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Springer Proceedings in Mathematics & Statistics

Andrea Matta

Evren Sahin

Jingshan Li

Alain Guinet

Nico J. Vandaele Editors

Health Care Systems Engineering for Scientists and Practitioners

HCSE, Lyon, France, May 2015

Volume169

Moreinformationaboutthisseriesat http://www.springer.com/series/10533

SpringerProceedingsinMathematics&Statistics

Thisbookseriesfeaturesvolumescomposedofselectcontributionsfromworkshops andconferencesinallareasofcurrentresearchinmathematicsandstatistics, includingORandoptimization.Inadditiontoanoverallevaluationoftheinterest, scientificquality,andtimelinessofeach proposalatthehandsofthepublisher, individualcontributionsareallrefereedtothehighqualitystandardsofleading journalsinthefield.Thus,thisseriesprovidestheresearchcommunitywith well-edited,authoritativereportsondevelopmentsinthemostexcitingareasof mathematicalandstatisticalresearchtoday.

AndreaMatta•EvrenSahin•JingshanLi AlainGuinet•NicoJ.Vandaele

Editors

HealthCareSystems EngineeringforScientists andPractitioners

HCSE,Lyon,France,May2015

Editors AndreaMatta

ShanghaiJiaoTongUniversity Shanghai,China

JingshanLi CollegeofEngineering UniversityofWisconsin Madison,Wisconsin,USA

NicoJ.Vandaele

KatholiekeUniversiteitLeuven Leuven,Belgium

EvrenSahin LaboratoireGénieIndustriel EcoleCentraleParis Châtenay-Malabry,France

AlainGuinet DISPLaboratory LyonUniversity,INSAdeLyon Villeurbanne,France

ISSN2194-1009ISSN2194-1017(electronic)

SpringerProceedingsinMathematics&Statistics

ISBN978-3-319-35130-8ISBN978-3-319-35132-2(eBook) DOI10.1007/978-3-319-35132-2

LibraryofCongressControlNumber:2016951313

©SpringerInternationalPublishingSwitzerland2016

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Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.

Printedonacid-freepaper

ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland

Preface

Thisvolumefeaturesselectedandpeer-reviewedcontributionsfromtheSecond InternationalConferenceonHealthCareSystemsEngineering(HCSE2015).This conferenceprovidesanopportunitytodiscussoperationsmanagementissuesin health-caresystems.Theemphasisisonquantitativemethodsfortheanalysis, designandmanagementofhealth-caresystems.

Theparticipantsarefaculties,students,medicaldoctorsandchiefnursesfrom severaldisciplines.Themainobjectiveisfosteringthecollaborationbetween operationsmanagementscientistsandclinicians.

Scientistsandpractitionershavetheopportunitytodiscussaboutnewideas, methodsandtechnologiesforimprovingtheoperationofhealth-careorganizations duringa3-daysinglesession.Theeventemphasizestheresearchinthefieldof health-caresystemsengineeringdevelopedinclosecollaborationwithclinicians.

ThissecondconferencetookplaceinLyon,France,between27and29May 2015intheSaint-Joseph/St-LucHospital.Alimitednumberofpaperswereselected underadouble-blindreviewprocess.IwouldliketothankalloftheScientific Committeeandtheanonymousreviewersfortheselectionoftheworks.Intotal, 19papersareincludedintheconference proceedings.Eachpaperwaspresentedat theconferencetoallparticipantsanddiscussedwithexpertsfromtheclinicalfield.

Iwouldliketoexpressmydeepgratitudetoourinvitedspeakers,Prof.Sally Brailsfordforagreeingtoaddresstheconferenceon“ModelingHumanBehaviorin HealthcareSystems:IsItPossibleandWhyShouldWeDoIt?”andProf.NicoVandaelewiththetopicof“TheVaccineSupplyChainDecathlon:TheReconciliation ofTechnology,EconomyandHumanity”.Theircontributionsareperfectlyinline withtheobjectiveoftheconferencewhichtriestoinitiatecollaborationsbetween scientistsandclinicians.

Iwouldliketothankallthespeakers,authorsanddiscussantsofthepapers togetherwiththeiraccompanyingpersonsfortheirparticipationinHCSE2015.

Igratefullyacknowledgethekeypeopleoftheorganizationofthisevent:Hélène Grange,MichelGreco,AndreaMatta,SylvieMeyran,JérémieLeynon,Anastasie Schiffer,SamuelVercraene

Myhopeisthatthisconferencewillserveasabiennialforumforresearchers, academicsandcliniciansinthebroadarea ofhealth-caresystemsengineeringto discusstheirmostrecentresearchfindingsandtoprovidethemwithopportunities forprojectdevelopmentandtechnologytransfer.

Lyon,FranceAlainGuinet

Contents

SystemsApproachforPreventingFallsinHospitals andNursingHomesUsingSensingDevicesSurrounding thePatient’sBed .................................................................1 M.Takanokura,M.Miyake,M.Kawakami,T.Yamada,S.Taki, andM.Kakehi

AMulti-objectivePatientAdmissionPlanningImproving ResourcesUtilisationUnderBedCapacityConstraints .....................13 AlainGuinet,NadineMeskens,andTaoWang

Multi-criteriaDecisionMakingApproachestoPrioritize SurgicalPatients .................................................................25

SamiraAbbasgholizadehRahimi,AfshinJamshidi, AngelRuiz,andDaoudAit-Kadi

BedManagers:ThePatient’sPersonalAssistant .............................35 SerenSchirra,GaelleOlleon,EstelleForestier,SylvieMeyran, EmmanuelBeaudry,andMarieLassaigne

AnOptimizationModelforSequenceDependentParallel OperatingRoomScheduling ....................................................41 JohanHolmgrenandMariePersson

AMean-FieldAnalysisfortheTwo-TieredHealthcareNetwork ThroughNonlinearMarkovProcesses .........................................53 Quan-LinLi,Rui-NaFan,andNaLi

SchedulingMagneticResonanceImagingExaminations: AnEmpiricalAnalysis ...........................................................65 FilippoVisintinandPaolaCappanera

AManagerialUseoftheVolume-OutcomeAssociation forHospitalPlanning ............................................................79 AriannaAlfieri,ElisabettaListorti,andAndreaMatta

ADiscreteEventSimulationModelfortheAdmissionof PatientstoaHomeCareRehabilitationService ..............................91

AzadehMaroufkhani,EttoreLanzarone,CecilyCastelnovo, andMariaDiMascolo

AmbulanceLocationProblemwithStochasticCallArrivals UnderNearestAvailableDispatchingPolicy ..................................101 InkyungSungandTaesikLee

ApproachtoClusteringClinicalDepartments ................................111 AlexanderHübner,ManuelWalther,andHeinrichKuhn

ManagementofBloodDonationSystem:LiteratureReview andResearchPerspectives ......................................................121 SedaBa¸s,GiulianaCarello,EttoreLanzarone,ZeynepOcak, andSemihYalçında ˘ g

StaffingRatioAnalysisinPrimaryCareRedesign: ASimulationApproach .........................................................133 XiangZhong,HyoKyungLee,MollyWilliams,SallyKraft, JefferySleeth,RichardWelnick,LoriHoschild,andJingshanLi DiseasePreventionandControlPlans:StateoftheArt andFutureResearchGuideline ................................................145 WanyingChen,AlainGuinet,andAngelRuiz

AGoal-ProgrammingApproachtotheMasterSurgical SchedulingProblem ..............................................................155 PaolaCappanera,FilippoVisintin,andCarloBanditori

HowDoMissingPatientsAggravateEmergencyDepartment Overcrowding?ARealCaseandaSimulationStudy ........................167 Yong-HongKuo,JannyM.Y.Leung,andColinA.Graham

SystemDynamicsModellingofEmergentandElectivePatientFlows .....179 PaoloLanda,MicheleSonnessa,ElenaTànfani,andAngelaTesti MarkovDecisionProcessModelforPatientAdmission DecisionatanEmergencyDepartmentinDisasters .........................193 Hyun-RokLeeandTaesikLee

CrisisManagementPlan:PreventiveMeasuresandLessons LearnedfromaMajorComputerSystemFailure ............................203 HélèneGrangeandJérémieLeynon

Contributors

DaoudAit-Kadi DepartmentofMechanicalEngineering,LavalUniversity,Ville deQuébec,QC,Canada

AriannaAlfieri PolitecnicodiTorino,Torino,Italy

CarloBanditori IBISLab,DipartimentodiIngegneriaIndustriale,Università degliStudidiFirenze,Firenze,Italy

SedaBa¸ s IndustrialandSystemsEngineeringDepartment,YeditepeUniversity, Istanbul,Turkey

EmmanuelBeaudry CentreHospitalierSt-Joseph/St-Luc,Lyon,France

PaolaCappanera IBISLab,DipartimentodiIngegneriadell’Informazione,UniversitàdegliStudidiFirenze,Firenze,Italy

GiulianaCarello DipartimentodiElettronica,InformazioneeBioingegneria, PolitecnicodiMilano,Milan,Italy

CecilyCastelnovo IstitutodiMatematicaApplicataeTecnologieInformatiche (IMATI),ConsiglioNazionaledelleRicerche(CNR),Milan,Italy

WanyingChen DISP,INSAdeLyon,Bât.JulesVerne,Villeurbanne,France

Rui-NaFan SchoolofEconomicsandManagementSciences,YanshanUniversity, Qinhuangdao,Hebei,China

EstelleForestier CentreHospitalierSt-Joseph/St-Luc,Lyon,France

ColinA.Graham AccidentandEmergencyMedicineAcademicUnit,Chinese UniversityofHongKong,HongKong,China

HélèneGrange CentreHospitalierSaint-Joseph/Saint-Luc,LyonCedex,France

AlainGuinet DISP,INSAdeLyon,Bât.JulesVerne,Villeurbanne,France

InstitutNationaldesSciencesAppliquéesdeLyon(INSAdeLyon),Villeurbanne, France

JohanHolmgren FacultyofTechnologyandSociety,DepartmentofComputer Science,MalmöUniversity,Malmö,Sweden

LoriHoschild UniversityofWisconsinMedicalFoundation,Middleton,WI,USA

AlexanderHübner DepartmentofOperations,CatholicUniversityofEichstättIngolstadt,Ingolstadt,Germany

AfshinJamshidi DepartmentofMechanicalEngineering,LavalUniversity,Ville deQuébec,QC,Canada

M.Kakehi DepartmentofIndustrialSystem,FacultyofSymbioticSystems Science,FukushimaUniversity,Fukushima,Japan

M.Kawakami FacultyofNursing,JichiMedical University,Shimotsuke-shi, Tochigi,Japan

SallyKraft Dartmouth-Hitchcock,Hanover,NH,USA

HeinrichKuhn CatholicUniversityofEichstätt-Ingolstadt,Ingolstadt,Germany

Yong-HongKuo StanleyHoBigDataDecisionAnalyticsResearchCentre, ChineseUniversityofHongKong,HongKong,China

PaoloLanda DepartmentofEconomicsandBusinessStudies,Universityof Genova,Genova,Italy

UniversityofExeterMedicalSchool,Exeter(UnitedKingdom)

EttoreLanzarone IstitutodiMatematicaApplicataeTecnologieInformatiche (IMATI),ConsiglioNazionaledelleRicerche(CNR),Milan,Italy

MarieLassaigne CentreHospitalierSt-Joseph/St-Luc,Lyon,France

Hyun-RokLee Industrial&SystemsEngineering,KAIST,Daejeon,Republicof Korea

HyoKyungLee UniversityofWisconsin-Madison,Madison,WI,USA

TaesikLee Industrial&SystemsEngineering,KAIST,Daejeon,RepublicofKorea

JannyM.Y.Leung DepartmentofSystemsEngineeringandEngineeringManagement,ChineseUniversityofHong,HongKong,China

JérémieLeynon CentreHospitalierSaint-Joseph/Saint-Luc,LyonCedex,France

JingshanLi UniversityofWisconsin-Madison,Madison,WI,USA

NaLi DepartmentofIndustrialEngineeringandManagement,ShanghaiJiaotong University,Shanghai,China

Quan-LinLi SchoolofEconomicsandManagementSciences,YanshanUniversity,Qinhuangdao,Hebei,China

ElisabettaListorti PolitecnicodiTorino,Torino,Italy

AzadehMaroufkhani UniversitéGrenobleAlpes,Grenoble,France

MariaDiMascolo UniversitéGrenobleAlpes,Grenoble,France

AndreaMatta ShanghaiJiaoTongUniversity,Shanghai,China

NadineMeskens UCLMons,LouvainSchoolofManagement,Mons,Belgium

SylvieMeyran CentreHospitalierSt-Joseph/St-Luc,Lyon,France

M.Miyake CourseofIndustrialEngineeringandManagement,GraduateSchool ofEngineering,KanagawaUniversity,Kanagawa-ku,Yokohama,Japan

ZeynepOcak IndustrialandSystemsEngineeringDepartment,YeditepeUniversity,Istanbul,Turkey

GaelleOlleon CentreHospitalierSt-Joseph/St-Luc,Lyon,France

MariePersson DepartmentofComputerScienceandEngineering,Blekinge InstituteofTechnology,Karlskrona,Sweden

SamiraAbbasgholizadehRahimi DepartmentofMechanicalEngineering,Laval University,VilledeQuébec,QC,Canada

AngelRuiz CIRRELT,UniversitéLAVAL,Québec,QC,Canada

FacultyofBusinessAdministration,UniversitéLaval,PavillonPalasis-Prince2325, ruedelaTerrasse,Local2423, Québec(Québec),G1V0A6,Canada

SerenSchirra CentreHospitalierSt-Joseph/St-Luc,Lyon,France

JefferySleeth UniversityofWisconsinMedicalFoundation,Middleton,WI,USA

MicheleSonnessa DepartmentofEconomicsandBusinessStudies,Universityof Genova,Genova,Italy

InkyungSung KAIST,Daejeon,RepublicofKorea

M.Takanokura DepartmentofIndustrialEngineeringandManagement,Faculty ofEngineering,KanagawaUniversity,Kanagawa-ku,Yokohama,Japan

S.Taki DepartmentofManagementInformationScience,FacultyofSocialSystemsScience,ChibaInstituteofTechnology,Narashino-shi,Chiba,Japan

ElenaTànfani DepartmentofEconomicsandBusinessStudies,Universityof Genova,Genova,Italy

AngelaTesti DepartmentofEconomicsandBusinessStudies,Universityof Genova,Genova,Italy

FilippoVisintin IBISLab,DipartimentodiIngegneriaIndustriale,Universitàdegli StudidiFirenze,Firenze,Italy

ManuelWalther CatholicUniversityofEichstätt-Ingolstadt,Ingolstadt,Germany

TaoWang InstitutNationaldesSciencesAppliquéesdeLyon(INSAdeLyon), Villeurbanne,France

RichardWelnick UniversityofWisconsinMedicalFoundation,Middleton,WI, USA

MollyWilliams UniversityofWisconsinMedicalFoundation,Middleton,WI, USA

SemihYalçinda ˘ g IndustrialandSystemsEngineeringDepartment,YeditepeUniversity,Istanbul,Turkey

T.Yamada DepartmentofInformatics,GraduateSchoolofInformationand Engineering,TheUniversityofElectro-Communications,Tokyo,Japan

XiangZhong UniversityofWisconsin-Madison,Madison,WI,USA

SystemsApproachforPreventingFalls inHospitalsandNursingHomesUsingSensing DevicesSurroundingthePatient’sBed

M.Takanokura,M.Miyake,M.Kawakami,T.Yamada,S.Taki,and M.Kakehi

Introduction

SocietiesinJapanandotherdevelopedcountrieshavebeenrapidlyaging.Aging societieshavealsobeenrecognizedasacriticalproblemindevelopingcountries. ThepopulationinJapanhasdecreasedsince2012,andtheelderlypopulation, especiallythose75yearsinageandolder,iscriticallyincreasing,whereasthe working-ageandyoungerpopulationiscontinuouslydecreasing(CabinetOfficein JapaneseGovernment 2015).Apublicnursinginsurancesystemwasadoptedin Japanin2000totakemeasuresagainstthenation’shighlyagingsociety.However,

M.Takanokura( )

DepartmentofIndustrialEngineeringandManagement,FacultyofEngineering, KanagawaUniversity,3-27-1Rokkakubashi,Kanagawa-ku,Yokohama221-8686,Japan e-mail: takanokura@kanagawa-u.ac.jp

M.Miyake

CourseofIndustrialEngineeringandManagement,GraduateSchoolofEngineering, KanagawaUniversity,3-27-1Rokkakubashi,Kanagawa-ku,Yokohama221-8686,Japan

M.Kawakami

FacultyofNursing,JichiMedicalUniversity,3311-159Yakushiji,Shimotsuke-shi, Tochigi329-0498,Japan

T.Yamada

DepartmentofInformatics,GraduateSchoolofInformaticsandEngineering,TheUniversity ofElectro-Communications,1-5-1Chofugaoka,Chofu-shi,Tokyo182-8585,Japan

S.Taki

DepartmentofManagementInformationScience,FacultyofSocialSystemsScience, ChibaInstitute ofTechnology,2-17-1Tsudanuma,Narashino-shi,Chiba275-0016,Japan

M.Kakehi

DepartmentofIndustrialSystem,FacultyofSymbioticSystemsScience, FukushimaUniversity,1Kanayagawa,Fukushima-shi,Fukushima960-1296,Japan

©SpringerInternationalPublishingSwitzerland2016

A.Mattaetal.(eds.), HealthCareSystemsEngineeringforScientists andPractitioners,SpringerProceedingsinMathematics&Statistics169, DOI10.1007/978-3-319-35132-2_1

thesocialsecuritycostsinJapanaregreatlyincreasing.Tosupplementtheshortage ofannualrevenueagainsttheexpansionofsocialsecuritycosts,thegovernmenthas decidedtoraisetheconsumptiontaxfrom5to8%in2014,andto10%in2017. Thistaxincreasewillbeusedforanenrichmentofbasicpensions,elderlymedical care,long-termcare,andchildcare(MinistryofHealth,LabourandWelfareJapan 2015).ElderlycareisasocialproblemwithhigherpriorityinJapan.

Animprovementinqualityoflife(QOL)hasbeenoneofthecriticalproblems forelderlycare.Nursesandcareworkers,referredtohereafterascare-givers,take careofpatientsandelderlypersons,alsoreferredtoascare-receivers,fortheirdaily activities.Adevotedcare-giverenhancestheQOLandwell-beingoftheircarereceivers.Althoughtheworking-agepopulationshouldbearesourceforelderly care,suchpopulationisdecreasinginJapan,aspreviouslystated.Thereisan absolutelackinhumanresourcesfortheelderlybecausethepopulationdistribution inJapanishighlydistorted.Inaddition,apoorworkingenvironmentforcare-givers isregardedasasocialproblem(e.g.,longworkinghours,lowincome,andhighrate ofcare-giversleavingtheirjobs).Althoughtheseproblemsshouldfirstbesolved politically,systemsengineeringcanbeapowerfultoolforasociallybasedsolution.

Afallisoneofthemostimpactfultypesofaccidentsthatcanoccurinahospital ornursinghome.ItreducestheQOLofcare-receiversanddeterioratestheprofessionaldutiesofcare-givers.Inaddition,fallsareregardedasanon-processtype accidentinhospitalsandnursinghomes.Itismoredifficultforcare-giverstoprevent anon-processtypeaccidentthanaprocesstypeonebecauseitcannotpredictthe occurrenceofaccidenteasily.SomeseriousaccidentshavebeenreportedinJapan. Forexample,anelderlymalefellaroundabedside,andhisneckwascaughtina spacebetweenbedrails.Unfortunately,hewasdiedafterthisaccident.Anelderly femalestoodupfromabedbygraspingabedrail,butitwasnotfixedsufficiently. Shelostbalanceandfell.Herarmwasfracturedbythisaccident.Preventingfalls shouldbeaccomplishedthroughtechnical,organizational,andsocialmeasures.As atechnicalmeasure,someproductsforpreventingfallsarecurrentlyavailablein Japan.Forexample,touchsensorscanbeplacedatabedsideoronthefloor,and cannotifycare-giversifacare-receivertouchesthemwhensittingonthebedsideor standingupfromthebedontothefloor.However,suchsensingdevicesmayissue afalsealarmowingtotheirsimplestructure.Inaddition,videomonitoringsystems cannotbeusedforpreventingfallsinhospitalsandnursinghomesbecauseofan inherentprivacyissue.Recently,somemicro-electro-mechanicalsystems(MEMSs) havebeendevelopedformonitoringdailyactivitiesandimprovingtheQOL.For example,theactivitymeterwithatri-axialaccelerometerhasbeenusedforthe preventionoflifestyle-relateddiseases(Ohshimaetal. 2012;Ohkawaraetal. 2011; TanakaandTanaka 2009)andaquantitativeunderstandingofchild-careactivities (Takanokuraetal. 2014;Yamadaetal. 2015).Nowadays,MEMSareregardedasan essentialsystemforanenhancementofQOLineverystageoflife.

Inthisstudy,wefocusonfallsaroundabed,butaccidentswilloccurinother environmentssuchaswalking.However, non-processtypeaccidentsarethecritical probleminhospitalsandnursinghomes,andtheyoccuratanytimezoneorat anyplacewithoutcare-giversespeciallyatnight.Patientsandelderlypersonsfirst

standupfromtheirbedbeforewalking.Accidentsinwalkingcanbepreventedby detectionofdangerousmotionswhilestandingup.

WefirstappliedMEMStechnologiesforpreventingfallsbydetectingdangerous motionswhilestandingupfromabedandwhenleavingaroominhospitalsand nursinghomes.Inaddition,asystemsapproachisnecessarytomanagesuchMEMS technologiesduringthenursingprocess.Thus,wediscusshowMEMStechnologies havebeenusedeffectivelyandefficiently forthepreventionoffallsinhospitalsand nursinghomes.

Objectives

First,wedescribethedevelopmentofour systemfordetectingandpredictingcarereceiver’smotionsusingamicro-computer andsensingdevices.Thissystemdetects thecare-receiver’smotionsonandaroundtheirbed,andpredictsdangerousmotions thatcanleadtoafall.Second,wediscusshowfallsshouldbepreventedbyusing measureddataobtainedfromthedevelopedsystemfromtheviewpointsofboth care-giversandcare-receivers.Finally,wemodelthenursingprocessesofcaregiversinahospitalandnursinghome,andconsiderhowthedevelopedsystem shouldbeutilizedintothenursing processforpreventingfalls.

DetectionandPredictionofDangerousMotions onandAroundaBed

Care-receivershaveahigheroccurrenceoffallswhilestandingupfromabedor whenleavingtheirroom.Beforeconductingsuchdangerousmotions,thecarereceiverraisestheirupperbodyfromthebed,andthensitsatthebedside.Such behaviormaybeasignofadangerousmotionthatcouldresultinafall.Before standingupfromthebed,thepressuredistributionaroundthecare-receivervaries basedontheirmotion.First,thecare-receiverraisestheirupperbodyfromthe bed,therebyloweringthepressurearoundtheirheadandshoulders.Then,thecarereceivermovestothebedsideandsits.Thepressurearoundthehipsandlegsisthen reduced.Wecandetectthistypeofbehaviorfromacontinuousvariationofpressure distributiononthebedbyusingaforcesensingresistor(FSR)(No.406,Interlink, USA)asasensortodetectthecare-receiver’smotion.AnFSRisrectangularin shapewith38.1-mmlongsides.Furthermore,afallwilllikelyoccurjustasthe care-receiverstandsupfromthebed.Forexample,acare-receiverwithcognitive impairmentmaylingeraroundtheirroom,whereasthosewithoutsuchimpairment maystandupfromthebedinthedarktousethetoiletduringthenight.Forsuch care-receivers,weconsiderstandingupfromthebedandsimplymovingtoward thedooroftheroomasdangerousmotions.Therefore,apassiveinfra-redhuman

Fig.1 Sensingdevicesandplacementofthemonandaroundabed.FSRsareplacedon (1)–(7) Humandetectionsensorsaresetat (A) and (B)

detectionsensor(NaPiOnspottype,Panasonic,Japan)isusedfordetectingthese typesofdangerousmotions.

TheplacementofthesensingdevicesonandaroundabedisshowninFig. 1. Whenacare-receiverliesontheirbed,theirbackandhipstouchthemattress.There aredifferentbehaviorsthatoccurwhenstanding-upfromabed,butapopularone isthecare-receiverfirstraisingtheirupperbodyfromthebed(sitting-up)andthen sittingatthebedside.Inthiscase,thepressureisfirstloweredaroundtheshoulders andthenaroundthehips.Therefore,fourFSRsareplacedunderbothsidesofthe shoulders,threeFSRsareplacedunderthehips(chest),andtwohuman-detection sensorsareplacedjustunderthebed(A)andinfrontofthedoor(B).Thefirstfour areactivatedwhenthecare-receiversitsat thebedside,andthelastoneisactivated whenheorsheopensthedoortoleavetheroom.

Thissystemforpreventingfallsusesan ArduinoUno,whichisatoolkitwith anAVRmicroprocessor;sevenFSRs;andtwohuman-detectionsensors.Asecure digital(SD)memorycardshieldisalsoinstalledinthesystem.Thedatameasured bythesensorsarefedintothe ArduinoUno andstoredontheSDcard.

Thevalidityofthedevelopedsystemwasexaminedexperimentally.Itshould beexaminedwithparticipationofcare-receiversinhospitalsornursinghomes. However,thedevelopedsystemisaprototypeforpreventionoffalls,andsafety ofthesystemisnotguaranteedsufficiently.Therefore,twohealthyyoungmalesare selectedastheparticipant.ParticipantAwas22yearsoldwithaheightof170cm andweightof58kg.ParticipantBwas22yearsoldwithaheightof180cmand weightof66kg.Theyfirstlieddownonthebed,andthenturnedontotheirleftor rightsideorsatupright.Aftermovingonthebed,theysatatthebedside,andthen stoodupfromthebed.Theythenmovedtothedoor.Figure 2 showsthemeasured signalsoftheFSRsandhuman-detectionsensorsforparticipantA.Period(1)was measuredwhentheparticipantlieddownonthebedanddidnotmove.Alldata measuredfromtheFSRswereequaltothemaximalvalue(1023arbitraryunit).The participantturnedontotheirrightsideduringPeriod(3),andpressureattheleft shoulderwasreduced.Similarly,thepressureattheirrightshoulderwasreduced

Fig.2 MeasuredsignalsfromtheFSRsandhumandetectionsensorsforparticipantA. C, RS,and LS indicatethepressuredatafromthechest(hips),rightshoulder,andleftshoulder,respectively. Thepressureismeasuredasanarbitraryunit. H_E and H_S indicatesignalsfromhumandetection sensorsplacedinfrontofthedoor(entrance)andunderthebedside,respectively.Sensorsdetecta valueof“1”(existence)or“0”(absence)

duringPeriod(4)becausetheparticipantturnedontotheirleftside.Theparticipant thensatupduringPeriod(2),andthusthemeasuredsignalswerereducedfrom bothsidesoftheshoulder.DuringPeriod(5),theparticipantsatatthebedsideand thenstoodupfromthebed.NoneoftheFSRsmeasuredanypressureonbed,and thehuman-detectionsensorunderthebedsidewasactivated.Afterstandingup,the participantwenttothedoorduringPeriod(6),andthehuman-detectionsensorin frontofthedoorwasactivated.

Asaresult,wecoulddetectandpredictdangerousmotionsthatcouldleadtoa fall.Inaddition,weproposeanalgorithmforpredictingfallsthroughtheuseofthe developedsystem,asshowninFig. 3.Thedevelopedsystemcanpredictdangerous motionsaroundabedsideandthedoorofaroom,asshowninFig. 3

Thesystemwillnotifythecare-giversorcare-receiversifalarm(A)or(B)isset off,asshowninFig. 3,buttheinformationprovideddependsonhowthedeveloped systemisused.Thisissueisconsideredinthenextsection.Inaddition,dangerous motionsarejudgedbythesystemwhenthethresholdpressurevalue(600or200in anarbitraryunit)isexceeded.Thisisasimplealgorithm,andweintendtoconsider otheralgorithmssuchasastatisticalmeasure,time-seriesanalysis,ormeta-heuristic approach(GA,neuralnetwork,etc.).

Inthisstudy,theparticipantsaretwo healthyyoungmalesbecauseofasafety reason.However,elderlypersonsinnursinghomeshaveadifferentbodyshape suchashighBodyMassIndexcomparedwithyoungmales.Inaddition,patients andelderlypersonsmovetheirbodyslowly.Therefore,weshouldconsiderproper thresholdvaluesorotheralgorithmsforpatientsandelderlypersons.

Start

Measure from FSR on bed and human detection sensors

FSR: C&LS&RS > 600

Patient lies on bed

FSR: RS(LS) >600 & C&LS(RS) < 200

Patient turns to right or left side

FSR: C > 600 & LS&RS <200

Patient sits up on bed

HS on bedside: detected

yes

Alarm (A)

End

HS on entrance: detected

yes no

Alarm (B)

Fig.3 Algorithmusedforpreventingafallthroughthedevelopedsystem. C, LS,and RS indicate thechest(hips),leftshoulder,andrightshoulder,respectively. HS indicatesahuman-detection sensor

FallPrevention:ViewpointsofCare-Givers andCare-Receivers

Thedevelopedsystemcanpredictdangerousmotionsthatmayleadtoafall.The care-giversorcare-receiversshouldbenotifiedaboutapredictedmotionbythe systembeforeafalloccurs.Forcare-givers,thenotificationshouldbethrough analarm.Forexample,ifacare-receiverwithcognitiveimpairmentintendsto standupneartheirbedandleavetheroom,thecare-giversarenotifiedofsuch motionsbythesystem.Afterreceivinganalarmfromthesystem,thecare-givers canreachthecare-receiverasquicklyas possibleandhelpthemrestraintheir dangerousmotions.Suchinformationshouldbeprovidedtothecare-giversthrough anemergencycall,orshouldberecordedasadailylogduringthenursingprocess. Recordedinformationisusefulforcareplanninginhospitalsandnursinghomes. However,suchinformationshouldbeprovidedtonotonlycare-giversbutalsoto care-receivers.Forexample,afteranysurgicaloperation,acare-receivershould restinbedandrefrainfromsittingupfromthebedorstanding.Iftheychoose suchadangerousmotion,theroomlightcanbeturnedonorawarningmessage suchas“Foryoursafety,donotstandup!”canbegiventothecare-receiver.To implementthisfunctionintothedevelopedsystem,weinstalledanLCDmodule (Fig. 4)andaloudspeakerintothesystem.Thealgorithmusedforfallprevention (Fig. 3)providesanalarm(A)asamessagetothecare-receiver.Atthesame time,care-giverswillreceiveawarningmessagefromthesystem.Alarm(B)is notrequiredforacare-receiverafteranoperation,butisneededforacare-receiver withacognitiveimpairment.Wecanpreventfallsfromoccurringbycombiningand judgingthesignalsfromthedevelopedsystem(alarms(A)and(B)inFig. 3)based onthesymptomsorconditionsofthecare-receivers.

ProcessModelingofCare-GiverstoCare-Receivers

Fallscanbepreventednotonlythroughtheuseoftechnicalproductsbutalsofrom organizationalmeasures.Asdescribedpreviously,wedevelopedthesystemtouse sensingdevices;however,thisisatechnicalmeasure.Itisalsoimportantforus tounderstandthenursingprocessofthecare-giverswhenapplyingorganizational measurestoenhancethequalityofhealthcare.Therefore,wemodeledthenursing processesofthecare-giverstothecare-receivers,andconsideredhowthedeveloped systemshouldbeutilizedintheseprocessesforpreventingfalls.Adiagramforthe processmodelingwasdevelopedthroughthedailylogsofcare-giverstakenatnight. Thesedailylogswereobtainedfromahospital(cerebralsurgeryward)andalongtermcarefacility.

Junetal.proposedanode-linkbasedcharacterizationofeightdifferentdiagrams (Junetal. 2009).A“flowchart”hasthehighestusabilityandutilityamongallofthe diagramsused.Thenursingprocessoverallcanbeunderstoodbasedonaflowchart. Althoughacare-giver’sdecisionswereincludedintheflowchart,theycouldnot beidentifiedfromdailylogs.The“processcontent”and“statetransition”diagrams haveahigherusabilityandutility.Theformercanbeusedtounderstandadetailed taskstructure,andthelattercanclarifya processinacare-receiver-centeredmanner. Therefore,wedevelopedtwodiagramsfromdailynursinglogstakenatnightina hospitalandalong-termcarefacility.

Figure 5 illustratesaprocesscontentdiagramofnursesworkinginahospital. Thenurseshavefourprocesses:treatment,confirmation,preparation,andmeetings. Theseprocessesaredividedintorespectivesub-processes.Theprocessesinthe

diagramaretheprincipalcare(treatment,confirmation,andpreparation)givento thepatientsandtheduty(meetings)ofthenurses.Along-termcarefacilitysimply takescareofelderlypersons,anddoesnotprovidehospitaltreatment,andthus the“treatment”processisnotincludedinthediagram.However,careworkersare devotedtothecareofelderlypersons,andthenursingprocesswasclassifiedinto confirmation,preparation,andmeetings.

Theprocesscontentdiagramexpressesthe“healthcareprocess,”whichwill enhancetheQOLofcare-receiversbasedonthedutiesofthecare-givers.However, theQOLisreducedfromunexpectedaccidentsthatoccurinhospitalsandnursing homes,whichwecall“non-processtypeaccidents.”Fallsaretypicalcasesofnonprocesstypeaccidents.Inaddition,“process-typeaccidents”suchasmedication errorsmayoccurinhospitalsandnursinghomes.Aprocesscontentdiagramisa powerfultoolfortheidentificationandpreventionofprocess-typeaccidents,but non-processtypeaccidentscannotbe foundfromthistypeofdiagram.

Figure 6 illustratesastatetransitiondiagramofnursesinahospitalatnight. Thenursingprocessisexpressedinacare-receiver-centeredmannerasapatient state.Duringthenight,patientsfirstreceivemedicalcarefromnurses(drip,insulin injection,ortubalfeeding)andanexamination(measurementofvitalsigns).These processesarecarriedoutduringtheearlyperiodofpatientcareatnight,andnon-

Fig.6 Statetransitiondiagraminahospitalatnight

10M.Takanokuraetal.

processtypeaccidentsdonotfrequentlyoccurbecausenursescanvisuallyconfirm thestateoftheirpatients.Nursescompletetheprocessofpatientcareonetime,and donotvisuallyconfirmthepatient’sstatefrequentlyafterchangingtheirpositionor givingthemmedicine.Non-processtypeaccidentsmaythereforeoccurduringthis periodofpatientcare.Thedevelopedsensing-devicebasedsystemcanbeusedfor preventingfallsaftertheearlyperiodofpatientcareatnightinbothhospitalsand nursinghomes.Althoughfallscanbeeffectivelyandefficientlypreventedusingthe developedsystem,weneedtoconsidertheimplementationofthedevelopedsystem asthenextstageofthisstudy.

Conclusion

Wedevelopedasystemforpreventingfallsfromoccurringaroundacare-receiver’s bedusingMEMStechnologies.Care-giversandcare-receiverscanbenotifiedboth visuallyandaurallyifthesystempredictsdangerousmotionsonoraroundabed.A statetransitiondiagramindicatesthatfallsmayoccurafteranearlycareperiodat nightbecausecare-giversareunabletofrequentlyconfirmthecare-receiver’sstate. Fallsareanon-processtypeaccident,andthedevelopedsystemcanbeutilized duringthisperiodforthepreventionofnon-processtypeaccidents.Thesystemcan improvetheQOLofcare-receiversandalleviatetheprofessionaldutiesofcaregiversthroughareductioninthenumberoffalls.However,weshouldimprovethe abilityofthesystemthroughtheuseofmorecomplexalgorithms,suchasametaheuristicapproach,andbyanalyzingthenursingprocessmorepreciselybeyondthe dailylogstakenathospitalsandnursinghomes.

Acknowledgements ThisresearchispartiallysupportedbyJSPSKAKENHI26282091.

References

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Jun,G.T.,etal.:Healthcareprocessmodelling:whichmethodwhen?Int.J.Qual.HealthCare 21, 214–224(2009)

MinistryofHealth,LabourandWelfareJapan:Thecomprehensivereformofsocialsecurity andtax—Whydowehavetobeartheburden? http://www.mhlw.go.jp/english/social_security/ kaikaku_2.html.Accessed30Jan2015

Ohkawara,K.,Oshima,Y.,Hikihara,Y.,Ishikawa-Tanaka,K.,Tabata,I.,Tanaka,S.:Real-time estimationofdailyphysicalactivityintensitybyatriaxialaccelerometerandagravity-removal classificationalgorithm.Br.J.Nutr. 105,1681–1691(2011)

Ohshima,Y.,Kawaguchi,K.,Tanaka,S.,Ohkawara,K.,Hikihara,Y.,Ishikawa-Takata,K.,Tabata, I.:Classifyinghouseholdandlocomotiveactivitiesusingatriaxial accelerometer.GaitPosture 31,370–374(2012)

Takanokura,M.,Ariizumi,K.,Imai,N.,Taki,S.,Yamada,T.:Identificationofhigh-intensity physicalactivitiesduringchildrearingusingatriaxial accelerometer.In:Jang,R.,Ahram,T. (eds.)AdvancesinPhysicalErgonomicsandHumanFactors:PartII,pp.93–102.AHFE Conference,USA(2014)

Tanaka,C.,Tanaka,S.:DailyphysicalactivityinJapanesepreschoolchildrenevaluatedby triaxialaccelerometry:therelationshipbetweenperiodofengagementinmoderate-to-vigorous physicalactivityanddailystepcounts.J.Physiol.Anthropol. 28,283–288(2009)

Takanokura,M.,Taki,S.,Sato,T.:Measurementandanalysismethodfor“Ikumen”activities includingchildcaretasksusingatriaxialaccelerometerJ.Jpn.Ind.Manage.Assoc. 66,161–168(2015)

AMulti-objectivePatientAdmissionPlanning ImprovingResourcesUtilisationUnderBed CapacityConstraints

AlainGuinet,NadineMeskens,andTaoWang

Introduction

Thehospitalmanagementtakesplaceinanincreasinglycompetitiveenvironment.

Lotsofhospitalshavethereforedecidedtocentretheirorganizationonabest resourcesaffectationandmoreprecisely ononespecificresource:thehospitalbeds.

Hospitalbedmanagementconsistsofmanagingasbestaspossiblethewelcomingcapacityinordertotakeintoaccountavolumeofactivity,whilerespecting qualitycriterion(qualityofcares,delays,duration ::: )underresourcesconstraints.

Hospitalbedmanagementhasanimpactontheoccupationratesofthedifferent careunitsallalongthepatients’trajectory:fromthebeginningofthehospitalstay, withanimpactontheemergencyservicetotheoperatingtheatre,onintensivecare unitsandevenonrehabilitationcentersandresthomes.Therefore,thehospitalthat wishestooptimizetheallocationofitsresourceswillhavetocleverlyschedulethe admissionofitspatients.Indeed,duringthepatientadmission,abedinawardis assignedandaseriesofresources(formedicaldiagnosisorasurgery)willbeused.

Inthispaper,wetrytoassignanadmissiondayandabedtoeachelected patientwhomustbeadmittedtoasurgeryoramedicalunit.Atimewindowfor hospitalizationandalengthofstayare associatedtoeachpatientadmission.The unitcapacityi.e.thenumberofbedsofthemedicalandsurgicalunitsmustbe respected.Internalpatienttransfersbetweencareunitsareallowed.

Wetakeintoaccounttwoobjectivefunctions.Withthefirstonethesmoothing ofresourcesissought.Wehavetwokindsofresources.Diagnosisresourcessuch

A.Guinet( )•T.Wang

InstitutNationaldesSciences AppliquéesdeLyon(INSALyon),DISP,Villeurbanne,France e-mail: alain.guinet@insa-lyon.fr

N.Meskens

UCLMons,LouvainSchoolofManagement,Mons,Belgium

©SpringerInternationalPublishingSwitzerland2016

A.Mattaetal.(eds.), HealthCareSystemsEngineeringforScientists andPractitioners,SpringerProceedingsinMathematics&Statistics169, DOI10.1007/978-3-319-35132-2_2

asX-raysandCT-scanscouldbeusedduringtheentirepatientstayforsurgical ormedicalunitadmissions.Otherresourcesareusedonlyonthedayofpatient surgerysuchasoperatingrooms.Wewillcallthem“surgicalresources.”Whatever theresource,wewilltrytoadmitpatientssothattheuseofresourcesisdistributed asfairlyaspossibleoverdays.Moreovertheresourcesmoothingisalsointeresting tostudytheresourceutilizationrate(overorunderutilization)andforresource sizing.

Regardingthesecondobjective,wewilltrytoallocateabedintheunitof carewherethesurgicalormedicaloperationwillbeperformedonthepatient. Apatientwhoisnotintheappropriatecareunitlinkedtohispathologyimplies anorganizationalproblemregardingthephysicians’visits,aproblemrelatedtothe nursingstaffcompetencies,notalwaysavailabletocareforspecificpathologies, andalsoproblemsregardingtheirinternaltransferthatcanbecostly,especiallyin termsofstretch-bearingactivities,administration,etc.Inaddition,hospitalsarein competition,satisfactionandpatientsafetybecomeparamountandthesearedirectly relatedtothequalityofcare.Thecaseswherethepatientisnotinasuitablecare unithappenwhenacareunithasnomorebedsavailable.Wealsofoundinreallife somecareunitsthatwereclosedduringweekends,alsothepatientsweretransferred toawardwherebedswerevacant.Internal patienttransfersarethusrealized.

Acostpenaltyperperiod(day)willbeusedwhenthepatientwillbeinaninadequatecareunit.Wewillthereforeseektoensurethatthepatientwillspendthe leastamountoftimeaspossibleinsuchacareunit.Furthermore,itmustbeensured thathospitalizationoccursduringthewindowofadmission.Indeedthephysician willgiveadatenotearlierthanandadatenotlaterthanfortheadmission.Whenthe hospitalizationwindowisnotrespected,penaltycostswillbetakenintoaccount.

Thispaperisstructuredintofourparts: aftertheintroductionthesecondpartof thispaperpresentsareviewofliteraturelinkedtothestudiedproblemsandhelpsus indevelopingthemodelwhichwasbuiltfordecisionmakers.Athirdsectionrelated tohospitalbedmanagementmodelfollowsit.Thefourthpartofthisdissertation presentsanexperimentofourmodelbasedonfielddata.

PreviousWorks

Decisionmakingregardingbedmanagement,couldbeclassifiedaccordingtothe scaleandthedetailsofdecisionproblemsmodelling:bedsizing,bedorganization, bedoccupancyandadmissionorbedplanning.

Alotofresearcherswereinterestedindeterminingtheoptimalnumberofbedsin hospitalsbutalsoinrecoveryrooms,intensivecareunitsoremergencydepartments. Letusquotesomeofthemasexamples:Lietal.(2009),MaandDemeulemeester (2012),DeBruinetal.(2009),CochranandRoche(2009),Kokangul(2008),and TroyandRosenberg(2009).

Lessnumerousarestudiesregardingbedorganizationsuchastheassessmentof differentstrategiesforbedreservation(KimandHorowitz 2002)ortoallocatebeds amonghospitalwardsinordertominimizetheover-booking(Holmetal. 2013)

Bedoccupancyisimportanttobestudiedbutisalsoimpactedbyarelevant managementofcareunitsorotherdepartmentssuchastheoperatingtheatre.Indeed, buildingcyclemastersurgeryschedules influencesthebedoccupancyrates.Some papersinvestigatedmodelsrelatedtobedoccupancyandconsideroperatingtheatre occupancyasaresourcethatneedstobetakenintoaccount(GuinetandChaabane 2003;Wargonetal. 2014).

Whenapatientisadmitted,abedinacareunitisgrantedforthem.Theproblem calledbedplanningisalsocalledadmissionplanningoradmissionscheduling.Bed planningconsistsmainlyofassigning abedtoeachelectivepatientandeachpostacutepatienti.e.apatientwholeavestheemergencydepartmentinordertobe hospitalized.Someauthorswereinterestedintheproblemofbedplanning:Hulshof etal.(2013),BenBachouchetal.(2012),andGartnerandKolisch(2014).

Inconclusionwehaveobservedthatmostofpreviousworksfocusonbed sizing.Bedoccupancyhasbeenmainlystudiedbyoperatingtheatremanagement investigations.Itcouldbeinterestingtodothereversei.e.tostudysurgicaland medicalresourcesbyabedmanagementapproach.Thatistheaimofthispaper.

Tothebestofourknowledgenoresearchhassoughttosmooththeresources usedthroughoutthepatient’sstayorhasmadeadistinctionbetweenthesurgical resourcesandthediagnosisresources.Furthermoreweseektooptimizetwo objectivefunctions.Thefirstistomaximizesatisfactionandpatientsafetybygiving themabedintheappropriatecareunitandduringtherequiredtimewindow.Internal patienttransfershavenotbeentakenintoaccountinbedplanningliterature.The secondobjectiveseekstosmooththeresources.Wehavenotfoundotherresearch thatrealizedthis.

Inconclusionaccordingtotheliteratureshortageonadmissionorbedplanning worthwhileinvestigationscouldbedoneinthisarea.

Model

Atthehospitalscale,thedifferentcare unitsareschematized withabedcapacity. ResourceslikeX-raysandCTscansneeded bythepatientsaredefinedbytheir availabletimes.Consideringthatapatientstaymaybesplitonmorethanone careunit,theaimofthemodelistobalancetheworkloadofresources(diagnosis resourcesandsurgicalresources)authorizingthelocationofthepatientinanother servicethantheintendedone.Indeed,wesearchtoreducethepossiblebottlenecks ofconflictresourcesusedatthesametimebyseveralpatients.

Wewilldifferentiatepatientswhoenterthehospitalforasurgery(calledsurgical patients)andthosewhogotoreceivecareinmedicine(calledmedicalpatients). Thesurgicalpatientsusediagnosisresourcesaswellassurgicalresources,whereas themedicalpatientsneedonlydiagnosisresources.

Toavoidtoomanywrongpatientlocations,anotherobjectivetriestominimize thenumberofdaysinaninadequatecareunit.Thisobjectivefunctionwillalso penalizeheavily(leadingtonon-feasibility)patientstaysthatwouldnotbeplanned inthehospitalizationtimewindow.Thepenaltyvalueisfixedat1foreachdayspent inaninadequatecareunitandatahighvalueifthehospitalizationwindowisnot respected.

Notation

ISetofpatients,i D 1,2, N D IS[IM

ISSetofsurgicalpatients,s D 1,2, S

IMSetofmedicalpatients,m D 1,2, M

JSetofcareunits,j D 1,2, U

Setofdiagnosisresources,k D 1,2, K

RSetofsurgicalresources,q D 1,2, Q

PSetofperiods(days),t D 1,2, T

ai Earlyhospitalizationdateforpatienti

bi Latesthospitalizationdateforpatienti

di Lengthofstayforpatienti(indays)

cui Careunitrequiredbypatienti

mjt Numberofbedsavailableincareunitjatperiodt

rssq Numberofminutesrequiredbyasurgicalpatientforsurgicalresourceq

rdik Numberofminutesrequiredbypatientifordiagnosisresourcek

cijt Penaltycostofhospitalizationstayincareunitjforpatientiduringperiodt:

C9999 Ift < ai ort > bi C di 1.ifadmissiondayisoutsidethe patientadmissionwindow/ 1 Ifj ¤ cui andai t bi C di 1.ifpatientisadmittedinthe non requiredcareunit/ 0 otherwise

DecisionVariables

zijt

Binaryvariableequalto1ifpatientiishospitalizedincareunitjduring periodt,0otherwise

xit Binaryvariableequalsto1ifpatientistartshisstayatt,0otherwise

Cmaxk Maximumnumberofminutesrequiredfordiagnosisresourcek

Cmaxq Maximumnumberofminutesrequiredforsurgicalresourceq

Model

Min .Z1 / D 2 4 0 @ K X k D1 C maxk C Q X qD1 C maxq 1 A 3 5 (1)

Min .Z2 / D " XN iD1 XU jD1 T X tD1 zijt cijt !# (2)

Subjectto 2 4 0 @ U X jD1 N X iD1 zijt rdik =di C maxk 1 A 3 5 8t 2 f1;:::; T g 8k 2 f1;:::; K g (3) 2 6 6 6 6 6 6 4 0 B B B B B B @ S X s D 1 s 2 IS xst 1 rssq C maxq 1 C C C C C C A 3 7 7 7 7 7 7 5 8t 2 f2;:::; T g ; 8q 2 f1;:::; Qg (4) " N X iD1 zijt mjt !# 8t 2 f1;:::; T g ; 8j 2 f1;:::; U g (5) 2 4 0 @ U X jD1 zijt 11 A 3 5 8i 2 f1;:::; N g ; 8t 2 f1;:::; T g (6) 2 4xit U X jD1 zijt U X jD1 zijt 1 3 5 8i 2 f1;:::; N g ; 8t 2 f1;:::; T g (7a) " T X tD1 xit D 1# 8i 2 f1;:::; N g (7b) zij0 D 0; 8i 2 f1;:::; N g ; 8j 2 f1;:::; U g (7c)