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Resolve Problem 4.19 with α = .1 and β =.8. Using MSE, determine which smoothing constants provide a better
To determine: Compute MSE using the given smoothing constants and find the better forecasting smoothing constants using trend-adjusted exponential smoothing method.
Introduction: A sequence of data points in successive order is known as time series. Time series forecasting is the prediction based on past events which are at uniform time interval. Moving average method and trend projections are two of the time series methods which use weights to prioritize past data.
Answer to Problem 20P
On comparing MSE from two smoothing constants (refer to equations (1) and (2)), it can be inferred that smoothing constant with α=0.1 and β=0.8 provides better forecast because it minimizes the error.
Explanation of Solution
Given information:
| Time period | Month | Income ($ in thousands) |
| 1 | February | 70 |
| 2 | March | 68.5 |
| 3 | April | 64.8 |
| 4 | May | 71.7 |
| 5 | June | 71.3 |
| 6 | July | 72.8 |
Formula to calculate the MSE &forecasted demand
Where,
Calculation of FIT and MSE usingα=0.1 andβ=0.2:
| Time period | Month | Income ($ in thousands) | Ft ($ in thousands) | Tt | FIT | Error | Sq. Error |
| 1 | February | 70 | 65 | 0 | 65 | 5 | 25 |
| 2 | March | 68.5 | 65.500 | 0.100 | 65.600 | 2.9 | 8.41 |
| 3 | April | 64.8 | 65.890 | 0.158 | 66.048 | -1.25 | 1.56 |
| 4 | May | 71.7 | 65.923 | 0.133 | 66.056 | 5.64 | 31.85 |
| 5 | June | 71.3 | 66.621 | 0.246 | 66.867 | 4.43 | 19.66 |
| 6 | July | 72.8 | 67.310 | 0.335 | 67.644 | 5.16 | 26.58 |
| Total | 113.05 | ||||||
| MSE | 18.84 | ||||||
Table 1
Excel worksheet:
Calculation of FIT for February:
To calculate FIT for February, compute F1 and T1. The forecast F1 is $ 65 and trend T1 is 0. The sum of both values gives FIT, which is equal to $ 65.
Calculation of FIT for March:
To calculate FIT for February, compute F2 and T2. The forecast F2 is $ 65.5 and trend T2 is 0.1. The sum of both values gives FIT, which is equal to $ 65.60.
Calculation of FIT for April:
To calculate FIT for March, compute F3 and T3. The forecast F3 is $ 65.890 and trend T3 is 0.158. The sum of both values gives FIT, which is equal to $ 66.048.
Calculation of FIT for May:
To calculate FIT for May, compute F4 and T4. The forecast F4 is $ 65.923 and trend T4 is 0.133. The sum of both values gives FIT, which is equal to $ 66.056.
Calculation of FIT for June:
To calculate FIT for June, compute F5 and T5. The forecast F5 is $ 66.621 and trend T5 is 0.246. The sum of both values gives FIT, which is equal to $ 66.867.
Calculation of FIT for July:
To calculate FIT for July, compute F6 and T6. The forecast F6 is $ 67.310 and trend T6 is 0.335. The sum of both values gives FIT, which is equal to $ 67.644.
Calculation of MSE:
MSE is obtained by dividing the summation value of the square of the difference between actual and forecasted sales with the number of years n; n=6.
Table 1provides the values for square of the difference between actual and forecasted sales.
MSE using α=0.1 and β=0.2 is 18.84.
Calculation of FIT and MSE using α=0.1 and β=0.8:
| Time period | Month | Income ($ in thousands) | Ft ($ in thousands) | Tt | FIT | Error | Sq. Error |
| 1 | February | 70 | 65 | 0 | 65 | 5 | 25 |
| 2 | March | 68.5 | 65.500 | 0.400 | 65.900 | 2.6 | 6.76 |
| 3 | April | 64.8 | 66.160 | 0.608 | 66.768 | -1.968 | 3.87 |
| 4 | May | 71.7 | 66.571 | 0.451 | 67.022 | 4.678 | 21.89 |
| 5 | June | 71.3 | 67.490 | 0.825 | 68.314 | 2.986 | 8.91 |
| 6 | July | 72.8 | 68.613 | 1.064 | 69.677 | 3.123 | 9.76 |
| Total | 76.19 | ||||||
| MSE | 12.70 | ||||||
Table 2
Excel worksheet:
Calculation of FIT for February:
To calculate FIT for February, compute F1 and T1. The forecast F1 is $ 65 and trend T1 is 0. The sum of both values gives FIT, which is equal to $ 65.
Calculation of FIT for March:
To calculate FIT for February, compute F2 and T2. The forecast F2 is $ 65.5 and trend T2 is 0.4. The sum of both values gives FIT, which is equal to $ 65.90.
Calculation of FIT for April:
To calculate FIT for March, compute F3 and T3. The forecast F3 is $ 66.160 and trend T3 is 0.608. The sum of both values gives FIT, which is equal to $ 66.786.
Calculation of FIT for May:
To calculate FIT for May, compute F4 and T4. The forecast F4 is $66.571 and trend T4 is 0.451. The sum of both values gives FIT, which is equal to $ 67.022.
Calculation of FIT for June:
To calculate FIT for June, compute F5 and T5. The forecast F5 is $ 67.490 and trend T5 is 0.825. The sum of both values gives FIT, which is equal to $ 68.314.
Calculation of FIT for July:
To calculate FIT for July, compute F6 and T6. The forecast F6 is $ 68.613 and trend T6 is 1.064. The sum of both values gives FIT, which is equal to $ 69.677.
Calculation of MSE:
MSE is obtained by dividing the summation value of the square of the difference between actual and forecasted sales with the number of years n=6.
Table 2provides the values for square of the difference between actual and forecasted sales.
MSE using α=0.1 and β=0.8 is 12.70.
On comparing MSE from two smoothing constants (refer to equations(1)&(2)), it can be inferred that smoothing constant with α=0.1 and β=0.8 provides better forecast because it minimizes the error.
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