Forecasting methods

Why it matters

Every production plan, inventory policy and capacity decision starts with a demand forecast. Over-forecasting ties up money in stock and idle capacity; under-forecasting causes shortages, overtime and lost customers. Engineers need to pick a method that suits the demand pattern, compute forecasts quickly and, just as important, measure and monitor forecast error.

Key ideas

Principles: forecasts are almost always wrong, so state the expected error too; aggregate forecasts (product families, monthly totals) are more accurate than item-level ones; accuracy falls as the horizon lengthens.

Qualitative (judgemental) methods, used when data are scarce, for new products or long-range decisions:

• Delphi method: a panel of experts answers questionnaires anonymously in several rounds, with a summary fed back each round, until the forecasts converge. Anonymity avoids dominance by senior members.
• Market surveys, sales-force composite, jury of executive opinion, and historical analogy.

Quantitative methods

• Time-series methods assume that the past pattern continues. Demand is made of a level, trend, seasonality, cyclical movement and random variation.
• Causal (associative) methods relate demand to drivers such as price, advertising or GDP, usually by regression.

Time-series techniques

• Naive: next forecast = last actual demand.
• Simple moving average (SMA): average of the last n actual demands. A larger n smooths more but responds more slowly. All n points get equal weight.
• Weighted moving average (WMA): weights summing to 1, usually largest on the most recent period.
• Simple exponential smoothing (SES): new forecast = old forecast + α × (latest error). Weights on past data decline geometrically, so the method needs only the last forecast and last demand. A large α (0.3–0.5) responds quickly but follows noise; a small α (0.05–0.2) is stable. An SES and an SMA have the same average age of data when α = 2/(n + 1).
• SMA and SES both lag behind a trend. Trend-adjusted (Holt's) smoothing adds a smoothed trend term with a second constant β; seasonal data use seasonal indices (actual/average for each season), applied by multiplication.
• Linear regression (least squares) fits a straight line D = a + b·t through the history and extends it. It minimises the sum of squared errors, and the fitted line always passes through the mean point (t̄, D̄).

Measuring error (error e = actual − forecast)

• MAD (mean absolute deviation): average |e|; same units as demand.
• MSE: average e², penalises large errors.
• MAPE: average of |e|/D × 100%, scale-free.
• Bias (cumulative forecast error, CFE = Σe): persistent positive bias means the method keeps under-forecasting.
• Tracking signal TS = CFE / MAD: if it drifts outside about ±4 (limits chosen between ±3 and ±8), the method is biased and should be changed or re-tuned.

Formulas

SMA: F(t+1) = (D(t) + D(t−1) + … + D(t−n+1)) / n

• F = forecast (units/period), D = actual demand (units/period), n = number of periods.

WMA: F(t+1) = Σ wᵢ · D(t−i+1), with Σ wᵢ = 1

SES: F(t+1) = α · D(t) + (1 − α) · F(t) = F(t) + α · (D(t) − F(t))

• α = smoothing constant, 0 ≤ α ≤ 1 (dimensionless).

Equivalent α for an n-period SMA: α = 2 / (n + 1)

Least-squares line: F = a + b·t, b = (n·Σt·D − Σt·ΣD) / (n·Σt² − (Σt)²), a = D̄ − b·t̄

• t = period number, a = intercept (units), b = trend (units per period).

MAD = Σ|eₜ| / N; MSE = Σeₜ² / N; MAPE = (100/N) · Σ|eₜ| / Dₜ; TS = Σeₜ / MAD

• N = number of periods compared.

Worked examples

Example 1 (standard): three methods on the same data. Demand for periods 1–4: 100, 120, 130, 150 units. Forecast period 5 by (a) 3-period SMA, (b) WMA with weights 0.5, 0.3, 0.2 (most recent first), (c) SES with α = 0.3, starting with F(1) = D(1) = 100.

1. SMA: F(5) = (120 + 130 + 150)/3 = 400/3 = 133.3 units.
2. WMA: F(5) = 0.5 × 150 + 0.3 × 130 + 0.2 × 120 = 75 + 39 + 24 = 138 units.
3. SES: F(2) = 0.3 × 100 + 0.7 × 100 = 100; F(3) = 0.3 × 120 + 0.7 × 100 = 106; F(4) = 0.3 × 130 + 0.7 × 106 = 39 + 74.2 = 113.2; F(5) = 0.3 × 150 + 0.7 × 113.2 = 45 + 79.24 = 124.2 units.
4. SES errors for periods 2–4: 20, 24, 36.8. MAD = 80.8/3 = 26.9 units; CFE = 80.8; TS = 80.8/26.9 = 3.0. All errors are positive and growing: with a rising trend, SES lags, which is the signal to use a trend method.

Example 2 (GATE level): linear trend by least squares. Demand for periods t = 1 to 5: 40, 44, 48, 50, 58 units. Forecast period 6.

1. n = 5, Σt = 15, ΣD = 240, Σt² = 55, Σt·D = 40 + 88 + 144 + 200 + 290 = 762.
2. b = (n·ΣtD − Σt·ΣD) / (n·Σt² − (Σt)²) = (5 × 762 − 15 × 240) / (5 × 55 − 225) = (3810 − 3600)/50 = 210/50 = 4.2 units per period.
3. a = D̄ − b·t̄ = 48 − 4.2 × 3 = 35.4 units.
4. F(6) = 35.4 + 4.2 × 6 = 60.6 units.
5. Check: fitted values 39.6, 43.8, 48.0, 52.2, 56.4; residuals 0.4, 0.2, 0, −2.2, 1.6 sum to zero, as they must for least squares.

Common mistakes

• Including the period being forecast in the moving average, or using the oldest data instead of the most recent n periods.
• Putting α on the old forecast and (1 − α) on the actual demand.
• Using SMA or SES on trending data and being surprised by a persistent lag.
• Defining error as forecast − actual in one place and actual − forecast in another; the sign of bias and tracking signal depends on it.
• Taking MAD of signed errors (they cancel) instead of absolute errors.
• In regression, forgetting that a must be found after b, from the means.

For GATE ME

This topic regularly gives quick numericals: an exponential smoothing forecast over two or three periods, a moving average, the slope or forecast from a least-squares line, or MAD, MSE and tracking signal from a small table. Conceptual questions cover the effect of α and n on responsiveness and the qualitative methods. Practise chaining SES by hand without rounding too early.

Quick check

1. With α = 0.4, F(t) = 200 and D(t) = 230, what is F(t + 1)?
2. Which α gives an SES with the same average data age as a 9-period SMA?
3. Errors over four periods are 5, −3, 4, 6. What are MAD and the tracking signal?
4. Does a larger n in a moving average make it more or less responsive?
5. Name one qualitative forecasting method that uses anonymous rounds of expert opinion.

Answers: 1. 212. 2. 0.2. 3. MAD = 4.5, TS = 12/4.5 = 2.67. 4. Less responsive. 5. Delphi method.