Question: The owner of Leisure Boutique in a local mall is forecasting this month's (October's) demand for one of her best-selling products based on the following historical data:
MONTH QUANTITY SOLD
April 100
May 140
June 110
July 150
August 120
September 160
What is this month's forecast using exponential smoothing with alpha = $.2$, if August's forecast was $145$?
1. **State the problem:** We want to forecast October's demand using exponential smoothing with smoothing constant $\alpha = 0.2$. Given August's forecast $F_{August} = 145$ and September's actual sales $A_{September} = 160$, we need to find October's forecast $F_{October}$.
2. **Formula for exponential smoothing:**
$$
F_{t} = \alpha A_{t-1} + (1 - \alpha) F_{t-1}
$$
where $F_t$ is the forecast for month $t$, $A_{t-1}$ is the actual sales for previous month, and $F_{t-1}$ is the forecast for previous month.
3. **Apply the formula for September's forecast (already given as 145 for August):**
We use the formula to find September's forecast $F_{September}$:
$$
F_{September} = \alpha A_{August} + (1 - \alpha) F_{August} = 0.2 \times 120 + 0.8 \times 145
$$
Calculate:
$$
F_{September} = 24 + 116 = 140
$$
4. **Calculate October's forecast $F_{October}$ using September's actual sales $A_{September} = 160$ and September's forecast $F_{September} = 140$:**
$$
F_{October} = \alpha A_{September} + (1 - \alpha) F_{September} = 0.2 \times 160 + 0.8 \times 140
$$
Calculate:
$$
F_{October} = 32 + 112 = 144
$$
5. **Final answer:** The forecast for October is $\boxed{144}$.
This corresponds to option (a).