Product and Inventory Management (35E00300) Forecasting Models Trend analysis

Exponential Smoothing Data Storage Shed Sales Period Actual Value(Y t ) Ŷ t-1 α Y t-1 Ŷ t-1 Ŷ t January 10 = 10 0.1 February 12 10 + 0.1 *( 10-10 ) = 10.000 March 16 10 + 0.1 *( 12-10 ) = 10.200 April 13 10.2 + 0.1 *( 16-10.2 ) = 10.780 May 17 10.78 + 0.1 *( 13-10.78 ) = 11.002 June 19 11.002 + 0.1 *( 17-11.002 ) = 11.602 July 15 11.602 + 0.1 *( 19-11.602 ) = 12.342 August 20 12.342 + 0.1 *( 15-12.342 ) = 12.607 September 22 12.607 + 0.1 *( 20-12.607 ) = 13.347 October 19 13.347 + 0.1 *( 22-13.347 ) = 14.212 November 21 14.212 + 0.1 *( 19-14.212 ) = 14.691 December 19 14.691 + 0.1 *( 21-14.691 ) = 15.322 2

Exponential Smoothing (Alpha =.419) 3

Exponential Smoothing Exponential Smoothing 25 20 15 Sheds Actual value Forecast 10 5 0 January February March April May June July August September October November December 4

Forecasting Seasonal Data: Quick Method Enrolment (in thousands) Quarter Year 1 Year 2 Fall 24 26 Winter 23 22 Spring 19 19 Summer 14 17 Total 80 84 Calculate average demand for each quarter or "Season" Year 1: 80/4 = 20 Year 2: 84/4 = 21 5

Forecasting Seasonal Data: Quick Method Compute a seasonal index for every season of every year for which you have data Enrolment (in thousands) Quarter Year 1 Year 2 Fall 24/20= 1,2 26/21= 1,238 Winter 23/20= 1,15 22/21= 1,048 Spring 19/20= 0,95 19/21= 0,905 Summer 14/20= 0,7 17/21= 0,810 Calculate average sesonal index for each index Quarter Fall (1,2+1,238)/2= 1,219 Winter (1,15+1,048)/2= 1,099 Spring (0,95+0,905)/2= 0,927 Summer (0,7+0,81)/2= 0,755 6

Forecasting Seasonal Data: Quick Method Calculate the average deamand per seson for next year (Next years annual number of enrolments is 9000) 90000/4 = 22500 Multiply next years average seasonal demand by each seasonal index Quarter Average demand Index Forecast (students) Fall 22500 1,219 27428,57 Winter 22500 1,099 24723,21 Spring 22500 0,927 20866,07 Summer 22500 0,755 16982,14 7

Trend & Seasonality Trend analysis Technique that fits a trend equation (or curve) to a series of historical data points Projects the equation into the future for medium and long term forecasts. Typically do not want to forecast into the future more than half the number of time periods used to generate the forecast Seasonality analysis Adjustment to time series data due to variations at certain periods. Adjust with seasonal index - ratio of average value of the item in a season to the overall annual average value. Examples: demand for coal in winter months; demand for soft drinks in the summer and over major holidays 8

Linear Trend Analysis Midwestern Manufacturing Sales Scatter Diagram Actual value (or) Y Period number (or) X 74 1995 79 1996 80 1997 90 1998 105 1999 142 2000 122 2001 9

Least Squares for Linear Regression Midwestern Manufacturing Least Squares Method Values of Dependent Variables Objective: Minimize the squared deviations! Time 10

Least Squares Method Y^ = a + bx Where Y^= predicted value of the dependent variable (demand) X = value of the independent variable (time) a = Y-axis intercept = Y - b* X b = Slope of the regression line = [ Y XY - n X X 2 _ 2 - n X ] 11

Linear Trend Data & Error Analysis Midwestern Manufacturing Company Forecasting Linear trend analysis Enter the actual values in cells shaded YELLOW. Enter new time period at the bottom to forecast Input Data Forecast Error Analysis Actual value Period number Absolute Squared Absolute Period (or) Y (or) X Forecast Error error error % error Year 1 74 1 67.250 6.750 6.750 45.563 9.12% Year 2 79 2 77.786 1.214 1.214 1.474 1.54% Year 3 80 3 88.321-8.321 8.321 69.246 10.40% Year 4 90 4 98.857-8.857 8.857 78.449 9.84% Year 5 105 5 109.393-4.393 4.393 19.297 4.18% Year 6 142 6 119.929 22.071 22.071 487.148 15.54% Year 7 122 7 130.464-8.464 8.464 71.644 6.94% Average 8.582 110.403 8.22% Intercept 56.714 MAD MSE MAPE Slope 10.536 Next period 141.000 8 12

Least Squares Graph 13

Quantitative Forecasting using Trend Extrapolation There are several tools available for using trend extrapolation to first plot a trend line to historical time-series data, and then extend this to future periods for the purpose of forecasting or predicting values for those periods. Some might be tempted to visually extend a trend line to future periods with a pencil on a printed graph, but algebraic techniques are more precise, more varied, and more powerful. 14

Quantitative Forecasting using Trend Extrapolation There are three general approaches to use algebraic techniques for trend line extrapolation. 1. Applying formulas to calculate a future period. 2. Make use of specialized functions within a spreadsheet program or another data analysis program. 3. Make use of a spreadsheet or data analysis program to construct a graph with a trend line, and to automatically extend that trend line to future periods. 15

Triple Exponential Smoothing 16

Holt-Winters (HW) method This method is used when the data shows trend and seasonality. To handle seasonality, we have to add a third parameter. A third equation is introduced to take care of seasonality. The resulting set of equations is called the Holt- Winters (HW) method after the names of the inventors. There are two main HW models, depending on the type of seasonality Multiplicative Seasonal Model Additive Seasonal Model The rest of the todays presentation focuses on these two models

Multiplicative Seasonal Model This model is used when the data exhibits multiplicative seasonality The multiplicative seasonal model is appropriate for a time series in which the amplitude of the seasonal pattern is proportional to the average level of the series, i.e. a time series displaying multiplicative seasonality. This section describes the forecasting equations used in the model along with the initial values to be used for the parameters. 18

Overview This model is used when the data exhibits Multiplicative seasonality. We assume that the time series is represented by the model yy tt = bb 1 + bb 2 tt SS tt + εε tt Where bb 1 is the base signal also called the permanent component bb 2 is a linear trend component SS tt is a multiplicative seasonal factor εε tt is the random error component 19

Length of the Season Let the length of the season be LL periods. The seasonal factors are defined so that they sum to the length of the season, i.e. SS tt = LL 1 tt LL If the trend component bb 2 (previous page) is deemed unnecessary, it may be deleted from the model. 20

Notations Used Let the current deseasonalized level of the process at the end of period T be denoted by RR TT. At the end of a time period t, let RR tt be the estimate of the deseasonalized level. GG tt be the estimate of the trend SS tt be the estimate of seasonal component (seasonal index) 21

Procedure for updating the estimates of model parameters Overall smoothing Smoothing of the trend factor Smoothing of the seasonal index 22

Overall Smoothing RR tt = yy tt + 1 RR SS tt 1 tt 1 + GG tt 1 where 0 1 is the first smoothing constant. Dividing yy tt by SS tt 1, which is the seasonal factor for period T computed one season (L periods) ago, deseasonalizes the data so that only the trend component and the prior value of the permanent component enter into the updating process for RR tt. 23

Smoothing the Trend Factor GG tt = ββ SS tt SS tt 1 + 1 ββ GG tt 1 where 0 ββ 1 is the second smoothing constant. The estimate of the trend component is simply the smoothed difference between two successive estimates of the deseasonalized level. 24

Smoothing of the seasonal index SS tt = γγ yy tt / RR tt + 1 γγ SS tt LL where 0 γγ 1 is the Third smoothing constant. 25

Value of Forecast 1. Forecast for the next period The forecast for the next period is given by: yy tt = RR tt 1 + GG tt 1 SS tt LL Note that the best estimate of the seasonal factor for this time period in the season is used, which was last updated LL periods ago. 26

Value of Forecast 2. Multiple-step-ahead forecasts (for T < q) The value of forecast T periods hence is given by: yy tt+tt = RR tt 1 + TT GG tt 1 SS tt+tt LL 27

Initial Values for Model Parameters As a rule of thumb, a minimum of two full seasons (or 2L periods) of historical data is needed to initialize a set of seasonal factors. Suppose data from m seasons are available and let xx jj, jj = 1,2,, mmmm denote the average of the observations during the j th season. 1. Estimation of trend component 2. Estimation of deseasonalized level 3. Estimation of seasonal components 28

Initial Value for Trend Component Estimate the trend component by: GG 0 = yy mm + yy 1 mm 29

Initial Value for Deseasonalized Level Estimate the deseasonalized level by: RR 0 = xx jj 30

Initial Values for Seasonal Components Seasonal factors are computed for each time period t = 1,2, mmmm as the ratio of actual observation to the average seasonally adjusted value for that season, further adjusted by the trend; that is, SS tt = yy ii RR 0 the t index is the position of the period t within the season. 31

Country Greeting Cards Triple Exponential Smoothing quarte level trend seasonal. year r period sales k forecast '(R) (G) (S) error abs.error %error 1 1 1 222 0,50 2 2 339 0,76 3 3 336 0,76 4 4 878 443,75 55,25 1,98 2 1 5 443 249,64 518,33 63,37 0,84 193,36 193,36 43,6% 2 6 413 444,39 579,64 62,51 0,72-31,39 31,39 7,6% 3 7 398 486,23 636,33 60,06 0,63-88,23 88,23 22,2% 4 8 1143 1377,87 690,45 57,57 1,67-234,87 234,87 20,5% 3 1 9 695 626,06 752,14 59,30 0,92 68,94 68,94 9,9% 2 10 698 580,24 819,67 62,76 0,84 117,76 117,76 16,9% 3 11 737 557,74 896,61 68,72 0,81 179,26 179,26 24,3% 4 12 1648 1613,64 966,36 69,15 1,70 34,36 34,36 2,1% 4 1 13 1141 952,34 1045,77 73,46 1,08 188,66 188,66 16,5% 2 14 1036 945,46 1124,59 75,72 0,92 90,54 90,54 8,7% 3 15 938 975,24 1198,02 74,75 0,78-37,24 37,24 4,0% 4 16 2168 2168,40 1272,76 74,75 1,70-0,40 0,40 0,0% 5 1 17 1 1458,67 BIAS= 40,06 2 18 2 1304,78 LS= 0,050 MAD= 105,42 3 19 3 1174,31 TS= 0,420 MSE= 18039,7 4 20 4 2677,33 SS= 0,950 MAPE= 14,7% 32

3000 2500 2000 sales forecast 1500 1000 500 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 5 year, quarter, period 33

Additive Seasonal Model This model is used when the data exhibits additive seasonality. The additive seasonal model is appropriate for a time series in which the amplitude of the seasonal pattern is independent of the average level of the series, i.e. a time series displaying additive seasonality. This section describes the forecasting equations used in the model along with the initial values to be used for the parameters. 34

Overview This model is used when the data exhibits Additive seasonality. We assume that the time series is represented by the model yy tt = bb 1 + bb 2 tt + SS tt + εε tt Where bb 1 is the base signal also called the permanent component bb 2 is a linear trend component SS tt is a additive seasonal factor εε tt is the random error component 35

Overall Smoothing RR tt = yy tt SS tt LL + 1 RR tt 1 + GG tt 1 where 0 1 is the first smoothing constant. Dividing yy tt by SS tt LL, which is the seasonal factor for period T computed one season (L periods) ago, deseasonalizes the data so that only the trend component and the prior value of the permanent component enter into the updating process for RR tt. 37

Smoothing of the Seasonal Index SS tt = γγ yy tt SS tt + 1 γγ SS tt LL where 0 γγ 1 is the third smoothing constant. The estimate of the seasonal component is a combination of the most recently observed seasonal factor given by the demand yy tt divided by the deseasonalized series level estimate RR tt and the previous best seasonal factor estimate for this time period. Since seasonal factors represent deviations above and below the average, the average of any L consecutive seasonal factors should always be 1. Thus, after estimating SS tt, it is good practice to renormalize the most recent seasonal factors such that tt ii=tt qq+1 SS ii = qq 39

Value of Forecast Forecast for the next period The forecast for the next period is given by: yy tt = RR tt 1 + GG tt 1 + SS tt LL Note that the best estimate of the seasonal factor for this time period in the season is used, which was last updated LL periods ago. 40

Thank you! 41