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

頁碼 作品 授權條件 作者 /來源

5本作品轉載自 CoolCLIPS 網站 (http://dir.coolclips.com/Popular/World_of_Industry/Food/Shopping_cart_full_of_groceries_vc012266.html ) ，瀏覽日期 2011/12/28 。依據著作權法第 46 、 52 、 65 條合理使用。

6 本作品轉載自 WIKIPEDIA(http://en.wikipedia.org/wiki/File:Palm-m505.jpg) ，瀏覽日期 2012/2/21 。

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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