© 2006 Prentice Hall, Inc.4 – 1  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

© 2006 Prentice Hall, Inc.4 – 2 Trend Seasonal Cyclical Random Time Series Components

© 2006 Prentice Hall, Inc.4 – 3 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

© 2006 Prentice Hall, Inc.4 – 4 Graph of Moving Average ||||||||||||JFMAMJJASONDJFMAMJJASOND||||||||||||JFMAMJJASONDJFMAMJJASOND Shed Sales – – – – – – – – – – – Actual Sales Moving Average Forecast

© 2006 Prentice Hall, Inc.4 – 5 Impact of Different  – – – – ||||||||| ||||||||| Quarter Demand  =.1 Actual demand  =.5

© 2006 Prentice Hall, Inc.4 – 6 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 ^

© 2006 Prentice Hall, Inc.4 – 7 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)

© 2006 Prentice Hall, Inc.4 – 8 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

© 2006 Prentice Hall, Inc.4 – 9 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 ^

© 2006 Prentice Hall, Inc.4 – 10 Least Squares Example ||||||||| – – – – – – – – – – – – Year Power demand Trend line, y = x ^

© 2006 Prentice Hall, Inc.4 – 11 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 ^

© 2006 Prentice Hall, Inc.4 – 12 Associative Forecasting Example SalesLocal Payroll ($000,000), y($000,000,000), x – 3.0 – 2.0 – 1.0 – ||||||| ||||||| Sales Area payroll

© 2006 Prentice Hall, Inc.4 – 13 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

© 2006 Prentice Hall, Inc.4 – 14 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 $600 million, then: Sales = (6) Sales = $325,

© 2006 Prentice Hall, Inc.4 – 15 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

© 2006 Prentice Hall, Inc.4 – 16 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

© 2006 Prentice Hall, Inc.4 – 17 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

© 2006 Prentice Hall, Inc.4 – 18 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 $30,600 in sales

© 2006 Prentice Hall, Inc.4 – 19  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

© 2006 Prentice Hall, Inc.4 – 20 Correlation Coefficient r = n  xy -  x  y [n  x 2 - (  x) 2 ][n  y 2 - (  y) 2 ]

© 2006 Prentice Hall, Inc.4 – 21 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

© 2006 Prentice Hall, Inc.4 – 22  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

© 2006 Prentice Hall, Inc.4 – 23 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

© 2006 Prentice Hall, Inc.4 – 24 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 = $300,000

© 2006 Prentice Hall, Inc.4 – 25  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

© 2006 Prentice Hall, Inc.4 – 26 Monitoring and Controlling Forecasts Tracking signal RSFEMAD= = ∑(actual demand in period i - forecast demand in period i)  ∑|actual - forecast|/n)

© 2006 Prentice Hall, Inc.4 – 27 Tracking Signal Tracking signal + 0 MADs – Upper control limit Lower control limit Time Signal exceeding limit Acceptable range

© 2006 Prentice Hall, Inc.4 – 28 Tracking Signal Example Cumulative AbsoluteAbsolute ActualForecastForecastForecast QtrDemandDemandErrorRSFEErrorErrorMAD

© 2006 Prentice Hall, Inc.4 – 29Cumulative AbsoluteAbsolute ActualForecastForecastForecast QtrDemandDemandErrorRSFEErrorErrorMAD Tracking Signal Example Tracking Signal (RSFE/MAD) -10/10 = /7.5 = -2 0/10 = 0 -10/10 = -1 +5/11 = /14.2 = +2.5 The variation of the tracking signal between -2.0 and +2.5 is within acceptable limits