scs to c inventory

7/29/2019 Scs to c Inventory

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Safety Inventories

Chapter 11 of Chopra


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Why to hold Safety Inventory?

Desire for quick product availability

Ease of search for another supplier

I want it now culture

Demand uncertainty

Short product life cycles

Safety inventory


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Measures

Measures of demand uncertainty Variance of demand

Ranges for demand

Delivery Lead Time, LMeasures of product availability

Stockout, what happens? Backorder (patient customer, unique product or big cost advantage) or

Lost sales. I. Cycle service level(CSL), % of cycles with no stockout

II. Product fill rate (fr), % of products sold from the shelf

Order fill rate, % of orders Equivalent to product fill rate if orders contain one product


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Service measures: CSL and fr are different

CSL is 0%, fill rate is almost 100%

CSL is 0%, fill rate is almost 0%

inventory

inventory

time

time

0

0


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Replenishment policies

When to reorder?

How much to reorder? Most often these decisions are related.

Continuous Review: Order fixed quantity when totalinventory drops below Reorder Point (ROP).

- ROP meets the demand during the lead time L.

- One has to figure out the ROP.

Information technology facilitates continuous review.


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DemandDuring Lead time

variance).(mean,or),,(Mostly

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Normal Density Function

Mean

95.44%

99.74%

x

)1,_,,(normdist:at x(cdf)functionCumulative

)0,_,,(normdist:at x(pdf)functionDensity

:functionslstatisticaExcel

devstmeanx

devstmeanx

normdist(x,.,.,0)

Prob

normdist(x,.,.,1)frequency



8

Cumulative Normal Density

)_,,(norminv:prob""atcdfoffunctionInverse

)1,_,,(normdist:at x(cdf)functionCumulative

:functionslstatisticaExcel

devstmeanprob

devstmeanx

0

1

x

normdist(x,mean,st_dev,1)

prob

norminv(prob,mean,st_dev)



9

Demand During Lead Time Determines ROP

Suppose that demands are identically and independently distributed.To mean identically and independently distributed, use iid.

),(then),(If

)(and)(

2

1

2

2

11

LLRNDRND

LDVarLRDE

L

i

ii

L

i

i

L

i

i

1,,,Normdist,;1

LLRaLLRaFaDPL

i

i

)(/)(:DofnvariatiooftCoefficien DEDVarcv

F is the cumulative density function of the demand in a single period,

say a day. The second equality above holds if demand is Normal.



10

Optimal Safety Inventory Levels

Lead Times

time

inventory

Shortage

An inventory cycle

ROP

Q



11

I. Cycle Service Level: ROP CSL

),;(LevelServiceCycle LLRROPFCSL

Cycle service level: percentage of cycles with stock out

ROP]timeleadduring[Demandinventory]t[Sufficien

inventorysufficienthascyclesingleay thatProbabilit0.7

7.0

otherwise1stockout,hascycleaif0Write10

1010111011

:cycles10considerexampleFor

CSL

CSL

CSL

ROP: Reorder point



12

I. Cycle Service Level for Normal Demands

ondistributinormalstandardObtaining)1,0(

StDevby theDividing),(

meanout theTaking),(

),(

notationvariance)(mean,Recall),,(

)1,0(

2

2

2

R

R

N

R

R

R

R

R

L

LRROPNP

L

LRROP

L

LRLLRNP

LRROPLRLLRNP

ROPLLRNP

NLLRROPFCSL

The last equality is a property of the Normal distribution.



13

Example: Finding CSL for given ROP

R = 2,500 /week; = 500

L = 4 weeks; Q = 10,000;ROP = 16,000

ss =ROPL R =

Cycle service level,

Average Inventory =

Average Flow Time =Average inventory/Thruput=

LtimeleadduringdemandofStdev

F ROP L R L( ; , )

If you wish to compute Average Inventory = Q/2 + ss



14

Safety Inventory: CSL ROP

L

LCSLFLCSLF

LRLLRCSLFss

LRROPss

LLRCSLFROPLLRROPFCSL

R

RR

L,0,1)Norminv(CS

)1,0;(),0;(

),,(

:demandddistributenormallyFor

:stockSafety

),,(or),,(

11

1

1

The last two equalities are by properties of the Normal distribution.

Very important remark: Safety inventory is a more general concept.

It exists without lead time. It is the stock held minus the expected demand.



15

Finding ROP for given CSL

R = 2,500/week; = 500

L = 4 weeks; Q = 10,000; CSL = 0.90

Factors driving safety inventory

Replenishment lead time

Demand uncertainty

ss F CSL L

ROP L R ss

1 0 1( ; , )



16

II. Fill rate:

Expected shortage per cycle

ESCis the expected shortage per cycle

ESC is not a percentage, it is the number of units, also see next page

ESC E

ESC x ROP f x dxx ROP

(max{

( ) ( )

Demand during lead time- ROP,0})

=

ROPDemand

ROPDemand

if

if

0

ROPDemandShortage



17

Inventory and Demand during Lead Time

0

ROP

Demand

During LT

LT

Inventory DLT: DemandDuring LT

0

0

ROPInventory=

ROP-DLTUpside

down



18

Shortage and Demand during Lead Time

0

ROP

Demand

During LT

LTShortage

DLT:Deman

dDuringLT0

0

ROP

Shortage=

DLT-ROP

Upside

down



19


First let us study shortage during the lead time

DLT.ofpdfisfwhere)()(

))max(,0(shortageExpected

D

ROPD

D dDDfROPD

ROPDLTE

Ex:

4

1

4

110)}-(11max{0,

4

210)}-(10max{0,

4

110)}-(9max{0,

)()}(d)}(max{0,shortageExpected

Shortage?Expected,

1/4probwith11

2/4probwith10

1/4probwith9

,10

11

10d

3

1i

33

22

11

dDPROPpROPd

pd

pd

pd

DROP

ii



20


If demand is normal:

Does ESC decrease or increase with ss, L?

Does ESC decrease or increase with expected value of demand?

2170-172

)10(102

1061)12(10

212

6110

261

61)10(shortageExpected

Shortage?Expected),12,6(,10

2212

10

212

10

D

DD

DD

dDD

UniformDROP

Ex:

ESC ss normdistss

LL normdist

ss

L

1 0 11 010

, , , , , ,



21

Fill Rate

Fill rate: Proportion of customer demand satisfied from stock

Q: Order quantity

Q

ESCfr 1



22

Finding the Fill Rate

ss fr

= 500;L = 2 weeks; ss=1000; Q = 10,000;

Fill Rate (fr) = ?

fr= (Q -ESC)/Q = (10,000 - 25.13)/10,000 = 0.9975.

ESC ss normdistss

LL normdist

ss

L

ESC normdist

nomdist

ESC

1 0 11 0 1 0

1000 1 1000 707 0 11

707 1000 707 0 1 0

2513

, , , , , ,

( ( / , , , )

( / , , , )

.


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Finding Safety Inventory for a Fill Rate:

fr ss

If desired fill rate isfr = 0.975, how much safety

inventory should be held?

Clearly ESC = (1 -fr)Q = 250

Try some values of ss or use goal seek of Excel to solve

0,1,0,7077071,1,0,7071250

ssnormdist

ssnormdistss


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Evaluating Safety Inventory For Given Fill Rate

Fill Rate Safety Inventory

97.5% 67

98.0% 18398.5% 321

99.0% 499

99.5% 767

Safety inventory is very sensitive to fill rate. Is fr=100% possible?


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Factors Affecting Fill Rate

Safety inventory: If Safety inventory is up,

Fill Rate is up

Cycle Service Level is up.

Lot size: If Lot size Q is up, Cycle Service Level does not change. Reorder point,

demand during lead time specify Cycle Service

Level.

Expected shortage per cycle does not change. Safetystock and the variability of the demand during the

lead time specify the Expected Shortage per Cycle.

Fill rate is up.


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To Cut Down the Safety Inventory

Reduce the Supplier Lead Time Faster transportation

Air shipped semiconductors from Taiwan

Better coordination, information exchange, advance retailerdemand information to prepare the supplier

Textiles; Obermeyer case

Space out orders equally as much as possible

Reduce uncertainty of the demand Contracts

Better forecasting to reduce demand variability


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Lead Time Variability

2222

11

:)()( L

L

i

i

L

i

i sRLDVarLRDE

Suppliers lead time may be uncertain:

timeleadofVariancetime.leadAverage 2 sL

22211)1,0;()1,0;( sRLCSLFCSLFss

L

0,1,0;1,1,0;1

L

L

L

ssnomdist

ssnormdistssESC

The formulae do not change:


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Impact of Lead Time Variability, s

R = 2,500/day; = 500

L = 7 days; Q = 10,000; CSL = 0.90

StDev of LT ss Jump in ss

0 1695 -

1 3625 1930

2 6628 3003

3 9760 31324 12927 3167

5 16109 3182

6 19298 3189


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Methods of Accurate Response to Variability

Centralization

Physical, Laura Ashley

Information

Virtual aggregation, Barnes&Nobles stores

Specialization, what to aggregate

Product substitution

Raw material commonality - postponement


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Centralization: Inventory Pooling

K

iji

ji

K

ii

CK

ii

CDDRR

11

22

1

),cov(2)(;

),(111

RD

),( CC

R

Which of two systems provides a higher level of service for a givensafety stock?

Consider locations and demands:

),(333 RD

),(222

RD

),(444 RD

With k locations centralized, mean and variance of

K

i

i

CDD

1


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Sum of Random Variables Are Less Variable

K

i

i

K

i

i

C

11

2

K

i

i

K

i

i

K

i

K

jii

jii

C

1

2

11 1

2 2

When they are independent,cov(Di,Dj)=0

When they are perfectly positively correlated,cov(Di,Dj)=i j

When they are perfectly negatively correlated,cov(Di,Dj)= - ij

K

i

i

K

i

i

K

i

K

jii

jii

C

11

2

1 1

22


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Factors Affecting Value of Aggregation

When to aggregate? Statistical checks: Positive correlation and

Coefficient of Variation.

Aggregation reduces variance almost always except when products are

positively correlated

Aggregation is not effective when there is little variance to begin with.When coefficient of variation of demand is relatively small (variance w.r.t.

the mean is small), do not bother to aggregate.

In real life,

Is the electricity demand in Arlington and Plano are positively or negativelycorrelated? Is there an underlying factor which affects both in the same

direction? Note that a big portion of electricity is consumed for

heating/cooling.

Are the Campbell soup sales over time positively or negatively correlated?

How many soups can you drink per day?


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Impact of Correlation on Aggregated Safety

Inventory (Aggregating 4 outlets)

Safety stocks are proportional to the StDev of the demand.

With four locations, we have total ss proportional to 4*

If four locations are all aggregated, ss proportional to 4* with correlation 1

ss proportional to 2* with correlation 0

Benefit=SS before - SS after / SS before


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Impact of Correlation on Aggregated

Safety Inventory (Aggregating 4 outlets)

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

Benefit

Benefit=(SS before - SS after) / SS before


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EX 11.8: W.W. Grainger a supplier of

Maintenance and Repair products

About 1600 stores in the US

Produces large electric motors and

industrial cleaners

Each motor costs $500; Demand is iidNormal(20,40x40) at each store

Each cleaner costs $30; Demand is iid

Normal(1000,100x100) at each store

Which demand has a larger coefficient ofvariation?

How much savings if motors/cleaners

inventoried centrally?


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Use CSL=0.95

Supply lead time L=4 weeks for motors and cleaners

L L,0,1)Norminv(CSss:demandddistributenormallyForFor a single store

Motor safety inventory=Norminv(0.95,0,1) 2 (40)=132

Cleaner safety inventory=Norminv(0.95,0,1) 2 (100)=329

Value of motor ss=1600(132)(500)=$105,600,000Value of cleaner ss=1600(329)(30)=$15,792,000

Standard deviation of demands after aggregating 1600 stores

Standard deviation of Motor demand=40(40)=1,600

Standard deviation of Cleaner demand=40(100)=4,000

For the aggregated store

Motor safety inventory=Norminv(0.95,0,1) 2 (1600)=5,264

Cleaner safety inventory=Norminv(0.95,0,1) 2 (4,000)=13,159

Value of motor ss=5264(500)=$2,632,000

Value of cleaner ss=13,159(30)=$394,770


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EX. 11.8: Specialization: Impact ofcv on

Benefit From 1600-Store Aggregation , h=0.25

Motors Cleaner

Mean demand/wk 20 1,000

SD of demand 40 100

Disaggregate cv 2 0.1

Value/Unit $500 $30

Disaggregate ss value $105,600,000 $15,792,000

Aggregate cv 0.05 0.0025

Aggregate ss value $2,632,000 $394,770

Inventory cost savings $102,968,000 $15,397,230

Holding Cost Saving $25,742,000 $3,849,308

Saving / Unit 2574200/(1600*20*52)=$15.47 $0.046


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Slow vs Fast Moving Items

Low demand = Slow moving items, vice versa.

Repair parts are typically slow moving items

Slow moving items have high coefficient of variation, vice versa.

Stock slow moving items at a central store

Buying a best seller at Amazon.com vs. a Supply Chain book vs. a Banach spacesbook, which has a shorter delivery time?

- Why cannot I find a driver-side-door lock cylinder for my 1994 ToyotaCorolla at Pep Boys?

- Your instructor on March 26 2005.

- Case Interview books are not in our s.k.u. list. You must check with ourcentral stores.

- Store keeper at Barnes and Nobles at Collin Creek, March 2002.


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Product Substitution

Manufacturer driven Customer driven

Consider: The price of the products substituted for each other and

the demand correlations

One-way substitution Army boots. What if your boot is large? Aggregate?

Two-way substitution:

Grainger motors; water pumps model DN vs IT. Similar products, can customer detect specifications.

If products are very similar, why not to eliminate one of them?


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Component Commonality. Ex. 11.9

Dell producing 27 products with 3 components

(processor, memory, hard drive)

No product commonality: A component is used in only 1

product. 27 component versions are required for eachcomponent. A total of 3*27 = 81 distinct components

are required.

Component commonality allows for component

inventory aggregation.


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Max Component Commonality

Only three distinct versions for each component. Processors: P1, P2, P3. Memories: M1, M2, M3. Hard drives: H1, H2, H3

Each combination ofcomponents is a distinct product. Acomponent is used in 9products.

Each way you can go from left to right is a product.

P1

P2

P3

M1

M2

M3

H1

H2

H3

Left Right


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Example 11.9: Value of Component Commonality

in Safety Inventory Reduction

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

1 2 3 4 5 6 7 8 9

SS

# of products a component is used in

Aggregation provides reduction in total standard deviation.


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Standardization

StandardizationExtent to which there is an absence of variety in a product,

service or process

The degree of Standardization?

Standardized products are immediately available to

customers

Who wants standardization?

The day we sell standard products is the day we lose asignificant portion of our profit

A TI manager on November 1, 2005


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Advantages of Standardization

Fewer parts to deal with in inventory & manufacturing Less costly to fill orders from inventory

Reduced training costs and time

More routine purchasing, handling, and inspectionprocedures

Opportunities for long production runs, automation

Need for fewer parts justifies increased expenditures on

perfecting designs and improving quality controlprocedures.


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Disadvantages of Standardization

Decreased variety results in less consumer appeal.

Designs may be frozen with too many imperfections remaining.

High cost of design changes increases resistance to improvements

Who likes optimal Keyboards?

Standard systems are more vulnerable to failure

Epidemics: People with non-standard immune system stop theplagues.

Computer security: Computers with non-standard software stop the

dissemination of viruses. Another reason to stop using Microsoft products!

I T i C


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InventoryTransportation Costs:

Eastern Electric Corporation: p.427

Major appliance manufacturer, buys motors from Westview motors in Dallas

Annual demand = 120,000 motors

Cost per motor = $120; Weight per motor 10 lbs.

Current order size = 3,000 motors

30,000 pounds = 300 cwt

1 cwt =centum weight = 100 pounds; Centum = 100 in Latin.

Lead time = 1 + the number of days in transit

Safety stock carried = 50% of demand during delivery lead time

Holding cost = 25%

Evaluate the mode of transportation for all unit discounting based on shipment

weight


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AM Rail proposal:

Over 20,000 lbs at 0.065 per lb in 5 days

For the appliance manufacturer No fixed cost of ordering besides the transportation cost

No reason to transport at larger lots than 2000 motors, whichmake 20,000 lbs.

Cycle inventory=Q/2=1,000 Safety inventory=(6/2)(120,000/365)=986

In-transit inventory

All motors shipped 5 days ago are still in-transit

5-days demand=(120,000/365)5=1,644

Total inventory held over an average day=3,630 motors Annual holding cost=3,630*120*0.25=$108,900

Annual transportation cost=120,000(10)(0.065)=$78,000

I t T t ti t d ff E t


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InventoryTransportation trade off: Eastern

Electric Corporation, see p.426-8 for details

Alternative(Lot size)

TransportCost

CycleInventory

SafetyInventory

TransitInventory

InventoryCost

TotalCost

AM Rail(2,000)

$78,000120000(0.65)

1,000 986 1,644120000(5/365)

$108,900 $186,900

NortheastTrucking(1,000)

$90,000 500 658 986 $64,320 $154,320

Golden(500)

$96,000120000(0.80)

250 658 986120000(3/365)

$56,820 $152,820

Golden(2,500)

$86,400 1,250 658 986 $86,820 $173,220

Golden(3,000)

$78,000 1,500 658 986 $94,320 $172,320

Golden(4,000)

$67,500 2,000 658 986 $109,320 $176,820

If fast transportation not justified cost-wise, need to consider rapid response

Ph i l I A i


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Physical Inventory Aggregation:

Inventory vs. Transportation cost: p.428

HighMed Inc. producer of medical equipment

sold directly to doctors

Located in Wisconsin serves 24 regions in USA

As a result of physical aggregation Inventory costs decrease

Inbound transportation cost decreases

Inbound lots are larger

Outbound transportation cost increases


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Inventory Aggregation at HighMed

Highval ($200, .1 lbs/unit) demand in each of 24 territories

H = 2, H = 5

Lowval ($30/unit, 0.04 lbs/unit) demand in each territory

L = 20, L = 5

UPS rate: $0.66 + 0.26x {for replenishments}

FedEx rate: $5.53 + 0.53x {for customer shipping}

Customers order 1 H + 10 L


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Inventory Aggregation at HighMed

CurrentScenario

Option A Option B

# Locations 24 24 1

Reorder Interval 4 weeks 1 week 1 week

Inventory Cost $54,366 $29,795 $8,474

Shipment Size 8 H + 80 L 2 H + 20 L 1 H + 10 L

Transport Cost $530 $1,148 $14,464

Total Cost $54,896 $30,943 $22,938

If shipment size to customer is 0.5H + 5L, total cost of option B

increases to $36,729.


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Summary of Cycle and Safety Inventory

Reduce Buffer Inventory

Economies of Scale

Supply / Demand

Variability

Seasonal

Variability

Cycle Inventory Safety Inventory Seasonal Inventory

Match Supply & Demand

Reduce fixed cost

Aggregate acrossproducts

Volume discounts

Promotion on Sell

thru

Quick Response measures

Reduce Info UncertaintyReduce lead time

Reduce supply

uncertainty

Accurate Response measuresAggregation

Component commonalityand postponement

M C t i ti


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Mass customization:

A strategy of producing standardized goods or services,

but incorporating some degree of customization

Modular design

Delayed differentiation

Mass Customization

M C t i ti I C t i


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Mass Customization I: Customize

Services Around Standardized Products

DEVELOPMENT PRODUCTION MARKETING DELIVERY

Deliver customized services as

well as standardized productsand services

Market customized services with standardized

products or services

Continue producing standardized products or services

Continue developing standardized products or services

Source: B. Joseph PineWarranty for contact lenses:

M C t i ti II C t


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Mass Customization II: Create

Customizable Products and Services


Deliver standard (but

customizable) productsor services

Market customizable products or services

Produce standard (but customizable) products or services

Develop customizable products or services

Customizing the look of screen with windows operating systemGillette sensor adjusting to the contours of the face

Mass C stomi ation III: Pro ide Q ick


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Mass Customization III: Provide Quick

Response Throughout Value Chain


Reduce Delivery Cycle Times

Reduce selection and order processing cycle

times

Reduce Production cycle time

Reduce development cycle time

Skiing parkas manufactured abroad vs. in the U.S.A.:

Mass Customization IV: Provide Point of


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Mass Customization IV: Provide Point of

Delivery Customization


Deliver standardize portion

Market customized products or services

Produce standardized portion centrally

Develop products where point of delivery customization is feasible

Point of delivery

customization

Paint mixingLenscrafters for glasses.

Mass Customization V: Modularize


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Mass Customization V: Modularize

Components to Customize End Products


Deliver customized product

Market customized products or services

Produce modularized components

Develop modularized products

Computer industry, Dell computers:


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Modular Design

Modular designis a form of standardization in which component parts aresubdivided into modules that are easily replaced or interchanged.

Goodexample: Dell uses same components to assemble variouscomputers.

Bad example: Earlier Ford SUVs shared the lower body with Ford cars. Ugly example:

It allows:

easier diagnosis and remedy of failures

easier repair and replacement

simplification of manufacturing and assembly

Types of Modularity for Mass


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Types of Modularity for Mass

Customization

Component Sharing Modularity, Dell

Cut-to-Fit Modularity,Gutters that do not require

seams

Bus Modularity, E-books

Mix Modularity, Paints

Sectional Modularity, LEGO

+ =


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Periodic Review

Order at fixed time intervals (T apart) to raise total inventory(on hand + on order) to Order up to Level (OUL)

Inventory

OUL

LT

T

LT

T+LT

OUL must cover

the Demand during


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Periodic Review Policy: Safety Inventory

T: Reorder interval

R: Standard deviation of demand per unit time

L+T: Standard deviation of demand during L+T periods

OUL: Order up to level

ssOUL

CSLFss

TL

RLT

R

R

LT

LT

LT

LT

)1,0;(

)(

1


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Example: Periodic Review Policy

R = 2,500/week; R = 500

L = 2 weeks; T= 4 weeks; CSL = 0.90

What is the required safety inventory?

Factors driving safety inventory Demand uncertainty

Replenishment lead time

Reorder interval

1570)1,0;(1

LTCSLFss


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Periodic vs Continuous Review

Periodic review ss covers the uncertainty over

[0,T+L], T periods more than ss in continuous case.

Periodic review ss is larger.

Continuous review is harder to implement, use it forhigh-sales-value per time products

scs to c inventory

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