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Mgt 2070 Assignment 2 – Solutions

4.13 As you can see in the following table, demand for heart transplant surgery at Washington General Hospital has increased steadily in the past few years:

|Year |1 |2 |3 |4 |5 |6 |
|Heart Transplants |45 |50 |52 |56 |58 |? |

The director of medical services predicted 6 years ago that demand in year 1 would be 41 surgeries.

a) Use exponential smoothing, first with a smoothing constant of .6 and then with one of .9, to develop forecasts for years 2 through 6.

Exponential Smoothing α = 0.6

|Year |A |Ft = Ft-1 + ((At-1 – Ft-1) |Ft || Error | |
|1 |45 |41 (given in question) |41 |4.0 |
|2 |50 |41.0 + 0.6(45 - 41) |43.4 |6.6 |
|3 |52 |43.4 + 0.6(50 – 43.4) |47.4 |4.6 |
|4 |56 |47.4 + 0.6(52 – 47.4) |50.2 |5.8 |
|5 |58 |50.2 + 0.6(56 – 50.2) |53.7 |4.3 |
|6 |? |53.7 + 0.6(58 – 53.7) |56.3 | |
| | | | |Σ = 25.3 |

MAD = 25.3 / 5 = 5.06

(5 marks)

Exponential Smoothing α = 0.9

|Year |A |Ft = Ft-1 + ((At-1 – Ft-1) |Ft || Error | |
|1 |45 |41 (given in question) |41 |4.0 |
|2 |50 |41.0 + 0.9(45 - 41) |44.6 |5.4 |
|3 |52 |44.6 + 0.9(50 – 44.6) |49.5 |2.5 |
|4 |56 |49.5 + 0.9(52 – 49.5) |51.8 |4.2 |
|5 |58 |51.8 + 0.9(56 – 51.8) |55.6 |2.4 |
|6 |? |55.6 + 0.9(58 – 55.6) |57.8 | |
| | | | |Σ = 18.5 |

MAD = 18.5 / 5 = 3.7

(5 marks)
b) Use a 3-year moving average to forecast demand in years 4, 5, and 6.

3-Year Moving Average

|Year |A |3-Year Moving Average || Error | |
|1 |45 | | |
|2 |50 | | |
|3 |52 | | |
|4 |56 |(45 + 50 + 52) / 3 = 49 |7 |
|5 |58 |(50 + 52 + 56) / 3 = 52.7 |5.3 |
|6 |? |(52 + 56 + 58) / 3 = 55.3 | |
| | | |Σ = 12.3 |

MAD = 12.3 / 2 = 6.2

(5 marks)

c) Use the trend-projection method to forecast demand in years 1 through 6.

|Year (X) |Demand (Y) |X2 |XY |
|1 |45 |1 |45 |
|2 |50 |4 |100 |
|3 |52 |9 |156 |
|4 |56 |16 |224 |
|5 |58 |25 |290 |
|Σ = 15 |Σ = 261 |Σ = 55 |Σ = 815 |

[pic]15 / 5 = 3 [pic]261 / 5 = 52.2

[pic]= [pic] = 3.2

[pic]= 52.2 – 3.2(3) = 42.6

Our equation is then Y = a + bX, ie Y = 42.6 + 3.2X

|Year (X) |Demand |Trend Projection |Forecast(Y) || Error | |
|1 |45 |Y = 42.6 + 3.2(1) |45.8 |0.8 |
|2 |50 |Y = 42.6 + 3.2(2) |49.0 |1.0 |
|3 |52 |Y = 42.6 + 3.2(3) |52.2 |0.2 |
|4 |56 |Y = 42.6 + 3.2(4) |55.4 |0.6 |
|5 |58 |Y = 42.6 + 3.2(5) |58.6 |0.6 |
|6 |? |Y = 42.6 + 3.2(6) |61.8 | |
| | | | |Σ = 3.2 |

MAD = 3.2 / 5 = 0.64

(5 marks)

4.14 Refer to problem 4.13. With MAD as the criterion, which of the four forecasting methods is best?

|Forecast Methodology |MAD |
|Exponential Smoothing, α = 0.6 |5.06 |
|Exponential Smoothing, α = 0.9 |3.7 |
|3-year Moving Average |6.2 |
|Trend Projection |0.6 |

Based on the Mean Absolute Deviation criterion, trend projection is the best fit.

(5 marks)

4.19 Consulting income at Dr. Thomas W. Jones Associates for the period February to July has been as follows. Use trend-adjusted exponential smoothing to forecast August’s income. Assume that the initial forecast for February is $65,000 and the initial trend adjustment is 0. The smoothing constants selected are ( = .1 and ( = .2.

Forecast Ft = ((At-1) + (1 - ()(Ft-1 + Tt-1), Trend Tt = ((Ft – Ft-1) + (1 - ()T t-1

|Mo |Inc |Forecast |Trend |FIT |Err2 |
|Feb |70.0 |65.0 |0 |65+0 = 65 |25.0 |
|Mar |68.5 |0.1(70)+0.9(65) = 65.5 |0.2(65.5–65)+(0.8)0 = 0.1 |65.5+0.1=65.6 |8.4 |
|Apr |64.8 |0.1(68.5)+0.9(65.6) = 65.89 |0.2(65.89-65.5)+(0.8)0.1 = 0.16 |65.89+0.16=66.05 |1.6 |
|May |71.7 |0.1(64.8)+0.9(66.05) = 65.93 |0.2(65.92-65.89)+(0.8)0.16 = 0.13 |65.93+0.13=66.06 |31.9 |
|Jun |71.3 |0.1(71.7)+0.9(66.06) = 66.62 |0.2(66.62-65.93)+(0.8)0.13 = 0.25 |66.62+0.25=66.87 |19.7 |
|Jul |72.8 |0.1(71.3)+0.9(66.87) = 67.31 |0.2(67.31-66.62)+(0.8)0.25 = 0.33 |67.31+0.33=67.64 |26.6 |
|Aug | |0.1(72.8)+0.9(67.64) = 68.16 |0.2(68.16-67.31)+(0.8)0.33 = 0.43 |68.16+0.43=68.60 | |
| | | | |Σ = |113.2 |

MSE = 113.2 / 6 = 18.87

(5 marks)

4.20 Resolve problem 4.19 with ( = .1 and ( = .8. Using MSE, which smoothing constants provide a better forecast?

|Mo |Inc |Forecast |Trend |FIT |Err2 |
|Feb |70.0 |65.0 |0 |65+0 = 65 |25.0 |
|Mar |68.5 |0.1(70)+0.9(65) = 65.5 |0.8(65.5–65)+(0.2)0 = 0.4 |65.5+0.4=65.9 |6.76 |
|Apr |64.8 |0.1(68.5)+0.9(65.9) = 66.16 |0.8(66.16-65.5)+(0.2)0.4 = 0.61 |66.16+0.61=66.77 |3.87 |
|May |71.7 |0.1(64.8)+0.9(66.77) = 66.57 |0.8(66.57-66.16)+(0.2)0.61 = 0.45 |66.57+0.45=67.02 |21.89 |
|Jun |71.3 |0.1(71.7)+0.9(67.02) = 67.49 |0.8(67.49-66.57)+(0.2)0.45 = 0.82 |67.49+0.82=68.31 |8.91 |
|Jul |72.8 |0.1(71.3)+0.9(68.31) = 68.61 |0.8(68.61-67.49)+(0.2)0.82 = 1.06 |68.61+1.06=69.68 |9.76 |
| | | | |Σ = |76.19 |

MSE = 76.19 / 6 = 12.70

Based upon the MSE criterion, exponential smoothing with ( = .1 and ( = .8 provides a better forecast.

(5 marks)

4.37 The accountant at Tick Wing Coal Distributors, Inc., in San Francisco notes that the demand for coal seems to be tied to an index of weather severity developed by the U.S. Weather Bureau. When weather was extremely cold in the U.S. over the past five years (and the index was thus high), coal sales were high. The accountant proposes that one good forecast of next year’s coal demand could be made by developing a regression equation and then consulting the Farmer’s Almanac to see how severe next year’s winter would be.

For the data in the following table, derive a least squares regression and compute the coefficient of correlation of the data. Also compute the standard error of the estimate.

|Coal Sales, y |4 |1 |4 |6 |5 |
|(in millions of tons) | | | | | |
|Weather Index, x |2 |1 |4 |5 |3 |

|X |Y |X2 |Y2 |XY |
|2 |4 |4 |16 |8 |
|1 |1 |1 |1 |1 |
|4 |4 |16 |16 |16 |
|5 |6 |25 |36 |30 |
|3 |5 |9 |25 |15 |
|Σ = 15 |Σ = 20 |Σ = 55 |Σ = 94 |Σ = 70 |

[pic]15 / 5 = 3 [pic]20 / 5 = 4

[pic]= [pic] = 1.0

[pic]= 4 – 1(3) = 1.0

Our equation is then Y = a + bX, ie Y = 1.0 + 1.0X

Correlation Coefficient:
[pic]= [pic] r = [pic] = [pic]= [pic]= 0.845

Standard Error of the Estimate:
[pic]= [pic]= [pic]= 1.15

(10 marks)

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