Chapter 6 Forecasting n Quantitative Approaches to Forecasting n The Components of a Time Series n...

Chapter 6Chapter 6ForecastingForecasting

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 Using Smoothing Methods in Forecasting Using Smoothing Methods in Forecasting Using Trend Projection in Forecasting Using Trend Projection in Forecasting Using Trend and Seasonal Components in Using Trend and Seasonal Components in

ForecastingForecasting Using Regression Analysis in ForecastingUsing Regression Analysis in Forecasting Qualitative Approaches to ForecastingQualitative Approaches to Forecasting

Quantitative Approaches to Quantitative Approaches to ForecastingForecasting

Quantitative methodsQuantitative methods are based on an analysis of are based on an analysis of historical data with one or more time series.historical data with one or more time 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 series 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 procedure is that we are trying to forecast, the procedure is called a called a causal methodcausal method. .

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

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

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

below the trend line is attributable to the below the trend line is attributable to the cyclical cyclical componentcomponent 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 that short-term, unanticipated and non-recurring factors that affect the values of the time series. One cannot attempt affect the values of the time series. One cannot attempt to predict its impact on the time series in advance.to predict its impact on the time series in advance.

Measures of Forecast AccuracyMeasures of Forecast Accuracy

Mean Squared Error - Mean Squared Error - The average of the squared 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 Deviation - Mean Absolute Deviation - The mean of the The mean of the absolute values of all forecast errors is calculated, absolute values of all forecast errors is calculated, and the forecasting method or parameter(s) which and the 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.

Smoothing MethodsSmoothing Methods

In cases in which the time series is fairly In cases in which the time series is fairly stable and has no significant trend, stable and has no significant trend, seasonal, or cyclical effects, one can use seasonal, or cyclical effects, one can use smoothing methodssmoothing methods to average out the to average out the irregular components of the time series. irregular 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


Moving Average MethodMoving Average Method

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

Example: Robert’s DrugsExample: Robert’s Drugs

During the past ten weeks, sales of cases During the past ten weeks, sales of cases of Comfort brand headache medicine at Robert's of Comfort brand headache medicine at Robert's Drugs have been as follows:Drugs have 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 a 3-period moving average If Robert's uses a 3-period moving average to forecast sales, what is the forecast for Week to forecast sales, what is the forecast for Week 11?11?


Excel Spreadsheet Showing Input DataExcel Spreadsheet Showing Input DataA B C

1 Robert's Drugs2

3 Week (t ) Salest Forect+1

4 1 1105 2 1156 3 1257 4 1208 5 1259 6 120

10 7 13011 8 11512 9 11013 10 130

Example: Robert’s DrugsExample: Robert’s Drugs Steps to Moving Average Using ExcelSteps to Moving Average Using Excel

• Step 1: Select the Step 1: Select the ToolsTools pull-down menu. pull-down menu.• Step 2: Select the Step 2: Select the Data AnalysisData Analysis option. option.• Step 3: When the Data Analysis Tools dialog Step 3: When the Data Analysis Tools dialog

appears, choose Mappears, choose Moving Averageoving Average..• Step 4: When the Moving Average dialog box Step 4: When the Moving Average dialog box

appears:appears:

Enter B4:B13 in the Enter B4:B13 in the Input RangeInput Range box. box.

Enter 3 in the Enter 3 in the IntervalInterval box. box.

Enter C4 in the Enter C4 in the Output RangeOutput Range box. box.

Select Select OKOK..


Spreadsheet Showing Results Using Spreadsheet Showing Results Using nn = 3 = 3A B C

1 Robert's Drugs2

3 Week (t ) Salest Forect+1

4 1 110 #N/A5 2 115 #N/A6 3 125 116.77 4 120 120.08 5 125 123.39 6 120 121.7

10 7 130 125.011 8 115 121.712 9 110 118.313 10 130 118.3


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 n n periods' data and associating it with the periods' data and associating it with the midpoint of the periods. For example, the midpoint of the periods. For example, the average for periods 5, 6, and 7 is associated average for periods 5, 6, and 7 is associated with period 6. This methodology is useful in with period 6. This methodology is useful in the process of computing season indexes.the process of computing season indexes.


Weighted Moving Average MethodWeighted Moving Average Method

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

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 period], )[the forecasted value for the current period],

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


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?


Exponential Smoothing Exponential Smoothing

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

Let: Let: YYtt = actual sales in week = actual sales in week tt

FFt t = forecasted sales in week = forecasted sales in week tt

FF11 = = YY11 = 110 = 110

For other weeks, For other weeks, FFtt+1+1 = .1 = .1YYtt + .9 + .9FFtt

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

For For = .1, 1 - = .1, 1 - = .9 = .9

FF11 = 110 = 110

FF2 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) = 110.5

FF44 = .1 = .1YY33 + .9 + .9FF33 = .1(125) + .9(110.5) = 111.95 = .1(125) + .9(110.5) = 111.95

FF55 = .1 = .1YY44 + .9 + .9FF44 = .1(120) + .9(111.95) = 112.76 = .1(120) + .9(111.95) = 112.76

FF66 = .1 = .1YY55 + .9 + .9FF55 = .1(125) + .9(112.76) = 113.98 = .1(125) + .9(112.76) = 113.98

FF77 = .1 = .1YY66 + .9 + .9FF66 = .1(120) + .9(113.98) = 114.58 = .1(120) + .9(113.98) = 114.58

FF88 = .1 = .1YY77 + .9 + .9FF77 = .1(130) + .9(114.58) = 116.12 = .1(130) + .9(114.58) = 116.12

FF99 = .1 = .1YY88 + .9 + .9FF88 = .1(115) + .9(116.12) = 116.01 = .1(115) + .9(116.12) = 116.01

FF1010= .1= .1YY99 + .9 + .9FF99 = .1(110) + .9(116.01) = 115.41 = .1(110) + .9(116.01) = 115.41

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

For For = .8, 1 - = .8, 1 - = .2 = .2

FF11 = 110 = 110

FF22 = .8(110) + .2(110) = 110 = .8(110) + .2(110) = 110

FF33 = .8(115) + .2(110) = 114 = .8(115) + .2(110) = 114

FF44 = .8(125) + .2(114) = 122.80 = .8(125) + .2(114) = 122.80

FF55 = .8(120) + .2(122.80) = 120.56 = .8(120) + .2(122.80) = 120.56

FF66 = .8(125) + .2(120.56) = 124.11 = .8(125) + .2(120.56) = 124.11

FF77 = .8(120) + .2(124.11) = 120.82 = .8(120) + .2(124.11) = 120.82

FF88 = .8(130) + .2(120.82) = 128.16 = .8(130) + .2(120.82) = 128.16

FF99 = .8(115) + .2(128.16) = 117.63 = .8(115) + .2(128.16) = 117.63

FF1010= .8(110) + .2(117.63) = 111.53= .8(110) + .2(117.63) = 111.53


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 the nine weeks of forecasts, weeks 2 through 10 by:10 by:

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


= .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

108.25108.25108.25108.25 94.1794.17 94.1794.17



1 Robert's Drugs23 Week Sales4 1 1105 2 1156 3 1257 4 1208 5 1259 6 120

10 7 13011 8 11512 9 11013 10 130

Example: Robert’s DrugsExample: Robert’s Drugs Steps to Exponential Smoothing Using ExcelSteps to Exponential Smoothing Using Excel

• Step 1: Select the Step 1: Select the ToolsTools pull-down menu. pull-down menu.• Step 2: Select the Step 2: Select the Data AnalysisData Analysis option. option.• Step 3: When the Data Analysis Tools dialog Step 3: When the Data Analysis Tools dialog

appears, choose appears, choose Exponential SmoothingExponential Smoothing..• Step 4: When the Exponential Smoothing dialog Step 4: When the Exponential Smoothing dialog

box box appears:appears:

Enter B4:B13 in the Enter B4:B13 in the Input RangeInput Range box. box.

Enter 0.9 (for Enter 0.9 (for = 0.1) in = 0.1) in Damping FactorDamping Factor box. box.

Enter C4 in the Enter C4 in the Output RangeOutput Range box. box.



Spreadsheet Showing Results Using Spreadsheet Showing Results Using = 0.1 = 0.1A B C

1 Robert's Drugs2 = 0.1

3 Week (t ) Salest Forect +1

4 1 110 #N/A5 2 115 110.06 3 125 110.57 4 120 112.08 5 125 112.89 6 120 114.0

10 7 130 114.611 8 115 116.112 9 110 116.013 10 130 115.4


Repeating the Process for a = 0.8Repeating the Process for a = 0.8• Step 4: When the Exponential Smoothing Step 4: When the Exponential Smoothing

dialog box dialog box appears:appears:

Enter B4:B13 in the Enter B4:B13 in the Input Input RangeRange box. box.

Enter 0.2 (for Enter 0.2 (for = 0.8) in = 0.8) in Damping FactorDamping Factor box. box.

Enter D4 in the Enter D4 in the Output RangeOutput Range box.box.



Spreadsheet Showing Results Using Spreadsheet Showing Results Using = 0.1 and = 0.1 and = 0.8 = 0.8

A B C D1 Robert's Drugs2 = 0.1 = 0.8

3 Week (t ) Salest Forect +1 Forect +1

4 1 110 #N/A #N/A5 2 115 110.0 110.06 3 125 110.5 114.07 4 120 112.0 122.88 5 125 112.8 120.69 6 120 114.0 124.1

10 7 130 114.6 120.811 8 115 116.1 128.212 9 110 116.0 117.613 10 130 115.4 111.5

Trend ProjectionTrend Projection

If a time series exhibits a linear trend, the method If a time series exhibits a linear trend, the method of of least squaresleast squares may be used to determine a may be used to determine a trend line (projection) for future forecasts. trend line (projection) for future forecasts.

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

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

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

bb1 1 = 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 - - t t YYtt bb00 = = YY - - bb11tt

nnt t 22 - ( - (t t ))22

where: where: YYtt = observed value of the time series at time period = observed value of the time series at time 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

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

The number of plumbing repair jobs The number of plumbing repair jobs performed by Auger's Plumbing Service in each performed by Auger's Plumbing Service in each of the last nine months are listed below.of the last nine months 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 Forecast the number of repair jobs Auger's will perform in December using the least squares perform in December using the least squares method. method.



(month) (month) tt YYtt tYtYtt t t 22

(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


Trend Projection (continued)Trend Projection (continued)

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

nntYtYtt - - t t YYtt (9)(17844) - (45)(3480) (9)(17844) - (45)(3480) bb11 = = = = = 7.4 = 7.4 nnt t 22 - ( - (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) = 423.667423.667423.667423.667



1 Auger's Plumbing Service23 Month Calls4 1 3535 2 3876 3 3427 4 3748 5 3969 6 409

10 7 39911 8 41212 9 40813


Steps to Trend Projection Using ExcelSteps to Trend Projection Using Excel• Step 1: Select an empty cell (B13) in the Step 1: Select an empty cell (B13) in the

worksheet.worksheet.• Step 2: Select the Step 2: Select the InsertInsert pull-down menu. pull-down menu.• Step 3: Choose the Step 3: Choose the FunctionFunction option. option.• Step 4: When the Paste Function dialog box Step 4: When the Paste Function dialog box

appears:appears:Choose Statistical in Function Category box.

Choose Forecast in the Function Name box.

Select OK.

more . . . . . . .


Steps to Trend Projecting Using Excel Steps to Trend Projecting Using Excel (continued)(continued)• Step 5: When the Forecast dialog box Step 5: When the Forecast dialog box

appears:appears:

Enter 10 in the Enter 10 in the xx box (for box (for month 10).month 10).

Enter B4:B12 in the Enter B4:B12 in the Known y’sKnown y’s box.box.

Enter A4:A12 in the Enter A4:A12 in the Known x’sKnown x’s box.box.



Spreadsheet Showing Trend Projection for Spreadsheet Showing Trend Projection for Month 10Month 10

A B C1 Auger's Plumbing Service23 Month Calls4 1 3535 2 3876 3 3427 4 3748 5 3969 6 409

10 7 39911 8 41212 9 40813 10 423.667 Projected

Example: Auger’s Plumbing Service Example: Auger’s Plumbing Service (B)(B)

Forecast for December (month 10) Forecast for December (month 10) using a three-period (using a three-period (nn = 3) weighted moving = 3) weighted moving average with weights of .6, .3, and .1. average with weights of .6, .3, and .1.

Then, compare this month 10 weighted Then, compare this month 10 weighted moving average forecast with the Month 10 moving average forecast with the Month 10 trend projection forecast.trend projection forecast.


Three-Month Weighted Moving Average Three-Month Weighted Moving Average

The forecast for December will be the The forecast for December will be the weighted average of the preceding three weighted average of the preceding three months: September, October, and November.months: September, 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)

= =


FF1010 = 423.7 (from earlier slide) = 423.7 (from earlier slide)408.3408.3408.3408.3


ConclusionConclusion

Due to the positive trend component in Due to the positive trend component in the time series, the trend projection the time series, the trend projection produced a forecast that is more in tune produced a forecast that is more in tune with the trend that exists. The weighted with the trend that exists. The weighted moving average, even with heavy (.6) placed moving average, even with heavy (.6) placed on the current period, produced a forecast on the current period, produced a forecast that is lagging behind the changing data. that is lagging behind the changing data.

Forecasting with TrendForecasting with Trendand Seasonal Componentsand Seasonal Components

Steps of Multiplicative Time Series ModelSteps of Multiplicative Time Series Model

1. 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 factors3. Determine the seasonal and irregular factors

4. Determine the average seasonal factors.4. Determine the average seasonal factors.

5. Scale the seasonal factors (5. Scale the seasonal factors (SSt t ).).

6. Determine the deseasonalized data.6. Determine the deseasonalized data.

7. Determine a trend line of the deseasonalized data.7. Determine a trend line of the deseasonalized data.

8. Determine the deseasonalized predictions.8. Determine the deseasonalized predictions.

9. Take into account the seasonality.9. Take into account the seasonality.

Example: Terry’s Tie ShopExample: Terry’s Tie ShopBusiness at Terry's Tie Shop can be viewed as Business at Terry's Tie Shop can be viewed as

falling into three distinct seasons: (1) Christmas falling into three distinct seasons: (1) Christmas (November-December); (2) Father's Day (late May - mid-(November-December); (2) Father's Day (late May - mid-June); and (3) all other times. Average weekly sales (in June); and (3) all other times. Average weekly sales (in $'s) during each of these three seasons during the past $'s) during each of these three seasons during the past four years has been as follows: 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 Determine a forecast for the average weekly sales in year 5 for each of the three seasons.in year 5 for each of the three seasons.

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

Year Season (Year Season (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

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 year. There are three distinct seasons in each year. Hence, take a three season moving average to Hence, take a three season moving average to eliminate seasonal and irregular factors. For eliminate seasonal and irregular factors. For example the first moving average is: (1856 + 2012 + example the first moving average is: (1856 + 2012 + 985)/3 =1617.67. 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 1 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 center Note that the moving averages from step 1 center themselves on integer-valued periods because themselves on integer-valued periods because nn is is an odd number.an odd number.


3. 3. Determine the seasonal and irregular factors (Determine the seasonal and irregular factors (SSt t ,,IIt t ).).

Isolate the trend and cyclical components. For each Isolate the trend and cyclical components. For each period period tt, this is given by:, this is given by:

YYt t /(Moving Average for period /(Moving Average for period t t ).).

4. 4. Determine the average seasonal factors.Determine the average seasonal factors.

Averaging all Averaging all SSt t IItt values corresponding to that values corresponding to that

season:season:

Season 1: (1.163 + 1.196 + 1.181) /3 = 1.180Season 1: (1.163 + 1.196 + 1.181) /3 = 1.180

Season 2: (1.244 + 1.242 + 1.224 + 1.244) /4 = 1.238Season 2: (1.244 + 1.242 + 1.224 + 1.244) /4 = 1.238

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


5. Scale the seasonal factors (5. Scale the seasonal factors (SSt t ).).

Divide each seasonal factor by the Divide each seasonal factor by the average of the seasonal factors. Then average of the seasonal factors. Then average the seasonal factors = (1.180 + average the seasonal factors = (1.180 + 1.238 + .587)/3 = 1.002.1.238 + .587)/3 = 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


6. Determine the deseasonalized data.6. Determine the deseasonalized data.

Divide the data point values, Divide the data point values, YYt t , by , by SSt t ..

7. Determine a trend line of the 7. Determine a trend line of the deseasonalized data.deseasonalized data.

Using the least squares method for Using the least squares method for tt = = 1, 2, ..., 12, gives:1, 2, ..., 12, gives:

TTtt = 1580.11 + 33.96 = 1580.11 + 33.96tt

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 Multiply each deseasonalized prediction by its seasonal factor to give the following forecasts for year 5: seasonal 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) =2382238223822382

25412541254125411225122512251225

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, is separated from the others and is anonymous, is asked to respond to a sequential series of 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 the entire group are made known to each of the other panel members so that they may revise other panel members so that they may revise their previous forecast response. their previous forecast response.

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


Scenario WritingScenario Writing• Scenario writing consists of developing a Scenario writing consists of developing a

conceptual scenario of the future based conceptual scenario of the future based on a well defined set of assumptions. on a well defined set of assumptions.

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


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 committees or panels seeking to develop new ideas or solve complex problems.new ideas or solve complex problems.

• They often involve "brainstorming They often involve "brainstorming sessions". sessions".

• It is important in such sessions that any It is important in such sessions that any ideas or opinions be permitted to be ideas or opinions be permitted to be presented without regard to its relevancy presented without regard to its relevancy and without fear of criticism.and without fear of criticism.

The End of Chapter 6The End of Chapter 6

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