MAD over months 4-8 is 12.0 (3-MA), 10.4 (WMA), 16.6 (exp. smoothing). Which is most accurate and why is exp. smoothing worst?

WMA is best; exp. smoothing with small alpha lags behind an upward trend

Exp. smoothing; it reacts quickly to the trend

How is the tracking signal defined?

Running sum of forecast errors divided by MAD

Sum of absolute errors divided by n

Sum of squared errors divided by n

Practice · 8 marks

Forecasting by moving average and smoothing

Short answer

Month 9 forecast: 155.3 (3-MA), 157.6 (wtd MA), 149.8 (exp. smoothing α=0.3). MAD over months 4–8: 12.0, 10.4, 16.6; weighted MA most accurate. MAD is mean absolute deviation, MSE is mean squared error, tracking signal is cumulative error divided by MAD, flags bias if outside ±4.

The monthly demand (units) of a product for the last 8 months is: 120, 132, 128, 140, 150, 146, 158, 162. (a) Forecast the demand for month 9 using a 3-month simple moving average, a 3-month weighted moving average with weights 0.5, 0.3, 0.2 (largest weight on the latest month) and exponential smoothing with alpha = 0.3 (forecast for month 1 = 120). (b) Compare the accuracy of the three methods over months 4 to 8 using mean absolute deviation (MAD). (c) Define MAD, MSE and tracking signal.

(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 $F_1 = 120$ :

Month	Demand	Forecast	Error
1	120	120.00	0.00
2	132	120.00	12.00
3	128	123.60	4.40
4	140	124.92	15.08
5	150	129.44	20.56
6	146	135.61	10.39
7	158	138.73	19.27
8	162	144.51	17.49

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

(b) MAD over months 4 to 8

Month	Demand	3-MA	Error	WMA	Error
4	140	126.7	13.3	127.6	12.4
5	150	133.3	16.7	134.8	15.2
6	146	139.3	6.7	142.6	3.4
7	158	145.3	12.7	146.0	12.0
8	162	151.3	10.7	152.8	9.2

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.

✓ Answer checked by Claude Opus 5.5 (high effort)

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Forecasting by moving average and smoothing

(a) Forecasts for month 9

(b) MAD over months 4 to 8

(c) Forecast error measures

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