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CHAPTER 4 : FORECASTING – Suggested Solutions Summer II, 2009
Question 4.
(a) 3-week moving average:
(b) 3-week weighted moving average:
Week of Pints used Weight Computations
August 31 360
September 7 389
September 14 410
September 21 381 0.1 381x0.1= 38.
September 28 368 0.3 368x0.3=110.
October 5 374 0.6 374x0.6=224.
October 12 Forecast 372.
(c) Exponential smoothing (with a smoothing constant, α = 0.2):
Week of Actual Forecast: Ft = Ft- 1 + α(At- 1 – Ft- 1 )
August 31 360 360.0 0 September 7 389 360. 00 = 360.0 0 + 0.2( 360 - 360.0 0 ) September 14 410 365. 80 = 360. 00 + 0.2(389 - 360. 00 ) September 21 381 374. 64 = 3 65. 80 + 0.2(410 - 365. 80 ) September 28 368 37 5.91 = 37 4. 64 + 0.2(381 - 374. 64 ) October 5 374 374. 33 = 37 5. 91 + 0.2(3 68 - 375. 91 ) October 12 374.26 = 374.33 + 0.2(374 - 374. 33 )
Question 4. (a) 2-year moving average: (b) Mean Absolute Deviation (MAD) Year Mileage Two-year Moving Average Error IErrorI 1 3000 2 4000 3 3400 3500 - 100 100 4 3800 3700 100 100 5 3700 3600 100 100 Totals 100 300 MAD = 300/3 = 100 (c) 2-Year Weighted Moving Average Year Mileage Two-year Weighted Moving Average Error IErrorI 1 3000 2 4000 3 3400 3600 - 200 200 4 3800 3640 160 160 5 3700 3640 60 60 Totals 20 420 MAD = 420/3 = 140 (d) Exponential Smoothing using α=0.5 and an initial forecast of 3000 for year 1. Year Mileage Forecast Forecast Error Error x 0.5 New Forecast 1 3000 3000 0 0 3000 2 4000 3000 1000 500 3500 3 3400 3500 - 100 - 50 3450 4 3800 3450 350 175 3625 5 3700 3625 75 38 3663 Total 1325 The forecast for year 6 is 3,663 miles.
Question 4.7 Weighted Moving Average. Assume that Present = Period (week) 6. So: Where: 1.0 = ∑ weights 0.333 + 0.25 + 0.25 + 0.167 or 1/3, ¼, ¼, 1/ Question 4. (a) Exponential smoothing, = 0.6: Exponential Absolute Year Demand Smoothing = 0.6 Deviation 1 45 41 4. 2 50 41.0 + 0.6(45–41) = 43.4 6. 3 52 43.4 + 0.6(50–43.4) = 47.4 4. 4 56 47.4 + 0.6(52–47.4) = 50.2 5. 5 58 50.2 + 0.6(56–50.2) = 53.7 4. 6? 53.7 + 0.6(58–53.7) = 56. = 25. MAD = 5. Exponential smoothing, = 0.9: Exponential Absolute Year Demand Smoothing = 0.9 Deviation 1 45 41 4. 2 50 41.0 + 0.9(45–41) = 44.6 5. 3 52 44.6 + 0.9(50–44.6 ) = 49.5 2. 4 56 49.5 + 0.9(52–49.5) = 51.8 4. 5 58 51.8 + 0.9(56–51.8) = 55.6 2. 6? 55.6 + 0.9(58–55.6) = 57. = 18. MAD = 3. (b) 3-year moving average: Three-Year Absolute Year Demand Moving Average Deviation 1 45 2 50 3 52 4 56 (45 + 50 + 52)/3 = 49 7 5 58 (50 + 52 + 56)/3 = 52.7 5. 6? (52 + 56 + 58)/3 = 55. = 12. MAD = 6.
1 1 1 1
3 4 4 6 1 1 1 1 (52) + (63) + (48) + (70) = 56.75 patients 3 4 4 6 F A A A A
(c) Trend projection: Absolute Year Demand Trend Projection Deviation 1 45 42.6 + 3.2 1 = 45.8 0. 2 50 42.6 + 3.2 2 = 49.0 1. 3 52 4 2.6 + 3.2 3 = 52.2 0. 4 56 42.6 + 3.2 4 = 55.4 0. 5 58 42.6 + 3.2 5 = 58.6 0. 6? 42.6 + 3.2 6 = 61. = 3. MAD = 0.
Y a bX XY nXY b X nX a Y bX X Y XY X^2 1 45 45 1 2 50 100 4 3 52 156 9 4 56 224 16 5 58 290 25 Then: X = 15, Y = 261, XY = 815, X^2 = 55, X = 3, Y = 52. Therefore:
815 – 5 3 52.
55 – 5 3 3 52.20 – 3.20 3 42. 42.6 3.2 6 61. b a Y (d) Comparing the results of the forecasting methodologies for parts (a), (b), and (c). Forecast Methodology MAD Exponential smoothing, = 0.6 5. Exponential smoothing, = 0.9 3. 3-year moving average 6. Trend projection 0. Based on a mean absolute deviation criterion, the trend projection is to be preferred over the exponential smoothing with = 0.6, exponential smoothing with = 0.9, or the 3-year moving average forecast methodologies.
Question 4.39 Raw data set up for trend analysis: Year X Patients Y X^2 Y^2 XY 1 36 1 1 , 296 36 2 33 4 1 , 089 66 3 40 9 1 , 600 120 4 41 16 1 , 681 164 5 40 25 1 , 600 200 6 55 36 3 , 025 330 7 60 49 3 , 600 420 8 54 64 2 , 916 432 9 58 81 3 , 364 522 10 61 100 3 , 721 610 55 478 385 23 , 892 2 , 900 Given: Y = a + bX where:
XY nXY b X nX a Y bX and X = 55, Y = 478, XY = 2900, X^2 = 385, Y^2 = 23892, X 5.5, Y 47.8, Then:
2900 10 5.5 47.8 2900 2629 271
385 10 5.5^385 302.5^ 82. 47.8 3.28 5.5 29. b a and Y = 29.76 + 3.28 X. For: 11: 29.76 3.28 11 65. 12: 29.76 3.28 12 69. X Y X Y Therefore: Year 11 65.8 patients Year 12 69.1 patients The model ―seems‖ to fit the data pretty well. One should, however, be more precise in judging the adequacy of the model. Two possible approaches are computation of (a) the correlation coefficient, or (b) the mean absolute deviation.
The correlation coefficient:
10 2900 55 478 10 385 55 10 23892 478 29000 26290 3850 3025 238920 228484 2710 2710
825 10436 2934.
n XY X Y r n X X n Y Y r The coefficient of determination of 0.853 is quite respectable—indicating our original judgment of a ―good‖ fit was appropriate. Year Patients Trend Absolute X Y Forecast Deviation Deviation 1 36 29.8 + 3.28 1 = 33.1 2.9 2. 2 33 29.8 + 3.28 2 = 36.3 – 3.3 3. 3 40 29.8 + 3.28 3 = 39.6 0.4 0. 4 41 29.8 + 3.28 4 = 42.9 – 1.9 1. 5 40 29.8 + 3.28 5 = 46.2 – 6.2 6. 6 55 29.8 + 3.28 6 = 49.4 5.6 5. 7 60 29.8 + 3.28 7 = 52.7 7.3 7. 8 54 29.8 + 3.28 8 = 56.1 – 2.1 2. 9 58 29.8 + 3.28 9 = 59.3 – 1.3 1. 10 61 29.8 + 3.28 10 = 62.6 – 1.6 1. = 32. MAD = 3. The MAD is 3.26—this is approximately 7% of the average number of patients and 10% of the minimum number of patients. We also see absolute deviations, for years 5, 6, and 7 in the range 5.6–7.3. The comparison of the MAD with the average and minimum number of patients and the comparatively large deviations during the middle years indicate that the forecast model is not exceptionally accurate. It is more useful for predicting general trends than the actual number of patients to be seen in a specific year.
