© 2008 Prentice Hall, Inc.4 – 1 Outline – Continued  Time-Series Forecasting  Decomposition of a Time Series  Naive Approach  Moving Averages  Exponential Smoothing  Exponential Smoothing with Trend Adjustment  Trend Projections  Seasonal Variations in Data  Cyclical Variations in Data

© 2008 Prentice Hall, Inc.4 – 2 Outline – Continued  Associative Forecasting Methods: Regression and Correlation Analysis  Using Regression Analysis for Forecasting  Standard Error of the Estimate  Correlation Coefficients for Regression Lines  Multiple-Regression Analysis

© 2008 Prentice Hall, Inc.4 – 3 Outline – Continued  Monitoring and Controlling Forecasts  Adaptive Smoothing  Focus Forecasting  Forecasting In The Service Sector

© 2008 Prentice Hall, Inc.4 – 4 Forecasting at Disney World  Global portfolio includes parks in Hong Kong, Paris, Tokyo, Orlando, and Anaheim  Revenues are derived from people – how many visitors and how they spend their money  Daily management report contains only the forecast and actual attendance at each park

© 2008 Prentice Hall, Inc.4 – 5 Forecasting at Disney World  Disney generates daily, weekly, monthly, annual, and 5-year forecasts  Forecast used by labor management, maintenance, operations, finance, and park scheduling  Forecast used to adjust opening times, rides, shows, staffing levels, and guests admitted

© 2008 Prentice Hall, Inc.4 – 6 Forecasting at Disney World  20% of customers come from outside the USA  Economic model includes gross domestic product, cross-exchange rates, arrivals into the USA  A staff of 35 analysts and 70 field people survey 1 million park guests, employees, and travel professionals each year

© 2008 Prentice Hall, Inc.4 – 7 Forecasting at Disney World  Inputs to the forecasting model include airline specials, Federal Reserve policies, Wall Street trends, vacation/holiday schedules for 3,000 school districts around the world  Average forecast error for the 5-year forecast is 5%  Average forecast error for annual forecasts is between 0% and 3%

© 2008 Prentice Hall, Inc.4 – 8 What is Forecasting?  Process of predicting a future event  Underlying basis of all business decisions  Production  Inventory  Personnel  Facilities ??

© 2008 Prentice Hall, Inc.4 – 9  Short-range forecast  Up to 1 year, generally less than 3 months  Purchasing, job scheduling, workforce levels, job assignments, production levels  Medium-range forecast  3 months to 3 years  Sales and production planning, budgeting  Long-range forecast  3 + years  New product planning, facility location, research and development Forecasting Time Horizons

© 2008 Prentice Hall, Inc.4 – 10 Seven Steps in Forecasting  Determine the use of the forecast  Select the items to be forecasted  Determine the time horizon of the forecast  Select the forecasting model(s)  Gather the data  Make the forecast  Validate and implement results

© 2008 Prentice Hall, Inc.4 – 11 The Realities!  Forecasts are seldom perfect  Most techniques assume an underlying stability in the system  Product family and aggregated forecasts are more accurate than individual product forecasts

© 2008 Prentice Hall, Inc.4 – 12 Forecasting Approaches  Used when situation is ‘stable’ and historical data exist  Existing products  Current technology  Involves mathematical techniques  e.g., forecasting sales of color televisions Quantitative Methods

© 2008 Prentice Hall, Inc.4 – 13 Overview of Quantitative Approaches 1.Naive approach 2.Moving averages 3.Exponential smoothing 4.Trend projection 5.Linear regression Time-Series Models Associative Model

© 2008 Prentice Hall, Inc.4 – 14  Set of evenly spaced numerical data  Obtained by observing response variable at regular time periods  Forecast based only on past values, no other variables important  Assumes that factors influencing past and present will continue influence in future Time Series Forecasting

© 2008 Prentice Hall, Inc.4 – 15 Trend Seasonal Cyclical Random Time Series Components

© 2008 Prentice Hall, Inc.4 – 16 Components of Demand Demand for product or service |||| 1234 Year Average demand over four years Seasonal peaks Trend component Actual demand Random variation Figure 4.1

© 2008 Prentice Hall, Inc.4 – 17  Persistent, overall upward or downward pattern  Changes due to population, technology, age, culture, etc.  Typically several years duration Trend Component

© 2008 Prentice Hall, Inc.4 – 18  Regular pattern of up and down fluctuations  Due to weather, customs, etc.  Occurs within a single year Seasonal Component Number of PeriodLengthSeasons WeekDay7 MonthWeek4-4.5 MonthDay28-31 YearQuarter4 YearMonth12 YearWeek52

© 2008 Prentice Hall, Inc.4 – 19  Repeating up and down movements  Affected by business cycle, political, and economic factors  Multiple years duration  Often causal or associative relationships Cyclical Component

© 2008 Prentice Hall, Inc.4 – 20  Erratic, unsystematic, ‘residual’ fluctuations  Due to random variation or unforeseen events  Short duration and nonrepeating Random Component MTWTFMTWTFMTWTFMTWTF

© 2008 Prentice Hall, Inc.4 – 21 Naive Approach  Assumes demand in next period is the same as demand in most recent period  e.g., If January sales were 68, then February sales will be 68  Sometimes cost effective and efficient  Can be good starting point

© 2008 Prentice Hall, Inc.4 – 22  MA is a series of arithmetic means  Used if little or no trend  Used often for smoothing  Provides overall impression of data over time Moving Average Method Moving average = ∑ demand in previous n periods n

© 2008 Prentice Hall, Inc.4 – 23 January10 February12 March13 April16 May19 June23 July26 Actual3-Month MonthShed SalesMoving Average ( )/3 = 13 2 / 3 ( )/3 = 16 ( )/3 = 19 1 / 3 Moving Average Example ( )/3 = 11 2 / 3

© 2008 Prentice Hall, Inc.4 – 24 Graph of Moving Average ||||||||||||JFMAMJJASONDJFMAMJJASOND||||||||||||JFMAMJJASONDJFMAMJJASOND Shed Sales – – – – – – – – – – – Actual Sales Moving Average Forecast

© 2008 Prentice Hall, Inc.4 – 25  Increasing n smooths the forecast but makes it less sensitive to changes  Do not forecast trends well  Require extensive historical data Potential Problems With Moving Average

© 2008 Prentice Hall, Inc.4 – 26  Form of weighted moving average  Weights decline exponentially  Most recent data weighted most  Requires smoothing constant (  )  Ranges from 0 to 1  Subjectively chosen  Involves little record keeping of past data Exponential Smoothing

© 2008 Prentice Hall, Inc.4 – 27 Exponential Smoothing New forecast =Last period’s forecast +  (Last period’s actual demand – Last period’s forecast) F t = F t – 1 +  (A t – 1 - F t – 1 ) whereF t =new forecast F t – 1 =previous forecast  =smoothing (or weighting) constant (0 ≤  ≤ 1)

© 2008 Prentice Hall, Inc.4 – 28 Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20

© 2008 Prentice Hall, Inc.4 – 29 Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20 New forecast= (153 – 142)

© 2008 Prentice Hall, Inc.4 – 30 Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20 New forecast= (153 – 142) = = ≈ 144 cars

© 2008 Prentice Hall, Inc.4 – 31 Effect of Smoothing Constants Weight Assigned to Most2nd Most3rd Most4th Most5th Most RecentRecentRecentRecentRecent SmoothingPeriodPeriodPeriodPeriodPeriod Constant(  )  (1 -  )  (1 -  ) 2  (1 -  ) 3  (1 -  ) 4  =  =

© 2008 Prentice Hall, Inc.4 – 32 Impact of Different  – – – – ||||||||| ||||||||| Quarter Demand  =.1 Actual demand  =.5

© 2008 Prentice Hall, Inc.4 – 33 Impact of Different  – – – – ||||||||| ||||||||| Quarter Demand  =.1 Actual demand  =.5  Chose high values of  when underlying average is likely to change  Choose low values of  when underlying average is stable

© 2008 Prentice Hall, Inc.4 – 34 Choosing  The objective is to obtain the most accurate forecast no matter the technique We generally do this by selecting the model that gives us the lowest forecast error Forecast error= Actual demand - Forecast value = A t - F t

© 2008 Prentice Hall, Inc.4 – 35 Common Measures of Error Mean Absolute Deviation (MAD) MAD = ∑ |Actual - Forecast| n Mean Squared Error (MSE) MSE = ∑ (Forecast Errors) 2 n

© 2008 Prentice Hall, Inc.4 – 36 Common Measures of Error Mean Absolute Percent Error (MAPE) MAPE = ∑ 100|Actual i - Forecast i |/Actual i n n i = 1

© 2008 Prentice Hall, Inc.4 – 37 Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =

© 2008 Prentice Hall, Inc.4 – 38 Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  = MAD = ∑ |deviations| n = 82.45/8 = For  =.10 = 98.62/8 = For  =.50

© 2008 Prentice Hall, Inc.4 – 39 Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  = MAD = 1,526.54/8 = For  =.10 = 1,561.91/8 = For  =.50 MSE = ∑ (forecast errors) 2 n

© 2008 Prentice Hall, Inc.4 – 40 Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  = MAD MSE = 44.75/8 = 5.59% For  =.10 = 54.05/8 = 6.76% For  =.50 MAPE = ∑ 100|deviation i |/actual i n i = 1

© 2008 Prentice Hall, Inc.4 – 41 Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  = MAD MSE MAPE5.59%6.76%

© 2008 Prentice Hall, Inc.4 – 42 Trend Projections Fitting a trend line to historical data points to project into the medium to long-range Linear trends can be found using the least squares technique y = a + bx ^ where y= computed value of the variable to be predicted (dependent variable) a= y-axis intercept b= slope of the regression line x= the independent variable ^

© 2008 Prentice Hall, Inc.4 – 43 Least Squares Method Time period Values of Dependent Variable Figure 4.4 Deviation 1 (error) Deviation 5 Deviation 7 Deviation 2 Deviation 6 Deviation 4 Deviation 3 Actual observation (y value) Trend line, y = a + bx ^

© 2008 Prentice Hall, Inc.4 – 44 Least Squares Method Time period Values of Dependent Variable Figure 4.4 Deviation 1 Deviation 5 Deviation 7 Deviation 2 Deviation 6 Deviation 4 Deviation 3 Actual observation (y value) Trend line, y = a + bx ^ Least squares method minimizes the sum of the squared errors (deviations)

© 2008 Prentice Hall, Inc.4 – 45 Least Squares Method Equations to calculate the regression variables b =  xy - nxy  x 2 - nx 2 y = a + bx ^ a = y - bx

© 2008 Prentice Hall, Inc.4 – 46 Least Squares Example b = = = ∑xy - nxy ∑x 2 - nx 2 3,063 - (7)(4)(98.86) (7)(4 2 ) a = y - bx = (4) = TimeElectrical Power YearPeriod (x)Demandx 2 xy ∑x = 28∑y = 692∑x 2 = 140∑xy = 3,063 x = 4y = 98.86

© 2008 Prentice Hall, Inc.4 – 47 Least Squares Example b = = =  xy - nxy  x 2 - nx 2 3,063 - (7)(4)(98.86) (7)(4 2 ) a = y - bx = (4) = TimeElectrical Power YearPeriod (x)Demandx 2 xy  x = 28  y = 692  x 2 = 140  xy = 3,063 x = 4y = The trend line is y = x ^

© 2008 Prentice Hall, Inc.4 – 48 Least Squares Example ||||||||| – – – – – – – – – – – – Year Power demand Trend line, y = x ^

© 2008 Prentice Hall, Inc.4 – 49 Least Squares Requirements 1.We always plot the data to insure a linear relationship 2.We do not predict time periods far beyond the database 3.Deviations around the least squares line are assumed to be random

© 2008 Prentice Hall, Inc.4 – 50 Seasonal Variations In Data The multiplicative seasonal model can adjust trend data for seasonal variations in demand

© 2008 Prentice Hall, Inc.4 – 51 Seasonal Variations In Data 1.Find average historical demand for each season 2.Compute the average demand over all seasons 3.Compute a seasonal index for each season 4.Estimate next year’s total demand 5.Divide this estimate of total demand by the number of seasons, then multiply it by the seasonal index for that season Steps in the process:

© 2008 Prentice Hall, Inc.4 – 52 Seasonal Index Example Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec DemandAverageAverage Seasonal Month MonthlyIndex

© 2008 Prentice Hall, Inc.4 – 53 Seasonal Index Example Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec DemandAverageAverage Seasonal Month MonthlyIndex Seasonal index = average monthly demand average monthly demand = 90/94 =.957

© 2008 Prentice Hall, Inc.4 – 54 Seasonal Index Example Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec DemandAverageAverage Seasonal Month MonthlyIndex

© 2008 Prentice Hall, Inc.4 – 55 Seasonal Index Example Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec DemandAverageAverage Seasonal Month MonthlyIndex Expected annual demand = 1,200 Janx.957 = 96 1, Febx.851 = 85 1, Forecast for 2008

© 2008 Prentice Hall, Inc.4 – 56 Seasonal Index Example – – – – – – – – ||||||||||||JFMAMJJASONDJFMAMJJASOND||||||||||||JFMAMJJASONDJFMAMJJASOND Time Demand 2008 Forecast 2007 Demand 2006 Demand 2005 Demand

© 2008 Prentice Hall, Inc.4 – 57 Associative Forecasting Used when changes in one or more independent variables can be used to predict the changes in the dependent variable Most common technique is linear regression analysis We apply this technique just as we did in the time series example

© 2008 Prentice Hall, Inc.4 – 58 Associative Forecasting Forecasting an outcome based on predictor variables using the least squares technique y = a + bx ^ where y= computed value of the variable to be predicted (dependent variable) a= y-axis intercept b= slope of the regression line x= the independent variable though to predict the value of the dependent variable ^

© 2008 Prentice Hall, Inc.4 – 59 Associative Forecasting Example SalesLocal Payroll ($ millions), y($ billions), x – 3.0 – 2.0 – 1.0 – ||||||| ||||||| Sales Area payroll

© 2008 Prentice Hall, Inc.4 – 60 Associative Forecasting Example Sales, y Payroll, xx 2 xy ∑y = 15.0∑x = 18∑x 2 = 80∑xy = 51.5 x = ∑x/6 = 18/6 = 3 y = ∑y/6 = 15/6 = 2.5 b = = =.25 ∑xy - nxy ∑x 2 - nx (6)(3)(2.5) 80 - (6)(3 2 ) a = y - bx = (.25)(3) = 1.75

© 2008 Prentice Hall, Inc.4 – 61 Associative Forecasting Example 4.0 – 3.0 – 2.0 – 1.0 – ||||||| ||||||| Sales Area payroll y = x ^ Sales = (payroll) If payroll next year is estimated to be $6 billion, then: Sales = (6) Sales = $3,250,

© 2008 Prentice Hall, Inc.4 – 62 Standard Error of the Estimate  A forecast is just a point estimate of a future value  This point is actually the mean of a probability distribution Figure – 3.0 – 2.0 – 1.0 – ||||||| ||||||| Sales Area payroll 3.25

© 2008 Prentice Hall, Inc.4 – 63 Standard Error of the Estimate wherey=y-value of each data point y c =computed value of the dependent variable, from the regression equation n=number of data points S y,x = ∑(y - y c ) 2 n - 2

© 2008 Prentice Hall, Inc.4 – 64 Standard Error of the Estimate Computationally, this equation is considerably easier to use We use the standard error to set up prediction intervals around the point estimate S y,x = ∑y 2 - a∑y - b∑xy n - 2

© 2008 Prentice Hall, Inc.4 – 65 Standard Error of the Estimate 4.0 – 3.0 – 2.0 – 1.0 – ||||||| ||||||| Sales Area payroll 3.25 S y,x = = ∑y 2 - a∑y - b∑xy n (15) -.25(51.5) S y,x =.306 The standard error of the estimate is $306,000 in sales

© 2008 Prentice Hall, Inc.4 – 66  How strong is the linear relationship between the variables?  Correlation does not necessarily imply causality!  Coefficient of correlation, r, measures degree of association  Values range from -1 to +1 Correlation

© 2008 Prentice Hall, Inc.4 – 67 Correlation Coefficient r = n  xy -  x  y [n  x 2 - (  x) 2 ][n  y 2 - (  y) 2 ]

© 2008 Prentice Hall, Inc.4 – 68 Correlation Coefficient r = n  xy -  x  y [n  x 2 - (  x) 2 ][n  y 2 - (  y) 2 ] y x (a)Perfect positive correlation: r = +1 y x (b)Positive correlation: 0 < r < 1 y x (c)No correlation: r = 0 y x (d)Perfect negative correlation: r = -1

© 2008 Prentice Hall, Inc.4 – 69  Coefficient of Determination, r 2, measures the percent of change in y predicted by the change in x  Values range from 0 to 1  Easy to interpret Correlation For the Nodel Construction example: r =.901 r 2 =.81

© 2008 Prentice Hall, Inc.4 – 70 Multiple Regression Analysis If more than one independent variable is to be used in the model, linear regression can be extended to multiple regression to accommodate several independent variables y = a + b 1 x 1 + b 2 x 2 … ^ Computationally, this is quite complex and generally done on the computer

© 2008 Prentice Hall, Inc.4 – 71 Multiple Regression Analysis y = x x 2 ^ In the Nodel example, including interest rates in the model gives the new equation: An improved correlation coefficient of r =.96 means this model does a better job of predicting the change in construction sales Sales = (6) - 5.0(.12) = 3.00 Sales = $3,000,000

© 2008 Prentice Hall, Inc.4 – 72  Measures how well the forecast is predicting actual values  Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)  Good tracking signal has low values  If forecasts are continually high or low, the forecast has a bias error Monitoring and Controlling Forecasts Tracking Signal

© 2008 Prentice Hall, Inc.4 – 73 Monitoring and Controlling Forecasts Tracking signal RSFEMAD= = ∑(Actual demand in period i - Forecast demand in period i)  ∑|Actual - Forecast|/n)

© 2008 Prentice Hall, Inc.4 – 74 Tracking Signal Tracking signal + 0 MADs – Upper control limit Lower control limit Time Signal exceeding limit Acceptable range