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Optimized allocation of police officers and Denver

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Optimized allocation of police officers and Denver
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Desautels, Alexa
Ebben, Christina
Gibala, Anna
Luginbill, Joshua
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Denver, CO
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University of Colorado Denver
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OptimizedAllocationofPoliceOcersinDenverCounty Authors:ChristinaEbben,JoshuaLuginbill,AnnaGibala,AlexaDesautels Abstract :Inthispaper,weproposetooptimizetheallocationofpoliceocersacrosspolicedistricts inDenverCounty.WerstconsideranIntegerLinearProgrammingformulationwhichtakesinto accountrealpopulation,crimedata,numberofocers,andbudgetdataforDenverCountyfrom theyear2014.Theresultsofthismodelshowaheavilyskeweddistributionofocersinfavorof thedistrictswithhighercrime.Next,weconsideranNonlinearProgrammingformulationunder theassumptionthateachadditionalocerplacedinadistrictgivesadiminishedchanceofsolving eachcrime.Theresultsofourlinearmodelareinteresting,buttheresultsofthenonlinearmodel showedamorebalanceddistributionofocerswheredistrictswithmorecrimewerestillallocated moreocers,butbyalowermargin. May16,2018 UniversityofColoradoDenver

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Contents 1Introduction 2 2Methodology 2 2.1DiscussionoftheCavadasetal.Paper..........................2 2.2OurWorkwiththeLinearModel.............................4 2.3OurWorkwiththeNonlinearModel...........................5 3ResultsandAnalysis6 3.1ResultsfromLinearModel.................................6 3.2ResultsfromNonlinearModel...............................8 4PolicyRecommendations9 5ConclusionandFutureWork10 ListofTables 1Population,Crime,CalculatedMinandIdeal,andAllocation.............6 2Population,Crime,CalculatedMinandIdeal,andAllocationWithoutBudgetConstraints...........................................7 3ResultsofNonlinearModelForDierent Values...................8 ListofFigures 1NonlinearObjectiveFunctionRepresentation......................5 2ResultsFromLinearModel................................7 3ResultsWithoutBudgetConstraints...........................8 4ResultsUsing =1.0...................................9 5ResultsUsing =0.005..................................9 6DenverCountyPoliceDistrictKey............................11 1

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1Introduction TheFederalBureauofInvestigationFBIdenesviolentcrimesasoenseswhichinvolveforceor threatofforce.Theseoensesarefurtherdividedintofourtypesofcrimes:murderandnonnegligent manslaughter,forciblerape,robbery,andaggravatedassault[1].Whenthesecrimesoccur,itis crucialthatlawenforcementocersareabletosolvethecrimesinordertogetviolentoenders oofthestreets. Inthispaper,wemaketheassumptionthatanocerinagivenareahasasmall"likelihoodof solving"foreachviolentcrimeoccurrenceinthatarea.Eachadditionalocerintheareaincreases thelikelihoodofsolving,butbyadiminishedamount.Sinceeverypolicedepartmentoperatesona limitedbudget,andthereforecanonlyemployanitenumberofocers,werstproposealinear optimizationmodel,followedbyanonlinearprogrammingformulationfortheoptimizationofpolice ocerallocationtosolvethegreatestnumberofviolentcrimes.Inthispaper,wedemonstratethis conceptbyconsidering,asresources,thetotalamountofpoliceocersinDenverCountytoallocate acrossthecounty'spolicedistricts,takingintoaccountviolentcrimeoccurrencesandpopulation ineachdistrictfortheyear2014. ThisworkwasmotivatedbytheUniversityofColoradoDenver'sDatatoPolicychallenge.The DenverOpenDataCatalogueDODC[3]providedinformationontheviolentcrimeoccurrences ineachpolicedistrictofDenverfor2014.TheDenverPoliceDepartment's2014AnnualReport providedinformationonthepopulationresidingineachpolicedistrictinDenverfor2014aswell asthetotalsizeofthepoliceforceinDenverCountyfor2014. 2Methodology Giventheviolentcrimeoccurrencesfor2014,weproposetooptimizetheallocationofpoliceocers acrossthepolicedistrictsofDenverCountytosolvethegreatestnumberofviolentcrimes. Weconsideredacertainnumberofocerstodistribute,takingintoaccountviolentcrimeoccurrencestoallocatemoreocerstodistrictswheremoreviolentcrimeoccurred.Theallocationis constrainedbyan"idealnumber"ofocersthateachdistrictwouldliketoreceive.Additionally, eachdistrictmustreceiveaminimumnumberofocersneededtoensurebasicpublicsafety. Giventhatthenumberofocersisanintegerquantity,werstusedanintegeroptimizationmodel. Welaterconsideredanonlinearobjectivefunction,butkepttheparametersandconstraintsthe sameasthelinearmodel. 2.1DiscussionoftheCavadasetal.Paper Ina2015papertitled"CrimePredictionUsingRegressionandResourcesOptimization,"Cavadas, Branco,andPereira[2]withtheUniversityofPorto,Portugual,proposedanintegerlinearprogrammingformulationfortheoptimizationofpoliceocerdistributionacrossthestatesofthe UnitedStatesofAmerica,whichweusedasabasisforourownwork. Animportantpartoftheirworkascrimepredictiondata.Thepaperourmodelisbasedoof usesinformationthatcombinessocio-economicalandlawenforcementdatafrom90'sCensus,1990 2

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LawEnforcementManagementandAdminStatssurveyandthe1995FBIUCR.Theauthors pre-processedthisdatabyonlyconsideringviolentcrimes,thusremoving17otherpossibletarget variables.Thisdatawasusedtosolvearegressionproblemthatpredictsthatnumberofviolent crimesper100kinhabitants. Theauthorswereveryconcernedwiththepossibilityofunder-predictingviolentcrime.Toaddress thisconcern,theyusedtheSmoteRalgorithm,thegoalofwhichistoover-sampleextremehigh valuesofviolentcrimesoastogeneratenewsyntheticexamples.Thusthelearningsystemsused fortheregressionproblemwouldbeforcedtofocusonsuchextremecaseswhichwouldbehard todowiththeoriginaldata.ThelearningalgorithmsusedareSupportVectorMachines,RFand MultivariateAdaptiveRegressionSplines. Theauthorsthenusedthesecrimepredictionstooptimizethedistributionofpoliceocersby stateusinganIntegerLinearProgram.Thisproblemisdenedby, maximize m X i =1 s i x i subjectto m X i =1 x i = N ; x i H i ; x i f i H i ; c i x i B i ; x i 2 N where m =46states, x i isthenumberofocersallocatedperstate, s i aretheviolentcrime predictionsperstate, H i isthenumberofidealocerstobeassignedperstate, f i isthefraction ofidealocersperstateneededforpublicsafety, c i isthecostofeachocerperstate,and B i is theavailablebudgetineachstate. Theidealnumberofocerstobeassignedperstate, H i ,wasdeterminedasafunctionofthe violentcrimepredictioninthestateandthestatepopulation, p i , H i = s i p i 100 Theminimumnumberofpoliceocers, f i wasthendeterminedtobeafractionoftheidealnumber. Upperandlowerboundsweredenedforthisfraction, u b and l b .Accordingtothetext,theauthors claimthatthisminimumnumberofocersisintheinterval[ v l ;v h ]andcalculatethenumberusing theformula, f i = s i )]TJ/F20 10.9091 Tf 10.909 0 Td [(v l v h )]TJ/F20 10.9091 Tf 10.909 0 Td [(v l u b )]TJ/F20 10.9091 Tf 10.909 0 Td [(l b + l b . Finally,theauthorsdeterminedabudgetforeachstateusingafunctionofstatepopulation, p s i , andpopulationdensity, d s i ,whichhavebeenrescaledtotherange[0,100]soastobegivenasimilar weight.Theformulafor B i isgivenby, B i = d s i + a p s i B T P m i =1 d s i + a p s i 3

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where B T isthetotalnationalbudgetand a> 0isaparametertotunetheweightofthepopulation andpopulationdensity.Theauthorsassignedcostofeachocerperstaterandomly,anduniformly, fromvaluesbetween5and15. Cavadas,Branco,andPereirausedsomerealdatafortheirpaper,butsyntheziseddataforthe numberofavailablepoliceocers,thetotalbudget,andthecostofpoliceocersbystate.The authorsusedtheirworktoshowaproofofconceptfortheirmethodsandsuggestthatfuturework shouldbedoneusingrealdata,andthatthemethodscouldbeappliedtodierentregions.This wasourmainmotivationforusingtheintegerlinearprogramwithrealdataforDenverCounty. 2.2OurWorkwiththeLinearModel WeusedthethelinearprogramproposedbyCavadasetal.asthebasisforourwork,butwe denedseveralofthemodel'sparametersdierently.Ourintegermodelcanbefoundbelow,as wellashowwehavedenedtheparametersofourprogram. max m X i =1 s i x i s.t. m X i =1 x i N c i x i b i f i x i h i x i 2 N For, i 2f 1 ;:::;m g m :=numberofpolicedistrictsinDenverCounty x i :=numberofpoliceocersassignedtodistrict i s i :=violentcrimeoccurancesbydistrict i N :=totalnumberofavailablepoliceocers f i :=theminimumnumberofpoliceocersneededindistrict i toensurepublicsafety h i :=idealnumberofpoliceocersfordistrict i Whencreatingourlinearmodel,westartedwiththeassumptionthatthenumberofpoliceocers perdistricthasalinearrelationshipwiththeamountofcrimeinDenver.Inotherwords,ifthere aremorecrimesoccurringinonedistrict,thereshouldbemoreocersallocatedinthatdistrict. Bythisassumption,wecanassumethatthemoreocerswecanhavesolvingacrimeandstationed inalocationincreasesthelikelihoodofcatchingthecriminalintheactorsolvingthecrime. Thelinearmodeldemonstratestheassumptionthatthenumberofpoliceocersperdistrict hasalinearrelationshipwiththeamountofcrimeinDenverthroughtheobjectivefunction max P m i =1 s i x i .Theobjectivefunctioncanbeunderstoodasmaximizingthequantityofocers 4

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allocatedtoapolicedistrictgiventhenumberofcrimesoccurrencesinthatparticulardistrict, andsubjecttoasetofconstraints.Therstconstraint, P m i =1 x i N ,statesthatthetotal numberofpoliceocerscannotexceedthetotalnumberofavailableocers.Thenextconstraint, f i x i h i ,isincludedasthesafetyconstraint,where f i istheminimumnumberofpoliceocers neededindistrict i toensurepublicsafety,and h i istheidealnumberofpoliceocersfordistrict i . TocalculatetheidealnumberofocersforeachdistrictinDenverCounty,wedeterminedthatno policedistrictinDenvershouldneedmoreocerspercapitathantheU.S.citywiththehighest tallyofocerspercapita.Thisassumptionwasmadebasedontheideathatcitieswithhigher violentcrimeoccurrencesrequireagreaternumberofocersso,sinceDenverhasarelatively moderatenumberofviolentcrimeoccurrencespercapita,nodistrictinDenvershouldsurpassthat number.UsingtheFBI'sUniformCrimeReportingUCRdata[3],andexcludingoneoutlier,we foundthatthehighestnumberofocersper10,000populationwas43.4[4].Therefore,wedened h i = 43 : 4 p i 10 ; 000 ,where p i isthepopulationofdistrict i inDenverCounty[6]. Likewise,tocalculatetheminimumnumberofocersforeachdistrict,wedeterminedthatno policedistrictinDenvershouldhavelessocerspercapitathantheU.S.citywiththelowest tallyofocerspercapita.Thisassumptionwasbasedontheideathatthecitieswiththelowest violentcrimeoccurrencesrequirelittlemorethanenoughocerstoguaranteebasicpublicsafety so,sinceDenverhasarelativelymoderatenumberofviolentcrimeoccurrencespercapita,each districtmusthaveatleastthatnumberofocerspercapita.UsingtheFBI'sUCRdata[3],and excludingthelowestoutlier,wefoundthatthelowestnumberofocersper10,000populationwas 7.1[4].Therefore,wedened f i = 7 : 1 p i 10 ; 000 ,againusing p i asthepopulationofdistrict i inDenver County[6]. 2.3OurWorkwiththeNonlinearModel Figure1:NonlinearObjective FunctionRepresentation Therstassumptionthatwasmentionedregardingthelinearmodel,assumedalinearrelationshipbetweenthenumber ofcrimesandthenumberofpolice.However,whenlooking moreintothewaypolicewerestudiedintheUCRdata[1], weneededtoadjustthismodeltoincorporatethefactthat atonepointtherewillbetoomanypoliceocersthanare neededandpotentiallyawasteofpoliceresources.Thatis,as youincreasethenumberofocersinanyonearea,theprobabilityofsolvingeachcrimeincreasesbyadiminishedamount andanincreaseinocersislessnecessarywitheachadditional ocerinthearea.Mathematically,thiscreatesmoreofacurvedexponentialgraphasrepresented inFigure1.Becauseofthisconcept,wedecidedtoalsotryanonlinearmodeltoaccommodatefor thefactthatafteracertainpoint,therateatwhichtoaddpolicedecreases.Thenonlinearmodel canbefoundbelow. 5

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max m X i =1 s i )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F20 10.9091 Tf 14.545 0 Td [(e )]TJ/F21 7.9701 Tf 6.587 0 Td [( x i s i s.t. m X i =1 x i N f i x i h i 0isaparametertotunetheweightofviolentcrimeoccurrencesonourocerallocation,and allotherparametersaredenedthesameasthelinearmodel. 3ResultsandAnalysis 3.1ResultsfromLinearModel Theobjectiveofourworkwastoassigntoeachpolicedistrictacertainamountofpoliceocers givenatotalnumberofpoliceocersinDenverCounty.Weassumedthatthecostperpoliceocer wasuniformacrossthecounty,sothatchanginghowocersweredistributedamongstthepolice districtswouldleavethetotalbudgetforthecountyunchanged.Thevaluesforthepopulationand thetotalnumberofocersarerealvalues[6].However,thevaluesfortheidealnumberofocers andtheminimumnumberofocersforeachdistrictweredenedbyus. Table1:Population,Crime,CalculatedMinandIdeal,andAllocation District Population ViolentCrime Min.Ocers IdealOcers OcerAllocation 1 84,467 624 56 344 130 2 100,225 677 72 442 153 3 189,878 595 121 736 195 4 120,850 715 92 563 179 5 78,593 525 44 296 106 6 44,908 861 41 252 141 Total 618,921 3,997 426 2,606 904 Table1,above,showstheresultsofourlinearmodelusingthebudget[5]constraintsforeach districtthatwecalculated.Table2,below,showstheresultsofourlinearmodelwhenthebudget constraintsoneachdistrictareloosened,representingashiftofresourcesamongstthedistricts withouthavingtoincreasethecounty'stotalbudget.OurresultswereobtainedusingAMPL software. 6

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Table2:Population,Crime,CalculatedMinandIdeal,andAllocationWithoutBudgetConstraints District Population ViolentCrime Min.Ocers IdealOcers OcerAllocation 1 84,467 624 56 344 56 2 100,225 677 72 442 72 3 189,878 595 121 736 121 4 120,850 715 92 563 363 5 78,593 525 44 296 44 6 44,908 861 41 252 252 Total 618,921 3,997 426 2,606 908 WecanseefromtheresultsincludedinTable1thatwhenweusedourbudgetcalculationsasa constraint,everydistrictwasconstrainedbyitsbudget,resultinginadistributionthatdidnotuse allavailableocers.Fromtheseresults,wedeterminedthattheoutcomewasnotverymeaningful, sowedecidedtoloosenthebudgetconstraintsonthedistricts.Whenwedidthis,wesawthatour modellledupthedistrictwiththehighestnumberofviolentcrimes,district6,untilthatdistrict hititsidealnumber.Anyadditionalocerswereplacedinthedistrictwiththenexthighest numberofviolentcrimes,district4.Weranoutofocersbeforedistrict4hititsidealnumber, buthadweincreasedournumberofocers,theywouldhavebeenplacedindistrict4untilits idealnumberhadbeenmet. Figure2:ResultsFromLinearModel Figure2showstheresultsofthelinearmodelwithbudgetconstraintsin amapofDenverCountypleasesee Figure6onpage11forakeyofthe districtsinDenvercounty,wheredistrict7hasbeengroupedintodistrict 5.Figure2visuallydepictstheresultsofourlinearmodel,including budgetconstraints,suchthatdarker goldisassociatedwithmoreviolent crimeoccurrencesandthenumber ineachdistrictcorrespondstothe numberofocersassignedtothat district.Thecolorassignedtothe numberofocersineachdistrictindicateswhichrestrictionlimitedthe numberofocers,withgreenindicatingthatthedistrictreceiveditsminimumnumber,redindicatingthatthedistrictreceiveditsidealnumber,andblueindicatingthat thedistrictreceivedamiddlenumberofocers,morethanitsminimumnumberbutlessthanits idealnumber. 7

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Figure3:ResultsWithoutBudgetConstraints Figure3showstheresults,visually depictedthesameasinFigure2, whenourbudgetconstraintoneach districtwasloosened.Thatis,when wedidnotrestricttheallocationof policeocersineachdistrictbythe budget.Sincethethesamenumber ofpoliceocers, N ,isstillanactive constraintandwehavenotchanged thecostofpoliceocer,theoverallbudgetforallofDenvercountyis maintained. Afterobtainingtheresultsofourlinearmodel,withandwithoutbudget restrictions,wedeterminedthatone extracrimeinadistrictshouldnot haveresultedinthatdistrictbeing prioritizedtoreceiveasmanyadditionalocersaswehadavailable.Itwasatthispointthat wedecidedtomovetoanonlinearobjectivefunction. 3.2ResultsfromNonlinearModel Table3,below,showstheresultsofdistributingthe908policeocersinDenverCountyamongst thepolicedistrictsusingdierentvaluesfor . Table3:ResultsofNonlinearModelForDierent Values District =1 : 0 =0 : 005 =0 : 0025 =0 : 00125 1 142 144 153 102 2 154 157 164 72 3 135 121 121 121 4 162 165 92 92 5 119 121 126 269 6 196 199 252 252 Total 908 907 908 908 8

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Figure4:ResultsUsing =1.0 FromTable3,wecanseethataswedecrease ourvaluefor ,theweightofviolentcrimeoccurrencesonourresultsincreases.Forinstance, when =1 : 0,ocersarespreadrelatively evenlyamongstthedistricts,withnodistrict obtainingitsidealnumberoritsminimumnumberofocers. Asexpected,thedistrictwiththemostviolent crimeoccurrences,district6,receivesthemost ocerswhilethedistrictwiththeleastviolent crimeoccurrences,district5,receivestheleast ocers.Whenwedecrease to0.005,wesee thatthenumberofocersindistrict3isreducedtoitsminimumnumberinordertoassignmoreocerstodistrictswithmoreviolent crimeoccurrences. Figure5:ResultsUsing =0.005 Figure4,above,showstheresultsofournonlinearmodelusing =1 : 0visuallydepictedin amapofDenverCounty,wheredarkergoldis associatedwithmoreviolentcrimeoccurrences andthenumberineachdistrictcorrespondsto thenumberofocersassignedtothatdistrict. Likewise,Figure5totheleftshowstheresults ofournonlinearmodelusing =0 : 005. 4PolicyRecommendations Furtherinsightwouldberequiredtochoosewhichvaluefor wouldbemosteective,thatis, todeterminehowmuchemphasisshouldbeplacedonviolentcrimeoccurrenceswhenallocating policeocerstodistricts.However,weproposethatocersshouldbeallocatedtodistrictsbased onthenumberofviolentcrimesthatoccurineachdistrict,ashavinganadditionalocerallocated toadistrictwithlessviolentcriminalitymaynotincreasethelikelihoodofsolvingeachcrimeas muchasthelikelihoodmayincreaseifthatocerwasallocatedtoadistrictwithahighernumber ofviolentcrimeoccurrences.Policeresourcesarescarce,andweproposethatourallocationmay helpinreducingwastedresources. 9

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5ConclusionandFutureWork Inthispaper,weproposedamethodforoptimallyallocatingpoliceocersinthescopeofDenver Countytoachievethegreatestlikelihoodofsolvingeachviolentcrimeoccurrence,takinginto accountrealviolentcrimedataandpopulationfrom2014. Forfutureworkinthisscope,weproposethatthisframeworkcouldbeimplementedusingall crimes,asopposedtostrictlyviolentcrimes,tondamorerealisticoptimizedallocationofocers inDenverCounty.Additionally,ifweweretodivideeachdistrictintoitsseparateneighborhoods, wecouldattainanevenmoreaccurateallocationofocerswhichcouldincreasethelikelihoodof solvingeachcrimeevenfurther.Moreover,weproposeexpandingthescopeofourworkoutsideof DenverCountycouldshowtheoptimalallocationofpoliceocersacrossdierentstates,countries, orregionsoftheworld. Wedoplanoncontinuingourworkwiththisproject.Wehavesteeredawayfromthelinear modelinfavorofthenonlinearmodeldiscussedinthispaper.Bycontinuingtoworkwiththe nonlinearmodel,webelievethatwecanobtain"better"policeocerallocationresults.Butmore importantly,wewanttousethenonlinearmodeltoobtainabettervaluefor h i .Thatis,wewould liketoobtainthenumberof"ideal"policeocersorganicallybysolvingthenonlinearmodel,rather thansynthesizingthe"ideal"numberfrompolicepresencedata.Recallthat h i iscurrentlydened as 43 : 4 p i 10 ; 000 where43.3isthehighestnumberofocersper10,000populationand p i isthepopulation perdistrict i inDenverCounty.Webelievethatwecanobtainbettervaluesfor h i organically throughimprovingour valuesandsolvingthenonlinearmodelitself. AnotherstudentoftheUniversityofColoradoDenverdepartmentofMathematicalandStatisticalSciencesiscurrentlyusingasimilaroptimizationmodelthatutilizesDiscreteWasserstein Barycenterstoimprovepoliceresponsetime.ThisworkwithDiscreteBarycentersisdirectedby Dr.SteenBorgwardt,whoisinterestedinincorporatingthemethodsforpoliceallocationused inthisprojectintohisongoingworktoimprovepoliceresponsetimes. Overall,thereareseveralwaysthatwecouldmoveforwardwithourworktooptimizepolice resourcesinDenverCounty.Webelievethatbyimprovingouroptimizationmodelwecangather valuableresultsthatcouldbeimplementedinrealworldpolicepractices. 10

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Figure6:DenverCountyPoliceDistrictKey Note:District7isgroupedintodistrict5. 11

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References [1] ViolentCrime .FBI,2012. [2]BrunoCavadas,PaulaBranco,andSrgioPereira.Crimepredictionusingregressionandresourcesoptimization. LectureNotesinComputerScience ,2015. [3]CityandDenverPoliceDepartment/DataAnalysisUnitCountyofDenver.Crime.2018. [4]Governing.Policeemployment,ocerspercapitaratesforu.s.cities. Governing ,2016. [5]MichaelB.Hancock. CityandCountyofDenverMayorsProposed2015Budget .Cityand CountyofDenver,2014. [6]ChiefRobertC.White.2014annualreportdenverpolicedepartment.pages6{27,2014. 12



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` Printing: Optimized Allocation of Police Officers in Denver County Authors: Alexa Desautels , Christina Ebben , Anna Gibala, Joshua Luginbill Abstract Police presence is known to be a key factor in reducing violent crime in an area. However, the question of where officers should be located, and in what quantity, in order to best reduce violent crime is rarely trivial. In this work, we propose an Integer Linear Programming formulation for the optimization of police officer allocation across police districts in Denver County. This allocation takes into account the population, budget, number of officers, and violent crime data for Denver County from 2014. Moreover, we demonstrate how our allocation will be affected by changes to the budget and number of officers employed. Objective Objective Function with Constraints Background to Policy project. Our methodology was inspired by the proof of concept proposed by Cavadas et. al. in which police officers were optimally distributed across the states of the United States of America. Methods Given the violent crime data for 2014, we propose to optimize the allocation of police officers across police districts. We considered a certain number of officers to distribute, taking into account violent crime data to allocate more officers to districts where more violent officers that each district would like to receive and a budget for each district. Additionally, each district must receive a minimum number of officers needed to ensure basic public safety. Results References Police employment, officers per capita rates for u.s. cities. Governing, 2016. Bruno Cavadas , Paula Branco, and S ´ ergio Pereira. Crime prediction using regression and resources optimization. Lecture Notes in Computer Science, 2015. City and Denver Police Department/Data Analysis Unit County of Denver. Crime. 2018. Budget. City and County of Denver, 2014. Chief Robert C. White. 2014 annual report denver police department. pages 6 27, 2014.6 27, 2014. Policy Recommendations Number of police officers assigned to district Number of violent crime predictions in district Total number of policer officers available Minimum number of officers needed in district Ideal number of officers in district Cost of police officer Budget for district Maximize the number of police officers in district relative to the number of crime occurrences The number of officers should be between the minimum and the ideal number The cost of officers cannot exceed the budget in district X is an integer albeit unrealistic, number. District 6 has the highest priority for police officer allocation. Officers should of officers has been met. At that point, any additional officers should be allocated to district 4 while districts 1, 2, 3, and 5 should receive only the minimum number of officers required to guarantee basic public safety. Below are the results of our model using data from 2014. The number of officers cannot exceed the total number of available District Budget Officer Allocation 1 8,5598,24.00 56 2 11,005,488.00 72 3 18,495,334.00 121 4 55,486,002.00 363 5 6,725,576.00 44 6 38,519,208.00 252 Total 138,791,432.00 908