slides prepared by john s. loucks st. edward’s university...slideslide 99 using smoothing methods...

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Slides Prepared bySlides Prepared byJOHN S. LOUCKSJOHN S. LOUCKS

St. Edward’s UniversitySt. Edward’s University

© 2002 South© 2002 South--Western/Thomson LearningWestern/Thomson Learning™™

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Chapter 18Chapter 18ForecastingForecasting

Time Series and Time Series MethodsTime Series and Time Series MethodsComponents of a Time SeriesComponents of a Time SeriesSmoothing MethodsSmoothing MethodsTrend ProjectionTrend ProjectionTrend and Seasonal ComponentsTrend and Seasonal ComponentsRegression AnalysisRegression AnalysisQualitative Approaches to ForecastingQualitative Approaches to Forecasting

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Time Series and Time Series MethodsTime Series and Time Series Methods

By reviewing historical data over time, we can better By reviewing historical data over time, we can better understand the pattern of past behavior of a variable understand the pattern of past behavior of a variable and better predict the future behavior.and better predict the future behavior.A A time seriestime series is a set of observations on a variable is a set of observations on a variable measured over successive points in time or over measured over successive points in time or over successive periods of time.successive periods of time.The objective of time series methods is to discover a The objective of time series methods is to discover a pattern in the historical data and then extrapolate the pattern in the historical data and then extrapolate the pattern into the future.pattern into the future.The forecast is based solely on past values of the The forecast is based solely on past values of the variable and/or past forecast errors.variable and/or past forecast errors.

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The Components of a Time SeriesThe Components of a Time Series

Trend ComponentTrend Component•• It represents a gradual shifting of a time series to It represents a gradual shifting of a time series to

relatively higher or lower values over time.relatively higher or lower values over time.•• Trend is usually the result of changes in the Trend is usually the result of changes in the

population, demographics, technology, and/or population, demographics, technology, and/or consumer preferences.consumer preferences.

Cyclical ComponentCyclical Component•• It represents any recurring sequence of points It represents any recurring sequence of points

above and below the trend line lasting more than above and below the trend line lasting more than one year.one year.

•• We assume that this component represents We assume that this component represents multiyear cyclical movements in the economy.multiyear cyclical movements in the economy.

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Seasonal ComponentSeasonal Component•• It represents any repeating pattern, less than one It represents any repeating pattern, less than one

year in duration, in the time series.year in duration, in the time series.•• The pattern duration can be as short as an hour, or The pattern duration can be as short as an hour, or

even less. even less. Irregular ComponentIrregular Component•• It is the “catchIt is the “catch--all” factor that accounts for the all” factor that accounts for the

deviation of the actual time series value from what deviation of the actual time series value from what we would expect based on the other components.we would expect based on the other components.

•• It is caused by the shortIt is caused by the short--term, unanticipated, and term, unanticipated, and nonrecurring factors that affect the time series.nonrecurring factors that affect the time series.

The Components of a Time SeriesThe Components of a Time Series

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Forecast AccuracyForecast Accuracy

Mean Squared Error (MSE)Mean Squared Error (MSE)•• It is the average of the sum of all the squared It is the average of the sum of all the squared

forecast errors.forecast errors.Mean Absolute Deviation (MAD)Mean Absolute Deviation (MAD)•• It is the average of the absolute values of all the It is the average of the absolute values of all the

forecast errors.forecast errors.

One major difference between MSE and MAD is thatOne major difference between MSE and MAD is thatthe MSE measure is influenced much more by largethe MSE measure is influenced much more by largeforecast errors than by small errors.forecast errors than by small errors.

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Moving AveragesMoving Averages•• We use the average of the most recentWe use the average of the most recent n n data data

values in the time series as the forecast for the next values in the time series as the forecast for the next period.period.

•• The average changes, or moves, as new The average changes, or moves, as new observations become available.observations become available.

•• The moving average calculation isThe moving average calculation is

Moving Average = Moving Average = ΣΣ(most recent (most recent nn data values)/data values)/nn

Using Smoothing Methods in ForecastingUsing Smoothing Methods in Forecasting

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Weighted Moving AveragesWeighted Moving Averages•• This method involves selecting weights for each of This method involves selecting weights for each of

the data values and then computing a weighted the data values and then computing a weighted mean as the forecast.mean as the forecast.

•• For example, a 3For example, a 3--period weighted moving average period weighted moving average would be computed as follows.would be computed as follows.

FFtt + 1 + 1 = = ww11((YYtt -- 22) + ) + ww22((YYtt -- 11) + ) + ww33((YYtt) )

where the sum of the weights (where the sum of the weights (w w values) is 1.values) is 1.


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Exponential SmoothingExponential Smoothing•• It is a special case of the weighted moving It is a special case of the weighted moving

averages method in which we select only the averages method in which we select only the weight for the most recent observation.weight for the most recent observation.

•• The weight placed on the most recent observation The weight placed on the most recent observation is the value of the is the value of the smoothing constantsmoothing constant, , αα..

•• The weights for the other data values are The weights for the other data values are computed automatically and become smaller at an computed automatically and become smaller at an exponential rate as the observations become older. exponential rate as the observations become older.

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Exponential SmoothingExponential Smoothing

FFtt + 1 + 1 = = ααYYtt + (1 + (1 -- αα))FFtt

where where FFtt + 1 + 1 = forecast value for period = forecast value for period tt + 1+ 1YYtt = actual value for period = actual value for period tt + 1+ 1FFtt = forecast value for period = forecast value for period ttαα = smoothing constant (0 = smoothing constant (0 << αα << 1)1)

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Example: Executive Seminars, Inc.Example: Executive Seminars, Inc.

Executive Seminars specializes in conductingExecutive Seminars specializes in conductingmanagement development seminars. In order to bettermanagement development seminars. In order to betterplan future revenues and costs, management would likeplan future revenues and costs, management would liketo develop a forecasting model for their “Timeto develop a forecasting model for their “TimeManagement” seminar.Management” seminar.

Enrollments for the past ten “TM” seminars are:Enrollments for the past ten “TM” seminars are:

(oldest)(oldest) (newest)(newest)SeminarSeminar 11 22 33 44 55 66 77 88 99 1010Enroll. Enroll. 3434 4040 3535 3939 4141 3636 3333 3838 4343 4040

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Exponential SmoothingExponential SmoothingLet Let αα = .2, = .2, FF1 1 = = YY1 1 = 34= 34

FF2 2 = = ααYY11 + (1 + (1 -- αα))FF11= .2(34) + .8(34)= .2(34) + .8(34)= 34= 34

FF3 3 = = ααYY22 + (1 + (1 -- αα))FF22= .2(40) + .8(34)= .2(40) + .8(34)= 35.20= 35.20

FF4 4 = = ααYY33 + (1 + (1 -- αα))FF33= .2(35) + .8(35.20)= .2(35) + .8(35.20)= 35.16 = 35.16

. . . and so on. . . and so on

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SeminarSeminar Actual EnrollmentActual Enrollment Exp. Sm. ForecastExp. Sm. Forecast11 3434 34.0034.0022 4040 34.0034.0033 3535 35.2035.2044 3939 35.1635.1655 4141 35.9335.9366 3636 36.9436.9477 3333 36.7636.7688 3838 36.0036.0099 4343 36.4036.40

1010 4040 37.7237.721111 Forecast for the next seminarForecast for the next seminar = = 38.1838.18

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Equation for Linear TrendEquation for Linear Trend

TTtt = = bb00 + + bb11tt

wherewhereTTtt = trend value in period = trend value in period ttbb00 = intercept of the trend line= intercept of the trend linebb1 1 = slope of the trend line= slope of the trend line

tt = time= timeNote: Note: tt is the independent variable.is the independent variable.

Using Trend Projection in ForecastingUsing Trend Projection in Forecasting

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Computing the Slope (Computing the Slope (bb11) and Intercept () and Intercept (bb00))

bb11 = = ΣΣtYtYtt -- ((ΣΣt t ΣΣYYtt)/)/nnΣΣt t 22 -- ((ΣΣt t ))22//nn

bb00 = (= (ΣΣYYtt//nn) ) -- bb11ΣΣtt//n n = = YY -- bb11tt

wherewhereYYtt = actual value in period= actual value in period ttn = n = number of periods in time seriesnumber of periods in time series

Using Trend Projection in ForecastingUsing Trend Projection in Forecasting

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Example: Sailboat Sales, Inc.Example: Sailboat Sales, Inc.

Sailboat Sales is a major marine dealer in Chicago. Sailboat Sales is a major marine dealer in Chicago. The firm has experienced tremendous sales growth in The firm has experienced tremendous sales growth in the past several years. Management would like to the past several years. Management would like to develop a forecasting method that would enable develop a forecasting method that would enable them to better control inventories.them to better control inventories.

The annual sales, in number of boats, for one The annual sales, in number of boats, for one particular sailboat model for the past five years are:particular sailboat model for the past five years are:

YearYear 11 22 33 44 55SalesSales 1111 1414 2020 2626 3434

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Linear Trend EquationLinear Trend Equation

tt YYtt tYtYtt t t 22

11 1111 1111 1122 1414 2828 4433 2020 6060 9944 2626 104104 161655 3434 170170 2525

TotalTotal 1515 105105 373373 5555


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Trend ProjectionTrend Projection

bb1 1 = 373 = 373 -- (15)(105)/5 = 5.8(15)(105)/5 = 5.855 55 -- (15)(15)22/5/5

bb00 = 105/5 = 105/5 -- 5.8(15/5) = 3.65.8(15/5) = 3.6

TTtt = 3.6 + 5.8= 3.6 + 5.8tt

TT66 = 3.6 + 5.8(6) = 38.4= 3.6 + 5.8(6) = 38.4


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Trend and Seasonal ComponentsTrend and Seasonal Componentsin Forecastingin Forecasting

Multiplicative ModelMultiplicative ModelCalculating the Seasonal IndexesCalculating the Seasonal IndexesDeseasonalizingDeseasonalizing the Time Seriesthe Time SeriesUsing the Using the DeseasonalizingDeseasonalizing Time SeriesTime Seriesto Identify Trendto Identify Trend

Seasonal AdjustmentsSeasonal AdjustmentsCyclical ComponentCyclical Component

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Multiplicative ModelMultiplicative Model

Using Using TTtt , , SSt t , and , and IItt to identify the trend, seasonal, to identify the trend, seasonal, and irregular components at time and irregular components at time tt, we describe the , we describe the time series value time series value YYtt by the following by the following multiplicative multiplicative time series modeltime series model::

YYtt = = TTtt xx SStt xx IItt

TTtt is measured in units of the item being forecast.is measured in units of the item being forecast.SStt and and IItt are measured in relative terms, with values are measured in relative terms, with values above 1.00 indicating effects above the trend and above 1.00 indicating effects above the trend and values below 1.00 indicating effects below the trend.values below 1.00 indicating effects below the trend.

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Calculating the Seasonal IndexesCalculating the Seasonal Indexes

1. Compute a series of 1. Compute a series of n n --period centered moving period centered moving averages, where averages, where n n is the number of seasons in the is the number of seasons in the time series.time series.

2. If 2. If nn is an even number, compute a series of 2is an even number, compute a series of 2--period period centered moving averages.centered moving averages.

3. Divide each time series observation by the 3. Divide each time series observation by the corresponding centered moving average to identify corresponding centered moving average to identify the seasonalthe seasonal--irregular effect in the time series.irregular effect in the time series.

4. For each of the 4. For each of the nn seasons, average all the computed seasons, average all the computed seasonalseasonal--irregular values for that season to eliminate irregular values for that season to eliminate the irregular influence and obtain an estimate of the the irregular influence and obtain an estimate of the seasonal influence, called the seasonal influence, called the seasonal indexseasonal index, for that , for that season.season.

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DeseasonalizingDeseasonalizing the Time Seriesthe Time Series

The purpose of finding seasonal indexes is to remove The purpose of finding seasonal indexes is to remove the seasonal effects from the time series.the seasonal effects from the time series.This process is called This process is called deseasonalizingdeseasonalizing the time series.the time series.By dividing each time series observation by the By dividing each time series observation by the corresponding seasonal index, the result is a corresponding seasonal index, the result is a deseasonalizeddeseasonalized time series.time series.With With deseasonalizeddeseasonalized data, relevant comparisons can data, relevant comparisons can be made between observations in successive periods.be made between observations in successive periods.

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Using the Using the DeseasonalizingDeseasonalizing Time SeriesTime Seriesto Identify Trendto Identify Trend

To identify the linear trend, we use the linear To identify the linear trend, we use the linear regression procedure covered earlier; in this case, the regression procedure covered earlier; in this case, the data are the data are the deseasonalizeddeseasonalized time series values.time series values.In other words, In other words, YYtt now refers to the now refers to the deseasonalizeddeseasonalizedtime series value at time time series value at time tt and not to the actual value and not to the actual value of the time series.of the time series.The resulting line equation is used to make trend The resulting line equation is used to make trend projections, as it was earlier.projections, as it was earlier.

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Seasonal AdjustmentsSeasonal Adjustments

The final step in developing the forecast is to use the The final step in developing the forecast is to use the seasonal index to adjust the trend projection.seasonal index to adjust the trend projection.The forecast for period The forecast for period tt, season , season ss, is obtained by , is obtained by multiplying the trend projection for periodmultiplying the trend projection for period t t by the by the seasonal index for season seasonal index for season ss..

YYt,st,s = = IIss[[bb00 + + bb11((t t )])]

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Example: Eastern Athletic SuppliesExample: Eastern Athletic Supplies

Management of EAS would like to develop aManagement of EAS would like to develop aquarterly sales forecast for one of their tennis quarterly sales forecast for one of their tennis

rackets. rackets. Sales of tennis rackets is highly seasonal and hence Sales of tennis rackets is highly seasonal and hence

ananaccurate quarterly forecast could aid substantially accurate quarterly forecast could aid substantially

ininordering raw material used in manufacturing.ordering raw material used in manufacturing.

The quarterly sales data (000 units) for the The quarterly sales data (000 units) for the previousprevious

three years is shown on the next slide.three years is shown on the next slide.

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Year QuarterYear Quarter SalesSales11 11 33

22 9933 6644 22

22 11 4422 111133 8844 33

33 11 5522 151533 111144 33


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YearYear QuarterQuarter SalesSales 44--CMACMA 22--CMACMA11 11 33

22 99 5.005.0033 66 5.255.25 5.135.1344 22 5.755.75 5.505.50

22 11 44 6.256.25 6.006.0022 1111 6.506.50 6.386.3833 88 6.756.75 6.636.6344 33 7.757.75 7.257.25

33 11 55 8.508.50 8.138.1322 1515 8.508.50 8.508.5033 111144 33


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YearYear QuarterQuarter Sales 2Sales 2--CMACMA SeasSeas--IrregIrreg11 11 33

22 9933 66 5.135.13 1.171.1744 22 5.505.50 0.360.36

22 11 44 6.006.00 0.670.6722 1111 6.386.38 1.721.7233 88 6.636.63 1.211.2144 33 7.257.25 0.410.41

33 11 55 8.138.13 0.620.6222 1515 8.508.50 1.761.7633 111144 33


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QuarterQuarter SeasSeas--IrregIrreg ValuesValues Seas. IndexSeas. Index11 0.67, 0.620.67, 0.62 0.650.6522 1.72, 1.761.72, 1.76 1.741.7433 1.17, 1.211.17, 1.21 1.191.1944 0.36, 0.410.36, 0.41 0.390.39

Total =Total = 3.973.97

Seas.IndexSeas.Index Adj. FactorAdj. Factor Adj.Seas.IndexAdj.Seas.Index0.650.65 4/3.97 4/3.97 .655 .655 1.741.74 4/3.97 4/3.97 1.7531.7531.191.19 4/3.97 4/3.97 1.1991.1990.390.39 4/3.97 4/3.97 .393.393

Total = 4.000 Total = 4.000


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YearYear QuarterQuarter SalesSales Seas.IndexSeas.Index Deseas.SalesDeseas.Sales11 11 33 .655.655 4.584.58

22 99 1.7531.753 5.135.1333 66 1.1991.199 5.005.0044 22 .393.393 5.095.09

22 11 44 .655.655 6.116.1122 1111 1.7531.753 6.276.2733 88 1.1991.199 6.676.6744 33 .393.393 7.637.63

33 11 55 .655.655 7.637.6322 1515 1.7531.753 8.568.5633 1111 1.1991.199 9.179.1744 33 .393.393 7.637.63


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Trend ProjectionTrend Projection

TTtt = 4.066 + .3933= 4.066 + .3933ttTT1313 = 4.066 + .3993(13) = 9.1789= 4.066 + .3993(13) = 9.1789

Using the trend component only, Using the trend component only, we would forecast we would forecast sales of 9,179 tennis rackets sales of 9,179 tennis rackets for period 13 (year 4, for period 13 (year 4, quarter 1).quarter 1).

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Seasonal AdjustmentsSeasonal Adjustments

PeriodPeriod TrendTrend SeasonalSeasonal QuarterlyQuarterlytt ForecForec. . IndexIndex ForecastForecast

13 9,17913 9,179 .655.655 6,0126,01214 9,57214 9,572 1.7531.753 16,78016,78015 9,96615 9,966 1.1991.199 11,94911,9491616 10,35910,359 .393.393 4,0714,071

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Models Based on Monthly DataModels Based on Monthly Data

Many businesses use monthly rather than quarterly Many businesses use monthly rather than quarterly forecasts.forecasts.The preceding procedures can be applied with minor The preceding procedures can be applied with minor modifications:modifications:•• A 12A 12--month moving average replaces the 4month moving average replaces the 4--

quarter moving average.quarter moving average.•• 12 monthly, rather than 4 quarterly, seasonal 12 monthly, rather than 4 quarterly, seasonal

indexes must be computed.indexes must be computed.•• Otherwise, the procedures are identical.Otherwise, the procedures are identical.

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The multiplicative model can be expanded to include The multiplicative model can be expanded to include a cyclical component that is expressed as a a cyclical component that is expressed as a percentage of trend.percentage of trend.

However, there are difficulties in including a cyclical However, there are difficulties in including a cyclical component:component:•• A cycle can span several (many) years and enough A cycle can span several (many) years and enough

data must be obtained to estimate the cyclical data must be obtained to estimate the cyclical component.component.

•• Cycles usually vary in length.Cycles usually vary in length.

Cyclical ComponentCyclical Component

t t t t tY T C S I= × × ×t t t t tY T C S I= × × ×

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Regression AnalysisRegression Analysis

One or more independent variables can be used to One or more independent variables can be used to predict the value of a single dependent variable.predict the value of a single dependent variable.The time series value that we want to forecast is the The time series value that we want to forecast is the dependent variable.dependent variable.The independent variable(s) might include any The independent variable(s) might include any combination of the following:combination of the following:•• Previous values of the time series variable itselfPrevious values of the time series variable itself•• Economic/demographic variablesEconomic/demographic variables•• Time variablesTime variables

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An An autoregressive modelautoregressive model is a regression model in is a regression model in which the independent variables are previous values which the independent variables are previous values of the time series being forecast.of the time series being forecast.A A causal forecasting modelcausal forecasting model uses other time series uses other time series related to the one being forecast in an effort to related to the one being forecast in an effort to explain the cause of a time series’ behavior.explain the cause of a time series’ behavior.

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For a function involving For a function involving kk independent variables, we independent variables, we use the following notation:use the following notation:

YYtt = value of the time series in period = value of the time series in period ttxx11tt = value of independent variable 1 in period = value of independent variable 1 in period ttxx22tt = value of independent variable 2 in period = value of independent variable 2 in period tt

xxktkt = value of independent variable = value of independent variable kk in period in period tt

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In forecasting sales of refrigerators, we might select In forecasting sales of refrigerators, we might select the following five independent variables:the following five independent variables:

xx11tt = price of refrigerator in period = price of refrigerator in period ttxx22tt = total industry sales in period = total industry sales in period tt -- 11xx33tt = number of new= number of new--house building permits house building permits

in period in period tt -- 1 1 xx44tt = population forecast for period = population forecast for period ttxx55tt = advertising budget for period = advertising budget for period tt

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The The nn periods of data necessary to develop the periods of data necessary to develop the estimated regression equation would appear as:estimated regression equation would appear as:

PeriodPeriod Time Series Value of Independent VariablesTime Series Value of Independent Variables((tt)) ((YYtt)) ((xx11tt) () (xx22tt) () (xx33tt) . . () . . (xxktkt))

11 YY11 xx1111 xx2121 xx3131 . . . . xxkk1122 YY22 xx1212 xx2222 xx3232 . . . . xxkk22.. .. . . . . . .. . . . . .. . .. . . . . . .. . . . . .nn YYnn xx11nn xx22nn xx33nn . . . . xxknkn

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Qualitative Approaches to ForecastingQualitative Approaches to Forecasting

Delphi MethodDelphi Method•• It is an attempt to develop forecasts through It is an attempt to develop forecasts through

“group consensus.”“group consensus.”•• The goal is to produce a relatively narrow spread The goal is to produce a relatively narrow spread

of opinions within which the majority of the panel of opinions within which the majority of the panel of experts concur.of experts concur.

Expert JudgmentExpert Judgment•• Experts individually consider information that Experts individually consider information that

they believe will influence the variable; then they they believe will influence the variable; then they combine their conclusions into a forecast.combine their conclusions into a forecast.

•• No two experts are likely to consider the same No two experts are likely to consider the same information in the same way.information in the same way.

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Qualitative Approaches to ForecastingQualitative Approaches to Forecasting

Scenario WritingScenario Writing•• This procedure involves developing several This procedure involves developing several

conceptual scenarios, each based on a wellconceptual scenarios, each based on a well--defined defined set of assumptions.set of assumptions.

•• The decision maker must decide how likely each The decision maker must decide how likely each scenario is and then make decisions accordingly.scenario is and then make decisions accordingly.

Intuitive ApproachesIntuitive Approaches•• A committee or panel seeks to develop new ideas A committee or panel seeks to develop new ideas

or solve complex problems through a series of or solve complex problems through a series of “brainstorming sessions.”“brainstorming sessions.”

•• Individuals are free to present any idea without Individuals are free to present any idea without being concerned about criticism or relevancy.being concerned about criticism or relevancy.

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End of Chapter 18End of Chapter 18

slides prepared by john s. loucks st. edward’s university...slideslide 99 using smoothing methods...

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