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, 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
inventory
time
CSL is 0%, fill rate is almost 0%
time
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 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
E ( Di ) Ri by the linearity of integratio n
L L i 1 L i 1
Var( Di ) cov(Di , D j ) cov(Di , D j )
i 1 i 1 j 1 i 1 2 i
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L
L
L
L
L
i 1 j 1 j i
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Normal Density Function
frequency
normdist(x,.,.,1) Prob x
normdist(x,.,.,0)
Mean 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)
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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)
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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.
E ( Di ) LR and Var( Di ) L 2
i 1 i 1
L
L
If Di N ( R, ) then Di N ( LR, L 2 )
2 i 1
L
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)
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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 10 CSL 0.7
Write0 if a cycle has stockout,1 otherwise
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 )
P N ( R L, L
P N ( R L, L R ) ROP
2 2 R
Recall N (mean,variance) notation
) R L ROP R L
Taking out the mean
2 N ( R L, L R ) R L ROP R L Dividing by theStDev P L R L R N ( 0 ,1) ROP R L Obtaining standard normal distribution P N (0,1) 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 Shortage 0 if Demand ROP Demand ROP
ESC E (max{Demand during lead time - ROP,0}) ESC =
x ROP
( x ROP) f ( x)dx
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Inventory and Demand during Lead Time
ROP
0 ROP
Inventory= ROP-DLT DLT: Demand During LT
0
LT
Inventory
Upside down
Demand During LT
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0
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Shortage and Demand during Lead Time
Shortage= DLT-ROP
0 Shortage
LT Demand During LT
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Upside down
ROP
0
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DLT: Demand During LT
ROP
0
Expected shortage per cycle
First let us study shortage during the lead time
Expected shortage E (0, max(DLT ROP))
Ex:
D ROP
( D ROP) f
D
( D)dD
wheref D is pdf of DLT.
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
Expected shortage max{0,(d i ROP)} pi
i 1
d 10
(d ROP)}P( D d )
19
11
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? 1 102 1 1 D2 1 122 Expected shortage ( D 10) dD 10D 10(12) 10(10) 6 2 6 6 2 6 2 D 10 D 10 172 - 170 2
12 D 12
If demand is normal:
ss ss ESC ss1 normdist ,0,11 L normdist , ,0,1,0 L L
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Does ESC decrease or increase with ss, L? Does ESC decrease or increase with expected value of demand? 20
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 ESC ss 1 normdist ,0,11 , L
ss L normdist ,0,1,0 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 97.5% 98.0% 98.5% 99.0% 99.5% Safety Inventory 67 183 321 499 767
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.
E ( Di ) LR
i 1 L
s 2 Variance of lead time
L
2 Var ( Di ) L 2 R 2 s 2 : L 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 ESC ss 1 normdist ;0,1,1 L nomdist ;0,1,0 L L
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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 0 ss 1695 Jump in ss -
1 2 3 4 5 6
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3625 6628 9760 12927 16109 19298
1930 3003 3132 3167 3182 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 ,
)
1
D2 ( R2 ,
2
)
( R , )
C C
D3 ( R3 , 3)
D4 ( R4 , 4)
With k locations centralized, mean and variance of
D C Di
i 1
K
R
C
Ri;
i 1
K
(
C 2
)
i 1
K
2 i
2 cov(Di , D j )
i j i 1
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K
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Sum of Random Variables Are Less Variable
When they are independent, C cov(Di,Dj)=0
cov(Di,Dj)=σi σj
i 1
K
2 i
i
i 1
K
When they are perfectly positively correlated,
K C i2 2 i j i i i 1 i 1 i 1 i 1 K K K i j 2
When they are perfectly negatively correlated,
cov(Di,Dj)= - σi σj
C
i2 2 i j
i 1 i 1 i j
K
K
i2 i
i 1 i 1
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K
K
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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
0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1
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before - SS after) / SS before
Benefit
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 36 Value of cleaner ss=13,159(30)=$394,770 utdallas.edu/~metin
EX. 11.8: Specialization: Impact of cv on Benefit From 1600-Store Aggregation , h=0.25
Motors Mean demand/wk SD of demand Disaggregate cv Value/Unit Disaggregate ss value Aggregate cv Aggregate ss value Inventory cost savings Holding Cost Saving Saving / Unit
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2574200/(1600*20*52)=
20 40 2 $500 $105,600,000 0.05 $2,632,000 $102,968,000 $25,742,000 $15.47
Cleaner 1,000 100 0.1 $30 $15,792,000 0.0025 $394,770 $15,397,230 $3,849,308 $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. 38
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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 Right
Left
P2 P3
M2
M3
H2
H3
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Example 11.9: Value of Component Commonality in Safety Inventory Reduction
450000 400000 350000 300000 250000 200000 150000 100000 50000 0 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
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 (Lot size) Cost AM Rail (2,000) Northeast Trucking (1,000) Golden (500) Golden (2,500) Golden (3,000) Golden (4,000) $78,000 120000(0.65) $90,000 Cycle Safety Inventory Inventory 1,000 500 986 Transit Inventory Inventory Total Cost Cost $108,900 $64,320 $186,900 $154,320
1,644 120000(5/365) 658 986
$96,000 120000(0.80) $86,400 $78,000 $67,500
250 1,250 1,500 2,000
658
986 120000(3/365) 658 986 658 658 986 986
$56,820 $86,820 $94,320 $109,320
$152,820 $173,220 $172,320 $176,820
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
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
HighMed
– 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
# Locations Reorder Interval Inventory Cost Shipment Size Transport Cost Total Cost Current Scenario 24 4 weeks $54,366 8 H + 80 L $530 $54,896 Option A 24 1 week $29,795 2 H + 20 L $1,148 $30,943 Option B 1 1 week $8,474 1 H + 10 L $14,464 $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
Economies of Scale
Supply / Demand Variability
Seasonal Variability Seasonal Inventory
Cycle Inventory
•Reduce fixed cost •Aggregate across products •Volume discounts •Promotion on Sell thru
Safety Inventory
•Quick Response measures •Reduce Info Uncertainty •Reduce lead time •Reduce supply uncertainty •Accurate Response measures
•Aggregation •Component commonality and postponement
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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 OUL
T
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LT
LT
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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
(T L) R L T ss
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ss F 1 (CSL;0,1) T L
OUL
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R
T L
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
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
Periodic
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