Question 3 of 6: Categories of Forecasting Technique, and Method Selection from Data

Question 3: Categories of Forecasting Technique, and Method Selection from Data (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.

Part (a) — the categories of forecasting technique. Forecasting methods fall into four families, and they differ in one respect above all: what they take as evidence.

Qualitative or judgmental techniques take expert opinion as evidence. Market research, panel consensus, the Delphi method, historical analogy and the grass-roots build-up of a sales force's own estimates all belong here. They are subjective and cannot be error-bounded statistically, but they are the only family available when there is no history to analyse — a new product, a new market, a technology substitution — and they are the only family that can anticipate a discontinuity, because a discontinuity is by construction absent from the data.

Time-series analysis takes the past of the series itself as evidence, and assumes the future is a continuation of the past. Moving averages, weighted moving averages, exponential smoothing in its simple, trend-adjusted and seasonal (Winters) forms, decomposition, and Box-Jenkins ARIMA models are the members. They are cheap, fully automatable over thousands of stock-keeping units, and accurate over the short to medium term; they cannot explain why demand moves, and they cannot see a turning point until after it has happened.

Causal or associative techniques take other, external variables as evidence, on the argument that demand is driven by identifiable factors. Simple and multiple regression, econometric systems, leading-indicator models and input-output models sit here. They can forecast a turning point, because the driver turns before the demand does; the price is that the drivers must themselves be known or forecastable, and that a relationship estimated over one regime need not hold in the next.

Simulation takes an explicit model of the system's behaviour as evidence: the demand process, the customers, the competitors, the capacity and the policies are represented dynamically and run forward, usually stochastically. It answers questions the other three cannot — what happens under a capacity constraint, or under a competitor's response, or across a distribution of outcomes rather than at a point — but it is expensive to build and only as good as the structural assumptions inside it.

The practical differences follow from the evidence each uses: cost and effort rise steeply from time series to simulation; data requirements rise the same way; the useful horizon lengthens from time series (short) to causal and qualitative (long); and only the causal and qualitative families can call a turning point. A working operations forecasting system usually runs time-series models for the routine short-horizon item-level forecasts and overlays judgment and causal models at the aggregate and long-horizon end.

Given. Twenty-four consecutive monthly demand observations for a component part, in thousands of units, covering two full years.

Find. A plot of the series, a diagnosis of which components of demand it contains, and a reasoned recommendation of the forecasting method to use — demonstrated by fitting that method and projecting a third year.

Approach. Plot the data first and read the components off the plot; then confirm each component numerically — the trend from the two annual totals, the season from ratio-to-average indices — and let the diagnosis select the method rather than the other way round.

1. Sub-part (a) — plot the observations and read them. The figure above plots all 24 months on one time axis. Three features are immediate. The series is strongly and regularly seasonal, falling to a minimum in month 4 and rising to a maximum in month 10 in both years, with the same shape repeated. The second year lies above the first at almost every month, so there is an upward trend. And the swing is very large relative to the mean, which rules out treating the season as noise.
2. Confirm the trend from the annual totals. Summing each year, $$\sum_{\text{year 1}} D = 162, \qquad \sum_{\text{year 2}} D = 223$$ in thousands, so the monthly averages are $162/12 = 13.50$ and $223/12 = 18.58$ thousand and the year-on-year growth is $$\frac{223}{162} - 1 = \boxed{0.377 \equiv 37.7\ \text{per cent}}$$ A trend of this size cannot be left to a smoothing model without trend correction, which would lag the series by roughly $(1-\alpha)/\alpha$ periods permanently.
3. Confirm the season with ratio-to-average indices. Dividing each observation by its own year's monthly average removes the level and the trend between years, leaving the seasonal factor plus noise; averaging the two years' ratios for each month gives the index $$S_m = \tfrac{1}{2}\left(\frac{D_{1,m}}{\bar{D}_1} + \frac{D_{2,m}}{\bar{D}_2}\right)$$ which yields, for months 1 to 12: 0.693, 0.592, 0.555, 0.363, 0.454, 0.791, 1.036, 1.430, 1.787, 1.962, 1.269 and 1.067. They sum to 12.000 as they must, so no normalisation is needed.
4. Quantify how strong the season is. The extremes are $$\boxed{S_{\max} = S_{10} = 1.962, \qquad S_{\min} = S_{4} = 0.363, \qquad \frac{S_{\max}}{S_{\min}} = 5.40}$$ October demand is more than five times April demand at the same trend level. Any method that does not carry seasonal indices — a simple moving average, single exponential smoothing, a plain regression on time — will be wrong by a factor of two in both directions every year.
5. Deseasonalise and fit the trend. Dividing each observation by its month's index and regressing the result on $t = 1,\dots,24$ by least squares gives $$\boxed{\hat{D}_t = 11.41 + 0.3775\,t \quad \text{(thousands, deseasonalised)}}$$ with a coefficient of determination of only $R^2 = 0.55$. That modest fit is worth reporting honestly rather than hiding: with just two seasonal cycles the ratio-to-yearly-average indices are estimated from two observations per month and absorb part of the trend, so a good deal of month-to-month scatter survives deseasonalisation. The direction and slope of the trend are nonetheless solid, being anchored on two full-year totals rather than on the noisy monthly values.
6. Sub-part (b) — recommend the method. The diagnosis is a series with a clear multiplicative season, a firm upward trend and no visible cycle, observed monthly and needed monthly. That is exactly the specification of Winters' triple exponential smoothing, which carries three updated states — level, trend and a vector of twelve seasonal indices — with smoothing constants alpha, beta and gamma, and which re-estimates all three every month as new demand arrives. Initialise it with the level and trend from the regression above and the seasonal indices just computed, then let it adapt. The classical decomposition model just fitted (trend line multiplied by seasonal index) is the correct simpler alternative and the right way to produce the initial estimates, but it does not update itself; use it for the annual plan and Winters for the rolling monthly forecast. A tracking signal should be run alongside, as in Question 1, because the trend estimate rests on only two years.
7. Demonstrate the recommendation on a third year. Applying $\hat{D}_t \times S_m$ for $t = 25$ to 36 gives, in thousands: 14.45, 12.57, 11.99, 7.99, 10.15, 17.98, 23.95, 33.59, 42.65, 47.56, 31.24 and 26.67. The third-year total is $$\boxed{\sum \hat{D} = 280.8\ \text{thousand}, \quad \text{26 per cent above year 2}}$$ and the month-10 peak of 47.6 thousand is the number that sizes the capacity and the inventory build.