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Inventory Management:

Safety Inventory ( I )

CC

3.0

Inventory Management: Safety Inventory ( I )

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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

Average

Inventory

Inventory

Time

Safety Inventory

Cycle Inventory

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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.

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Measuring Demand Uncertainty

k

i=1

DiP=

CV= s/m

P=KDDks=W

Coefficient of variation

The total demand during kperiod is normally distributed with a mean of Pand a standard deviation of W :

If demand in each period is independent and normally distributed with a

mean of D and a standard deviation of sD, then

W=

si2

+2 Cov(i,j)i=1 i>j

k=

si2

+2rsisji=1 i>j

k

Uncertainty within lead time

Assume that demand for each period i, i=1,.,k is normally distributedwith a mean Di and standard deviation si .

si2

i=1

k

+2rsisji>j

Dks

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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 beingmet

The CSLis equal to the probability of not having a stockout in a replenishmentcycle

A CSLof 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 bein stock for an order to be filled

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Product fill rate ( fr)

Order fill rate

Cycle service level (CSL)

Measuring Product Availability -- Page 5

On-handinventory

Orderreceived

Unfilled

demand

Filled

demand0

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

MicrosoftMicrosoft6
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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 Qis 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

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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 positionOH= on-hand inventory

SR= scheduled receipts (open orders)

BO= units backordered or allocated

IP = OH+SR-BO

Whenever a withdrawal brings IPdown to the reorder point (ROP), placean order for Q(fixed) units.

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Time

On-hand

inventoryOrderreceived

ROP

OH

IP

TBO2 TBO3L2 L3

Orderreceived

OH

Q

IP

Order

placed

ROP= average demand during lead time + safety stock

Continuous Review

System

ROP

Order

placed

L1

TBO1

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Time

On-hand

inventory

TBO2 TBO3L2 L3

Orderreceived

OH

Q

IP

Order

placed


Continuous Review

System

Order

placed

L1

TBO1

FIX

Orderreceived

ROP

OH

IP

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Time

On-hand

inventory

TBO2 TBO3L2 L3


Continuous Review

System

L1

TBO1

Orderreceived

OH

Q

IP

Order

placed

Order

placed

Orderreceived

ROP

IP

OH

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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

=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

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Evaluating Cycle Service Level and Safety Inventory

DLL LDLD ss == and

CSL=Function ( ROP,DL,sL)

CSL= Prob (Demand during lead time of L weeks ROP)

z=Fs-1(CSL)

ss=z LsD

Demand during lead time is normally distributed with a mean of DLand a

standard deviation of sL

ROP=DL+Z LsD

CSL

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Finding Safety Stock with a Normal Probability

Distribution for an 85 Percent SL

Safety stock = zsL

Averagedemand

duringlead time

Probability of stockout

(1.0 - 0.85= 0.15)

ROP

CSL= 85%?

zsL

1

2

3

4:->ROP

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Evaluating Cycle Service Level and Safety Inventory

DLL LDLD ss == and

CSL=Function ( ROP,DL,sL)

CSL= Prob (Demand during lead time of L weeks ROP)

z=Fs-1(CSL)

ss=z LsD

Demand during lead time is normally distributed with a mean of DLand a

standard deviation of sL

ROP=DL+Z LsD

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Example


Q= 10,000

ROP= 6,000

L= 2 weeks

D=2,500/week, D=500

2x2,500=5,000 DL=DL=

=F(ROP, DL, sL)=F(6000,5000,707)

=NORMDIST(6000,5000,707,1)=0.92

= 2 x500=707

CSL=Proability of not stocking out in a cycle

sL= L sD

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Normal Distribution in Excel

Commands Page 12)

)(NORMINV)(

)0,1,0,(NORMDIST)(

)1,1,0,(NORMDISTor)(NORMDIST)(

NormalStandard

1 ppF

xxf

xxxF

s

s

s

=

=

=

),,(NORMINV),,(

)0,,,(NORMDIST),,(

)1,,,(NORMDIST),,(

1smsm

smsm

smsm

ppF

xxf

xxF

=

=

=

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Normal Distribution in Excel

Demo)

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Example


Q= 10,000

ROP= 6,000

L= 2 weeks

D=2,500/week, sD=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)

sL= L sD

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Example


D=2,500/week

sD=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)xsL=NORMDIST(CSL)xsL

sL= L sD

ROP= 2x2,500+906=5,906

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Example


D=2,500/week

sD=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)xsL=NORMDIST(CSL)xsL

sL= L sD

ROP= 2x2,500+906=5,906

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Periodic Review System

Other names are:

fixed interval reorder system or periodic reorder system.

Decision Rule

Review the items inventory position IPeveryTtime periods. Place anorder equal to (OUL-IP) where OULis the target inventory, that is, the

desired IPjust after placing a new order.

The periodic review system has two parameters: TandOUL .

Here Qvaries, and time between orders (TBO) is fixed.

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On-hand

inventory

Periodic Review System

OUL

Time

Orderplaced

IP

L

T

L L

Order

received

OH

Q2

IP

Orderplaced

Q1Q3

Orderplaced

TProtection interval

OHIP1

IP3

IP2

OUL

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The new order must be large enough to make the inventory position,IP,

last not only beyond the next review, which is Tperiods from now, butalso for one lead time (L) after the next review. IPmust be enough to

cover demand over a protection interval of T + L.

OUL=

Finding

OUL

+Safety stock forprotection interval

D

1

s LT)CSL(FD)LT( s=

Average demand

during protectioninterval

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Administratively convenient (such as each Friday)

weeks)52(D

EOQT=

weeks4orweeks4.3)52(1200

100==T

Selecting the Reorder Interval

T

)

Example: Suppose D= 1200 /year and EOQ= 100

Approximation of EOQ

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Example


D=2,500/week

sD=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)xsT+L=Fs-1(0.9)xsT+L

OUL=DT+L+ss = 1,5000+1,570=16,570

DT+L= T+L sD= (4+2) x500=1,225

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Periodic System versus Continuous System

FeatureContinuous review

systemPeriodic review system

Order quantity Q-constant Q-variable

When to placeorder

When quantity on handdrops to the reorder level

When the review periodarrives

Recordkeeping Each time a withdrawal oraddition is made

Counted only at review period

Size of inventory Less than periodic system Larger than continuous system

Factors drivingsafety inventory

Demand uncertaintyReplenishment lead time

Demand uncertaintyReplenishment lead timeReorder interval

Type of items Higher-priced, critical, orimportant items

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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 policy

Expected shortage per replenishment cycle (ESC) is the average units

of demand that are not satisfied from inventory in stock per cycle

Q

Q-ESC

Q

ESC

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f (x) is the density function of demand


fr=1

- =


ESCX=ROP

=(X-ROP) f(x)dx




ESC

Q

Q-ESC

Q

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fr=1

- =


ESCX=ROP

=(X-ROP) f(x)dx




ESC

Q

Q-ESC

Q

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010

1101

1

,,,/

,,,/

LL

L

L

sL

L

s

ssNORMDIST

ssNORMDISTss

ssf

ssFssESC

ss

ss

s

s

=

=



fr=1- =


ESCX=ROP

=(X-ROP) f(x)dx




ESC

Q

Q-ESC

Q

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Proof

= =

=ROPx

dxxfROPxESC )()(

=

ssDx

Dx

L

L

L

dxLLessDx 2

1)(

22 2)( s

s

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sLdz

Proof

= =

=ROPx

dxxfROPxESC )()(

Substituting Z=(X-DL)/sL and dx=sLdz , we have

=

=

Lssz

z

L dzesszESC

s s

/

2/2

2

1)(

=

ssDx

Dx

L

L

L

dxLLessDx 2

1)(

22 2)( s

s

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Proof

=

=

Lssz

z

L dzesszESC

s s

/

2/2

2

1)(

= =

=ROPx

dxxfROPxESC )()(

Substituting Z=(X-DL)/sL and dx=sLdz , we have

=

ssDx

Dx

L

L

L

dxLLessDx 2

1)(

22 2)( s

s

=

=

Lssz

z dzss es /

2/2

2

1

=

Lssz

z

L dzz e

s s

/

2/2

2

1

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Proof

=

=

Lssz

z

L dzesszESC

s s

/

2/2

2

1)(

=

=

Lssz

z dzss es /

2/2

2

1

=

Lssz

z

L dzz e

s s

/

2/2

2

1

36

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Proof

=

=

Lssz

z

L dzesszESC

s s

/

2/2

2

1)(

=

Lssz

z

L dzz e

s s

/

2/2

2

1

= )]/(1[ Ls ssFss s

=

=

Lssz

z dzss es /

2/2

2

1

37

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Proof

=

=

Lssz

z

L dzesszESC

s s

/

2/2

2

1)(

=

Lssz

z

L dzz e

s s

/

2/2

2

1

= )]/(1[ Ls ssFss s

=

=

Lssz

z dzss es /

2/2

2

1

s=

s2L

2 2/ssw

w

L dw

2

1e

]2

1[)]/(1[

2

2

1

= Lss

LLs essFss s

ss

)2/:(note 2zw=

dw=2zdz/2

dw=zdz

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Proof

=

=

Lssz

z

L dzesszESC

s s

/

2/2

2

1)(

=

Lssz

z

L dzz e

s s

/

2/2

2

1

= )]/(1[ Ls ssFss s

=

=

Lssz

z dzss es /

2/2

2

1

=

22/2

2

1

Lssw

w

L dwe

s

s

]2

1[)]/(1[

2

2

1

= Lss

LLs essFss s

ss

)2/:(note 2zw=

ESC derivation

s

de

Lss

= 22/2 2

1

0

=22/22

1

Lss

es

2)/(21

2

1 Lsse

s

=

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Proof

=

=

Lssz

z

L dzesszESC

s s

/

2/2

2

1)(

=

Lssz

z

L dzz e

s s

/

2/2

2

1

= )]/(1[ Ls ssFss s

=

=

Lssz

z dzss es /

2/2

2

1

s=

s2L2/ssw

w

L

2

dw

2

1e )2/:(note

2zw=

]2

1[)]/(1[

2

2

1

= Lss

LLs essFss s

ss

)/()]/(1[ LsLLs ssfssFss sss =

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010

1101

1

,,,/

,,,/

LL

L

L

sL

L

s

ssNORMDIST

ssNORMDISTss

ssf

ssFssESC

ss

ss

s

s

=

=



fr=1- =


ESCX=ROP

=(X-ROP) f(x)dx




ESC

Q

Q-ESC

Q

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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)]

fr

= =0.997510,000

10,000-25

+707xNORMDIST(1000/707,0,1,1)

=25

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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)]

fr

= =0.997510,000

10,000-25

+707xNORMDIST(1000/707,0,1,1)

=25

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Factors Affecting Fill Rate

Safety inventory

Fill rate increases if safety inventory is increased. This also increases thecycle service level.

Lot size

Fill rate increases with the increase of the lot size even though cycle

service level does not change.

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Factors Affecting Fill Rate -- Page 42

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Ls CSLFss s=

)(1

)/()]/(1[ LsLLs ssfssFssESC sss =

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.

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Evaluating Safety Inventory

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Given Desired Fill Rate

==

L

sL

L

s

ssssFssESC f s

ss

1250

= 0,1,1,1250L

ssNORMDISTL

L

ssNORMSDISTsss

ss

If desired fill rate is fr= 0.975, how much safety inventory should be held?

ESC= (1 - fr)Q= 250Solve

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Excel-Demo

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47

Evaluating Safety Inventory

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Given Desired Fill Rate

==

L

sL

L

s

ssssFssESC f s

ss

1250

= 0,1,1,1250L

ssNORMDISTL

L

ssNORMSDISTsss

ss

If desired fill rate is fr= 0.975, how much safety inventory should be held?

ESC= (1 - fr)Q= 250Solve

48

Evaluating Safety Inventory Given Fill

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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).

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Two Managerial Levers to Reduce Safety Inventory

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Safety inventory increases with an increase in the lead time and

the standard deviation of periodic demand.

Reduce the underlying uncertainty of demand ( sD)

Reduce the supplier lead time (L)

k

If lead time decreases by a factor of k, safety inventory in the retailerdecreases by a factor of .

If sDis reduced by a factor of k, safety inventory decreases by afactor of k.

The reduction in sDcan 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 tothe supplier.

50

Impact of Supply Lead time) Uncertainty on

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Safety Inventory

222

LDL SDL = ss

Assume demand per period and replenishment lead time are normally distributed

D:Average demand per periodsD: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)

DLDL =

51

Example

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550,17725005007

500,177500,2

9.0)days(7)days(7500500,2

222222 ======

=====

LDL

L

LD

SDL

DLD

CSLSLD

ss

s

491,221 == Ls

CSLFss s

SL L 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

1,695

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/

5

CoolCLIPS

(http://dir.coolclips.com/Popular/World_of_Industry/Food/Shopping_cart_full_of_gr

oceries_vc012266.html)2011/12/28465265

6WIKIPEDIA(http://en.wikipedia.org/wiki/File:Palm-m505.jpg)2012/2/21

6Microsoft Office 2007Microsoft

465265

19

19

22

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