Forecasting

Forecasting is the estimation of future demand or other variables from past data and judgement. Production and inventory plans, capacity, purchasing and manpower all depend on it. A forecast is never exact, so it should be given with an error estimate, and forecasts for groups of products are more accurate than for single items.

Qualitative (judgemental) techniques

They are used when past data are scarce or unreliable, for example for a new product.

1. Executive (jury) opinion: senior managers from marketing, production and finance give their estimates, which are averaged or discussed. Quick and cheap, but may be dominated by a strong personality.
2. Delphi method: a panel of experts answers questionnaires anonymously in several rounds; after each round the summary is circulated and experts revise their views until agreement. It avoids group pressure but is slow.
3. Sales force composite: each salesperson estimates sales in their territory; the estimates are added up and adjusted. Uses close customer knowledge but may be biased.
4. Market survey (customer opinion): questionnaires, interviews or test marketing to find buying intentions. Good for new products but costly and may not reflect actual buying.
5. Historical analogy: the demand of a similar earlier product is used.

Qualitative versus quantitative

Qualitative methods depend on opinion and experience, are subjective, and suit new products and long-term technology forecasts. Quantitative methods use numbers from the past (time series or causal models) and suit existing products with stable patterns.

(a) Forecasts for month 9

3-month moving average:

$$F_9 = \frac{146 + 158 + 162}{3} = \frac{466}{3} = 155.3\ \text{units}$$

Weighted moving average:

$$F_9 = 0.5(162) + 0.3(158) + 0.2(146) = 81.0 + 47.4 + 29.2 = 157.6\ \text{units}$$

Exponential smoothing, $F_{t+1} = F_t + \alpha (D_t - F_t)$ with $F_1 = 120$:

$$F_9 = 144.51 + 0.3(162 - 144.51) = 149.76 \approx 150\ \text{units}$$

(b) MAD over months 4 to 8

• MAD (3-month MA) $= 60.0/5 = 12.0$
• MAD (weighted MA) $= 52.2/5 = 10.4$
• MAD (exp. smoothing) $= (15.08+20.56+10.39+19.27+17.49)/5 = 16.6$

The weighted moving average has the smallest error. Demand has an upward trend, and exponential smoothing with a small alpha lags behind a trend, so it is least accurate here (a larger alpha or a trend method would do better).

(c) Forecast error measures

• MAD $= \dfrac{\sum |A_t - F_t|}{n}$: average size of error.
• MSE $= \dfrac{\sum (A_t - F_t)^2}{n}$: penalises large errors.
• Tracking signal $= \dfrac{\sum (A_t - F_t)}{MAD}$ (running sum of forecast errors divided by MAD). It stays near zero for an unbiased forecast; values outside about $\pm 4$ show bias and a need to revise the model.

Answer: Month 9 forecast: 155.3 (3-MA), 157.6 (WMA), 149.8 (exp. smoothing); MAD = 12.0, 10.4, 16.6, so WMA is best.

(a) Least squares line

Take $x$ = year number, $n = 6$.

$$b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} = \frac{6(1164) - 21(312)}{6(91) - 441} = \frac{6984 - 6552}{105} = 4.114$$ $$a = \bar{y} - b\bar{x} = 52 - 4.114(3.5) = 37.6$$

Trend line: $y = 37.6 + 4.114x$ (demand rises by about 4.1 thousand units per year).

(b) Forecasts

• Year 7: $37.6 + 4.114(7) = 66.4$ thousand units
• Year 8: $37.6 + 4.114(8) = 70.5$ thousand units

(c) Forecast error of the fit

• MAD $= 4.34/6 = 0.72$ thousand units
• MSE $= 3.77/6 = 0.63$
• MAPE $= 1.4\%$

The errors are small and change sign, so the straight line fits the data well (correlation coefficient $r \approx 0.994$). Forecasting two years ahead is reasonable, but the line should be revised as new data arrive.

Causal methods

Causal (associative) techniques forecast a variable from other variables that influence it, for example demand of cement from construction activity, using simple or multiple regression and correlation, or econometric models. Time-series methods (moving average, exponential smoothing, trend) use only the past values of the variable itself.

Answer: $y = 37.6 + 4.114x$; forecasts 66.4 (year 7) and 70.5 (year 8) thousand units; MAD = 0.72, MSE = 0.63, MAPE = 1.4%.