Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Technical Examinations — December 2015 — 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, (b) exponential smoothing, (c) linear regression; (d) 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, fit the regression line against each observation's true month index (skipping $t=3$ rather than closing the gap), then compare in-sample fit and the recent-data pattern to justify a single recommended forecast.
(a) Naive. The naive forecast is simply the most recent actual value:
$$\boxed{F_{Oct}=A_{Sep}=800\ \text{units}}.$$
(b) Exponential smoothing. 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") finds $\alpha=0.99$ as the SSE-minimizing constant (SSE $\approx329{,}179$) — the series is volatile enough month to month that the best-fitting smoothing constant is nearly fully reactive, close to the naive method itself. 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 visibly worse in-sample fit) and gives $F_{Oct}\approx814$ — both cluster close to the naive value, unlike the regression forecast below.
(c) Regression line. Fitting $y=b_0+b_1t$ by least squares on the seven known $(t,\text{sales})$ pairs (April's index $t=3$ excluded, not interpolated):
$$b_1=86.4\ \text{units/month},\qquad b_0=341.2,\qquad R^2=0.64.$$
Extrapolating to October ($t=9$):
$$F_{Oct}=341.2+86.4(9)\approx\boxed{1{,}119\ \text{units}}.$$
(d) Comparison and recommendation. In-sample mean absolute deviation is lowest for the regression line (MAD $\approx128$, vs. $\approx192$ for naive and exponential smoothing), which on its own would favour regression. But the regression's upward slope is driven mainly by the steep February–June run-up, while the four most recent months (June–September: 1000, 950, 1000, 800) have plateaued in an 800–1000 band with no further growth — a pattern the naive and exponential-smoothing forecasts (both $\approx800$) pick up directly because they weight recent data, but a straight trend line fit across all seven points does not. The recommended point forecast is therefore the exponential-smoothing value ($\approx\boxed{800\text{–}810}$ units), with the regression's $\approx1{,}119$ reported as an upside scenario if the earlier growth trend resumes rather than as the primary forecast.
Exponential smoothing ($\alpha=0.3$, conventional, for comparison)
814 units
Linear regression ($R^2=0.64$)
1,119 units
Recommended forecast
≈800–810 units (exponential smoothing/naive)
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 (seven points give a weak basis for a long-run slope); 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.
