balanced flowtime and makespan Generating outpatient chemotherapy appointment templates with European Journal of Operational Research

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European Journal of Operational Research

journalhomepage:www.elsevier.com/locate/ejor

Innovative Applications of O.R.

Generating outpatient chemotherapy appointment templates with balanced flowtime and makespan

Alireza F. Hesaraki

^∗

, Nico P. Dellaert , Ton de Kok

School of Industrial Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands

a rt i c l e i nf o

Article history:

Received 7 October 2017 Accepted 12 November 2018 Available online 15 November 2018 Keywords:

OR in health services Integer programming Scheduling

Outpatient chemotherapy

a b s t ra c t

Westudy theproblemofschedulingoutpatientchemotherapyinfusionappointmentsatoncologyclin- ics.Patientsarepreparedduringafixedinitialperiodoftheirinfusionappointments.Intheremainderof theirappointments,thepatientsaremonitoredbynursesandifneeded,takencareof.Duringpreparation andthesetting-upoftheinfusiondevice,onenursemustbefullyassignedtothepatient.Nurseswho areneitheronabreaknorbusywithpreparingpatients,simultaneouslymonitoruptoacertainnumber ofpatientswhoarealreadyreceiving infusion.The prescribedinfusiondurationcansignificantlydiffer frompatienttopatient.Theobjectiveofthisstudyistogenerateanarrangementofvacantappointment slots,i.e.,atemplate,subjecttothenursingconstraints.Thisisdonewhilereachingabalancebetween startingtheappointmentsasearlyaspossibleandcompletingthelastappointmentsoftheday,asearly aspossible.Wesolvethisproblemusingintegerprogramming.Byadjustingtwoparametersintheobjec- tivefunction,thesolutioncanbetunedbetweenminimizingtheweightedflowtimeandminimizingthe makespan.Thus,someappointments canbeprioritizedforstartingasearly aspossible.Ournumerical resultsshowthatthemodelcanbesolvedtooptimalitywithshortcomputationtimesforlargerealistic sizeinstancesusingcommercialsolversoftware.Thegeneratedtemplate isintended toserveasalink betweenplanningonatacticallevelandonlineschedulingonanoperationallevel.

© 2018ElsevierB.V.Allrightsreserved.

1. Introduction

Chemotherapyisoneofthemostcommonproceduresforcan- cer treatment next to surgery and radiotherapy. For every pa- tient, chemotherapy is planned accordingto treatment protocols that specifythe chemotherapeutic agents (drugs),doses, delivery method(topical,oral,injection,infusion),durationofinfusion,days ofinfusioninacycle(consecutivenumberofdays),flexibilitywin- dows(tolerancedaysaroundnominalinfusiondays),andthenum- ber ofcycles. Onthe tactical level,resource planning mostlyde- pendsonthedistributionofpatientsamongtheprotocols,e.g.,the relativefrequencyhistogramofinfusiondurations,andtherelative frequencyhistogramofthenumberofinfusiondaysperpatient.

Ontheoperationallevel,sinceithasmultiplesteps,chemother- apyadministrationiscomplicatedby precedenceandinterdepen- denceconstraintsimposedontasksandresources.Themajorsteps requiredforaninfusion sessionarethe labtest(including taking vitals),oncologistvisit,pharmacydrugpreparation,setting-upthe

∗ Corresponding author. Tel.: +31 40 247 8858

E-mail addresses: a.hesaraki@tue.nl (A.F. Hesaraki), n.p.dellaert@tue.nl (N.P. Del- laert), a.g.d.kok@tue.nl (T. de Kok).

infusion device, and monitoring drug infusion (Turkcan, Zeng, &

Lawley,2012);thesestepsaredemonstratedinFig.1.Thelabtests areintendedtocontroltheefficacyandsideeffectsofchemother- apy.Ifthelabresultsarenotsatisfactory,theinfusionispostponed (e.g.,foroneweek)orthetreatmentmightcontinuewithanother protocol. The mostsignificant steps froman operationalperspec- tive,however,arethelasttwosteps,whicharecarriedoutbythe nurses and comprise the infusion appointment. Besides the fact that the actualtreatment takes placeduringthesesteps, the du- rationofinfusionissignificantlylongerthanpriorsteps,andthere areextraconstraintsonnursingduringthesetwosteps.Moreover, the infusion duration varies significantly frompatient to patient dependingontheprescribedtreatmentprotocols,rangingfromfif- teenminutestoeighthours(Hahn-Goldbergetal.,2014b;Turkcan etal.,2012).

Duringtheinfusionappointments,nursesfullyattendtheirpa- tients for a fixed amount of time at the beginning of each ap- pointment to preparethepatient and set-up theinfusion device.

Fortheremainder ofthe appointment, they monitortheinfusion progress and attend the patient if action is needed, e.g., when the patient needs help or the infusion is hindered due to some malfunction.Thus,eachnursemaybesharedamongmultiplepa- tients.Aftersetup,anursewatchesthepatient,besidesanumber https://doi.org/10.1016/j.ejor.2018.11.028

0377-2217/© 2018 Elsevier B.V. All rights reserved.

Lab Oncologist

Paent Pharmacy Setup Monitoring

Infusion and Nursing

Post-pone

(this paper) Fig. 1. Operationally significant steps in outpatient chemotherapy.

of other patients who have already been connected to the infu- sion devices andare simultaneously receiving treatment. Besides a nurse, a combinationof a chairor bedand an infusion device isrequiredto carryout theinfusionforeachpatient. Wereferto thiscombinationasastation,andweassumethat allstationsare identical.Everypatientisassignedonestationforthedurationof the infusion appointment. Thus, one type of resource (station)is exclusivelyassignedtoindividualpatients,whiletheothertypeof resource(nurse)ismostlysharedamongmultiplepatients.

Responding to appointment requests can be divided into two majorapproaches:onlineschedulingandofflinescheduling.Inthe former case,one ormore (butnot all)ofthe patientsofthe day tobe scheduledareknown andtheappointmentdayandtimeis giventothepatienteitherimmediatelyaftertherequest,orwithin a short time period, e.g., 24 hours. In the lattercase all the pa- tientsofthedaytobescheduledareknown,buttheappointment time isdetermined later—e.g.,few daysbeforethe appointment—

thoughthedaymaybegivenimmediately.Inonlinescheduling,it is commonpractice to useappointment templates (e.g.,Erdogan, Gose, & Denton, 2015;Liu, van de Ven, & Zhang, 2018) that are arrangementsofvacantappointmentslotsoptimizedforsomecri- teriausingthehistoryofdemand.Offlinescheduling,ontheother hand,takesadvantageofaprioriinformationtooptimizethetim- ing, basedondesiredobjectives.Infact,inonlinescheduling,the offline method is used on an “imaginary” set of patients drawn fromthehistoryofdemandtogeneratethetemplate.

Because of the setting-up and monitoring constraints, a first- come-first-serveddisciplinemayresultinovertimeorlowresource utilization. Using a template forappointment schedulingis anal- ogous to the Rhythm Wheelused forcyclic planning in produc- tion.WithRhythmWheelsavarietyofproductscanbeefficiently planned whilethe capacityloadis smoothened.This reducesthe dependence on accurate forecasting (Packowski & Francas, 2013).

Similarly, designingatemplate isasteptowardcapacitymanage- mentforappointments.Thetemplateisintendedtoserveasalink betweenplanningona tacticallevelandonlineschedulingonan operationallevel.

Inthispaper,weconsideronlytheappointmenttimesonaday.

Vacantappointmentslots are arrangedoffline fora knownsetof imaginarypatientsonthatday.Inthiscontext,wedefinetheready time asthe earliesttime thatpatientsofspecificprotocolswould be available for drug administration, i.e., prior steps have been completed for them. For example,the appointmentof a specific protocolmaybe limitedbythe priorvisit tothe treatingoncolo- gistwhoisavailableforaverylimitedtimeonthesamedayasin- fusion.Thus,theappointmentsforthoseprotocolsmayhaveready timesin themiddle ofthework-shift, ratherthan thebeginning.

Flowtime andmakespanare twocommon criteriafor scheduling.

Flowtime isdefinedasthetime betweentheready timeandthe completion time ofthe scheduled appointment. Makespan is the latestcompletiontimeofallscheduledappointmentsontheday.

Flowtimeminimizationimpliesfasterresponsetojobrequests.

However, inthecontext ofappointmentscheduling,whereactual

requests arrive days before the scheduled appointment day, the flowtimeontheappointmentdaycorrespondstotheprioritygiven to the job for starting it as close as possible to its ready time.

Makespanminimization isanobjectiveforbetterutilizationofre- sources.Usingbothflowtimeandmakespanintheobjectivefunc- tionresultsinsolutionswithmoredesirableproperties,e.g.,least flowtimewithin theminimummakespanratherthan anarbitrary solutionwiththeminimummakespan.

Flowtimereductionandmakespanreductionaretosomeextent alignedwitheachother:bothdependoncompletiontimesofjobs.

However,minimizingbothofthemisconflictinginthechemother- apyappointmentproblem. Thus, preferencemustbe balanced by adjustingtheirweightsintheobjectivefunction.

Chemotherapy-trainednursesaredifficulttoreplace(Dikken&

Smith,2006).Hence,thefocusofthispaperisonoptimalschedul- ingofthelast twosteps(setting-upandmonitoring)accordingto thetimelimitationsandpreferencespertainingtopatientsandthe clinic.Thiscanbe realizedasatrade-off betweenflowtimemini- mizationandmakespanminimization.

Inthispaper,weassumethatpatientsareavailableforinfusion beforetheirscheduledappointments:theyarriveontime,theirlab resultsaresatisfactory,andtherearenooperationaldelays.Despite the precedence that exists among the steps (lab, oncologist, and pharmacy before infusion), the model we propose can be incor- poratedindifferentsettings.The threestepspriorto theinfusion appointments bookedin the template can be scheduled through backpropagation.

Manyoutpatientchemotherapydrugsare preparedontheday ofadministration. This is because ofstability limitationsand the riskofwasting expensivemedicineduetono-shows andunsatis- factorylabresults(Aboumateretal.,2008).However,withthead- ventofroboticautomateddrugpreparationforchemotherapy, the thirdstep isexpectedtobe lesscriticalinthe future.Automated drugpreparationsystemsarealreadyinuseindifferentcountries includingDenmark, Germany,Italy, Japan, Spain, Turkey, and the UnitedStates(Masinietal.,2014).We assumethat thepharmacy canreserve a sufficientnumberofservers (techniciansorrobots) tomatchthestarttimesofthebookedinfusionappointments.

Wenotethatsomeclinicshavethelabtestandoncologistvisit on the day before infusion (Dobish, 2003; Holmes et al., 2010;

Sadki,Xie,&Chauvin, 2010b;Sevinc,Sanli,& Goker, 2013), while othershavethemonthesamedayasinfusion(Liang,Turkcan,Cey- han,&Stuart,2015).Theclinicmayadoptamixedpolicyforthese twosteps,i.e.,schedulingthemonthedaybeforeorthesameday asinfusion,dependingon individual patient requirements. More- over, at some clinics, oncology interns (Mazier & Xie, 2009) or evenspecializednurses(Hahn-Goldbergetal., 2014a)areallowed to authorize infusion based on the lab results.However, even in caseswhereoncologistsmustauthorize infusiononthesameday, it is still possible to schedule their appointments based on the bookedinfusion appointments.Severalinfusionappointmentscan bestartedduringthefirst60–90minutesofthework-shift when nursesare not yetbusymonitoring patientsandareavailable for

setup.Thus,duringthat initialperiodofthework-shift,manypa- tients needto have their oncologistvisits completed beforehand.

During that early period,oncologists would be scheduled to au- thorizeinfusion.Fortherestoftheday,oncologistscantake care ofother tasks besidesauthorizinginfusion, e.g., seeingnewly re- ferred patients, radiotherapytreatments, medical team meetings, etc.Hence,abackpropagationapproachforoutpatientchemother- apyschedulingwouldnotresultinidletimefortheoncologists.

We consider non-negative ready times for appointments, i.e., within the work-shift. A portion of the flowtime is betweenthe readytime and theappointment start time.We call thisthe de- ferringtime. Thedifference betweenflowtimeanddeferringtime is the appointment length, which is constant in a deterministic model,i.e.,independentoftheschedule(solution).Thedirectwait- ingtime isthetime thatthepatient isattheclinic butdoesnot receiveanyservice.Ifthepatientisalreadyattheclinicbecauseof apriorappointment,thedeferringtimeisawaitingtimeaswell.

Thecontributionofourstudyissummarizedasfollows:

• Althoughwe focusonthe lasttwo stepsinFig.1,we have taken theprecedence ofthe prior steps into consideration.

As ourobjective,we attempttohavethestarttimesofap- pointmentsasearlyaspossiblewhilehavingthemakespan asshort aspossible.Thus, a trade-off canbe madeforthe time ofthreesetsofstakeholders:(1)startingtheappoint- mentsaccordingtothetimelimitationsofthepharmacyand oncologists, (2)completing thelast appointmentson a day as early as possible for the nurses, (3) having less direct waiting time for the patients by coordinating the tasks of theotherstakeholders.

• Our makespan minimization approach is different from other relevant studies: Turkcan et al. (2012) and Hahn- Goldberg et al. (2014a). Rather than treating makespan as a variable in an integer programming formulation, we use makespan suppression cost coefficients in the objec- tive function. Thus, the model is kept linear in all binary variables. Moreover,withan integervariable formakespan, there isnodistinction amongthenumberofappointments being completedat themakespan momentor runninglate inthework-shift.However,withourmakespansuppression approach,asolutionthathasfewerappointmentscompleted at the makespan moment corresponds to a better (lower) objectivevalue.

• With only two parameters in the objective function we can tune the objective between minimum weighted flow- time andminimum makespan. However, makespan is lim- ited to thework-shift duration.This limitsthefreedom for minimizing the weightedflowtime whethera reduction in makespan is desirable.Hence, makespancan be viewed as the primary objectivequantity betweenthe two. However, withinthework-shift,ourapproachisgearedtowardpriori- tizingonecriterionovertheother.

2. Relatedliterature

The application of operations research in healthcare related problems is increasing. Brailsford and Vissers (2011) state three reasonsfor thisincrease: (1) healthcare hasbecome a major in- dustry,(2)costofhealthcarehas increaseddueto newtechnolo- gies and the aging population, and(3) patients have higher ex- pectationsfor the quality ofservice. The OR methodologiesused forappointment scheduling includequeueing models, simulation methods,mathematicalprogramming,andheuristics(Deceuninck, Fiems,&DeVuyst,2018).Inthispaper,weusemathematicalpro- gramming and propose a simple heuristic for nurse assignment

that canbe used afterthe solutionfortheappointment schedul- ingproblemisfound.

Ahmadi-Javid, Jalali, andKlassen (2017) review several recent studies on optimization of outpatient appointment scheduling.

However, there are only a few studies in the literature since 2009 that are focused on the planning and scheduling of out- patient chemotherapy appointments using optimization methods (Condotta &Shakhlevich, 2014;Gocgun&Puterman,2014;Hahn- Goldbergetal., 2014a;2014b;Liang& Turkcan,2016;Liang etal., 2015; Mazier & Xie, 2009; Sadki, Xie,& Chauvin, 2010a; 2010b;

2013; Santibáñezet al., 2012; Sevinc et al., 2013; Turkcan et al., 2012). These studies differ significantly in the aspects of the chemotherapy service that they consider and theobjectives they define.Thereasonforthisistwofold:(1)thechemotherapyservice iscomplicated,withmanysteps,resources,andinterdependencies, whichmakesan optimizationproblemlargeanddifficulttosolve, and(2)thepreferencesanddetails oftheproceduredifferamong clinics.

MazierandXie(2009),Sadkietal.(2010b),Sadkietal.(2010a), Sadki, Xie,and Chauvin (2013), andLiang etal. (2015) focus on, orinclude,theoncologistsinthestudy.Mazier, Billaut,andTour- namille(2010), Masinietal.(2014),andTurkcanet al.(2012)fo- cuson, orinclude,thepharmacy intheirstudy.Studies thatlook atschedulingnursingincludeTurkcanetal.(2012),Hahn-Goldberg etal.(2014a),Hahn-Goldbergetal.(2014b),andLiangandTurkcan (2016).

Gocgun and Puterman (2014) focus on planning appointment days,whereas Turkcan etal.(2012), Santibáñezetal.(2012), and Condotta and Shakhlevich (2014) consider both the day and the timeofappointmentsintwoconsecutivestages.

Turkcanetal.(2012),Hahn-Goldbergetal.(2014b),andHahn- Goldbergetal.(2014a)use makespanastheschedulingobjective forthenursingstage.However,withintheplanningstage,Turkcan etal.(2012)considerthe objectivetominimize overtimeandthe idletimeofthestaff.Somestudiesincludeworkloadintheobjec- tive,e.g.,CondottaandShakhlevich(2014),andLiangandTurkcan (2016).

There are a few recent studies in the literature on multi- objective scheduling in healthcare, e.g., Saremi, Jula, ElMekkawy, and Wang (2015), and Rezaeiahari and Khasawneh (2017). How- ever,they donot conform to ourpurposeoftemplate generation for the nursing step of chemotherapy. Even though some stud- ies considerwaitingtime and completion time,in thecontext of chemotherapythey translatetoincludingall fivestepsinasingle optimizationproblem.Inthebackpropagationapproach,wegener- ate thetemplate independent of schedules forthe steps prior to infusion.Forthis,wedrawuponthemachineschedulingliterature toformulatetheobjectivefortemplategeneration.

Severalstudiesinthemachineschedulingliteraturehavecom- binedflowtimeandmakespanorotherperformance measuresfor the optimization objective: Gupta and Dudek (1971), Fry, Arm- strong,andLewis(1989),DileepanandSen(1988),Framinan,Leis- ten,andRuiz-Usano(2002),Sivrikaya-¸Serifo˘gluandUlusoy(1998), Gupta, Neppalli, and Werner(2001),Van Oyen and Pichitlamken (2000),McCormickandPinedo(1995),andEckandPinedo(1993). GuptaandDudek(1971)statethatageneraloptimalitycriterion for scheduling problems is the minimum total opportunity cost, andthiscanincludemultiple objectives.Ina survey onbicriteria schedulingofsinglemachines, DileepanandSen(1988)view two classesof bicriteriaapproaches. Inthe first class,one criterion is usedintheobjectivefunction,whiletheothercriterionisusedin the constraints.In thesecond class, both criteriaare used inthe objective function. Framinan et al. (2002) propose two heuristic algorithms for the flowshopsequencing problemwith the objec- tive ofminimizing bothmakespan andflowtime. Thefirstheuris- tic is an a priori approach: the weights that the decision maker

preferstogivetothetwocriteriaareknowninadvance.Thesec- ond heuristic is basedon a posterioriinformation: aset ofsolu- tions withdifferentweightsbetweenmakespanandflowtimeare generated.Theaposteriorialgorithmgeneratesbetterschedulesat theexpenseoflongercomputationtime.

In our approach for bicriterion optimization of infusion ap- pointmentswe use the mainideasfrom multi-objectivemachine scheduling,e.g.,givingweightstothetwocriteriaintheobjective function,limitingonecriterionintheconstraints,andtreatingone criterionastheprimaryandtheotherasthesecondary.Moreover, within the primaryandsecondary significanceof thecriteria, we considermultipleconditionswhereminimizingoneofthetwocri- teriaisprioritizedovertheother.

3. Modelsandmethods

WeconsideranoutpatientchemotherapyclinicwithKidentical stationsandNnurses.Atemplateisgeneratedbyofflineschedul- ing of a given set of P appointment durations that conform to the history of demand. The template specifies the available ap- pointmentslotsforasingleday(work-shift).Theinfusionappoint- ment durations lp (p=1,...,P) are deterministic; it is assumed that slack is already taken into account in the given set of ap- pointment durations, i.e., extra time at the endof appointments to avoid delays while not resulting in, significant idletime. The appointments are givenpriority weights wp. We assume no late arrivals orno-shows. Preemption is not allowed, and thereis no precedenceamongappointments.Timeisdividedintotimeslotsof afixednumberoftimeunits,e.g.,fifteenminutes.Thedurationof the work-shift is T timeslots(t=1,...,T);timeslot t isthe time interval(^t−1,t].Unlessotherwisestated,eachappointmenthasa zeroreadytimeandaninfiniteduetime:rp=0, dp=∞.

Moreover, all appointment durations are integer numbers of timeslots. The first timeslot ofevery appointmentis always used for setup, and the subsequent timeslots are used for monitoring theinfusionprogress.Anappointmentthathasa durationofonly onetimeslot hasonlysetupinthetemplateandnomonitoring.It is assumedthat removing a patient from a stationatthe end of theappointmenttakesasmallfractionofatimeslotduration and itcanbedoneasamonitoringtask.

Onlyone stationisassignedto eachpatient fortheentireap- pointment.Onenursemustsetthestationupinthefirsttimeslot ofthe appointment.At every timeslot,a nursewho isneitheron abreaknorbusysetting-upanystation,maysimultaneouslymon- itorup to Mpatientsplacedat Malreadyset-up stations.At any timeslotduringbothsetting-upastationandmonitoringapatient, only one nurse is assignedfor the task.The requirementthat at every momenta nursewho isnottakinga coffeeorlunchbreak canset-uponlyonestationormonitoronlyuptoMstations—but notbothatthesametime— isahardconstraint,incontrasttosoft constraints,whichare preferencesthat couldbe violatedto some extent.

Certain periods (consecutive timeslots) of the work-shift are designated forcoffee or lunchbreaks forthe nurses. We assume that each coffee or lunchbreak is divided into two half-periods.

Forinstance,the lunchperiodmaybe fourconsecutivetimeslots.

Halfofthenursestakethebreakinthefirsttwotimeslotsandthe other nurses in the second half-period.The number of available nurses, whichvaries atdifferenttimeslots becauseofthe breaks, is indicated by N_t, which is an input to the model. When there areanoddnumberofnursesNallpresentduringtheentirework- shift,thedecisionmakermaykeepthenumberofnursesat

^N/2

duringbreaksbygivingdoublebreakstoonenurse,orkeep

^N/2

nursesduringonehalfofthebreakperiodand

^N/2

^nursesinthe

otherhalf.

Asmentioned earlier,oneaspectofthe outpatientchemother- apyschedulingproblemistheexplicitconstraintsimposedonthe nursesinadditiontothestations.Theseconstraintslimiteconom- icalinvestmentintheclinic:thecapacityparametersN,K,andM havetobematchedtoavoididleresources.Obviously,thenumber of stations cannot be fewer than the number of nurses, because thenatevery momentone ormorenurses wouldbe idleevenif themonitoring capacityis loweredto M=1.Onthe other hand, ifthenumberofstations ismorethan a certainnumber, thenat everymomentoneormorestationswouldbeidle.Themaximum numberof stations that can simultaneously be used depends on thenumberofnursesandthemonitoringcapacity,anditisgiven inthefollowingproposition:

Proposition1. At a clinicwith N nurses and a monitoring capac- ityMpernurse,amaximumofKmax=(^N−1)^M+1stationscanbe usedsimultaneously.

Proof. WefirstestablishanupperlimitonKthroughproofbycon- tradiction.ThenweshowaconditionunderwhichthevalueforK canreachtheupperlimit.

When every station isbeing set-up all M unitsof monitoring areconsumed.Therefore,tohavethemostnumberofstationsoc- cupied,allnursesmustbebusymonitoring.Supposethatatsome timeslott_b inFig.2a thereare (^N−1)^M+2patientsbeingmon- itoredatK=(^N−1)^M+2stations. Wenote that ateverytimes- lot,eachnursecanatmostmonitorMstationsorset-uponesta- tion.Movingbackintimetillonesetuptakesplaceatsometimes- lotta whileK−1 stationsarebeingmonitored,we canwrite the followingforthe minimumnumberof nurses Nta neededatthat timeslot:

min

(

^Nta

)

=

_K−1

M

+1

=

(

^N−1

)

^M+1 M

+1

=N+1

>N

Thiscontradictsthe initialassumptionthat thenumberofnurses workingattheclinicisN;therefore,

K<

(

^N−1

)

^{M+2. (1)}

On the other hand, for any given number of nurses N with a monitoring capacity M we can place patients one by one un- tiltherearenomorenursesavailableforanothersetup.Moreover, weassumethatappointmentsarelongenoughthatnoinfusionis completeduntilthelast nursehassetastationup.Thiscondition isshownin Fig.2b.Under thiscondition, thefollowing recursive relationholdsbetweenthenumberofsetups,Ps,tandthenumber ofmonitors,Pm,t ateachtimeslott:

_P

m,0=0, and Ps,0=0

Pm,t=Pm,t−1+Ps,t−1, and Ps,t=N−

_Pm

,t

M

For the timeslot t at which no more nurses are available for setup(t=5inFig.2b)wecanwritethefollowing:

Ps,t=0

⇒

(

^N−1

)

^M<Pm,tN·M

However,fromEq.(1)weknowthatPm,t<(^N−1)^M+2;there- fore,P_m,t=(^N−1)^M+1,andthisconcludestheproof.

Proposition1setsanupperboundontheinstantaneousutiliza- tionofthenursingcapacitywhennosetuptakesplace:

ρ

^UBnosetup,t= Kmax

N·M =

(

^N−1

)

^M+1 N·M

Fig. 2. Proposition 1 . The gray station-timeslot blocks are setups and the white ones are monitors . (a) One station is redundant when K = (N −1)M + 2 . (b) K max= (N − 1)M + 1 = 5 stations for N = 2 and M = 4 .

Thisupperboundcangetcloserto100%atlargerclinicswithmore nurses,stations,andpatients. Theinstantaneous utilizationofthe nursingcapacityis100%att=1whenallNnursesaresetting-up andoccasionallyatlatertimeslotssuchast=4inFig.2b.

The following inequalities are sufficient for the limiting re- sourcetoalternatebetweennursesandstationsduringthework- shift:

min

(

^Nt

)

·M<K<max

(

^Nt

)

·M

Weassumethatonly

^N/2

^{ofthenursesareavailableduringthe}

breaks.

min

(

^Nt

)

=

^N/2

N/2 max

(

^Nt

)

=N

M>1⇒

(

^N−1

)

^M+1<N·M

Hence,for thelimitingresource to alternate betweennurses and stations,itsufficestohave

(

^N/2

)

^M<K

(

^N−1

)

^{M+1 (2)}

3.1.Variablesandstationassignmentrule

The overall problem space has five dimensions: activity (setting-upormonitoring),patient,nurse,station,andtime.These couldberepresentedbyfiveindexesforbinarydecisionvariables ˆ

x_p,t,k,n,a=

_{1, activityadonebynursenforpatientpat}

stationkandtimeslott 0, otherwise

However,toreducethecomplexityofproblemformulation,we eliminatethreeoftheindexes.First,weknowthatsetting-upand monitoring are the only two activities and when a setup takes place the remaining timeslots of the corresponding appointment areusedformonitoring.Thus,itsufficestoknowwhenthesetups occur,i.e., the timeslotsatwhichappointments start,ratherthan

thetype ofactivity. Then,we note thatan appointmentschedule showswhenandwherethepatientsreceiveinfusion.Ithasthree dimensions:time,station,andpatient.Thenursesareimplicitlyac- countedforbythe nursingcapacityviathesetting-up andmoni- toring constraints.Nurses are assignedto patientsin a roster af- ter a solution forthe appointmentschedule isfound. Hence, the nurseindexisremovedfromthevariables.Last,sinceall stations areidentical,withoutlossofgeneralitywedefineastationassign- mentrule:amongappointmentsstartingatthesametime,alower indexedappointmentisplacedatalowerindexedavailablestation.

Thisrulelets usdiscardthestationindexfromthedecisionvari- ables.Hence, we definethe decisionvariableswithonlyappoint- mentandtimeindexes:

xp,t=

1, appointment pstartsat timeslott 0, otherwise

wherep=1,...,P,andt=1,...,T. 3.2. Objectivefunction

Inourmodel,theearliesttimeduringthework-shiftthatwould bepossibletostartappointmentpisitsreadytimerp,i.e.,theend oftimeslot r_p; itis zeroby default. However, theremaybe infu- sionappointmentsthat cannotbe scheduledbeforecertaintimes.

Forexample,becauseofpreparationofa certaindrugonlyinthe afternoonorlimitedavailabilityofanoncologistwhenasame-day policy is used. In such cases, the ready time would have a pos- itive value within the work-shift (0<rpT−lp). Let sp be the setuptimeslotscheduledforappointmentp.Theappointmentstart time is then sp−1.We define the time betweenthe ready time and the start time as the deferring time

τ

^{p, i.e., the time that}

the appointment had the opportunity to start butwas deferred:

τ

^p=sp−rp−1.Ontheotherhand,someappointmentsmayhave alimitingfiniteduetimedpT−1<∞,e.g.,becauseofafollow- up procedureonthesameday. Thesetimeslotsaredemonstrated inFig.3.

Moreover, becauseof secondary managerialguidelines, it may bepreferredtohavetheappointmentsofcertainpatientcategories scheduledwithhigherprioritythanothers.However,thesearesoft constraints.Weconsiderthreeprioritylevelsfortheappointments:

high,mid,andlow.Theseareimposedbytheweightswp.Thepri- orities indicate the requirement for the appointment to start as close as possible to its ready time. The appointment start time (sp−1) ratherthanits completiontime(Cp)isthereforemorein- dicativeofrespectingtheappointmentpriority.

As explained in the literature review, two commonly used schedulingobjectivefunctionsaretheflowtimeandthemakespan.

Herealso,minimizingthemakespanisdesirableforbetterutiliza- tion of the resources. However, because of the appointmentpri- orities(forstartingclosertothereadytimes)andthelargerange

Fig. 3. Deferring time and flowtime. The difference between the completion time ( C p) and the ready time ( r p) is the flowtime. The difference between the start time ( s p−1 ) and the ready time is the deferring time ( τp).

ofappointmentdurations,theaverage(ortotal)deferringtimeisa moreinformativeindicatorofperformancethantheaverage(orto- tal)flowtime.Wealsonotethatthereisonlyaconstantdifference betweentheaverageflowtimeandtheaveragedeferringtimethat is independent of the solution. Forthe flowtimeCp−rp andthe deferringtime

τ

^{p of}appointmentp,wecanwritethefollowing:

Cp−rp=

τ

^p+lp

⇒ 1 P

P

p=1

(

^Cp−rp

)

=

τ

¯+1 P

P

p=1

lp

⇒ averageflowtime=averagedeferringtime+constant where

τ

¯ ^{istheaveragedeferringtimecalculatedoverthePsched-} uledappointments.

Similar to GuptaandDudek (1971),our overallobjective isto minimize opportunity costs: deferring appointments after ready times and keeping resources idle. Thus, a trade-off is made be- tween theweightedaverage deferringtime(orflowtime)andthe makespan.

A weak lower bound for the makespan, C_max,LB, can be cal- culated by assuming that at the first timeslot of every appoint- ment, a nurse does not have to fully attend the patient who is beingpreparedforinfusion.Onlyforthisboundcalculationweas- sume that a nursemay be sharedamong Mpatientsalso during setups.Thus, everynursemaysimultaneouslytakecareofamax- imum ofMpatients, i.e., M·Nt isthe maximumnursing capacity of all nurses at timeslot t. Besides the availability of a nurse, a station isalsoneededfor eachappointment. The effectivecapac- ityatanytimeslottisthereforecapt=min

{

^K,M·Nt

}

^.Thetimeat

whichtheareaundertheeffectivecapacityfunctionisequaltothe sumofappointmentdurations,isthusaweaklowerboundforthe makespan:

h

(

^y

)

=Nt for y∈R⁺ andt−1<yt C_max,LB=

v

^v

0

min

{

^K,M·h

(

^y

) }

^dy= P

p=1

lp

.

ThislowerboundcalculationforthemakespanisshowninFig.4a. Inour model,makespaniscontrolled by introducingpenalties intheobjectivefunction.Thepenaltiesincreaseexponentiallywith timetomakealongermakespanlesslikelyinthesolution.Thein- crementbaseis

η

∈R⁺

\

(⁰,1].Toavoidinterferingwiththeother aspectsoftheobjectivefunction—weightedaveragedeferringtime (flowtime) minimization—makespan suppression is imposed after the above calculated weak lower bound forthe makespan. Thus, the makespan suppression penalty (cost) coefficients are defined asfollows:

ct=

η

^{t−Cmax,LB−1, t>C}max,LB

0, tCmax,LB

wheret=1,...,T.Thesepenalty coefficientsare demonstratedin Fig.4b.

The numberof appointmentsrunning atevery moment,Pt,is thesumofPs,tstationsbeingset-upandPm,tpatientsbeingmon- itoredatthat time.However, Pt can beexpressedintermsofthe binarydecisionvariables.Appointmentpthatisrunningattcould havestartedonlyatatimeslot intherangefrommax

{

¹,t−(^lp− 1)

}

^tot:

Pt= P

p=1

t

t=max{¹,t−lp+1} x_p,t

Thethreeprioritylevelsareimposedbyweightingthedeferring time inthe objective function. A constant weight ratio q∈Z⁺ is usedfortheweights:w_high=q³,w_mid=q²,andw_low=q.Atrade- off between minimizing the average weighted deferring time of

Fig. 4. Makespan suppression (a) makespan lower bound calculation: C max,LB= ^{9 . 4 = 10 for N = 2 , M = 4 , K = 5 , and l} p= 45 . (b) makespan suppression penalty coefficients c tfor η= 2 .

appointmentsandminimizingthemakespanisreachedbyadding thetotalweighteddeferringtimez₁andthemakespansuppression penalties z₂ inthe objectivefunction. The makespansuppression termz2isthesumofpenaltiesimposedforappointmentsrunning afterC_max,LB.Ateverytimeslottheimposedpenaltyistheproduct ofthenumberofappointmentsrunningPtandthepenaltycoeffi- cientct.

Thus, the objective function for scheduling the infusion ap- pointmentsisasfollows:

z=z1+z2 (3a)

=

P p=1

wp·

τ

p

+

T t=1

ct·Pt

(3b)

=

_P

p=1

wp·

^T

t=1

(

^t−1−rp

)

·xp,t

+

_T

t=1

ct·

^P

p=1

t

t=max{¹,t−lp+1} x_p,t

(3c)

= P

p=1

T

t=1

wp·

(

^t−1−rp

)

·x_p,t+ct

t

t=max{¹,t−lp+1} xp,t

(3d)

Hence,theproblemisformulatedwiththefollowingobjective:

min

(

^z

)

=min

P p=1

T

t=1

wp·

(

^t−1−rp

)

·xp,t+ct

t

t=max{¹,t−lp+1} xp,t

(4)

Table 1

Different combinations of parameter values corresponding to trade-offs between minimizing the weighted flowtime and minimizing the makespan.

# η q Corresponding equivalent objective

1 η^{= 0 q = 1 min} P

p=1C p

within T

2 η= 0 q 1 min P

p=1w pC p

within T

3 η1 q = 0 min ( C max) within T

4 η1 q = 1 min P

p=1C p _{min (C}

max) within T

5 η¹ η^{q 1 min} P

p=1w pC p _{min (C}

max) within T 6 η1 q ≈η min ( C max) traded-off for min P

p=1w pC p

within T

7 η1 q η min (C max) _min P p=1w pC p

within T

Thetrade-off isadjustedbytheconstant valueschosenfor

η

^and

q, whichare embedded inthe costs ct andthe weights wp,e.g., when

η

^{issettozero,i.e.,ct}=0,

∀

^t,nomakespansuppressionis imposedbesidesT,theendofthework-shift.

As there is only a constant difference between the total weightedflowtimeandthetotalweighteddeferringtime,theterm min(z₁) in the objective function corresponds to weighted flow- timeminimization.Hence,differentvaluesofthetwoparameters

η

andqcorrespondtoapreferencebetweenweightedflowtimemin- imizationandmakespanminimization.Theseconditionsarelisted inTable 1, where’’signifies sufficiently larger. Thus,with only twoparameterswecanselectfrommultipleobjectivepreferences.

3.3.Binaryintegerprogrammingformulation

The optimal appointment schedule is found by determining the setup timeslots of the appointments, using the binary deci- sion variables x_p,t. In the following constraints, we have P=

{

¹,...,P

}

^andT =

{

¹,...,T

}

^.

Theremustbeoneandonlyonestationsetuptimeslotforeach appointment:

T

t=1

xp,t=1,

∀

^p∈P (setupconstraints) (5) Noappointmentcanstartbeforeitsreadytime:

ifrp>0, then

rp

t=1

x_p,t=0,

∀

^p∈P (readytimeconstraints)

(6) Eachappointmentmust becompleted beforeitsduetime and theendofthework-shift:

Cpmin

{

^dp,T

}

,

∀

^p∈P

⇒ T t=1

(

^t+lp−1

)

^xp,tmin

{

^dp,T

}

^,

∀

^p∈P (completiontimeconstraints) (7) Thenumberofappointmentsrunningateverymomentcannot bemorethanthenumberofstations:

PtK,

∀

^t∈T

⇒ P

p=1

t

t=max{¹,t−lp+1}

x_p,t K,

∀

^t∈T (stationconstraints) (8)

Setting-up a station takesas much nursingcapacity asmoni- toring M patients. Moreover, the nursing capacityusage at every momentcannotexceedtheinstantaneousnursingcapacity:

M·Ps,t+Pm,tM·Nt,

∀

^t∈T

⇒ M·P_s,t+Pt−P_s,tM·Nt,

∀

^t∈T

⇒

(

^M−1

)

^Ps,t+PtM·Nt,

∀

^t∈T

⇒ P

p=1

1− 1 M

xp,t+ 1 M

t

t=max{¹,t−lp+1} x_p,t

Nt,

∀

^t∈T (monitorconstraints) (9) Eqs. (4)–(9), constitute the binary integer programming (BIP) formulationthatweconsiderfortemplategeneration.

3.4. Integerprogrammingformulationusingmakespan

A more straightforward approach is to simply use makespan (Cmax) as an integer variable in the formulation rather than makespansuppressionusingpenalties.Thiscanbedonebyreplac- ingthemakespansuppressiontermz₂inEq.(3a)withthecostof makespan z₃=c·Cmax. Thus, theobjective function could be for- mulatedasfollows:

min

(

^z

)

=min

(

^z1+z3

)

^(10a)

=min

^P

p=1

wp·

τ

^p

+c·Cmax

(10b)

=min

c·Cmax+ P

p=1

T

t=1

wp·

(

^t−1−rp

)

·xp,t

(10c)

where Cmax∈Z⁺ is an integer variable in the integer program- ming formulation, and c∈Z⁺ is the cost of one timeslot of makespan.Sincenoappointmentcanrunafterthemakespan,and themakespaniswithinthework-shift,C_max shouldalsoreplaceT in the right-hand-sideof the constraintsshown in Eq.(7). How- ever,asthefunctionmin(•)isnonlinearforthenewlyintroduced integervariable C_max,we replaceEq.(7)withthe followingthree setsofconstraints:

CmaxT, (work-shiftconstraint) (11) T

t=1

(

^t+lp−1

)

^xp,tCmax,

∀

^p∈P (makespanconstraints) (12)

T

t=1

(

^t+lp−1

)

^xp,tdp,

∀

^p∈P (duetimeconstraints) (13) Eqs. (10c), (11), (12),and (13),together with Eqs.(5), (6),(8), and(9)constitutetheintegerprogramming(IP)formulationalter- nativetotheBIPformulationpresentedinSubsection3.3. 3.5. Nurseassignment

As we explainedin Subsection3.1,stations, nurses,andactiv- ities are excluded from the binary decision variables of the BIP model.Thestationsareassignedusingthestationassignmentrule.

The nursesare onlytakeninto accountfortheir nursing capacity in the constraints. The activities at stations are known fromthe appointmentdurationsandtheappointmentstarttimes.Afterthe state(setup,monitor,idle)ofeachstation-timeslotblockoftheap- pointment schedule is determined, we can assign the nurses to the fixedsetup andmonitorstation-timeslot blocks.However,this dependson the specific requirements ofthe clinic, e.g., balanced workload, vicinity ofstations assigned to one nurse,fewer nurse changesduringappointments,etc.

Since nurse assignment depends on the specific preferences of the clinic, in thispaper we focus on the hard constraints for setting-up and monitoring. We assume that the setting-up and monitoring tasks haveworkloads proportionate to their required nursing capacity,i.e., Mand1, respectively,andall appointments have similar levels of instantaneous workload. We also assume thatallnurses arepresentduringtheentirework-shift,exceptfor their coffee andlunch breaks duringwhich, only

^N/2

^{of them}

are available. With theseassumptions, we propose the following heuristictogeneratearosterwithalmostequallydistributedwork- loadamongthenurses:

1. Assign

^P/N

^setups,butnotmorethanonesetupatatimes- lot, to the first nurse (A), starting at the earliest not yet assigned setup and not during the nurse’s break timeslots.

Repeatthisforallothernurses.Forthenursesattheendof the list,theremaybe fewer than

^P/N

^{timeslotsleft with}

setupsandsomesetupsmaynotyetbeassignedinthisstep.

2. At every timeslot with not yet assigned setups, assign the setups in alphabetic order to the nurses who are not on a breakandnotbusywithasetup atthattimeslot. Thiscon- cludesthesetupassignment.

3. The monitorsareassignedasfollows:the monitorsatevery timeslot are equallydistributed amongthenurses who are not ona breakandnot busy withasetup atthat timeslot.

Thestationsareassignedtothenursesinreversealphabetic ordertobettermaintainspatialvicinityofmonitorsassigned tothesamenursethroughouttheroster.

4. Numericalillustration

Inthissectionwedemonstratetheresultsofournumericalex- perimentsonlargeinstances.ForsolvingtheBIPandIPmodelswe usedGurobi7.0.2(GurobiOptimization,2016)intheJuliaprogram- minglanguage(Bezanson,Karpinski,Shah,&Edelman,2012)using its mathematicalprogrammingpackageJuMP(Dunning,Huchette,

&Lubin,2017;Lubin&Dunning,2015)ona64-bitcomputerwith Windows10,Intelprocessori7-6700HQ(2.6–3.5GHz,6 MBcache, 4cores),and16GBofRAM.

Ournumerical experimentsare designed fora largeclinic ex- amplewithaten-hourwork-shift from8:30AMtill6:30PM.The clinic treats one hundred patients per day, which is considered a large number for realistic instances. Turkcan et al. (2012) and Hahn-Goldberg et al. (2014a) solve the chemotherapy schedul- ing problem for minimizing makespan at outpatient chemother- apyclinics that havearound 50 and100 appointmentseach day,

Fig. 5. Relative frequency histogram of infusion appointment durations at Amphia Hospital in Breda, The Netherlands.

respectively. In our problem, a nurse takes a fifteen-minute cof- feebreakinthemorningaround 10:30AM,athirty-minutelunch breakaround1:00PM,andafifteen-minutecoffeebreakintheaf- ternoonaround 4:00 PM. The timeslotsare fifteen minutes long.

Thus,throughoutournumericalexperiments,thework-shiftisT= 40timeslots,andtimeslots8,9,17,18,19,20,30,and31aredesig- natedforcoffeeandlunchbreaks.Thesearedemarcatedwiththe sixverticallinesintheplottedschedulesandroster.

Throughoutthe experiments we use a monitoring capacity of M=4forsets ofP=100 patients. The appointments ineach set arenumberedinascendingorderoftheirdurations.Appointments 18,45,93,and98aregivenhighpriority,andappointments12,41, 88,and94aregivenlowpriority. Thus,foreachpriority,two ap- pointmentshaveratherlongdurationsandtwoappointmentshave rathershortdurations.

We set the number ofstations atthree times the number of nurses: K=3N.Thus, whenM=4and N≥3,the numberofsta- tions K fallsinthe rangegivenin Eq.(2), wherethe limitingre- sourcealternatesbetweennurses andstations.During thebreaks thenumberofnursesdropsto

^N/2

^.

Theappointmentdeferringtimes

τ

^{pareweightedintheobjec-}

tivefunctiontoimposethepriorities.However,theaveragedefer- ringtimes(

τ

¯ ^and

τ

^¯priority)that are reportedasperformance indi- catorsinthissectionarenotweighted.

4.1. Appointmentscheduleandnurseroster

Inthissubsectionwedemonstratenumericalresultsofapplying themodels andmethods presentedin Section 3.Throughout,we useasingleinstanceofonehundredappointmentswithdurations exactlymatchingtherelativefrequencyhistogramofappointment durations at Amphia Hospital in Breda, The Netherlands, which is shown in Fig. 5 (Menting, 2014). We refer to this instance as thehistogram-instance. It is assumedthat all appointments have zeroready times andinfinitedue timesin thisinstance. The in- stancehasa totalappointmentdurationof

lp=1091timeslots, amakespanlowerboundofC_max,LB=33timeslotsandaminimum makespanofmin(^Cmax)=37 timeslotswhen usingN=12nurses atK=36stations.

Fig.6showstheappointmentschedulecorrespondingto

η

=0 andq1(thesecondconditionlistedinTable1).Appointments18 and98both have highprioritiesandzero readytimes.However, the duration is significantly different between the two. The fact thatbothappointmentsstartattimeslott=1isideal.Thiswould not be clear from the significantly different flowtimes (4 versus 22),i.e., the appointmentduration in the flowtimeconceals how welltheprioritiesaremet.Thedeferringtimeisthereforemorein- formativeinthatregard.Theappointmentprioritiesarerespected within the work-shift,i.e., dark-gray andlight-gray appointments appearatthetwoextremesofthework-shift,andCmax=40.

In Fig. 7a the appointment schedule corresponding to

η

1 and

η

q1 (the fifth condition in Table 1) is shown for the histogram-instance. In comparison to the schedule in Fig. 6, the

Fig. 6. Appointment schedule for the histogram-instance under the second condition ( η= 0 and q = 10 ) in Table 1 with N = 12 nurses, P = 100 appointments, zero ready time and infinite due time for all appointments. The dark-gray and light-gray appointments have high and low priorities, respectively. The vertical intercepts demarcate the break periods.

makespanis reduced to 37 timeslots at theexpense of violating priorities: not all high-priority (dark-gray) appointments start at t=1.

The nurse roster inFig. 7bis derived using theheuristic out- linedinSubsection3.5.Thenumberofnursesdropsto

¹²/2

=6 duringthebreaks. Boththe appointmentschedulein Fig.7a and the nurse roster in Fig. 7b include information about time, sta- tion,and activity ofthe solution. The schedule alsoincludes the appointmentinformation,whilethe roster includes thenurse in- formation.Togetherthey determine a solutioninthe fivedimen- sionalspace.Thexˆ_{P×T×K×N×A}arrayforthehistogram-instance has 100×40×36×12×2=3,456,000 binary elements, of which 1091 areonesandtherestarezerosinthesolutionshowninFig.7aand 7b.The1091onescanbefoundfromthenon-idleelementsinthe two K×T arrays ofstation-timeslot blocks plotted inthe figures, e.g.,setupisdonebynurseBatstation14andtimeslot10forap- pointment90,i.e.,xˆ_{90,10,14,B,setup}=1.

The appointmentschedule in Fig. 8 isgenerated using the IP formulation of Subsection 3.4 with c=10⁵ and q=10. The BIP- generatedschedule inFig. 7a hasonly one appointment running atthemakespan,whereasthisIP-generatedschedulehasmanyap- pointmentsrunningatthemakespan,thoughbothscheduleshave the sameCmax=37. Moreover, as shown in Fig. 9, different val- ues for c result in more than one appointment running at the makespanoftheIP-generatedschedule.However,theIP-generated schedule(Fig.8)haslowerdeferringtimesforhigh-priority(dark- gray)appointmentsthantheBIP-generatedschedule(Fig.7a).Nev- ertheless,respectingtheprioritiesmaybeimprovedbyincreasing qintheBIPformulationattheexpenseofrunningmoreappoint- mentsatthemakespan.

Next, we examine the seven conditions of Table 1 on the histogram-instance. Although under the first, third, and fourth conditions there are no priorities—because of all identical

weights—the average deferring times

τ

¯priority of the three differ- entlylabeled setsofappointments(high, mid,andlow priorities) are calculatedtodemonstratehow theweightsaffecttheir defer- ringtimes.Table2showstheresults.Forcomparisonwehavealso included the conditionwhere both

η

^{andq are zero. Underthis}

conditiontheobjectivefunctionisalwayszero,andGurobigener- ates a feasible solution subject to the constraints.The two main performance indicators, Cmax and

τ

¯, have the worst values un- der thiscondition.Under thenext two conditionswherenosup- pression is imposed (

η

=0), the deferring time (flowtime) is the shortest and the makespan is the longest. Under the conditions thatsuppressionisimposed(

η

1),becauseofthelargeexponen- tialcosts,theaveragedeferringtimeiscompromisedanditshifts to a higher levelwhen minimizing the objectivefunction. Under thoseconditions,largeenoughweights(e.g.,q=1000)havetobe usedtocounteractthesuppressionfortheprioritiestobemet,i.e.,

τ

¯low>

τ

¯mid>

τ

¯high=0.

WhenasetupisequivalenttoMmonitors,thetotalworkloadof aninstancecanbeexpressedintermsofthetotalequivalentnum- berofmonitoringstation-timeslotblocksitrequires.Nurseutiliza- tion

ρ

^{canthusbedefinedforthetotalnursingcapacityallocated}

forthatworkloadasfollows:

ρ

:= workload[equivalentnumberofmonitors] nursingcapacity[equivalentnumberofmonitors]

=

(

^M−1

)

^P+^P

p=1

lp

N·M·T

Thetotalworkloadintheappointmentscheduleisequivalentto 1391monitors.Theworkloadassignedtothetwelvenurses—using theheuristicinSubsection3.5—islistedinTable3.Thenumbersin eachnursecolumn(A,...,L)arethepercentageofthetotalwork- load of the appointment schedule assigned to the nurse, under

Fig. 7. Appointment schedule and nurse roster for the histogram-instance.

Fig. 8. Appointment schedule for the histogram-instance generated using the IP formulation with c = 10 ⁵, q = 10 , N = 12 nurses, P = 100 appointments, zero ready time and infinite due time for all appointments. The dark-gray and light-gray appointments have high and low priorities, respectively. The vertical intercepts demarcate the break periods.

Fig. 9. The makespan and number of makespan appointments when the IP formulation is applied to the histogram-instance for different values of the makespan cost c , with q = 10 , N = 12 , and K = 36 .

Table 2

Makespan ( C max) and average deferring time ( τ¯) for different combinations of ηand q for the histogram-instance.

# η q C max τ¯high τ¯mid τ¯low τ¯ gap MIP(%) t CPU( sec )

0 0 0 40 19.25 16.38 19.50 16.62 0 . 00 0

1 0 1 40 6.75 9.22 11.50 9.21 0 . 38 300

2 0 10 40 0.00 9.46 24.00 9.66 0 . 55 300

3 100 0 37 19.00 13.05 15.00 13.37 0 . 00 10

4 100 1 37 21.75 13.10 9.25 13.29 0 . 00 12

5 100 10 37 3.50 12.88 22.50 12.89 0 . 00 24

6 100 100 37 1.00 12.98 22.25 12.87 0 . 00 26

7 100 10 0 0 38 0.00 12.48 19.25 12.25 0 . 01 158

different conditions for makespan suppression and prioritization (

η

^{andq).With12nurses,theaverageworkloadpernurseis8.3%}

ofthe totalworkload. The workload,is to agood extent, equally distributedamongthenurses:thestandarddeviationamongthem (lastcolumninTable3)islessthan1%whensuppressionorprior- itizationisimposed.

Tosee theimpact ofthe numberof nurses onmakespan, de- ferring time, and nurse utilization, we solved the BIP for the histogram-instance withN=12,...,17 nurses at K=3N stations with

η

=100andq=100.TheinstanceisinfeasibleforN≤11.The resultsarelistedinTable4.Withmorenurses,bothmakespanand averagedeferringtimedecreaseatthecost ofmoreexpensesand

Table 3

Workload division and its standard deviation ( σ) among the twelve nurses ( A, . . . , L ) for the histogram-instance under dif- ferent conditions. Both the workload share and its standard deviation are given in percentages of the 1391 total equivalent monitoring station-timeslot blocks required for