1. Chapter 7 Demand Forecasting in a Supply Chain Forecasting -4 Adaptive Trend Adjusted Exponential Smoothing Ardavan Asef-Vaziri References: Supply Chain Management; Chopra and Meindl USC Marshall School of Business Lecture Notes

2. Ardavan Asef-Vaziri Data With Trend Trend and Seasonality: Adaptive -2  The problem: exponential smoothing (and also moving average) lags the trend.  The solution: we require another forecasting method.  Linear Regression  Double exponential smoothing

3. Ardavan Asef-Vaziri Trend Adjusted Exponential Smoothing: Holt’s Model Appropriate when there is a trend in the systematic component of demand. Trend and Seasonality: Adaptive -3 F t+1 = ( L t + T t ) = forecast for period t+1 in period t F t+l = ( L t + lT t ) = forecast for period t+l in period t L t = Estimate of level at the end of period t T t = Estimate of trend at the end of period t F t = Forecast of demand for period t (made at period t-1 or earlier) D t = Actual demand observed in period t

4. Ardavan Asef-Vaziri General Steps in adaptive Forecasting 0- Initialize: Compute initial estimates of level, L 0, trend,T 0 using linear regression on the original set of data; L 0 = b 0, T 0 = b 1. No need to remove seasonality, because there is no seasonality. 1- Forecast: Forecast demand for period t+1 using the general equation, F t+1 = L t +T t 2- Modify estimates: Modify the estimates of level, L t+1 and trend, T t+1. Repeat steps 1, 2, and 3 for each subsequent period Trend and Seasonality: Adaptive -4

5. Ardavan Asef-Vaziri Trend-Corrected Exponential Smoothing (Holt’s Model) In period t, the forecast for future periods is expressed as follows F t+1 = L t + T t F t+l = L t + lT t F 1 = L 0 + T 0 What about F 2 ? Trend and Seasonality: Adaptive -5 L t+1 =  D t+1 + (1-  ) (L t + T t ) T t+1 =  ( L t+1 – L t ) + (1-  ) T t  = smoothing constant for level  = smoothing constant for trend

6. Ardavan Asef-VaziriTrend and Seasonality: Adaptive -6 Holt’s Model Example (continued) Using linear regression on the original set of data, L 0 = 12015 (linear intercept) T 0 = 1549 (linear slope) Example: Tahoe Salt demand data. Forecast demand for period 1 using Holt’s model (trend corrected exponential smoothing)

7. Ardavan Asef-Vaziri Holt’s Model Example (continued) Forecast for period 1: F1 = L0 + T0 = 12015 + 1549 = 13564 Observed demand for period 1 = D1 = 8000 E1 = F1 - D1 = 13564 - 8000 = 5564 Assume  = 0.1,  = 0.2 L1 =  D1 + (1-  )(L0+T0) = (0.1)(8000) + (0.9)(13564) = 13008 T1 =  (L1 - L0) + (1-  )T0 = (0.2)(13008 - 12015) + (0.8)(1549) = 1438 F2 = L1 + T1 = 13008 + 1438 = 14446 Trend and Seasonality: Adaptive -7

8. Ardavan Asef-Vaziri Holt’s Model Example (continued) Trend and Seasonality: Adaptive -8 F13 = L12 + T12 = 30445 + 1542 = 31987 F18 = L12 + 5T12 = 30445 + 7710 = 38155

9. Ardavan Asef-Vaziri Example : L0 = 100, T0 = 10,  = 0.2 and  = 0.3 Trend and Seasonality: Adaptive -9 L t+1 =  D t+1 + (1-  ) (L t + T t ) T t+1 =  ( L t+1 – L t ) + (1-  ) T t L 1 =  D 1 + 0.8 (L 0 + T 0 ) T 1 =  ( L 1 – L 0 ) + 0.7 T 0 L 1 =  (115) + 0.8 (110) = 111 T 1 =  ( 111-100 ) + 0.7 (10) = 10.3 L 0 = 100, T 0 = 10 F 1 = L 0 + T 0 = 100 +10 =110 D 1 =115

10. Ardavan Asef-Vaziri Double Exponential Smoothing:  = 0.2 and  = 0.3 Trend and Seasonality: Adaptive -10 L 2 =  D 2 + 0.8 (L 1 + T 1 ) T 2 =  ( L 2 – L 1 ) + 0.7 T 1 L 2 =  (125) + 0.8 (121.3) = 122.04 T 2 =  ( 122.04-111 ) + 0.7 (10.3) = 10.52 F 3 = L 2 + T 2 = 122.04 +10.52 =132.56 L 1 = 111, T 1 = 10.3 F 2 = L 1 + T 1 = 111 +10.3 =121.3 D 2 =125

11. Ardavan Asef-Vaziri Varying Trend Example Trend and Seasonality: Adaptive -11

12. Ardavan Asef-Vaziri Varying Trend Example Trend and Seasonality: Adaptive -12

13. Ardavan Asef-Vaziri Double Exponential Smoothing Trend and Seasonality: Adaptive -13  Basic idea - introduce a trend estimator that changes over time  Similar to single exponential smoothing  If the underlying trend changes, over-shoots may happen  Issues to choose two smoothing rates,  and    close to 1 means quicker responses to trend changes, but may over-respond to random fluctuations   close to 1 means quicker responses to level changes, but again may over-respond to random fluctuations