4 Forecasting Demand in a Supply Chain BIA

4. Forecasting Demand in a Supply Chain BIA 674 - Supply Chain Analytics

Learning Objectives 1. Understand the role of forecasting for both an enterprise and a supply chain. 2. Identify the components of a demand forecast. 3. Forecast demand in a supply chain given historical demand data using time-series and regression methodologies. 4. Analyze demand forecasts to estimate forecast error.

Overview of Forecasting Methods

Role of Forecasting in a Supply Chain The basis for all planning decisions in a supply chain Used for both push and pull processes Production scheduling, inventory, aggregate planning Sales force allocation, promotions, new production introduction Plant/equipment investment, budgetary planning Workforce planning, hiring, layoffs All of these decisions are interrelated

Types of Forecasting Methods Depend on Time frame (how far into the future) Short- to mid-range forecast (immediate future - up to two years) Long-range forecast Demand behavior Causes of behavior

Types of Forecasting Methods Demand Behavior Trend Random variations movements in demand that do not follow a pattern Cycle a gradual, long-term up or down movement of demand an up-and-down repetitive movement in demand Seasonal pattern an up-and-down repetitive movement in demand occurring periodically

Types of Forecasting Methods Forms of Forecast Movement

Components and Methods Companies must identify the factors that influence future demand then ascertain the relationship between these factors and future demand, for example: Past demand Lead time of product replenishment Planned advertising or marketing efforts Planned price discounts State of the economy Actions that competitors have taken

Components and Methods 1. Qualitative Primarily subjective Rely on judgment 2. Time Series Use historical demand only Best with stable demand 3. Causal – Regression Relationship between demand some other factor 4. Simulation Imitate consumer choices that give rise to demand

Components of an Observation Observed demand (O) = systematic component (S) + random component (R) Systematic component – expected value of demand Level (current deseasonalized demand) Trend (growth or decline in demand) Seasonality (predictable seasonal fluctuation) Random component – part of forecast that deviates from systematic component (size and variability) Forecast error – difference between forecast

Time-Series Methods Use historical data only Relate the forecast to the time only • Linear Trend Line method • Static Methods • Additive Methods

Time-Series Techniques Double Exponential Smoothing 28 26 24 22 20 18 16 14 12 10 Actual Observations Demand Forecast DES Best -> Forecast DES: MAPE 8. 177 1 2 3 4 5 6 7 Time 8 9 10 11 12

Time-Series Forecasting Methods Assumption: what has occurred in the past will continue to occur in the future Relate the forecast to only one factor - time Linear trend line method Additive methods Static methods Three ways to calculate the systematic component: – – – Multiplicative S = level x trend x seasonal factor Additive S = level + trend + seasonal factor Mixed S = (level + trend) x seasonal factor

Linear Trend Line Method S = level + trend y = a + bx where a = intercept coefficient b = slope of the line x = time period (variable coefficient) y = forecast for demand for period x xy - nxy = b 2 x - nx 2 a = y - b x where n = number of periods x x = n = mean of the x values y y = n = mean of the y values

Linear Trend Line Method x(PERIOD) y(DEMAND) xy x 2 1 37 37 1 2 40 80 4 3 41 123 9 4 37 148 16 5 45 225 25 6 50 300 36 7 43 301 49 8 47 376 64 9 56 504 81 10 52 520 100 11 55 605 121 12 54 648 144 78 557 3867 650 Least Squares Example

Linear Trend Line Method Least Squares Example (cont. ) 78 12 557 12 x = = 6. 5 y = = 46. 42 xy - nxy 3867 - (12)(6. 5)(46. 42) b = =1. 72 x 2 - nx 2 650 - 12(6. 5)2 a = y - bx = 46. 42 - (1. 72)(6. 5) = 35. 2

Linear trend line y = 35. 2 + 1. 72 x Forecast for period 13 y = 35. 2 + 1. 72(13) = 57. 56 units 70 – 70 Demand 60 – 60 Actual 50 – 50 40 – 40 Linear trend line 30 – 30 20 – 20 10 – 10 0 – 0 | 1 | 2 | 3 | 4 | 5 | | 6 7 Period | 8 | 9 | 10 | 11 | 12 | 13

Adaptive Forecasting Methods The estimates of level, trend, and seasonality are adjusted after each demand observation Estimates incorporate all new data that are observed. Methods: Moving Averages + Weighted Moving Averages Simple Exponential Smoothing Trend-Corrected Exponential Smoothing (Holt’s model) Trend- and Seasonal-Corrected Exponential

Adaptive Forecasting Methods where estimate of level at the end of Period t estimate of trend at the end of Period t estimate of seasonal factor for Period t forecast of demand for Period t (made Period t – 1 or earlier) Dt = actual demand observed in Period t Et = Ft – Dt = forecast error in Period t Lt Tt St Ft = =

Steps in Adaptive Forecasting Initialize Forecast demand for period t + 1 Estimate error Compute initial estimates of level (L 0), trend (T 0), and seasonal factors (S 1, …, Sp) Compute error Et+1 = Ft+1 – Dt+1 Modify estimates Modify the estimates of level (Lt+1), trend (Tt+1), and seasonal factor (St+p+1), given the error Et+1

Moving Average Assumption: no observable trend or seasonality Systematic component of demand = level The level in period t is the average demand over the last N periods Lt = (Dt + Dt-1 + … + Dt–N+1) / N Ft+1 = Lt and Ft+n = Lt After observing the demand for period t + 1, revise the estimates Lt+1 = (Dt+1 + Dt + … + Dt-N+2) / N, Ft+2 = Lt+1

Moving Average Example I A supermarket has experienced weekly demand of milk of D 1 = 120, D 2 = 127, D 3 = 114, and D 4 = 122 gallons over the past four weeks What is the forecast demand for Period 5 using a 4 -period moving average? What is the forecast error if demand in Period 5 turns out to be 125 gallons?

Moving Average Example I L 4 = (D 4 + D 3 + D 2 + D 1)/4 = (122 + 114 + 127 + 120)/4 = 120. 75 Forecast demand for Period 5: F 5 = L 4 = 120. 75 gallons Error if demand in Period 5 = 125 gallons E 5 = F 5 – D 5 = 125 – 120. 75 = 4. 25 Revised demand L 5 = (D 5 + D 4 + D 3 + D 2)/4 = (125 + 122 + 114 + 127)/4 = 122

Moving Average Example II 3 -month Simple Moving Average MONTH Jan Feb Mar Apr May June July Aug Sept ORDERS PER MONTH 120 90 100 75 110 50 75 130 110 Oct Nov 90 - MOVING AVERAGE – – – 103. 3 88. 3 95. 0 78. 3 85. 0 105. 0 110. 0

Moving Average Example II 5 -month Simple Moving Average MONTH Jan Feb Mar Apr May June July Aug Sept ORDERS PER MONTH 120 90 100 75 110 50 75 130 110 Oct Nov 90 - MOVING AVERAGE – – – 99. 0 85. 0 82. 0 88. 0 95. 0 91. 0

Moving Average Example II Smoothing Effects 5 -month 3 -month

Weighted Moving Average Enhancement: Adjusts the moving average method to more closely reflect data fluctuations The level in period t is the weighted average demand over the last N periods Lt = Wt Dt + Wt-1 Dt-1 + … + Wt–N+1 Dt–N+1 Ft+1 = Lt and Ft+n = Lt Wt is the weight for period t, between 0 and 100 percent All weights must add up to 1.

Weighted Moving Average Example MONTH WEIGHT August September October 17% 33% 50% DATA 130 110 90 November Forecast = (0. 50)(90) + (0. 33)(110) + (0. 17)(130) = 103. 4 orders

Simple Exponential Smoothing q Assumption: No observable trend or seasonality Systematic component of demand = level q Initial estimate of level L 0 is assumed to be the average of all historical data (this is optional)

Simple Exponential Smoothing Given data for Periods 1 to n (this step is optional) Current forecast Revised forecast using smoothing constant 0 < a < 1 Thus

Simple Exponential Smoothing Use supermarket data to forecast period 5 using a=0. 1 E 1 = F 1 – D 1 = 120. 75 – 120 = 0. 75

Simple Exponential Smoothing If we continue the calculations for the next periods (2, 3 & 4), we obtain F 5 = 120. 72 a Period 0 1 2 3 4 5 0. 1 Dt 120 127 114 122 Lt 120. 75 120. 68 121. 31 120. 58 120. 72 Ft 120. 75 120. 68 121. 31 120. 58 120. 72 Et 0. 75 -6. 33 7. 31 -1. 42

Simple Exponential Smoothing Effect of Smoothing Constant 0. 0 1. 0 If = 0. 20, then Lt +1 = 0. 20 Dt + 0. 80 Lt If = 0, then Lt +1 = 0 Dt + 1 Lt = Lt Forecast does not reflect recent data If = 1, then Lt +1 = 1 Dt + 0 Lt = Dt Forecast based only on most recent data

Simple Exponential Smoothing (α=0. 30) Period 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Dt 37 40 41 37 45 50 43 47 56 52 55 54 Lt 46. 42 43. 59 42. 51 42. 06 40. 54 41. 88 44. 32 43. 92 44. 84 48. 19 49. 33 51. 03 51. 92 Ft 46. 42 43. 59 42. 51 42. 06 40. 54 41. 88 44. 32 43. 92 44. 84 48. 19 49. 33 51. 03 51. 92 Et 9. 42 3. 59 1. 51 5. 06 -4. 46 -8. 12 1. 32 -3. 08 -11. 16 -3. 81 -5. 67 -2. 97

Simple Exponential Smoothing (α=0. 50) Period 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Dt 37 40 41 37 45 50 43 47 56 52 55 54 Lt 46. 42 41. 71 40. 85 40. 93 38. 96 41. 98 45. 99 44. 50 45. 75 50. 87 51. 44 53. 22 53. 61 Ft 46. 42 41. 71 40. 85 40. 93 38. 96 41. 98 45. 99 44. 50 45. 75 50. 87 51. 44 53. 22 53. 61 Et 9. 42 1. 71 -0. 15 3. 93 -6. 04 -8. 02 2. 99 -2. 50 -10. 25 -1. 13 -3. 56 -0. 78

Simple Exponential Smoothing

Trend-Corrected Exponential Smoothing (Holt’s Model) Appropriate when the demand is assumed to have a level and trend in the systematic component of demand but no seasonality Systematic component of demand = level + trend

Trend-Corrected Exponential Smoothing (Holt’s Model) Obtain initial estimate of level and trend by running a linear regression Dt = at + b T 0 = a, L 0 = b In Period t, the forecast for future periods is Ft+1 = Lt + Tt and Ft+n = Lt + n. Tt Revised estimates for Period t (β is a smoothing constant factor) Lt+1 = a. Dt+1 + (1 – a)(Lt + Tt) Tt+1 = b(Lt+1 – Lt) + (1 – b)Tt

Trend-Corrected Exponential Smoothing (Holt’s Model) MP 3 player demand D 1 = 8, 415, D 2 = 8, 732, D 3 = 9, 014, D 4 = 9, 808, D 5 = 10, 413, D 6 = 11, 961 a = 0. 1, b = 0. 2 Using regression analysis (line trend method) L 0 = 7, 367 and T 0 = 673 Forecast for Period 1 F 1 = L 0 + T 0 = 7, 367 + 673 = 8, 040

Trend-Corrected Exponential Smoothing (Holt’s Model) Revised estimate L 1 = a. D 1 + (1 – a)(L 0 + T 0) = 0. 1 x 8, 415 + 0. 9 x 8, 040 = 8, 078 T 1 = b(L 1 – L 0) + (1 – b)T 0 = 0. 2 x (8, 078 – 7, 367) + 0. 8 x 673 = 681 With new L 1 F 2 = L 1 + T 1 = 8, 078 + 681 = 8, 759 Continuing F 7 = L 6 + T 6 = 11, 399 + 673 = 12, 072

Trend-Corrected Exponential Smoothing (Holt’s Model) Period January February March April May June July August September October November December 0 1 2 3 4 5 6 7 8 9 10 11 12 Demand L 35. 212 37 36. 97 40 39. 35 41 41. 14 37 40. 01 45 43. 00 50 47. 29 43 46. 34 47 47. 37 56 52. 33 52 53. 36 55 55. 17 54 55. 55 b a 0. 300 0. 500 T Forecast 1. 7238 1. 73 36. 94 1. 93 38. 70 1. 89 41. 28 0. 98 43. 03 1. 58 41. 00 2. 40 44. 58 1. 39 49. 69 1. 28 47. 74 2. 39 48. 65 1. 98 54. 71 1. 93 55. 33 1. 46 57. 10 57. 01 MAD 3. 167 MSE 16. 649 MAPE 6. 826 |Et| 0. 06 1. 30 0. 28 6. 03 4. 00 5. 42 6. 69 0. 74 7. 35 2. 71 0. 33 3. 10 Et 2 0. 00 1. 69 0. 08 36. 31 16. 04 29. 37 44. 71 0. 54 54. 01 7. 35 0. 11 9. 58 (|Et|/Dt)% 0. 17 3. 25 0. 68 16. 29 8. 90 10. 84 15. 55 1. 57 13. 12 5. 21 0. 61 5. 73

Trend-Corrected Exponential Smoothing (Holt’s Model)

Trend- and Seasonality-Corrected Exponential Smoothing Appropriate when the systematic component of demand is assumed to have a level, trend, and seasonal factor Systematic component = (level + trend) x seasonal factor Ft+1 = (Lt + Tt)St+1 and Ft+l = (Lt + l. Tt)St+l

Trend- and Seasonality-Corrected Exponential Smoothing q After observing demand for period t + 1, revise estimates for level, trend, and seasonal factors Lt+1 = a(Dt+1/St+1) + (1 – a)(Lt + Tt) Tt+1 = b(Lt+1 – Lt) + (1 – b)Tt St+p+1 = g(Dt+1/Lt+1) + (1 – g)St+1 a = smoothing constant for level b = smoothing constant for trend g = smoothing constant for seasonal factor

Winter’s Model q Tahoe Salt Example L 0 = 18, 439 T 0 = 524 S 1= 0. 47, S 2 = 0. 68, S 3 = 1. 17, S 4 = 1. 67 F 1 = (L 0 + T 0)S 1 = (18, 439 + 524)(0. 47) = 8, 913 The observed demand for Period 1 = D 1 = 8, 000 Forecast error for Period 1 = E 1 = F 1 – D 1 = 8, 913 – 8, 000 = 913 Year, Qtr 1, 2 1, 3 1, 4 2, 1 2, 2 2, 3 2, 4 3, 1 3, 2 3, 3 3, 4 4, 1 Period t 1 2 3 4 5 6 7 8 9 10 11 12 Demand Dt 8, 000 13, 000 23, 000 34, 000 10, 000 18, 000 23, 000 38, 000 12, 000 13, 000 32, 000 41, 000

Winter’s Model q Assume a = 0. 1, b = 0. 2, g = 0. 1; revise estimates for level and trend for period 1 and for seasonal factor for Period 5 L 1 = a(D 1/S 1) + (1 – a)(L 0 + T 0) = 0. 1 x (8, 000/0. 47) + 0. 9 x (18, 439 + 524) = 18, 769 T 1 = b(L 1 – L 0) + (1 – b)T 0 = 0. 2 x (18, 769 – 18, 439) + 0. 8 x 524 = 485 S 5 = g(D 1/L 1) + (1 – g)S 1 = 0. 1 x (8, 000/18, 769) + 0. 9 x 0. 47 = 0. 47 F 2 = (L 1 + T 1)S 2 = (18, 769 + 485)0. 68 = 13, 093

Winter’s Model Period t Demand Dt Level Lt Trend Tt 18, 439 524 1 8, 000 18, 769 485 2 13, 000 19, 240 482 3 23, 000 19, 716 481 4 34, 000 20, 214 484 5 10, 000 20, 776 500 6 18, 000 21, 797 604 7 23, 000 22, 127 549 8 38, 000 22, 683 551 9 12, 000 23, 479 600 10 13, 000 23, 543 493 11 32, 000 24, 399 565 12 41, 000 24, 921 557 13 14 15 16 alpha Beta Gamma 0. 1 0. 2 0. 1 Seasonal Factor Absolute Error St Forecast Ft Error Et At 0. 47 8, 913 913 0. 68 13, 093 93 93 1. 17 23, 076 76 76 1. 67 33, 730 -270 0. 47 9, 637 -363 0. 68 14, 458 -3, 542 1. 17 26, 202 3, 202 1. 67 37, 898 -102 0. 47 10, 855 -1, 145 0. 69 16, 715 3, 715 1. 16 27, 801 -4, 199 1. 67 41, 731 731 0. 47 12, 015 0. 68 17, 703 1. 17 31, 167 1. 67 45, 307 Mean Squared Error MSEt 832, 857 420, 727 282, 393 230, 049 210, 321 2, 265, 821 3, 406, 502 2, 981, 997 2, 796, 424 3, 896, 961 5, 145, 678 4, 761, 357 MADt % Error MAPEt 913 11 11. 41 503 1 6. 06 360 0 4. 15 338 1 3. 31 343 4 3. 37 876 20 6. 09 1, 208 14 7. 21 1, 070 0 6. 34 1, 078 10 6. 70 1, 342 29 8. 89 1, 602 13 9. 27 1, 529 2 8. 65 TSt 1. 00 2. 00 3. 00 2. 40 1. 31 -3. 53 0. 09 0. 01 -1. 06 1. 92 -1. 01 -0. 58

Winter’s Model

Time Series Models Forecasting Method Applicability Moving average No trend or seasonality Simple exponential smoothing No trend or seasonality Holt’s model Trend but no seasonality Winter’s model Trend and seasonality

Static Forecasting Methods Static methods assumes that the estimates of level, trend, and seasonality within the systematic component do not vary as new demand is observed where L T St Dt Ft = = = estimate of level at t = 0 estimate of trend estimate of seasonal factor for Period t actual demand observed in Period t forecast of demand for Period t

Static Forecasting Methods - Tahoe Salt Example Year Quarter Period, t Demand, Dt 1 2 1 8, 000 1 3 2 13, 000 1 4 3 23, 000 2 1 4 34, 000 2 2 5 10, 000 2 3 6 18, 000 2 4 7 23, 000 3 1 8 38, 000 3 2 9 12, 000 3 3 10 13, 000 3 4 11 32, 000 4 1 12 41, 000

Static Forecasting Methods - Tahoe Salt Example Observe that the demand is seasonal, the lowest occurs every second quarter of each year, and the overall demand pattern repeats every year.

Static Forecasting Methods - Tahoe Salt Example How to estimate level, trend, and seasonality factors? Basic Steps: 1. Deseasonalize demand run the linear trend line method to estimate level and trend To ensure that each season is given equal weight when deseasonalizing the demand, we must take the average of p consecutive periods of demand. 2. Estimate seasonal factors The seasonal factor of a period t is the ratio between the actual demand the deseasonalized demand

Static Forecasting Methods - Tahoe Salt Example Estimate Level and Trend Deseasonalized Demand formulas: Periodicity p = 4, t = 3

Static Forecasting Methods - Tahoe Salt Example Estimate Level and Trend

Static Forecasting Methods - Tahoe Salt Example Estimate Level and Trend A linear relationship exists between the deseasonalized demand time based on the change in demand over time

Static Forecasting Methods - Tahoe Salt Example Estimate Level and Trend The values of L and T for the deseasonalized demand can be estimated using the linear trend line method: See also later slides on Linear Regression

Static Forecasting Methods - Tahoe Salt Example Estimating Seasonal Factors

Static Forecasting Methods - Tahoe Salt Example Estimating Seasonal Factors Given r seasonal cycles in the data, for all periods of the form pt+1, 1<=i<=p, we should obtain the seasonal factor for each period: For r = 3 (seasonal cycles) and p=4 (periods) we get:

Static Forecasting Methods - Tahoe Salt Example Forecast for the next 4 periods Given the level, trend all seasonal factors calculated earlier, using the following formula we obtain the forecast for the next 4 quarters:

Static Forecasting Methods - Tahoe Salt Example

Forecast Errors

Measures of Forecast Error Declining alpha

Forecast Accuracy Forecast error difference between forecast and actual demand MAD mean absolute deviation MAPD mean absolute percent deviation Cumulative error Average error or bias

Mean Absolute Deviation (MAD) Dt - Ft MAD = n where t = period number Dt = demand in period t Ft = forecast for period t n = total number of periods = absolute value

MAD Example PERIOD 1 2 3 4 5 6 7 8 9 10 11 12 DEMAND, Dt 37 40 41 37 45 50 43 47 56 52 55 54 557 Ft ( =0. 3) (Dt - Ft) 37. 00 37. 90 38. 83 38. 28 40. 29 43. 20 43. 14 44. 30 47. 81 49. 06 50. 84 – 3. 00 3. 10 -1. 83 6. 72 9. 69 -0. 20 3. 86 11. 70 4. 19 5. 94 3. 15 – 3. 00 3. 10 1. 83 6. 72 9. 69 0. 20 3. 86 11. 70 4. 19 5. 94 3. 15 49. 31 53. 39 |Dt - Ft| MAD = 53. 39/11 = 4. 85

Other Accuracy Measures Mean absolute percent deviation (MAPD) |Dt - Ft| MAPD = Dt Cumulative error E = et Average error et E= n

Forecast Control Tracking signal monitors the forecast to see if it is biased high or low 1 MAD ≈ 0. 8 б Control limits of 2 to 5 MADs are used most frequently (Dt - Ft) E Tracking signal = = MAD

Tracking Signal Values PERIOD 1 2 3 4 5 6 7 8 9 10 11 12 DEMAND Dt 37 40 41 37 45 50 43 47 56 52 55 54 FORECAST, Ft ERROR Dt - Ft E = (Dt - Ft) 37. 00 – – 37. 00 37. 90 3. 10 6. 10 38. 83 -1. 83 4. 27 38. 28 6. 72 10. 99 Tracking signal for period 3 40. 29 9. 69 20. 68 43. 20 -0. 20 20. 48 6. 10 43. 14 3. 86 24. 34 TS 3 = = 2. 00 3. 05 44. 30 11. 70 36. 04 47. 81 4. 19 40. 23 49. 06 5. 94 46. 17 50. 84 3. 15 49. 32 TRACKING MAD SIGNAL – – 3. 00 1. 00 3. 05 2. 00 2. 64 1. 62 3. 66 3. 00 4. 87 4. 25 4. 09 5. 01 4. 06 6. 00 5. 01 7. 19 4. 92 8. 18 5. 02 9. 20 4. 85 10. 17

Statistical Control Charts = (Dt - Ft)2 n - 1 Using we can calculate statistical control limits for the forecast error Control limits are typically set at 3

Statistical Control Charts 18. 39 – UCL = +3 12. 24 – Errors 6. 12 – 0 – -6. 12 – -12. 24 – -18. 39 – | 0 LCL = -3 | 1 | 2 | 3 | 4 | 5 | 6 Period | 7 | 8 | 9 | 10 | 11 | 12

Selecting the Best Smoothing Constants

Selecting the Best Smoothing Constant

Selecting the Best Smoothing Constant

Application at Tahoe Salt

Forecasting Demand at Tahoe Salt Moving average Simple exponential smoothing Trend-corrected exponential smoothing Trend- and seasonality-corrected exponential smoothing

Forecasting Demand at Tahoe Salt

Forecasting Demand at Tahoe Salt Moving average L 12 = 24, 500 F 13 = F 14 = F 15 = F 16 = L 12 = 24, 500 s = 1. 25 x 9, 719 = 12, 148

Forecasting Demand at Tahoe Salt

Forecasting Demand at Tahoe Salt Single exponential smoothing L 0 = 22, 083 L 12 = 23, 490 F 13 = F 14 = F 15 = F 16 = L 12 = 23, 490 s = 1. 25 x 10, 208 = 12, 761

Forecasting Demand at Tahoe Salt

Forecasting Demand at Tahoe Salt Trend-Corrected Exponential Smoothing L 0 = 12, 015 and T 0 = 1, 549 L 12 = 30, 443 and T 12 = 1, 541 F 13 = L 12 + T 12 = 30, 443 + 1, 541 = 31, 984 F 14 = L 12 + 2 T 12 = 30, 443 + 2 x 1, 541 = 33, 525 F 15 = L 12 + 3 T 12 = 30, 443 + 3 x 1, 541 = 35, 066 F 16 = L 12 + 4 T 12 = 30, 443 + 4 x 1, 541 = 36, 607 s = 1. 25 x 8, 836 = 11, 045

Forecasting Demand at Tahoe Salt

Forecasting Demand at Tahoe Salt Trend- and Seasonality-Corrected L 0 = 18, 439 T 0 =524 S 1 = 0. 47 S 2 = 0. 68 S 3 = 1. 17 S 4 = 1. 67 L 12 = 24, 791 T 12 = 532 F 13 = (L 12 + T 12)S 13 = (24, 791 + 532)0. 47 = 11, 940 F 14 = (L 12 + 2 T 12)S 13 = (24, 791 + 2 x 532)0. 68 = 17, 579 F 15 = (L 12 + 3 T 12)S 13 = (24, 791 + 3 x 532)1. 17 = 30, 930 F 16 = (L 12 + 4 T 12)S 13 = (24, 791 + 4 x 532)1. 67 = 44, 928 s = 1. 25 x 1, 469 = 1, 836

Forecasting Demand at Tahoe Salt

Forecasting Demand at Tahoe Salt Forecasting Method MAD MAPE (%) TS Range Four-period moving average 9, 719 49 – 1. 52 to 2. 21 Simple exponential smoothing 10, 208 59 – 1. 38 to 2. 15 Holt’s model 8, 836 52 – 2. 15 to 2. 00 Winter’s model 1, 469 8 – 2. 74 to 4. 00

Regression Methods

Regression Methods Linear regression a mathematical technique that relates a dependent variable to an independent variable in the form of a linear equation Correlation a measure of the strength of the relationship between independent and dependent variables

Linear Regression Example (WINS) x (ATTENDANCE) y xy x 2 4 6 6 8 6 7 5 7 36. 3 40. 1 41. 2 53. 0 44. 0 45. 6 39. 0 47. 5 145. 2 240. 6 247. 2 424. 0 264. 0 319. 2 195. 0 332. 5 16 36 36 64 36 49 25 49 49 346. 7 2167. 7 311

Linear Regression Example (cont. ) 49 x = = 6. 125 8 346. 9 y = = 43. 36 8 xy - nxy 2 b= 2 2 x - nx (2, 167. 7) - (8)(6. 125)(43. 36) = 2 (311) - (8)(6. 125) = 4. 06 a = y - bx = 43. 36 - (4. 06)(6. 125) = 18. 46

Linear Regression Example (cont. ) 60, 000 – 60, 000 Attendance forecast for 7 wins y = 18. 46 + 4. 06(7) = 46. 88, or 46, 880 Attendance, y 50, 000 – 50, 000 40, 000 – 40, 000 30, 000 – 30, 000 Linear regression line, y = 18. 46 + 4. 06 x 20, 000 – 20, 000 10, 000 – 10, 000 | 1 | 2 | 3 | 4 | | 5 6 Wins, x | 7 | 8 | 9 | 10

Correlation and Coefficient of Determination Correlation, r Measure of strength of relationship Varies between -1. 00 and +1. 00 Coefficient of determination, r 2 Percentage of variation in dependent variable resulting from changes in the independent variable

Computing Correlation r= r= n xy - x y [n x 2 - ( x)2] [n y 2 - ( y)2] (8)(2, 167. 7) - (49)(346. 9) [(8)(311) - (49)2] [(8)(15, 224. 7) - (346. 9)2] r = 0. 947 Coefficient of determination r 2 = (0. 947)2 = 0. 897

Correlation Coefficient y y x x (a) Perfect negative correlation y (e) Perfect positive correlation y y x x (b) Negative correlation (d) Positive correlation x (c) No correlation High | | – 1. 0 – 0. 8 Moderate | – 0. 6 | Low – 0. 4 – 0. 2 0 0. 2 Correlation coefficient values | Moderate | 0. 4 | 0. 6 High 0. 8 | 1. 0

Standard Error of the Estimate ► A forecast is just a point estimate of a future value This point is actually the mean of a probability distribution Nodel’s sales (in$ millions) ► 4. 0 – 3. 25 3. 0 – Regression line, 2. 0 – 1. 0 – 0 | 1 | 2 | 3 | 4 | 5 Area payroll (in $ billions) | 6 | 7

Standard Error of the Estimate where y = y-value of each data point yc = computed value of the dependent variable, from the regression equation n = number of data points

Standard Error of the Estimate Computationally, this equation is considerably easier to use We use the standard error to set up prediction intervals around the point estimate

The standard error of the estimate is $306, 000 in sales Nodel’s sales (in$ millions) Standard Error of the Estimate 4. 0 – 3. 25 3. 0 – 2. 0 – 1. 0 – 0 | 1 | 2 | 3 | 4 | 5 Area payroll (in $ billions) | 6 | 7

Regression Analysis with Excel

Regression Analysis with Excel Maosplace. xls – Linear Regression

Multiple Regression Study the relationship of demand to two or more independent variables y = 0 + 1 x 1 + 2 x 2 … + kxk where 0 = the intercept 1, … , k = parameters for the independent variables x 1, … , xk = independent variables

Multiple Regression with Excel

Example of Multi-Regression for Demand Forecasting Day Temperature Flyer Sales 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 63 70 73 75 80 82 85 88 90 91 92 75 98 100 92 87 84 88 80 82 76 ✓ 152 168 180 235 236 225 268 330 314 306 374 192 340 388 317 283 258 310 226 214 198 ✓ ✓ ✓ Can we predict customer reactions or the demand for a product (ice cream)? ✓ : Flyer was included in the local daily newspaper

Do you see any relationships in these data? Day Temperature Flyer Sales 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 63 70 73 75 80 82 85 88 90 91 92 75 98 100 92 87 84 88 80 82 76 0 0 0 1 0 0 0 152 168 180 235 236 225 268 330 314 306 374 192 340 388 317 283 258 310 226 214 198 Two parameters possibly affecting sales: temperature and flyer advertisement in the local daily newspaper Flyer = 0: No flyer included in the local daily newspaper Flyer = 1: Otherwise

Sales Forecast – Single Regression Use as predictor variable the temperature, assuming a linear relationship between sales and temperature Can I use this model to forecast the demand for a day in 2 months from now? Can I make forecasts outside the above temperature range? Should I cleanup the historical data (e. g. days with sold outs)?

Sales Forecast - Multiple Regression Can I increase predictability by considering flyer promotions on particular days? The relationship with TWO explanatory (predictor) variables can be formalized as follows: Sales(t) = B 0 + B 1*Temperature(t)+ B 2*Flyer(t)

Sales Forecast - Multiple Regression Better predictability than the original regression (R Square was 0. 8928) SUMMARY OUTPUT Regression Statistics Multiple R 0. 969768733 R Square 0. 940451395 Adjusted R Square 0. 933834884 Standard Error 14. 95415498 Observations 21 ANOVA Regression Residual Total Intercept Temperature Flyer df 2 18 20 SS MS F Significance F 63571. 28991 31785. 64 142. 1370421 9. 41552 E-12 4025. 281521 223. 6268 67596. 57143 Coefficients Standard Error t Stat P-value -250. 4242164 31. 33367525 -7. 99218 2. 48511 E-07 6. 027360446 0. 380548578 15. 83861 5. 16976 E-12 3. 000080948 8. 093034548 0. 370699 0. 715188186 B 0; B 1; B 2 Very small chance (<0. 05) the independent values are not useful Lower 95% Upper 95% Lower 95. 0% Upper 95. 0% -316. 2538254 -184. 5946075 5. 227857551 6. 82686334 -14. 00275371 20. 0029156 Significance

Sales Forecast - Multiple Regression

Application in Practice

Demand Forecasting Extrapolate past demand data into the future using time- series forecasting methods: • Select a preferred forecasting method from a set of techniques • Quantify the forecasting errors to be experienced when forecasting the future Use regression analysis to build predictive models that will relate targeted customer segments with casual variable such as price, inventory, and so on.

Basic Approach 1. Understand the objective of forecasting. 2. Integrate demand planning and forecasting throughout the supply chain. 3. Identify the major factors that influence the demand forecast. 4. Forecast at the appropriate level of aggregation. 5. Establish performance and error measures for the forecast.

Forecasting In Practice 1. Identify the purpose of forecast 2. Collect historical data 6. Check forecast accuracy with one or more measures 5. Develop/compute forecast for period of historical data 7. Is accuracy of forecast acceptable? No 3. Plot data and identify patterns 4. Select a forecast model that seems appropriate for data 8 b. Select new forecast model or adjust parameters of existing model Yes 8 a. Forecast over planning horizon 9. Adjust forecast based on additional qualitative information and insight 10. Monitor results and measure forecast accuracy

Summary of Learning Objectives 1. Understand the role of forecasting for both an enterprise and a supply chain 2. Identify the components of a demand forecast 3. Forecast demand in a supply chain given historical demand data using time-series methodologies 4. Analyze demand forecasts to estimate forecast error