Tablet-Computer Sales Forecast (Naive, Moving Average): 23-Ind-A4 Production Management, May 2018, Q6

Question 6: Tablet-Computer Sales Forecast (Naive, Moving Average) (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.

Month	Index $t$	Sales
February	1	450
March	2	300
April	3	Missing
May	4	740
June	5	1,000
July	6	950
August	7	1,000
September	8	800

Find. October sales forecasts by (a) naïve and moving average, with an appropriately chosen window length; (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 any window that spans it): pick a moving-average window that uses only consecutive KNOWN months, so the gap need not be bridged at all.

Naïve. The naïve forecast is simply the most recent actual value: $$\boxed{F_{Oct}=A_{Sep}=800\ \text{units}}.$$
Moving average. A $k$-period moving average needs $k$ consecutive known actuals ending at September. The known months after the gap run May–September (5 consecutive values), so any $k\le5$ avoids the missing April ($k=6$ would need it). May's 740 is still on the launch ramp, below the June–September level of about 950, so $k=4$ (June–September) is chosen: it is the longest window that sits wholly on the current sales level (a 5-month average would be pulled down to 898 by the ramp month): $$F_{Oct}=\frac{A_{Jun}+A_{Jul}+A_{Aug}+A_{Sep}}{4}=\frac{1000+950+1000+800}{4}=\boxed{937.5\approx938\ \text{units}}.$$
Comparison and recommendation (part b). The two forecasts differ by about 17% (800 vs. 938) because naïve reacts only to September's single value, while the 4-period average reflects the higher June–August plateau (1000, 950, 1000) that September's dip may not represent going forward. The recommended forecast is the $\boxed{938\text{-unit moving-average estimate}}$, since it damps the risk that September's 800 is a one-off dip rather than a genuine step down from the ~1000-unit plateau; naïve would understate October sales if the dip proves temporary.

Method	October forecast
Naïve	800 units
4-period moving average (Jun–Sep)	937.5 ≈ 938 units
Recommended forecast	≈938 units (4-period moving average)

Improving the forecast. Recover the true April figure rather than dropping the observation, so the moving-average window is not forced to skip over a gap in the series; extend the history beyond eight months to distinguish a genuine trend or seasonal pattern from short-term volatility; and compare a small grid of window lengths ($k=2,3,4$) against in-sample forecast error (MAD or SSE) rather than fixing $k=4$ by judgment alone, since the retailer's own data can indicate whether a shorter (more responsive) or longer (smoother) window fits better.

Concept & Theory

Topic. Comparing naïve and moving-average forecasts on a short time series with one missing observation.

Key relations. Naïve $F_{t+1}=A_t$; $k$-period moving average $F_{t+1}=\frac{1}{k}\sum_{i=t-k+1}^{t}A_i$; in-sample MAD $=\frac{1}{n}\sum|A_t-F_t|$ used to screen candidate window lengths.

Why this works. A moving average is an unweighted average of the last $k$ observations, so it damps single-period noise more than naïve, which reacts fully and only to the single most recent value; a shorter window shifts the method's behaviour toward naïve, a longer window toward a long-run average, and picking $k$ to land on a consecutive run of known months sidesteps the missing-data problem entirely rather than papering over it.

Common pitfall. Silently interpolating the missing April observation before fitting, which manufactures a data point the retailer never actually observed; and choosing a moving-average window length that straddles the gap (e.g. $k=6$, reaching back to April) without a documented rule for handling the missing term.

Source. Nahmias & Olsen, Production and Operations Analysis, Ch. 2 (forecasting: naive, moving average).

Question 6 of 8: Tablet-Computer Sales Forecast (Naive, Moving Average)

Question 6: Tablet-Computer Sales Forecast (Naive, Moving Average) (20 marks)