# Safety Inventories Chapter 11 of Chopra.

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Safety Inventories Chapter 11 of Chopra

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

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

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

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.

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

Cumulative Normal Density
1 prob normdist(x,mean,st_dev,1) x norminv(prob,mean,st_dev)

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.

Optimal Safety Inventory Levels
An inventory cycle Q ROP time Lead Times Shortage

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

I. Cycle Service Level for Normal Demands
Notes: The last equality is a property of the Normal distribution.

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

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.

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

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:

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

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

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

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?

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

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:

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:

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

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:

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

The formulae do not change:

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

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.

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

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

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?

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

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

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?

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

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

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.

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?

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.

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

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.

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

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.

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!

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:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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:

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:

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