chopra_scm7_inppt_12.ppt

Supply Chain Management: Strategy,
Planning, and Operation
Seventh Edition
Chapter 12
Managing Uncertainty in a
Supply Chain Safety
Inventory
Copyright © 2019, 2016, 2013 Pearson Education, Inc. All Rights Reserved

Learning Objectives (1 of 2)
12.1 Understand the role of safety inventory in a supply
chain.
12.2 Identify factors that influence the required level of
safety inventory.
12.3 Evaluate the appropriate level of safety inventory for a
supply chain.
12.4 Discuss the impact of supply uncertainty on safety
inventory.

Learning Objectives (2 of 2)
12.5 Understand how aggregation helps reduce the
required safety inventory in a supply chain.
12.6 Determine the impact of replenishment policies on
safety inventory.
12.7 Improve the management of safety inventory in a
multiechelon supply chain.
12.8 Identify managerial levers that lower safety inventory
without hurting product availability.

The Role of Safety Inventory (1 of 3)
• Safety inventory is carried to satisfy demand that
exceeds the amount forecasted
– Raising the level of safety inventory increases product
availability and thus the margin captured from
customer purchases
– Raising the level of safety inventory increases
inventory holding costs

Figure 12-1 Inventory Profile with Safety Inventory

• Three key questions
1. What is the appropriate level of product availability?
2. How much safety inventory is needed for the desired
level of product availability?
3. What actions can be taken to reduce safety inventory
without hurting product availability?

Summary of Learning Objective 1
Safety inventory helps a supply chain provide customers
with a high level of product availability in spite of supply
and demand uncertainty. It is carried just in case demand
exceeds the amount forecasted or supply arrives later than
expected.

Factors Affecting the Level of Safety
Inventory
• The desired level of product availability
• The uncertainty of demand
• The uncertainty of supply
• Inventory replenishment policies

Measuring Product Availability
1. Product fill rate (fr)
– Fraction of product demand satisfied from
product in inventory
2. Order fill rate
– Fraction of orders filled from available inventory
3. Cycle service level (CSL)
– Fraction of replenishment cycles that end with
all customer demand being met
– Replenishment cycle – the interval between two
successive replenishment deliveries

Measuring Demand Uncertainty
D = Average demand per period
σD = Standard deviation of demand (forecast error) per
period
Lead time (L) is the gap between when an order is placed
and when it is received

Evaluating Demand Distribution over L
Periods
i
    
 
  
  
L L
L i L i j i j
i i i j
D D 2
1 1 >
2
 
 
L L D
D DL L
The coefficient of variation
 

cv /

Measuring Supply Uncertainty
Lead time (L) is normally distributed with
L = Average lead time
σL = Standard deviation of lead time

Replenishment Policies
1. Continuous review
– Inventory is continuously tracked
– Order for a lot size Q is placed when the inventory
declines to the reorder point (ROP)
2. Periodic review
– Inventory status is checked at regular periodic
intervals
– Order is placed to raise the inventory level to a
specified threshold

Safety inventory is influenced by the desired product
availability, demand uncertainty, replenishment lead times,
and lead time variability. Product availability is measured
using the fill rate or cycle service level. Demand uncertainty
is measured by the forecast error. For lead time one
measures both the mean and the standard deviation. The
required safety inventory is also influenced by the inventory
policy implemented. Continuous review policies order a
fixed quantity after variable replenishment intervals.
Periodic review policies order variable quantities after fixed
replenishment intervals.

Determining the Appropriate Level of
Safety Inventory (1 of 8)
• Evaluating Safety Inventory Given a Reorder Point
Expected demand during lead time = D × L
Safety inventory, ss = ROP − D × L

Average demand per week, D = 2,500
Standard deviation of weekly demand, D = 500
Average lead time for replenishment, L = 2 weeks
Reorder point, ROP = 6,000
Average lot size, Q = 10,000
Safety inventory, ss = ROP −DL = 6,000 −5,000 = 1,000
  
10,000
Cycle inventory 5,000
2 2
Q

Average inventory = cycle inventory + safety inventory
= 5,000 + 1,000 = 6,000
  
average inventory 6,000
Average flow time 2.4 weeks
throughput 2,500

• Evaluating Cycle Service Level Given a Reorder Point


CSL Prob P
L RO
ddlt of weeks
( )
(ddlt = demand during lead time)
CSL = F(ROP, DL, σL) = NORMDIST(ROP, DL, σL, 1)

Q = 10,000, ROP = 6,000, L = 2 weeks
D = 2,500/week, σD = 500
 
    
   
L
L D
D D L
L
2 2,500 5,000
2 500 707
CSL= F(ROP, DL, σL) = NORMDIST(ROP, DL, σL, 1)
= NORMDIST(6,000, 5,000, 707, 1) = 0.92

• Evaluating Required Safety Inventory Given a Desired
Cycle Service Level
Desired cycle service level = CSL
Mean demand during lead time = DL
Standard deviation of demand during lead time = σL
Probability(demand during lead time )
L
D ss CSL
  
• Identify safety inventory ss so that
F(DL + ss, DL, sL) = CSL

  
L L L L L
D ss F CSL,D ,σ NORMINV CSL,D ,σ
–1
( ) ( )
or
– –
 
L L L L L L
ss F CSL,D ,σ D NORMINV CSL,D ,σ D
–1
( ) ( )
 

   
 
– –
S L S D
D
ss F CSL F CSL L
NORMSINV CSL L
1 1
( ) ( )
( )


   
D
D CSL L
2,500 / week, 500, 0.9, 2 weeks
 
   
   
L
L D
D DL
L
2 2,500 5,000
2 500 707
 
   
  
–1
s L L
ss F CSL NORMSINV CSL
NORMSINV(
( ) ( )
0.90) 707 906

Evaluating Fill Rate Given a Reorder
Point (1 of 4)
• Expected shortage per replenishment cycle (ESC) is
the average units of demand that are not satisfied from
inventory in stock per replenishment cycle
• Product fill rate
–
–
 
ESC Q ESC
fr
Q Q
(
1
)

Point (2 of 4)
–


 x ROP
ESC x ROP f x dx
( ) ( )
– 1– s 
 
 
   
 
 
   
   
 
L s
L L
ss ss
ESC ss F f
–
  
  
L
L L
ESC ss NORMDIST ss
NORMDIST ss
[1 ( / ,0,1,1)]
( / ,0,1,0)

Point (3 of 4)
Lot size, Q = 10,000
Average demand during lead time, DL =5,000
Standard deviation of demand during lead time, s L
= 707
Safety inventory, ss = ROP − DL = 6,000−5,000 = 1,000
– –
1,000[1 (1,000 707,0,1,1)]
707 (1,000 707,0,1,0) 25
ESC NORMDIST
NORMDIST

 
( –
   
fr Q Q
ESC) 110,000 252 10,000 0.9975

Point (4 of 4)
Figure 12-2 Excel Solution of Example 12-4

Evaluating Safety Inventory Given Desired
Fill Rate (1 of 4)
• Expected shortage per replenishment cycle is
ESC = (1 − fr)Q
• No equation for ss
• Try values or use GOALSEEK in Excel

Fill Rate (2 of 4)
Desired fill rate, fr = 0.975
Lot size, Q = 10,000 boxes
Standard deviation of ddlt,    
L 2 500 707
ESC = (1−fr)Q = (1 − 0.975)10,000 = 250

Fill Rate (3 of 4)
–
250 1 s L s
L L
ss ss
ESC ss F f

 
 
   
   
 
   
   
 
– –
1 707
707 707
s s
ss ss
ss F f
 
   
 
   
 
   
 
 
– –
( )
250 1 707,0,1,1
707 707,0,1,0
ss NORMDIST ss
NORMDIST ss
  
 

• Use GOALSEEK to find safety inventory ss = 67 boxes

Fill Rate (4 of 4)
Figure 12-3 Spreadsheet to Solve for ss Using GOALSEEK

Impact of Desired Product Availability,
Lead Time, and Demand Uncertainty
• As desired product availability goes up the required
safety inventory increases
Table 12- Required Safety Inventory for Different Values of Fill Rate
Fill Rate Safety Inventory
97.5% 67
98.0% 183
98.5% 321
99.0% 499
99.5% 767

Impact of Desired Product Availability and
Uncertainty
• Goal is to reduce the level of safety inventory required in
a way that does not adversely affect product availability
1. Reduce the supplier lead time L
2. Reduce the underlying uncertainty of demand
(represented by σD )

Benefits of Reducing Lead Time and
Demand Uncertainty
D = 2,500/week σD, CSL = 0.95
( )
( 95) 9 800 3,948
D
ss NORMSINV CSL L
NORMSINV .

 
   
• If lead time is reduced to one week
(.95) 1 800 1,316
ss NORMSINV
   
• If standard deviation is reduced to 400
(.95) 9 400 1,974
ss NORMSINV
   

Adjusting Safety Inventory for Demand
Lumpiness and Seasonality
• Orders typically in large lots
– Demand at various stages in the supply chain tends
to be lumpy
– Raise safety inventory by half the average size of a
customer order
• Demand is often seasonal
– Fixing a ROP may lead to stockouts
– Keep ROP constant in terms of days of demand

Summary of Learning Objective 3 (1 of 2)
Given a desired cycle service level CSL, a lead time L, and
a standard deviation of periodic demand σD , the required
safety inventory ss for a continuous review policy is given
by ( ) D
ss NORMSINV CSL L
  Given a reorder point
ROP, a lead time L, a standard deviation of periodic
demand σD, and periodic demand D, the resulting cycle
service level is given by
( , , ,1)
D
CSL NORMDIST ROP D L L
 

Summary of Learning Objective 3 (2 of 2)
Given a level of safety inventory, one can evaluate the
resulting fill rate. Given a desired fill rate, one can evaluate
the required safety inventory. The required safety inventory
increases with an increase in desired product availability,
lead time, and uncertainty of periodic demand. In practice,
it is best to evaluate safety inventory in terms of days of
demand to account for seasonality of demand.

Impact of Supply Uncertainty on Safety
Inventory
• We incorporate supply uncertainty by assuming that lead
time is uncertain
D : Average demand per period
σL : Standard deviation of demand per period
L : Average lead time for replenishment
sL : Standard deviation of lead time

  
L L D L
D DL Lσ D s
2 2 2

Impact of Lead Time Uncertainty on Safety
Inventory (1 of 3)
Average demand per period, D = 2,500
Standard deviation of demand per period, σD = 500
Average lead time for replenishment, L = 7 days
Standard deviation of lead time, sL = 7 days
Mean ddlt, DL = DL = 2,500 × 7 = 17,500
 
 
   

L D L
L D s
2 2 2
2 2 2
Standard deviation of ddlt
7 500 2,500 7
17,500

Inventory (2 of 3)
• Required safety inventory
 
   
 

–
S L L
NORMSINV
1
( ) ( )
(0.90) 17,500
22,491tablets

Inventory (3 of 3)
Table 12-2 Required Safety Inventory as a Function of Lead Time
Uncertainty
sL σL ss (units) ss (days)
6 15,058 19,298 7.72
5 12,570 16,109 6.44
4 10,087 12,927 5.17
3 7,616 9,760 3.90
2 5,172 6,628 2.65
1 2,828 3,625 1.45
0 1,323 1,695 0.68

An increase in supply uncertainty significantly increases the
amount of safety inventory required for a given level of
product availability. Lead time uncertainty has a more
significant impact on the required safety inventory than lead
time itself. A reduction in supply uncertainty can help to
dramatically reduce the required safety inventory without
hurting product availability.

Impact of Aggregation on Safety
Inventory (1 of 5)
• How does aggregation affect forecast accuracy and
safety inventories
Di: Mean periodic demand in region i, i = 1, …, k
σi : Standard deviation of periodic demand in region i, i =,
…, k
ρij: Correlation of periodic demand for regions i, j,   
i j k
1

Inventory (2 of 5)
Total safety inventory in
decentralized option ( )
k
–1
S i
i=1
F CSL L 
  

 
 
 
2
–1
2
1
2
var 2 ;
var
k
k
C C
i i ij i j
i
i 1 i> j
C C
D
C C
D
D D ; D
D
D kD k k k
   

  


  

  
  
Simplified to
C
D D
k
 


Inventory (3 of 5)
Require safety inventory on aggregation
–


  

k
C
S D
i 1
F CSL L
1
( )
Holding – cost savings on aggregation per unit sold
–
1
1
( )
– k
C
S
i D
C
i
F CSL L H
D
 

   
  
 


Inventory (4 of 5)
• The safety inventory savings on aggregation increase with the
desired cycle service level CSL
replenishment lead time L
holding cost H
coefficient of variation of demand ( )
D D

The safety inventory savings on aggregation decrease as the
correlation coefficients increase

Impact of Correlation on Value of
Aggregation (1 of 3)
Standard deviation of weekly demand, σD =5
Replenishment, L = 2 weeks; Decentralized CSL = 0.9
Total required safety inventory,  
   
–
s D
ss k F CSL L σ
1
 
 
   
    
–
s
F
NORMSINV
1
4 0.9 2 5
4 0.9 2 5 36.25 cars
Aggregate ρ = 0
Standard deviation of weekly
demand at central outlet,   
C
D
σ 4 5 10
   
      
– C
s D
ss F L σ NORMSINV
1
0.9 0.9 2 10 18.12

Table 12-3 Safety Inventory in the Disaggregate and Aggregate
Options
Rho
Disaggregate
Safety Inventory
Aggregate
Safety Inventory
0 36.25 18.12
0.2 36.25 22.93
0.4 36.25 26.88
0.6 36.25 30.33
0.8 36.25 33.42
1.0 36.25 36.25
ρ

Inventory (5 of 5)
Figure 12-4 Square-Root Law

• Two possible disadvantages to aggregation
1. Increase in response time to customer order
2. Increase in transportation cost to customer

Trade-Offs of Physical Centralization (1 of 2)
• Use four regional or one national distribution center

   
1,000 week, 300, L 4weeks, 0.95
D
D CSL
• Four regional centers
Total required
safety inventory, 1
4 ( )
s D
ss F CSL L 

   
 
    
4 0.95 4 300 3,948
NORMSINV

Trade-Offs of Physical Centralization (2 of 2)
• One national distribution center,  = 0
Standard deviation
of weekly demand,
4 300 600
C
D
   
  

   C
s D
ss F L
1
0.95
 
   
NORMSINV 0.95 4 600 1,974
Decrease in holding costs ( )
3,948 1,974 $1,100 0.2
   
$394,765

Decrease in facility costs $150,000

Increase in transportation    
52 1,000 (13 10)
$624,000


Information Centralization
• Online systems that allow customers or stores to locate
stock
• Improves product availability without adding to
inventories
• Reduces the amount of safety inventory

Specialization (1 of 2)
• Inventory is carried at multiple locations
• Should all products should be stocked at every location?
– Required level of safety inventory
– Affected by coefficient of variation of demand
– Low demand, slow-moving items, typically have a
high coefficient of variation
– High demand, fast-moving items, typically have a
low coefficient of variation

Impact of Coefficient of Variation on Value
of Aggregation (1 of 2)
Table 12-4 Value of Aggregation at W.W. Grainger
Blank Motors Cleaner
Inventory is stocked in each store Blank Blank
Mean weekly demand per store 20 1,000
Standard deviation 40 100
Coefficient of variation 2.0 0.1
Safety inventory per store 132 329
Total safety inventory 211,200 526,400
Value of safety inventory $105,600,000 $15,792,000
Inventory is aggregated at the DC Blank Blank
Mean weekly aggregate demand 32,000 1,600,000
Standard deviation of aggregate demand 1,600 4,000
Coefficient of variation 0.05 0.0025
Aggregate safety inventory 5,264 13,159
Value of safety inventory $2,632,000 $394,770

Impact of Coefficient of Variation on Value
of Aggregation (2 of 2)
Table 12-4 [Continued]
Blank Motors Cleaner
Savings Blank Blank
Total inventory saving on aggregation $102,968,000 $15,397,230
Total holding cost saving on aggregation $25,742,000 $3,849,308
Holding cost saving per unit sold $15.47 $0.046
Savings as a percentage of product cost 3.09% 0.15%

Specialization (2 of 2)
Item Type Centralized Inventories Decentralized Inventories
Fast Moving Predictable
{Low Value}
Customer willing to pay
premium?
Low cost
Slow Moving
Unpredictable {High
Value}
Low cost Customer willing to pay
premium?
Figure 12-5 Specialization of Inventory Based on Product Type

Product Substitution
• The use of one product to satisfy demand for a different
product
1. Manufacturer-driven substitution
▪ Allows aggregation of demand
▪ Reduce safety inventories
▪ Influenced by the cost differential, correlation of
demand
2. Customer-driven substitution
▪ Allows aggregation of safety inventory

Component Commonality
• Without common components
– Uncertainty of demand for a component is the same
as for the finished product
– Results in high levels of safety inventory
• With common components
– Demand for a component is an aggregation of the
demand for the finished products
– Component demand is more predictable
– Component inventories are reduced

Value of Component Commonality (1 of 3)
27 servers, 3 components, 3 × 27 = 81 distinct components
Monthly demand = 5,000
Standard deviation = 3,000
Replenishment lead time = 1 month
CSL = 0.95
Total safety
inventory required
   

81 (0.95) 1 3,000
399,699units
NORMSINV
Safety inventory per
common component
 
   

NORMSINV 0.95 1 9 3,000
14,803.68units

• With component commonality
• Nine distinct components
Total safety inventory required = 9´14,803.68 =133,233

Table 12-5 Marginal Benefit of Component Commonality
Number of Finished Products
per Component
Safety
Inventory
Marginal Reduction in
Safety Inventory
Total Reduction in
Safety Inventory
1 399,699 Blank Blank
2 282,630 117,069 117,069
3 230,766 51,864 168,933
4 199,849 30,917 199,850
5 178,751 21,098 220,948
6 163,176 15,575 236,523
7 151,072 12,104 248,627
8 141,315 9,757 258,384
9 133,233 8,082 266,466

Postponement (1 of 2)
• Delay product differentiation or customization until closer
to the time the product is sold
– Have common components in the supply chain for
most of the push phase
– Move product differentiation as close to the pull phase
of the supply chain as possible
– Inventories in the supply chain are mostly aggregate

Postponement (2 of 2)
Figure 12-6 Supply Chain Flows without and with Postponement

Value of Postponement
100 different paint colors, D = 30/week, D = 10, L = 2
weeks, CSL = 0.95
Total required
safety inventory,  
1
100 s D
ss F CSL L 

   
 
100 0.95 2 10 2,326
NORMSINV
    
Standard deviation of base paint weekly demand,
100 10 100
C
D
   
 
1
(0.95) 2 100 233
D
C
s
ss F CSL L NORMSINV


      

Aggregation reduces the required safety inventory as long
as the demand across the aggregated regions is not
perfectly, positively correlated. The safety inventory savings
on aggregation increase with the desired C S L, the
replenishment lead time, the product holding cost, and the
coefficient of variation of demand. The safety inventory
savings on aggregation decrease as the correlation of
demand across regions increases. Firms can aggregate
inventories through physical aggregation, information
centralization, product substitution, component
commonality, and postponement.

Impact of Replenishment Policies on Safety
Inventory (1 of 4)
• Continuous Review Policies
σD : Standard deviation of demand per period
Mean demand during lead time, DL = D×L
Standard deviation of demand during lead time, L D
L
 

–1
( ) ( )
S L D L
ss F CSL NORMSINV CSL L ,ROP D ss
 
     

Inventory (2 of 4)
• Periodic Review Policies
– Lot size determined by prespecified order-up-to level
(OUL)
σD : Standard deviation of demand per period
T : Review interval
CSL : Desired cycle service level

Inventory (3 of 4)
Probability   
L T OUL CSL
(demand during )
Mean demand during T+L periods,   
T L
D T L D
( )
Std dev demand during T+L periods,  
  
T L D
T L
T L
OUL D ss

 
1
1
( ) ( )
Average lot size,
s D T L
T
Q D D T
 

 
   
  

Inventory (4 of 4)
Figure 12-7 Inventory Profile for Periodic Review Policy with
L = 4, T = 7

Evaluation Safety Inventory for a Periodic
Review Policy
D = 2,500, σD= 500, L = 2 weeks, T = 4 weeks
Mean demand during T + L periods, ( )
(2 4)2,500 15,000
T L
D T L D
  
  
Std dev demand
during T + L periods,
( 4 2)500 1,225
T L D
T L
 
  
  
   
 
1
0.90 1,225 1,570 boxes
15,000 1,570 16,570
S D L T L
T L
NORMSINV
OUL D ss
 

 

   
  
    

Whereas the required safety inventory is proportional to for
a continuous review policy, the required safety inventory
for a periodic review replenishment policy is proportional to
L T
 where T is the reorder interval. As a result, periodic
review replenishment policies require more safety
inventory than continuous review policies for the same
lead time and level of product availability.

Managing Safety Inventory in a
Multiechelon Supply Chain
• In multiechelon supply chains, stages often do not know
demand and supply distributions
• Inventory between a stage and the final customer is
called the echelon inventory
• Reorder points and order-up-to levels at any stage should
be based on echelon inventory
• Decisions must be made about the level of safety
inventory carried at different stages

In a multiechelon supply chain, it is important to manage
safety inventory across stages in a coordinated manner.
Increasing safety inventory at upstream stages allows
downstream stages to decrease the amount of safety
inventory they carry.

Managerial Levers to Reduce Safety
Inventory
• Reduction of supply uncertainty
– Sharing information
– Coordinated demand
• Reduction of lead times
– Delays contribute more to lead time than production
and transportation time
• Reduction of demand uncertainty
– Reduce information distortion through sharing
– Aggregate demand

The required level of safety inventory may be reduced and
product availability may be improved if a supply chain can
reduce demand uncertainty, replenishment lead times, and
the variability of lead times. A switch from periodic
monitoring to continuous monitoring can also help reduce
inventories. Another key managerial lever to reduce the
required safety inventories is to exploit aggregation. This
may be achieved by physically aggregating inventories,
virtually aggregating inventories using information
centralization, specializing inventories based on demand
volume, exploiting substitution, using component
commonality, and postponing product differentiation.

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