Question 3 of 7: Categories of Forecasting Techniques and a Two-Year Demand Series

Question 3: Categories of Forecasting Techniques and a Two-Year Demand Series (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 techniques fall into four broad categories, and the useful way to tell them apart is to ask what each one treats as the source of information about the future.

Qualitative (judgemental) techniques take human opinion as the data. Market surveys, panels of executive opinion, sales-force composites, the Delphi method and historical analogy all belong here. They require no numerical history at all, which is precisely why they are used for a genuinely new product, for a long-range capacity or technology decision, or where a structural change (a competitor's entry, a regulatory shift) makes the past a poor guide. Their weakness is that they are expensive in senior time, are not reproducible, and carry whatever optimism or anchoring the participants bring. Delphi exists specifically to strip out the dominance effects of a committee by keeping the respondents anonymous and iterating.

Time-series analysis takes the history of the series itself as the only data and projects its internal pattern forward. Moving averages, weighted moving averages, exponential smoothing (simple, trend-adjusted and seasonally-adjusted Winters models), classical decomposition into trend, seasonal, cyclical and random components, and Box–Jenkins ARIMA models all sit in this group. They are cheap, entirely mechanical and therefore well suited to the thousands of short-term item-level forecasts a manufacturing plant must produce every month. Their defining limitation is that they cannot anticipate anything not already present in the history: a time-series model always turns late.

Causal (associative or econometric) techniques relate demand to one or more other variables that are believed to drive it — simple and multiple regression, leading-indicator models, input–output and econometric systems. The gain is that the model can explain and can anticipate a turn, because the independent variable moves first; the cost is that the driving variables must themselves be forecast, and a relationship fitted over one economic regime may not hold in the next. These are the natural tools for medium-range aggregate planning and for capital decisions.

Simulation models the demand-generating system directly and lets it run: a Monte Carlo model of a distribution network, or a dynamic system model of a supply chain. It is the only category that copes comfortably with feedback, queueing and policy interactions, and it produces a distribution of outcomes rather than a point forecast, which is what a risk assessment needs. It is also the most expensive to build and validate.

The differences that matter in practice are therefore four: the data each requires (opinion, own history, related series, a structural model); the horizon each suits (qualitative for long range, time series for short, causal for medium, simulation for policy questions); the cost and skill demanded; and, above all, whether the technique can turn ahead of the series or only follow it. A working forecasting system almost always blends them — a time-series model to generate the routine item forecasts, a causal or judgemental overlay to catch the structural changes the time-series model is blind to, and a tracking signal to say when the overlay is needed.

Part (b) — the two-year series

Given. Twenty-four consecutive monthly observations of demand for component parts, in thousands, tabulated above. First-year total 162 thousand (average 13.5 per month), second-year total 223 thousand (average 18.58 per month).

Find. A plot of the observations, and a reasoned recommendation of the forecasting method or methods that suit this series, supported by a fitted model and a year-3 forecast.

Approach. Read the plot first and let it choose the model class: a series with a repeating annual shape riding on a rising level calls for a multiplicative decomposition (or the equivalent Winters seasonal exponential smoothing), so compute seasonal indices, deseasonalise, fit a trend to what is left, and recombine.

1. Part (b)(i) — plot the observations, then read the plot before fitting anything. The 24 points are drawn in sequence in the figure above. Three features are visible and each one rules a method in or out. The shape repeats with a period of twelve months, with the minimum in month 4 and the maximum in month 10 in both years; the general level rises, the annual total going from 162 to 223 thousand, a 37.7 per cent increase; and the amplitude of the seasonal swing grows with the level, which is the signature of a multiplicative rather than an additive seasonal. There is no evidence of a longer business cycle in only two years of data, and the residual scatter is modest.
2. Compute seasonal indices by the ratio-to-yearly-average method. For each month, divide the observation by that year's monthly average and average the two years: $$I_m = \frac{1}{2}\left(\frac{y_{1,m}}{\bar{y}_1} + \frac{y_{2,m}}{\bar{y}_2}\right), \qquad \bar{y}_1 = 13.500,\ \bar{y}_2 = 18.583$$ For month 10, for example, $I_{10} = \tfrac{1}{2}(29/13.500 + 33/18.583) = \tfrac{1}{2}(2.1481+1.7758) = 1.962$. The twelve indices come out as follows.

   Month	1	2	3	4	5	6	7	8	9	10	11	12
   Seasonal index $I_m$	0.693	0.592	0.555	0.363	0.454	0.791	1.036	1.430	1.787	1.962	1.269	1.067

3. Check the indices before using them. The twelve indices sum to 12.000 as they must, so no normalising step is needed. The peak-to-trough ratio is $1.962/0.363 = 5.40$: October demand runs more than five times April demand. A seasonal that large is the decisive fact of this problem, because it means a simple moving average or simple exponential smoothing — neither of which carries a seasonal — cannot possibly track the series.
4. Deseasonalise and fit the trend. Dividing each observation by its index gives a series in which only trend and noise remain; least squares on $t = 1 \dots 24$ gives $$D_t = 11.407 + 0.3775\,t \qquad (R^2 = 0.551)$$ so the underlying level is growing by about 378 units per month, roughly 4.5 thousand per year on a base of about 13 thousand. The modest $R^2$ is honest and expected: with only two seasonal cycles the ratio-to-yearly-average indices absorb part of the trend, leaving real month-to-month scatter in the deseasonalised series. The trend is nonetheless unmistakable and significant.
5. Recombine to forecast year 3. The forecast for month $m$ of the third year is the trend value at $t = 24+m$ multiplied back by that month's index, $F_{24+m} = (11.407 + 0.3775(24+m))\,I_m$. For month 10, for instance, $F_{34} = (11.407+0.3775 \times 34)(1.962) = 47.6$ thousand. $$\boxed{\ \text{Year-3 forecast: } 14.4,\ 12.6,\ 12.0,\ 8.0,\ 10.2,\ 18.0,\ 24.0,\ 33.6,\ 42.7,\ 47.6,\ 31.2,\ 26.7\ \text{thousand}\ }$$ totalling 281 thousand, about 26 per cent above year 2.
6. Part (b)(ii) — estimate the best methods of forecasting future demands. The recommendation follows from the three features read off the plot. Multiplicative decomposition, as worked above, or equivalently Winters' three-parameter seasonally-adjusted exponential smoothing (level, trend and seasonal smoothing constants), is the right routine method: it carries both of the features the data actually has, updates cheaply every month, and is standard in any production planning package. If a manual method is wanted, a seasonally-adjusted moving average with a trend correction reproduces most of the accuracy. What must be avoided is equally clear: a plain 3- or 12-month moving average or simple exponential smoothing will lag badly — the last three months of year 1 average 21.0 against an actual of 12 in month 13, an error of 75 per cent — and a bare linear regression on time ignores the seasonal entirely. Finally, because only two cycles are available the indices are provisional; carry a tracking signal and refit the indices as the third year accumulates, and overlay judgement if a known structural change (a new customer, a product substitution) is expected.