1. **Problem Statement:** Forecast the number of participating households for May 2024 using various methods based on data from July 2023 to April 2024. 2. **Given Data:** | Month | Households | |-------|------------| | Jul-23 | 420 | | Aug-23 | 430 | | Sep-23 | 460 | | Oct-23 | 465 | | Nov-23 | 470 | | Dec-23 | 480 | | Jan-24 | 475 | | Feb-24 | 495 | | Mar-24 | 500 | | Apr-24 | 510 | 3. **Methods and Formulas:** - One-month Moving Average (Naive): $F_{t+1} = A_t$ - Three-month Moving Average: $F_{t+1} = \frac{A_t + A_{t-1} + A_{t-2}}{3}$ - Four-month Moving Average: $F_{t+1} = \frac{A_t + A_{t-1} + A_{t-2} + A_{t-3}}{4}$ - Weighted Average: $F_{t+1} = \frac{3A_t + 2A_{t-1} + 1A_{t-2}}{3+2+1}$ - Exponential Smoothing: $F_{t+1} = \alpha A_t + (1-\alpha)F_t$ - Linear Regression: $Y = a + bX$ where $X$ is time index - MAD: $\text{MAD} = \frac{1}{n} \sum |A_t - F_t|$ 4. **Step-by-step Calculations:** **1) One-month Moving Average (Naive):** Forecast for May 2024 is the actual value of April 2024: $$F_{May} = A_{Apr} = 510$$ **2) Three-month Moving Average:** Use Feb, Mar, Apr data: $$F_{May} = \frac{495 + 500 + 510}{3} = \frac{1505}{3} = 501.67$$ **3) Four-month Moving Average:** Use Jan, Feb, Mar, Apr data: $$F_{May} = \frac{475 + 495 + 500 + 510}{4} = \frac{1980}{4} = 495$$ **4) Weighted Average (weights 3,2,1):** Use Apr, Mar, Feb data: $$F_{May} = \frac{3 \times 510 + 2 \times 500 + 1 \times 495}{6} = \frac{1530 + 1000 + 495}{6} = \frac{3025}{6} = 504.17$$ **5) Exponential Smoothing:** Initialize $F_{Jul} = A_{Jul} = 420$ Calculate forecasts for Aug to Apr: - For $\alpha=0.5$: $$F_{Aug} = 0.5 \times 430 + 0.5 \times 420 = 425$$ $$F_{Sep} = 0.5 \times 460 + 0.5 \times 425 = 442.5$$ $$F_{Oct} = 0.5 \times 465 + 0.5 \times 442.5 = 453.75$$ $$F_{Nov} = 0.5 \times 470 + 0.5 \times 453.75 = 461.88$$ $$F_{Dec} = 0.5 \times 480 + 0.5 \times 461.88 = 470.94$$ $$F_{Jan} = 0.5 \times 475 + 0.5 \times 470.94 = 472.97$$ $$F_{Feb} = 0.5 \times 495 + 0.5 \times 472.97 = 483.99$$ $$F_{Mar} = 0.5 \times 500 + 0.5 \times 483.99 = 491.99$$ $$F_{Apr} = 0.5 \times 510 + 0.5 \times 491.99 = 500.99$$ Forecast for May: $$F_{May} = 0.5 \times 510 + 0.5 \times 500.99 = 505.50$$ - For $\alpha=0.3$: $$F_{Aug} = 0.3 \times 430 + 0.7 \times 420 = 423$$ $$F_{Sep} = 0.3 \times 460 + 0.7 \times 423 = 436.1$$ $$F_{Oct} = 0.3 \times 465 + 0.7 \times 436.1 = 445.27$$ $$F_{Nov} = 0.3 \times 470 + 0.7 \times 445.27 = 453.69$$ $$F_{Dec} = 0.3 \times 480 + 0.7 \times 453.69 = 463.58$$ $$F_{Jan} = 0.3 \times 475 + 0.7 \times 463.58 = 467.51$$ $$F_{Feb} = 0.3 \times 495 + 0.7 \times 467.51 = 474.26$$ $$F_{Mar} = 0.3 \times 500 + 0.7 \times 474.26 = 481.98$$ $$F_{Apr} = 0.3 \times 510 + 0.7 \times 481.98 = 491.39$$ Forecast for May: $$F_{May} = 0.3 \times 510 + 0.7 \times 491.39 = 496.97$$ **6) Linear Regression (Trend Line):** Assign $X=1$ for Jul-23, $X=10$ for Apr-24. Data points: $(X_i, Y_i)$ for $i=1$ to $10$. Calculate: $$\bar{X} = \frac{1+2+...+10}{10} = 5.5$$ $$\bar{Y} = \frac{420+430+460+465+470+480+475+495+500+510}{10} = 464.5$$ Calculate slope $b$: $$b = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2}$$ Calculate numerator: $$(1-5.5)(420-464.5) + (2-5.5)(430-464.5) + ... + (10-5.5)(510-464.5) = 1392.5$$ Calculate denominator: $$(1-5.5)^2 + (2-5.5)^2 + ... + (10-5.5)^2 = 82.5$$ So, $$b = \frac{1392.5}{82.5} = 16.88$$ Calculate intercept $a$: $$a = \bar{Y} - b \bar{X} = 464.5 - 16.88 \times 5.5 = 464.5 - 92.84 = 371.66$$ Forecast for May ($X=11$): $$F_{May} = a + b \times 11 = 371.66 + 16.88 \times 11 = 371.66 + 185.68 = 557.34$$ **7) MAD for Exponential Smoothing:** Calculate absolute deviations for $\alpha=0.5$ and $\alpha=0.3$ from Aug to Apr. - For $\alpha=0.5$: $$|430-425| + |460-442.5| + |465-453.75| + |470-461.88| + |480-470.94| + |475-472.97| + |495-483.99| + |500-491.99| + |510-500.99| = 5 + 17.5 + 11.25 + 8.12 + 9.06 + 2.03 + 11.01 + 8.01 + 9.01 = 81.99$$ MAD: $$\frac{81.99}{9} = 9.11$$ - For $\alpha=0.3$: $$|430-423| + |460-436.1| + |465-445.27| + |470-453.69| + |480-463.58| + |475-467.51| + |495-474.26| + |500-481.98| + |510-491.39| = 7 + 23.9 + 19.73 + 16.31 + 16.42 + 7.49 + 20.74 + 18.02 + 18.61 = 147.22$$ MAD: $$\frac{147.22}{9} = 16.36$$ **Summary of Forecasts for May 2024:** - One-month Moving Average: 510 - Three-month Moving Average: 501.67 - Four-month Moving Average: 495 - Weighted Average: 504.17 - Exponential Smoothing ($\alpha=0.5$): 505.50 - Exponential Smoothing ($\alpha=0.3$): 496.97 - Linear Regression: 557.34 **MAD Comparison:** - $\alpha=0.5$ MAD = 9.11 (better accuracy) - $\alpha=0.3$ MAD = 16.36 Hence, exponential smoothing with $\alpha=0.5$ is more accurate based on MAD.