chapter 7 demand forecasting in a supply chain forecasting -5 adaptive trend and seasonality...

Chapter 7Demand Forecastingin a Supply Chain

Forecasting -5Adaptive Trend and Seasonality Adjusted Exponential Smoothing

Ardavan Asef-Vaziri

References: Supply Chain Management; Chopra and MeindlUSC Marshall School of Business Lecture Notes

Ardavan Asef-Vaziri

Monthly US Electric Power Consumption

Trend and Seasonality: Adaptive -2

14

0.0

17

5.0

21

0.0

24

5.0

28

0.0

1 40 79 118 157

Plot of Power

Time

Po

we

r

Ardavan Asef-Vaziri

Trend and Seasonality


200.0

400.0

600.0

800.0

1000.0

1994.9 1996.6 1998.4 2000.1 2001.9

sales Forecast Plot

Time

sa

les

Ardavan Asef-Vaziri

Trend & Seasonality-Corrected Exponential Smoothing


The estimates of level, trend, and seasonality are adjusted after each demand observation. Assume periodicity p

Ft+1 = ( Lt + Tt )St+1 = forecast for period t+1 in period t

Ft+l = ( Lt + lTt )St+l = 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

St = Estimate of seasonal factor for period t

Ft = Forecast of demand for period t (made at period t-1 or earlier)

Dt = Actual demand observed in period t

Ardavan Asef-Vaziri

General Steps in Adaptive Forecasting

0- Initialize: Compute initial estimates of level, L0, trend ,T0, and seasonal factors, S1,…,Sp. As in static forecasting.

1- Forecast: Forecast demand for period t+1 using the general equation, Ft+1 = (Lt+Tt )×St+1

2- Estimate error: Compute error Et+1 = Ft+1- Dt+1

3- Modify estimates: Modify the estimates of level, Lt+1, trend, Tt+1, and seasonal factor, St+p+1, given the error Et+1 in the forecast

Repeat steps 1, 2, and 3 for each subsequent period


Ardavan Asef-Vaziri 7-2-6

After observing demand for period t+1, revise estimates for level, trend, and seasonal factors as follows:

Lt+1 = a(Dt+1/St+1) + (1-a)(Lt+Tt)

Tt+1 = b(Lt+1 - Lt) + (1-b)Tt

St+p+1 = g(Dt+1/Lt+1) + (1-g)St+1

a = smoothing constant for level

b = smoothing constant for trend

g = smoothing constant for seasonal factor




t Dt 4PAverage1 80002 130003 23000 197504 34000 206255 10000 212506 18000 217507 23000 225008 38000 221259 12000 22625

10 13000 2412511 3200012 41000

Regression StatisticsMultiple R $0.96R Square $0.92 L0= 18439Adjusted R Square $0.90 T0= 524Standard Error $414.50Observations 8

ANOVAdf SS MS F Significance F

Regression 1 11523810 11523810 67 0.0002Residual 6 1030878 171813Total 7 12554688

CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Intercept 18439 441 41.83 1.2E-08 17360.37 19517.61X Variable 1 524 64 8.19 1.8E-04 367.31 680.31

1 0.472 0.683 1.174 1.66

Example: Tahoe Salt data. Forecast demand for period 1 using Winter’s model. Initial estimates of level, trend, and seasonal factors are obtained as in the static forecasting case

L0 = 18439 T0 = 524 S1=0.47, S2=0.68, S3=1.17, S4=1.66

F1 = (L0 + T0)S1 = (18439+524)(0.47) = 18963(0.47)= 8913

The observed demand for period 1 = D1 = 8000.

Assume a = 0.1, b=0.2, g=0.1


L1 = a(Actual Surrogate) + (1-a)(Forecast Surrogate)

Forecast Surrogate for L1 = L0+T0

Actual Surrogate for L1 = D1/S1

L1 = a(D1/S1) + (1-a)(L0+T0)

L1 = 0.1(D1/S1) + 0.9(L0+T0)

L1 =(0.1)(8000/0.47)+(0.9)(18439+524)=18769

T1 = b(Actual Surrogate) + (1-b)(Forecast Surrogate)

Forecast Surrogate for T1 = T0

Actual Surrogate for T1 = D1-D0

T1 = 0.2(L2-L1) + 0.8(T0)

T1 = (0.2)(18769-18439)+(0.8)(524) = 485



S5 = g(Actual Surrogate) + (1-g)(Forecast Surrogate)

Forecast Surrogate for S5 = S1

Actual Surrogate for S5 = D1/L1

S5 = g (D1/L1) + (1-g)(S1)

S5 = 0.1 (D1/L1) + 0.9(S1)

S5 = (0.1)(8000/18769)+(0.9)(0.47) = 0.47

F2 = (L1+T1)S2 = (18769 + 485)(0.68) = 13093



L1 = 18769, T1 = 485, S2 = 0.68, D2 = 13000.

L2 = 0.1(D2/S2) + 0.9(L1+T1)

D2/S2 = 13000/0.68 = 19118

L1+T1 = 18769+485 = 19254

L2 = 0.1(19118) + 0.9(19254) = 19240

T2 = 0.2(L2-L1) + 0.8(T1)

T1 = (0.2)(19240-18769)+(0.8)(485) = 482

S5 = g(Actual Surrogate) + (1-g)(Forecast Surrogate)

S6 = 0.1 (D2/L2) + 0.9(S2)

S5 = (0.1)(13000/19240)+(0.9)(0.68) = 0.68

F3 = (L2+T2)S3 = (19240 + 482)(0.68) = 13411



Forecasting in Practice

Collaborate in building forecasts The value of data depends on where you are in the

supply chain Be sure to distinguish between demand and sales

Ardavan Asef-Vaziri

Practice: Given L0 = 11, T0 = 1, S1 to S4 =0.5,1.0,1.5,1.0


Quarter Demand

Forecast

Level Trend Seasonal

0 11 1

1 6 6 0.5

2 1.0

3 1.5

4 1.0

5

Forecast 1 = (11+1)*0.5

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L1, T1, F2, S5


Quarter Demand

Forecast


0 11 1

1 6 6 12 1 0.5

2 13 1.0

3 1.5

4 1.0

5 0.5New level = 0.25(6/0.5)+0.75(11+1)=12New trend = 0.25(12-11)+0.75(1)=1New seasonal = 0.25(6/12)+0.75(0.5)=0.5New Forecast = (12+1)*1=13

Ardavan Asef-Vaziri

L2, T2, F3, S6


Quarter Demand

Forecast


0 11 1

1 6 6 12 1 0.5

2 14 13 1.0

3 1.5

4 1.0

5 0.5New level = 0.25(14/1)+0.75*(12+1)=13.25New trend = 0.25(13.25-12)+0.75(1)=1.06New seasonal = 0.25(14/13.25)+0.75*1=1.014New Forecast = (13.25+1.06)*1.5=21.45


Practice: α = 0.05, β = 0.1, δ = 0.1

1111

11

111

1001

)1()/(

))(1()(

))(1()/(

)(

tttpt

tttt

ttttt

SLDS

TLLT

TLSDL

STLF

t Dt Lt Tt St Ft

18439 5241 8000 18863 514 0.47 89442 13000 19058 482 0.68 132423 23000 19713 499 1.17 228764 34000 20901 568 1.66 336415 10000 20896 511 0.47 214696 18000 21237 494 0.68 214077 23000 21794 500 1.17 217318 38000 23079 579 1.66 222949 12000 23075 521 0.47 23658

10 13000 23066 468 0.70 2359611 32000 23957 510 1.16 2353412 41000 25294 593 1.66 24467

Alpha= 0.05 Beta= 0.1 Gamma= 0.1

Ardavan Asef-Vaziri

Assignment


0

10

20

30

0 2 4 6 8 10 12 14

Demand

Each cycle is 4 periods long. Periodicity = 4. There are three cycles. Compute b0, b1, S1, S2, S3, S4 using

static method and forecast using trend and seasonality adjusted method for α= β = δ = 0.25

Quarter Demand 1 42 93 154 115 66 147 238 169 810 1711 2712 19

Ardavan Asef-Vaziri

Using Static Model We Can Compute Seasonality


Quarter Demand Average Regression SeasIndex1 4 8.11 0.493392072 9 9.18 0.980544747

3 15 10.00 10.25 1.4634146344 11 10.88 11.32 0.971608833

5 6 12.50 12.39 0.4841498566 14 14.13 13.46 1.0397877987 23 15.00 14.54 1.5823095828 16 15.63 15.61 1.025171625

9 8 16.50 16.68 0.47965738810 17 17.38 17.75 0.957746479

11 27 18.82 1.43453510412 19 19.89 0.955116697

b0= 7.04 0.5b1= 1.07 1

1.51

b0 (Level) and b1 (Trend) are computed exactly the same as static method by applying regression on deseasonalized data.

Initial average seasonality indices are also computed in the same way.

Ardavan Asef-Vaziri

Practice; α=β= γ = 0.25


t Dt Ft Lt Tt St0 7.04 1.071 4 0.5

)5.0)(07.104.7(

)(

)(

1

1001

11

F

STLF

STLF tttt t Dt Ft Lt Tt St

1 4 4.055 0.52 9 1

L1 = α(D1/S1)+(1-α)(L0+T0)= 8.08T1 = β(L1-L0)+(1- β)T0= 1.06S5=* (D1/L1)+(1-γ)S1= 0.50

F2=(L1+T1)S2= 9.15

t Dt Ft Lt Tt St

1 4 4.06 8.08 1.06 0.52 9 9.15 13 15 1.54 115 6 0.50

Ardavan Asef-Vaziri

Practice; α=β= γ = 0.25


L2 = α(D2/S2)+(1-α)(L1+T1)= 9.11T2= β(L2-L1)+(1- β)T1= 1.05

S6=* (D2/L2)+(1-γ)S2= 1.00F3=(L2+T2)S3= 15.24

t Dt Ft Lt Tt St

1 4 5.02 9.53 0.51 0.52 9 10.04 9.11 1.05 13 15 15.24 1.54 11 1

L3= α(D3/S3)+(1-α)(L2+T2)= 10.12T3= β(L3-L2)+(1- β)T2= 1.04

S7=* (D3/L3)+(1-γ)S3= 1.50F4=(L3+T3)S4= 11.17

1 4 5.02 9.53 0.51 0.52 9 10.04 9.78 0.45 13 15 15.34 10.12 1.04 1.54 11 11.17 15 6 0.48

L4= α(D4/S4)+(1-α)(L3+T3)= 11.12T4= β(L4-L3)+(1- β)T3= 1.03

S8=* (D4/L4)+(1-γ)S4= 1.00F5=(L4+T4)S5= 5.84

chapter 7 demand forecasting in a supply chain forecasting -5 adaptive trend and seasonality...

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