Question 6 of 8: Tablet-Computer Sales Forecast (Naive, Moving Average), Two Months Ahead
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Technical Examinations — December 2018 — 17-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); candidates do two questions from Section A and three from Section B, and only the first five questions appearing in the answer book are marked. 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/EPQ) and aggregate planning; Sipper & Bulfin, Production: Planning, Control, and Integration — production scheduling, JIT/kanban 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 and days-off workforce scheduling; Hopp & Spearman, Factory Physics (3rd ed.) — variability, buffering, and production scheduling; Liker, The Toyota Way, and Shingo, A Revolution in Manufacturing: The SMED System — 5S, Five Whys, SMED and lean root-cause analysis.
The question gives an 8-month sales table (including the missing April value) and asks for forecasts for both October and November, so a second forecast step (using each method's own forecast for October in place of the not-yet-observed actual) is derived 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 AND November sales forecasts by (a) naïve and moving average, with an appropriately chosen window length; (b) the best-justified forecast and how it can be improved.
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. Since no actual sales figure will exist for October at the time the November forecast is made, roll each method's own October forecast forward as the stand-in for October's (not-yet-observed) actual — the standard convention for a multi-period-ahead forecast made from a single origin with no intervening data.
Naïve, October. The naïve forecast is simply the most recent actual value:
$$\boxed{F_{Oct}=A_{Sep}=800\ \text{units}}.$$
Naïve, November. With no new actual observed between October and November, the naïve method carries the same last-known value forward (its own October forecast stands in for the unobserved October actual):
$$\boxed{F_{Nov}=F_{Oct}=800\ \text{units}}.$$
Moving average, October. A $k$-period moving average needs $k$ consecutive known actuals ending at September. May–September is the longest run of consecutive known months (5 months), so any $k\le5$ avoids the missing April; only $k=6$ would reach it. $k=4$ is chosen: it averages the June–September plateau and leaves out May's 740, which is still on the rising launch ramp. It also leaves May–August to test a one-step forecast of September (the $k=5$ alternative would give $(740+1000+950+1000+800)/5=898$ for October):
$$F_{Oct}=\frac{A_{Jun}+A_{Jul}+A_{Aug}+A_{Sep}}{4}=\frac{1000+950+1000+800}{4}=\boxed{937.5\approx938\ \text{units}}.$$
Moving average, November. Rolling the window forward by one month, using the just-computed October forecast in place of the (not yet observed) October actual:
$$F_{Nov}=\frac{A_{Jul}+A_{Aug}+A_{Sep}+F_{Oct}}{4}=\frac{950+1000+800+937.5}{4}=\boxed{921.9\approx922\ \text{units}}.$$
Comparison and recommendation (part b). Naïve stays perfectly flat at 800 for both months (it has no mechanism to revert toward the June–August plateau once September dips), while the moving average drifts from 937.5 toward 921.9 as September's lower value works its way into the rolling window — a mild downward correction rather than a snap back to 800. The one month both methods can be scored on is September: the 4-period average of May–August forecast $922.5$ (error $122.5$), while naïve forecast August's $1{,}000$ (error $200$). The recommended forecast is the $\boxed{\text{4-period moving average}}$ for both months, since it damps the risk that September's 800 is a one-off dip rather than a genuine step down from the $\approx$1000-unit plateau, while still responding (more than naïve does) to the lower recent value; naïve would understate both months if the dip proves temporary, while the moving average's slight downward drift correctly acknowledges the extra uncertainty of forecasting two months out from the same data origin.
Method
October forecast
November forecast
Naïve
800 units
800 units
4-period moving average
937.5 ≈ 938 units
921.9 ≈ 922 units
Recommended forecast
4-period moving average (≈938 Oct, ≈922 Nov)
Improving the forecast. Recover the true April figure rather than dropping the observation, so the moving-average window is not forced to skip over a gap in the series; extend the history beyond eight months to distinguish a genuine trend or seasonal pattern from short-term volatility (tablet sales are plausibly seasonal around a fourth-quarter shopping period, which neither method here can detect from eight months of data); compare a small grid of window lengths ($k=2,3,4$) against in-sample forecast error (MAD or SSE) rather than fixing $k=4$ by the availability argument alone; and, specifically for the two-month-ahead forecast, use a method (e.g. simple exponential smoothing or a trend-aware model) that produces an explicit, growing prediction interval for the second step, rather than silently treating the rolled-forward forecast as if it were as reliable as an observed actual.
