Question 6 of 8: 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 2017 — 98-Ind-A4 Production Management. Three-hour, closed-book exam; Casio or Sharp approved calculators only. Format: eight questions, each worth 20 marks (sub-part weights 10/10 as tabulated on the front-page marking scheme); only the first five questions appearing in the answer book are marked, so candidates effectively choose 5 of 8. All eight 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 scheduling and shop-floor implementation gaps; 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, buffering, and production scheduling; Liker, The Toyota Way, and the Toyota Production System literature — 5S, Five Whys, and lean root-cause analysis.
The known-sales sequence and naïve forecast are reused; the exponential-smoothing computation and part-(b) comparison are worked fresh 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 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.
Naïve. The naïve forecast is simply the most recent actual value:
$$\boxed{F_{Oct}=A_{Sep}=800\ \text{units}}.$$
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:
$$F_{Mar}=450,\quad F_{Apr}=0.3(300)+0.7(450)=405,\quad F_{May}=405\ (\text{carried, April actual missing}),$$
$$F_{Jun}=0.3(740)+0.7(405)=505.5,\quad F_{Jul}=0.3(1000)+0.7(505.5)=653.85,$$
$$F_{Aug}=0.3(950)+0.7(653.85)=742.70,\quad F_{Sep}=0.3(1000)+0.7(742.70)=819.89,$$
$$F_{Oct}=0.3(800)+0.7(819.89)=\boxed{813.9\approx814\ \text{units}}.$$
Choosing $\alpha$ from the data (part a, continued). $\alpha=0.3$ is only a starting choice, so check it against the in-sample one-step-ahead errors. Six months have both a known actual and a prior forecast (March and May–September). Each method carries its last forecast across the missing April.
Method
MAD
SSE
Mean error (bias)
October forecast
ES, $\alpha=0.3$
258.8
533,561
+202.2
813.9
ES, $\alpha=0.5$
232.8
416,725
+138.7
866.1
ES, $\alpha=0.9$
190.1
335,430
+68.4
819.5
Naïve (= ES with $\alpha=1$)
191.7
328,700
+58.3
800
With $\alpha=0.3$ the forecast lags the steep February–June ramp: it under-forecasts every month from May to August, by 257 to 495 units, for a mean bias of +202 units. SSE falls steadily as $\alpha$ rises, all the way to $\alpha=1$, which is the naïve method. MAD is essentially flat between $\alpha=0.9$ (190.1) and naïve (191.7). The appropriate smoothing constant for this series is therefore high, about $\alpha\approx0.9$, which gives $F_{Oct}\approx820$ units.
Choice of best forecast (part b). The naïve forecast has the lowest SSE and the smallest bias, and its MAD is within 2 units of the best smoothing constant. It is the best forecast here:
$$\boxed{F_{Oct}=800\ \text{units (naïve)}}.$$
Smoothing with a high $\alpha$ (about 820 units) is a practically equivalent alternative. The conventional $\alpha=0.3$ result (814 units) should not be chosen. Its October value lands near the naïve one only because its lag happens to offset September's drop, and its in-sample errors show it under-forecasting the whole ramp. On a short series that has just ramped up to a new level, the method that tracks the latest level wins.
800 units (naïve: lowest SSE 328,700 and lowest bias)
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; and, as the history grows, fit a trend-adjusted (Holt) or seasonal model by in-sample MAD/SSE instead of a level-only smoother. Tracking old-model and new-model sales separately would also stop product-transition effects from being read as demand noise.
