Dennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa. Forecasting demand 02/06/03 page 1 of 34
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1 demand Dennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa Forecasting demand 02/06/03 page 1 of 34
2 Forecasting is very difficult. especially about the future! ----??? Forecasting the demand for an item. time-series methods (based upon historical demand data) causal methods-- using regression analysis with independent variables such as state of the economy, weather, price, marketing expenditures, etc. qualitative methods (subjective methods, based upon the experience of experts such as marketing and sales staff) Forecasting demand 02/06/03 page 2 of 34
3 Forecasting techniques Qualitative methods Delphi method Jury of executive option Sales force composite Consumer market survey Time series methods Moving average Exponential smoothing Trend projections Seasonality adjustments Causal methods Regression analysis Multiple regression Forecasting demand 02/06/03 page 3 of 34
4 Time-series forecasting methods 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) Forecast: demands for m future periods: D 1, D 2, D m Forecasting demand 02/06/03 page 4 of 34
5 Time-series analysis demand Forecasting demand 02/06/03 page 5 of 34
6 Sales Period Demand Demand Time Two of the observations might be classified as outliers Forecasting demand 02/06/03 page 6 of 34
7 Sales Demand Time Two observations might be classified as outliers and ignored. Forecasting demand 02/06/03 page 7 of 34
8 Linear regression analysis: a statistical technique Given an independent variable (in our case, T, time) a dependent variable (in our case, D, demand) a set of N pairs of observations (, ) 0 (where T=0 is today ) D T, t= 1-N, 2-T, -2, -1, Find parameters a (trend) and b (intercept) such that t ε 2 t is minimized, where the error ε t is defined by [ ] ε = D b+ a T t t t (least-squared-error minimization) t t Forecasting demand 02/06/03 page 8 of 34
9 Regression analysis, using T (time) as single independent variable D D = T T Forecasting demand 02/06/03 page 9 of 34
10 Forecast: Set T=1 for forecast one period into future: ( ) D 1 = = From the regression analysis, we obtain: Standard Error Therefore, we assume that the demand D 1 has distribution N( , ) when setting a value for safety stock & re-order point. Forecasting demand 02/06/03 page 10 of 34
11 Advantages of regression analysis it is well understood, software is available & easy to use can handle nonlinearities (e.g., Disadvantages of regression analysis: 2 t, sin ( t ), 1 t, etc.) when developing the forecast, the same weight or importance is given to old data as to the most recent data independence of demands is assumed fitting the past & forecasting the future are confused-- a good fit to the far past may not lead to a good forecast of the future! Forecasting demand 02/06/03 page 11 of 34
12 DS for Windows forecasting module time-series models in DS for Windows: moving average weighted moving average exponential smoothing double exponential smoothing (with trend) linear regression seasonal decomposition Forecasting demand 02/06/03 page 12 of 34
13 Moving average The average of a fixed number, n, of the most recent demands, is used as an estimate for the next demand. 1 D = D + D + + D + D n ( ) 1 1 n 2 n 1 0 Each time a new demand D 0 is recorded, the new demand is included in the average the oldest demand is discarded (hence the moving average) the error in the forecast ε 0 = D0 D 0 is recorded. Requires storing only the most recent n historical values! Forecasting demand 02/06/03 page 13 of 34
14 The MAD (mean absolute deviation) is recomputed each time another demand is recorded: ε t Dt D t t all t MAD = = # of forecasts # of forecasts The relationship between MAD and σ is π σ= MAD 1.25 MAD 2 ( ) so that the distribution N D π MAD 1, 2 can be used in calculating safety stock & re-order point. Forecasting demand 02/06/03 page 14 of 34
15 4-period Moving Average Period Demand Forecast D D 1 = ( ) = 114 n 1 = ( ) = n Forecasting demand 02/06/03 page 15 of 34
16 Period Demand Forecast Error Absolute Total Average = = n The forecast for the next period is D ( ) Forecasting demand 02/06/03 page 16 of 34
17 Note the effect of the outliers as they enter and leave the set of four demands to be averaged! Forecasting demand 02/06/03 page 17 of 34
18 4-period Moving Average Forecasting demand 02/06/03 page 18 of 34
19 Period Demand Forecast Error Absolute Total Average Forecasting demand 02/06/03 page 19 of 34
20 Compare with 4-period average: using a longer period in the average results in a smoother set of forecasts (but more error!) Forecasting demand 02/06/03 page 20 of 34
21 Weighted Moving Average We can give the more recent demand data a higher weight in the average than the older data. Forecasting demand 02/06/03 page 21 of 34
22 Period Demand Weights Forecast Error Absolute Total Average Forecasting demand 02/06/03 page 22 of 34
23 Forecasting demand 02/06/03 page 23 of 34
24 Exponential Smoothing simpler, requires less storage of data than moving averages approximates weighted moving averages with all past demands included in the average parameter: α where 0 < α < 1 ( 1 ) D = α D +αd (new forecast) = (1 α) (old forecast) + α (new demand) The larger the value of α, the more weight is given to more recent demand data, & the more responsive the forecast to changes in the demand pattern. Forecasting demand 02/06/03 page 24 of 34
25 Exponential Smoothing ( α = 1/4) Forecasting demand 02/06/03 page 25 of 34
26 Period Demand Forecast Error Absolute Total Average D = 0.75 D D = = D = 0.75 D D = = Forecasting demand 02/06/03 page 26 of 34
27 Forecasting demand 02/06/03 page 27 of 34
28 The moving average and (single) exponential smoothing methods have the implicit assumption that demand pattern is essentially level, with no trend. Forecasting demand 02/06/03 page 28 of 34
29 Double Exponential Smoothing Includes a trend b in the forecast: D = a+ b t t In particular, the forecast one period into the future is 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 Forecasting demand 02/06/03 page 29 of 34
30 Double exponential smoothing Update level & trend using formula That is, ( 1 ) a0 =α D0 + α D 0 b =β ( D D ) + ( 1 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 forecasts, D 0 D and the old trend, 1 b 1 Then compute new forecast, D t = a0 + b0 t, in particular, D = a + b Forecasting demand 02/06/03 page 30 of 34
31 Double Exponential Smoothing Here, we have made the smoothing constant alpha for level larger than the smoothing constant beta for trend, so as to be less responsive in making updates to trend. Forecasting demand 02/06/03 page 31 of 34
32 Forecasting demand 02/06/03 page 32 of 34
33 Forecasting demand 02/06/03 page 33 of 34
34 Double exponential smoothing Forecasting demand 02/06/03 page 34 of 34
Dennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa. Exponential Smoothing 02/13/03 page 1 of 38
demand -5-4 -3-2 -1 0 1 2 3 Dennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa Exponential Smoothing 02/13/03 page 1 of 38 As with other Time-series forecasting methods,
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