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© 2000 South-Western College Publishing/ITP© 2000 South-Western College Publishing/ITPSlides Prepared by JOHN Slides Prepared by JOHN LOUCKSLOUCKS

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Chapter 16Chapter 16ForecastingForecasting

Quantitative Approaches to ForecastingQuantitative Approaches to Forecasting The Components of a Time SeriesThe Components of a Time Series Measures of Forecast AccuracyMeasures of Forecast Accuracy Forecasting Using Smoothing MethodsForecasting Using Smoothing Methods Forecasting Using Trend ProjectionForecasting Using Trend Projection Forecasting with Trend and Seasonal Forecasting with Trend and Seasonal

ComponentsComponents Forecasting Using Regression ModelsForecasting Using Regression Models Qualitative Approaches to ForecastingQualitative Approaches to Forecasting

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Quantitative Approaches to ForecastingQuantitative Approaches to Forecasting Quantitative methodsQuantitative methods are based on an analysis of are based on an analysis of

historical data concerning one or more time historical data concerning one or more time series.series.

A A time seriestime series is a set of observations measured at is a set of observations measured at successive points in time or over successive successive points in time or over successive periods of time.periods of time.

If the historical data used are restricted to past If the historical data used are restricted to past values of the series that we are trying to forecast, values of the series that we are trying to forecast, the procedure is called a the procedure is called a time series methodtime series method..

If the historical data used involve other time If the historical data used involve other time series that are believed to be related to the time series that are believed to be related to the time series that we are trying to forecast, the series that we are trying to forecast, the procedure is called a procedure is called a causal methodcausal method. .

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Time Series MethodsTime Series Methods Three time series methods are: Three time series methods are:

• smoothingsmoothing• trend projectiontrend projection• trend projection adjusted for seasonal trend projection adjusted for seasonal

influenceinfluence

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Components of a Time SeriesComponents of a Time Series The The trend componenttrend component accounts for the gradual accounts for the gradual

shifting of the time series over a long period of time.shifting of the time series over a long period of time. Any regular pattern of sequences of values above Any regular pattern of sequences of values above

and below the trend line is attributable to the and below the trend line is attributable to the cyclical componentcyclical component of the series. of the series.

The The seasonal componentseasonal component of the series accounts for of the series accounts for regular patterns of variability within certain time regular patterns of variability within certain time periods, such as over a year.periods, such as over a year.

The The irregular componentirregular component of the series is caused by of the series is caused by short-term, unanticipated and non-recurring factors short-term, unanticipated and non-recurring factors that affect the values of the time series. One cannot that affect the values of the time series. One cannot attempt to predict its impact on the time series in attempt to predict its impact on the time series in advance.advance.

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Measures of Forecast AccuracyMeasures of Forecast Accuracy Mean Squared ErrorMean Squared Error

In this measure the average of the squared In this measure the average of the squared forecast errors for the historical data is calculated. forecast errors for the historical data is calculated. The forecasting method or parameter(s) which The forecasting method or parameter(s) which minimize this mean squared error is then selected.minimize this mean squared error is then selected.

Mean Absolute DeviationMean Absolute DeviationIn this measure, the mean of the absolute values In this measure, the mean of the absolute values of all forecast errors is calculated, and the of all forecast errors is calculated, and the forecasting method or parameter(s) which forecasting method or parameter(s) which minimize this measure is selected. The mean minimize this measure is selected. The mean absolute deviation measure is less sensitive to absolute deviation measure is less sensitive to individual large forecast errors than the mean individual large forecast errors than the mean squared error measure.squared error measure.

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Smoothing MethodsSmoothing Methods In cases in which the time series is fairly stable In cases in which the time series is fairly stable

and has no significant trend, seasonal, or and has no significant trend, seasonal, or cyclical effects, one can use cyclical effects, one can use smoothing smoothing methodsmethods to average out the irregular to average out the irregular components of the time series. components of the time series.

Four common smoothing methods are:Four common smoothing methods are:• Moving averagesMoving averages• Centered moving averagesCentered moving averages• Weighted moving averagesWeighted moving averages• Exponential smoothingExponential smoothing

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Smoothing MethodsSmoothing Methods Moving Average MethodMoving Average Method

The The moving average methodmoving average method consists of consists of computing an average of the most recent computing an average of the most recent nn data data values for the series and using this average for values for the series and using this average for forecasting the value of the time series for the forecasting the value of the time series for the next period.next period.

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Smoothing MethodsSmoothing Methods Centered Moving Average MethodCentered Moving Average Method

The The centered moving average methodcentered moving average method consists of computing an average of consists of computing an average of nn periods' periods' data and associating it with the midpoint of the data and associating it with the midpoint of the periods. For example, the average for periods 5, periods. For example, the average for periods 5, 6, and 7 is associated with period 6. This 6, and 7 is associated with period 6. This methodology is useful in the process of methodology is useful in the process of computing season indexes.computing season indexes.

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Smoothing MethodsSmoothing Methods Weighted Moving Average MethodWeighted Moving Average Method

In the In the weighted moving average methodweighted moving average method for computing the average of the most recent for computing the average of the most recent nn periods, the more recent observations are periods, the more recent observations are typically given more weight than older typically given more weight than older observations. For convenience, the weights observations. For convenience, the weights usually sum to 1.usually sum to 1.

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Smoothing MethodsSmoothing Methods Exponential SmoothingExponential Smoothing

• Using Using exponential smoothingexponential smoothing, the forecast for , the forecast for the next period is equal to the forecast for the the next period is equal to the forecast for the current period plus a proportion (current period plus a proportion () of the ) of the forecast error in the current period.forecast error in the current period.

• Using exponential smoothing, the forecast is Using exponential smoothing, the forecast is calculated by: calculated by:

[the actual value for the current period] [the actual value for the current period] ++

(1- (1- )[the forecasted value for the current )[the forecasted value for the current period], period],

where the smoothing constant, where the smoothing constant, , is a number , is a number between 0 and 1.between 0 and 1.

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Trend ProjectionTrend Projection If a time series exhibits a linear trend, the If a time series exhibits a linear trend, the

method of method of least squaresleast squares may be used to may be used to determine a trend line (projection) for future determine a trend line (projection) for future forecasts. forecasts.

Least squares, also used in regression Least squares, also used in regression analysis, determines the unique analysis, determines the unique trend line trend line forecastforecast which minimizes the mean square which minimizes the mean square error between the trend line forecasts and the error between the trend line forecasts and the actual observed values for the time series.actual observed values for the time series.

The independent variable is the time period The independent variable is the time period and the dependent variable is the actual and the dependent variable is the actual observed value in the time series.observed value in the time series.

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Trend ProjectionTrend Projection Using the method of least squares, the formula for the Using the method of least squares, the formula for the

trend projection is: trend projection is: TTtt = = bb00 + + bb11tt. . where: where: TTtt = trend forecast for time period = trend forecast for time period tt bb11= slope of the trend line= slope of the trend line

bb00 = trend line projection for time 0 = trend line projection for time 0

bb11 = = nntYtYtt - - ttYYtt bb00 = = YY - - bb11tt

nntt22 - ( - (tt))22

where: where: YYtt = observed value of the time series at time = observed value of the time series at time period period tt YY = average of the observed values for = average of the observed values for YYtt tt = average time period for the = average time period for the nn observations observations

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Example: Roberts’s DrugsExample: Roberts’s Drugs

During the past ten weeks, sales of cases of Comfort During the past ten weeks, sales of cases of Comfort brand headache medicine at Robert's Drugs have brand headache medicine at Robert's Drugs have been as follows:been as follows:

Week Sales Week SalesWeek Sales Week Sales 1 110 6 1201 110 6 120 2 115 7 1302 115 7 130 3 125 8 1153 125 8 115 4 120 9 1104 120 9 110 5 125 10 1305 125 10 130

If Robert's uses exponential smoothing to forecast If Robert's uses exponential smoothing to forecast sales, which value for the smoothing constant sales, which value for the smoothing constant , , = = .1 or .1 or = .8, gives better forecasts? = .8, gives better forecasts?

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Example: Robert’s DrugsExample: Robert’s Drugs Exponential Smoothing Exponential Smoothing

To evaluate the two smoothing constants, To evaluate the two smoothing constants, determine how the forecasted values would determine how the forecasted values would compare with the actual historical values in each compare with the actual historical values in each case. case.

Let: Let: YYtt = actual sales in week = actual sales in week ttFFt t = forecasted sales in week = forecasted sales in week tt

FF11 = = YY11 = 110 = 110For other weeks, For other weeks, FFtt+1+1 = .1 = .1YYtt + .9 + .9FFtt

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Example: Robert’s DrugsExample: Robert’s Drugs Exponential SmoothingExponential Smoothing

For For = .1, 1 - = .1, 1 - = .9 = .9FF11 = 110 = 110FF2 2 = .1= .1YY11 + .9 + .9FF11 = .1(110) + .9(110) = 110 = .1(110) + .9(110) = 110 FF33 = .1 = .1YY22 + .9 + .9FF22 = .1(115) + .9(110) = 110.5 = .1(115) + .9(110) = 110.5FF44 = .1 = .1YY33 + .9 + .9FF33 = .1(125) + .9(110.5) = 111.95 = .1(125) + .9(110.5) = 111.95FF55 = .1 = .1YY44 + .9 + .9FF44 = .1(120) + .9(111.95) = 112.76 = .1(120) + .9(111.95) = 112.76FF66 = .1 = .1YY55 + .9 + .9FF55 = .1(125) + .9(112.76) = 113.98 = .1(125) + .9(112.76) = 113.98FF77 = .1 = .1YY66 + .9 + .9FF66 = .1(120) + .9(113.98) = 114.58 = .1(120) + .9(113.98) = 114.58FF88 = .1 = .1YY77 + .9 + .9FF77 = .1(130) + .9(114.58) = 116.12 = .1(130) + .9(114.58) = 116.12FF99 = .1 = .1YY88 + .9 + .9FF88 = .1(115) + .9(116.12) = 116.01 = .1(115) + .9(116.12) = 116.01FF1010= .1= .1YY99 + .9 + .9FF99 = .1(110) + .9(116.01) = 115.41 = .1(110) + .9(116.01) = 115.41

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Example: Robert’s DrugsExample: Robert’s Drugs Exponential SmoothingExponential Smoothing

For For = .8, 1 - = .8, 1 - = .2 = .2FF11 = 110 = 110FF22 = .8(110) + .2(110) = 110 = .8(110) + .2(110) = 110FF33 = .8(115) + .2(110) = 114 = .8(115) + .2(110) = 114FF44 = .8(125) + .2(114) = 122.80 = .8(125) + .2(114) = 122.80FF55 = .8(120) + .2(122.80) = 120.56 = .8(120) + .2(122.80) = 120.56FF66 = .8(125) + .2(120.56) = 124.11 = .8(125) + .2(120.56) = 124.11FF77 = .8(120) + .2(124.11) = 120.82 = .8(120) + .2(124.11) = 120.82FF88 = .8(130) + .2(120.82) = 128.16 = .8(130) + .2(120.82) = 128.16FF99 = .8(115) + .2(128.16) = 117.63 = .8(115) + .2(128.16) = 117.63FF1010= .8(110) + .2(117.63) = 111.53= .8(110) + .2(117.63) = 111.53

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Example: Robert’s DrugsExample: Robert’s Drugs Mean Squared ErrorMean Squared Error

In order to determine which smoothing In order to determine which smoothing constant gives the better performance, constant gives the better performance, calculate, for each, the mean squared error for calculate, for each, the mean squared error for the nine weeks of forecasts, weeks 2 through 10 the nine weeks of forecasts, weeks 2 through 10 by:by:

[([(YY22--FF22))22 + ( + (YY33--FF33))22 + ( + (YY44--FF44))22 + . . . + ( + . . . + (YY1010--FF1010))22]/9]/9

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Example: Robert’s DrugsExample: Robert’s Drugs = .1 = .1 = .8 = .8

Week Week YYtt FFtt ( (YYtt - - FFtt))22 FFt t ((YYtt - - FFtt))22

1 110 1 110 2 115 110.00 25.00 110.00 25.002 115 110.00 25.00 110.00 25.00 3 125 110.50 210.25 114.00 121.003 125 110.50 210.25 114.00 121.00 4 120 111.95 64.80 122.80 7.844 120 111.95 64.80 122.80 7.84 5 125 112.76 149.94 120.56 19.715 125 112.76 149.94 120.56 19.71 6 120 113.98 36.25 124.11 16.916 120 113.98 36.25 124.11 16.91 7 130 114.58 237.73 120.82 84.237 130 114.58 237.73 120.82 84.23 8 115 116.12 1.26 128.16 173.308 115 116.12 1.26 128.16 173.30 9 110 116.01 36.12 117.63 58.269 110 116.01 36.12 117.63 58.26 10 130 115.41 212.87 111.53 341.2710 130 115.41 212.87 111.53 341.27

Sum 974.22 Sum 847.52Sum 974.22 Sum 847.52 MSE Sum/9 Sum/9MSE Sum/9 Sum/9

108.25108.25 94.1794.17

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Example: Auger’s Plumbing ServiceExample: Auger’s Plumbing Service

The number of plumbing repair jobs performed by The number of plumbing repair jobs performed by Auger's Plumbing Service in each of the last nine months Auger's Plumbing Service in each of the last nine months are listed below.are listed below.

MonthMonth JobsJobs MonthMonth JobsJobs MonthMonth JobsJobs

March 353 June 374 September 399March 353 June 374 September 399 April 387 July 396 October 412April 387 July 396 October 412 May 342 August 409 November 408May 342 August 409 November 408

Forecast the number of repair jobs Auger's will perform Forecast the number of repair jobs Auger's will perform in December using the least squares method. Also, in December using the least squares method. Also, forecast for December using a three-period weighted forecast for December using a three-period weighted moving average with weights of .6, .3, and .1. Then, moving average with weights of .6, .3, and .1. Then, compare the two forecasts.compare the two forecasts.

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Example: Auger’s Plumbing ServiceExample: Auger’s Plumbing Service Trend ProjectionTrend Projection (month) (month) tt YYtt tYtYtt tt22

(Mar.) 1 353 353 1 (Mar.) 1 353 353 1 (Apr.) 2 387 774 4(Apr.) 2 387 774 4 (May) 3 342 1026 9(May) 3 342 1026 9 (June) 4 374 1496 16(June) 4 374 1496 16 (July) 5 396 1980 25(July) 5 396 1980 25 (Aug.) 6 409 2454 36(Aug.) 6 409 2454 36 (Sep.) 7 399 2793 49(Sep.) 7 399 2793 49 (Oct.) 8 412 3296 64(Oct.) 8 412 3296 64 (Nov.) 9(Nov.) 9 408408 36723672 8181

Sum 45 3480 17844 285Sum 45 3480 17844 285

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Example: Auger’s Plumbing ServiceExample: Auger’s Plumbing Service Trend Projection (continued)Trend Projection (continued)

tt = 45/9 = 5 = 45/9 = 5 YY = 3480/9 = 386.667 = 3480/9 = 386.667

nntYtYtt - - ttYYtt (9)(17844) - (45)(3480) (9)(17844) - (45)(3480) bb11 = = = = = 7.4 = 7.4 nntt22 - ( - (tt))22 (9)(285) - (45) (9)(285) - (45)22 bb00 = = YY - - bb11tt = 386.667 - 7.4(5) = 349.667 = 386.667 - 7.4(5) = 349.667

TT1010 = 349.667 + (7.4)(10) = = 349.667 + (7.4)(10) = 423.667423.667

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Example: Auger’s Plumbing ServiceExample: Auger’s Plumbing Service Three-Month Weighted Moving Average Three-Month Weighted Moving Average

The forecast for December will be the weighted The forecast for December will be the weighted average of the preceding three months: September, average of the preceding three months: September, October, and November.October, and November. FF1010 = .1 = .1YYSep.Sep. + .3 + .3YYOct.Oct. + .6 + .6YYNov.Nov. = .1(399) + .3(412) + .6(408) = .1(399) + .3(412) + .6(408) = = Due to the positive trend component in the time Due to the positive trend component in the time series, the least squares method produced a forecast series, the least squares method produced a forecast that is more in tune with the trend that exists. The that is more in tune with the trend that exists. The weighted moving average, even with heavy (.6) weighted moving average, even with heavy (.6) placed on the current period, produced a forecast placed on the current period, produced a forecast that is lagging behind the changing data. that is lagging behind the changing data.

408.3408.3

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Forecasting with TrendForecasting with Trendand Seasonal Componentsand Seasonal Components

Steps of Multiplicative Time Series ModelSteps of Multiplicative Time Series Model1. Calculate the centered moving averages (CMAs).1. Calculate the centered moving averages (CMAs).2. Center the CMAs on integer-valued periods.2. Center the CMAs on integer-valued periods.3. Determine the seasonal and irregular factors 3. Determine the seasonal and irregular factors

((SSttIItt).).4. Determine the average seasonal factors.4. Determine the average seasonal factors.5. Scale the seasonal factors (5. Scale the seasonal factors (SStt).).6. Determine the deseasonalized data.6. Determine the deseasonalized data.7. Determine a trend line of the deseasonalized 7. Determine a trend line of the deseasonalized

data.data.8. Determine the deseasonalized predictions.8. Determine the deseasonalized predictions.9. Take into account the seasonality.9. Take into account the seasonality.

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Example: Terry’s Tie ShopExample: Terry’s Tie Shop

Business at Terry's Tie Shop can be viewed as falling Business at Terry's Tie Shop can be viewed as falling into three distinct seasons: (1) Christmas into three distinct seasons: (1) Christmas (November-December); (2) Father's Day (late May - (November-December); (2) Father's Day (late May - mid-June); and (3) all other times. Average weekly mid-June); and (3) all other times. Average weekly sales (in $'s) during each of these three seasons sales (in $'s) during each of these three seasons during the past four years has been as follows:during the past four years has been as follows:

YearYear SeasonSeason 11 22 33 44 1 1856 1995 2241 22801 1856 1995 2241 2280 2 2012 2168 2306 24082 2012 2168 2306 2408 3 985 1072 1105 11203 985 1072 1105 1120

Determine a forecast for the average weekly sales in Determine a forecast for the average weekly sales in year 5 for each of the three seasons.year 5 for each of the three seasons.

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Example: Terry’s Tie ShopExample: Terry’s Tie Shop Dollar Moving Scaled Dollar Moving Scaled

Year Season Sales (Year Season Sales (YYtt) Average ) Average SSttIItt SStt YYtt//SStt 1 1 1856 1.178 15761 1 1856 1.178 1576 2 2012 1617.67 1.244 1.236 16282 2012 1617.67 1.244 1.236 1628 3 985 1664.00 .592 .586 16813 985 1664.00 .592 .586 1681 2 1 1995 1716.00 1.163 1.178 16942 1 1995 1716.00 1.163 1.178 1694 2 2168 1745.00 1.242 1.236 17542 2168 1745.00 1.242 1.236 1754 3 1072 1827.00 .587 .586 18293 1072 1827.00 .587 .586 1829 3 1 2241 1873.00 1.196 1.178 19023 1 2241 1873.00 1.196 1.178 1902 2 2306 1884.00 1.224 1.236 18662 2306 1884.00 1.224 1.236 1866 3 1105 1897.00 .582 .586 18863 1105 1897.00 .582 .586 1886 4 1 2280 1931.00 1.181 1.178 19354 1 2280 1931.00 1.181 1.178 1935 2 2408 1936.00 1.244 1.236 19482 2408 1936.00 1.244 1.236 1948 3 1120 .586 19113 1120 .586 1911

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Example: Terry’s Tie ShopExample: Terry’s Tie Shop 1. Calculate the centered moving averages.1. Calculate the centered moving averages.

There are three distinct seasons in each There are three distinct seasons in each year. Hence, take a three season moving year. Hence, take a three season moving average to eliminate seasonal and irregular average to eliminate seasonal and irregular factors. For example the first moving average is: factors. For example the first moving average is: (1856 + 2012 + 985)/3 =1617.67. (1856 + 2012 + 985)/3 =1617.67.

2. Center the CMAs on integer-valued periods.2. Center the CMAs on integer-valued periods.The first moving average computed in step The first moving average computed in step

1 (1617.67) will be centered on season 2 of year 1 (1617.67) will be centered on season 2 of year 1. Note that the moving averages from step 1 1. Note that the moving averages from step 1 center themselves on integer-valued periods center themselves on integer-valued periods because because nn is an odd number. is an odd number.

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Example: Terry’s Tie ShopExample: Terry’s Tie Shop 3. Determine the seasonal and irregular factors (3. Determine the seasonal and irregular factors (SStt,,IItt). ).

Isolate the trend and cyclical components. For Isolate the trend and cyclical components. For each period each period tt, this is given by:, this is given by: YYtt/(Moving Average for period /(Moving Average for period tt).).

4. Determine the average seasonal factors. 4. Determine the average seasonal factors. Averaging all Averaging all SSttIItt values corresponding to that season:values corresponding to that season:

Season 1: (1.163 + 1.196 + 1.181) /3 = Season 1: (1.163 + 1.196 + 1.181) /3 = 1.1801.180

Season 2: (1.244 + 1.242 + 1.224 + 1.244) /4 = Season 2: (1.244 + 1.242 + 1.224 + 1.244) /4 = 1.2381.238

Season 3: (.592 + .587 + .582) /3 = Season 3: (.592 + .587 + .582) /3 = .587.587

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Example: Terry’s Tie ShopExample: Terry’s Tie Shop

5. Scale the seasonal factors (5. Scale the seasonal factors (SStt).).Divide each seasonal factor by the average Divide each seasonal factor by the average

of the seasonal factors. Then average the of the seasonal factors. Then average the seasonal factors = (1.180 + 1.238 + .587)/3 = seasonal factors = (1.180 + 1.238 + .587)/3 = 1.002.1.002.

Season 1: 1.180/1.002 = 1.178Season 1: 1.180/1.002 = 1.178 Season 2: 1.238/1.002 = 1.236Season 2: 1.238/1.002 = 1.236 Season 3: .587/1.002 = Season 3: .587/1.002 = .586 .586

Total = 3.000Total = 3.000

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Example: Terry’s Tie ShopExample: Terry’s Tie Shop 6. Determine the deseasonalized data.6. Determine the deseasonalized data.

Divide the data point values, Divide the data point values, YYtt, by , by SStt.. 7. Determine a trend line of the deseasonalized 7. Determine a trend line of the deseasonalized

data.data. Using the least squares method for Using the least squares method for tt = 1, = 1,

2, ..., 12, gives:2, ..., 12, gives: TTtt = 1580.11 + 33.96 = 1580.11 + 33.96tt

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Example: Terry’s Tie ShopExample: Terry’s Tie Shop 8. Determine the deseasonalized predictions.8. Determine the deseasonalized predictions.

Substitute Substitute tt = 13, 14, and 15 into the least squares = 13, 14, and 15 into the least squares equation:equation:

TT1313 = 1580.11 + (33.96)(13) = 2022 = 1580.11 + (33.96)(13) = 2022 TT1414 = 1580.11 + (33.96)(14) = 2056 = 1580.11 + (33.96)(14) = 2056 TT1515 = 1580.11 + (33.96)(15) = 2090 = 1580.11 + (33.96)(15) = 2090 9. Take into account the seasonality.9. Take into account the seasonality.

Multiply each deseasonalized prediction by its seasonal Multiply each deseasonalized prediction by its seasonal factor to give the following forecasts for year 5:factor to give the following forecasts for year 5:

Season 1: (1.178)(2022) =Season 1: (1.178)(2022) = Season 2: (1.236)(2056) =Season 2: (1.236)(2056) = Season 3: ( .586)(2090) =Season 3: ( .586)(2090) =

23822382

25412541

12251225

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Qualitative Approaches to ForecastingQualitative Approaches to Forecasting Delphi ApproachDelphi Approach

• A panel of experts, each of whom is physically A panel of experts, each of whom is physically separated from the others and is anonymous, separated from the others and is anonymous, is asked to respond to a sequential series of is asked to respond to a sequential series of questionnaires. questionnaires.

• After each questionnaire, the responses are After each questionnaire, the responses are tabulated and the information and opinions of tabulated and the information and opinions of the entire group are made known to each of the entire group are made known to each of the other panel members so that they may the other panel members so that they may revise their previous forecast response. revise their previous forecast response.

• The process continues until some degree of The process continues until some degree of consensus is achieved.consensus is achieved.

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Qualitative Approaches to ForecastingQualitative Approaches to Forecasting Scenario WritingScenario Writing

• Scenario writing consists of developing a Scenario writing consists of developing a conceptual scenario of the future based on a conceptual scenario of the future based on a well defined set of assumptions. well defined set of assumptions.

• After several different scenarios have been After several different scenarios have been developed, the decision maker determines developed, the decision maker determines which is most likely to occur in the future and which is most likely to occur in the future and makes decisions accordingly.makes decisions accordingly.

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Qualitative Approaches to ForecastingQualitative Approaches to Forecasting Subjective or Interactive ApproachesSubjective or Interactive Approaches

• These techniques are often used by These techniques are often used by committees or panels seeking to develop new committees or panels seeking to develop new ideas or solve complex problems.ideas or solve complex problems.

• They often involve "brainstorming sessions". They often involve "brainstorming sessions". • It is important in such sessions that any ideas It is important in such sessions that any ideas

or opinions be permitted to be presented or opinions be permitted to be presented without regard to its relevancy and without without regard to its relevancy and without fear of criticism.fear of criticism.

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The End of Chapter 16The End of Chapter 16