Dennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa. Exponential Smoothing 02/13/03 page 1 of 38
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1 demand Dennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa Exponential Smoothing 02/13/03 page 1 of 38
2 As with other Time-series forecasting methods, We are given: historical demands (observations) for the past N periods: D, D, D, D, D N 1 N 2 N 1 0 (where now = time of most recent demand = period #0) We wish to forecast: demands for m future periods: D 1, D 2, D m Exponential Smoothing 02/13/03 page 2 of 38
3 (Single) Exponential Smoothing simpler, & requires less storage of data than moving averages parameter: α where 0 < α < 1 new forecast (& MAD--mean absolute deviation) are a mixture of the old forecast (or MAD) and the new observation: ( 1 ) ( 1 ) D 1 = α D 0 +αd0 MAD = α MAD +α D D (new forecast) = (1 α) (old forecast) + α (new demand) (new MAD) = (1 α) (old MAD) + α new error Exponential Smoothing 02/13/03 page 3 of 38
4 approximates weighted moving averages with all past demands being included in the average: D =α D + α D ( 1 ) ( 1 ) ( 1 ) =α D + α α D + α D ( ) ( 1 ) ( 1 ) 2 =α D + α α D + α D ( 1 ) ( 1 ) ( 1 ) =α D + α α D + α α D + α D ( ) ( 1 ) ( 1 ) ( 1 ) 2 3 =α D + α α D + α α D + α αd ( 1 ) ( 1 ) ( 1 ) ( 1 ) =α D + α α D + α α D + α α D + + α αd n n Exponential Smoothing 02/13/03 page 4 of 38
5 Exponential smoothing as a weighted moving average: ( 1 ) ( 1 ) 2 ( 1 ) 3 ( 1 ) n D1 =α D0 + α α D 1+ α α D 2 + α α D α αd n For example, if α = 0.2, the weights are Period etc. Weight Exponential Smoothing 02/13/03 page 5 of 38
6 The larger the value of α, the more weight is given to more recent demand data, & the more sensitive the forecast to changes in the demand pattern. Exponential Smoothing 02/13/03 page 6 of 38
7 Exponential Smoothing ( α = 1/4) D = 0.75 D D To begin, we need an initial forecast-- for this we will use a simple average of past actual demands. Exponential Smoothing 02/13/03 page 7 of 38
8 Exponential Smoothing We use an initial value of D = = and an initial MAD (mean absolute deviation) of =17. Period Demand Forecast ? We advance one period, update the calendar, and record an actual demand of 137. D 1 = = MAD 1 = = Exponential Smoothing 02/13/03 page 8 of 38
9 Exponential Smoothing Period Demand Forecast ? We again advance one period, update the calendar, and record an actual demand of 162. D 1 = = MAD = 0.75 MAD D D = = Exponential Smoothing 02/13/03 page 9 of 38
10 Period Demand Forecast Error Absolute Total Average D = 0.75 D D = = Exponential Smoothing 02/13/03 page 10 of 38
11 Exponential Smoothing 02/13/03 page 11 of 38
12 Period Demand We take the previous data, with some changes: The outliers in periods -13 & -16 were data entry errors, and have been corrected. In period -6 & later, a quantity of 50 was added to each demand. Exponential Smoothing 02/13/03 page 12 of 38
13 Exponential Smoothing 02/13/03 page 13 of 38
14 Both the moving average and (single) exponential smoothing methods have the implicit assumption: demand pattern is essentially level, with no trend. Exponential Smoothing 02/13/03 page 14 of 38
15 Double Exponential Smoothing Includes both level a & trend b in the forecast: D = a+ b t In particular, the forecast one period into the future is t D = a+ b Uses 2 smoothing parameters, both between 0 and 1: α for smoothing of the level, a β for smoothing of the trend, b 1 Exponential Smoothing 02/13/03 page 15 of 38
16 Double exponential smoothing Update level & trend using formula That is, ( 1 ) ( ) ( 1 ) a0 =α D0 + α D0 b =β a a + β b the level a is updated by a weighted average of the new demand D 0 and the old forecast D 0. the trend b is updated by a weighted average of the change in level, a0 a 1 and the old trend, b 1 Likewise, the Mean Absolute Deviation is updated by ( 1 ) MAD = α MAD +α D D Exponential Smoothing 02/13/03 page 16 of 38
17 Then compute new forecast, D t = a0 + b0 t, in particular, D = a + b Exponential Smoothing 02/13/03 page 17 of 38
18 Example Initially, estimates are level = a = 100 trend = b = 0.5 MAD = 1.0 Our smoothing parameters are α = 0.5 β = 0.25 Suppose that the next demand which we observe is 105. Exponential Smoothing 02/13/03 page 18 of 38
19 Then we update the level, trend, and MAD: a0 =α D0 + ( 1 α ) D 0 = ( ) = b0 =β ( a0 a 1) + ( 1 β ) b 1 = 0.25 ( ) = and MAD0 = ( 1 α ) MAD 1+α D0 D 0 σ 1.25 MAD = = The forecast for the next period is therefore D 1 = a0+ b0 = = Therefore we assume the next period s demand has distribution N(103.8, 3.44) when computing safety stock, etc. Exponential Smoothing 02/13/03 page 19 of 38
20 Suppose that we observe the next demand to be 98. a0 =α D0 + ( 1 α ) D 0 = ( 103.8) = Then, b0 =β ( a0 a 1) + ( 1 β ) b 1 = 0.25 ( ) = and MAD0 = ( 1 α ) MAD 1+α D0 D 0 σ 1.25 MAD = = The forecast for the next period is therefore a + b 1 = = and we assume the demand has distribution N(101.24, 5.35) when computing safety stock, etc. Exponential Smoothing 02/13/03 page 20 of 38
21 Exponential Smoothing 02/13/03 page 21 of 38
22 Suppose that the next demand that we observe is 104. The updated level & trend are a0 =α D0 + ( 1 α ) D 0 = ( ) = b0 =β ( a0 a 1) + ( 1 β ) b 1 = 0.25 ( ) = and MAD0 = ( 1 α ) MAD 1+α D0 D 0 σ 1.25 MAD = = so that the new forecast is = Therefore, we assume the next period s demand has distribution N(103.32, 4.4) Exponential Smoothing 02/13/03 page 22 of 38
23 Exponential Smoothing 02/13/03 page 23 of 38
24 Suppose that the next demand is 110 Then the updated level, trend, and MAD are and ( 1 ) ( ) ( ) ( 1 ) ( ) a0 =α D0 + α D0 = = b0 =β a0 a 1 + β b 1 = = 1.52 ( ) MAD0 = 1 α MAD 1+α D0 D 0 σ 1.25 MAD = = The new forecast is therefore = and the distribution is ~ N(108.17, 6.39) Exponential Smoothing 02/13/03 page 24 of 38
25 Exponential Smoothing 02/13/03 page 25 of 38
26 DS for Windows Forecasting Module Double Exponential Smoothing Here, we have made the smoothing constants alpha for level and beta for trend both equal to Exponential Smoothing 02/13/03 page 26 of 38
27 Exponential Smoothing 02/13/03 page 27 of 38
28 Exponential Smoothing 02/13/03 page 28 of 38
29 At period 0, the estimated level a is and the estimated trend b is , so that the forecast one period into the future is D a b 1 = + 1 = = Exponential Smoothing 02/13/03 page 29 of 38
30 Here we have corrected the outliers which were mistakes in data entry, and added 50 to demands at periods 6 and later. At t=0, level a = 205.5, trend b = , forecast = Exponential Smoothing 02/13/03 page 30 of 38
31 Exponential Smoothing 02/13/03 page 31 of 38
32 Choice of smoothing parameters α & β There is a trade-off between stability (small α & β) and being responsive to changes in demand pattern (large α & β) In general, several values should be tried, using historical data, in order to establish reasonable values. Suggestion: Single exponential smoothing α ranging between 0.01 and 0.3, with 0.1 being a reasonable compromise. Exponential Smoothing 02/13/03 page 32 of 38
33 Double Exponential Smoothing with Seasonal Factors Example: Shown is a plot of the quarterly sales for product EDM-617 for the past four years: Exponential Smoothing 02/13/03 page 33 of 38
34 Exponential Smoothing 02/13/03 page 34 of 38
35 Super-imposing the demand patterns for the four years, we see a seasonal pattern-- high demands in quarters 2 & 3, low demand in quarters 1 & 4: Exponential Smoothing 02/13/03 page 35 of 38
36 Exponential smoothing for seasonal (multiplicative) model: t ( ) D = a+ bt F where a= level, b=trend, F t = seasonal index for period t t Exponential Smoothing 02/13/03 page 36 of 38
37 F 1 = F 2 = F 3 = F 4 = For example, the forecast for period 19 is computed by: D 19 = ( a+ bt) Ft = = unadjusted forecast seasonal factor Exponential Smoothing 02/13/03 page 37 of 38
38 Exponential Smoothing 02/13/03 page 38 of 38
Dennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa. Forecasting demand 02/06/03 page 1 of 34
demand -5-4 -3-2 -1 0 1 2 3 Dennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa Forecasting demand 02/06/03 page 1 of 34 Forecasting is very difficult. especially about the
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