To solve this problem, firstly, let's understand the concept of exponential smoothing with trend. This is a forecasting method that calculates a forecast with a trend (T) by combining the previous forecast, trend, and actual value. It involves two smoothing equations, one for the level (average) and one for the trend.
The equations are:
1. F(t + 1) = A(t) + T(t) : Forecast equation
2. A(t) = αD(t) + (1 - α)(A(t - 1) + T(t - 1)) : Level (or Smoothing) equation
3. T(t) = β(A(t) - A(t - 1)) + (1 - β)T(t - 1) : Trend equation
Where:
- F(t + 1) is the forecast for the next period
- D(t) is the actual value at time 't'
- A(t) is the level at time 't'
- T(t) is the trend at time 't'
- F(t + 1) is the forecast at time 't + 1'
- α is the level smoothing coefficient
- β is the trend smoothing coefficient
Mean Absolute Deviation (MAD) is a common measure of forecast error and is calculated by computing the average absolute differences between the actual (real) and the forecast values.
Let's resolve the question step by step:
(Important: As the question doesn't provide an actual dataset, I'll describe a guide to how you would approach it. You can replace the variables with your actual values to perform the calculations.)
1. Initialize the level and trend from the first actual data. Generally, we can set initial level A(1) as the first data point and initial trend T(1) as the difference of the second and the first data point.
2. Calculate the level ('A') and trend ('T') for each year using the level and trend equations, respectively. Use the provided values α = 0.01, β = 0.3 for the first case and α = 0.1, β = 0.05 for the second case.
3. Use the derived level and trend in the forecast equation to get the forecast for each subsequent year.
4. Now, you calculate the forecast errors, which are the differences between actual data and the forecast data.
5. Compute the Mean Absolute Deviation (MAD) by taking the average of the absolute forecast errors.
After you have calculated the MAD for both cases, choose the case with the lower MAD value as the better one. The reason is that a lower MAD indicates a lower average absolute discrepancy between the actual and forecasted numbers of sold cars, which implies a more accurate forecasting model.
Remember, because we want to apply exponential smoothing with a trend, we need to forecast not just a single value, but also a trend (a direction upwards or downwards or stable). And in our case, the trend for the 2012 year was assumed to be zero.