Understanding Exponential Smoothing Forecast

Exponential smoothing is a time series forecasting method where the forecast for the next period is calculated based on the forecast for the current period, the actual sales for the current period, and a smoothing constant (alpha, $\alpha$). It is a simple and widely used technique in business forecasting.

The formula for exponential smoothing is:

$\qquad F_{t+1} = F_t + \alpha (A_t - F_t)$

Where:

• $F_{t+1}$ is the forecast for the next period
• $F_t$ is the forecast for the current period
• $A_t$ is the actual sales for the current period
• $\alpha$ is the exponential smoothing constant, where $0 \le \alpha \le 1$

The term $(A_t - F_t)$ is also known as the forecast error for the current period.

Applying Exponential Smoothing to the Problem

We are given the following information for December 2008:

• Forecast sales for December 2008 ($F_{Dec}$) = 25
• Actual sales for December 2008 ($A_{Dec}$) = 20
• Exponential smoothing constant ($\alpha$) = 0.2

We need to calculate the forecast sales for January 2009 ($F_{Jan}$).

Using the exponential smoothing formula, with $t$ representing December 2008 and $t+1$ representing January 2009:

$\qquad F_{Jan} = F_{Dec} + \alpha (A_{Dec} - F_{Dec})$

Step-by-Step Forecast Calculation

Let's substitute the given values into the formula:

Step 1: Identify the values for the current period (December 2008).

• $F_{Dec} = 25$
• $A_{Dec} = 20$
• $\alpha = 0.2$

Step 2: Calculate the forecast error for the current period.

Forecast error = $A_{Dec} - F_{Dec} = 20 - 25 = -5$

Step 3: Apply the exponential smoothing formula.

$\qquad F_{Jan} = F_{Dec} + \alpha (A_{Dec} - F_{Dec})$

$\qquad F_{Jan} = 25 + 0.2 \times (-5)$

$\qquad F_{Jan} = 25 - 1$

$\qquad F_{Jan} = 24$

Alternatively, we can use the equivalent form of the formula:

$\qquad F_{t+1} = \alpha A_t + (1 - \alpha) F_t$

Substituting the values:

$\qquad F_{Jan} = \alpha A_{Dec} + (1 - \alpha) F_{Dec}$

$\qquad F_{Jan} = 0.2 \times 20 + (1 - 0.2) \times 25$

$\qquad F_{Jan} = 0.2 \times 20 + 0.8 \times 25$

$\qquad F_{Jan} = 4 + 20$

$\qquad F_{Jan} = 24$

Both methods yield the same result.

Therefore, the forecast sales for January 2009 would be 24.

Summary of Calculation