To solve this exponential smoothing problem, we need to calculate the forecast for year 8 using the data provided and an alpha (smoothing constant) of 0.30.

Steps to Solution:

1. Calculate Initial Forecast (F4) for Year 4:
   The problem states to use the average demand for years 1 to 3 as the initial forecast. $\text{Initial Forecast (F4)} = \frac{321 + 407 + 449}{3} = 392.33 \approx 392$

2. Apply Exponential Smoothing for Each Subsequent Year: The exponential smoothing formula is: $F_{t+1} = \alpha \times D_t + (1 - \alpha) \times F_t$ where:

   • $F_{t+1}$ is the forecast for the next period,
   • $D_t$ is the actual demand for the current period,
   • $F_t$ is the forecast for the current period, and
   • $\alpha = 0.30$.

   Calculate forecasts from year 4 to year 8.

   • Forecast for Year 5 (F5): $F_5 = 0.30 \times 507 + 0.70 \times 392 = 152.1 + 274.4 = 426.5 \approx 427$

   • Forecast for Year 6 (F6): $F_6 = 0.30 \times 499 + 0.70 \times 427 = 149.7 + 298.9 = 448.6 \approx 449$

   • Forecast for Year 7 (F7): $F_7 = 0.30 \times 551 + 0.70 \times 449 = 165.3 + 314.3 = 479.6 \approx 480$

   • Forecast for Year 8 (F8): $F_8 = 0.30 \times 461 + 0.70 \times 480 = 138.3 + 336 = 474.3 \approx 474$

Final Answer:

The forecast for the 8th year is approximately 474 cases of Merlot wine.

Would you like more details or have any questions about this solution?

Related Questions:

1. How does the smoothing constant ($\alpha$) affect the forecast in exponential smoothing?
2. What would the forecast be for year 9 if the demand in year 8 was 427?
3. Can exponential smoothing be used for trend-based data, and if so, how?
4. How would the forecast change if a different initial forecast was chosen?
5. Why might a company prefer exponential smoothing over other forecasting methods?

Tip:

Exponential smoothing gives more weight to recent data, making it ideal for time series data without strong seasonal or trend patterns.