© 2006 Prentice Hall, Inc.4 – 1 Operations Management Chapter 4 - Forecasting Chapter 4 - Forecasting © 2006 Prentice Hall, Inc. PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 6e Operations Management, 8e

© 2006 Prentice Hall, Inc.4 – 2 Outline  Global Company Profile: Tupperware Corporation  What Is Forecasting?  Forecasting Time Horizons  The Influence of Product Life Cycle  Types Of Forecasts

© 2006 Prentice Hall, Inc.4 – 3 Outline – Continued  The Strategic Importance Of Forecasting  Human Resources  Capacity  Supply-Chain Management  Seven Steps In The Forecasting System

© 2006 Prentice Hall, Inc.4 – 4 Outline – Continued  Forecasting Approaches  Overview of Qualitative Methods  Overview of Quantitative Methods

© 2006 Prentice Hall, Inc.4 – 5 Outline – Continued  Time-series Forecasting  Decomposition of a Time Series  Naïve Approach  Moving Averages  Exponential Smoothing  Exponential Smoothing with Trend Adjustment  Trend Projections  Seasonal Variations in Data  Cyclical Variations in Data

© 2006 Prentice Hall, Inc.4 – 6 Outline – Continued  Associative Forecasting Methods: Regression And Correlation Analysis  Using Regression Analysis to Forecast  Standard Error of the Estimate  Correlation Coefficients for Regression Lines  Multiple-Regression Analysis

© 2006 Prentice Hall, Inc.4 – 7 Outline – Continued  Monitoring And Controlling Forecasts  Adaptive Smoothing  Focus Forecasting  Forecasting In The Service Sector

© 2006 Prentice Hall, Inc.4 – 8 Learning Objectives When you complete this chapter, you should be able to : Identify or Define:  Forecasting  Types of forecasts  Time horizons  Approaches to forecasts

© 2006 Prentice Hall, Inc.4 – 9 Learning Objectives When you complete this chapter, you should be able to : Describe or Explain:  Moving averages  Exponential smoothing  Trend projections  Regression and correlation analysis  Measures of forecast accuracy

© 2006 Prentice Hall, Inc.4 – 10 Forecasting at Tupperware  Each of 50 profit centers around the world is responsible for computerized monthly, quarterly, and 12-month sales projections  These projections are aggregated by region, then globally, at Tupperware’s World Headquarters  Tupperware uses all techniques discussed in text

© 2006 Prentice Hall, Inc.4 – 11 Tupperware’s Process

© 2006 Prentice Hall, Inc.4 – 12 Three Key Factors for Tupperware  The number of registered “consultants” or sales representatives  The percentage of currently “active” dealers (this number changes each week and month)  Sales per active dealer, on a weekly basis

© 2006 Prentice Hall, Inc.4 – 13 Forecast by Consensus  Although inputs come from sales, marketing, finance, and production, final forecasts are the consensus of all participating managers  The final step is Tupperware’s version of the “jury of executive opinion”

© 2006 Prentice Hall, Inc.4 – 14 What is Forecasting?  Process of predicting a future event  Underlying basis of all business decisions  Production  Inventory  Personnel  Facilities ??

© 2006 Prentice Hall, Inc.4 – 15  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 – 16 Distinguishing Differences  Medium/long range forecasts deal with more comprehensive issues and support management decisions regarding planning and products, plants and processes  Short-term forecasting usually employs different methodologies than longer-term forecasting  Short-term forecasts tend to be more accurate than longer-term forecasts

© 2006 Prentice Hall, Inc.4 – 17 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

© 2006 Prentice Hall, Inc.4 – 18 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 Internet Flat-screen monitors Sales DVD CD-ROM Drive-through restaurants Fax machines 3 1/2” Floppy disks Color printers Figure 2.5

© 2006 Prentice Hall, Inc.4 – 19 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 Less rapid 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

© 2006 Prentice Hall, Inc.4 – 20 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 product

© 2006 Prentice Hall, Inc.4 – 21 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 advance

© 2006 Prentice Hall, Inc.4 – 22 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

© 2006 Prentice Hall, Inc.4 – 23 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

© 2006 Prentice Hall, Inc.4 – 24 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

© 2006 Prentice Hall, Inc.4 – 25 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

© 2006 Prentice Hall, Inc.4 – 26 Overview of Qualitative Methods  Jury of executive opinion  Pool opinions of high-level executives, sometimes augment by statistical models  Delphi method  Panel of experts, queried iteratively

© 2006 Prentice Hall, Inc.4 – 27 Overview of Qualitative Methods  Sales force composite  Estimates from individual salespersons are reviewed for reasonableness, then aggregated  Consumer Market Survey  Ask the customer

© 2006 Prentice Hall, Inc.4 – 28  Involves small group of high-level managers  Group estimates demand by working together  Combines managerial experience with statistical models  Relatively quick  ‘Group-think’ disadvantage Jury of Executive Opinion

© 2006 Prentice Hall, Inc.4 – 29 Sales Force Composite  Each salesperson projects his or her sales  Combined at district and national levels  Sales reps know customers’ wants  Tends to be overly optimistic

© 2006 Prentice Hall, Inc.4 – 30 Delphi Method  Iterative group process, continues until consensus is reached  3 types of participants  Decision makers  Staff  Respondents Staff (Administering survey) Decision Makers (Evaluate responses and make decisions) Respondents (People who can make valuable judgments)

© 2006 Prentice Hall, Inc.4 – 31 Consumer Market Survey  Ask customers about purchasing plans  What consumers say, and what they actually do are often different  Sometimes difficult to answer

© 2006 Prentice Hall, Inc.4 – 32 Overview of Quantitative Approaches 1.Naive approach 2.Moving averages 3.Exponential smoothing 4.Trend projection 5.Linear regression Time-Series Models Associative Model

© 2006 Prentice Hall, Inc.4 – 33  Set of evenly spaced numerical data  Obtained by observing response variable at regular time periods  Forecast based only on past values  Assumes that factors influencing past and present will continue influence in future Time Series Forecasting

© 2006 Prentice Hall, Inc.4 – 34 Trend Seasonal Cyclical Random Time Series Components

© 2006 Prentice Hall, Inc.4 – 35 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 – 36  Persistent, overall upward or downward pattern  Changes due to population, technology, age, culture, etc.  Typically several years duration Trend Component

© 2006 Prentice Hall, Inc.4 – 37  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

© 2006 Prentice Hall, Inc.4 – 38  Repeating up and down movements  Affected by business cycle, political, and economic factors  Multiple years duration  Often causal or associative relationships Cyclical Component

© 2006 Prentice Hall, Inc.4 – 39  Erratic, unsystematic, ‘residual’ fluctuations  Due to random variation or unforeseen events  Short duration and nonrepeating Random Component MTWTFMTWTFMTWTFMTWTF

© 2006 Prentice Hall, Inc.4 – 40 Naive Approach  Assumes demand in next period is the same as demand in most recent period  e.g., If May sales were 48, then June sales will be 48  Sometimes cost effective and efficient

© 2006 Prentice Hall, Inc.4 – 41  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

© 2006 Prentice Hall, Inc.4 – 42 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

© 2006 Prentice Hall, Inc.4 – 43 Graph of Moving Average ||||||||||||JFMAMJJASONDJFMAMJJASOND||||||||||||JFMAMJJASONDJFMAMJJASOND Shed Sales – – – – – – – – – – – Actual Sales Moving Average Forecast

© 2006 Prentice Hall, Inc.4 – 44  Used when trend is 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

© 2006 Prentice Hall, Inc.4 – 45 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 3Last month 2Two months ago 1Three months ago 6Sum of weights

© 2006 Prentice Hall, Inc.4 – 46  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

© 2006 Prentice Hall, Inc.4 – 47 Moving Average And Weighted Moving Average – – – – – 5 5 – Sales demand ||||||||||||JFMAMJJASONDJFMAMJJASOND||||||||||||JFMAMJJASONDJFMAMJJASOND Actual sales Moving average Weighted moving average Figure 4.2

© 2006 Prentice Hall, Inc.4 – 48  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

© 2006 Prentice Hall, Inc.4 – 49 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)

© 2006 Prentice Hall, Inc.4 – 50 Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20

© 2006 Prentice Hall, Inc.4 – 51 Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20 New forecast= (153 – 142)

© 2006 Prentice Hall, Inc.4 – 52 Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20 New forecast= (153 – 142) = = ≈ 144 cars

© 2006 Prentice Hall, Inc.4 – 53 Effect of Smoothing Constants Weight Assigned to Most2nd Most3rd Most4th Most5th Most RecentRecentRecentRecentRecent SmoothingPeriodPeriodPeriodPeriodPeriod Constant(  )  (1 -  )  (1 -  ) 2  (1 -  ) 3  (1 -  ) 4  =  =

© 2006 Prentice Hall, Inc.4 – 54 Impact of Different  – – – – ||||||||| ||||||||| Quarter Demand  =.1 Actual demand  =.5

© 2006 Prentice Hall, Inc.4 – 55 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

© 2006 Prentice Hall, Inc.4 – 56 Common Measures of Error Mean Absolute Deviation (MAD) MAD = ∑ |actual - forecast| n Mean Squared Error (MSE) MSE = ∑ (forecast errors) 2 n

© 2006 Prentice Hall, Inc.4 – 57 Common Measures of Error Mean Absolute Percent Error (MAPE) MAPE = 100 ∑ |actual i - forecast i |/actual i n n i = 1

© 2006 Prentice Hall, Inc.4 – 58 Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =

© 2006 Prentice Hall, Inc.4 – 59 Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  = MAD = ∑ |deviations| n = 84/8 = For  =.10 = 100/8 = For  =.50

© 2006 Prentice Hall, Inc.4 – 60 Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  = MAD = 1,558/8 = For  =.10 = 1,612/8 = For  =.50 MSE = ∑ (forecast errors) 2 n

© 2006 Prentice Hall, Inc.4 – 61 Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  = MAD MSE = 45.62/8 = 5.70% For  =.10 = 54.8/8 = 6.85% For  =.50 MAPE = 100 ∑ |deviation i |/actual i n i = 1

© 2006 Prentice Hall, Inc.4 – 62 Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  = MAD MSE MAPE5.70%6.85%

© 2006 Prentice Hall, Inc.4 – 63 Exponential Smoothing with Trend Adjustment When a trend is present, exponential smoothing must be modified Forecast including (FIT t ) = trend exponentiallyexponentially smoothed (F t ) +(T t )smoothed forecasttrend

© 2006 Prentice Hall, Inc.4 – 64 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

© 2006 Prentice Hall, Inc.4 – 65 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

© 2006 Prentice Hall, Inc.4 – 66 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

© 2006 Prentice Hall, Inc.4 – 67 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

© 2006 Prentice Hall, Inc.4 – 68 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 1 FIT 2 = = units Step 3: Calculate FIT for Month 2

© 2006 Prentice Hall, Inc.4 – 69 Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t Table

© 2006 Prentice Hall, Inc.4 – 70 Exponential Smoothing with Trend Adjustment Example Figure 4.3 ||||||||| ||||||||| Time (month) Product demand – – – – – – 5 5 – 0 0 – Actual demand (A t ) Forecast including trend (FIT t )

© 2006 Prentice Hall, Inc.4 – 71 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 ^

© 2006 Prentice Hall, Inc.4 – 72 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 – 73 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 – 74 Least Squares Method Equations to calculate the regression variables b =  xy - nxy  x 2 - nx 2 y = a + bx ^ a = y - bx

© 2006 Prentice Hall, Inc.4 – 75 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 – 76 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 – 77 Least Squares Example ||||||||| – – – – – – – – – – – – Year Power demand Trend line, y = x ^

© 2006 Prentice Hall, Inc.4 – 78 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

© 2006 Prentice Hall, Inc.4 – 79 Seasonal Variations In Data The multiplicative seasonal model can modify trend data to accommodate seasonal variations in demand 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

© 2006 Prentice Hall, Inc.4 – 80 Seasonal Index Example Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec DemandAverageAverage Seasonal Month MonthlyIndex

© 2006 Prentice Hall, Inc.4 – 81 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

© 2006 Prentice Hall, Inc.4 – 82 Seasonal Index Example Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec DemandAverageAverage Seasonal Month MonthlyIndex

© 2006 Prentice Hall, Inc.4 – 83 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 2006

© 2006 Prentice Hall, Inc.4 – 84 Seasonal Index Example – – – – – – – – ||||||||||||JFMAMJJASONDJFMAMJJASOND||||||||||||JFMAMJJASONDJFMAMJJASOND Time Demand 2006 Forecast 2005 Demand 2004 Demand 2003 Demand

© 2006 Prentice Hall, Inc.4 – 85 San Diego Hospital 10,200 10,200 – 10,000 10,000 – 9,800 9,800 – 9,600 9,600 – 9,400 9,400 – 9,200 9,200 – 9,000 9,000 – |||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNovDec Month Inpatient Days Figure 4.6 Trend Data

© 2006 Prentice Hall, Inc.4 – 86 San Diego Hospital – – – – – – – 0.92 – |||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNovDec Month Index for Inpatient Days Figure 4.7 Seasonal Indices

© 2006 Prentice Hall, Inc.4 – 87 San Diego Hospital 10,200 10,200 – 10,000 10,000 – 9,800 9,800 – 9,600 9,600 – 9,400 9,400 – 9,200 9,200 – 9,000 9,000 – |||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNovDec Month Inpatient Days Figure Combined Trend and Seasonal Forecast

© 2006 Prentice Hall, Inc.4 – 88 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

© 2006 Prentice Hall, Inc.4 – 89 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 – 90 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 – 91 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 – 92 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 – 93 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 – 94 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 – 95 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 – 96 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 – 97  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 – 98 Correlation Coefficient r = n  xy -  x  y [n  x 2 - (  x) 2 ][n  y 2 - (  y) 2 ]

© 2006 Prentice Hall, Inc.4 – 99 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 – 100  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 – 101 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 – 102 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 – 103  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 – 104 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 – 105 Tracking Signal Tracking signal + 0 MADs – Upper control limit Lower control limit Time Signal exceeding limit Acceptable range

© 2006 Prentice Hall, Inc.4 – 106 Tracking Signal Example Cumulative AbsoluteAbsolute ActualForecastForecastForecast QtrDemandDemandErrorRSFEErrorErrorMAD

© 2006 Prentice Hall, Inc.4 – 107Cumulative 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

© 2006 Prentice Hall, Inc.4 – 108 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

© 2006 Prentice Hall, Inc.4 – 109 Focus Forecasting Developed at American Hardware Supply, focus forecasting is based on two principles: 1.Sophisticated forecasting models are not always better than simple models 2.There is no single techniques that should be used for all products or services This approach uses historical data to test multiple forecasting models for individual items The forecasting model with the lowest error is then used to forecast the next demand

© 2006 Prentice Hall, Inc.4 – 110 Forecasting in the Service Sector  Presents unusual challenges  Special need for short term records  Needs differ greatly as function of industry and product  Holidays and other calendar events  Unusual events

© 2006 Prentice Hall, Inc.4 – 111 Fast Food Restaurant Forecast 20% 20% – 15% 15% – 10% 10% – 5% 5% – (Lunchtime)(Dinnertime) Hour of day Percentage of sales Figure 4.12