4 - 1© 2011 Pearson Education, Inc. publishing as Prentice Hall 4 4 Forecasting PowerPoint presentation to accompany Heizer and Render Operations Management, 10e Principles of Operations Management, 8e PowerPoint slides by Jeff Heyl

4 - 2© 2011 Pearson Education, Inc. publishing as Prentice Hall What is Forecasting?  Process of predicting a future event  Underlying basis of all business decisions  Production  Inventory  Personnel  Facilities ??

4 - 3© 2011 Pearson Education, Inc. publishing as Prentice Hall  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

4 - 4© 2011 Pearson Education, Inc. publishing as Prentice Hall Distinguishing Differences  Medium/long range  Medium/long range forecasts deal with more comprehensive issues and support management decisions regarding planning and products, plants and processes  Short-term  Short-term forecasting usually employs different methodologies than longer-term forecasting  Short-term  Short-term forecasts tend to be more accurate than longer-term forecasts

4 - 5© 2011 Pearson Education, Inc. publishing as Prentice Hall Influence of Product Life Cycle  Introduction and growth require longer forecasts than maturity and decline  As product passes through life cycle, forecasts are useful in projecting  Staffing levels  Inventory levels  Factory capacity Introduction – Growth – Maturity – Decline

4 - 6© 2011 Pearson Education, Inc. publishing as Prentice Hall Product Life Cycle Best period to increase market share R&D engineering is critical Practical to change price or quality image Strengthen niche Poor time to change image, price, or quality Competitive costs become critical Defend market position Cost control critical IntroductionGrowthMaturityDecline Company Strategy/Issues Figure 2.5 Internet search engines Sales Drive-through restaurants CD-ROMs Analog TVs iPods Boeing 787 LCD & plasma TVs Twitter Avatars Xbox 360

4 - 7© 2011 Pearson Education, Inc. publishing as Prentice Hall Product Life Cycle Product design and development critical Frequent product and process design changes Short production runs High production costs Limited models Attention to quality IntroductionGrowthMaturityDecline OM Strategy/Issues Forecasting critical Product and process reliability Competitive product improvements and options Increase capacity Shift toward product focus Enhance distribution Standardization Fewer product changes, more minor changes Optimum capacity Increasing stability of process Long production runs Product improvement and cost cutting Little product differentiation Cost minimization Overcapacity in the industry Prune line to eliminate items not returning good margin Reduce capacity Figure 2.5

4 - 8© 2011 Pearson Education, Inc. publishing as Prentice Hall Types of Forecasts  Economic forecasts  Address business cycle – inflation rate, money supply, housing starts, etc.  Technological forecasts  Predict rate of technological progress  Impacts development of new products  Demand forecasts  Predict sales of existing products and services

4 - 9© 2011 Pearson Education, Inc. publishing as Prentice Hall Strategic Importance of Forecasting  Human Resources – Hiring, training, laying off workers  Capacity – Capacity shortages can result in undependable delivery, loss of customers, loss of market share  Supply Chain Management – Good supplier relations and price advantages

4 - 10© 2011 Pearson Education, Inc. publishing as Prentice Hall Forecasting Approaches  Used when situation is vague and little data exist  New products  New technology  Involves intuition, experience  e.g., forecasting sales on Internet Qualitative Methods

4 - 11© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 12© 2011 Pearson Education, Inc. publishing as Prentice Hall Overview of Quantitative Approaches 1.Naive approach 2.Moving averages 3.Exponential smoothing 4.Trend projection 5.Linear regression time-series models associative model

4 - 13© 2011 Pearson Education, Inc. publishing as Prentice Hall  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

4 - 14© 2011 Pearson Education, Inc. publishing as Prentice Hall Trend Seasonal Cyclical Random Time Series Components

4 - 15© 2011 Pearson Education, Inc. publishing as Prentice Hall Components of Demand Demand for product or service |||| 1234 Time (years) Average demand over 4 years Trend component Actual demand line Random variation Figure 4.1 Seasonal peaks

4 - 16© 2011 Pearson Education, Inc. publishing as Prentice Hall  Persistent, overall upward or downward pattern  Changes due to population, technology, age, culture, etc.  Typically several years duration Trend Component

4 - 17© 2011 Pearson Education, Inc. publishing as Prentice Hall  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

4 - 18© 2011 Pearson Education, Inc. publishing as Prentice Hall  Repeating up and down movements  Affected by business cycle, political, and economic factors  Multiple years duration  Often causal or associative relationships Cyclical Component

4 - 19© 2011 Pearson Education, Inc. publishing as Prentice Hall  Erratic, unsystematic, ‘residual’ fluctuations  Due to random variation or unforeseen events  Short duration and nonrepeating Random Component MTWTFMTWTF

4 - 20© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 21© 2011 Pearson Education, Inc. publishing as Prentice Hall  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

4 - 22© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 23© 2011 Pearson Education, Inc. publishing as Prentice Hall Graph of Moving Average ||||||||||||JFMAMJJASOND||||||||||||JFMAMJJASOND Shed Sales 30 – 28 – 26 – 24 – 22 – 20 – 18 – 16 – 14 – 12 – 10 – Actual Sales Moving Average Forecast

4 - 24© 2011 Pearson Education, Inc. publishing as Prentice Hall  Used when some trend might be present  Older data usually less important  Weights based on experience and intuition Weighted Moving Average Weighted moving average = ∑ (weight for period n) x (demand in period n) ∑ weights

4 - 25© 2011 Pearson Education, Inc. publishing as Prentice Hall January10 February12 March13 April16 May19 June23 July26 Actual3-Month Weighted MonthShed SalesMoving Average [(3 x 16) + (2 x 13) + (12)]/6 = 14 1 / 3 [(3 x 19) + (2 x 16) + (13)]/6 = 17 [(3 x 23) + (2 x 19) + (16)]/6 = 20 1 / 2 Weighted Moving Average [(3 x 13) + (2 x 12) + (10)]/6 = 12 1 / 6 Weights AppliedPeriod 3 3Last month 2 2Two months ago 1 1Three months ago 6Sum of weights

4 - 26© 2011 Pearson Education, Inc. publishing as Prentice Hall  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

4 - 27© 2011 Pearson Education, Inc. publishing as Prentice Hall Moving Average And Weighted Moving Average 30 – 25 – 20 – 15 – 10 – 5 – Sales demand ||||||||||||JFMAMJJASOND||||||||||||JFMAMJJASOND Actual sales Moving average Weighted moving average Figure 4.2

4 - 28© 2011 Pearson Education, Inc. publishing as Prentice Hall  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

4 - 29© 2011 Pearson Education, Inc. publishing as Prentice Hall 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)

4 - 30© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20

4 - 31© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20 New forecast= (153 – 142)

4 - 32© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20 New forecast= (153 – 142) = = ≈ 144 cars

4 - 33© 2011 Pearson Education, Inc. publishing as Prentice Hall Effect of Smoothing Constants Weight Assigned to Most2nd Most3rd Most4th Most5th Most RecentRecentRecentRecentRecent SmoothingPeriodPeriodPeriodPeriodPeriod Constant(  )  (1 -  )  (1 -  ) 2  (1 -  ) 3  (1 -  ) 4  =  =

4 - 34© 2011 Pearson Education, Inc. publishing as Prentice Hall Impact of Different  225 – 200 – 175 – 150 – ||||||||| ||||||||| Quarter Demand  =.1 Actual demand  =.5

4 - 35© 2011 Pearson Education, Inc. publishing as Prentice Hall Impact of Different  225 – 200 – 175 – 150 – ||||||||| ||||||||| 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

4 - 36© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 37© 2011 Pearson Education, Inc. publishing as Prentice Hall Common Measures of Error Mean Absolute Deviation (MAD) MAD = ∑ |Actual - Forecast| n Mean Squared Error (MSE) MSE = ∑ (Forecast Errors) 2 n

4 - 38© 2011 Pearson Education, Inc. publishing as Prentice Hall Common Measures of Error Mean Absolute Percent Error (MAPE) MAPE = ∑ 100|Actual i - Forecast i |/Actual i n i = 1

4 - 39© 2011 Pearson Education, Inc. publishing as Prentice Hall Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =

4 - 40© 2011 Pearson Education, Inc. publishing as Prentice Hall Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  = MAD = ∑ |deviations| n = 82.45/8 = For  =.10 = 98.62/8 = For  =.50

4 - 41© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 42© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 43© 2011 Pearson Education, Inc. publishing as Prentice Hall Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  = MAD MSE MAPE5.59%6.76%

4 - 44© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment When a trend is present, exponential smoothing must be modified Forecast including (FIT t ) = trendExponentially smoothed (F t ) +smoothed(T t ) forecasttrend

4 - 45© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment F t =  (A t - 1 ) + (1 -  )(F t T t - 1 ) T t =  (F t - F t - 1 ) + (1 -  )T t - 1 Step 1: Compute F t Step 2: Compute T t Step 3: Calculate the forecast FIT t = F t + T t

4 - 46© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t Table 4.1

4 - 47© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t Table 4.1 F 2 =  A 1 + (1 -  )(F 1 + T 1 ) F 2 = (.2)(12) + (1 -.2)(11 + 2) = = 12.8 units Step 1: Forecast for Month 2

4 - 48© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t Table 4.1 T 2 =  (F 2 - F 1 ) + (1 -  )T 1 T 2 = (.4)( ) + (1 -.4)(2) = = 1.92 units Step 2: Trend for Month 2

4 - 49© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t Table 4.1 FIT 2 = F 2 + T 2 FIT 2 = = units Step 3: Calculate FIT for Month 2

4 - 50© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t Table

4 - 51© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Figure 4.3 ||||||||| ||||||||| Time (month) Product demand 35 – 30 – 25 – 20 – 15 – 10 – 5 – 0 – Actual demand (A t ) Forecast including trend (FIT t ) with  =.2 and  =.4

4 - 52© 2011 Pearson Education, Inc. publishing as Prentice Hall 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 ^

4 - 53© 2011 Pearson Education, Inc. publishing as Prentice Hall 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 ^

4 - 54© 2011 Pearson Education, Inc. publishing as Prentice Hall 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 ^ Least squares method minimizes the sum of the squared errors (deviations)

4 - 55© 2011 Pearson Education, Inc. publishing as Prentice Hall Least Squares Method Equations to calculate the regression variables b =  xy - nxy  x 2 - nx 2 y = a + bx ^ a = y - bx

4 - 56© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 57© 2011 Pearson Education, Inc. publishing as Prentice Hall 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 = Least Squares Example The trend line is y = x ^

4 - 58© 2011 Pearson Education, Inc. publishing as Prentice Hall Least Squares Example ||||||||| – 150 – 140 – 130 – 120 – 110 – 100 – 90 – 80 – 70 – 60 – 50 – Year Power demand Trend line, y = x ^

4 - 59© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 60© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Variations In Data The multiplicative seasonal model can adjust trend data for seasonal variations in demand

4 - 61© 2011 Pearson Education, Inc. publishing as Prentice Hall 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:

4 - 62© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Index Example Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec DemandAverageAverage Seasonal Month MonthlyIndex

4 - 63© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 64© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Index Example Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec DemandAverageAverage Seasonal Month MonthlyIndex

4 - 65© 2011 Pearson Education, Inc. publishing as Prentice Hall 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 2010

4 - 66© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Index Example 140 – 130 – 120 – 110 – 100 – 90 – 80 – 70 – ||||||||||||JFMAMJJASOND||||||||||||JFMAMJJASOND Time Demand 2010 Forecast 2009 Demand 2008 Demand 2007 Demand

4 - 67© 2011 Pearson Education, Inc. publishing as Prentice Hall San Diego Hospital 10,200 – 10,000 – 9,800 – 9,600 – 9,400 – 9,200 – 9,000 – |||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNovDec Month Inpatient Days Figure 4.6 Trend Data

4 - 68© 2011 Pearson Education, Inc. publishing as Prentice Hall San Diego Hospital 1.06 – 1.04 – 1.02 – 1.00 – 0.98 – 0.96 – 0.94 – 0.92 – |||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNovDec Month Index for Inpatient Days Figure 4.7 Seasonal Indices

4 - 69© 2011 Pearson Education, Inc. publishing as Prentice Hall San Diego Hospital 10,200 – 10,000 – 9,800 – 9,600 – 9,400 – 9,200 – 9,000 – |||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNovDec Month Inpatient Days Figure Combined Trend and Seasonal Forecast

4 - 70© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 71© 2011 Pearson Education, Inc. publishing as Prentice Hall 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 ^

4 - 72© 2011 Pearson Education, Inc. publishing as Prentice Hall Associative Forecasting Example SalesArea Payroll ($ millions), y($ billions), x – 3.0 – 2.0 – 1.0 – ||||||| ||||||| Sales Area payroll

4 - 73© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 74© 2011 Pearson Education, Inc. publishing as Prentice Hall Associative Forecasting Example y = x ^ Sales = (payroll) If payroll next year is estimated to be $6 billion, then: Sales = (6) Sales = $3,250, – 3.0 – 2.0 – 1.0 – ||||||| ||||||| Nodel’s sales Area payroll 3.25

4 - 75© 2011 Pearson Education, Inc. publishing as Prentice Hall 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 – ||||||| ||||||| Nodel’s sales Area payroll 3.25

4 - 76© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 77© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 78© 2011 Pearson Education, Inc. publishing as Prentice Hall Standard Error of the Estimate 4.0 – 3.0 – 2.0 – 1.0 – ||||||| ||||||| Nodel’s 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

4 - 79© 2011 Pearson Education, Inc. publishing as Prentice Hall  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

4 - 80© 2011 Pearson Education, Inc. publishing as Prentice Hall Correlation Coefficient r = n  xy -  x  y [n  x 2 - (  x) 2 ][n  y 2 - (  y) 2 ]

4 - 81© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 82© 2011 Pearson Education, Inc. publishing as Prentice Hall  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

4 - 83© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 84© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

4 - 85© 2011 Pearson Education, Inc. publishing as Prentice Hall  Measures how well the forecast is predicting actual values  Ratio of cumulative forecast errors 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

4 - 86© 2011 Pearson Education, Inc. publishing as Prentice Hall Monitoring and Controlling Forecasts Tracking signal Cumulative error MAD = Tracking signal = ∑(Actual demand in period i - Forecast demand in period i)  ∑|Actual - Forecast|/n)

4 - 87© 2011 Pearson Education, Inc. publishing as Prentice Hall Tracking Signal Tracking signal + 0 MADs – Upper control limit Lower control limit Time Signal exceeding limit Acceptable range

4 - 88© 2011 Pearson Education, Inc. publishing as Prentice Hall Tracking Signal Example CumulativeAbsolute ActualForecastCummForecastForecast QtrDemandDemandErrorErrorErrorErrorMAD

4 - 89© 2011 Pearson Education, Inc. publishing as Prentice Hall CumulativeAbsolute ActualForecastCummForecastForecast QtrDemandDemandErrorErrorErrorErrorMAD Tracking Signal Example Tracking Signal (Cumm Error/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

4 - 90© 2011 Pearson Education, Inc. publishing as Prentice Hall Adaptive Forecasting  It’s possible to use the computer to continually monitor forecast error and adjust the values of the  and  coefficients used in exponential smoothing to continually minimize forecast error  This technique is called adaptive smoothing