Cell-Phone Sales Forecast — Naive and Exponential Smoothing: 23-Ind-A4 Production Management, Undated paper, Q6

Question 6: Cell-Phone Sales Forecast — Naive and Exponential Smoothing (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.

Check — March’s low value is a real data point

This sitting’s eight-month table is a retailer’s monthly sales, with April misplaced. The unusually low March figure (30, versus 450 in February and 740 in May) is printed as “30”, so it is treated as a genuine data point (a real, sharp one-month dip).

Given.

Month	Index $t$	Sales
February	1	450
March	2	30
April	—	Missing
May	3	740
June	4	1,000
July	5	950
August	6	1,400
September	7	1,800

Find. October AND November sales forecasts by (a) naive and exponential smoothing, with an appropriately chosen smoothing constant; (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 the smoothing recursion); run naive and exponential smoothing over the seven known observations in sequence, skipping April entirely rather than bridging it; check the smoothing constant $\alpha$ by a grid search over $0<\alpha\le1$ minimizing in-sample squared one-step error (the standard way to “choose an appropriate parameter”; note that $\alpha=1$ is the naïve method); and, since neither method carries a trend term, roll each method's one-step-ahead forecast forward unchanged for the second (November) step, since no new actual will exist for October at the time the November forecast is made.

Naïve, October and November. The naïve forecast is simply the most recent actual value, carried forward unchanged since no new actual arrives before November either: $$\boxed{F_{Oct}=F_{Nov}=A_{Sep}=1{,}800\ \text{units}}.$$
Exponential smoothing — choosing $\alpha$ (part a). With $F_1=A_1=450$ (February) and $F_{t+1}=\alpha A_t+(1-\alpha)F_t$ run over the seven known months in sequence (Feb, Mar, May, Jun, Jul, Aug, Sep — April skipped), in-sample SSE falls steadily as $\alpha$ rises and is lowest at $\alpha=1$ (SSE $=1{,}113{,}100$), which is exactly the naïve forecast. The data carry a strong uptrend (740 to 1,800 in five months), and any smoothing lags behind it. For a genuine smoothing forecast choose a high but less-than-one constant, $\alpha=0.8$ (SSE $\approx1{,}165{,}944$); a conventional $\alpha=0.3$ fits far worse (SSE $\approx2{,}009{,}040$).
Exponential smoothing — October and November forecasts. Running the recursion with $\alpha=0.8$ through September ($F_{Sep}\approx1{,}308.9$) and applying one more step with $A_{Sep}=1{,}800$: $$F_{Oct}=0.8(1{,}800)+0.2(1{,}308.9)\approx\boxed{1{,}702\ \text{units}}.$$ Simple exponential smoothing has no trend term, so its multi-step-ahead forecast is flat beyond the first step: with no new actual observed before November, $F_{Nov}=\alpha F_{Oct}+(1-\alpha)F_{Oct}=F_{Oct}$, i.e. $$\boxed{F_{Nov}=F_{Oct}\approx1{,}702\ \text{units}}.$$ For comparison, $\alpha=0.3$ gives $F_{Oct}\approx1{,}184$, badly underreacting to the June–September uptrend.
Comparison and recommendation (part b). Naïve has the lowest in-sample error of every option: SSE $1{,}113{,}100$ and MAD $\approx382$, against SSE $\approx1{,}165{,}944$ and MAD $\approx401$ for $\alpha=0.8$. The grid search confirms it, since the SSE-optimal smoothing constant is $\alpha=1$. The reason is the trend: sales have risen from May (740) to September (1,800), so the latest value is the best available predictor and any weight on older, lower months pulls the forecast down. The recommended forecast is therefore $\boxed{\text{naïve: }1{,}800\ \text{units for both October and November}}$. It is still likely to be low if the uptrend continues, which is the main point for improvement below.

Method	October forecast	November forecast
Naïve	1,800 units	1,800 units
Exponential smoothing ($\alpha=0.8$)	≈1,702 units	≈1,702 units
Exponential smoothing ($\alpha=0.3$, conventional, for comparison)	≈1,184 units	—
Recommended forecast	1,800 units (naïve; lowest SSE and MAD, equal to smoothing with the SSE-optimal $\alpha=1$), both months

Improving the forecast. Recover the true April figure rather than dropping the observation, so the smoothing recursion is not forced to skip over a gap in the series; extend the history beyond eight months to confirm whether the June–September run is a genuine sustained uptrend (plausible ahead of a fourth-quarter shopping season) or a short-term rebound from March's anomalous dip that will not continue into October and November; investigate what actually caused the March figure (a stockout, a data-entry error, a one-off promotion cannibalizing a competitor month) before treating it as ordinary noise in future forecasts; and, specifically because neither method here carries a trend component, fit a trend-aware model (e.g. double exponential smoothing / Holt's method, or a simple regression against the month index) as a cross-check, since a genuine uptrend would make both naïve and simple exponential smoothing systematically understate October and November once the trend continues past September.

Concept & Theory

Topic. Comparing naive and exponential-smoothing forecasts, two months ahead, on a short time series with one missing observation and a sharp single-month dip.

Key relations. Naive $F_{t+1}=A_t$; simple exponential smoothing $F_{t+1}=\alpha A_t+(1-\alpha)F_t$; SSE-optimal $\alpha$ found by grid search over in-sample one-step forecast error; because simple ES has no trend term, its $h$-step-ahead forecast for $h\ge1$ from a single origin is flat, $F_{t+h}=F_{t+1}$.

Why this works. Naïve assumes next month looks like this month; exponential smoothing blends recent history with weight $\alpha$. On a series with a strong trend, every bit of weight on older months pulls the forecast toward the past, so SSE keeps falling as $\alpha\to1$ and the best simple-smoothing forecast collapses to naïve. Neither model has a trend term, which is why both lag.

Common pitfall. Silently interpolating a missing observation, or “correcting” an unusual real data point (here, March) to look more like a typical value, instead of checking it against the printed data; assuming a multi-step-ahead exponential-smoothing forecast should differ from the one-step forecast when the model has no trend term to make it do so; and stopping an $\alpha$ grid at 0.99, which hides that $\alpha=1$ (naïve) fits best.

Source. Nahmias & Olsen, Production and Operations Analysis, Ch. 2 (forecasting: naive, moving average, exponential smoothing).

Question 6 of 8: Cell-Phone Sales Forecast — Naive and Exponential Smoothing

Question 6: Cell-Phone Sales Forecast — Naive and Exponential Smoothing (20 marks)