CHAPTER 4: FORECASTING Suggested Solutions Summer II, 2009

Transcription

1 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 Septemer Septemer Septemer x0.1= 38.1 Septemer x0.3=110.4 Octoer x0.6=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 Septemer = ( ) Septemer = ( ) Septemer = ( ) Septemer = ( ) Octoer = ( ) Octoer = ( ) 1 BUS P301:01

2 Question 4.5 (a) -year moving average: () Mean Asolute Deviation (MAD) Year Mileage Two-year Moving Average Error IErrorI Totals MAD = 300/3 = 100 (c) -Year Weighted Moving Average Year Mileage Two-year Weighted Moving Average Error IErrorI 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 Total 135 The forecast for year 6 is 3,663 miles. BUS P301:01

3 Question 4.6 Raw data set up for trend projections Y Sales X Period X XY January Feruary March April May June July August Septemer Octoer Novemer Decemer Sum Average (a) Plotting the monthly sales data: () [i] Naïve method: The coming January = Decemer = 3 [ii] 3-month moving average: ( )/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 (0 18) (0 18.6) (1 19.0) (3 19.6) [v] Trend x 78, x 6.5, y = 18, y a 1474 (1)(6.5)(18.) (6.5) (6.5) Forecast = (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

4 Question 4.7 Weighted Moving Average. Assume that Present = Period (week) 6. So: F7 A6 A5 A4 A (5) + (63) + (48) + (70) = patients Where: 1.0 = weights or 1/3, ¼, ¼, 1/6 Question 4.13 (a) Exponential smoothing, = 0.6: Exponential Asolute Year Demand Smoothing = 0.6 Deviation (45 41) = ( ) = (5 47.4) = (56 50.) = ? ( ) = 56.3 = 5.3 MAD = 5.06 Exponential smoothing, = 0.9: Exponential Asolute Year Demand Smoothing = 0.9 Deviation (45 41) = ( ) = (5 49.5) = ( ) = ? ( ) = 57.8 = 18.5 MAD = 3.7 () 3-year moving average: Three-Year Asolute Year Demand Moving Average Deviation ( )/3 = ( )/3 = ? ( )/3 = 55.3 = 1.3 MAD = 6. 4 BUS P301:01

5 (c) Trend projection: Asolute Year Demand Trend Projection Deviation = = = = = ? = 61.8 = 3. MAD = 0.64 Y a X a Y X XY nxy X nx X Y XY X Then: X = 15, Y = 61, XY = 815, X = 55, X = 3, Y = 5. Therefore: a Y (d) Comparing the results of the forecasting methodologies for parts (a), (), and (c). Forecast Methodology MAD Exponential smoothing, = Exponential smoothing, = 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

6 Question 4.14 Week Actual Method 1 Error IErrorI Error^ Method Error IErrorI Error^ Totals MAD 0.15 <<Better MAD 0.18 MSE 0.01 MSE <<Better 6 BUS P301:01

7 Question 4.39 Raw data set up for trend analysis: Year X Patients Y X Y XY , , , , , , , , , , ,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 and Y = X. For: X X 11: Y : Y Therefore: Year patients Year 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

8 The correlation coefficient: r n XY X Y n X X n Y Y r The coefficient of determination of is quite respectale indicating our original judgment of a good fit was appropriate. Year Patients Trend Asolute X Y Forecast Deviation Deviation = = = = = = = = = = = 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 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

9 Question 4.40 Raw data set up for trend analysis: Crime Patients Year Rate X Y X Y XY , ,96, ,733. 1,089, , ,600, , ,681 3, ,577. 1,600 3, ,91.0 3,05 4, ,1. 3,600 6, ,987.0,916 5, , ,364 5, ,50.4 3,71 7,088. Column Totals ,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 = , X = , Y = 389, X = 85.4, Y = Then: a and Y = X For: X 131. : Y (131.) 7.7 X 90.6 : Y (90.6) 50.6 Therefore: Crime rate = 131. Crime rate = patients 50.6 patients Note that rounding differences occur when solving with Excel. 9 BUS P301:01