Team#32879Page21of35
3.6OctagonalSectorSearchModel
Thelastsearchmodelthatwillbepresentedisanoctagonal“sectorsearch,”whichcanbevisualizedinthefollowing?gure:
Figure17:“SectorSearch”inaregion[12].
Althoughtheoutlininggeometryisnolongerasquare:westillseekarelationshipbetweenzandl,wherelisthesidelengthoftheregularoctagon:
Figure18:Parametersforthe“SectorSearch”.
Here,thepathlengthzisindependentofthechoiceofgridsizeW.Inonefullsearch,theplanetravelsalongtheperimeteronceandeveryinteriorpathtwice,meaningthatthecenterispassedovereighttimes.Therelationshipbetweenlandz,derivedfrombasicgeometry,isshownhere.
zsin(67.5?)l=
8(cos(67.5?)+1)
也许大学四年，我们会一直在迷茫中度过，因为生活总是难以言说。赛氪APP与您相伴！
(25)
21
Team#32879Page22of35
Theoptimizationforthismethodisslightlydi?erentduetotheinteriorsectionsofthepatternleftunsearched.Forthesquareorrectangularpaths,theprobabilityoverthefullareacouldbecalculatedusingasimpledoublesum.Forthispath,however,interiorsectionsarenotsearchedsoitwouldnotbevalidtooptimizethedoublesumofthefullsquare.Instead,thesearchlocationisprioritizedbasedonthecentralarea,asitispassedovereighttimes.
Whenlookingforthebestgridsquareonwhichtocenterthesearch,themodellooksforthehighest3-by-3doublesumofprobabilities,whichwouldtheoreticallyrepresenttheoptimalcenterofthegrid.However,sincecertainsearchareasaremuchfurtherfromtherunway,theremaybelessusablesearchrange,whichmakesthatsearchlesse?cient.Tobalancetheusablerangeforthewholesearchwiththehighestprobabilitysumforthecentergrid,theproductofthesetwoquantitiesisoptimized.Thismethodisrudimentaryatbest,asitoperatesnaivelyontheassumptionthatasimpleproductofthesetwoquantitiestrulymodelstheactuale?ciencyofthesearchpath.Inthefuture,amorerealisticoptimizationmodelshouldbeimplemented.
Theprobabilityfunctionafter?vedaysandthecumulativeprobabilityovertimearebothshownbelow:
Figure19:ProbabilityDensityDistributionaftera5-daysearch.
也许大学四年，我们会一直在迷茫中度过，因为生活总是难以言说。赛氪APP与您相伴！
22
Team#32879
Single Plane Octagonal Sector Search ModelCumulative Probability of SuccessDaily Probability of SuccessPage23of35
0.60.50.4Probability0.30.20.1002468Search Day101214161820Figure20:CumulativeProbabilityofSuccessaftera20-daysearch.
Thismodelisalsoverycomputationallye?cient.Theweaknessesofthismodelhowever,aresigni?cant.Asdiscussedpreviously,theoptimizationisonlybasedonthecentralarea,whichdoesnotgloballyoptimizethesearchareaandlocation.
3.7ModelVariationandComparison
Withenoughresources,asearchoperationforalostaircraftmayconsistofdozensofplanesoverthecourseofweeksormonths,ifnotlonger.Itisveryimportanttooptimizetheuseofalloftheresourcesallottedtothesearchoperation.Inthecontextofourmodel,thismeanssimulatingmanydi?erenttypesofscenarios,fromnumberofdaysspentsearchingtotypeofsearchplaneavailable,andevenvariationsinsearchpatternschosenfordi?erentplanesbaseduponthatplane’scapabilities.Thee?ciencyofeachofthedi?erentmodelswillbecomparedthroughtheircumulativesuccessprobabilities.Theoptimalsearchpatternandplanecombinationshouldideallyresultinthegreatestaccumulatedprobabilityof?ndingthemissingaircraftonanygivenday.
Inthefollowingsections,theresultsfordi?erentvariationsofourmodelarepresented:?Onesearchaircraft?Multiplesearchaircraft?Highprobabilityofstall?Shorter-rangesearchaircraft
?Utilizingmultiplesearchpatternsfordi?erentsearchaircraft
也许大学四年，我们会一直在迷茫中度过，因为生活总是难以言说。赛氪APP与您相伴！
23
Team#32879Page24of35
?Searchaircraftthathavedi?erente?ectiveness(sensors,electronics,numberofsearchers)?Missingaircraftthatareeasier/hardertosee?Di?erentparametersforthemissingaircraft
Thesevariationsdemonstrateoneofthemajorstrengthsofourmodel:theeaseofmodi?-cation.3.7.1
SinglePlaneSearchModelComparison
Resultsfromeachofthefourmodelshavebeenpresentedseparately,butitismorehelpfultodirectlycomparethee?ectivenessofeachmodel.Shownbelowarethecumulativeproba-bilitiesofsuccessforasingleplanesearchoverthecourseof20days.Thisisthebestdirectcomparisonofthe?ightpathsbecauseitmodelsthesimplestcase.0.9Single Plane Model Comparison - Cumulative Success ProbabilitySimple Square ModelSpiral Square ModelOctagonal Sector ModelRectangular Model0.80.70.6Probability0.50.40.30.20.1002468Search Day101214161820Figure21:DirectModelComparison.
Therectangularsearchpatternisthebestpatternconsistently,followedverycloselybythesquaresweepandthesquarespiral.Therectangleisageneralizedcaseofthesquare,sotheprobabilityofsuccessshouldalwaysbethesameifnotbetterthanforthesquare.Thesquarespiralisslightlylesse?cientthanthesquaresweepduetotheincreasedsearchlengthrequiredtosearchthesamearea.Theoctagonalsectorsearchismuchlesse?ective,partiallyduetothenatureofthesearchandpartiallyduetotheshortcomingsofthemodel.Thesearchleaveslargetrianglesun-searched,andonsubsequentdays,thereisnowayto
也许大学四年，我们会一直在迷茫中度过，因为生活总是难以言说。赛氪APP与您相伴！
24
Team#32879Page25of35
e?ectivelysearchthesetriangles.Also,themodelislikelynotidentifyingtheexactoptimalsearchpath.Therearemanyine?cienciesassociatedwiththissearchpaththatmakeitmuchlesse?ective.3.7.2
FivePlaneModelComparison
Todeterminehowthenumberofsearchplanesa?ectstheresults,themodelswererunusing?veplanesperdayinsteadofone.Ine?ect,thisisnodi?erentthansearchingwithoneplanefor?vetimesthenumberofdays.Itmakessensethatthesameresultsarefoundfor?veplanesasforoneplane:therectangularsearchisbest,closelyfollowedbythetwosquarepaths.ThisisshownbelowinFigure22:1Five Plane Model Comparison - Cumulative Success ProbabilitySimple Square ModelSpiral Square ModelOctagonal Sector ModelRectangular Model0.90.80.7Probability0.60.50.40.30.202468Search Day101214161820Figure22:ComparisonofSearchPatternsfor5LargePlanesover20Days.
3.7.3HighLikelihoodofStall
Todemonstratethee?ectofadi?erentinitialprobabilitydistribution,wemodelacasewherethereishighlikelihoodofstall.Theinitialprobabilitydistributioncouldbevariedbasedonadditionalinformationaboutthemissingaircraft.Inthiscase,weconsiderahighlikelihoodofstallbasedonpastoccurrencesofstallinthemissingaircraft.Intheprobabilitydensityfunctionforθ,pcrashisweightedmoreheavily,resultinginthefollowingpriorprobabilitydistribution:
也许大学四年，我们会一直在迷茫中度过，因为生活总是难以言说。赛氪APP与您相伴！
25