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
DeliveryLead Time, L
Measures 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
inventory
CSL is 0%, fill rate is almost 100%
0 time
inventory
CSL is 0%, fill rate is almost 0%
0 time
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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 total
inventory 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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Demand During Lead time
Di demand in period i.
Mostly Di Normal ( Ri , i2 ), or N (mean,variance).
f i , Fi probabilit y density and cumulative density functions for D i
Ri E( Di ) Di f i ( Di )dDi Var( Di ) i2 E{(Di Ri ) 2 } ( Di Ri ) 2 f i (Di )dDi
cov(Di , D j ) i2, j E{(Di Ri )(D j R j )}
( Di Ri )(D j R j ) f i , j ( Di , D j )dDi dD j
i2, j /( i j ) correlation coefficient
L L
E ( Di ) Ri by the linearity of integratio n
i 1 i 1
L L L L L L
Var( Di ) cov(Di , D j ) cov(Di , D j ) i
2
i 1 i 1 j 1 i 1 i 1 j 1 6
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Normal Density Function
frequency
normdist(x,.,.,1) normdist(x,.,.,0)
Prob
Mean x
95.44%
99.74%
Excel statistical functions :
Density function (pdf) at x : normdist( x, mean, st _ dev,0)
Cumulative function (cdf)at x : normdist( x, mean, st _ dev,1) 7
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Cumulative Normal Density
1
prob
normdist(x,mean,st_dev,1)
0 x
norminv(prob,mean,st_dev)
Excel statistical functions :
Cumulative function (cdf)at x : normdist( x, mean, st _ dev,1)
Inversefunction of cdf at " prob": norminv( prob, mean, st _ dev) 8
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Demand During Lead Time Determines ROP
Suppose that demands are identically and independently distributed.
To mean identically and independently distributed, use iid.
L L
E ( Di ) LR and Var( Di ) L 2
i 1 i 1
L
If Di N ( R, ) then Di N ( LR, L 2 )
2
i 1
L
P Di a F a; LR, L Normdist a, LR, L ,1
i 1
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.
Coefficient of variation of D : cv Var( D) / E( D) 9
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Optimal Safety Inventory Levels
inventory
An inventory cycle
Q
ROP
time
Lead Times
Shortage
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I. Cycle Service Level: ROP CSL
Cycle service level: percentage of cycles with stock out
For example consider10 cycles:
11 0 111 0 1 0 1
CSL Write0 if a cycle has stockout,1 otherwise
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CSL 0.7
CSL 0.7 Probabilit y that a single cycle has sufficient inventory
[Sufficient inventory] [Demand during lead time ROP]
ROP: Reorder point
CSL Cycle Service Level F ( ROP ; R L, L )
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I. Cycle Service Level for Normal Demands
CSL F ( ROP, R L, L R ) Recall N (mean,variance) notation
P N ( R L, L R ) ROP
2
P N ( R L, L ) R L ROP R L
2
R Taking out the mean
N ( R L, L R ) R L ROP R L Dividing by theStDev
2
P
L R L R
N ( 0 ,1)
ROP R L
P N (0,1)
Obtaining standard normal distribution
L R
The last equality is a property of the Normal distribution.
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Example: Finding CSL for given ROP
R = 2,500 /week; = 500
L = 4 weeks; Q = 10,000; ROP = 16,000
Stdev of demand during lead time L
ss = ROP – L R =
Cycle service level, F ( ROP; L R, L )
If you wish to compute Average Inventory = Q/2 + ss
Average Inventory =
Average Flow Time =Average inventory/Thruput=
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Safety Inventory: CSL ROP
CSL F ( ROP, R L, L R ) or ROP F 1 (CSL, R L, L R )
Safety stock ss : ROP R L
For normally distributed demand :
ss F 1 (CSL, R L, L R ) R L
F 1 (CSL;0, L ) F 1 (CSL;0,1) L
Norminv(CSL,0,1) L
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.
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Finding ROP for given CSL
R = 2,500/week; = 500
L = 4 weeks; Q = 10,000; CSL = 0.90
ss F 1 (CSL;0,1) L
ROP L R ss
Factors driving safety inventory
– Replenishment lead time
– Demand uncertainty
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II. Fill rate:
Expected shortage per cycle
ESC is the expected shortage per cycle
ESC is not a percentage, it is the number of units, also see next page
Demand ROP if Demand ROP
Shortage
0 if Demand ROP
ESC E (max{Demand during lead time - ROP,0})
ESC = ( x ROP) f ( x)dx
x ROP
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Inventory and Demand during Lead Time
ROP 0
Inventory=
ROP ROP-DLT
Upside
0 Inventory down
DLT: Demand
During LT
LT
Demand 0
During LT 17
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Shortage and Demand during Lead Time
ROP 0
DLT: Demand During LT
Shortage=
DLT-ROP
Upside ROP
0 down
Shortage
LT
Demand 0
During LT 18
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Expected shortage per cycle
First let us study shortage during the lead time
Expected shortage E (0, max(DLT ROP))
( D ROP) f
D ROP
D ( D)dD wheref D is pdf of DLT.
Ex:
d1 9 with prob p1 1/4
ROP 10, D d 2 10 with prob p2 2/4, Expected Shortage?
d 11 with prob p 1/4
3 3
3 11
Expected shortage max{0,(d i ROP)} pi (d ROP)}P( D d )
i 1 d 10
1 2 1 1
max{0,(9 - 10)} max{0,(10 - 10)} max{0,(11- 10)}
4 4 4 4
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Expected shortage per cycle
Ex:
ROP 10, D Uniform(6,12), Expected Shortage?
D 12
1 D2 1 122 1 102
12
1
Expected shortage ( D 10) dD 10D 10(12) 10(10)
D 10
6 6 2
D 10
6 2
6 2
172 - 170 2
If demand is normal:
ss ss
ESC ss1 normdist ,0,11 L normdist
, ,0,1,0
L L
Does ESC decrease or increase with ss, L?
Does ESC decrease or increase with expected value of demand?
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Fill Rate
Fill rate: Proportion of customer demand satisfied from stock
Q: Order quantity
ESC
fr 1
Q
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Finding the Fill Rate
ss fr
= 500; L = 2 weeks; ss=1000; Q = 10,000;
Fill Rate (fr) = ?
ss ss
ESC ss 1 normdist ,0,11
, L normdist ,0,1,0
L L
ESC 1000(1 normdist (1000 / 707,0,11)
,
707nomdist (1000 / 707,0,1,0)
ESC 2513
.
fr = (Q - ESC)/Q = (10,000 - 25.13)/10,000 = 0.9975.
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Finding Safety Inventory for a Fill Rate:
fr ss
If desired fill rate is fr = 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
ss ss
250 ss 1 normdist ,0,1,1 707normdist ,0,1,0
707 707
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Evaluating Safety Inventory For Given Fill Rate
Fill Rate Safety Inventory
97.5% 67
98.0% 183
98.5% 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. Safety
stock 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 retailer
demand 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
Supplier’s lead time may be uncertain:
L Average lead time. s 2 Variance of lead time
L L
E ( Di ) LR Var ( Di ) L 2 R 2 s 2 : L
2
i 1 i 1
The formulae do not change:
ss F 1 (CSL;0,1) L F 1 (CSL;0,1) L 2 R 2 s 2
ss ss
;0,1,1 L nomdist ;0,1,0
ESC ss 1 normdist
L L 27
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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 3132
4 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
Which of two systems provides a higher level of service for a given
safety stock?
Consider locations and demands:
D1 ( R1 , ) D2 ( R2 , )
( R , )
1 2
C C
D3 ( R3 , 3) D4 ( R4 , 4)
K
With k locations centralized, mean and variance of D C Di
i 1
K K K
Ri; ) 2 cov(Di , D j )
C C 2 2
R i 1
(
i 1
i
i j
i 1 30
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Sum of Random Variables Are Less Variable
K K
When they are independent, C
cov(Di,Dj)=0
i 1
i
2
i
i 1
When they are perfectly positively correlated,
cov(Di,Dj)=σi σj 2
K K
KK
C i2 2 i j i i
i 1 i 1 i 1 i 1
i j
When they are perfectly negatively correlated,
cov(Di,Dj)= - σi σj K K K K
C i2 2 i j
i 1 i 1
i2 i
i 1 i 1
i j
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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 negatively
correlated? 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? 32
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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*σ
If four locations are all aggregated,
ss proportional to 4*σ with correlation 1
ss proportional to 2*σ with correlation 0
Benefit=SS before - SS after / SS before
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Impact of Correlation on Aggregated
Safety Inventory (Aggregating 4 outlets)
Benefit=(SS before - SS after) / SS before
0.6
0.5
0.4
0.3 Benefit
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1
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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 iid
Normal(20,40x40) at each store
Each cleaner costs $30; Demand is iid
Normal(1000,100x100) at each store
Which demand has a larger coefficient of
variation?
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
For normally distributed demand : ss Norminv(CS L,0,1) L
For 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)=$105,600,000
Value 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
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Value of cleaner ss=13,159(30)=$394,770
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EX. 11.8: Specialization: Impact of cv 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 $15.47
2574200/(1600*20*52)= $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 spaces
book, which has a shorter delivery time?
- Why cannot I find a “driver-side-door lock cylinder” for my 1994 Toyota
Corolla at Pep Boys?
- Your instructor on March 26 2005.
- “Case Interview books” are not in our s.k.u. list. You must check with our
central 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 each
component. 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 of components is a distinct product. A
component is used in 9 products.
Each way you can go from left to right is a product.
P1 M1 H1
Left P2 H2 Right
M2
P3 M3 H3 41
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Example 11.9: Value of Component Commonality
in Safety Inventory Reduction
450000
400000
350000
300000
250000
SS
200000
150000
100000
50000
0
1 2 3 4 5 6 7 8 9
# of products a component is used in
Aggregation provides reduction in total standard deviation. 42
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Standardization
Standardization
– Extent 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 a
significant 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 inspection
procedures
Opportunities for long production runs, automation
Need for fewer parts justifies increased expenditures on
perfecting designs and improving quality control
procedures.
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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 the
plagues.
– Computer security: Computers with non-standard software stop the
dissemination of viruses.
Another reason to stop using Microsoft products!
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Inventory–Transportation 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
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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, which
make 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
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Inventory–Transportation trade off: Eastern
Electric Corporation, see p.426-8 for details
Alternative Transport Cycle Safety Transit Inventory Total
(Lot size) Cost Inventory Inventory Inventory Cost Cost
AM Rail $78,000 1,000 986 1,644 $108,900 $186,900
(2,000) 120000(0.65) 120000(5/365)
Northeast $90,000 500 658 986 $64,320 $154,320
Trucking
(1,000)
Golden $96,000 250 658 986 $56,820 $152,820
(500) 120000(0.80) 120000(3/365)
Golden $86,400 1,250 658 986 $86,820 $173,220
(2,500)
Golden $78,000 1,500 658 986 $94,320 $172,320
(3,000)
Golden $67,500 2,000 658 986 $109,320 $176,820
(4,000)
If fast transportation not justified cost-wise, need to consider rapid response
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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
Current Option A Option B
Scenario
# 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
Match Supply & Demand
Reduce Buffer Inventory
Supply / Demand Seasonal
Economies of Scale Variability Variability
Cycle Inventory Safety Inventory Seasonal Inventory
•Reduce fixed cost •Quick Response measures
•Aggregate across •Reduce Info Uncertainty
products •Reduce lead time
•Volume discounts •Reduce supply
•Promotion on Sell uncertainty
thru •Accurate Response measures
•Aggregation
•Component commonality 52
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Mass Customization
Mass customization:
– A strategy of producing standardized goods or services,
but incorporating some degree of customization
– Modular design
– Delayed differentiation
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Mass Customization I: Customize
Services Around Standardized Products
Warranty for contact lenses: Source: B. Joseph Pine
DEVELOPMENT PRODUCTION MARKETING DELIVERY
Deliver customized services as
well as standardized products
and services
Market customized services with standardized
products or services
Continue producing standardized products or services
Continue developing standardized products or services
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Mass Customization II: Create
Customizable Products and Services
Customizing the look of screen with windows operating system
Gillette sensor adjusting to the contours of the face
DEVELOPMENT PRODUCTION MARKETING DELIVERY
Deliver standard (but
customizable) products
or services
Market customizable products or services
Produce standard (but customizable) products or services
Develop customizable products or services
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Mass Customization III: Provide Quick
Response Throughout Value Chain
Skiing parkas manufactured abroad vs. in the U.S.A.:
DEVELOPMENT PRODUCTION MARKETING DELIVERY
Reduce Delivery Cycle Times
Reduce selection and order processing cycle
times
Reduce Production cycle time
Reduce development cycle time
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Mass Customization IV: Provide Point of
Delivery Customization
Paint mixing
Lenscrafters for glasses.
DEVELOPMENT PRODUCTION MARKETING DELIVERY
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
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Mass Customization V: Modularize
Components to Customize End Products
Computer industry, Dell computers:
DEVELOPMENT PRODUCTION MARKETING DELIVERY
Deliver customized product
Market customized products or services
Produce modularized components
Develop modularized products
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Modular Design
Modular design is a form of standardization in which component parts are
subdivided into modules that are easily replaced or interchanged.
– Good example: Dell uses same components to assemble various
computers.
– 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
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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 must cover
the Demand during
T+LT T
OUL
LT LT 61
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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
R T L
(T L) R
T L
L T
ss F 1 (CSL;0,1) T L
OUL R T L
ss
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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?
ss F 1 (CSL;0,1) T L 1570
Factors driving safety inventory
– Demand uncertainty
– Replenishment lead time
– Reorder interval
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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 for
high-sales-value per time products
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