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Engineering and Technology Management Student Projects Engineering and Technology Management

Winter 2018

Minimizing Commute Distance for Small Groups: A Minimizing Commute Distance for Small Groups: A

Linear Programming Approach Linear Programming Approach

Kevin Payne Portland State University

Kritika Kumari Portland State University

Levi Huddleston Portland State University

Rabi Hassan Portland State University

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Citation Details Citation Details Payne, Kevin; Kumari, Kritika; Huddleston, Levi; and Hassan, Rabi, "Minimizing Commute Distance for Small Groups: A Linear Programming Approach" (2018). Engineering and Technology Management Student Projects. 2104. https://pdxscholar.library.pdx.edu/etm_studentprojects/2104

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ETMOFFICEUSEONLYReportNo.:Type: StudentProjectNote:

MinimizingCommuteDistanceforSmallGroups:ALinearProgrammingApproach

CourseTitle:OperationsResearch

CourseNumber:ETM540

Instructor:TimothyAnderson

Term:Winter

Year:2018

Author(s):KevinPayne,KritikaKumari,LeviHuddleston,RabiHassan

2

Abstract:Thispaperaimstominimizetotaldrivetimebetweenmembersandtheirrespectivegroupleader.Givenalimitongroupsizeanddaysavailable,howcanaformulationofagroupoccursuchthatthesumofthetotaldrivetimeisminimized.ToaccomplishthistaskaLinearProgram(LP)isimplementedthatincludesthreesetsofbinarydecisionsvariablessummingto4100variablesandavarietyofconstraintssummingbetween4200and4341dependingontheconstraintsenforced.For200membersand15leaderstheminimizedaveragecommutingtimewasfoundtobebetween4.99and5.36minutesdependingonthedistributionofavailabilityassumed.

Keywords:LinearProgramming

3

TableofContents

Abstract......................................................................................................................................................2

TableofContents…………………………………………………………………………………………………………………………………..3

Introduction...............................................................................................................................................4

LiteratureReview…....................................................................................................................................5

ProjectObjective........................................................................................................................................7

Results…………………………………....................................................................................................................10

Conclusion.................................................................................................................................................12

Appendix………….........................................................................................................................................14

Rcode…………….........................................................................................................................................19

4

Introduction

Oneofthekeyfunctionsofachurchistoprovideamemberwithavenuetobuildasenseof

communitywithotherchurchmembers.Oneoftheprimaryreasonswhycongregantswillselectasmall

churchtoattendratherthanalargechurchistheclose-knitcommunitythatmanysmallchurcheslend

themselvesto.Becauseregularchurch-goersoflargechurchesrarelyknow–letalonerecognize–all

theothercongregants,largechurcheswilloftenofferspiritually-relatedspecialinterestgroupsor

connectcongregantsintosmallgroupswheremembersareencouragedtoengagewiththeirfaith.

Whilechurchesofmoderatetolargesizecanoftenaccommodateamanualsortingprocessof

connectinginterestedmembertoavailablesmallgroups,theprocesscanbecomecumbersomefor

churcheswithweeklyworshippersabovetwothousand.Theprocessisoftencomplicatedbythefact

thatmemberswishtoattendasmallgrouplocatedclosetotheirresidence.Smallgroupsaretypically

hostedbyamemberofthesmallgroupattheirprivateresidence,howeveritisnotunusualthatthe

grouplocationalternatesbetweenmembers’residences.

Churchleadersaretaskedwithconnectingnewcongregantswithexistinggroups,formingnew

groups,and/orencouragingmaturecongregantstoformagroupoftheirown.Manychurchestakea

directhandinthemanagementofthesmallgroupsthatformbetweenthemembersoftheirchurch.

Becausethewell-beingofchurch-goers–especiallyregularchurch-goerswhoareinvolvedbeyondthe

Sundayservice–isvitaltothesuccessofachurch,itisimportantthattheleadersofthechurchensure

thatmembersaresatisfiedwithintheirsmallgroupsandthatthefundamentalprinciplesoffaithheldby

thechurcharenotcompromised.Therearemanyalternativepurposesbehindencouragingsmall

groups.Thesepurposesincludebutarenotlimitedtoconstructinganavenuetobuildupmoreleaders

forthechurch,providinganavenueforchurch-memberstoservetheirlocalcommunityorasocialneed

thattheyfeelpassionateabout,andincreasingthehealthofamember’sspirituallife.Whilethe

selectionofmembersforagroupdoesnothaveadeterministiceffectonthefulfillmentofthechurch’s

goals,factorssuchasthedistributionofageandgenderwithinagroup,thegeographicproximityof

groupmemberstoeachother,andthespiritualmaturityofgroupmembersdoplayanimportantrolein

thedegreeofsuccessintheabovegoalsthatthegroupachieves.

Theanalysisdoneinthispaperistoclustermembersofachurchintomultiplegroupssuchthat

thecharacteristicsofthegroupareoptimalforsuccess.AninterviewwasconductedwithCollin

5

Mayjack(PastorofCommunities)andGavinBennett(DirectorofCommunities)fromtheBridgetown

Churchinwhichthemanualsortingprocessforconnectingcongregantswithsmallgroupswasdescribed

andtheformulationofthisprojectbegan.Allchurch-goerswhowishtojoinasmallgroup(knownasa

BridgetownCommunity)mustfilloutaquestionnaire.Thesequestionswillultimatelyserveasourdata

forourmodel.Forinformationprivacyreasons,dataonaddressesofcongregantscouldnotbe

obtained.However,adatasetofaddressesinthePortlandareawasobtainedandtransformedtothe

sameformatasthespreadsheettemplateprovidedbyBridgetownChurch.

Inadditiontohomeaddress,thespreadsheettemplatealsoincludedfieldsforage,gender,

scheduleavailabilityanddesiretoleadasmallgroup.Thedistributionforage,gender,geographic

proximity,andavailabilityofmembers’scheduleswasdiscussedBridgetownleaders.Thedemographics

informationcouldbeusedforfutureworkbutwasnotincludedinourinitialanalysis.Themostrecent

Basicscohortconsistedofapproximately200members,with20membersexpressinginterestinleading

agroup.Anidealgroupsizeconsistsof10-15people,butbecausemembersmayoptoutoftheirgroup,

theoptimizationproblemallowsformationofgroupssized12-18people.Thereisalsoanon-binding

constraintthatlimitsthemaximumnumberofleadersto15.

LiteratureReview

Theproblemofgroupingmembersbyleaderis,atbase,aclusteringproblem.However,

becausetheprobleminvolvesassigningmemberstopre-specifiedleaders,typicaldatamining

algorithmssuchasK-meansclusteringcouldnotbeused.Rather,nonlinearoptimizationisthetool

usedtoselectaspecifiednumberofleadersfromalargergroupofpotentialleaderssuchthatthe

distancefrommemberstoleadersisminimizedandavailabilityconstraintsofmembersaresatisfied.

Ingeneral,optimizationproblemshavethefollowingcomponents:

•Anobjective–expressedasafunctiontomaximizeorminimize.

•Decisionvariables–elementsoftheproblemathandthatcanbechanged.

•Parameters–characteristicsoftheproblemthatarefixed.

•Constraints–limitsontheobjectiveordecisionvariables.

6

AlfredWeber(1970)wasthepioneerinresearchinlocationanalysis,afieldconcernedwith

locatingfacilitiessuchthatcostanddistanceareminimized.Manyfacilitylocationproblemsaremulti-

objectiveinnature,thetoptwoobjectivesbeingcost/distanceminimizationanddemandcoverage.

Profitmaximizationandenvironmentalconcernsarealsooccasionallyfactoredintolocationanalysis

objectivefunctions,buttraditionallyminimizingcost/timehasbeentheprimaryobjectiveofsuch

models.1

Facilitylocationproblemscanalsobeclassifiedaccordingtonumberoflevels.Bi-levelproblems

aresignificantlymoredifficulttoconstruct,asonemustformulatethemodelsuchthatthereisaleader

andafollowerindecisionmakingandwhereinformationavailabilityisdifferentatdifferentstagesof

decisionmaking.2Ourmodelinvolvesonlyonelevelofdecision-making:whichmemberstoassignto

whichleadersonagivenday.

Ouroptimizationproblemcouldbecategorizedfurtherasaschedulingproblem,asthe

availabilityofeachleaderandmembermustmatchinordertoformgroups.Examiningtheliteraturein

thisfield,wedistinguishedthreemaingroupsofschedulingproblems:shiftscheduling,daysoff

schedulingandtourscheduling,whichcombinesthefirsttwotypes.Ourprojectfallsunderthesecond

category(daysoffor“day-of-week”scheduling),astheavailabilityconstraintinourmodelforcesthe

optimizationofmemberandleaderschedulessuchthatallmembersofagroupcanmeetwiththe

leaderonatleastonedayoftheweek.3

Theschedulingproblemwithinourmodelcanbeclassifiedmorespecificallyasaspecialformof

coveringproblemwhereonefollowershouldbegroupedwithoneleader.Schillingetal.(1993)4classify

suchmodelsthatusetheconceptofcoveringintwocategories:(1)SetCoveringProblems(SCP)where

coverageisrequiredand(2)MaximalCoveringLocationProblems(MCLP)wherecoverageisoptimized.

OurprojecttakestheSetCoveringProblem(SCP)approach.

1Current,J.,Min,H.,&Schilling,D.(1990).Multiobjectiveanalysisoffacilitylocationdecisions.EuropeanJournalofOperationalResearch,49(3),295-307.2Caunhye,A.M.,Nie,X.,&Pokharel,S.(2012).Optimizationmodelsinemergencylogistics:Aliteraturereview.Socio-economicplanningsciences,46(1),4-13.3Baker,K.R.(1976).Workforceallocationincyclicalschedulingproblems:Asurvey.JournaloftheOperationalResearchSociety,27(1),155-167.4Schilling,D.A.(1993).Areviewofcoveringproblemsinfacilitylocation.LocationScience,1,25-55.

7

RevelleandHogan(1989)introducerandomnessinavailabilityandconsiderstraveltimeas

deterministic.5Ouranalysislikewiseconsidersmemberavailabilitytoberandomandusesonenon-

rush-hourestimateofdrivetimebetweeneverymemberwitheachpossibleleader.

Intheirnurseschedulingproblem,Bruckeretal.[56]constructshiftsequencesforeachnurse,

withthedifferingskilllevelsofnursesbeingcodedashardconstraints.6Whiletheavailabilityof

membersisconstructedasahardconstraintinourmodel,themostprudentmethodofincorporating

genderandagetargetswouldbetoformulatethesegoalsassecondaryobjectives.Settingappropriate

weightstothesesecondaryobjectiveswouldbeequivalenttoforcingasoftconstraintontheobjective

ofminimizingdistance.Thus,similartotheeffectofavailabilityontheobjectivevalueofourmodel,

incorporatingageandgenderintotheobjectivefunctionwouldresultinasolutioninwhichmembers

arematchedwithleadersthatarepotentiallyfurtherawaythanotherwise.Formulatingmember

characteristicsashardconstraints(asinBruckeretal.[56])wouldgreatlyincreasetheaveragedistance

betweenmemberandleader.BecausethemainconcernofBridgetownChurchistominimizedistance

betweenmembersandleaders,itwasdeterminedthatfactoringingenderandagewouldunnecessarily

complicatethemodel.

ProjectObjective

Thisoptimizationproblemselectsleaders,andmemberstobeinagroupwiththoseleaders,

suchthattheoveralldrivetimetraveledisminimized.Thefirsttaskafterattainingthedatasetwhich

includedthosevariableslistedabove,istofindthedrivetimetraveledbetweenmembersandleaders.

Thisisthecruxoftheoptimizationproblem,sincethemainobjectiveistominimizedrivetimesubjectto

avarietyofconstraints.TofindthesetimestraveleditmaybetemptingtousesomeformofEuclidian

distanceasameasureanddividebyaveragespeed,butoftentimesthiscanbemisleadingsinceactual

drivetimesmaybemuchdifferent.Tocounteractthis,thisresearchpullsdrivetimedatafromGoogle

Map’sAPIinseconds,aswellasdistance.Thetotalnumberofsecondsdrivenforallmemberstotheir

5ReVelle,C.,&Hogan,K.(1989).Themaximumavailabilitylocationproblem.TransportationScience,23(3),192-200.6VandenBergh,J.,Beliën,J.,DeBruecker,P.,Demeulemeester,E.,&DeBoeck,L.(2013).Personnelscheduling:Aliteraturereview.EuropeanJournalofOperationalResearch,226(3),367-385.

8

selectedleaderswillbetheobjectivevalueforourminimizationproblem.Algebraicallytheformulation

isasfollows:

!"#"$"%&{ )"$& $, + ∗ -[$, +]}

122

345

12

645

7ℎ&9&)"$& $, + :&;"#&<=ℎ&:9">&�"$&?&=@&&#$&$?&9$A#:+&A:&9+, A#:

- $, + "<&BCA+=D1";$&$?&9$"<A<<"F#&:=D+&A:&9+, 0D=ℎ&9@"<&

Therearemanyconstraintsneededinthismodelsothattheassignmentsofindividualsto

leadersisbothrealisticandoptimal.Thefirstmostobviousconstraintisthateverymemberneedstobe

assignedaleader.Thisisshownbelow.

- $, + = 1∀!&$?&9<$JD#<=9A"#=K

12

645

Next,itmustbeimposedthatifamemberisassignedtoaleader,thenitmustbetruethattheleader

hasbeenhasbeenchosenoutofthepoolofpotentialleaders.Itisimportanttorecallthatnotall

individualswhospecifythattheywishtoleadagroupwillendupleading.Forthisreason,itisnecessary

toimposethefollowingcondition.

- $, + ≤ M + ∀!&$?&9<$A#:N&A:&9<+JD#<=9A"#=KK

7ℎ&9&M + "<&BCA+=D1";OD=&#="A++&A:&9�"<A<<"F#&:=D?&A9&A+"%&:+&A:&9

Theaboveconditionrestrictsthevalueof- $, + suchthatnomembermwillbeassignedtoagroup

leaderlifthegroupleaderdoesnotexist.Next,arestraintmustbeinplacetolimitthetotalnumberof

leaders.Otherwiseallleaderswilllikelybeselectedsinceitcouldreducetotaldrivetime.Thefollowing

constraintrestrictsthetotalnumberofleadersallowable.

M[+]

12

645

≤ 20 − RJD#<=9A"#=KKK

7ℎ&9&R"<AO9&:&;"#&:OA9A$&=&9?M=ℎ&C<&9

9

Thelastconstraintthatwillmaketheoptimizationproblemrealistic,isthatthetotalsizeofeverygroup

needstobegreaterorequaltotwelvebutlessthanorequaltoeighteen.Thisconstraintcanbe

implementedasfollows.

- $, + ≤ 18A#: - $, + ≥ 12 ∗ M + ∀

122

345

122

345

N&A:&9<+JD#<=9A"#=KU

Theseconstraintsinsurethatforallleadersthemaximumnumberofmembersmustbelessthanor

equalto18.Inaddition,iftheleadersareselected,thentheirgroupmembersshouldsumgreaterthan

orequalto12.Theseconstraintswouldensurethatrealisticgroupingswouldbemade.

Moreconstraintsareneededinordertotakeintoaccounttheaddeddimensionofavailability.If

twoindividualsdonothavematchingschedulesthenthesetwoindividualsshouldnotbeplacedinthe

samegroupevenifthetwoindividualsarecloseinproximity.Wecanextendthissothatifamemberis

assignedagroupthenallmembersinthatgroupmusthaveacommondayavailability.Thisensuresthat

therearenoschedulingconflictsbetweenmembersinthesamegroup.Thisconstraintcanbe

implementedasfollows.

- $, + ∗ V>A"+A?"+"=M $, +, W

122

345

≥ - $, +

122

345

− X − 18 ∗ 9 +, W − M + + 1 JD#<=9A"#=U

∀N&A:&9<+A#:WAM<W

V#:

9 W, +

Z

5

≤ 3∀N&A:&9<+JD#<=9A"#=UK

7ℎ&9&V>A"+A?"+"=M"<&BCA+=D1";$&$?&�$A#:+&A:&9+\A#$&&=D#WAMW, 0D=ℎ&9@"<&

9 +, W "<&BCA+=D1";N&A:&9+D#WAMW"<#D=<&+&\=&:

X"<AO9&:&;"#&:OA9A$&=&9=ℎA=<D;=&#<=ℎ&\D#=<9A"#=

TheabovestatementsinConstraintsVandVIensurethatifamemberisassignedtoagroup

thenthetotalavailabilitymustbegreaterthanorequaltothetotalnumberofindividualsinthatgroup

foratleastonedayD.Since9 +, W needstobelessthanthreeinConstraintVI,thisensuresonlyoneof

10

thethreedayswillbeabindingconstraint.Inaddition,theincludedM + inConstraintVIprovidesa

measuresuchthatifthememberisnotcreatedthenthegroupavailabilityconstraintwilleasilybemet.

Kisequalto0iftheconstraintishardorischosenasequaling2iftheconstraintissoft.Thetotal

population’savailabilityincludingbothmembersandleadersintermsofpercentagesfollowstwo

distributions.Thefirstisoftenreferredinthispaperasthe‘MoreAvailability’matrixandfollowsthe

followingdistributionwith1percentofindividualsareavailableforonly1day,1percentofindividuals

areavailableforonly2days,30percentofindividualsareavailablefor3days,and68percentof

individualsareavailableforallfourdays.Theseconddistributionoftenreferredtoasthe‘Fewer

Availability’matrixhasadistributionsuchthat5%areavailableforonlyoneday,25%fortwo,40%for

threeand30%forfour.Theseconstraintstogetherwithanavailabilitymatrixwillprovidefora

minimizeddrivetimetraveledthatwillplacemembersintogroupsandselectrespectiveleaderforthose

groupssuchthatthereisatleastonedaywhereeveryonecanmeet.

Results

Asnotedintheabstract,theoptimalaveragecommutetimewasbetween4.99and5.36

minutesforthe200membersdependingonavailability.InRstudiotheLPsolver“symphony”and

“GLPK”wereusedtosolvetheseproblems.The“symphony”solversolutionwasultimatelyused.Figure

1intheappendixdisplaystheinitialproblemsetwithmemberscodedinthecolorblueandleaders

codedinwiththecolorred.Figure2displaysthesameaddresses,yetcolorcodeseachmember’s

addressbasedonwhotheirrespectiveleaderisfortheirgroup.Thisoptimizationusesallconstraints

andthe‘MoreAvailability’matrixwithKsettozerotofindanobjectivevalueof59870secondsor4.99

minutesonaverageperperson.Anaccessibleversion,withinteractivetoolsforbothmapshasbeen

madeaccessiblethroughthefollowinglinks.

Figure1:https://drive.google.com/open?id=1r8m2RUSJWpFgb6FZSjkWKiVrWt7e4-Fy&usp=sharing

Figure2:https://drive.google.com/open?id=1t7aPwC66Bbo38NjuCMkVC7_Xb7skBDmv&usp=sharing

Inadditiontotheinitialproblem,avarietyofotherproblemswerealsosolved.Oneofthese

problemswastoassesstheimpactsoftheavailability,groupsize,andleadersizeconstraintsonthe

optimalsolution.Itcanbeobservedinthedescriptionofthe‘MoreAvailability’matrixthat98percent

11

ofallindividualsinthepopulationareavailableformorethanthreedaysaweek.Asaresult,itwouldbe

expectedthattheremovalofsuchaneasilysatisfiedconstraintmaynotdrasticallyaffecttheoptimal

solution.Removingtheotherconstraintswoulddecreasecommutetime,butthesizeoftheeffectmay

bevaguer.Afterremovingallconstraintsdealingwithgroupsize,numberofleaderandavailability,time

traveleddecreasedbyabout18secondsperperson.Althoughthisamountmayseemsmallonan

individuallevel,thetotalincreaseintimetraveledduetoconstraintsis3,642seconds–aboutanhour

ofextracommutetimeaggregatedacrossallmemberseachweek.

Whileitmaybeinterestingtoanalyzetheimpactofthegroupsizeandleadersizeconstraintson

commutetime,theseconstraintsareessentialcomponentsofthesolutiondesiredbythechurch.Using

anavailabilitymatrixthatismorealignedwithactualreportedavailabilityofchurchmembers(the‘Less

Availability’matrix),the‘symphony’solverreportsaninfeasiblesolution.Tofindasolution,Kissetto

twosothatatleasttwomembersinagroupareallowedtobeunavailable.Underthisformulation,the

LPproblemisfeasibleandtheoptimalsolutionforthe‘LessAvailability’matrixis60449seconds(on

average5.36minutespermember).Comparingthissolutiontothepreviousresultwherethe‘More

Availability’matrixwasused,wefindthatreducingtheavailabilityofmembersandleadersaddsatotal

of60449–59870=579secondsor9.65minutestototalcommutetimeperweek.Thisisparticularly

interesting,sinceavailabilitydecreasesquitedramatically,whilethesolutiononlyincreasesslightlyin

termsoftime.ThelaxingoftheavailabilityconstraintwithKequaltotwo,allowsforthismodest

increase.TheclusteringofthisLPcanbeseeninFigure3,orinthelinkprovidedbelow.

Figure3:https://drive.google.com/open?id=1LDiGu6W57vkdD0wkQ9IKojFlFmpvEYjd&usp=sharing

Figures4and5giveasummaryofthefindingswithdifferentconstraints.Figure4presentsthe

objectivevalueforavarietyofdifferentconstraints.Asisexpectedandwillalwaysbetheresultof

imposingmorestringentconstraints,theobjectivevalueincreaseswitheachadditionofanew

constraint.SpecialattentionshouldbegiventotheimpactofhavingKequalstotwoontheobjective

function.NotonlydoesitallowtheLPtofindasolution,butitonlymarginallyincreasestheobjective

value.Figure5displayshowmanymemberseachleaderisassignedfordifferinglevelsofconstraints.

Withnoconstraintsimposed,numberofmembersperleaderisvolatile,withsomeleadershaving31

memberswhileothersonlyhave1member.Forformulationsinvolvinglimitsongroupsize,however,

thedifferencesinnumberofmembersassignedtoeachleadervarylittlewithchangesintheavailability

matrix.Onenotableexceptionisthemodelusingthe‘LessAvailability’matrix,whichassignsno

memberstoleader19.ItshouldalsobenotedthatFigure5onlycomparesthenumberofmembers

12

assignedtoeachleaderunderdifferentformulationsofthemodelbutdoesnotinformusastowhether

individualmembersareconsistentlyassignedtothesamegroup.Theconsistencyofindividualmember-

groupassignmentsacrossmodelscanbeverifiedbycrosscheckingtheinteractivemapsofFigure1,2

and3.

Althoughnotdiscussedindetailhere,thesameoptimizationswereconductedonwalkingtime.

ThemapsfortheseLPproblemsareinthesameorderintermsofconstraintsasthepreviousproblems

withdrivetimeandcanbeviewedinthefollowinglinks.

https://drive.google.com/open?id=1gLjbDIXJA98N1PH0wtssUmbJVjMExBt-&usp=sharing

https://drive.google.com/open?id=1e4BNlbWVow32TxFuMnnltxZkfn-CjqQm&usp=sharing

https://drive.google.com/open?id=1LviR-TMPC2i0czUoTkBSvsrkUe2tuSFA&usp=sharing

Conclusion

Theanalysisdoneinthispaperlaysthegroundworkforautomatingtheassigningofmembersto

leadersatBridgetownChurch.Whilethedataweregeneratedtomatchatemplateprovidedbythe

churchandthesolutionismerelyaproofofconcept,thepotentialusefulnessofthemodelframeworkis

significant.Ourmodelgeneratesthemostefficientgroupspossiblesubjecttotheavailabilityofevery

memberandleader.

Thoughourprojectissolvingaproblemofeffectivegroupformationforachurch,themodelwe

proposedtofindtheoptimumsolutionthatreducesthetraveltimefortheparticipantsinthegroupcan

easilybereplicabletosolveamyriadofproblems.Forexample,consideraretailbusinessmodel,where

firmsareconstantlytryingtoreachouttoasmanycustomersaspossiblewithoutincurringlargecosts

relatedtotheconstructionoffacilities.Ourmodelwouldminimizethedistancebetweencustomersand

stores,makingthedecisionofwheretobuildverysimple,givenalistofpotentialconstructionsites.

Similarly,whensettingupanewhospitalsorbank,decisionmakersarechieflyconcernedwithhowthey

couldmaketheserviceprovidedmoreaccessibletoallcustomers.Ourmodelagainwouldmakethe

decisionclearastowhattheoptimalallocationwouldbe.Additionalconstraintsorweightscouldbe

added,sothatdifferentareasmightbeassociatedwithahigherpriorityrelativetootherareas.

13

Ouroriginaldatasethasdemographicinformationofallthemembers.Itisnaturaland

compellingthatanextensionofourmodelwouldincludesuchfactorswhencreatinggroups.Thistype

ofimprovementwouldmakethemodelmoreinclusiveandnotonlysolvethedistanceminimization

problembutwouldaddyetanotherlayerontopoftheavailabilityconstraint.Suchamodelwould

undoubtedlyproveofinteresttomanybusinesses,sinceoptimalallocation,mayneedtoincludesome

referenceastowhattypeofpeoplelivearoundthearea.

Otherextensionsofourresearchworkcouldbetestingtheeffectivenessofthecreatedgroups.

ForBridgetownChurchthiscouldbeillustratedbytheturnoverrateineachgroupundertheleadership

ofaparticularleader.Indoingthisonecouldcreateasetofefficiencyindicesthatcouldmeasurethe

performanceofthosegroupsincomparisontoothergroups.Thiscouldpotentiallyuseadata

envelopmentanalysismethodashasbeensuggestedbysomepapersingroupleadershiptheory.

Whiletherearemanymoreextensionsthatcomenaturally,thismodelsolvestheissueof

creatinggroups,whileaccountingfornotonlydistancebutforavailabilityaswell.Thismethodof

clusteringwillprovidevalueaddedtotheBridgetownChurch,asitwillnowhavetheoptionofcreating

groupswithcompatibleschedules,asopposedtotheprevioustechniquewhereitwasunaccountedfor

intheirmethod.Thisvalueisembodiedinthetimesavedinselectinggroupsthatthechurchleadership

goesthrough,aswellasthetimesavedforeachmemberintermsofdrivetimeeachweek.

14

Appendix

Figure1:InitialMinimizationProblem

15

Figure2:GroupAssignmentsforDriveTime‘MoreAvailability’

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Figure3:GroupAssignmentsforDriveTime‘LessAvailability’

17

Figure4:ObjectiveValueforDifferentConstraint

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

DriveTimeNoConstraints DriveTimeAllConstraintsMoreAvailability

DriveTimeAllConstraintsFewerAvailability(Soft)

DriveTimeAllConstraintsFewerAvailability(Hard)Infeasible

Minutes

ObjectiveValueforDifferentConstraints

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MEMBERSASSIGNEDBYCONSTRAINTTYPEDriveTimeNoConstraints DriveTimeAllConstraintsFewerAvailability(Soft) DriveTimeAllConstraintsMoreAvailability

19

RCodeKritikaKumari,RabiHassan,LeviHuddlestonandKevinPayne

3/9/2018

#Model

install.packages("ompr.roi")install.packages("ROI.plugin.glpk")install.packages("ROI.plugin.symphony")install.packages("slam")install.packages("devtools")install.packages("data.table")install.packages("tidyr")install.packages("ompr")install.packages("magrittr")install.packages("dplyr")install.packages("dplyr")install_url("https://cran.r-project.org/src/contrib/Archive/slam/slam_0.1-37.tar.gz")

require(slam)require(ompr.roi)require(ROI.plugin.glpk)require(ROI.plugin.symphony)require(devtools)library(data.table)library(tidyr)library(ompr)library(magrittr)library(dplyr)library(stringr)library(readr)setwd("C:/Users/Levi/Documents/PortlandState/OperationsResearch/Project")#save(results_distance,file="distance")#save(availability_leaders_matrix,file="availability_leaders")#save(availability_members_matrix,file="availability_members")#save(availability_leaders_matrix,file="availability_leaders_sparse")#5,25,40,30#save(availability_members_matrix,file="availability_members__sparse")#5,25,40,30

#load(file="distance")#Drivetimeresults_distance<-read_csv("~/PortlandState/OperationsResearch/Project/distance_walk.csv")load(file="availability_leaders")availability_leader=availability_leaders_matrixload(file="availability_members")

20

availability_member=availability_members_matrixdistance<-as.data.table(results_distance[,1:3])rm(results_distance,availability_leaders_matrix,availability_members_matrix)#distance=distance[order(Time.de)]#distance<-transform(distance,id=match(Time.de,unique(Time.de)))a=200b=20k=5#b-k=actualnumberofgroupsuplimit=18lowlimit=12availabilty_soft=2

distance<-spread(distance,Time.de,Time.Time)setnames(distance,"Time.or","MemberLocations")distance=as.matrix(distance)distance=na.omit(distance)distance=distance[1:a,1:(b+1)]availability_leader=availability_leader[1:b,]availability_member=availability_member[1:a,]

availability=matrix(0,nrow=a*b,ncol=4)for(min1:a){for(dayin1:4){for(lin1:b){availability[b*(m-1)+l,day]=availability_leader[l,day]*availability_member[m,day]}}}colnames(availability)=colnames(availability_leader)availability=as.data.table(availability)availability$L=rep(1:b,a)

member_stuff=NULLfor(iin1:a){temp=rep(i,b)dim(temp)=c(b,1)

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member_stuff=rbind(member_stuff,temp)}availability$M=member_stuffrm(temp,member_stuff,availability_leader,availability_member)

#FORMULATINGTHEMODELmodel<-MIPModel()%>%#1iffmembermgetsassignedtoleaderladd_variable(x[m,l],m=1:a,l=1:b,type="binary")%>%

#1iffleaderlischoosenadd_variable(y[l],l=1:b,type="binary")%>%#1ifonDayDforleaderLthedayisnotchosenasthemeetingadd_variable(r[D,l],D=1:4,l=1:b,type="binary")%>%

#Minimizethedistanceset_objective(sum_expr(as.numeric(distance[m,l+1])*x[m,l],m=1:a,l=1:b),"min")%>%

#everymemberneedstobeassignedtoaleaderadd_constraint(sum_expr(x[m,l],l=1:b)==1,m=1:a)%>%

#ifamemberisassignedtoaleader,#thenthisleadermustbetheactualleaderofthegroup#notjustthepotentialleaderadd_constraint(x[m,l]<=y[l],m=1:a,l=1:b)%>%

#Lessleadersthanthetotaladd_constraint(sum_expr(y[l],l=1:b)<=(b-k))%>%#Numberofmembersineachgroupneedstobebetween12and18add_constraint(sum_expr(x[m,l],m=1:a)<=(uplimit),l=1:b)%>%add_constraint(sum_expr(x[m,l],m=1:a)>=(lowlimit)*y[l],l=1:b)%>%

#Availability,atleastonedayintheweek,theyallcanmeetonthesamedayadd_constraint(sum_expr(x[m,l]*as.numeric(availability[M==m&L==l,1]),m=1:a)>=(sum_expr(x[m,l],m=1:a)-availabilty_soft)-uplimit*(r[1,l]-y[l]+1),l=1:b)%>%

add_constraint(sum_expr(x[m,l]*as.numeric(availability[M==m&L==l,2]),m=1:a)>=(sum_expr(x[m,l],m=1:a)-availabilty_soft)-uplimit*(r[2,l]-y[l]+1),l=1:b)%>%

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add_constraint(sum_expr(x[m,l]*as.numeric(availability[M==m&L==l,3]),m=1:a)>=(sum_expr(x[m,l],m=1:a)-availabilty_soft)-uplimit*(r[3,l]-y[l]+1),l=1:b)%>%

add_constraint(sum_expr(x[m,l]*as.numeric(availability[M==m&L==l,4]),m=1:a)>=(sum_expr(x[m,l],m=1:a)-availabilty_soft)-uplimit*(r[4,l]-y[l]+1),l=1:b)%>%

add_constraint(sum_expr(r[D,l],D=1:4)<=3,l=1:b)model

result<-solve_model(model,with_ROI(solver="symphony",time_limit=1200,gap_limit=.1))result

#result<-solve_model(model,with_ROI(solver="glpk",verbose=TRUE))#result

matching<-result%>%get_solution(x[i,j])%>%filter(value>.9)%>%select(i,j)#%>%#arrange(i)

matching=as.data.table(matching)setnames(matching,"i","MemberID")setnames(matching,"j","LeaderID")

groups=data.table(NULL)for(pin1:b){if(p%in%unique(matching$`LeaderID`)){groups=rbind(groups,cbind(distance[matching[`LeaderID`==p]$`MemberID`,1],colnames(distance)[p+1]))}}setnames(groups,"V1","MemberLocations")setnames(groups,"V2","LeaderLocation")tabulate(matching$`LeaderID`)

###Levi'sCode

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List=cbind(matching[1:200,1],groups[1:200,1],matching[1:200,2],groups[1:200,2])write.csv(List,"List.csv")