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Adjusted Exponential Smoothing

Adjusted Exponential Smoothing. AF t +1 = F t +1 + T t +1 where T = an exponentially smoothed trend factor T t +1 =  ( F t +1 - F t ) + (1 -  ) T t where T t = the last period trend factor  = a smoothing constant for trend. Adjusted Exponential Smoothing Example.

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Adjusted Exponential Smoothing

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  1. Adjusted Exponential Smoothing AFt +1 = Ft +1 + Tt +1 where T = an exponentially smoothed trend factor Tt +1 = (Ft +1 - Ft) + (1 - ) Tt where Tt= the last period trend factor = a smoothing constant for trend

  2. Adjusted Exponential Smoothing Example

  3. PERIOD MONTH DEMAND 1 Jan 37 2 Feb 40 3 Mar 41 4 Apr 37 5 May 45 6 Jun 50 7 Jul 43 8 Aug 47 9 Sep 56 10 Oct 52 11 Nov 55 12 Dec 54 Adjusted Exponential Smoothing Example

  4. PERIOD MONTH DEMAND 1 Jan 37 2 Feb 40 3 Mar 41 4 Apr 37 5 May 45 6 Jun 50 7 Jul 43 8 Aug 47 9 Sep 56 10 Oct 52 11 Nov 55 12 Dec 54 T3 = (F3 - F2) + (1 - ) T2 = (0.30)(38.5 - 37.0) + (0.70)(0) = 0.45 AF3 = F3 + T3 = 38.5 + 0.45 = 38.95 T13 = (F13 - F12) + (1 - ) T12 = (0.30)(53.61 - 53.21) + (0.70)(1.77) = 1.36 AF13 = F13 + T13 = 53.61 + 1.36 = 54.96 Adjusted Exponential Smoothing Example

  5. Adjusted Exponential Smoothing Example FORECAST TREND ADJUSTED PERIOD MONTH DEMAND Ft +1 Tt +1 FORECAST AFt +1 1 Jan 37 37.00 – – 2 Feb 40 37.00 0.00 37.00 3 Mar 41 38.50 0.45 38.95 4 Apr 37 39.75 0.69 40.44 5 May 45 38.37 0.07 38.44 6 Jun 50 38.37 0.07 38.44 7 Jul 43 45.84 1.97 47.82 8 Aug 47 44.42 0.95 45.37 9 Sep 56 45.71 1.05 46.76 10 Oct 52 50.85 2.28 58.13 11 Nov 55 51.42 1.76 53.19 12 Dec 54 53.21 1.77 54.98 13 Jan – 53.61 1.36 54.96

  6. 70 – 60 – 50 – 40 – 30 – 20 – 10 – 0 – Demand | | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 13 Period Adjusted Exponential Smoothing Forecasts

  7. 70 – 60 – 50 – 40 – 30 – 20 – 10 – 0 – Actual Demand Forecast ( = 0.50) | | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 13 Period Adjusted Exponential Smoothing Forecasts

  8. 70 – 60 – 50 – 40 – 30 – 20 – 10 – 0 – Adjusted forecast ( = 0.30) Actual Demand Forecast ( = 0.50) | | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 13 Period Adjusted Exponential Smoothing Forecasts

  9. Linear Trend Line y = a + bx where a = intercept (at period 0) b = slope of the line x = the time period y = forecast for demand for period x

  10. xy - nxy x2- nx2 b = a = y - b x where n = number of periods x = = mean of the x values y = = mean of the y values x n y n Linear Trend Line y = a + bx where a = intercept (at period 0) b = slope of the line x = the time period y = forecast for demand for period x

  11. x(PERIOD) y(DEMAND) 1 73 2 40 3 41 4 37 5 45 6 50 7 43 8 47 9 56 10 52 11 55 12 54 78 557 Least Squares Example

  12. x(PERIOD) y(DEMAND) xy x2 1 73 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

  13. x(PERIOD) y(DEMAND) xy x2 557 12 78 12 1 73 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 x = = 6.5 y = = 46.42 b = = = 1.72 a = y - bx = 46.42 - (1.72)(6.5) = 35.2 xy - nxy x2 - nx2 3867 - (12)(6.5)(46.42) 650 - 12(6.5)2 Least Squares Example

  14. Linear trend line x(PERIOD) y(DEMAND) xy x2 y = 35.2 + 1.72x 557 12 78 12 1 73 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 x = = 6.5 y = = 46.42 b = = = 1.72 a = y - bx = 46.42 - (1.72)(6.5) = 35.2 xy - nxy x2 - nx2 3867 - (12)(6.5)(46.42) 650 - 12(6.5)2 Least Squares Example

  15. x(PERIOD) y(DEMAND) xy x2 Linear trend line 557 12 78 12 1 73 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 x = = 6.5 y = = 46.42 b = = = 1.72 a = y - bx = 46.42 - (1.72)(6.5) = 35.2 y = 35.2 + 1.72x Forecast for period 13 y = 35.2 + 1.72(13) xy - nxy x2 - nx2 3867 - (12)(6.5)(46.42) 650 - 12(6.5)2 y = 57.56 units Least Squares Example

  16. 70 – 60 – 50 – 40 – 30 – 20 – 10 – 0 – Demand | | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 13 Period Linear Trend Line

  17. 70 – 60 – 50 – 40 – 30 – 20 – 10 – 0 – Actual Demand | | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 13 Period Linear Trend Line

  18. 70 – 60 – 50 – 40 – 30 – 20 – 10 – 0 – Actual Demand Linear trend line | | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 13 Period Linear Trend Line

  19. Seasonal Adjustments • Repetitive increase/ decrease in demand • Use seasonal factor to adjust forecast

  20. Di D Seasonal factor = Si = Seasonal Adjustments • Repetitive increase/ decrease in demand • Use seasonal factor to adjust forecast

  21. Seasonal Adjustment

  22. DEMAND (1000’S PER QUARTER) YEAR 1 2 3 4 Total 1999 12.6 8.6 6.3 17.5 45.0 2000 14.1 10.3 7.5 18.2 50.1 2001 15.3 10.6 8.1 19.6 53.6 Total 42.0 29.5 21.9 55.3 148.7 Seasonal Adjustment

  23. DEMAND (1000’S PER QUARTER) YEAR 1 2 3 4 Total 1999 12.6 8.6 6.3 17.5 45.0 2000 14.1 10.3 7.5 18.2 50.1 2001 15.3 10.6 8.1 19.6 53.6 Total 42.0 29.5 21.9 55.3 148.7 42.0 148.7 29.5 148.7 55.3 148.7 21.9 148.7 D1 D D2 D D4 D D3 D S1 = = = 0.28 S2 = = = 0.20 S4 = = = 0.37 S3 = = = 0.15 Seasonal Adjustment

  24. DEMAND (1000’S PER QUARTER) YEAR 1 2 3 4 Total 1999 12.6 8.6 6.3 17.5 45.0 2000 14.1 10.3 7.5 18.2 50.1 2001 15.3 10.6 8.1 19.6 53.6 Total 42.0 29.5 21.9 55.3 148.7 Si 0.28 0.20 0.15 0.37 Seasonal Adjustment

  25. DEMAND (1000’S PER QUARTER) YEAR 1 2 3 4 Total 1999 12.6 8.6 6.3 17.5 45.0 2000 14.1 10.3 7.5 18.2 50.1 2001 15.3 10.6 8.1 19.6 53.6 Total 42.0 29.5 21.9 55.3 148.7 Si 0.28 0.20 0.15 0.37 Seasonal Adjustment For 2002 y = 40.97 + 4.30x = 40.97 + 4.30(4) = 58.17

  26. DEMAND (1000’S PER QUARTER) YEAR 1 2 3 4 Total 1999 12.6 8.6 6.3 17.5 45.0 2000 14.1 10.3 7.5 18.2 50.1 2001 15.3 10.6 8.1 19.6 53.6 Total 42.0 29.5 21.9 55.3 148.7 Si 0.28 0.20 0.15 0.37 Seasonal Adjustment For 2002 y = 40.97 + 4.30x = 40.97 + 4.30(4) = 58.17 SF1 = (S1) (F5) SF3 = (S3) (F5) = (0.28)(58.17) = 16.28 = (0.15)(58.17) = 8.73 SF2 = (S2) (F5) SF4 = (S4) (F5) = (0.20)(58.17) = 11.63 = (0.37)(58.17) = 21.53

  27. Forecast Accuracy • Error = Actual - Forecast • Find a method which minimizes error • Mean Absolute Deviation (MAD) • Mean Absolute Percent Deviation (MAPD) • Cumulative Error (E)

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

  29. PERIOD DEMAND, DtFt ( =0.3) 1 37 37.00 2 40 37.00 3 41 37.90 4 37 38.83 5 45 38.28 6 50 40.29 7 43 43.20 8 47 43.14 9 56 44.30 10 52 47.81 11 55 49.06 12 54 50.84 557 MAD Example

  30. PERIOD DEMAND, DtFt ( =0.3) (Dt - Ft) |Dt - Ft| 1 37 37.00 – – 2 40 37.00 3.00 3.00 3 41 37.90 3.10 3.10 4 37 38.83 -1.83 1.83 5 45 38.28 6.72 6.72 6 50 40.29 9.69 9.69 7 43 43.20 -0.20 0.20 8 47 43.14 3.86 3.86 9 56 44.30 11.70 11.70 10 52 47.81 4.19 4.19 11 55 49.06 5.94 5.94 12 54 50.84 3.15 3.15 557 49.31 53.39 MAD Example

  31. PERIOD DEMAND, DtFt ( =0.3) (Dt - Ft) |Dt - Ft| 1 37 37.00 – – 2 40 37.00 3.00 3.00 3 41 37.90 3.10 3.10 4 37 38.83 -1.83 1.83 5 45 38.28 6.72 6.72 6 50 40.29 9.69 9.69 7 43 43.20 -0.20 0.20 8 47 43.14 3.86 3.86 9 56 44.30 11.70 11.70 10 52 47.81 4.19 4.19 11 55 49.06 5.94 5.94 12 54 50.84 3.15 3.15 557 49.31 53.39 • Dt - Ft n MAD = = = 4.85 53.39 11 MAD Example

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

  33. FORECAST MAD MAPD E (E) Exponential smoothing (= 0.30) 4.85 9.6% 49.31 4.48 Exponential smoothing (= 0.50) 4.04 8.5% 33.21 3.02 Adjusted exponential smoothing 3.81 8.1% 21.14 1.92 (= 0.50, = 0.30) Linear trend line 2.29 4.9% – – Comparison of Forecasts

  34. Forecast Control • Reasons for out-of-control forecasts • Change in trend • Appearance of cycle • Weather changes • Promotions • Competition • Politics

  35. (Dt - Ft) MAD E MAD Tracking signal = = Tracking Signal • Compute each period • Compare to control limits • Forecast is in control if within limits Use control limits of +/- 2 to +/- 5 MAD

  36. DEMAND FORECAST, ERROR E = PERIOD DtFtDt - Ft(Dt - Ft) MAD 1 37 37.00 – – – 2 40 37.00 3.00 3.00 3.00 3 41 37.90 3.10 6.10 3.05 4 37 38.83 -1.83 4.27 2.64 5 45 38.28 6.72 10.99 3.66 6 50 40.29 9.69 20.68 4.87 7 43 43.20 -0.20 20.48 4.09 8 47 43.14 3.86 24.34 4.06 9 56 44.30 11.70 36.04 5.01 10 52 47.81 4.19 40.23 4.92 11 55 49.06 5.94 46.17 5.02 12 54 50.84 3.15 49.32 4.85 Tracking Signal Values

  37. DEMAND FORECAST, ERROR E = PERIOD DtFtDt - Ft(Dt - Ft) MAD 1 37 37.00 – – – 2 40 37.00 3.00 3.00 3.00 3 41 37.90 3.10 6.10 3.05 4 37 38.83 -1.83 4.27 2.64 5 45 38.28 6.72 10.99 3.66 6 50 40.29 9.69 20.68 4.87 7 43 43.20 -0.20 20.48 4.09 8 47 43.14 3.86 24.34 4.06 9 56 44.30 11.70 36.04 5.01 10 52 47.81 4.19 40.23 4.92 11 55 49.06 5.94 46.17 5.02 12 54 50.84 3.15 49.32 4.85 6.10 3.05 Tracking signal for period 3 TS3 = = 2.00 Tracking Signal Values

  38. DEMAND FORECAST, ERROR E = TRACKING PERIOD DtFtDt - Ft(Dt - Ft) MAD SIGNAL 1 37 37.00 – – – – 2 40 37.00 3.00 3.00 3.00 1.00 3 41 37.90 3.10 6.10 3.05 2.00 4 37 38.83 -1.83 4.27 2.64 1.62 5 45 38.28 6.72 10.99 3.66 3.00 6 50 40.29 9.69 20.68 4.87 4.25 7 43 43.20 -0.20 20.48 4.09 5.01 8 47 43.14 3.86 24.34 4.06 6.00 9 56 44.30 11.70 36.04 5.01 7.19 10 52 47.81 4.19 40.23 4.92 8.18 11 55 49.06 5.94 46.17 5.02 9.20 12 54 50.84 3.15 49.32 4.85 10.17 Tracking Signal Values

  39. 3 – 2 – 1 – 0 – -1 – -2 – -3 – Tracking signal (MAD) | | | | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 11 12 Period Tracking Signal Plot

  40. 3 – 2 – 1 – 0 – -1 – -2 – -3 – Exponential smoothing ( = 0.30) Tracking signal (MAD) | | | | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 11 12 Period Tracking Signal Plot

  41. 3 – 2 – 1 – 0 – -1 – -2 – -3 – Exponential smoothing ( = 0.30) Tracking signal (MAD) Linear trend line | | | | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 11 12 Period Tracking Signal Plot

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

  43. 18.39– 12.24– 6.12– 0– -6.12– -12.24– -18.39– | | | | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 11 12 Period Statistical Control Charts Errors

  44. 18.39– 12.24– 6.12– 0– -6.12– -12.24– -18.39– UCL = +3 LCL = -3 | | | | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 11 12 Period Statistical Control Charts Errors

  45. Causal Modeling with Linear Regression • Study relationship between two or more variables • Dependent variable y depends on independent variable xy = a + bx

  46. a = y - b x b = where a = intercept (at period 0) b = slope of the line x = = mean of the x data y = = mean of the y data xy - nxy x2- nx2 x n y n Linear Regression Formulas

  47. x y (WINS) (ATTENDANCE) xyx2 4 36.3 145.2 16 6 40.1 240.6 36 6 41.2 247.2 36 8 53.0 424.0 64 6 44.0 264.0 36 7 45.6 319.2 49 5 39.0 195.0 25 7 47.5 332.5 49 49 346.7 2167.7 311 Linear Regression Example

  48. x y (WINS) (ATTENDANCE) xyx2 4 36.3 145.2 16 6 40.1 240.6 36 6 41.2 247.2 36 8 53.0 424.0 64 6 44.0 264.0 36 7 45.6 319.2 49 5 39.0 195.0 25 7 47.5 332.5 49 49 346.7 2167.7 311 x = = 6.125 y = = 43.36 b= = = 4.06 a= y - bx = 43.36 - (4.06)(6.125) = 18.46 49 8 xy - nxy2 x2 - nx2 346.9 8 (2,167.7) - (8)(6.125)(43.36) (311) - (8)(6.125)2 Linear Regression Example

  49. x y (WINS) (ATTENDANCE) xyx2 4 36.3 145.2 16 6 40.1 240.6 36 6 41.2 247.2 36 8 53.0 424.0 64 6 44.0 264.0 36 7 45.6 319.2 49 5 39.0 195.0 25 7 47.5 332.5 49 49 346.7 2167.7 311 x = = 6.125 y = = 43.36 b= = = 4.06 a= y - bx = 43.36 - (4.06)(6.125) = 18.46 49 8 Regression equation xy - nxy2 x2 - nx2 346.9 8 y = 18.46 + 4.06x Attendance forecast for 7 wins (2,167.7) - (8)(6.125)(43.36) (311) - (8)(6.125)2 y = 18.46 + 4.06(7) = 46.88, or 46,880 Linear Regression Example

  50. Attendance, y | | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 Wins, x Linear Regression Line 60,000 – 50,000 – 40,000 – 30,000 – 20,000 – 10,000 –

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