1. 1 7. Managing Flow Variability: Safety Inventory Chapter 7 Managing Flow Variability: Safety Inventory 7.1 Demand Forecasts and Forecast Errors 7.2 Safety Inventory and Service Level 7.3 Optimal Service Level – The Newsvendor Problem 7.4 Lead Time Demand Variability 7.5 Pooling Efficiency through Aggregation 7.6 Shortening the Forecast Horizon 7.7 Levers for Reducing Safety Inventory

2. 2 7. Managing Flow Variability: Safety Inventory 7.1 Demand Forecast and Forecast Errors In review, we have 3 stages of a process: 1. Input (e.g. raw materials) 2. Process 3. Output (finished goods) This is important to forecasting because it will allow us to more closely match outputs to inputs and vice versa.

3. 3 7. Managing Flow Variability: Safety Inventory 7.1 Demand Forecasts and Forecast Errors We have previously assumed demand is known and is constant.  Demand varies in predictable and unpredictable ways.  Unpredictable, random factors affecting demand is referred to as “noise”. “As a process of predicting future demand, forecasting is, among other things, an effort to deal with NOISE.”

4. 4 7. Managing Flow Variability: Safety Inventory 7.1 Demand Forecast and Forecast Errors Why do we forecast? We forecast so that we can make decisions about the future. We need to make rational decisions about process inventory. * How to spend money and how not to spend money. * When to buys more widgets. * When to hire more workers. * How to avoid stockouts (upset customers = business losses) * How to avoid holding excess inventory (= $ lost)

5. 5 7. Managing Flow Variability: Safety Inventory 7.1 Demand Forecasts and Forecast Errors Forecasting methods  Subjective – Based on judgement and experience Surveys and expert judgements  Objective – Based on data analysis Causal models - Forecast methods that assume that in addition to data, there are other factors that influence demand (eg. Consumer prices.) Time series analyses - Methods that rely solely on past data.

6. 6 7. Managing Flow Variability: Safety Inventory 7.1 Demand Forecasts and Forecast Errors 4 Characteristics of Forecasts  Forecasts are usually wrong. Because of random noise – forecasts are inaccurate.  Forecasts should be accompanied by a measure of forecast error. A measure of forecast error quantifies the manager’s degree of confidence in the forecast.  Aggregate forecasts are more accurate than individual forecasts. Aggregate forecasts reduce the amount of variability – relative to the aggregate mean demand.  Long-range forecasts are less accurate than short-range forecasts. Precise forecasting of events far out in the future are much more difficult to predict than something that will occur in a matter of moments from now.

7. 7 7. Managing Flow Variability: Safety Inventory 7.1 Demand Forecasts and Forecast Errors Forecasts should incorporate hard quantitaive data as well as qualitative factors such as managerial judgement, intuition, and market savvy. Forecasting is as much art as science.

8. 8 7. Managing Flow Variability: Safety Inventory 7.1 Demand Forecast and Forecast Errors Safety Inventory cushions the process against supply disruptions or surges in demand. Having adequate Safety Inventory reduces the uncertainty in supply and demand. Ensuring reliable suppliers and stable demand eliminates the need for Safety Inventory.

9. 9 7. Managing Flow Variability: Safety Inventory 7.2 Safety Inventory and Service Level Objective: Review of common terms and a discussion of Service Level Where: SL = f (Safety Inventory, I safety ) And some math using Excel…

10. 10 7. Managing Flow Variability: Safety Inventory 7.2 Safety Inventory and Service Level SL: Service Level I safety :Safety Inventory (or Safety Stock) I cycle :Cycle Inventory LTD:Lead Time Demand ROP:Re-order Point L: Replenishment Lead Time Q: Order Size NORMDIST: Standard Normal Tables NORMSINV: Standard Normal Tables NORMINV: Standard Normal Tables

11. 11 7. Managing Flow Variability: Safety Inventory 7.2 Safety Inventory and Service Level Inventory, I (t) ROP 0 Safety Inventory (I safety ) LTD, # of Units used during lead time ORDER LL Time, t

12. 12 7. Managing Flow Variability: Safety Inventory 7.2 Safety Inventory and Service Level I safety = ROP – LTD 0 ROP LTD, # of Units used during lead time Safety Inventory (I safety ) Inventory, I (t)

13. 13 7. Managing Flow Variability: Safety Inventory 7.2 Safety Inventory and Service Level An Inventory with an Order Size = Q Average Inventory = Q/2 I cycle = Q/2 I = I cycle + I safety = Q/2 + I safety Average Flow Rate = R Average Flow Time as expressed by Little’s Law T = I /R = (Q/2 + I safety ) R

14. 14 7. Managing Flow Variability: Safety Inventory 7.2 Safety Inventory and Service Level The Service Level for a given ROP is given by: SL = Prob (LTD < ROP) To calculate SL, recall first that if LTD is normally distributed with mean LTD and standard deviation  LTD then I safety = z x  LTD, where z is a multiple of  LTD Or the number of standard deviations

15. 15 7. Managing Flow Variability: Safety Inventory 7.2 Safety Inventory and Service Level Example: At GE Lighting’s Paris warehouse, LTD (average Lead Time Demand) = 20,000 lamps Actual Demand varies daily and  LTD = 5,000 The warehouse re-orders whenever ROP = 24,000 Therefore, I safety = ROP – LTD = 24,000 – 20,000 = 4,000 And: z = I safety /  LTD = 4,000 / 5,000 = 0.8 And: SL= Prob (Z< 0.8) from Appendix II SL= 0.7881

16. 16 7. Managing Flow Variability: Safety Inventory 7.2 Safety Inventory and Service Level EXCEL Service Level

17. 17 7. Managing Flow Variability: Safety Inventory 7.2 Safety Inventory and Service Level EXCEL Safety Inventory

18. 18 7. Managing Flow Variability: Safety Inventory 7.2 Safety Inventory and Service Level EXCEL Reorder Point

19. 19 7. Managing Flow Variability: Safety Inventory 7.3 Optimal Service Level: The Newsvendor Problem So Far… Safety inventory has been defined for a desired level of customer service. But… How do we choose what level of service a firm should offer? Examples: Newspapers / Magazines Perishables (fish, produce, bread, milk, etc.) Seasonal Items (Summer & Winter Apparel)

20. 20 7. Managing Flow Variability: Safety Inventory 7.3 Optimal Service Level: The Newsvendor Problem Cost of Holding Extra Inventory Improved Service Optimal Service Level?? The Newsvendor Problem Decision making under uncertainty whereby the decision maker balances the expected costs of ordering too much with the expected costs of ordering too little to determine the optimal order quantity.

21. 21 7. Managing Flow Variability: Safety Inventory 7.3 Optimal Service Level: The Newsvendor Problem Cost: Price: Salvage: $1,800 $2,500 $1,700 DemandProbability Cumulative Probability Complementary Cumulative Probability rProb(R = r)Prob(R ≤ r)Prob(R > r) 1000.02 0.98 1100.050.070.93 1200.080.150.85 1300.090.240.76 1400.110.350.65 1500.160.510.49 1600.20.710.29 1700.150.860.14 1800.080.940.06 1900.050.990.01 2000.0110 Predicted Demand for HDTV’s Profit: Loss: $700 $100 Mean: Std. Dev: 151.6 22.44

22. 22 7. Managing Flow Variability: Safety Inventory 7.3 Optimal Service Level: The Newsvendor Problem 100110120130140150160170180190200 100$70,000 110$69,000$77,000 120$68,000$76,000$84,000 130$67,000$75,000$83,000$91,000 140$66,000$74,000$82,000$90,000$98,000 150$65,000$73,000$81,000$89,000$97,000$105,000 160$64,000$72,000$80,000$88,000$96,000$104,000$112,000 170$63,000$71,000$79,000$87,000$95,000$103,000$111,000$119,000 180$62,000$70,000$78,000$86,000$94,000$102,000$110,000$118,000$126,000 190$61,000$69,000$77,000$85,000$93,000$101,000$109,000$117,000$125,000$133,000 200$60,000$68,000$76,000$84,000$92,000$100,000$108,000$116,000$124,000$132,000$140,000 =IF($A3>B$1,B$1*700-($A3-B$1)*100,$A3*700) Demand Quantity Ordered 100 x $700 – (110-100) x $100 = $69,000

23. 23 7. Managing Flow Variability: Safety Inventory 7.3 Optimal Service Level: The Newsvendor Problem 100110120130140150160170180190200 100$70,000 110$69,000$77,000 120$68,000$76,000$84,000 130$67,000$75,000$83,000$91,000 140$66,000$74,000$82,000$90,000$98,000 150$65,000$73,000$81,000$89,000$97,000$105,000 160$64,000$72,000$80,000$88,000$96,000$104,000$112,000 170$63,000$71,000$79,000$87,000$95,000$103,000$111,000$119,000 180$62,000$70,000$78,000$86,000$94,000$102,000$110,000$118,000$126,000 190$61,000$69,000$77,000$85,000$93,000$101,000$109,000$117,000$125,000$133,000 200$60,000$68,000$76,000$84,000$92,000$100,000$108,000$116,000$124,000$132,000$140,000 $69,000(0.02) + $77,000(0.05) + …+ $77,000(0.01) = $76,840 Demand Quantity Ordered

24. 24 7. Managing Flow Variability: Safety Inventory 7.3 Optimal Service Level: The Newsvendor Problem Order Quantity (Q)Expected Profit 100$70,000 110$76,840 120$83,280 130$89,080 140$94,160 150$98,360 160$101,280 170$102,600 180$102,720 190$102,200 200$101,280

25. 25 7. Managing Flow Variability: Safety Inventory 7.3 Optimal Service Level: The Newsvendor Problem Net Marginal Benefit: Net Marginal Cost: MB = p – c MC = c - v MB = $2,500 - $1,800 = $700 MC = $1,800 - $1,700 = $100 We receive Marginal Benefit when R > Q, therefore at any order quantity Q, Expected MB = MB x Prob(R > Q) We receive Marginal Cost when R ≤ Q, therefore at any order quantity Q, Expected MC = MC x Prob(R ≤ Q) MC x Prob(R ≤ Q*) ≥ MB x Prob(R > Q*)

26. 26 7. Managing Flow Variability: Safety Inventory 7.3 Optimal Service Level: The Newsvendor Problem Time for Algebra… MC x Prob(R ≤ Q*) ≥ MB x Prob(R > Q*) Since, Prob(R > Q) = 1 – Prob(R ≤ Q) We can write, MC x Prob(R ≤ Q*) ≥ MB x [1 – Prob(R ≤ Q*)] After rearranging, Prob(R ≤ Q*) ≥ Newsvendor formula: SL* = Prob(R ≤ Q*) =

27. 27 7. Managing Flow Variability: Safety Inventory 7.3 Optimal Service Level: The Newsvendor Problem Going back to the example… So what quantity corresponds to this service level ? If we assume demand is normally distributed then,

28. 28 7. Managing Flow Variability: Safety Inventory 7.3 Optimal Service Level: The Newsvendor Problem

29. 29 7. Managing Flow Variability: Safety Inventory 7.4 Lead Time Demand Variability Average Lead Time Demand: Variability in Periodic Demand: Variability in Lead Time: Variability in Demand and Lead Time:

30. 30 7. Managing Flow Variability: Safety Inventory 7.5: Pooling Efficiency through Aggregation Third characteristic of forecasts Aggregation: pooling demand for several similar products Aggregate sales Safety Inventory: Uncertain demand Assume Decentralized: Warehouses operates independently Imbalance of inventory - Customer demand not satisfied

31. 31 7. Managing Flow Variability: Safety Inventory 7.5: Pooling Efficiency through Aggregation Physical Centralization: that the firm can consolidate all its stock in one location from which is can serve all its customers. ELIMINATES stock imbalance BETTER customer service SAME total inventory LESS inventory Location 1Location 2 Lead times demands: LTD 1 LTD 2 Mean of LTD Standard Deviation

32. 32 7. Managing Flow Variability: Safety Inventory 7.5: Pooling Efficiency through Aggregation LTD 1 and LTD 2: statistically identically distributed To provide desired level of service, SL each location must carry Safety Inventory : Z determined by the desired service level Each facility: Identical demand and service levels Total safety inventory decentralized system:

33. 33 7. Managing Flow Variability: Safety Inventory 7.5: Pooling Efficiency through Aggregation Independent Demands Centralizing the two locations in one location when lead time demands at the two locations are independent. LTD = LTD 1 + LTD 2 Centralized Pool The mean of total lead time demand is: LTD + LTD = 2 LTD Variance is: Standard Deviation is:

34. 34 7. Managing Flow Variability: Safety Inventory 7.5: Pooling Efficiency through Aggregation Comparing safety inventories of decentralized and centralized systems. Safety Inventory in Centralized system is in a 2 location decentralized system by a factor of

35. 35 7. Managing Flow Variability: Safety Inventory 7.5: Pooling Efficiency through Aggregation Centralization of N locations: Safety Inventory needed is Centralization will reduce inventory by factor of

36. 36 7. Managing Flow Variability: Safety Inventory 7.5: Pooling Efficiency through Aggregation Example GE lighting operating 7 warehouses Consolidated in to one centralized warehouse Replenishment lead time remain at 10 days What will be the impact of accepting the task force recommendations?

37. 37 7. Managing Flow Variability: Safety Inventory 7.5: Pooling Efficiency through Aggregation A warehouse with average lead time demand of 20,000 units with a standard deviation of 5,000 units needs to carry a safety inventory to provide a 95% service level. Total safety inventory across 7 warehouses: Task force accepted, single central warehouse will face total lead time demand with mean and standard deviation of:

38. 38 7. Managing Flow Variability: Safety Inventory 7.5: Pooling Efficiency through Aggregation 95% service level, the central warehouse must carry a safety inventory: Safety inventory with the single central warehouses is 35,894 less than that required under the current decentralized network of 7 warehouses. Decrease in safety inventory by a factor of

39. 39 7. Managing Flow Variability: Safety Inventory 7.5: Pooling Efficiency through Aggregation Square Root Law States that the total safety inventory required to provide a specific level of service increases by the square root of the number of locations in which it is held. Previous example Correlated Demands Does centralization offer similar benefits when demands in multiple locations are correlated?

40. 40 7. Managing Flow Variability: Safety Inventory 7.5: Pooling Efficiency through Aggregation LTD 1 and LTD 2 are statistically identically distributed but correlated. Correlation between two locations with coefficient Mean of total lead time: LTD + LTD = 2 LTD Variance is: Total safety in centralized system is:

41. 41 7. Managing Flow Variability: Safety Inventory 7.5: Pooling Efficiency through Aggregation The total safety inventory in the decentralized system: The safety inventory in the two-location decentralized system is larger than in the centralized system by a factor of If demand is positively correlated centralization offers no benefits in the reduction of safety inventory

42. 42 7. Managing Flow Variability: Safety Inventory 7.5: Pooling Efficiency through Aggregation Advantages 1. Centralized systems as the demand on the two locations become negatively correlated. 2. Centralized systems diminishes as the demand in the two locations become positively correlated Disadvantages of Centralization 1. Response time to Customers 2. Shipping Cost

43. 43 7. Managing Flow Variability: Safety Inventory 7.5.2: Principle of Aggregation and Pooling Inventory Statistical Principle Principle Aggregation: the standard deviation of the sum of random variables is less than the sum of the individual standard deviations. Pooling inventory: available inventory is shared among various sources of demand Pooling inventory applied in other ways other than physical centralization

44. 44 7. Managing Flow Variability: Safety Inventory 7.5.2: Principle of Aggregation and Pooling Inventory Virtual Centralization Specialization Component Commonality Product Substitute

45. 45 7. Managing Flow Variability: Safety Inventory 7.5.2: Principle of Aggregation and Pooling Inventory Virtual Centralization Distribution System Location A Location B Exceeds Available stock Available 1. Information about product demand and availability must be available at both locations 2. Shipping the product from one location to a customer at another location must be fast and cost effective

46. 46 7. Managing Flow Variability: Safety Inventory 7.5.2: Principle of Aggregation and Pooling Inventory Correlation is less than one – Pooling is Effective Inventory Decentralized instead of physically consolidated Virtual Centralization: is a system in which inventory pooling in a network of locations is facilitated using information regarding availability of goods and subsequent transshipment of goods between locations to satisfy demand.

47. 47 7. Managing Flow Variability: Safety Inventory 7.5.2: Principle of Aggregation and Pooling Inventory Specialization Each product only one specialized warehouse EXAMPLE Location A Location B P1P2 Safety Inventory is reduced because each inventory is now centralized at one location

48. 48 7. Managing Flow Variability: Safety Inventory 7.5.2: Principle of Aggregation and Pooling Inventory Component Commonality Aggregating demand across various products. Computer companies with models that vary. Make-to-stock: produce in anticipation of product demand Make-to-Order: Produce in response to customer orders Reduce inventory investment maintaining the same level of service and product variety

49. 49 7. Managing Flow Variability: Safety Inventory 7.5.2: Principle of Aggregation and Pooling Inventory Disadvantage Make-to-Order Strategy Customer must wait for firm to produce product Advantage Make-to-Stock Strategy Available for immediate consumption

50. 50 7. Managing Flow Variability: Safety Inventory 7.6: Shortening the Forecast Horizon through Postponement Postponement (or Delayed Differentiation): More Effective Short-Range forecast more accurate Two Alternative processes (both two weeks) Process A: Coloring the fabric, assembling Process B: Assembling T-shirts, coloring Does one have the advantage over the other?

51. 51 7. Managing Flow Variability: Safety Inventory 7.6: Shortening the Forecast Horizon through Postponement By Reversing: assembling and dyeing process Process B postponed the color difference until one week closer to the time of sale Postponement: the practice of delaying part of a process in order to reduce the need for safety inventory

52. 52 7. Managing Flow Variability: Safety Inventory 7.6: Shortening the Forecast Horizon through Postponement Process B has the advantage Aggregation Reduces Variability 1.Aggregates demands by color in the first phase 2.Requires shorter-range forecasts of individual T-shirts needed by color in the second phase. Less Demand Variability Less Total Safety Inventory

53. 53 7. Managing Flow Variability: Safety Inventory 7.7: Levers for Reducing Safety Inventory Levers for Reducing Flow Variability and the Required Safety Inventory 1. Reduce demand variability through improved forecasting 2. Reduce replenishment lead time 3. Reduce variability in replenishment lead time 4. Pool safety inventory for multiple locations or products

54. 54 7. Managing Flow Variability: Safety Inventory 7.7: Levers for Reducing Safety Inventory 5. Exploit product substitution 6. Use common components 7. Postpone product-differentiation processing until closer to the point of actual demand