Law (Cont’d)
Example Suppose that there is $1,000,000 of inventory at 3
stocking points for a total of $3,000,000. If it were all
consolidated into 1 location, we can expect:
IT = 1,000,000 3 = $1,732,051
If we wish to consolidate from 3 to 2 warehouses, the
level of inventory in each warehouse would be:
I
Ii = T = $1,732,051= $1,224,745
1.41
2
For a total system inventory of n x I = 2 x $1,224,745 =
$2,449,490.
CR (2004) Prentice Hall, Inc.
9-98
Square Root Law (Cont’d)
More simply
n2
I2 = I1 n
1
where
I2 = system inventory in n2 locations
I1 = system inventory in n1 locations
n2 = no. of new locations
n1 = no. of previous locations
So,
I2 = 3,000,000 2 = 2,449,490
3
CR (2004) Prentice Hall, Inc.
9-99
Warehouse average inventory, Ii ($000s)
Inventory-Throughput
Curve
3000
2500
2000
Ii = 1.57Di0.72
R = 0.85
1500
1000
500
0
0
CR (2004) Prentice Hall, Inc.
10000
20000
30000
40000
Annual warehouse throughput, Dj ($000s)
50000
9-100
Inventory-Throughput Curve
Example
Suppose two warehouses with 390,000 lb. and
770,000 lb. of throughput respectively are to be
consolidated into a single warehouse with 390,000 +
770,000 = 1,160,000 lb. of annual throughput. How
much inventory is likely to be in the single warehouse?
The inventory-throughput curve developed from the
company’s multiple warehouse stocks is shown in the
figure below.
Note Reading from the plot, the inventory in the
consolidated warehouse has dropped to 262,000 lb.
from 132,000 + 203,000 = 335,000 lb. in the two
warehouses.
CR (2004) Prentice Hall, Inc.
9-101
Inventory-Throughput Curve
450
Average inventory level, AIL i (000 lb.)
400
350
AILi = 3.12Di0.628
R2 = 0.77
300
262
250
203
200
150
132
100
Current
warehouse
50
Consolidated
warehouse
0
0
200
400
600
800
1000
1200
1400
1600
1800
Annual warehouse throughput,D i (000 lb.)
CR (2004) Prentice Hall, Inc.
9-102
Turnover Ratio
Annual sales
Turnover ratio =
Average inventory
A fruit grower stocks its dried fruit products in 12 warehouses
around the country. What is the turnover ratio for the
distribution system?
Warehouse
no.
1
2
3
4
5
6
Annual
Average
warehouse
inventory
throughput, $
level, $
21,136,032
2,217,790
16,174,988
2,196,364
78,559,012
9,510,027
17,102,486
2,085,246
88,228,672 11,443,489
40,884,400
5,293,539
TO ratio =
$425,295,236
= 9.7
$43,701,344
CR (2004) Prentice Hall, Inc.
Warehouse
no.
7
8
9
10
11
12
Totals
Annual
warehouse
throughput, $
43,105,917
47,136,632
24,745,328
57,789,509
16,483,970
26,368,290
425,295,236
Average
inventory
level, $
6,542,079
5,722,640
2,641,138
6,403,076
1,991,016
2,719,330
43,701,344
$ are at cost
9-103
Areas under
Standardized
Normal
Distribution
CR (2004) Prentice Hall, Inc.
A table entry is the proportion of the
area under the curve from a z of 0 to a
positive value of z. To find the area
from a z of 0 to a negative z, subtract the
tabled value from 1.
z
0.0
0.1
0.2
0.3
0.4
.00
0.5000
0.5398
0.5793
0.6179
0.6554
.01
0.5040
0.5438
0.5832
0.6217
0.6591
.02
0.5080
0.5478
0.5871
0.6255
0.6628
.03
0.5120
0.5517
0.5910
0.6293
0.6664
.04
0.5160
0.5557
0.5948
0.6331
0.6700
.05
0.5199
0.5596
0.5987
0.6368
0.6736
.06
0.5239
0.5636
0.6026
0.6406
0.6772
.07
0.5279
0.5675
0.6064
0.6443
0.6808
.08
0.5319
0.5714
0.6103
0.6480
0.6844
.09
0.5359
0.5753
0.6141
0.6517
0.6879
0.5
0.6
0.7
0.8
0.9
0.6915
0.7257
0.7580
0.7881
0.8159
0.6950
0.7291
0.7611
0.7910
0.8186
0.6985
0.7324
0.7642
0.7939
0.8212
0.7019
0.7357
0.7673
0.7967
0.8238
0.7054
0.7389
0.7704
0.7995
0.8264
0.7088
0.7422
0.7734
0.8023
0.8289
0.7123
0.7454
0.7764
0.8051
0.8315
0.7157
0.7486
0.7794
0.8078
0.8340
0.7190
0.7517
0.7823
0.8106
0.8365
0.7224
0.7549
0.7852
0.8133
0.8389
1.0
1.1
1.2
1.3
1.4
0.8413
0.8643
0.8849
0.9032
0.9192
0.8438
0.8665
0.8869
0.9049
0.9207
0.8461
0.8686
0.8888
0.9066
0.9222
0.8485
0.8708
0.8907
0.9082
0.9236
0.8508
0.8729
0.8925
0.9099
0.9251
0.8531
0.8749
0.8944
0.9115
0.9265
0.8554
0.8770
0.8962
0.9131
0.9279
0.8577
0.8790
0.8980
0.9147
0.9292
0.8599
0.8810
0.8997
0.9162
0.9306
0.8621
0.8830
0.9015
0.9177
0.9319
1.5
1.6
1.7
1.8
1.9
0.9332
0.9452
0.9554
0.9641
0.9713
0.9345
0.9463
0.9564
0.9649
0.9719
0.9357
0.9474
0.9573
0.9656
0.9726
0.9370
0.9484
0.9582
0.9664
0.9732
0.9382
0.9495
0.9591
0.9671
0.9738
0.9394
0.9505
0.9599
0.9678
0.9744
0.9406
0.9515
0.9608
0.9686
0.9750
0.9418
0.9525
0.9616
0.9693
0.9756
0.9429
0.9535
0.9625
0.9699
0.9761
0.9441
0.9545
0.9633
0.9706
0.9767
2.0
2.1
2.2
2.3
2.4
0.9772
0.9821
0.9861
0.9893
0.9918
0.9778
0.9826
0.9864
0.9896
0.9920
0.9783
0.9830
0.9868
0.9898
0.9922
0.9788
0.9834
0.9871
0.9901
0.9925
0.9793
0.9838
0.9875
0.9904
0.9927
0.9798
0.9842
0.9878
0.9906
0.9929
0.9803
0.9846
0.9881
0.9909
0.9931
0.9808
0.9850
0.9884
0.9911
0.9932
0.9812
0.9854
0.9887
0.9913
0.9934
0.9817
0.9857
0.9890
0.9916
0.9936
2.5
2.6
2.7
2.8
2.9
0.9938
0.9953
0.9965
0.9974
0.9981
0.9940
0.9955
0.9966
0.9975
0.9982
0.9941
0.9956
0.9967
0.9976
0.9982
0.9943
0.9957
0.9968
0.9977
0.9983
0.9945
0.9959
0.9969
0.9977
0.9984
0.9946
0.9960
0.9970
0.9978
0.9984
0.9948
0.9961
0.9971
0.9979
0.9985
0.9949
0.9962
0.9972
0.9979
0.9985
0.9951
0.9963
0.9973
0.9980
0.9986
0.9952
0.9964
0.9974
0.9981
0.9986
3.0
3.1
3.2
3.3
3.4
0.9987
0.9990
0.9993
0.9995
0.9997
0.9987
0.9991
0.9993
0.9995
0.9997
0.9987
0.9991
0.9994
0.9995
0.9997
0.9988
0.9991
0.9994
0.9996
0.9997
0.9988
0.9992
0.9994
0.9996
0.9997
0.9989
0.9992
0.9994
0.9996
0.9997
0.9989
0.9992
0.9994
0.9996
0.9997
0.9989
0.9992
0.9995
0.9996
0.9997
0.9990
0.9993
0.9995
0.9996
0.9997
0.9990
0.9993
0.9995
0.9997
0.9998
9-104
Unit Normal
Loss
Integrals
CR (2004) Prentice Hall, Inc.
9-105