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18-1Chapter18ForecastingLearningObjectives1.Understandthatthelong-runsuccessofanorganizationisoftencloselyrelatedtohowwellmanagementisabletopredictfutureaspectsoftheoperation.2.Knowthevariouscomponentsofatimeseries.3.Beabletousesmoothingtechniquessuchasmovingaveragesandexponentialsmoothing.4.Beabletousetheleastsquaresmethodtoidentifythetrendcomponentofatimeseries.5.Understandhowtheclassicaltimeseriesmodelcanbeusedtoexplainthepatternorbehaviorofthedatainatimeseriesandtodevelopaforecastforthetimeseries.6.Beabletodetermineanduseseasonalindexesforatimeseries.7.Knowhowregressionmodelscanbeusedinforecasting.8.Knowthedefinitionofthefollowingterms:timeseriesmeansquarederrorforecastmovingaveragestrendcomponentweightedmovingaveragescyclicalcomponentsmoothingconstantseasonalcomponentseasonalconstantirregularcomponentChapter1818-2Solutions:1.a.WeekTime-SeriesValueForecastForecastError(Error)2182133154171252551615116916-74975Forecastforweek7is(17+16+9)/3=14b.MSE=75/3=25c.Smoothingconstant=.3.WeektTime-SeriesValueYtForecastFtForecastErrorYt-FtSquaredError(Yt-Ft)2182138.005.0025.003159.006.0036.0041710.206.8046.2451611.564.4419.716912.45-3.4511.90138.85138.85Forecastforweek7is.2(9)+.8(12.45)=11.76d.Forthe=.2exponentialsmoothingforecastMSE=138.85/5=27.77.Sincethethree-weekmovingaveragehasasmallerMSE,itappearstoprovidethebetterforecasts.e.Smoothingconstant=.4.WeektTime-SeriesValueYtForecastFtForecastErrorYt-FtSquaredError(Yt-Ft)2182138.05.025.0031510.05.025.0041712.05.025.0051614.02.04.006914.8-5.833.64112.64MSE=112.64/5=22.53.Asmoothingconstantof.4appearstoprovidebetterforecasts.Forecastforweek7is.4(9)+.6(14.8)=12.48Forecasting18-32.a.WeekTime-SeriesValue4-WeekMovingAverageForecast(Error)25-WeekMovingAverageForecast(Error)211722131942351820.004.0061620.2518.0619.6012.9672019.001.0019.400.3681819.251.5619.201.4492218.0016.0019.009.00102019.001.0018.801.44111520.0025.0019.2017.64122218.7510.5619.009.0077.1851.84b.MSE(4-Week)=77.18/8=9.65MSE(5-Week)=51.84/7=7.41c.Forthelimiteddataprovided,the5-weekmovingaverageprovidesthesmallestMSE.3.a.WeekTime-SeriesValueWeightedMovingAverageForecastForecastError(Error)211722131942319.333.6713.4751821.33-3.3311.0961619.83-3.8314.6772017.832.174.7181818.33-0.330.1192218.333.6713.47102020.33-0.330.11111520.33-5.3328.41122217.834.1717.39103.43b.MSE=103.43/9=11.49Prefertheunweightedmovingaveragehere.c.Youcouldalwaysfindaweightedmovingaverageatleastasgoodastheunweightedone.Actuallytheunweightedmovingaverageisaspecialcaseoftheweightedoneswheretheweightsareequal.Chapter1818-44.WeekTime-SeriesValueForecastError(Error)211722117.004.0016.0031917.401.602.5642317.565.4429.5951818.10-0.100.0161618.09-2.094.3772017.882.124.4981818.10-0.100.0192218.093.9115.29102018.481.522.31111518.63-3.6313.18122218.273.7313.91101.72101.72MSE=101.72/11=9.25=.2providedalowerMSE;therefore=.2isbetterthan5.a.F13=.2Y12+.16Y11+.64(.2Y10+.8F10)=.2Y12+.16Y11+.128Y10+.512F10F13=.2Y12+.16Y11+.128Y10+.512(.2Y9+.8F9)=.2Y12+.16Y11+.128Y10+.1024Y9+.4096F9F13=.2Y12+.16Y11+.128Y10+.1024Y9+.4096(.2Y8+.8F8)=.2Y12+.16Y11+.128Y10+.1024Y9+.08192Y8+.32768F8b.Themorerecentdatareceivesthegreaterweightorimportanceindeterminingtheforecast.Themovingaveragesmethodweightsthelastndatavaluesequallyindeterminingtheforecast.6.a.MonthYt3-MonthMovingAveragesForecast(Error)2=2Forecast(Error)218028280.004.0038480.4012.9648382.001.0081.123.5358383.000.0081.502.2568483.330.4581.804.8478583.332.7982.247.6288484.000.0082.791.4698284.335.4383.031.06108383.670.4582.830.03118483.001.0082.861.30128383.000.0083.090.0111.1239.06MSE(3-Month)=11.12/9=1.24Forecasting18-5MSE(=.2)=39.06/11=3.55Use3-monthmovingaverages.b.(83+84+83)/3=83.37.a.MonthTime-SeriesValue3-MonthMovingAverageForecast(Error)24-MonthMovingAverageForecast(Error)219.529.339.449.69.400.0459.89.430.149.450.1269.79.600.019.530.0379.89.700.019.630.03810.59.770.539.730.5999.910.000.019.950.00109.710.070.149.980.08119.610.030.189.970.14129.69.730.029.920.101.081.09MSE(3-Month)=1.08/9=.12MSE(4-Month)=1.09/8=.14Use3-Monthmovingaverages.b.Forecast=(9.7+9.6+9.6)/3=9.63c.Forthelimiteddataprovided,the5-weekmovingaverageprovidesthesmallestMSE.8.a.MonthTime-SeriesValue3-MonthMovingAverageForecast(Error)2=.2Forecast(Error)212402350240.0012100.003230262.001024.004260273.33177.69255.6019.365280280.000.00256.48553.196320256.674010.69261.183459.797220286.674444.89272.952803.708310273.331344.69262.362269.579240283.331877.49271.891016.9710310256.672844.09265.511979.3611240286.672178.09274.411184.0512230263.331110.89267.531408.5017,988.5227,818.49MSE(3-Month)=17,988.52/9=1998.72MSE(=.2)=27,818.49/11=2528.95Chapter1818-6BasedontheaboveMSEvalues,the3-monthmovingaveragesappearsbetter.However,exponentialsmoothingwaspenalizedbyincludingmonth2whichwasdifficultforanymethodtoforecast.Usingonlytheerrorsformonths4to12,theMSEforexponentialsmoothingisrevisedtoMSE(=.2)=14,694.49/9=1632.72Thus,exponentialsmoothingwasbetterconsideringmonths4to12.b.Usingexponentialsmoothing,F13=Y12+(1-)F12=.20(230)+.80(267.53)=2609.a.Sm
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