chapter 7 demand forecasting in a supply chain
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Forecasting -4 Adaptive Trend Adjusted Exponential Smoothing Ardavan Asef-Vaziri References: Supply Chain Management; Chopra and Meindl USC Marshall School of Business Lecture Notes. Chapter 7 Demand Forecasting in a Supply Chain. Data With Trend. - PowerPoint PPT PresentationTRANSCRIPT
Chapter 7Demand Forecastingin a Supply Chain
Forecasting -4Adaptive Trend Adjusted Exponential
Smoothing
Ardavan Asef-Vaziri
References: Supply Chain Management; Chopra and MeindlUSC Marshall School of Business Lecture Notes
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
Forecast (α=0.2, α=0.5)Period Demand 0.2 0.5
1 12 2 1.0 1.03 3 1.2 1.54 4 1.6 2.15 5 2.0 2.86 6 2.6 3.57 7 3.3 4.38 8 4.0 5.29 9 4.8 6.0
10 10 5.7 6.911 11 6.5 7.812 12 7.4 8.8
Ardavan Asef-Vaziri
Trend Adjusted Exponential Smoothing: Holt’s ModelAppropriate when there is a trend in the systematic
component of demand.
Trend and Seasonality: Adaptive -3
Ft+1 = ( Lt + Tt ) = forecast for period t+1 in period t Ft+l = ( Lt + lTt ) = 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 (made at
period t-1 or earlier)Dt = Actual demand observed in period t
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General Steps in adaptive Forecasting0- Initialize: Compute initial estimates of level, L0,
trend ,T0 using linear regression on the original set of data; L0= b0 , T0 = b1. 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
Trend and Seasonality: Adaptive -4
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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 + lTt
F1 = L0 + T0
What about F2 ?
Trend and Seasonality: Adaptive -5
Lt+1 = a Dt+1 + (1-a) (Lt + Tt)Tt+1 = b ( Lt+1 – Lt ) + (1-b) Tt
a = smoothing constant for level b = smoothing constant for trend
Ardavan Asef-Vaziri Trend and Seasonality: Adaptive -6
Holt’s Model Example (continued)
t Dt
1 80002 130003 230004 340005 100006 180007 230008 380009 1200010 1300011 3200012 41000
Multiple R 0.48R Square 0.23Adjusted R Square 0.15Standard Error 10666.88Observations 12
ANOVAdf SS MS F Significance F
Regression 1 343092657 343092657 3.02 0.11Residual 10 1137824009 113782401
Total 11 1480916667
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 12015 6565 1.83 0.10 (2612.61) 26642.91X Variable 1 1549 892 1.74 0.11 (438.57) 3536.47
Using linear regression on the original set of data,L0 = 12015 (linear intercept)T0 = 1549 (linear slope)
Example: Tahoe Salt demand data. Forecast demand for period 1 using Holt’s model (trend corrected exponential smoothing)
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Holt’s Model Example (continued)Forecast for period 1:F1 = L0 + T0 = 12015 + 1549 = 13564Observed demand for period 1 = D1 = 8000E1 = F1 - D1 = 13564 - 8000 = 5564Assume a = 0.1, b = 0.2L1 = aD1 + (1-a)(L0+T0) = (0.1)(8000) + (0.9)(13564) = 13008T1 = b(L1 - L0) + (1-b)T0 = (0.2)(13008 - 12015) + (0.8)(1549) = 1438F2 = L1 + T1 = 13008 + 1438 = 14446
Trend and Seasonality: Adaptive -7
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Holt’s Model Example (continued)t Dt Lt Tt Ft
12015 15491 8000 13008 1438 135642 13000 14301 1409 144463 23000 16439 1555 157104 34000 19595 1875 179945 10000 20323 1646 214706 18000 21572 1567 219697 23000 23125 1564 231398 38000 26020 1830 246899 12000 26265 1513 27850
10 13000 26300 1217 2777811 32000 27965 1307 2751712 41000 30445 1542 29272
ttlt
ttt
tttt
tttt
lTLF
TLF
TLLTTLDL
TLF
1
11
11
001
))(1()())(1(
bbaa
Alpha= 0.1 Beta= 0.2
Trend and Seasonality: Adaptive -8
F13 = L12 + T12 = 30445 + 1542 = 31987F18 = L12 + 5T12 = 30445 + 7710 = 38155
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Example : L0 = 100, T0 = 10, a = 0.2 and b = 0.3
Trend and Seasonality: Adaptive -9
Lt+1 = a Dt+1 + (1-a) (Lt + Tt)Tt+1 = b ( Lt+1 – Lt ) + (1-b) Tt
L1 = 0.2 D1 + 0.8 (L0 + T0)T1 = 0.3( L1 – L0 ) + 0.7 T0
L1 = 0.2 (115) + 0.8 (110) = 111T1 = 0.3( 111-100 ) + 0.7 (10) = 10.3
L0 = 100, T0 = 10F1 = L0 + T0 = 100 +10 =110D1 =115
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Double Exponential Smoothing: a = 0.2 and b = 0.3
Trend and Seasonality: Adaptive -10
L2 = 0.2 D2 + 0.8 (L1 + T1)T2 = 0.3( L2 – L1 ) + 0.7 T1
L2 = 0.2 (125) + 0.8 (121.3) = 122.04T2 = 0.3( 122.04-111 ) + 0.7 (10.3) = 10.52F3 = L2 + T2 = 122.04 +10.52 =132.56
L1 = 111, T1 = 10.3F2 = L1 + T1 = 111 +10.3 =121.3D2 =125
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Varying Trend Example
Trend and Seasonality: Adaptive -11
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0
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61 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101
Trend
Series
Ardavan Asef-Vaziri
Varying Trend Example
Trend and Seasonality: Adaptive -12
-1
0
1
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61 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101
Trend
Series Data
Single Smoothing
Double smoothing-1
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1 6
11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
101
TrendSeries DataSingle Smoothing
Double smoothing
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, a and b.
b 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