1. Safety Inventories
Chapter 11 of Chopra

2. 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

3. Measures Measures of demand uncertainty Delivery Lead Time, L
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

4. Service measures: CSL and fr are different
inventory
CSL is 0%, fill rate is almost 100%
time
inventory
CSL is 0%, fill rate is almost 0%
time

5. 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.

6. Demand During Lead time

7. Normal Density Function
frequency
normdist(x,.,.,1)
normdist(x,.,.,0)
Prob
Mean x




95.44%
99.74%

8. Cumulative Normal Density1
prob
normdist(x,mean,st_dev,1) x
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.
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
An inventory cycle Q
ROP
time
Lead Times
Shortage

11. I. Cycle Service Level: ROP  CSL
Cycle service level: percentage of cycles with stock out
ROP: Reorder point

12. I. Cycle Service Level for Normal Demands
Notes:
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 = ROP – L R =
Cycle service level,
Average Inventory =
Average Flow Time =Average inventory/Thruput=
Notes:
If you wish to compute Average Inventory = Q/2 + ss

14. Safety Inventory: CSL  ROP
The last two equalities are by properties of the Normal distribution.
Notes:
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

16. 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
Notes:

17. Inventory and Demand during Lead Time
ROP
Inventory=
ROP-DLT
ROP
Upside
down
Inventory
DLT: Demand
During LT LT
Demand
During LT

18. Shortage and Demand during Lead Time
ROP
Shortage=
DLT-ROP
ROP
Upside
down
DLT: Demand During LT
Shortage LT
Demand
During LT

19. Expected shortage per cycle
First let us study shortage during the lead time
Ex:

20. Expected shortage per cycle
If demand is normal:
Does ESC decrease or increase with ss, L?
Does ESC decrease or increase with expected value of demand?

21. Fill Rate
Fill rate: Proportion of customer demand satisfied from stock
Q: Order quantity
Notes:

22. Finding the Fill Rate ss  fr
= 500; L = 2 weeks; ss=1000; Q = 10,000;
Fill Rate (fr) = ?
fr = (Q - ESC)/Q = (10, )/10,000 =
Notes:

23. 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
Notes:

24. Evaluating Safety Inventory For Given Fill Rate
Safety inventory is very sensitive to fill rate. Is fr=100% possible?

25. 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.
Notes:

26. 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

27. Lead Time Variability Supplier’s lead time may be uncertain:
The formulae do not change:

28. 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
1695 - 1
3625
1930 2
6628
3003 3
9760
3132 4
12927
3167 5
16109
3182 6
19298
3189

29. 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
Notes: Mention williams Sonoma and Laura Ashley as examples of Physical centralization. Mail order companies with multiple warehouses as examples of Information centralization. Also Wal-Mart and The Gap allow for Information Centralization cutting the level of safety stock carried.
Mention Bennetton for raw material commonality. The major benefit here is that since greige goods are to be ordered, the estimate is much more accurate since it aggregates across all colors. Also mention the HP Laser Jet example.

30. Centralization: Inventory Pooling
Which of two systems provides a higher level of service for a given safety stock?
Consider locations and demands:
Notes:
With k locations centralized, mean and variance of

31. Sum of Random Variables Are Less Variable
When they are independent,
cov(Di,Dj)=0
When they are perfectly positively correlated,
cov(Di,Dj)=σi σj
When they are perfectly negatively correlated,
cov(Di,Dj)= - σi σj

32. 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?

33. 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

34. Impact of Correlation on Aggregated Safety Inventory (Aggregating 4 outlets)
Benefit=(SS before - SS after) / SS before

35. 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?

36. Use CSL=0.95 Supply lead time L=4 weeks for motors and cleaners
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
Value of cleaner ss=13,159(30)=$394,770

37. EX. 11.8: Specialization: Impact of cv on Benefit From 1600-Store Aggregation , h=0.25

38. 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
“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.

39. 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?

40. 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.

41. 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. H1 P1 M1
Left H2
Right P2 M2 H3 P3 M3

42. Example 11.9: Value of Component Commonality in Safety Inventory Reduction
# of products a component is used in
Aggregation provides reduction in total standard deviation.

43. Standardization Standardization The degree of 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

44. 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.

45. 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!

46. 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 weight
Notes:

47. 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

48. Inventory–Transportation trade off: Eastern Electric Corporation, see p.426-8 for details
Notes:
Safety stock = 3 days demand for rail and 2 days demand for truck.
Daily demand = 120,000/365 = 329 motors.
Transit inventory = 120,000*5/365 = 986 for rail
120,000*3/365 = 658 for truck
Case discussion:
Mention role of incentives in choice of mode. Stress importance of considering beyond mere transportation cost.
If fast transportation not justified cost-wise, need to consider rapid response

49. Physical Inventory Aggregation: Inventory vs. Transportation cost: p
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

50. 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: $ x {for replenishments}
FedEx rate: $ x {for customer shipping}
Customers order 1 H + 10 L

51. Inventory Aggregation at HighMed
If shipment size to customer is 0.5H + 5L, total cost of option B
increases to $36,729.

52. Summary of Cycle and Safety Inventory
Reduce Buffer Inventory
Economies of Scale
Supply / Demand
Variability
Seasonal
Cycle Inventory
Safety Inventory
Seasonal Inventory
Match Supply & Demand
Reduce fixed cost
Aggregate across products
Volume discounts
Promotion on Sell
thru
Quick Response measures
Reduce Info Uncertainty
Reduce lead time
Reduce supply uncertainty
Accurate Response measures
Aggregation
Component commonality and postponement

53. Mass Customization Mass customization:
A strategy of producing standardized goods or services, but incorporating some degree of customization
Modular design
Delayed differentiation

54. 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
Notes:
Contact lenses with different warranty.
On one extreme is the IBM system/360 mainframe where the system was customized for each customer. This is very expensive and soon disappeared. An effort was made to develop a standardized product that filled most of the needs at a lower cost.
One may build many views of a database of cases and readings to satisfy marketing professionals, operations, new product etc. Here design of the data base becomes important since the kinds of views that are easily feasible will depend upon the way the data base has been designed. For example if articles can be sorted and searched based on key words that are assigned, it may be much easier to design different views.
Personalized newspapers and web profiles which only show a certain view of the web.
Here we have “postponed” differentiation to the point of delivery.
Market customized services with standardized
products or services
Continue producing standardized products or services
Continue developing standardized products or services

55. 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
Notes:
HP deskjets can be configured to be black and white or color by customer
Gillette sensor which “automatically adjusts to the contours of your face.”
ATMs
The key here is to identify the most personal, most individual characteristics. These are then embedded within the product or service. This is a lot of effort in terms of the development phase. Here we have “postponed” product differentiation to the customer.
Market customizable products or services
Produce standard (but customizable) products or services
Develop customizable products or services

56. Mass Customization III: Provide Quick Response Throughout Value Chain
Skiing parkas manufactured abroad vs. in the U.S.A.:
DEVELOPMENT
PRODUCTION
MARKETING
DELIVERY
Notes:
The idea here is to shrink all times so that a significant part of the chain can be postponed. Discuss in detail with the Sport Obermeyer story. Mention the apparel industry in the US. Mention that markdowns are the key problem in the apparel industry.
Quick response allows the manufacturer to provide products that are in tune with customer needs since there is little in the supply chain. It brings the customer closer to the development making the loop much quicker.
Reduce Delivery Cycle Times
Reduce selection and order processing cycle
times
Reduce Production cycle time
Reduce development cycle time

57. Mass Customization IV: Provide Point of Delivery Customization
Paint mixing
Lenscrafters for glasses.
DEVELOPMENT
PRODUCTION
MARKETING
DELIVERY
Point of delivery
customization
Notes:
Paint mixing
Mention HP in this case. Lenscrafters for glasses.
This method works when there are a few inherently individual characteristics in an otherwise standardized product. The rest is produced centrally in advance. All build-to-order computer manufacturers Dell, Micron, Compaq is moving to it, essentially do this form of customization.
In this form of customization, modularity plays an important role.
Deliver standardize portion
Market customized products or services
Produce standardized portion centrally
Develop products where point of delivery customization is feasible

58. Mass Customization V: Modularize Components to Customize End Products
Computer industry, Dell computers:
DEVELOPMENT
PRODUCTION
MARKETING
DELIVERY
Notes: Computer industry
Deliver customized product
Market customized products or services
Produce modularized components
Develop modularized products

59. 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

60. Types of Modularity for Mass Customization
Component Sharing Modularity, Dell
Cut-to-Fit Modularity,
Gutters that do not require seams
Bus Modularity, E-books
Notes:
Component sharing modularity: HP, Dell, Create a book (individualizes books using personal information on a child)
Cut-to-fit: National bicycle. Sized to fit individual customers.
Bus modularity: Individualized magazines based on Selectronic binding by R.R. Donnelly. Key here is the presence of a bus (superset) of components that are selected among to get different products. Product design is key here
Mix modularity: Paint mixed in the store itself. Fertilizer mixed. Anything with a recipe. Key design factor is the mixing device sine that will usually be at point of sale or delivery.
Sectional modularity: LEGO. Agfa’s Shared Document Management Sytem (Xerox is doing the same). “Document objects” can be any size and any type (tables etc.) and can be put together in any way desired by the user. Interfaces are key here. Lego has simple interfaces but in general that is not true. Once again this allows for pooling. + =
Mix Modularity, Paints
Sectional Modularity, LEGO

61. 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

62. 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
Notes:

63. 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
Notes:

64. 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