美賽(mcm)論文LaTeX模板

\documentclass[12pt]{article}\usepackage{CJK}\usepackage{titlesec,titletoc}\usepackage{indentfirst}\usepackage{graphicx}\usepackage{caption2}\usepackage{subfigure}\usepackage{longtable}\usepackage{slashbox}\usepackage{fancyhdr}\usepackage{times}\usepackage{amsmath}\special{papersize=21cm,29.7cm}\setlength{\textwidth}{15cm}\setlength{\textheight}{23cm}\setlength{\evensidemargin}{0.46cm}\setlength{\oddsidemargin}{0.46cm}\setlength{\topmargin}{-1.84cm}\setlength{\headheight}{2.9cm}\setlength{\headsep}{0.4cm}\newcommand{\chuhao}{\fontsize{42pt}{\baselineskip}\selectfont}\newcommand{\xiaochuhao}{\fontsize{36pt}{\baselineskip}\selectfont}\newcommand{\yihao}{\fontsize{26pt}{\baselineskip}\selectfont}\newcommand{\xiyihao}{\fontsize{24pt}{\baselineskip}\selectfont}\newcommand{\erhao}{\fontsize{22pt}{\baselineskip}\selectfont}\newcommand{\xiaoerhao}{\fontsize{18pt}{\baselineskip}\selectfont}\newcommand{\sanhao}{\fontsize{16pt}{\baselineskip}\selectfont}\newcommand{\xiaosanhao}{\fontsize{15pt}{\baselineskip}\selectfont}\newcommand{\sihao}{\fontsize{14pt}{\baselineskip}\selectfont}\newcommand{\xiaosihao}{\fontsize{12pt}{\baselineskip}\selectfont}\newcommand{\wuhao}{\fontsize{10.5pt}{\baselineskip}\selectfont}\newcommand{\xiaowuhao}{\fontsize{9pt}{\baselineskip}\selectfont}\newcommand{\liuhao}{\fontsize{7.5pt}{\baselineskip}\selectfont}\newcommand{\xiaoliuhao}{\fontsize{6.5pt}{\baselineskip}\selectfont}\newcommand{\qihao}{\fontsize{5.5pt}{\baselineskip}\selectfont}\newcommand{\bahao}{\fontsize{5pt}{\baselineskip}\selectfont}%頁眉的設(shè)置,要用到fancyhdr宏包\pagestyle{fancy}\fancyhead{}\fancyfoot{}\fancyhead[L]{\footnotesizeTeam\#189}\fancyhead[R]{\footnotesizePage\\thepage\of\42}\fancypagestyle{plain}{%\fancyhead[L]{\footnotesizeTeam\#189}\fancyhead[R]{\footnotesizePage\\thepage\of\42}}\setcounter{secnumdepth}{4}%更改\theparagraph的編號(hào)樣式\makeatletter\renewcommand{\theparagraph}{\@arabic\c@paragraph}\makeatother%章節(jié)格式的設(shè)置\titleformat{\section}{\erhao\bf}{}{0em}{}[]\titleformat{\subsection}{\xiaoerhao\bf}{}{0em}{}[]\titleformat{\subsubsection}{\sanhao\bf}{}{0em}{}[]\titleformat{\paragraph}[hang]{\vspace*{0.5ex}\sihao\bf}{\hspace*{1em}\theparagraph)}{0.5em}{}[\vspace*{-0.5ex}]%更改插圖的標(biāo)題\renewcommand{\figurename}{\wuhao\bf\sfFigure}\renewcommand{\captionlabeldelim}{\}%更改表格的標(biāo)題\renewcommand{\tablename}{\wuhao\bf\sfTable}%更改圖形或表格與其標(biāo)題的間距\setlength{\abovecaptionskip}{10pt}\setlength{\belowcaptionskip}{10pt}%定義產(chǎn)生不浮動(dòng)圖形和表格的標(biāo)題的命令\figcaption和\tabcaption\makeatletter\newcommand\figcaption{\def\@captype{figure}\caption}\newcommand\tabcaption{\def\@captype{table}\caption}\makeatother%自定義的可以調(diào)整粗細(xì)的水平線命令,用于繪制表格,調(diào)用格式\myhline{0.5mm}.\makeatletter\def\myhline#1{%\noalign{\ifnum0=`}\fi\hrule\@height#1\futurelet\reserved@a\@xhline}\makeatother%第一層列表序號(hào)為帶圈的阿拉伯?dāng)?shù)字\renewcommand{\labelenumi}{\textcircled{\arabic{enumi}}}%更改腳注設(shè)置\renewcommand{\thefootnote}{\fnsymbol{footnote}}\begin{document}\begin{CJK*}{GBK}{song}\CJKtilde\title{\bf\yihaoAviationBaggageScreening\\{\&}FlightSchedule}\author{}\date{}\maketitle\section{Introduction}FollowingtheterroristattacksonSeptember11,2001,thereisintenseinterestinimprovingthesecurityscreeningprocessforairlinepassengersandtheirbaggage.Airlinesandairportsareconsideredhigh-threattargetsforterrorism,soaviationsecurityiscrucialtothesafetyoftheair-travellingpublic.Bombsandexplosiveshavebeenknowntobeintroducedtoaircraftbyholdbaggageandcargo,carriedonbypassengers,andhiddenwithinaircraftsupplies.AtpresentToScreenorNottoScreen,thatisaHobson'schoice.USCurrentlawsmandate100{\%}screeningofallcheckedbagsatthe429passengerairportsthroughoutthenationbyexplosivedetectionsystems(EDS)bytheendoftheDec312003.However,becausethemanufacturersarenotabletoproducetheexpectednumberofEDSrequiredtomeetthefederalmandate,soitissignificanttodeterminethecorrectnumberofdevicesdeployateachairport,andtotakeadvantageofthemeffectively.TheTransportationSecurityAdministration(TSA)needsacomplicatedanalysisonhowtoallocatelimiteddeviceandhowtobestusethem.OurpapercontainsthemathematicalmodelstodeterminethenumberofEDSsandflightschedulesforallairportsinMidwestRegion.WealsodiscusstheETDdevicesastheadditionalsecuritymeasuresandthefuturedevelopmentofthesecuritysystems.\section{AssumptionandHypothesis}\begin{itemize}\itemThepassengerswhowillgetonthesameairplanewillarriveuniformly,namelythedistributionisflat.\itemThedetectionsystems,bothEDSandETD,operateallthetimeduringpeakhour,exceptdowntime.\itemTheairlinechecksthepassengersrandomly,accordingtoitsclaim.\itemThepassengers,whoarejustlandingandleaveout,donothavetobecheckedthroughEDSorETD.\itemAccordingtotheliterature,theaircraftloadsapproximatelyequalamongthesetsofdepartingflightduringthepeakhour.\itemThelandingflightdidnotaffectthedepartureoftheplane.\itemOnceapassengerarrives,hecangotoEDStobechecked,excepthehastowaitinline.\itemOncepassengersfinishscreening,theycanbroadontheplaneinnotime.\itemDuringpeakhours,asetofflightsdepartsatthesametimeeverythesameminutes.\itemAlltherunwaysareusedasmuchaspossibleduringpeakhours.\itemThemaximumnumberofthebaggageistwo,whichapassengercancarryonplane.[L]\itemThedetectionmachineexaminesthebagsatthesamespeed.\itemEDScannotmakemistakesthatitdetectanormalobjectasanexplosive.\end{itemize}\section{VariableandDefinition}\begin{longtable}{p{100pt}p{280pt}}\caption{Variables}\\%第一頁表頭的標(biāo)題\endfirsthead%第一頁的標(biāo)題結(jié)束\caption{(continued)}\\%第二頁的標(biāo)題\endhead%第二頁的標(biāo)題結(jié)束\hline\hline\textbf{Symbol}&\textbf{Description}\\\hline$n_{ij}$&Theairplanenumberofthe$i^{\mathrm{th}}$typeinthe$j^{\mathrm{th}}$flightset\\\hline${NP}_i\:(i=1,2,\ldots)$&Thenumberofpassengersoneachairplanesofthesametype.\\\hline$\xi_{ij}\:(i,j=1,2,\cdots)$&Thenumberofbaggageoneachairplaneofthe$j^{\mathrm{th}}$flights\\\hline$a$&Themaximalnumberofairplanestype\\\hline$B_j^{set}$&Thetotalbaggagenumberofeachsetofflight\\\hline${NF}_i$&Numberofairplanesofeachtype\\\hline$\bar{\rho}$&Themeanvalueofpassengers'baggagecomingperminuteineveryflightset\\\hline$N_{set}$&Thenumberofflightsets\\\hline$B_{total}$&Thetotalnumberofcheckedbaggageduringthepeakhour\\\hline$H_{peak}$&Thelengthofthepeakhour\\\hline$T_{set}$&Thetimelengthduringwhicheachflightset'spassengerswaittobechecked\\\hline$\Deltat$&Thetimeintervalbetweentwoconsecutiveflightset\\\hline$N_{EDS}$&ThenumberofalltheEDSs\\\hline$N_{shadow}$&Thenumberofflightsetswhosepassengerswillbemixedupbeforebeingchecked\\\hline$v_{EDS}$&ThenumberofbaggagecheckingbyoneEDSperminute\\\hline$\rho_j$&Thenumberofpassengers'baggagecomingperminuteinoneflightset\\\hline$N_{runway}$&Thenumberofanairport'srunway\\\hline\\*[-2.2ex]${\bar{B}}^{set}$&Themeanvalueofcheckedbaggagenumberofeveryflightset\\\hline$M$&Thesecuritycost\\\hline\hline\label{tab1}\end{longtable}\subsubsection{Definition:}\begin{description}\item[Flightset]Agroupofflightstakeoffatthesametime\item[Thelengthofpeakhour]Thetimebetweenthefirstsetofflightandthelastset\end{description}\section{BasicModel}Duringapeakhour,manyplanesandmanypassengerswoulddepartfromairports.Therefore,Itisdifficulttoarrangeforthepassengerstoenterairports.IftherearenotenoughEDSsforpassengers'baggagetocheck,itwilltaketoolongtimeforthemtoenter.Thatwouldresultinthedelayofairplanes.Onthecontrary,iftherearetoomanyEDSs,itwillbeawaste.ItisourtasktofindasuitablenumberofEDSsforairport.Inordertoreachthisobjective,weusethelinearprogrammingmethodtosolveit.\subsection{Baseanalysis}Theairplanesareoccupiedatleastpartly.Thepassengers'baggagewouldbecheckedbyEDSsbeforetheygetontheairplanes.Wehaveassumedthateverypassengercarrytwobaggages.Thisassumptionwouldsimplifytheproblem.Accordingtothedatafromtheproblemsheet,wecanobtaintheusefulinformationthatairlinesclaim20{\%}ofthepassengersdonotcheckanyluggage,20{\%}checkonebag,andtheremainingpassengerschecktwobags.Therefore,wecangainthetotalnumberofpassengers'baggagethatshouldbecarriedononeplane:$\xi_{ij}$.Moreover,wecangettheequationthatcalculate$\xi_{ij}$:\[\xi_{ij}={NP}_i\times20\%+{NP}_i\times60\%\times2\]Wedefinethematrixbelowasairplanebaggagenumbermatrix:\[\overset{\rightharpoonup}{\xi}_j=\left[\xi_{1j}\quad\xi_{2j}\quad\cdots\quad\xi_{ij}\quad\cdots\right]\]Wedefinethematrixbelowasflightschedulematrix:\[\left[\begin{array}{llcl}n_{11}&n_{12}&\cdots&n_{1N_{set}}\\n_{21}&n_{22}&\cdots&n_{2N_{set}}\\\multicolumn{4}{c}\dotfill\\n_{a1}&n_{a2}&\cdots&n_{aN_{set}}\end{array}\right]\]Inthismatrix,$n_{ij}$istheairplanenumberofthe$i^{\mathrm{th}}$typeinthe$j^{\mathrm{th}}$flightsetwhichwilltakeoff.Apparently,thisvalueisaninteger.Wedefinethematrixbelowasflightsetbaggagenumbermatrix:\[\left[B_1^{set}\quadB_2^{set}\quad\cdots\quadB_j^{set}\quad\cdots\quadB_a^{set}\right]\]Itisclearthattheymeettherelationbelow:\begin{equation}\begin{array}{cl}&\left[\xi_{1j}\quad\xi_{2j}\quad\cdots\quad\xi_{ij}\quad\cdots\right]\cdot\left[\begin{array}{llcl}n_{11}&n_{12}&\cdots&n_{1N_{set}}\\n_{21}&n_{22}&\cdots&n_{2N_{set}}\\\multicolumn{4}{c}\dotfill\\n_{a1}&n_{a2}&\cdots&n_{aN_{set}}\end{array}\right]\\=&\left[B_1^{set}\quadB_2^{set}\quad\cdots\quadB_j^{set}\quad\cdots\quadB_a^{set}\right]\end{array}\label{Flight:baggage}\end{equation}Then,weknow:\[B_j^{set}=\sum\limits_{i=1}^a\xi_{ij}\timesn_{ij}\]Therearesomeconstraintstotheequation(\ref{Flight:baggage}).First,foreachsetofflight,thetotalnumberofairplanesshouldbelessthanthenumberofrunways.Second,thetotalairplanenumberofthesametypelistedintheequation(\ref{Flight:baggage})fromeverysetofflightshouldbeequaltotheactualairplanenumberofthesametypeduringthepeakhour.Wecanexpressthemlikethese:\[\sum\limits_{i=1}^an_{ij}\leN_{runway}\quad\quad\sum\limits_{j=1}^bn_{ij}={NF}_i\]Weshouldresolvethenumberofflightsets.Accordingtoourassumptions,duringthepeakhour,theairlinesshouldmakethebestuseoftherunways.Thengetthenumberofflightsetsapproximatelybasedonthenumberofalltheairplanesduringthepeakhourandthatoftherunways.Weuseanequationbelowtoexpressthisrelation:\begin{equation}N_{set}=\left\lceil\frac{\sum\limits_{j=1}^{N_{set}}\sum\limits_{i=1}^an_{ij}}{N_{runway}}\right\rceil\label{sets:number}\end{equation}Thecheckedbaggagenumbersofeachflightsetareequaltoeachotheraccordingtoourassumption.Wemakeitbasedonliterature.Itcanalsosimplifyourmodel.Wedefine$\bar{B}^{set}$asthemeanvalueofcheckedbaggagenumberofeveryflightset.Moreover,Wedefine$\bar{\rho}$asthemeanvalueofcheckedbaggagenumberofeveryflightsetperminute:\[\bar{B}^{set}=\frac{B_{total}}{N_{set}}\]\[\bar{\rho}=\frac{\bar{B}^{set}}{T_{set}}=\frac{B_{total}}{T_{set}N_{set}}=\frac{B_{total}\Deltat}{T_{set}H_{peak}}\]Thecourseofpassengers'arrivalandenteringairportisimportantforustodecidethenumberofEDSsandtomaketheflightsschedule.Therefore,weshouldanalyzethisprocesscarefully.Passengerswillarrivebetweenforty-fiveminutesandtwohourspriortothedeparturetime,andthepassengerswhowillgetonthesameairplanewillarriveuniformly.Thenwecangettheflowdensityofallcheckedbaggageatanytimeduringpassengers'entering.Thisvalueisthesumofnumbersofpassengers'checkedbaggagecomingperminute.Tocalculatethisvalue,firstly,weshouldobtainflowdensityofeachflightset'scheckedbaggage.Wedefine$\rho_j$,namelythenumberofcheckedbaggageperminuteofoneflightset:\[\rho_j=\frac{B_j^{set}}{T_{set}}\]Secondly,wedrawgraphictohelpustounderstand.Weuserectangletoexpressthetimelengthforallthepassengersofoneflightsettocomeandcheckbags.Inthegraphic,theblackpartistheperiodforthemtocome.Duringthewhitepart,nopassengersforthisflightsetcome.Accordingtotheproblemsheet,theformeris75minute,andthelatteris45minute.Thelengthofrectangleis120minute.$T_{set}$istheperiodduringwhichallpassengersofoneflightsetwaittobechecked.Sincewehaveassumedthateachtimeintervalbetweentwoconsecutiveflightsetissamevalue,wedefine$\Deltat$asit.Observethesectionthatvaluewewanttosolveis$\sum\limits_j\rho_j$.Moreover,wecangetanotherimportantequationfromthegraphicbelow:\begin{equation}N_{set}=\frac{H_{peak}}{\Deltat}\label{PeakHour}\end{equation}\begin{figure}[hbtp]\centering\includegraphics[width=298.2pt,totalheight=141.6pt]{fig01.eps}\caption{}\label{fig1}\end{figure}EachEDShascertaincapacity.IfthenumberofEDSsis$N_{EDS}$andoneEDScancheckcertainnumberofbaggageperminute(Thatischeckingvelocity,markedby$v_{EDS}$),thetotalcheckingcapacityis$N_{EDS}\cdot\frac{v_{EDS}}{60}$.$v_{EDS}$isbetween160and210.Nowwecaneasilydecideinwhatconditionthepassengerscanbecheckedwithoutdelay:\[\sum\limits_j\rho_j\lev_{EDS}\]Thepassengershavetoqueuebeforebeingchecked:$\sum\limits_j\rho_j>v_{EDS}$Wellthen,howcanwedecidehowmany$\rho_j$?Itdependsonhowmanyflightsetswhosepassengerswillbemixedupbeforebeingchecked.Wenoteitas$N_{shadow}$.ReturntotheFigure\ref{fig1},wecanknow:\[N_{shadow}=\left\lfloor\frac{T_{set}}{\Deltat}\right\rfloor\]\begin{figure}%[htbp]\centering\includegraphics[width=240pt,totalheight=131.4pt]{fig02.eps}\caption{}\label{fig2}\end{figure}FromFigure\ref{fig1}andFigure\ref{fig2},wecangettheresultasfollows:\begin{enumerate}\itemIf$N_{shadow}\leN_{set}$,namely$H_{peak}>T_{set}$,then$\sum\limits_{j=1}^{N_{shadow}}\rho_j\leN_{EDS}\frac{v_{EDS}}{60}$\renewcommand{\theequation}{\arabic{equation}a}Thatis:\begin{equation}N_{EDS}\ge\frac{60}{v_{EDS}}\sum\limits_{j=1}^{N_{shadow}}\rho_j\approx\frac{60}{v_{EDS}}N_{shadow}\bar{\rho}=\frac{60B_{total}\Deltat}{v_{EDS}T_{set}H_{peak}}N_{shadow}\label{EDS:number:a}\end{equation}\itemIf$N_{shadow}>N_{set}$,namely$H_{peak}\leT_{set}$,then$\sum\limits_{j=1}^{N_{set}}\rho_j\leN_{EDS}\frac{v_{EDS}}{60}$\setcounter{equation}{3}\renewcommand{\theequation}{\arabic{equation}b}Thatis:\begin{equation}N_{EDS}\ge\frac{60}{v_{EDS}}\sum\limits_{j=1}^{N_{set}}\rho_j\approx\frac{60}{v_{EDS}}N_{set}\bar{\rho}=\frac{60B_{total}\Deltat}{v_{EDS}T_{set}H_{peak}}N_{set}\label{EDS:number:b}\end{equation}\end{enumerate}\subsection{ThenumberofEDSs}ThenwebegintoresolvethenumberofEDSsassistedbythelinearprogrammingmethod.EDSisoperationalabout92{\%}ofthetime.Thatistosay,wheneveritisduringapeakhour,therearesomeEDSsstoppingworking.ThentheworkingefficiencyofalltheEDSsislessthanthelevelwehaveexpected.Therefore,theairlinehastoaddmoreEDSstodothework,whichcanbedonewithlessEDSswithoutdowntime.Weusebinomialdistributiontosolvethisproblem.$N$isthenumberofactualEDSswithdowntimeand$k$isthenumberofEDSswithoutdowntime.Ifprobabilityis$P$,wecangettheequationbelow:\[\left(\begin{array}{c}N\\k\end{array}\right)\cdot98\%^k\cdot(1-98\%)^{N-k}=P\]Wecanobtain$N$whenwegive$P$acertainvalue.Inthispaper,$P$is95{\%}.The$N_{EDS}$istheactualnumberweobtainthroughtheequationabove.Nowwehaveassumedthatpassengerscanbecheckedunlessbedelayedbythepeoplebeforehimoncehearrivesatairport.Apparently,ifthetimelengthbetweentwosetsofflightisshort,thedensityofpassengerswillbegreat.Itwillbringgreatstresstosecuritycheckandmayevenmakesomepassengersmisstheirflight.Toresolvethisquestion,theairlinehastoinstallmoreEDSstomeetthedemand.However,thismeasurewillcostmuchmoremoney.Consequently,wehavetosetasuitabletimeintervalbetweentwosetofflight.Basedonthebaseanalysisabove.Wecanusetheequation(\ref{sets:number})todecidethenumberofflightsets$N_{set}$assumingweknowthenumberofrunwaysofacertainairport.Thenbasedontheequation(\ref{PeakHour}),wecandecidethepeakhourlength$H_{peak}$whenweassumeatimeintervalbetweentwoconsecutiveflightsets.Thenweuse\textcircled{1}and\textcircled{2}todecidewhichtochoosebetweenequation(\ref{EDS:number:a})andequation(\ref{EDS:number:b}).Inconsequence,wecanobtaintheminimumofEDSsnumber.Ifwechoosedifferentnumbersofrunwaysandthetimeintervalsbetweentwoflightsets,wecangetdifferentEDSsnumbers.Inthispaperthatfollowed,wegainatableofsomevalueof$N_{runway}$and$\Deltat$withthecorrespondingEDSsnumbers.Moreover,wedrawsomefiguretoreflecttheirrelation.Foracertainairport,itsnumberofrunwayisknown.Givencertaintimeinterval($\Deltat$),wecangetthelengthofthepeakhour($H_{peak}$).Whenthe$N_{runway}$isfewenough,perhaps$H_{peak}$istoolongtobeadopted.However,foracertainairline,theycandecidethetimeintervaloftheirownpeakhour.Inthisgiventimeinterval,theycouldfindtheminimumof$N_{runway}$throughtheFigure\ref{fig3}.Wedrawasketchmaptodescribeoursteps.\begin{figure}[hbtp]\centering\includegraphics[width=352.8pt,totalheight=214.2pt]{fig03.eps}\caption{}\label{fig3}\end{figure}\subsection{TheFlightSchedule}Accordingtothebaseanalysis,wecanknowthattheflightschedulematrixand$\Deltat$isoneformofflighttimetable.In``ThenumberofEDSs'',wecangetsuitable$\Deltat$.Thenweshouldresolvetheflightschedulematrix.Becausewehaveassumedthatthecheckedbaggagenumbersofeachflightsetareequaltoeachother.Itcanbedescribedasfollows:\[\left\{\begin{array}{l}\rho_j\approx\bar{\rho}\\B_j^{set}\approx\bar{B}^{set}\end{array}\right.\begin{array}{*{20}c}\hfill&{j=1,2,\cdots,N_{set}}\hfill\end{array}\]Theflightschedulematrixsubjecttothisgroup:\[\left\{\begin{array}{ll}\sum\limits_{j=1}^{N_{set}}n_{ij}={NF}_i&i=1,2,\cdots\\\sum\limits_{i=1}^an_{ij}\leN_{runway}&j=1,2,\cdots,N_{set}\\n_{ij}\ge0,&\mathrm{and}\:n_{ij}\:\mathrm{is}\:\mathrm{a}\:\mathrm{Integer}\end{array}\right.\]Inordertomakethebestuseofrunway,weshouldmake$\sum\limits_{i=1}^an_{ij}$asgreataswecanunlessitexceed$N_{runway}$.Thenwecanseethathowtoresolvetheflightschedulematrixisaproblemofdivideamongagroupofintegers.Thisgroupisallthenumbersofeachflightpassengers'baggageinoneflightset.WeprogramforthisproblemusingMATLABandwegetatleastonesolutionintheend.However,thematrixelementswehaveobtainedarenotinteger,wehavetoadjustthemtobeintegersmanually.\subsection{ResultsandInterpretationforAirportAandB}Thenumberofpassengersinacertainflight(${NP}_i$),thetimelengthofsecuritychecking($T_{set}$),thecheckingvelocityofEDS($v_{EDS}$),andthenumberofbaggagecarriedbyonepassengerarerandom.\subsubsection{DataAssumption:}\begin{itemize}\item$T_{set}$is110minutes,whichisreasonableforairline.\itemTosimplifytheproblem,weassumethateverypassengercarry2baggage.Ifsomeofthepassengerscarryonebaggage,thesolutionbasedon2baggagesperpassengermeetstherequirement.\itemThenumberofrunwaysinairportAandairportBis5.\end{itemize}\subsubsection{AirportA:}Oncethenumberofrunwayandthenumberoftheflightsaredecided,theflightschedulematrixisdecided,too.WeproducethismatrixusingMATLAB.Thismatrixcompaniedby$\Deltat$istheflightscheduleforairportA.$\Deltat$willbecalculatedin(\ref{Flight:baggage}),(\ref{sets:number})and(\ref{PeakHour}).Wecalculate$N_{EDS}$andmaketheflighttimetableinthreeconditions.Thethreeconditionsandthesolutionarelistedasfollowed:\paragraph{Everyflightarefullyoccupied}ThecheckingspeedofEDSis160bags/hour.\begin{table}[htbp]\centering\caption{}\begin{tabular}{*{11}c}\myhline{0.4mm}$\mathbf{\Deltat(\min)}$&\textbf{2}&\textbf{4}&\textbf{6}&\textbf{8}&\textbf{10}&\textbf{12}&\textbf{14}&\textbf{16}&\textbf{18}&\textbf{20}\\\myhline{0.4mm}$N_{EDS}(\ge)$&31&31&31&31&31&29&24&22&20&17\\\hline$H_{peak}(\min)$&20&40&60&80&100&120&140&160&180&200\\\myhline{0.4mm}\end{tabular}\label{tab2}\end{table}Weassumethatthesuitablevalueof$H_{peak}$is120minutes.Thenthesuitablevalueof$\Deltat$isabout12minutes,and$N_{EDS}$is29judgedfromFigure\ref{fig4}.Certainly,wecanwork$\Deltat$and$N_{EDS}$outthroughequation.\begin{figure}[htbp]\centering\includegraphics[width=294.6pt,totalheight=253.2pt]{fig04.eps}\caption{}\label{fig4}\end{figure}\paragraph{Everyflightisoccupiedbytheminimalnumberofpassengersonstatisticsinthelongrun.}ThecheckingspeedofEDSis210bags/hour.\begin{table}[htbp]\centering\caption{}\begin{tabular}{*{11}c}\myhline{0.4mm}$\mathbf{\Deltat(\min)}$&\textbf{2}&\textbf{4}&\textbf{6}&\textbf{8}&\textbf{10}&\textbf{12}&\textbf{14}&\textbf{16}&\textbf{18}&\textbf{20}\\\myhline{0.4mm}$N_{EDS}(\ge)$&15&15&15&15&15&14&13&12&10&7\\\hline$H_{peak}(\min)$&20&40&60&80&100&120&140&160&180&200\\\myhline{0.4mm}\end{tabular}\label{tab3}\end{table}Weassumethatthesuitablevalueof$H_{peak}$is120minutes.Thenthesuitablevalueof$\Deltat$isabout12minutes,and$N_{EDS}$is14judgingfromFigure\ref{fig5}.Certainly,wecanwork$\Deltat$and$N_{EDS}$outthroughequation.\begin{figure}[htbp]\centering\includegraphics[width=294.6pt,totalheight=253.2pt]{fig05.eps}\caption{}\label{fig5}\end{figure}\paragraph{${NP}_i$and$v_{EDS}$arerandomvalueproducedbyMATLAB.}\begin{table}[htbp]\centering\caption{}\begin{tabular}{*{11}c}\myhline{0.4mm}$\mathbf{\Deltat(\min)}$&\textbf{2}&\textbf{4}&\textbf{6}&\textbf{8}&\textbf{10}&\textbf{12}&\textbf{14}&\textbf{16}&\textbf{18}&\textbf{20}\\\myhline{0.4mm}$N_{EDS}(\ge)$&15&22&21&21&15&17&21&16&13&14\\\hline$H_{peak}(\min)$&20&40&60&80&100&120&140&160&180&200\\\myhline{0.4mm}\end{tabular}\label{tab4}\end{table}Weassumethatthesuitablevalueof$H_{peak}$is120minutes.Thenthesuitablevalueof$\Deltat$isabout12minutes,and$N_{EDS}$is17judgingfromFigure\ref{fig6}.Certainly,wecanwork$\Deltat$and$N_{EDS}$outthroughequation.\begin{figure}[htbp]\centering\includegraphics[width=294.6pt,totalheight=249.6pt]{fig06.eps}\caption{}\label{fig6}\end{figure}\subsubsection{Interpretation:}Byanalyzingtheresultsabove,wecanconcludethatwhen$N_{EDS}$is29,and$\Deltat$is12,theflightschedulewillmeetrequirementatanytime.Theflightscheduleis:\\[\intextsep]\begin{minipage}{\textwidth}\centering\tabcaption{}\begin{tabular}{c|*{8}c|c|c}\myhline{0.4mm}\backslashbox{\textbf{Set}}{\textbf{Type}}&\textbf{1}&\textbf{2}&\textbf{3}&\textbf{4}&\textbf{5}&\textbf{6}&\textbf{7}&\textbf{8}&\textbf{NumbersofBags}&\textbf{NumbersofFlights}\\\myhline{0.4mm}1&2&0&0&0&2&1&0&0&766&5\\\hline2&2&0&2&0&2&0&0&0&732&4\\\hline3&0&1&1&1&2&0&0&0&762&4\\\hline4&0&1&0&0&2&1&0&0&735&4\\\hline5&0&1&0&0&2&1&0&0&735&5\\\hline6&2&0&0&0&1&0&0&1&785&5\\\hline7&2&0&0&0&2&0&1&0&795&5\\\hline8&0&1&0&0&2&1&0&0&735&4\\\hline9&2&0&0&0&2&1&0&0&766&5\\\hline10&0&0&0&2&2&0&0&0&758&5\\\hlineTotal&10&4&3&3&19&5&1&1&7569&46\\\myhline{0.4mm}\end{tabular}\label{tab5}\end{minipage}\\[\intextsep]Wehaveproducedrandomvaluefor${NP}_i$and$v_{EDS}$.Onthiscondition,thenumberofEDSsis17,whichislessthan29thatwedecidefortheairportA.Thatistosayoursolutioncanmeettherealrequirement.\subsubsection{AirportB:}\paragraph{Thepassengerloadis100{\%}}ThecheckingspeedofEDSis160bags/hour.\begin{table}[htbp]\centering\caption{}\begin{tabular}{*{11}c}\myhline{0.4mm}$\mathbf{\Deltat(\min)}$&\textbf{2}&\textbf{4}&\textbf{6}&\textbf{8}&\textbf{10}&\textbf{12}&\textbf{14}&\textbf{16}&\textbf{18}&\textbf{20}\\\myhline{0.4mm}$N_{EDS}(\ge)$&33&33&33&33&33&30&27&23&21&19\\\hline$H_{peak}(\min)$&20&40&60&80&100&120&140&160&180&200\\\myhline{0.4mm}\end{tabular}\label{tab6}\end{table}Weassumethatthesuitablevalueof$H_{peak}$is120minutes.Thenthesuitablevalueof$\Deltat$isabout12minutes,and$N_{EDS}$is30judgingfromFigure\ref{fig7}.Certainly,wecanwork$\Deltat$and$N_{EDS}$outthroughequation.\begin{figure}[htbp]\centering\includegraphics[width=294.6pt,totalheight=249pt]{fig07.eps}\caption{}\label{fig7}\end{figure}\paragraph{Here,weonlyproducedrandomvalueof${NP}_i$and$v_{EDS}$.$\Deltat$willbecalculatedbelow.}\begin{minipage}{\textwidth}\centering\tabcaption{}\begin{tabular}{*{11}c}\myhline{0.4mm}$\mathbf{\Deltat(\min)}$&\textbf{2}&\textbf{4}&\textbf{6}&\textbf{8}&\textbf{10}&\textbf{12}&\textbf{14}&\textbf{16}&\textbf{18}&\textbf{20}\\\myhline{0.4mm}$N_{EDS}(\ge)$&23&22&24&21&22&23&20&19&15&15\\\hline$H_{peak}(\min)$&20&40&60&80&100&120&140&160&180&200\\\myhline{0.4mm}\end{tabular}\label{tab7}\end{minipage}\\[\intextsep]Weassumethatthesuitablevalueof$H_{peak}$is120minutes.Thenthesuitablevalueof$\Deltat$isabout12minutes,and$N_{EDS}$is23judgingfromFigure\ref{fig8}.Certainly,wecanwork$\Deltat$and$N_{EDS}$outthroughequation.\begin{figure}[htbp]\centering\includegraphics[width=294.6pt,totalheight=250.8pt]{fig08.eps}\caption{}\label{fig8}\end{figure}\subsubsection{Interpretation:}Byanalyzingtheresultsabove,wecanconcludethatwhen$N_{EDS}$is30,and$\Deltat$is12,theflightschedulewillmeetrequirementatanytime.Theflightscheduleis:\\[\intextsep]\begin{minipage}{\textwidth}\centering\tabcaption{}\begin{tabular}{c|*{8}c|c|c}\myhline{0.4mm}\backslashbo