第六單元：Inventory Management: Safety Inventory ( I )
Inventory Management:
Safety Inventory ( I )
郭瑞祥教授
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1
Safety Inventory
► Safety Inventory is inventory carried for the purpose of satisfying
demand that exceeds the amount forecasted for a given period.
► Purposes of holding safety inventory
–Demand uncertainty
–Supply uncertainty
Inventory
Average
Inventory
Cycle Inventory
Safety Inventory
Time
2
Planning Safety Inventory
► Appropriate level of safety inventory is determined by
》 Uncertainty of both demand and supply
– Uncertainty increases, then safety inventory increases.
》 Desired level of product availability Desired level of product availability
– increases, then safety inventory increases.
► Actions to improve product availability while reducing safety inventory
3
Measuring Demand Uncertainty
► Uncertainty within lead time
– Assume that demand for each period i, i=1,….,k is normally distributed
with a mean Di and standard deviation si .
– The total demand during k period is normally distributed with a mean of P
and a standard deviation of W :
W=
k
P=
2Di
s
i=1
i +2
k
i=1
Cov(i,j) =
i>j
k
si2 +2 rsisj
i=1
i>j
–If demand in each period is independent and normally distributed with a
mean of D and a standard deviation of sD , then
P=KD
W = k sD
 Coefficient of variation
CV= s/m
4
k
si2 +2 rsisj
i=1
k sD
i>j
Measuring Product Availability
► Product fill rate ( fr )
–
–
CoolCLIPS網站
The fraction of product demand that is satisfied from product in inventory
It is equivalent to the probability that product demand is supplied from available
inventory
► Order fill rate
– The fraction of orders that are filled from available inventory
– Order fill rates tend to be lower than product fill rates because all products must be
in stock for an order to be filled
► Cycle service level (CSL)
–
–
–
5
The fraction of replenishment cycles that end with all the customer demand being
met
The CSL is equal to the probability of not having a stockout in a replenishment
cycle
A CSL of 60 percent will typically result in a fill rate higher than 60%
Measuring Product Availability -- Page 5
► Product fill rate ( fr )
– An order for a total of 100 palms and has 90 in inventory → fill rate of 90%
► Order fill rate
– Customer may order a palm along with a calculator. The order is filled only if both
products are available.
► Cycle service level (CSL)
Order
received
– Don't run out of inventory in 6 out of 10
replenishment cycles → CSL = 60%
– In the 40% of the cycles where a stockout
does occur, most of the customer demand
is satisfied from inventory
On-hand
inventory
0
→ fill rate > 60%
6
Filled
demand
Cycle
Microsoft。
Microsoft。
Unfilled
demand
Replenishment Policies
► A replenishment policy consists of decisions regarding
– When to reorder
– How much to reorder.
► Continuous review
Q
– Inventory is continuously tracked and an order for a lot size Q is placed
when the inventory declines to the reorder point (ROP).
► Periodic review
P
– Inventory status is checked at regular periodic intervals and an order is
placed to raise the inventory level to a specified threshold, i.e. order up
to level (OUL) .
7
Replenishment Policies
► A replenishment policy consists of decisions regarding
– When to reorder
– How much to reorder.
► Continuous review
Q
– Inventory is continuously tracked and an order for a lot size Q is placed
when the inventory declines to the reorder point (ROP).
► Periodic review
P
– Inventory status is checked at regular periodic intervals and an order is
placed to raise the inventory level to a specified threshold.
8
Continuous Review System
► The remaining quantity of an item is reviewed each time a withdrawal is
►
made from inventory, to determine whether it is time to reorder.
► Other names are: Reorder point system, fixed order quantity system
► Inventory position
IP = OH+SR-BO
》IP = inventory position
》OH = on-hand inventory
》SR = scheduled receipts (open orders)
》BO = units backordered or allocated
► Decision rule
– Whenever a withdrawal brings IP down to the reorder point (ROP), place
an order for Q (fixed) units.
9
Continuous Review System
ROP = average demand during lead time + safety stock
IP
On-hand
inventory
IP
Order
received
Order
received
Q
OH
OH
ROP
Order
placed
Order
placed
L1
TBO1
10
L3
L2
TBO2
TBO3
Time
Continuous Review System
ROP = average demand during lead time + safety stock
IP
On-hand
inventory
IP
Order
received
Order
received
Q
OH
OH
ROP
Order
placed
Order
placed
L1
TBO1
11
FIX
TBO2
L3
L2
TBO3
Time
Continuous Review System
ROP = average demand during lead time + safety stock
IP
On-hand
inventory
IP
Order
received
Order
received
Q
OH
OH
ROP
Order
placed
Order
placed
L1
TBO1
12
L3
L2
TBO2
TBO3
Time
Example
Given the following data





Average demand per week, D = 2,500
Standard deviation of weekly demand, sD =500
Average lead time for replacement, 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
► Cycle inventory =Q/2=10,000/2=5,000
► Average inventory =5,000+1,000=6,000
► Average flow time = Average inventory / Throughput=6,000/2,500
=2.4weeks
13
Evaluating Cycle Service Level and Safety Inventory
CSL= Prob (Demand during lead time of L weeks  ROP)
Demand during lead time is normally distributed with a mean of DL and a
standard deviation of sL
DL = DL and s L = Ls D
► CSL=Function ( ROP,DL,sL )
ROP=DL+Z LsD
ss=z LsD
z=Fs-1(CSL)
14
CSL
Finding Safety Stock with a Normal Probability
Distribution for an 85 Percent CSL
?
CSL = 85%
1
Average
demand
during
lead time
Probability of stockout
(1.0 - 0.85= 0.15)
4:->ROP
ROP
3
2
z sL
Safety stock = z sL
15
Evaluating Cycle Service Level and Safety Inventory
CSL= Prob (Demand during lead time of L weeks  ROP)
Demand during lead time is normally distributed with a mean of DL and a
standard deviation of sL
DL = DL and s L = Ls D
 CSL=Function ( ROP,DL,sL )
ROP=DL+Z LsD
ss=z LsD
z=Fs-1(CSL)
16
Example
Given the following data




Q = 10,000
ROP = 6,000
L = 2 weeks
D=2,500/week, σD=500
► DL=DL= 2x2,500=5,000
► sL= L sD = 2 x500=707
► CSL=Proability of not stocking out in a cycle
=F(ROP, DL, sL )=F(6000,5000,707)
=NORMDIST(6000,5000,707,1)=0.92
17
Normal Distribution in Excel
Commands (Page 12)
► F ( x, m,s ) = NORMDIST ( x, m,s ,1)
f ( x, m,s ) = NORMDIST ( x, m,s ,0)
F 1( p, m,s ) = NORMINV ( p, m,s )
► Standard Normal
Fs ( x ) = NORMDIST ( x ) or NORMDIST ( x,0,1,1)
fs ( x ) = NORMDIST ( x,0,1,0)
Fs1( p ) = NORMINV ( p )
18
Normal Distribution in Excel
(Demo)
19
臺灣大學 郭瑞祥老師
Example
Given the following data





Q = 10,000
ROP = 6,000
L = 2 weeks
D=2,500/week, sD=500
CSL=0.9
► DL=DL= 2x2,500=5,000
► sL= L sD = 2 x500=707
► ss=Fs-1(CSL)
=F(ROP, DL, sL )=F(6000,5000,707)
=NORMDIST(6000,5000,707,1)=0.92
20
Example
Given the following data





D=2,500/week
sD=500
L = 2 weeks
Q = 10,000,
CSL=0.9
► DL=DL= 2x2,500=5,000
► sL= L sD = 2 x500=707
► ss=Fs-1(CSL)xsL=NORMDIST(CSL)xsL
=1.282x707=906
► ROP= 2x2,500+906=5,906
21
Example
Given the following data
 D=2,500/week

sD=500
 L = 2 weeks
 Q = 10,000,
 CSL=0.9
 DL=DL= 2x2,500=5,000
 sL= L sD = 2 x500=707
 ss=Fs-1(CSL)xsL=NORMDIST(CSL)xsL
=1.282x707=906
 ROP= 2x2,500+906=5,906
22
臺灣大學 郭瑞祥老師
Periodic Review System
► Other names are:
fixed interval reorder system or periodic reorder system.
► Decision Rule
Review the item’s inventory position IP every T time periods. Place an
order equal to (OUL-IP) where OUL is the target inventory, that is, the
desired IP just after placing a new order.
► The periodic review system has two parameters: T and
OUL.
► Here Q varies, and time between orders (TBO) is fixed.
23
Periodic Review System
OUL
OUL
IP
On-hand
inventory
Order
placed
IP
Order
received
Order
placed
Q1
Q3
Q2
IP1
OH
OH
IP3
Order
placed
IP2
L
T
Protection interval
24
L
L
T
Time
Finding OUL
► The new order must be large enough to make the inventory position, IP,
last not only beyond the next review, which is T periods from now, but
also for one lead time (L) after the next review. IP must be enough to
cover demand over a protection interval of T + L.
► OUL =
=
25
Average demand
+
during protection
interval
(T  L)D
Safety stock for
protection interval
 Fs1 (CSL)  T  L  sD
Selecting the Reorder Interval (T )
► Administratively convenient (such as each Friday)
► Approximation of EOQ
EOQ
T=
(52) weeks
D
► Example: Suppose D = 1200 /year and EOQ = 100
100
T=
(52) = 4.3 weeks or 4 weeks
1200
26
Example
Given the following data





D=2,500/week
sD=500
L = 2 weeks
T= 4weeks
CSL=0.9
► DT+L=(T+L)D= (4+2)x2,500=15,000
► DT+L= T+L sD= (4+2) x500=1,225
► ss=Fs-1(CSL)xsT+L=Fs-1(0.9)xsT+L =1,570
► OUL=DT+L+ss = 1,5000+1,570=16,570
27
Periodic System versus Continuous System
Feature
Order quantity
When to place
order
Recordkeeping
Q-constant
When quantity on hand
drops to the reorder level
Each time a withdrawal or
addition is made
Q-variable
When the review period
arrives
Counted only at review period
Size of inventory
Less than periodic system Larger than continuous system
Factors driving
safety inventory
Demand uncertainty
Replenishment lead time
Type of items
28
Continuous review
Periodic review system
system
Higher-priced, critical, or
important items
Demand uncertainty
Replenishment lead time
Reorder interval
Evaluating Fill Rate Given a Replenishment Policy
► For a continuous review policy
Expected shortage per replenishment cycle (ESC) is the average units
of demand that are not satisfied from inventory in stock per cycle

ESC =  (X-ROP) f(x)dx
X=ROP
f (x) is density function of demand
distribution during the lead time
fr=1- ESC = Q-ESC
Q
Q
► In the case of normal distribution, we have
29
Evaluating Fill Rate Given a Replenishment Policy
► For a continuous review policy
Expected shortage per replenishment cycle (ESC) is the average units
of demand that are not satisfied from inventory in stock per cycle

ESC =  (X-ROP) f(x)dx
X=ROP
f (x) is the density function of demand
distribution during the lead time
fr=1- ESC = Q-ESC
Q
Q
► In the case of normal distribution, we have
30
Evaluating Fill Rate Given a Replenishment Policy
► For a continuous review policy
Expected shortage per replenishment cycle (ESC) is the average units
of demand that are not satisfied from inventory in stock per cycle

ESC =  (X-ROP) f(x)dx
X=ROP
f (x) is density function of demand
distribution during the lead time
fr=1- ESC = Q-ESC
Q
Q
► In the case of normal distribution, we have
31
Evaluating Fill Rate Given a Replenishment Policy
► For a continuous review policy
Expected shortage per replenishment cycle (ESC) is the average units
of demand that are not satisfied from inventory in stock per cycle

ESC =  (X-ROP) f(x)dx
X=ROP
f (x) is density function of demand
distribution during the lead time
fr=1- ESC = Q-ESC
Q
Q
► In the case of normal distribution, we have

 ss 
 ss 
ESC = ss 1  Fs    s Lfs  
 s L 
sL 

= ss1  NORMDIST ss / s L , 0 , 1 , 1
 s LNORMDIST ss / s L , 0 , 1 , 0
32
Proof
( x D )2 2s 2
1
L dx
ESC =  ( x  ROP )f ( x )dx =  ( x  DL  ss )
e L
x =ROP
x =DL  ss
2 s L


WIKIPEDIA
33
WIKIPEDIA
Proof
( x D )2 2s 2
1
L dx
ESC =  ( x  ROP )f ( x )dx =  ( x  DL  ss )
e L
x =ROP
x =DL  ss
2 s L


Substituting Z=(X-DL)/sL and dx=sLdz , we have

ESC =
s
z =ss /
34
L
1 z2 / 2
( zs L  ss )
e
dz
2
sLdz
Proof
( x D )2 2s 2
1
L dx
ESC =  ( x  ROP )f ( x )dx =  ( x  DL  ss )
e L
x =ROP
x =DL  ss
2 s L




2 /2
1 L)/s
Substituting
Z=(X-D
z 2 / 2L and dx=sLdz 1, we zhave
e dz
= ss 
e dz  s L  z
z =ss / s
2
2
 z =ss / s L
L
1 z2 / 2
ESC =  ( zs L  ss )
e
dz
2
z =ss / s
L
35
Proof

ESC =
s
z =ss /
= ss
36
L
1 z2 / 2
( zs L  ss )
e
dz
2


z =ss / s
L
1
2
e
z2 / 2

1
dz  s L  z
z =ss / s
2
L
e
z2 / 2
dz
Proof

ESC =
s
z =ss /
= ss
L
1 z2 / 2
( zs L  ss )
e
dz
2


z =ss / s
L
1
2
e
z2 / 2
= ss[1  Fs (ss / s L )] 
37

1
dz  s L  z
z =ss / s
2
L
e
z2 / 2
dz
Proof

ESC =
s
z =ss /
= ss
L
1 z2 / 2
( zs L  ss )
e
dz
2


z =ss / s
L
1
2
e
z2 / 2

1
dz  s L  z
z =ss / s
2
L

= ss[1  Fs (ss / s L )]  s L
1 w
e dw
2

w =ss2 / 2 s 2L
1
= ss[1  Fs (ss / s L )]  s L [
2
38
e
1  ss
 
2 s
 L
e
z2 / 2




2
]
dz
(note : w = z 2 / 2)
dw=2zdz/2
dw=zdz
Proof
ESCderivation 1
0
z2 / 2
1(ss / s )2
ESC =  ( zs L  ss )
e
dz


L
1
1  2 1

2
z =ss / s
=
e
e d =
e
 L
2
2
ss / 2s L
2
2
 =ss2 / 2s L2 2

1
1
z2 / 2
z2 / 2
e dz
= ss 
e dz  s L  z
z =ss / s
2
z =ss / s
L
L 2
= ss[1  Fs (ss / s L )]  s L


w =ss2 / 2s L2
1
= ss[1  Fs (ss / s L )]  s L [
2
39
1
2
e
1  ss
 
2 s
 L




2
]
e
w
2
(note
:
w
=
z
/ 2)
dw
Proof

ESC =
s
z =ss /
= ss
L
1 z2 / 2
( zs L  ss )
e
dz
2


z =ss / s
L
1
2
e
z2 / 2

1
dz  s L  z
z =ss / s
2
L

= ss[1  Fs (ss / s L )]  s L
1 w
e dw
2

w =ss2 / 2 s 2L
= ss[1  Fs (ss / s L )]  s L [
1
2
e
1  ss
 
2 s
 L




2
= ss[1  Fs (ss / s L )]  s Lfs (ss / s L )
40
e
z2 / 2
]
dz
(note : w = z 2 / 2)
Evaluating Fill Rate Given a Replenishment Policy
► For a continuous review policy
Expected shortage per replenishment cycle (ESC) is the average units
of demand that are not satisfied from inventory in stock per cycle

ESC =  (X-ROP) f(x)dx
X=ROP
f (x) is density function of demand
distribution during the lead time
fr=1- ESC = Q-ESC
Q
Q
► In the case of normal distribution, we have

 ss 
 ss 
ESC = ss 1  Fs    s Lfs  
 s L 
sL 

= ss1  NORMDIST ss / s L , 0 , 1 , 1
 s LNORMDIST ss / s L , 0 , 1 , 0
41
Example
For a continuous review system with the following data



Lot size ,Q=10,000
DL=5,000
sL = 707
► ss=ROP-DL=6,000-5,000=1,000
► ESC= -1,000[1-NORMDIST(1000/707,0,1,1)]
+707xNORMDIST(1000/707,0,1,1)
=25
10,000-25
fr= 10,000
42
=0.9975
Excel-Demo
For a continuous review system with the following data
 Lot size ,Q=10,000


DL=5,000
sL = 707
 ss=ROP-DL=6,000-5,000=1,000
 ESC= -1,000[1-NORMDIST(1000/707,0,1,1)]
+707xNORMDIST(1000/707,0,1,1)
=25
10,000-25
fr= 10,000
43
=0.9975
臺灣大學 郭瑞祥老師
Factors Affecting Fill Rate
► Safety inventory
Fill rate increases if safety inventory is increased. This also increases the
cycle service level.
► Lot size
Fill rate increases with the increase of the lot size even though cycle
service level does not change.
44
Factors Affecting Fill Rate -- Page 42
► Safety inventory
Fill rate increases if safety inventory is increased. This also increases the
cycle service level.
ss  , ESC  , fr  ,CSL 
ESC = ss[1  Fs (ss / s L )]  s Lfs (ss / s L )
fr = 1- ESC/Q
ss = Fs1(CSL)  s L
► Lot size
Fill rate increases on increasing the lot size even though cycle service
level does not change.
fr = 1- ESC/Q
CSL = F(ROP, DL, sL) is independent of Q
45
Evaluating Safety Inventory
Given Desired Fill Rate
 If desired fill rate is fr = 0.975, how much safety inventory should be held?
ESC = (1 - fr)Q = 250
Solve

 ss 
 ss 
ESC = 250 = ss 1  Fs    s L f s  
 s L 
sL 






 ss 
 ss

250 = ss 1 NORMSDIST 
,1,1,0 
  s NORMDIST 
L
 s 
s


 L 
 L

46
Excel-Demo
47
臺灣大學 郭瑞祥老師
Evaluating Safety Inventory
Given Desired Fill Rate
 If desired fill rate is fr = 0.975, how much safety inventory should be held?
ESC = (1 - fr)Q = 250
Solve

 ss 
 ss 
ESC = 250 = ss 1  Fs    s L f s  
 s L 
sL 






 ss 
 ss

250 = ss 1 NORMSDIST 
,1,1,0 
  s NORMDIST 
L
 s 
s


 L 
 L

48
Evaluating Safety Inventory Given Fill
Rate
The required safety inventory grows rapidly with an
increase in the desired product availability (fill rate).
49
Fill Rate
Safety Inventory
97.5%
67
98.0%
183
98.5%
321
99.0%
499
99.5%
767
Two Managerial Levers to Reduce Safety Inventory
Safety inventory increases with an increase in the lead time and
the standard deviation of periodic demand.
► Reduce the supplier lead time (L)
– If lead time decreases by a factor of k, safety inventory in the retailer
decreases by a factor of k .
– It is important for the retailer to share some of the resulting benefits to
the supplier.
► Reduce the underlying uncertainty of demand ( sD )
– If sD is reduced by a factor of k, safety inventory decreases by a
factor of k.
– The reduction in sD can be achieved by reducing forecast uncertainty,
such as by sharing demand information through the supply chain.
50
Impact of Supply (Lead time) Uncertainty on
Safety Inventory
► Assume demand per period and replenishment lead time are normally distributed
D:Average demand per period
sD:Standard deviation of demand per period (demand uncertainty)
L: Average lead time for replenishment
SL:Standard deviation of lead time (supply uncertainty)
► Consider continuous review policy, we have:
Demand during the lead time is N(DL,sL2)
DL = DL
s
51
L
s
= L
2
D
 D 2SL2
Example
► Suppose we have
D = 2,500
s
D
= 500
L = 7(days ) SL = 7(days ) CSL = 0.9
DL = DL = 2,500  7 = 17,500
s
L
s
= L
2
D
 D 2SL2 = 7  5002  25002  72 = 17,550
Required safety inventory, ss = Fs1CSL 
s
L
= 22,491
► A reduction in lead time uncertainty can help reduce safety inventory
52
SL
σL
6
ss(days)
15,058
ss(units)
19,298
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
7.72
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本作品轉載自CoolCLIPS網站
(http://dir.coolclips.com/Popular/World_of_Industry/Food/Shopping_cart_full_of_gr
oceries_vc012266.html )，瀏覽日期2011/12/28。依據著作權法第46、52、65條
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本作品轉載自 WIKIPEDIA(http://en.wikipedia.org/wiki/File:Palm-m505.jpg)，瀏
覽日期2012/2/21。
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本作品轉載自Microsoft Office 2007多媒體藝廊，依據Microsoft 服務合約及著
作權法第46、52、65條合理使用。
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臺灣大學 郭瑞祥老師
19
臺灣大學 郭瑞祥老師
22
臺灣大學 郭瑞祥老師
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臺灣大學 郭瑞祥老師
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本作品轉載自
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瀏覽日期2012/2/21。
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臺灣大學 郭瑞祥老師
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臺灣大學 郭瑞祥老師
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臺灣大學 郭瑞祥老師