Sales Forecast for iPad Tablets: 23-Ind-A4 Production Management, December 2014, Q3

Question 3: Sales Forecast for iPad Tablets (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. Eight months of actual sales, February through September (see table); no other demand drivers (promotions, launch dates, pricing) are supplied.

Month	Sales (units)
February	450
March	300
April	400
May	740
June	500
July	100
August	1000
September	800

Find. A justified point forecast for October sales, and a discussion of the forecast's reliability and how it could be improved.

Approach. Test the series for a usable linear trend before trusting one, then compare a naive (last-value) forecast against a short moving average to pick the method that best matches the data's actual behaviour, rather than defaulting to whichever method is easiest to compute.

Check for a linear trend. Regressing sales $y$ against month index $t=1,\dots,8$ gives slope $\approx57.3$ units/month, intercept $\approx278.6$, but the coefficient of determination is $$R^2\approx\boxed{0.23},$$ meaning a straight line explains under a quarter of the month-to-month variation — the swings from 740 down to 100 then up to 1000 are mostly volatility a linear model cannot capture, not a steady climb. A naive trend-line forecast (which would give $\approx794$ units for October) is therefore not defensible as the primary method.
Compare candidate forecasts. Naive (last actual value, September) gives $F_{Oct}=\boxed{800}$; this reacts fully to the most recent month but ignores the very low July month (100) that came right before the August/September rebound — it would have forecast August from July's 100 and been off by a factor of ten. A 3-month moving average (Jul, Aug, Sep) smooths the recent volatility: $$F_{Oct}=\frac{100+1000+800}{3}=\boxed{633\ \text{units}}.$$
(a) Select and justify. With $R^2\approx0.23$ there is no reliable trend or visible seasonal cycle in only eight points, and the series alternates between a near-zero month (July, 100) and 700–1000-unit months, which looks like promotion- or launch-driven spikes rather than smooth demand. The 3-month moving average is the more defensible forecast because it damps the effect of any single unusual month (the July dip or the August spike) while still tracking the recent (Aug–Sep) high-sales regime, giving $\boxed{F_{Oct}\approx633\ \text{units}}$ as the point forecast, bracketed by the naive estimate (800, an upper case if September's pace continues) and the pure trend line (794) as a plausibility check.

Method	October forecast
Naive (last value)	800 units
Linear trend ($R^2\approx0.23$, unreliable)	794 units
3-month moving average (selected)	633 units

(b) Discussion and improvement. The forecast is weak: eight data points cannot establish seasonality (a real yearly cycle needs at least two full years of data to separate from noise), the near-zero July month (100 units) suggests a stock-out, a discontinued old model, or a lull ahead of a new-model launch rather than organic demand decay, and the August/September rebound (1000, 800) could equally be a one-time launch spike that will not repeat in October. To improve the forecast I would: (1) collect at least 24 months of history to test for a genuine annual seasonal pattern (back-to-school, holiday, or new-model-launch cycles are common for consumer electronics); (2) bring in causal information the retailer already has — Apple's announced launch calendar, planned promotions, and competitor pricing — since a launch-driven spike is a known event, not a random one, and should be modelled explicitly rather than smoothed away; (3) track forecast error (MAD or MSE) month over month and switch to exponential smoothing with a low smoothing constant once enough history exists, so the model adapts without over-reacting to a single outlier month; and (4) disaggregate "iPad sales" into old-model and new-model units, since combining them (as the table does) can hide a simple explanation — e.g. the August spike may be a new-model launch cannibalizing what would otherwise have been steady old-model demand.

Concept & Theory

Topic. Time-series demand forecasting with limited history: naive, moving-average, and trend methods, and the diagnostic role of $R^2$.

Key relations. Naive forecast $F_{t+1}=A_t$; $k$-period moving average $F_{t+1}=\frac{1}{k}\sum_{i=0}^{k-1}A_{t-i}$; coefficient of determination $R^2=1-SS_{res}/SS_{tot}$ measuring the fraction of variance a fitted trend explains.

Why this works. A moving average is a low-pass filter: it damps random period-to-period noise while still tracking a genuine shift in level, which is exactly the trade-off needed here since the series shows only a weak trend ($R^2\approx0.23$) but does show a level shift into the Aug–Sep regime. Choosing the window length ($k$) trades responsiveness (small $k$) against noise suppression (large $k$); $k=3$ is a reasonable default with only eight observations.

Common pitfall. Fitting a trend line to a short, volatile series and reporting the trend forecast with false confidence — the $R^2$ value must always be checked and reported alongside any trend-based forecast, and a low $R^2$ (even one as high as 0.23) should push the analyst toward a smoothing method instead.

Source. Nahmias & Olsen, Production and Operations Analysis, Ch. 2 (forecasting methods, moving averages, and trend/seasonal decomposition).

Question 3 of 7: Sales Forecast for iPad Tablets

Question 3: Sales Forecast for iPad Tablets (20 marks)