Team#32879Page26of35
Figure23:InitialProbabilityDistributionifaStallisMoreLikely.
Eachofthefourmodelswasrunwiththisinitialprobabilitydistribution,yieldingthefollowingresults:0.9Stall Likely Model Comparison - Cumulative Success ProbabilitySimple Square ModelSpiral Square ModelOctagonal Sector ModelRectangular Model0.80.70.6Probability0.50.40.30.20.1002468Search Day101214161820Figure24:ModelComparisonforLikelyStallScenario.
Thesametrendspersistinthismodel:therectangularsearchisthemoste?ectiveandtheoctagonalsectorsearchistheleaste?ective,whilethetwosquarepatternslieinthemiddle.
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Team#328793.7.4
ShortRangeSearchAircraft
Page27of35
Next,shortrangesearchaircraftareconsidered.InsteadofthebasecaseC-130searchaircraft,aV-22Ospreyisconsidered,whichhasarangeofonly1011miles[13].Forasingleoneoftheseaircraft,thecumulativesuccessprobabilitiesforeachsearchpatternareshownbelow,?rstover20daysandthenzoomedtoshowonlythe?rst?vedays:
0.20.180.160.140.035Small Plane Model Comparison - Cumulative Success ProbabilitySimple Square ModelSpiral Square ModelOctagonal Sector ModelRectangular Model0.05Small Plane Model Comparison - Cumulative Success ProbabilitySimple Square ModelSpiral Square ModelOctagonal Sector ModelRectangular Model0.0450.040.12ProbabilityProbability0.030.10.080.0250.020.060.040.0200.0150.0102468Search Day1012141618200.00511.522.5Search Day33.544.55(a)ComparisonofSearchPatternsforSingleSmallPlaneover20Days.(b)ComparisonofSearchPatternsforSingleSmallPlaneover5Days.
Theseplotsshowseveralinterestingtrends:
?Thecumulativesuccessprobabilityismuchmorelinearthanwhenusinglongerrangesearchaircraft.Thisisbecausethesearchareasaremuchsmaller,sothesearchoneachdaycanbenearlyase?ectiveasthesearchonthepreviousday.
?Foranincreasednumberofdaysthesimplesquaresearchpathisdemonstratedtooutperformallotherpatterns.Inthe?rstfewdays,therectangularsearchisoptimal,butastimecontinues,therectanglesbecomelesse?cientbecauseprecedingrectangularsearcheshavepartitionedtheprobabilitydistributioninawaythatfuturerectangleshavedi?cultycovering.Thisphenomenaseemsstrange,andshouldbeexploredfurtherinfuturedevelopments.
Theresultsfromtheprevioussectionsdescribeafewveryimportanttrendsthatpersistthroughouteachofoursimulationsregardlessoftypeofsearchplane,initialdistribution,ornumberofsearchplanes:
1.Theoptimizedrectangle,simplesquare,andspiralsquareareallrelativelyconsistentandsimilarintheirabilitytomaximizecumulativeprobabilityof?ndingtheplane.2.Theoptimizedrectangleisconsistentlythemostsuccessfulforthelargeplane(largerange)searches,followedcloselybythesimplesquareandspiralsquare,inorderofdecreasingsuccess.Theoctagonalsectorsearchisfarlesse?ective.3.Itisalsousefultonotethatallsearchesdoapproachacumulativeprobabilityof1,suggestingthatgivenenoughtimetheplanewouldinevitablybefound.
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Team#328793.7.5
ComparisonofVariedSearchPatterns
Page28of35
Wenowmovebeyondthecomparisonofdi?erentsearchpatternsandthedi?erentsearchair-crafttoinvestigatethee?ciencyofdi?erentsearchpatternsbeingemployedsimultaneouslybytwoaircraft.Theplotbelowdisplaysthecumulativeprobabilityofatwoplanesearchovertwentydays.Bothscenariosutilizealongrange(C-130)andashortrange(Osprey)searchplane.Inonescenariobothplanesutilizedthesimplesquaresearchmethod,whileintheotherscenariothelongrangeplaneperformedasimplesquaresearchwhiletheshortrangeplaneperformedanoctagonalsectorsearch.0.9Two Plane Model Comparison - Same vs. Different PatternsTwo Square PathsOne Square and One Octagonal Path0.80.70.6Probability0.50.40.30.20.102468Search Day101214161820Figure26:ComparisonofSameandDi?erentPaths.
Afterobservinghowrelativelyine?ectivetheoctagonalsectorsearchis,itmayseemasurprisingresultthatthecombinationofasquareandsectorsearchisnearlycomparabletothatofbothsquares.Thiscanbeexplainedbyobservingwhatoccurswiththeprobabilitydistributionmodeloverthecourseofmultiplesearchdaysandhoweachsearchpatternworks.Thesimplesquareseeksoutthelargestpossiblesquareofgreatestprobabilitytosearch,whilethesectorpatternlooksforasmallconcentrationofhighprobabilityandbuildsawideperimeteraboutthisconcentrationtosearch.Byusingthesetwopatternsinconjunction,thesquaresearchpatternisabletoreducelargeuniformregions,andthesectorpatternwilltargetregionsofhighprobabilitybutsmallarea.Thus,usingtwosquaresearchpathsisstillpreferabletoonesquarepathandoneoctagonalsectorpath,butonlyslightly.Thisisour?rstattemptattestingcombinationsofsearchpaths;inthefuture,wewouldliketotestmorecombinationsusingdi?erentquantitiesandtypesofaircraft.
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Team#328793.7.6
TheE?ectivenessParameter
Page29of35
Afteranalyzingthevariouscombinationsofsearchplanerangesandsearchpatterns,weintroducedanotherparametertocreateamorerobustmodel:thee?ectivenessparameter.This“e?ectivenessparameter”isusefulinmodifyingthesimulationtomorepreciselymodelarealisticscenario.Thee?ectivenessparameterλisde?nedbelowinEquation26astheproductoftwootherparameterswhicharedeterminedbythespeci?csearch.
λ=α?β
(26)
Theαparameter,avaluecenteredaround1,describestheeaseof?ndingalostplane.Forinstance,amissingplanesuchasthelargeBoeing747wouldhavealargerαthanaCessnabecauseitistheoreticallyeasierto?nd.Theβparameter,alsocenteredaround1,describesthee?ectivenessofthesearchaircraft.Forinstance,asmallplanewithasinglesearcherperformingavisualscanforsignsofthelostplanewouldhavealowerβvaluethanaC-130Herculesequippedwithmultipleobserversandelectronicsensorssuchassonarandinfra-reddetection.
Usingthede?nitionofλ,wecanre-deriveamodi?cationofEquation14.Becauseλrepresentshowwellanareaissearched,thismodi?esEquation11:
λWh
(27)
A
Thehigherthevalueofλis,themorelikelytheplanecouldbefoundinthatincrementalarea.Wethenplugthisexpressionforg(h)backintoEquation12toget
g(h)=
λW
(28)
A
Thesolutiontothisdi?erentialequationisnodi?erentthanfrombefore,exceptthattheexponentnowcontainsthee?ectivenessparameter:
b??(z)=[1?b(z)]q=1?e
?λzWA(29)
Inessence,wehavebeenusingane?ectivenessparameterofoneforalloftheprecedingmodels.
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Team#32879Page30of35
Theplotbelowshowstheresultofdoublingthee?ectivenessparameterbetweentwosquaresearcheswhereeveryotherparameterisheldconstant:
1Effectiveness Model Comparison - Cumulative Success ProbabilityUnit Effectiveness Square SearchDouble Effectiveness Square Search0.90.80.7Probability0.60.50.40.30.20.102468Search Day101214161820Figure27:ComparisonofDi?eringE?ectivenessParameters.
Thisillustratesthefollowingresultsfordoublingthee?ectivenessparameter:
?Thereisanoticeableincreaseofcumulativeprobabilityoverthecourseofa20daysearch.
?Thegreatere?ectivenessparameterdoesnotfullydoublethecumulativeprobabilityofthesearchwithlowere?ciency.
Theseresultsareconsistentwithwhatwouldbeexpected.TheLawofDiminishingReturnsstatesthatifonlyonefactorisincreasedcontinually,thereturnratewilldecreaseovertimeduetotheincrementalincreaseofthisfactor[8].AsshownaboveinEquation29,qdecreasesnon-linearlywithanincreaseofzorλ.Thismeansthatthelongerthepath(orthelongerthesearchduration),thelessprobabilityofsuccesswillbeaccumulatedonanygivendayofsearching.
Tomodelarealworldapplicationofthee?ectivenessparameter,weconsiderthedisap-pearanceofa747vs.thedisappearanceofaCessna172.TheCessnahasacruisealtitudeofonly13000ftandalift-to-dragratioof7.5[14].Thisnotonlychangesthee?ectivenessparameter,butalsotheinitialprobabilitydistribution.Ane?ectivenessparameterof1isusedforthe747searchand0.5isusedfortheCessnasearch,basedonthehypothesisthataCessnawouldbeabouttwiceashardforsearcherstosee.Thecumulativeprobabilityofsuccessforbothofthesecasesisshownbelow:
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！
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