Question 6 of 8: Tablet-Computer Sales Forecast (Naive, Exponential Smoothing)

Question 6: Tablet-Computer Sales Forecast (Naive, Exponential Smoothing) (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Find. October sales forecasts by (a) naïve and exponential smoothing, with an appropriately chosen smoothing constant; (b) the best-justified forecast and how to improve it.

Approach. Treat the missing April observation as genuinely absent rather than interpolated (inventing a value would corrupt every method that touches it): for exponential smoothing, initialize the forecast at the first known observation and carry the forecast forward unchanged across the missing month (no update is possible without an actual to compare against), then apply the standard recursion through the remaining known months.

1. Naïve. The naïve forecast is simply the most recent actual value: $$\boxed{F_{Oct}=A_{Sep}=800\ \text{units}}.$$
2. Exponential smoothing (part a). Simple exponential smoothing updates $F_{t+1}=\alpha A_t+(1-\alpha)F_t$. With $\alpha=0.3$ (a moderate, commonly-used starting choice for a series without a strong trend) and $F_{Feb}=A_{Feb}=450$ as the initializing forecast, and carrying $F$ forward unchanged across the missing April observation: $$F_{Mar}=450,\quad F_{Apr}=0.3(300)+0.7(450)=405,\quad F_{May}=405\ (\text{carried, April actual missing}),$$ $$F_{Jun}=0.3(740)+0.7(405)=505.5,\quad F_{Jul}=0.3(1000)+0.7(505.5)=653.85,$$ $$F_{Aug}=0.3(950)+0.7(653.85)=742.70,\quad F_{Sep}=0.3(1000)+0.7(742.70)=819.89,$$ $$F_{Oct}=0.3(800)+0.7(819.89)=\boxed{813.9\approx814\ \text{units}}.$$
3. Choosing $\alpha$ from in-sample error. $\alpha=0.3$ is only a conventional starting value, so test it against the six one-step-ahead forecasts that have an actual to compare with (March, May, June, July, August, September). The naïve forecast of May uses March, the last known actual.

   Method	MAD	SSE	October forecast
   ES, $\alpha=0.3$	258.8	533,561	813.9
   ES, $\alpha=0.5$	232.8	416,725	866.1
   ES, $\alpha=0.9$	190.1	335,430	819.5
   Naïve ($=$ ES with $\alpha=1$)	191.7	328,700	800

   Across the grid $\alpha=0.1,\dots,1.0$, MAD is lowest at $\alpha=0.9$ (190.1) and SSE keeps falling all the way to $\alpha=1$, which is the naïve method. The series stepped up from about 300–450 to about 1,000 (plausibly as the new model's sales took off), and a low $\alpha$ lags that step badly: $\alpha=0.3$ has about 35% more average error than naïve.
4. Comparison and recommendation (part b). The appropriate parameter is a high smoothing constant, $\alpha\approx0.9$, giving $F_{Oct}\approx820$. That is effectively tied with naïve (MAD 190.1 vs 191.7; naïve has the lower SSE), and both clearly beat $\alpha=0.3$. Recommend the naïve forecast, $\boxed{F_{Oct}=800\ \text{units}}$: it has the lowest squared error, it is the simplest, and the high-$\alpha$ ES value (820) differs by only 20 units. Only six error terms exist, so treat the range 800–820 as the real answer rather than claiming a precise winner.

Improving the forecast. Recover the true April figure rather than dropping the observation, so exponential smoothing is not forced to carry an un-updated forecast across a gap in the series; extend the history beyond eight months to distinguish a genuine trend or seasonal pattern from short-term volatility; re-fit $\alpha$ as more months arrive (once the ~1,000-unit plateau is established a lower, smoother $\alpha$ may win); track old- and new-model sales separately, since the combined series mixes a declining product with a growing one; and consider a trend-adjusted method (Holt) if growth resumes.