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Chapter 4

Chapter 4. Forecasting. Ch. 4: What is covered?. Moving Average Weighted Moving Average Exponential Smoothing Trend Projections Seasonal Index/Adjusted-Forecast MAD, CE, Bias, MSE, MAPD Linear Regression, r, r 2. Ch. 4: What is not covered?. Trend Adjusted Exponential Smoothing

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Chapter 4

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  1. Chapter 4 Forecasting

  2. Ch. 4: What is covered? • Moving Average • Weighted Moving Average • Exponential Smoothing • Trend Projections • Seasonal Index/Adjusted-Forecast • MAD, CE, Bias, MSE, MAPD • Linear Regression, r, r2

  3. Ch. 4: What is not covered? • Trend Adjusted Exponential Smoothing • Tracking Signal • By-Hand computation of anything done by Excel • a and b using Least Squares • MAD, CE, Bias, MSE, MAPD, r and r2 • Multiple Regression

  4. Forecasting • Predicting future events • Usually demand behavior over a time frame • Qualitative methods • Based on subjective methods • Quantitative methods • Based on mathematical formulas

  5. Demand Behavior • Trend • gradual, long-term up or down movement • Cycle • up & down movement repeating over long time frame • Seasonal pattern • periodic oscillation in demand which repeats • Random movements follow no pattern

  6. Demand Demand Random movement Time (a) Trend Time (b) Cycle Demand Demand Time (c) Seasonal pattern Time (d) Trend with seasonal pattern Forms of Forecast Movement

  7. n i= 1  Di MAn = n where n = number of periods in the moving average Di= demand in period i Moving Average • Average several periods of data • Dampen, smooth out changes • Use when demand is stable with no trend or seasonal pattern

  8. 3 i= 1 ORDERS MONTH PER MONTH  Di MA3 = Jan 120 Feb 90 Mar 100 Apr 75 May 110 June 50 July 75 Aug 130 Sept 110 Oct 90 3 90 + 110 + 130 3 = Simple Moving Average = 110 orders for Nov

  9. ORDERS THREE-MONTH MONTH PER MONTH MOVING AVERAGE Jan 120 – Feb 90 – Mar 100 – Apr 75 103.3 May 110 88.3 June 50 95.0 July 75 78.3 Aug 130 78.3 Sept 110 85.0 Oct 90 105.0 Nov – 110.0 Simple Moving Average

  10. 5 i= 1 ORDERS THREE-MONTH MONTH PER MONTH MOVING AVERAGE  Di MA5 = Jan 120 – Feb 90 – Mar 100 – Apr 75 103.3 May 110 88.3 June 50 95.0 July 75 78.3 Aug 130 78.3 Sept 110 85.0 Oct 90 105.0 Nov – 110.0 5 90 + 110 + 130 + 75 + 50 5 = Simple Moving Average = 91 orders for Nov

  11. ORDERS THREE-MONTH FIVE-MONTH MONTH PER MONTH MOVING AVERAGE MOVING AVERAGE Jan 120 – – Feb 90 – – Mar 100 – – Apr 75 103.3 – May 110 88.3 – June 50 95.0 99.0 July 75 78.3 85.0 Aug 130 78.3 82.0 Sept 110 85.0 88.0 Oct 90 105.0 95.0 Nov – 110.0 91.0 Simple Moving Average

  12. 150 – 125 – 100 – 75 – 50 – 25 – 0 – 5-month Orders 3-month Actual | | | | | | | | | | | Jan Feb Mar Apr May June July Aug Sept Oct Nov Month Smoothing Effects

  13. Wi Di WMAn = i = 1 where Wi = the weight for period i, between 0 and 100 percent  Wi= 1.00 Weighted Moving Average • Adjusts moving average method to more closely reflect data fluctuations

  14. MONTH WEIGHT DATA August 17% 130 September 33% 110 October 50% 90 Weighted Moving Average Example

  15. MONTH WEIGHT DATA August 17% 130 September 33% 110 October 50% 90 November forecast 3 i = 1  WMA3 = Wi Di = (0.50)(90) + (0.33)(110) + (0.17)(130) = 103.4 orders Weighted Moving Average Example

  16. Exponential Smoothing Ft +1 = Dt + (1 - )Ft where Ft +1 = forecast for next period Dt= actual demand for present period Ft= previously determined forecast for present period = weighting factor, smoothing constant • Averaging method • Weights most recent data more strongly • Reacts more to recent changes • Widely used, accurate method

  17. Effect of Smoothing Constant 0.0  1.0If = 0.20, then Ft +1 = 0.20Dt + 0.80 Ft If = 0, then Ft+1 = 0Dt + 1 Ft0 = FtForecast does not reflect recent data If = 1, then Ft +1 = 1Dt + 0 Ft=DtForecast based only on most recent data

  18. 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 F2 = D1 + (1 - )F1 = (0.30)(37) + (0.70)(37) = 37 F3 = D2 + (1 - )F2 = (0.30)(40) + (0.70)(37) = 37.9 F13 = D12 + (1 - )F12 = (0.30)(54) + (0.70)(50.84) = 51.79 Exponential Smoothing

  19. FORECAST, Ft + 1 PERIOD MONTH DEMAND ( = 0.3) 1 Jan 37 – 2 Feb 40 37.00 3 Mar 41 37.90 4 Apr 37 38.83 5 May 45 38.28 6 Jun 50 40.29 7 Jul 43 43.20 8 Aug 47 43.14 9 Sep 56 44.30 10 Oct 52 47.81 11 Nov 55 49.06 12 Dec 54 50.84 13 Jan – 51.79 Exponential Smoothing

  20. FORECAST, Ft + 1 PERIOD MONTH DEMAND ( = 0.3) ( = 0.5) 1 Jan 37 – – 2 Feb 40 37.00 37.00 3 Mar 41 37.90 38.50 4 Apr 37 38.83 39.75 5 May 45 38.28 38.37 6 Jun 50 40.29 41.68 7 Jul 43 43.20 45.84 8 Aug 47 43.14 44.42 9 Sep 56 44.30 45.71 10 Oct 52 47.81 50.85 11 Nov 55 49.06 51.42 12 Dec 54 50.84 53.21 13 Jan – 51.79 53.61 Exponential Smoothing

  21. 70 – 60 – 50 – 40 – 30 – 20 – 10 – 0 – Actual Orders | | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 13 Month Exponential Smoothing Forecasts

  22. 70 – 60 – 50 – 40 – 30 – 20 – 10 – 0 – Actual Orders  = 0.30 | | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 13 Month Exponential Smoothing Forecasts

  23. 70 – 60 – 50 – 40 – 30 – 20 – 10 – 0 – Actual  = 0.50 Orders  = 0.30 | | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 13 Month Exponential Smoothing Forecasts

  24. 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

  25. 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

  26. 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

  27. 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 = = = 4.45 53.39 12 MAD Example

  28. e2t n et n Average error = bias Mean squared error = E = E = Other Accuracy Measures

  29. 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

  30. 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

  31. x(PERIOD) y(DEMAND) 1 37 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

  32. x(PERIOD) y(DEMAND) xy x2 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

  33. x(PERIOD) y(DEMAND) xy x2 557 12 78 12 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 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

  34. 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

  35. x(PERIOD) y(DEMAND) xy x2 Linear trend line 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 y = 35.2 + 1.72x Forecast for period 13 y = 35.2 + 1.72(13) y = 57.56 units Least Squares Example

  36. 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

  37. 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

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

  39. 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

  40. 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

  41. 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

  42. 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

  43. 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) (F4)SF3 = (S3) (F4) = (0.28)(58.17) = 16.28 = (0.15)(58.17) = 8.73 SF2 = (S2) (F4)SF4 = (S4) (F4) = (0.20)(58.17) = 11.63 = (0.37)(58.17) = 21.53

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

  45. Correlation • Correlation, r • Measure of strength of relationship • Varies between -1.00 and +1.00 • 1.00 => an increase in the independent variable results in a linear increase in the dependent • -1.00 => an increase in the independent variable results in a linear decrease in the dependent • 0.0 => there does not seem to be a linear relationship between the

  46. Coefficient of Determination • Coefficient of determination, r2 • Percentage of variation in dependent variable resulting from changes in the independent variable

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