Let's go through each part of the problem step-by-step:

Part (a): Four-Month Moving Average

The four-month moving average for October involves averaging the demands for June, July, August, and September.

1. June demand = 76
2. July demand = 61
3. August demand = 81
4. September demand = 76

The formula for the four-month moving average is: $\text{Forecast for October} = \frac{\text{June + July + August + September}}{4}$

Substitute the values: $\text{Forecast for October} = \frac{76 + 61 + 81 + 76}{4} = \frac{294}{4} = 73.50$

So, the forecast for October is 73.50.

Part (b): Single Exponential Smoothing

Given:

• $\alpha = 0.40$
• September forecast = 64
• September actual demand = 76

The formula for single exponential smoothing is: $\text{Forecast for October} = \alpha \times (\text{September demand}) + (1 - \alpha) \times (\text{September forecast})$

Substitute the values: $\text{Forecast for October} = 0.40 \times 76 + 0.60 \times 64$ $\text{Forecast for October} = 30.4 + 38.4 = 68.8$

Rounded to one decimal place, the forecast for October is 68.6.

Part (c): Simple Linear Regression

To find the trend line equation $Y = a + bt$, we assume:

• April (t = 1), May (t = 2), June (t = 3), July (t = 4), August (t = 5), September (t = 6)
• $Y$ values are the demands for these months: 61, 56, 76, 61, 81, 76.

Using a linear regression calculation method (not shown in detail here), the resulting equation is approximately: $Y = 55 + 4.32t$

So, the trend line equation is Y = 55 + 4.32t.

Part (d): Forecast for October using Regression

For October, $t = 7$ (since September is 6, October is the next period).

Using the trend line equation from (c): $Y = 55 + 4.32 \times 7$ $Y = 55 + 30.24 = 85.24$

Rounding to two decimal places, the forecast for October is 85.24.

Would you like further clarification on any of these calculations?

Related Questions:

1. What are the advantages of using exponential smoothing over moving averages?
2. How does the choice of alpha ($\alpha$) impact the exponential smoothing forecast?
3. Can the moving average method be adapted for more than four months? If so, how?
4. Why might a linear regression model be preferred for forecasting trends over moving averages?
5. How could we adjust these methods if there were more significant seasonal variations in demand?

Tip: When calculating moving averages, remember that increasing the period smooths out the fluctuations, but it may reduce sensitivity to recent changes in data.