CHAPTER 4: FORECASTING Suggested Solutions Summer II, 2009

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CHAPTER 4: FORECASTING Suggested Solutions Summer II, 009 Question 4.1 (a) 3-week moving average: () 3-week weighted moving average: Week of Pints used Weight Computations August 31 360 Septemer 7 389 Septemer 14 410 Septemer 1 381 0.1 381x0.1= 38.1 Septemer 8 368 0.3 368x0.3=110.4 Octoer 5 374 0.6 374x0.6=4.4 Octoer 1 Forecast 37.9 (c) Exponential smoothing (with a smoothing constant, α = 0.): Week of Actual Forecast: F t = F t-1 + α(a t-1 F t-1 ) August 31 360 360.00 Septemer 7 389 360.00 = 360.00 + 0.(360-360.00) Septemer 14 410 365.80 = 360.00 + 0.(389-360.00) Septemer 1 381 374.64 = 365.80 + 0.(410-365.80) Septemer 8 368 375.91 = 374.64 + 0.(381-374.64) Octoer 5 374 374.33 = 375.91 + 0.(368-375.91) Octoer 1 374.6 = 374.33 + 0.(374-374.33) 1 BUS P301:01

Question 4.5 (a) -year moving average: () Mean Asolute Deviation (MAD) Year Mileage 1 3000 4000 Two-year Moving Average Error IErrorI 3 3400 3500-100 100 4 3800 3700 100 100 5 3700 3600 100 100 Totals 100 300 MAD = 300/3 = 100 (c) -Year Weighted Moving Average Year Mileage 1 3000 4000 Two-year Weighted Moving Average Error IErrorI 3 3400 3600-00 00 4 3800 3640 160 160 5 3700 3640 60 60 Totals 0 40 MAD = 40/3 = 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 4000 3000 1000 500 3500 3 3400 3500-100 -50 3450 4 3800 3450 350 175 365 5 3700 365 75 38 3663 Total 135 The forecast for year 6 is 3,663 miles. BUS P301:01

Question 4.6 Raw data set up for trend projections Y Sales X Period X XY January 0 1 1 0 Feruary 1 4 4 March 15 3 9 45 April 14 4 16 56 May 13 5 5 65 June 16 6 36 96 July 17 7 49 119 August 18 8 64 144 Septemer 0 9 81 180 Octoer 0 10 100 00 Novemer 1 11 11 31 Decemer 3 1 144 76 Sum 18 78 650 1474 Average 18.17 6.5 (a) Plotting the monthly sales data: () [i] Naïve method: The coming January = Decemer = 3 [ii] 3-month moving average: (0 + 1 + 3)/3 = 1.33 [iii] 6-m weighted [(.1 17)+(.1 18)+(.1 0)+(. 0)+(. 1)+ (.3 3)]/1.0 = 0.6 [iv] Exponential smoothing with alpha, α = 0.3 F F F F Oct Nov Dec Jan 18 0.3(0 18) 18.6 18.6 0.3(0 18.6) 19.0 19.0 0.3(1 19.0) 19.6 19.6 0.3(3 19.6) 0.6 1 [v] Trend x 78, x 6.5, y = 18, y 18.17 a 1474 (1)(6.5)(18.) 54.4 650 1(6.5) 143 18. 0.38(6.5) 15.73 0.38 Forecast = 15.73 +.38(13) = 0.67, where next January is the 13th month. (c) Only trend provides an equation that can extend eyond one month. 3 BUS P301:01

Question 4.7 Weighted Moving Average. Assume that Present = Period (week) 6. So: 1 1 1 1 F7 A6 A5 A4 A3 3 4 4 6 1.0 1 1 1 1 (5) + (63) + (48) + (70) = 56.75 patients 3 4 4 6 Where: 1.0 = weights 0.333 + 0.5 + 0.5 + 0.167 or 1/3, ¼, ¼, 1/6 Question 4.13 (a) Exponential smoothing, = 0.6: Exponential Asolute Year Demand Smoothing = 0.6 Deviation 1 45 41 4.0 50 41.0 + 0.6(45 41) = 43.4 6.6 3 5 43.4 + 0.6(50 43.4) = 47.4 4.6 4 56 47.4 + 0.6(5 47.4) = 50. 5.8 5 58 50. + 0.6(56 50.) = 53.7 4.3 6? 53.7 + 0.6(58 53.7) = 56.3 = 5.3 MAD = 5.06 Exponential smoothing, = 0.9: Exponential Asolute Year Demand Smoothing = 0.9 Deviation 1 45 41 4.0 50 41.0 + 0.9(45 41) = 44.6 5.4 3 5 44.6 + 0.9(50 44.6 ) = 49.5.5 4 56 49.5 + 0.9(5 49.5) = 51.8 4. 5 58 51.8 + 0.9(56 51.8) = 55.6.4 6? 55.6 + 0.9(58 55.6) = 57.8 = 18.5 MAD = 3.7 () 3-year moving average: Three-Year Asolute Year Demand Moving Average Deviation 1 45 50 3 5 4 56 (45 + 50 + 5)/3 = 49 7 5 58 (50 + 5 + 56)/3 = 5.7 5.3 6? (5 + 56 + 58)/3 = 55.3 = 1.3 MAD = 6. 4 BUS P301:01

(c) Trend projection: Asolute Year Demand Trend Projection Deviation 1 45 4.6 + 3. 1 = 45.8 0.8 50 4.6 + 3. = 49.0 1.0 3 5 4.6 + 3. 3 = 5. 0. 4 56 4.6 + 3. 4 = 55.4 0.6 5 58 4.6 + 3. 5 = 58.6 0.6 6? 4.6 + 3. 6 = 61.8 = 3. MAD = 0.64 Y a X a Y X XY nxy X nx X Y XY X 1 45 45 1 50 100 4 3 5 156 9 4 56 4 16 5 58 90 5 Then: X = 15, Y = 61, XY = 815, X = 55, X = 3, Y = 5. Therefore: 815 5 3 5. 3. 55 5 3 3 a 5.0 3.0 3 4.6 Y 6 4.6 3. 6 61.8 (d) Comparing the results of the forecasting methodologies for parts (a), (), and (c). Forecast Methodology MAD Exponential smoothing, = 0.6 5.06 Exponential smoothing, = 0.9 3.7 3-year moving average 6. Trend projection 0.64 Based on a mean asolute deviation criterion, the trend projection is to e preferred over the exponential smoothing with = 0.6, exponential smoothing with = 0.9, or the 3-year moving average forecast methodologies. 5 BUS P301:01

Question 4.14 Week Actual Method 1 Error IErrorI Error^ Method Error IErrorI Error^ 1 0.7 0.90-0.0 0.0 0.04 0.80-0.10 0.10 0.010 1.0 1.05-0.05 0.05 0.00 1.0-0.0 0.0 0.040 3 1.0 0.95 0.05 0.05 0.00 0.90 0.10 0.10 0.010 4 1.0 1.0-0.0 0.0 0.04 1.11-0.11 0.11 0.01 Totals 0.50 0.085 0.510 0.07 MAD 0.15 <<Better MAD 0.18 MSE 0.01 MSE 0.018 <<Better 6 BUS P301:01

Question 4.39 Raw data set up for trend analysis: Year X Patients Y X Y XY 1 36 1 1,96 36 33 4 1,089 66 3 40 9 1,600 10 4 41 16 1,681 164 5 40 5 1,600 00 6 55 36 3,05 330 7 60 49 3,600 40 8 54 64,916 43 9 58 81 3,364 5 10 61 100 3,71 610 55 478 385 3,89,900 Given: Y = a + X where: and Then: a Y X XY nxy X nx X = 55, Y = 478, XY = 900, X = 385, Y = 389, X 5.5, Y 47.8, a 900 10 5.5 47.8 900 69 71 385 10 5.5 385 30.5 8.5 47.8 3.8 5.5 9.76 3.8 and Y = 9.76 + 3.8X. For: X X 11: Y 9.76 3.8 11 65.8 1: Y 9.76 3.8 1 69.1 Therefore: Year 11 65.8 patients Year 1 69.1 patients The model seems to fit the data pretty well. One should, however, e more precise in judging the adequacy of the model. Two possile approaches are computation of (a) the correlation coefficient, or () the mean asolute deviation. 7 BUS P301:01

The correlation coefficient: r n XY X Y n X X n Y Y 10 900 55 478 10 385 55 10 389 478 r 9000 690 3850 305 3890 8484 710 710 85 10436 934.3 0.94 0.853 The coefficient of determination of 0.853 is quite respectale indicating our original judgment of a good fit was appropriate. Year Patients Trend Asolute X Y Forecast Deviation Deviation 1 36 9.8 + 3.8 1 = 33.1.9.9 33 9.8 + 3.8 = 36.3 3.3 3.3 3 40 9.8 + 3.8 3 = 39.6 0.4 0.4 4 41 9.8 + 3.8 4 = 4.9 1.9 1.9 5 40 9.8 + 3.8 5 = 46. 6. 6. 6 55 9.8 + 3.8 6 = 49.4 5.6 5.6 7 60 9.8 + 3.8 7 = 5.7 7.3 7.3 8 54 9.8 + 3.8 8 = 56.1.1.1 9 58 9.8 + 3.8 9 = 59.3 1.3 1.3 10 61 9.8 + 3.8 10 = 6.6 1.6 1.6 = 3.6 MAD = 3.6 The MAD is 3.6 this is approximately 7% of the average numer of patients and 10% of the minimum numer of patients. We also see asolute deviations, for years 5, 6, and 7 in the range 5.6 7.3. The comparison of the MAD with the average and minimum numer 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 numer of patients to e seen in a specific year. 8 BUS P301:01

Question 4.40 Raw data set up for trend analysis: Crime Patients Year Rate X Y X Y XY 1 58.3 36 3,398.9 1,96,098.8 61.1 33 3,733. 1,089,016.3 3 73.4 40 5,387.6 1,600,936.0 4 75.7 41 5,730.5 1,681 3,103.7 5 81.1 40 6,577. 1,600 3,44.0 6 89.0 55 7,91.0 3,05 4,895.0 7 101.1 60 10,1. 3,600 6,066.0 8 94.8 54 8,987.0,916 5,119. 9 103.3 58 10,670.9 3,364 5,991.4 10 116. 61 13,50.4 3,71 7,088. Column Totals 854.0 478 76,19.9 3,89 4,558.6 Given: Y = a + X where XY nxy X nx a Y X and X = 854, Y = 478, XY = 4558.6, X = 7619.9, Y = 389, X = 85.4, Y = 47.8. Then: a 4558.6 10 85.4 47.8 4558.6 4081. 7619.9 10 85.4 7619.9 7931.6 1737.4 0.543 3197.3 47.8 0.543 85.4 1.43 and Y = 1.43 + 0.543X For: X 131. : Y 1.43 0.543(131.) 7.7 X 90.6 : Y 1.43 0.543(90.6) 50.6 Therefore: Crime rate = 131. Crime rate = 90.6 7.7 patients 50.6 patients Note that rounding differences occur when solving with Excel. 9 BUS P301:01

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