Question 3 of 7: Tablet-Computer Sales Forecast (Naive, Exponential Smoothing)
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Technical Examinations — May 2016 — 98-Ind-A4 Production Management. Three-hour, closed-book exam; Casio or Sharp approved calculators only. Format: seven questions, each worth 20 marks (sub-part weights as tabulated on the front page); only the first five questions appearing in the answer book are marked, so candidates effectively choose 5 of 7. All seven are solved below for completeness. The paper asks for point-form answers wherever possible; the solutions below use full working for clarity.
Reference texts: Nahmias & Olsen, Production and Operations Analysis (7th ed., Waveland/McGraw-Hill) — forecasting, inventory (EOQ) and aggregate planning; Sipper & Bulfin, Production: Planning, Control, and Integration — production-management systems; Hillier & Lieberman, Introduction to Operations Research (11th ed.) — LP formulation and project scheduling (CPM/PERT); Pinedo, Scheduling: Theory, Algorithms, and Systems (5th ed.) — parallel-machine scheduling, makespan and tardiness; Hopp & Spearman, Factory Physics (3rd ed.) — variability and production-system inefficiency; Niebel & Freivalds, Methods, Standards, and Work Design — division of labour and work-design history; ISO 9001:2015 and the Toyota Production System literature — quality management, 5S/lean and TPM.
Find. October sales forecasts by (a) naive and exponential smoothing; (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); apply the naive and exponential-smoothing methods to the seven known observations in sequence, then compare their in-sample fit and recent-data behaviour to justify a single recommended forecast.
Naive. The naive forecast is simply the most recent actual value:
$$\boxed{F_{Oct}=A_{Sep}=800\ \text{units}}.$$
Exponential smoothing (part a). With $F_1=A_1=450$ and $F_{t+1}=\alpha A_t+(1-\alpha)F_t$, a grid search over $\alpha$ minimizing the in-sample sum of squared one-step errors (the standard way to "choose an appropriate parameter") shows SSE falling steadily as $\alpha$ rises (533,561 at $\alpha=0.3$, 346,605 at 0.8, 329,179 at 0.99), so the best constant inside the usual range $0<\alpha<1$ is $\alpha=0.99$ — the series is volatile enough month to month that the best-fitting smoothing constant is nearly fully reactive. The limit $\alpha=1$ is the naive method, and its SSE (328,700) is lower still. Running the recursion to September gives $F_{Sep}\approx999.5$, so
$$F_{Oct}=0.99(800)+0.01(999.5)\approx\boxed{802\ \text{units}}.$$
For comparison, a conventional $\alpha=0.3$ reacts more slowly (SSE $\approx533{,}561$, a much worse in-sample fit) and gives $F_{Oct}\approx814$ — both cluster close to the naive value.
Comparison and recommendation (part b). The two fit measures cannot separate the methods: SSE slightly favours naive (328,700 vs. 329,179) while MAD slightly favours $\alpha=0.99$ smoothing (191.5 vs. 191.7), and their October point forecasts agree within 2–14 units of each other (800 vs. 802–814) — both methods are reading the same signal: the four most recent months (June–September: 1000, 950, 1000, 800) have plateaued in an 800–1000 band after the earlier February–June run-up. The recommended forecast is the naive value, $\boxed{\approx800\ \text{units}}$: it has the lowest in-sample squared error of any smoothing constant (it is the $\alpha=1$ limit the SSE search is heading towards), it is the simplest method, and the best smoothing forecast (802) confirms it to within 2 units. A lower $\alpha$ would only make sense if the recent drop to 800 were believed to be noise around a higher level, which eight months of data cannot show.
Method
October forecast
Naive
800 units
Exponential smoothing ($\alpha=0.99$, best fit for $0<\alpha<1$)
802 units
Exponential smoothing ($\alpha=0.3$, conventional, for comparison)
814 units
Recommended forecast
≈800 units (naive; lowest in-sample SSE, confirmed by smoothing at 802)
Improving the forecast. Recover the true April figure rather than dropping the observation, so no method loses a degree of freedom; extend the history beyond eight months to distinguish a genuine trend from short-term volatility; and track forecast error month to month with a tracking signal or control chart so a genuine turning point (a resumed uptrend, or a new plateau) is caught quickly rather than assumed away. A regression line against the month index would also be worth fitting as a cross-check even though this sitting does not request it, since it can reveal whether the recent plateau is noise around a continuing uptrend or a genuine change in level.
