1. **Problem Statement:** We have monthly demand data for May to September and need to forecast demand for October, November, and December using three methods: 5-month Moving Average, 3-month Weighted Moving Average, and Exponential Smoothing. --- 2. **Given Data:** | Month | Demand | |------------|--------| | May (t=1) | 435 | | June | 385 | | July | 297 | | August | 250 | | September | 316 | --- 3. **Method 1: 5-month Moving Average** - The 5-month moving average forecast for October uses demands from May to September. Calculating October's forecast: $$ \text{Forecast}_{Oct} = \frac{435 + 385 + 297 + 250 + 316}{5} = \frac{1683}{5} = 336.6 $$ For November, use June to October (we need October's actual demand, which we do not have), so we cannot calculate November or December forecasts by 5-month moving average yet. Thus, only October forecast is computable. --- 4. **Method 2: 3-month Weighted Moving Average** Weights: - Most recent (weight 0.50) - Next most recent (weight 0.20) - Oldest of the three (weight 0.30) For October forecast, the most recent three months are September, August, and July demands: $$ \text{Forecast}_{Oct} = (0.50 \times 316) + (0.20 \times 250) + (0.30 \times 297) $$ Calculating: $$ = 158 + 50 + 89.1 = 297.1 $$ For November forecast, the three most recent demand months would be October (unknown), September, August, so we cannot forecast November without October's actual demand. Similarly, December forecast requires November or October demands, which are unknown. --- 5. **Method 3: Exponential Smoothing for October** Given: - Smoothing constant $\alpha = 0.2$ - Forecast for May $F_{May} = 475$ - Actual demand for May $A_{May} = 435$ Formula: $$ F_{t+1} = \alpha A_t + (1 - \alpha) F_t $$ Step-by-step: - Calculate forecast for June: $$ F_{June} = 0.2 \times 435 + 0.8 \times 475 = 87 + 380 = 467 $$ - Forecast for July: $$ F_{July} = 0.2 \times 385 + 0.8 \times 467 = 77 + 373.6 = 450.6 $$ - Forecast for August: $$ F_{August} = 0.2 \times 297 + 0.8 \times 450.6 = 59.4 + 360.48 = 419.88 $$ - Forecast for September: $$ F_{Sept} = 0.2 \times 250 + 0.8 \times 419.88 = 50 + 335.9 = 385.9 $$ - Forecast for October: $$ F_{Oct} = 0.2 \times 316 + 0.8 \times 385.9 = 63.2 + 308.72 = 371.92 $$ --- **Final answers:** - 5-month Moving Average: October forecast = $336.6$ - 3-month Weighted Moving Average: October forecast = $297.1$ - Exponential Smoothing (\(\alpha=0.2\)): October forecast = $371.92$