Inventory Management: Safety Inventory ( I )
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Inventory Management: Safety Inventory ( I )
Inventory Management: Safety Inventory ( I )
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第六單元: Inventory Management: Safety Inventory ( I )
郭瑞祥教授
1
Safety Inventory
–Demand uncertainty–Supply uncertainty
► 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
AverageInventory
Inventory
Time
Safety InventorySafety Inventory
Cycle InventoryCycle Inventory
2
Planning Safety Inventory
► Appropriate level of safety inventory is determined by
► Actions to improve product availability while reducing safety inventory
》 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.
3
Measuring Demand Uncertainty
k
i=1DiP=
CV=
P=KDDk
Coefficient of variation
– The total demand during k period is normally distributed with a mean of P and a standard deviation of :
–If demand in each period is independent and normally distributed with a
mean of D and a standard deviation of D , then
i2 +2 Cov(i,j)
i=1 i>j
k i
2 +2 iji=1 i>j
k
► Uncertainty within lead time
– Assume that demand for each period i, i=1,….,k is normally distributed with a mean Di and standard deviation i .
i2
i=1
k+2 ij
i>j
Dk
4
Measuring Product Availability
► Order fill rate
► Product fill rate ( fr )
► Cycle service level (CSL)
– 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%
– 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
– 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
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5
► Product fill rate ( fr )
► Order fill rate
► Cycle service level (CSL)
Measuring Product Availability -- Page 5
On-hand inventory
Orderreceived
Unfilled demand
Filled demand
0
– Don't run out of inventory in 6 out of 10
replenishment cycles
– An order for a total of 100 palms and has 90 in inventory
– Customer may order a palm along with a calculator. The order is filled only if both products are available.
→ CSL = 60%
→ fill rate > 60%
→ fill rate of 90%
– In the 40% of the cycles where a stockout
does occur, most of the customer demand
is satisfied from inventory
Cycle
Microsoft 。Microsoft 。6
► A replenishment policy consists of decisions regarding– When to reorder– How much to reorder.
► Continuous review– Inventory is continuously tracked and an order for a lot size Q is placed
when the inventory declines to the reorder point (ROP).
Replenishment Policies
► Periodic review– 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) .
Q
P
7
► A replenishment policy consists of decisions regarding– When to reorder– How much to reorder.
► Continuous review– Inventory is continuously tracked and an order for a lot size Q is placed
when the inventory declines to the reorder point (ROP).
Replenishment Policies
► Periodic review– Inventory status is checked at regular periodic intervals and an order is
placed to raise the inventory level to a specified threshold.
Q
P
8
Continuous Review System
► Other names are: Reorder point system, fixed order quantity system
► Decision rule
► The remaining quantity of an item is reviewed each time a withdrawal is
► made from inventory, to determine whether it is time to reorder.
► Inventory position
》IP = inventory position》OH = on-hand inventory》SR = scheduled receipts (open orders)》BO = units backordered or allocated
IP = OH+SR-BO
– Whenever a withdrawal brings IP down to the reorder point (ROP), place an order for Q (fixed) units.
9
Time
On-hand inventory Order
received
ROP
OH
IP
TBO2 TBO3
L2 L3
Orderreceived
OH
Q
IP
Orderplaced
ROP = average demand during lead time + safety stock
Continuous Review System
ROPOrderplaced
L1
TBO1
10
Time
On-hand inventory
TBO2 TBO3
L2 L3
Orderreceived
OH
Q
IP
Orderplaced
ROP = average demand during lead time + safety stock
Continuous Review System
Orderplaced
L1
TBO1
FIX
Orderreceived
ROP
OH
IP
11
Time
On-hand inventory
TBO2 TBO3
L2 L3
ROP = average demand during lead time + safety stock
Continuous Review System
L1
TBO1
Orderreceived
OH
Q
IP
Orderplaced
Orderplaced
Orderreceived
ROP
IP
OH
12
ExampleGiven 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
=ROP-DL=6,000-5,000=1,000► Safety inventory,ss
► Cycle inventory
► Average inventory
► Average flow time
=Q/2=10,000/2=5,000
=5,000+1,000=6,000
= Average inventory / Throughput=6,000/2,500
=2.4weeks
13
Evaluating Cycle Service Level and Safety Inventory
DLL LDLD and
► CSL=Function ( ROP,DL,L )
CSL= Prob (Demand during lead time of L weeks ROP)
z=Fs-1(CSL)
ss=z LD
Demand during lead time is normally distributed with a mean of DL and a
standard deviation of L
ROP=DL+Z LD
CSL
14
Finding Safety Stock with a Normal Probability Distribution for an 85 Percent
CSL
Safety stock = z L
Averagedemand
duringlead time
Probability of stockout(1.0 - 0.85= 0.15)
ROP
CSL = 85%?
z L
1
23
4:->ROP
15
Evaluating Cycle Service Level and Safety Inventory
DLL LDLD and
CSL=Function ( ROP,DL,L )
CSL= Prob (Demand during lead time of L weeks ROP)
z=Fs-1(CSL)
ss=z LD
Demand during lead time is normally distributed with a mean of DL and a
standard deviation of L
ROP=DL+Z LD
16
ExampleGiven the following data
Q = 10,000 ROP = 6,000 L = 2 weeks D=2,500/week, σD=500
2x2,500=5,000► DL=DL=
=F(ROP, DL, L )=F(6000,5000,707)
=NORMDIST(6000,5000,707,1)=0.92
= 2 x500=707 ► CSL=Proability of not stocking out in a cycle
► L= L D
17
Normal Distribution in Excel Commands (Page 12)
►
►
)(NORMINV)(
)0,1,0,(NORMDIST)(
)1,1,0,(NORMDIST or )(NORMDIST)(
Normal Standard
1 ppF
xxf
xxxF
s
s
s
),,(NORMINV),,(
)0,,,(NORMDIST),,(
)1,,,(NORMDIST),,(
1
ppF
xxf
xxF
18
Normal Distribution in Excel (Demo)
臺灣大學 郭瑞祥老師臺灣大學 郭瑞祥老師19
ExampleGiven the following data
Q = 10,000 ROP = 6,000 L = 2 weeks D=2,500/week, D=500 CSL=0.9
2x2,500=5,000► DL=DL=
=F(ROP, DL, sL )=F(6000,5000,707)
=NORMDIST(6000,5000,707,1)=0.92
= 2 x500=707
► ss=Fs-1(CSL)
► L= L D
20
ExampleGiven the following data
D=2,500/week
D=500 L = 2 weeks Q = 10,000, CSL=0.9
2x2,500=5,000► DL=DL=
=1.282x707=906
= 2 x500=707 ► ss=Fs
-1(CSL)xL=NORMDIST(CSL)xL
► L= L D
► ROP= 2x2,500+906=5,906
21
ExampleGiven the following data
D=2,500/week
D=500
L = 2 weeks Q = 10,000, CSL=0.9
2x2,500=5,000 DL=DL=
=1.282x707=906
= 2 x500=707 ss=Fs
-1(CSL)xL=NORMDIST(CSL)xL
L= L D
ROP= 2x2,500+906=5,906
臺灣大學 郭瑞祥老師臺灣大學 郭瑞祥老師22
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
On-hand inventory
Periodic Review SystemOUL
Time
Orderplaced
IP
L
T
L L
Orderreceived
OH
Q2
IP
Orderplaced
Q1Q3
Orderplaced
TProtection interval
OHIP1
IP3
IP2
OUL
24
► 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 =
Finding OUL
+Safety stock forprotection interval
D1
s LT)CSL(FD)LT(
Average demand during protection interval
25
► Administratively convenient (such as each Friday)
weeks)52(D
EOQT
weeks4 or weeks4.3 )52(1200
100T
Selecting the Reorder Interval (T )
► Example: Suppose D = 1200 /year and EOQ = 100
► Approximation of EOQ
26
ExampleGiven the following data
D=2,500/week D=500 L = 2 weeks T= 4weeks CSL=0.9
(4+2)x2,500=15,000► DT+L=(T+L)D=
=1,570► ss=Fs-1(CSL)xT+L=Fs
-1(0.9)xT+L
► OUL=DT+L+ss = 1,5000+1,570=16,570
► DT+L= T+L D= (4+2) x500=1,225
27
Periodic System versus Continuous System
Feature Continuous review
system Periodic review system
Order quantity Q-constant Q-variable
When to place order
When quantity on hand drops to the reorder level
When the review period arrives
Recordkeeping Each time a withdrawal or addition is made
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
Demand uncertainty Replenishment lead time Reorder interval
Type of items Higher-priced, critical, or important items
28
Evaluating Fill Rate Given a Replenishment Policy
f (x) is density function of demand distribution during the lead time
fr=1- =
► In the case of normal distribution, we have
ESCX=ROP(X-ROP) f(x)dx
► For a continuous review policyExpected shortage per replenishment cycle (ESC) is the average units of demand that are not satisfied from inventory in stock per cycle
QQ-ESC
QESC
29
Evaluating Fill Rate Given a Replenishment Policy
f (x) is the density function of demand distribution during the lead time
fr=1- =
► In the case of normal distribution, we have
ESCX=ROP(X-ROP) f(x)dx
► For a continuous review policyExpected shortage per replenishment cycle (ESC) is the average units of demand that are not satisfied from inventory in stock per cycle
ESCQ
Q-ESCQ
30
Evaluating Fill Rate Given a Replenishment Policy
f (x) is density function of demand distribution during the lead time
fr=1- =
► In the case of normal distribution, we have
ESCX=ROP(X-ROP) f(x)dx
► For a continuous review policyExpected shortage per replenishment cycle (ESC) is the average units of demand that are not satisfied from inventory in stock per cycle
ESCQ
Q-ESCQ
31
Evaluating Fill Rate Given a Replenishment Policy
010
1101
1
,,,/
,,,/
LL
L
LsL
Ls
ssNORMDIST
ssNORMDISTss
ssf
ssFssESC
f (x) is density function of demand distribution during the lead time
fr=1- =
► In the case of normal distribution, we have
ESCX=ROP(X-ROP) f(x)dx
► For a continuous review policyExpected shortage per replenishment cycle (ESC) is the average units of demand that are not satisfied from inventory in stock per cycle
ESCQ
Q-ESCQ
32
Proof
ROPx
dxxfROPxESC )()(
ssDx
Dx
L
L
L
dxLLessDx 2
1)(
22 2)(
WIKIPEDIA
WIKIPEDIA33
Ldz
Proof
ROPx
dxxfROPxESC )()(
Substituting Z=(X-DL)/L and dx=Ldz , we have
Lssz
zL dzesszESC
/
2/2
2
1)(
ssDx
Dx
L
L
L
dxLLessDx 2
1)(
22 2)(
34
Proof
Lssz
zL dzesszESC
/
2/2
2
1)(
ROPx
dxxfROPxESC )()(
Substituting Z=(X-DL)/L and dx=Ldz , we have
ssDx
Dx
L
L
L
dxLLessDx 2
1)(
22 2)(
Lssz
z dzss e /
2/2
2
1
Lssz
zL dzz e
/
2/2
2
1
35
Proof
Lssz
zL dzesszESC
/
2/2
2
1)(
Lssz
z dzss e /
2/2
2
1
Lssz
zL dzz e
/
2/2
2
1
36
Proof
Lssz
zL dzesszESC
/
2/2
2
1)(
Lssz
zL dzz e
/
2/2
2
1
)]/(1[ Ls ssFss
Lssz
z dzss e /
2/2
2
1
37
Proof
Lssz
zL dzesszESC
/
2/2
2
1)(
Lssz
zL dzz e
/
2/2
2
1
)]/(1[ Ls ssFss
Lssz
z dzss e /
2/2
2
1
2L
2 2/ssw
wL dw
2
1 e
]2
1[)]/(1[
2
2
1
L
ss
LLs essFss
)2/ :(note 2zw
dw=2zdz/2dw=zdz
38
Proof
Lssz
zL dzesszESC
/
2/2
2
1)(
Lssz
zL dzz e
/
2/2
2
1
)]/(1[ Ls ssFss
Lssz
z dzss e /
2/2
2
1
22/2
2
1
Lssw
wL dwe
]2
1[)]/(1[
2
2
1
L
ss
LLs essFss
)2/ :(note 2zw
ESC derivation
deLss
22/2 2
1
0
22/22
1
Lsse
2)/(21
2
1 Lsse
39
Proof
Lssz
zL dzesszESC
/
2/2
2
1)(
Lssz
zL dzz e
/
2/2
2
1
)]/(1[ Ls ssFss
Lssz
z dzss e /
2/2
2
1
2L2/ssw
wL
2
dw2
1 e )2/ :(note 2zw
]2
1[)]/(1[
2
2
1
L
ss
LLs essFss
)/()]/(1[ LsLLs ssfssFss
40
Evaluating Fill Rate Given a Replenishment Policy
010
1101
1
,,,/
,,,/
LL
L
LsL
Ls
ssNORMDIST
ssNORMDISTss
ssf
ssFssESC
f (x) is density function of demand distribution during the lead time
fr=1- =
► In the case of normal distribution, we have
ESCX=ROP(X-ROP) f(x)dx
► For a continuous review policyExpected shortage per replenishment cycle (ESC) is the average units of demand that are not satisfied from inventory in stock per cycle
ESCQ
Q-ESCQ
41
ExampleFor a continuous review system with the following data
Lot size ,Q=10,000
DL=5,000
L = 707
► ss=ROP-DL=6,000-5,000=1,000
► ESC= -1,000[1-NORMDIST(1000/707,0,1,1)]
fr= =0.997510,00010,000-25
+707xNORMDIST(1000/707,0,1,1) =25
42
Excel-DemoFor a continuous review system with the following data
Lot size ,Q=10,000
DL=5,000
L = 707
ss=ROP-DL=6,000-5,000=1,000
ESC= -1,000[1-NORMDIST(1000/707,0,1,1)]
fr= =0.997510,00010,000-25
+707xNORMDIST(1000/707,0,1,1) =25
臺灣大學 郭瑞祥老師43
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
Ls CSLFss )(1
)/()]/(1[ LsLLs ssfssFssESC
fr = 1- ESC/Q
fr = 1- ESC/Q
CSL = F(ROP, DL, sL) is independent of Q
► Safety inventory
Fill rate increases if safety inventory is increased. This also increases the cycle service level.
, f, , CSLESCss r
► Lot size Fill rate increases on increasing the lot size even though cycle service level does not change.
45
Evaluating Safety Inventory Given Desired Fill Rate
LsL
Ls
ssssFssESC f
1250
0,1,1,1250
L
ssNORMDISTLL
ssNORMSDISTss
If desired fill rate is fr = 0.975, how much safety inventory should be held?
ESC = (1 - fr)Q = 250
Solve
46
Excel-Demo
臺灣大學 郭瑞祥老師臺灣大學 郭瑞祥老師47
Evaluating Safety Inventory Given Desired Fill Rate
LsL
Ls
ssssFssESC f
1250
0,1,1,1250
L
ssNORMDISTLL
ssNORMSDISTss
If desired fill rate is fr = 0.975, how much safety inventory should be held?
ESC = (1 - fr)Q = 250
Solve
48
Evaluating Safety Inventory Given Fill Rate
Fill Rate Safety Inventory
97.5% 67
98.0% 183
98.5% 321
99.0% 499
99.5% 767
The required safety inventory grows rapidly with an increase in the desired product availability (fill rate).
49
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 underlying uncertainty of demand ( D )
► Reduce the supplier lead time (L)
k– If lead time decreases by a factor of k, safety inventory in the retailer
decreases by a factor of .
– If D is reduced by a factor of k, safety inventory decreases by a factor of k.
– The reduction in D can be achieved by reducing forecast uncertainty, such as by sharing demand information through the supply chain.
– It is important for the retailer to share some of the resulting benefits to the supplier.
50
Impact of Supply (Lead time) Uncertainty on Safety Inventory
222LDL SDL
► Assume demand per period and replenishment lead time are normally distributed
D:Average demand per period
D: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,L2)
DLDL
51
Example
550,17725005007
500,177500,2
9.0)days(7)days(7500500,2
222222
LDL
L
LD
SDL
DLD
CSLSLD
491,221 Ls CSLFss
SL σL ss(units) ss(days)
6 15,058 7.72
5 12,570 6.44
4 10,087 5.17
3 7,616 3.90
2 5,172 2.65
1 2,828 1.45
0 1,323 0.68
► Suppose we have
Required safety inventory,
► A reduction in lead time uncertainty can help reduce safety inventory
19,298
16,109
12,927
9,760
6,628
3,625
1,69552
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6 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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