1. 1 Managing Flow Variability: Safety Inventory Forecasts Depend on: (a) Historical Data and (b) Market Intelligence. Demand Forecasts and Forecast Errors Safety Inventory and Service Level Optimal Service Level – The Newsvendor Problem Lead Time Demand Variability Pooling Efficiency through Aggregation Shortening the Forecast Horizon Levers for Reducing Safety Inventory

2. 2 Managing Flow Variability: Safety Inventory Four Characteristics of Forecasts Forecasts are usually (always) inaccurate (wrong). Because of random noise. Forecasts should be accompanied by a measure of forecast error. A measure of forecast error (standard deviation) 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. StdDev of sum of two variables is less than sum of StdDev of the two variables. Long-range forecasts are less accurate than short-range forecasts. Forecasts further into the future tends to be less accurate than those of more imminent events. As time passes, we get better information, and make better prediction.

3. 3 Managing Flow Variability: Safety Inventory Demand During Lead Time is Variable N(μ,σ) Demand of sand during lead time has an average of 50 tons. Standard deviation of demand during lead time is 5 tons Assuming that the management is willing to accept a risk no more that 5%.

4. 4 Managing Flow Variability: Safety Inventory Forecast and a Measure of Forecast Error Forecasts should be accompanied by a measure of forecast error

5. 5 Managing Flow Variability: Safety Inventory Time Inventory Demand During Lead Time Demand during LT Lead Time

6. 6 Managing Flow Variability: Safety Inventory LT ROP when Demand During Lead Time is Fixed

7. 7 Managing Flow Variability: Safety Inventory LT Demand During Lead Time is Variable

8. 8 Managing Flow Variability: Safety Inventory Inventory Time Demand During Lead Time is Variable

9. 9 Managing Flow Variability: Safety Inventory Average demand during lead time A large demand during lead time ROP Time Quantity Safety stock reduces risk of stockout during lead time Safety Stock Safety stock LT

10. 10 Managing Flow Variability: Safety Inventory ROP Time Quantity Safety Stock LT

11. 11 Managing Flow Variability: Safety Inventory Re-Order Point: ROP Demand during lead time has Normal distribution. We can accept some risk of being out of stock, but we usually like a risk of less than 50%. If we order when the inventory on hand is equal to the average demand during the lead time; then there is 50% chance that the demand during lead time is less than our inventory. However, there is also 50% chance that the demand during lead time is greater than our inventory, and we will be out of stock for a while. We usually do not like 50% probability of stock out

12. 12 Managing Flow Variability: Safety Inventory RO P Risk of a stockout Service level Probability of no stockout Safety stock 0z Quantity z-scale Safety Stock and ROP Each Normal variable x is associated with a standard Normal Variable z Average demand x is Normal (Average x, Standard Deviation x)  z is Normal (0,1)

13. 13 Managing Flow Variability: Safety Inventory z Values SLz value 0.91.28 0.951.65 0.99 2.33 RO P Risk of a stockout Service level Probability of no stockout Safety stock 0z Quantity z-scale Average demand There is a table for z which tells us a)Given any probability of not exceeding z. What is the value of z b)Given any value for z. What is the probability of not exceeding z

14. 14 Managing Flow Variability: Safety Inventory μ and σ of Demand During Lead Time Demand of sand during lead time has an average of 50 tons. Standard deviation of demand during lead time is 5 tons. Assuming that the management is willing to accept a risk no more that 5%. Find the re-order point. What is the service level. Service level = 1-risk of stockout = 1-.05 =.95 Find the z value such that the probability of a standard normal variable being less than or equal to z is.95 Go to normal table, look inside the table. Find a probability close to.95. Read its z from the corresponding row and column.

15. 15 Managing Flow Variability: Safety Inventory The table will give you z Given a 95% SL 95% Probability Page 319: Normal table Up to the first digit after decimal Second digit after decimal Probability z 1.6 0.05 Z = 1.65 z Value using Table

16. 16 Managing Flow Variability: Safety Inventory The standard Normal Distribution F(z) F(z) z 0 F(z) = Prob( N(0,1) < z)

17. 17 Managing Flow Variability: Safety Inventory Relationship between z and Normal Variable x RO P Risk of a stockout Service level Probability of no stockout Safety stock 0z Quantity z-scale Average demand z = (x-Average x)/(Standard Deviation of x) x = Average x +z (Standard Deviation of x) μ = Average x σ = Standard Deviation of x  x = μ +z σ

18. 18 Managing Flow Variability: Safety Inventory Relationship between z and Normal Variable ROP RO P Risk of a stakeout Service level Probability of no stockout Safety stock 0z Quantity z-scale Average demand LTD = Lead Time Demand ROP = Average LTD +z (Standard Deviation of LTD) ROP = LTD+zσ LTD  ROP = LTD + I safety

19. 19 Managing Flow Variability: Safety Inventory Demand During Lead Time is Variable N(μ,σ) Demand of sand during lead time has an average of 50 tons. Standard deviation of demand during lead time is 5 tons Assuming that the management is willing to accept a risk no more that 5%. Compute safety stock I safety = zσ LTD I safety = 1.64 (5) = 8.2 ROP = LTD + I safety ROP = 50 + 1.64(5) = 58.2 z = 1.65

20. 20 Managing Flow Variability: Safety Inventory Service Level for a given ROP SL = Prob (LTD ≤ ROP) LTD is normally distributed  LTD = N(LTD,  LTD ). ROP = LTD + zσ LTD  ROP = LTD + I safety  I safety = z  LTD At GE Lighting’s Paris warehouse, LTD = 20,000,  LTD = 5,000 The warehouse re-orders whenever ROP = 24,000 I safety = ROP – LTD = 24,000 – 20,000 = 4,000 I safety = z  LTD  z = I safety /  LTD = 4,000 / 5,000 = 0.8 SL= Prob (Z ≤ 0.8) from Appendix II  SL= 0.7881

21. 21 Managing Flow Variability: Safety Inventory Excel: Given z, Compute Probability

22. 22 Managing Flow Variability: Safety Inventory Excel: Given Probability, Compute z

23. 23 Managing Flow Variability: Safety Inventory Demand of sand has an average of 50 tons per week. Standard deviation of the weekly demand is 3 tons. Lead time is 2 weeks. Assuming that the management is willing to accept a risk no more that 10%. Compute the Reorder Point μ and σ of demand per period and fixed LT

24. 24 Managing Flow Variability: Safety Inventory μ and σ of demand per period and fixed LT R: demand rate per period (a random variable) R: Average demand rate per period σ R : Standard deviation of the demand rate per period L: Lead time (a constant number of periods) LTD: demand during the lead time (a random variable) LTD: Average demand during the lead time σ LTD : Standard deviation of the demand during lead time

25. 25 Managing Flow Variability: Safety Inventory μ and σ of demand per period and fixed LT A random variable R: N(R, σ R ) repeats itself L times during the lead time. The summation of these L random variables R, is a random variable LTD If we have a random variable LTD which is equal to summation of L random variables R LTD = R 1 +R 2 +R 3 +…….+R L Then there is a relationship between mean and standard deviation of the two random variables

26. 26 Managing Flow Variability: Safety Inventory Demand of sand has an average of 50 tons per week. Standard deviation of the weekly demand is 3 tons. Lead time is 2 weeks. Assuming that the management is willing to accept a risk no more that 10%. μ and σ of demand per period and fixed LT I safety = zσ LTD = 1.28(4.24) = 5.43 ROP = 100 + 5.43 z = 1.28, R = 50, σ R = 3, L = 2

27. 27 Managing Flow Variability: Safety Inventory Lead Time Variable, Demand fixed Demand of sand is fixed and is 50 tons per week. The average lead time is 2 weeks. Standard deviation of lead time is 0.5 week. Assuming that the management is willing to accept a risk no more that 10%. Compute ROP and I safety.

28. 28 Managing Flow Variability: Safety Inventory μ and σ of lead time and fixed Demand per period L: lead time (a random variable) L: Average lead time σ L : Standard deviation of the lead time R L RLRL R: Demand per period (a constant value) LTD: demand during the lead time (a random variable) LTD: Average demand during the lead time σ LTD : Standard deviation of the demand during lead time

29. 29 Managing Flow Variability: Safety Inventory μ and σ of demand per period and fixed LT A random variable L: N(L, σ L ) is multiplied by a constant R and generates the random variable LTD. If we have a random variable LTD which is equal to a constant R times a random variables L LTD = RL Then there is a relationship between mean and standard deviation of the two random variables R L RLRL

30. 30 Managing Flow Variability: Safety Inventory Variable R fixed L …………….Variable L fixed R R L RL R RR RR L R+R+R+R+RR+R+R+R+R

31. 31 Managing Flow Variability: Safety Inventory Lead Time Variable, Demand fixed Demand of sand is fixed and is 50 tons per week. The average lead time is 2 weeks. Standard deviation of lead time is 0.5 week. Assuming that the management is willing to accept a risk no more that 10%. Compute ROP and I safety. z = 1.28, L = 2 weeks, σ L = 0.5 week, R = 50 per week I safety = zσ LTD = 1.28(25) = 32 ROP = 100 + 32

32. 32 Managing Flow Variability: Safety Inventory Both Demand and Lead Time are Variable R: demand rate per period R: Average demand rate σ R : Standard deviation of demand L: lead time L: Average lead time σ L : Standard deviation of the lead time LTD: demand during the lead time (a random variable) LTD: Average demand during the lead time σ LTD : Standard deviation of the demand during lead time

33. 33 Managing Flow Variability: Safety Inventory Optimal Service Level: The Newsvendor Problem Cost of Holding Extra Inventory Improved Service Optimal Service Level under uncertainty The Newsvendor Problem 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. How do we choose what level of service a firm should offer?

34. 34 Managing Flow Variability: Safety Inventory Optimal Service Level: The Newsvendor Problem Cost =1800, Sales Price = 2500, Salvage Price = 1700 Underage Cost = 2500-1800 = 700, Overage Cost = 1800-1700 = 100 What is probability of demand to be equal to 130? What is probability of demand to be less than or equal to 140? What is probability of demand to be greater than 140? What is probability of demand to be equal to 133?

35. 35 Managing Flow Variability: Safety Inventory Optimal Service Level: The Newsvendor Problem What is probability of demand to be equal to 116? What is probability of demand to be less than or equal to 160? What is probability of demand to be greater than 116? What is probability of demand to be equal to 13.3?

36. 36 Managing Flow Variability: Safety Inventory Optimal Service Level: The Newsvendor Problem What is probability of demand to be equal to 130? What is probability of demand to be less than or equal to 140? What is probability of demand to be greater than 140? What is probability of demand to be equal to 133?

37. 37 Managing Flow Variability: Safety Inventory Compute the Average Demand Average Demand = +100×0.02 +110×0.05+120×0.08 +130×0.09+140×0.11 +150×0.16 +160×0.20 +170×0.15 +180×0.08 +190×0.05+200×0.01 Average Demand = 151.6 How many units should I have to sell 151.6 units (on average)? How many units do I sell (on average) if I have 100 units?

38. 38 Managing Flow Variability: Safety Inventory Suppose I have ordered 140 Unities. On average, how many of them are sold? In other words, what is the expected value of the number of sold units? When I can sell all 140 units? I can sell all 140 units if  R≥ 140 Prob(R≥ 140) = 0.76 The the expected number of units sold –for this part- is (0.76)(140) = 106.4 Also, there is 0.02 probability that I sell 100 units  2 units Also, there is 0.05 probability that I sell 110 units  5.5 Also, there is 0.08 probability that I sell 120 units  9.6 Also, there is 0.09 probability that I sell 130 units  11.7 106.4 + 2 + 5.5 + 9.6 + 11.7 = 135.2

39. 39 Managing Flow Variability: Safety Inventory Suppose I have ordered 140 Unities. On average, how many of them are salvaged? In other words, what is the expected value of the number of sold units? 0.02 probability that I sell 100 units. In that case 40 units are salvaged  0.02(40) =.8 0.05 probability to sell 110  30 salvage  0.05(30)= 1.5 0.08 probability to sell 120  30 salvage  0.08(20) = 1.6 0.09 probability to sell 130  30 salvage  0.09(10) =0.9 0.8 + 1.5 + 1.6 + 0.9 = 4.8 Total number Solved 135.2 @ 700 = 94640 Total number Salvaged 4.8 @ -100 = -480 Expected Profit = 94640 – 480 = 94,160

40. 40 Managing Flow Variability: Safety Inventory Cumulative Probabilities

41. 41 Managing Flow Variability: Safety Inventory Number of Units Sold, Salvages

42. 42 Managing Flow Variability: Safety Inventory Total Revenue for Different Ordering Policies

43. 43 Managing Flow Variability: Safety Inventory Net Marginal Benefit: Net Marginal Cost: MB = p – c MC = c - v MB = $2,500 - $1,800 = $700 MC = $1,800 - $1,700 = $100 Analytical Solution for the Optimal Service Level Suppose I have ordered Q units. What is the expected cost of ordering one more units? What is the expected benefit of ordering one more units? If I have ordered one unit more than Q units, the probability of not selling that extra unit is if the demand is less than or equal to Q. Since we have P( R ≤ Q). The expected marginal cost =MC× P( R ≤ Q) If I have ordered one unit more than Q units, the probability of selling that extra unit is if the demand is greater than Q. We know that P(R>Q) = 1- P( R ≤ Q). The expected marginal benefit = MB× [1-Prob.( R ≤ Q)]

44. 44 Managing Flow Variability: Safety Inventory As long as expected marginal cost is less than expected marginal profit we buy the next unit. We stop as soon as: Expected marginal cost ≥ Expected marginal profit MC×Prob(R ≤ Q*) ≥ MB× [1 – Prob(R ≤ Q*)] Prob(R ≤ Q*) ≥ In a continuous model: SL* = Prob(R ≤ Q*) = Analytical Solution for the Optimal Service Level If we assume demand is normally distributed, What quantity corresponds to this service level ?

45. 45 Managing Flow Variability: Safety Inventory Analytical Solution for the Optimal Service Level -4-3-201234 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Probability Less than Upper Bound is 0.87493 Density Critical Value (z) z = 1.15

46. 46 Managing Flow Variability: Safety Inventory Aggregate Forecast is More Accurate than Individual Forecasts

47. 47 Managing Flow Variability: Safety Inventory Physical Centralization Physical Centralization: the firm consolidates all its warehouses in one location from which is can serve all customers. Example: Two warehouses. Demand in the two ware houses are independent. Both warehouses have the same distribution for their lead time demand. LTD 1 : N(LTD, σ LTD ) LTD 2 : N(LTD, σ LTD ) Both warehouses have identical service levels To provide desired SL, each location must carry I safety = zσ LTD z is determined by the desired service level The total safety inventory in the decentralized system is

48. 48 Managing Flow Variability: Safety Inventory Independent Lead time demands at two locations LTD C = LTD 1 + LTD 2  LTD C = LTD + LTD = 2 LTD  GE lighting operating 7 warehouses. A warehouse with average lead time demand of 20,000 units with a standard deviation of 5,000 units and a 95% service level needs to carry a safety inventory of I safety = 1.65×5000= 8250 Decrease in safety inventory by a factor of Centralization reduced the safety inventory by a factor of 1/√2

49. 49 Managing Flow Variability: Safety Inventory independent Lead time demands at N locations Centralization of N locations: Independent demand in N locations: Total safety inventory to provide a specific SL increases not by N but by √N If centralization of stocks reduces inventory, why doesn’t everybody do it? – Longer response time – Higher shipping cost – Less understanding of customer needs – Less understanding of cultural, linguistics, and regulatory barriers These disadvantages my reduce the demand.

50. 50 Managing Flow Variability: Safety Inventory Dependent Demand Does centralization offer similar benefits when demands in multiple locations are correlated? LTD 1 and LTD 2 are statistically identically distributed but correlated with a correlation coefficient of ρ. No Correlation: ρ close to 0

51. 51 Managing Flow Variability: Safety Inventory + Correlation, + Perfect Correlation Negative Correlation: ρ close to -1 Positive Correlation: ρ close to 1 Perfect Negative Correlation: ρ = -1 Perfect Positive Correlation: ρ = +1

52. 52 Managing Flow Variability: Safety Inventory Correlation The safety inventory in the two-location decentralized system is larger than in the centralized system by a factor of If demand is positively fully correlated, ρ = 1, centralization offers no benefits in the reduction of safety inventory Benefits of centralization increases as the demand on the two locations become negatively correlated. The best case is  = -1, where we do not need safety inventory at all

53. 53 Managing Flow Variability: Safety Inventory Principle of Aggregation and Pooling Inventory Inventory benefits  due to principle of aggregation. Statistics: Standard deviation of sum of random variables is less than the sum of the individual standard deviations. Physical consolidation is not essential, as long as available inventory is shared among various locations  Pooling Inventory –Virtual Centralization –Specialization –Component Commonality –Delayed Differentiation –Product Substitution

54. 54 Managing Flow Variability: Safety Inventory Virtual Centralization Location A Exceeds Available stock 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 Location B Less than Available stock Virtual Centralization: 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. Pooling is achieved by keeping the inventories at decentralized locations.

55. 55 Managing Flow Variability: Safety Inventory Specialization, Substitution Demand for both products exist in both locations. But a large portion of demand for P1 is in location A, while a large portion of demand for P2 is in location B. Both locations keep average inventory. Safety inventory is kept only in the specialized warehouse Location A Product P1 Location B Product P2 One other possibility to deal with variability is product substitution.

56. 56 Managing Flow Variability: Safety Inventory Component Commonality Up to now we have discussed aggregating demand across various geographic locations, either physical or virtual Aggregating demand across various products has the same benefits. Computer manufacturers: offer a wide range of models, but few components, CPU, RMA, HD, CD/DVD drive, are used across product lines. Replace Make-to-stock with make Make-to-Order Commonality + MTO: Commonality: Safety inventory of the common components much less than safety inventory of unique components stored separately. MTO: Inventory cost is computed in terms of WIP cost not in terms of finished good cost (which is higher).

57. 57 Managing Flow Variability: Safety Inventory Postponement (Delayed Differentiation) Forecasting Characteristic: Forecasts further into the future tends to be less accurate than those of more imminent events. Since shorter-range forecasts are more accurate, operational decisions will be more effective if supply is postponed closer to the point of actual demand. Two Alternative processes (each activity takes one week)  Alternative A: (1) Coloring the fabric, (2) assembling T-shirts  Alternative B: (1) Assembling T-shirts, (2) coloring the fabric No changes in flow time. Alternative B postponed the color difference until one week closer to the time of sale. Takes advantage of the forecasting characteristic: short-Range forecast more accurate.

58. 58 Managing Flow Variability: Safety Inventory Postponement (Delayed Differentiation) Two advantages: Taking advantage of two demand forecasting characteristics  Commonality Advantage: At week 0; Instead of forecast for each individual item, we forecast for aggregates item – uncolored T- shirt. Forecast for aggregate demand is more accurate than forecast for individual item. It is easier to more accurately forecast total demand for different colored T-shirts for next week than the week after the next.  Postponement Advantage: Instead of forecasting for each individual items two weeks ahead, we do it at week 1. Shorter rang forecasts are more accurate. It is easier to more accurately forecast demand for different colored T-shirts for next week than the week after the next.

59. 59 Managing Flow Variability: Safety Inventory Lessons Learned Levels for Reducing Safety Capacity  Reduce demand variability through improved forecasting  Reduce replenishment lead time  Reduce variability in replenishment lead time  Pool safety inventory for multiple locations or products  Exploit product substitution  Use common components  Postpone product-differentiation processing until closer to the point of actual demand