Forecasting 4 Trend Adjusted Exponential Smoothing Ardavan AsefVaziri

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

Data With Trend Problem: Exponential smoothing (and also moving average) lags the trend. Solution: We need a trend included forecasting method. § Linear Regression § Trend Adjusted (Double) exponential smoothing Ardavan Asef-Vaziri Trend and Seasonality: Adaptive -2

Trend Adjusted Exponential Smoothing: Holt’s Model Appropriate when there is a trend in the systematic component of demand. Ft+1 = ( Lt + Tt ) = forecast for period t+1 in period t Ft+l = ( Lt + l. Tt ) = forecast for period t+l in period t Lt = Estimate of level at the end of period t Tt = Estimate of trend at the end of period t Ft = Forecast of demand for period t At = Actual demand observed in period t Ardavan Asef-Vaziri Trend and Seasonality: Adaptive -3

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, Ft+1 = Lt+Tt. 2 - Modify estimates: Modify the estimates of level, Lt+1 and trend, Tt+1. Repeat steps 1, 2, and 3 for each subsequent period. Ardavan Asef-Vaziri Trend and Seasonality: Adaptive -4

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

Example : L 0 = 100, T 0 = 10, a = 0. 2 and b = 0. 3 L 0 = 100, T 0 = 10 F 1 = L 0 + T 0 = 100 +10 =110 A 1 =115 Lt = a At + (1 -a) Ft L 1 = 0. 2 A 1 + 0. 8 F 1 L 1 = 0. 2 (115) + 0. 8 (110) = 111 Tt = ( Lt – Lt-1 ) + (1 - ) Tt-1 T 1 = 0. 3( L 1 – L 0 ) + 0. 7 T 0 T 1 = 0. 3( 111 -100 ) + 0. 7 (10) = 10. 3 Ardavan Asef-Vaziri Trend and Seasonality: Adaptive -6

Example : L 0 = 100, T 0 = 10, a = 0. 2 and b = 0. 3 L 1 = 111, T 1 = 10. 3 F 2 = L 1 + T 1 = 111 +10. 3 =121. 3 A 2 =125 Lt = a At + (1 -a) Ft L 2 = 0. 2 A 2 + 0. 8 F 2 L 2 = 0. 2 (125) + 0. 8 (121. 3) = 122 Tt = ( Lt – Lt-1 ) + (1 - ) Tt-1 T 2 = 0. 3( L 2 – L 1 ) + 0. 7 T 1 T 2 = 0. 3( 122 -111 ) + 0. 7(10. 3) = 10. 5 F 3 = L 2 + T 2 = 122 +10. 5 =132. 5 Ardavan Asef-Vaziri Trend and Seasonality: Adaptive -7

Holt’s Model Example 2 Tahoe Salt demand data. Forecast demand for period 1 using Holt’s model (trend adjusted exponential smoothing). Assume a = 0. 1, b = 0. 2. Using linear regression on the original set of data, L 0 = 12015 (linear intercept) T 0 = 1549 (linear slope) Ardavan Asef-Vaziri Trend and Seasonality: Adaptive -8

Holt’s Model Example (continued) Forecast for period 1: F 1 = L 0 + T 0 = 12015 + 1549 = 13564 A 1 = 8000 L 1 = a. A 1 + (1 -a)F 1 = (0. 1)(8000) + (0. 9)(13564) = 13008 T 1 = b(L 1 - L 0) + (1 -b)T 0 = (0. 2)(13008 - 12015) + (0. 8)(1549) = 1438 F 2 = L 1 + T 1 = 13008 + 1438 = 14446 A 2 = 13000 L 2 = a. A 2 + (1 -a)F 2 = (0. 1)(13000) + (0. 9)(14446) = 14301 T 2 = b(L 2 – L 1) + (1 -b)T 1 = (0. 2)(14301 - 13008) + (0. 8)(1438) = 1409 F 3 = L 2 + T 2 = 14301+ 1409 = 15710 Ardavan Asef-Vaziri Trend and Seasonality: Adaptive -9

Holt’s Model Example (continued) F 13 = L 12 + T 12 = 30445 + 1542 = 31987 F 18 = L 12 + 5 T 12 = 30445 + 7710 = 38155 Ardavan Asef-Vaziri Trend and Seasonality: Adaptive -10

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

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

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