--- title: "Forecasting by moving average and smoothing" subject: "Industrial Engineering and Management" chapter: 6 key: "semester_6/industrial_engineering_and_management:moving-average-and-exponential-smoothing" source: https://www.bctengineer.com/q/semester_6/industrial_engineering_and_management/moving-average-and-exponential-smoothing site: BCT Engineering --- # Forecasting by moving average and smoothing Industrial Engineering and Management · Chapter 6: Forecasting - Asked in: year not given, 8 marks - Video: [Numerical on Sales Estimation| 3 Months Moving Average| Weighted Moving Avg| Exponential Smoothing](https://www.youtube.com/watch?v=n-iY9eDTL44) — NV Academy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/industrial_engineering_and_management/moving-average-and-exponential-smoothing - Whole chapter: https://www.bctengineer.com/md/semester_6/industrial_engineering_and_management/chapter-6 ## Question 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. ## 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. ## Answer #### (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.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the 3-month simple moving average forecast for month 9? A. 155.3 B. 157.6 C. 150 D. 162 Answer: A. (146 + 158 + 162)/3 = 466/3 = 155.3 units. (Key point: 3-month moving average) 2. With weights 0.5, 0.3, 0.2 (largest on latest), what is the month 9 WMA forecast? A. 155.3 B. 157.6 C. 149.8 D. 160.0 Answer: B. 0.5(162) + 0.3(158) + 0.2(146) = 81.0 + 47.4 + 29.2 = 157.6. (Key point: Weighted moving average) 3. 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? A. Exp. smoothing; it reacts quickly to the trend B. 3-MA; it uses equal weights C. WMA is best; exp. smoothing with small alpha lags behind an upward trend D. All are equal Answer: C. Lowest MAD is best (WMA). Exponential smoothing with a small alpha lags a trend. (Key point: Best method by MAD) 4. How is the tracking signal defined? A. Sum of absolute errors divided by n B. Sum of squared errors divided by n C. MAD divided by the mean demand D. Running sum of forecast errors divided by MAD Answer: D. Tracking signal = sum(A - F)/MAD; values beyond about +-4 indicate bias. 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