Vol. 43, No. 5, September–October 2013, pp. 449–461ISSN 0092-2102 (print) � ISSN 1526-551X (online) http://dx.doi.org/10.1287/inte.2013.0695

© 2013 INFORMS

Improving Patient Access to ChemotherapyTreatment at Duke Cancer Institute

Jonathan C. WoodallDuke Medicine, Durham, North Carolina 27710, jonathan.woodall@duke.edu

Tracy Gosselin, Amy BoswellDuke Cancer Institute, Durham, North Carolina 27710

{tracy.gosselin@duke.edu, amy.suggs@duke.edu}

Michael MurrEdward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University,

Raleigh, North Carolina 27695, mrmurr@ncsu.edu

Brian T. DentonDepartment of Industrial and Operations Engineering, University of Michigan, Ann Arbor, Michigan 48109,

btdenton@umich.edu

This paper describes how we applied simulation and optimization in combination to improve patient flowwithin the Duke Cancer Institute, a large cancer center. We first developed a discrete-event simulation modelto predict patient waiting time and resource utilization throughout various parts of the center, including theoutpatient clinic, radiology, the pharmacy, laboratory services, and the oncology treatment facility. Simulationmodel studies showed that nurse unavailability during oncology treatment causes a serious bottleneck in patientflow. Next, we developed a mixed-integer programming model to relieve the bottleneck by optimizing weeklyand monthly scheduling of different types of nurses. Finally, we developed a novel simulation-optimizationmodel to further relieve the bottleneck by optimizing the starting times of nurse shifts. Our paper summarizesour main findings and the resulting recommendations that Duke Cancer Institute implemented.

Key words : simulation; optimization; integer programming; healthcare; nurse scheduling.

In many countries, cancer is a leading cause ofdeath. As a result, patient demand for cancer ser-

vices, which has been increasing steadily, is expectedto continue to increase (Erikson et al. 2007). From thepatient’s perspective, these increases in demand cancause long waiting times, either at the cancer centeror waiting for the day of a scheduled appointment.From the cancer center’s perspective, they can resultin higher-than-normal resource utilization, requiringovertime and causing congestion at the center. Fur-thermore, in a high-demand environment, variationsin patient mix and patient-flow patterns can resultin overutilization in some areas of the cancer centerduring some times of the day, and underutilizationin others.

In this paper, we describe how we developed andimplemented discrete-event simulation and mixed-integer programming (MIP) models to improve

patient care at Duke Cancer Institute in Durham,North Carolina. Moreover, the insights we drew fromour project are applicable to other cancer centers.We begin by describing a conceptual model of a can-cer center. Next, we describe the various parts of thediscrete-event simulation model and how we com-bined simulation and optimization methods to iden-tify bottlenecks within the cancer center, optimizenurse staffing within the chemotherapy infusion cen-ter, and plan for future capacity expansion to meetpatient needs. Finally, we summarize the benefits thatDuke Cancer Institute gained from implementing ourmodels and discuss opportunities for future research.

Cancer Center Backgroundand ChallengesPatients visit cancer centers for many reasons, suchas referrals from primary care physicians because of

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suspicion of cancer, second opinions, consultationsabout treatment, and follow-up consultations aftercompleting treatment. A typical cancer center is orga-nized into five departments (i.e., locations): clinic,radiology, central laboratory (lab), oncology treatmentcenter (OTC), and pharmacy (see Figure 1).

Patients visit oncologists (cancer-specialist physi-cians) at the clinic, which is usually organized bytype of cancer (e.g., breast, prostate, lung). Radiol-ogy performs imaging scans. The central lab performslab tests, including blood tests—an important part ofcancer diagnosis and treatment monitoring that mustbe reviewed before a patient receives chemotherapy.At the OTC, patients receive chemotherapy treat-ment as either an injection or an infusion, in whichmedicine is dripped intravenously into the patient(i.e., an IV). The pharmacy is the central locationwhere the chemotherapy drugs are mixed prior topatient treatment. Because of the high cost of can-cer drugs, the pharmacy typically does not mix themuntil the patient has checked into (i.e., registered at)the OTC and the patient’s oncologist has reviewedthe appropriate lab results. Thus, the services within acancer center have many dependencies that can influ-ence patient flow.

Clinic Treatmentcenter

Radiology

Centrallabs

Pharmacy

Patient arrives

Patient radiologyscan

Labs sent tocentral labs

Patient travels totreatment center

Results sentto pharmacy

Results sent totreatment center

Mixed drugsent to

treatmentcenter

Results sent backto provider to signoff on OTC order

Figure 1: This figure depicts the patient and information flow among locations in a cancer center.

Most patients begin by registering at a clinic, havelab tests taken and processed, see an oncologist, andfinish at the OTC. However, patient flow through aclinic varies. For example, some patients visit the can-cer center for a clinic visit only. Of these patients,some require lab work and (or) radiology services;others do not. Some patients have previously vis-ited the cancer center; others are there for the firsttime, and therefore tend to spend more time in theclinic. Some patients go from the clinic to the OTC onthe same day; others return for treatment on anotherday. These variations can contribute to uncertainty(and therefore delays) in utilizing the downstreamresources, such as the OTC, from day to day.

We developed detailed descriptions of the patientflow through each major area in the cancer center.However, because much of this paper focuses onchanges within the OTC, we show the flow throughthis area in Figures 2 and 3.

Figure 2 illustrates the steps before treatmentbegins. First, the charge nurse—the nurse with over-all responsibility for the OTC—reviews the patient’schart and medical information and determines if thedrug order is complete. If the order is incomplete,the charge nurse contacts the patient’s oncologist to

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Begin treatment

OTC nursechecks and

reviews chart

Pharmacy mixesdrug

Contact oncologistfor approval

(pharmacist orcharge nurse)

Contact oncologistto obtain

appropriateinformation

Samples OK fortreatment?

Drug ordercomplete and

signed byphysician?

Charge nursereviews patient

chart

Patient arrives atOTC

Patient checks in;Receptionist

brings chart tocharge nurse

Patient sits inwaiting roomuntil called for

treatment

Nurse obtainsdrug, calls back

patient

YY

N N

Figure 2: This flowchart of the OTC illustrates the patient, charge nurse (i.e., nurse with overall OTC respon-sibility), OTC nurse (i.e., nurse directly taking care of patient), and pharmacy activities prior to administeringchemotherapy treatment in the OTC.

obtain the appropriate information. The charge nursealso reviews the lab results to ensure that treatingthe patient is appropriate, and contacts the oncologistif the lab results are abnormal. Once the pharmacyverifies the drug order and labs, it mixes the drug.Finally, before the patient is called back for treatment,the OTC nurse also reviews the chart and lab results.

Nurse calls backpatient to begin

treatment

Injection orInfusion?

Injection

Infusion

Nurseadministers

injection

Nurse hooksup IV Infusion time

Nurse unhooks IV

Patientdischarged

Figure 3: This flowchart shows the OTC flow when the patient requires either an injection or an infusion.

Figure 3 illustrates the steps of treatment. Oncethe aforementioned process described in Figure 2is complete, the OTC nurse caring for the patientbrings the patient back to a treatment chair (i.e., achair in which the patient will sit for the durationof the chemotherapy treatment). Of the two treat-ment types—injections and infusions—injections take

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much less time. Prior to injection, the nurse reviewsand discusses the appropriate medical history withthe patient and gives the patient relevant informa-tion about the injection and managing symptoms.Once the injection is complete, the patient can bedischarged.

For a chemotherapy infusion, a nurse reviews thepatient’s medical history, discusses necessary med-ical information with the patient, connects the IV,takes the patient’s vital signs, and begins the infu-sion. The nurse can then prepare other patients, upto a maximum of four, concurrently. When the infu-sion has completed, the nurse disconnects the IV anddischarges the patient.

Although patients are generally punctual in arriv-ing at the cancer center, most visits involve a clinicappointment, lab tests, and other activities (e.g., radi-ology) that can delay their arrival time at the OTC.After the patient arrives at the OTC, additional delaysmay occur before treatment can begin. For exam-ple, delays may arise in receiving an order from thepharmacy, obtaining lab results, or gaining physicianapproval of orders.

Prior Work on Cancer CenterPlanning and SchedulingSantibáñez et al. (2009) examine a cancer center atthe BC Cancer Agency in Canada. They focus onthe interaction of cancer clinics and study resourceallocation decisions. The authors present scenariosthat include changes to operational factors (e.g., clinicstart time and faculty, such as residents and fellows),appointment scheduling (e.g., sequence of appoint-ment types during the day, scheduling of urgentpatients that arises on short notice), and resourceallocation (e.g., pooled clinic resources or desig-nated resources). They find that to obtain significantimprovements, multiple changes to the existing sys-tems are required.

Sepúlveda et al. (1999) present a model for the MDAnderson Cancer Center in Orlando, Florida, includ-ing the oncology clinic, OTC, and pharmacy. Theirsimulation model examines several scenarios involv-ing changes to the layout and scheduling policieswithin this facility. The authors test policy changes inwhich they increase the number of short-term patients

during slow times of the day and decrease this num-ber during busier times.

Turkcan et al. (2010) examine patient-schedulingdecisions in the setting of a cancer center. The authorscombine two MIP models to plan patient chemother-apy treatment over a specified length of time, suchthat the same patient returns for multiple treatmentsover a sequence of days. The first MIP determinesthe resources required for the patient; the second MIPdetermines the best time to schedule the patient fortreatment, subject to the constraint that the nurse doesnot exceed a specified acuity level for the day, wherethe acuity level is a measure of effort required forpatient care. These authors also examine staffing lev-els to establish the optimal allocation of resources.

Project ContributionsWe developed and implemented the project describedin this paper from 2010 to 2011. It resulted in a num-ber of findings that are transferrable to other cancercenters and other areas of the healthcare delivery sys-tem; however, it differs from the works cited previ-ously in several ways. First, in contrast to Santibáñezet al. (2009) and Sepúlveda et al. (1999), our focuswas on the OTC and in optimally designing nurseschedules to match daily provider (i.e., cancer cen-ter) supply and patient demand. Because of a nationalshortage of skilled nurses, efficiently allocating nurseresources is a high priority at most cancer centers.Second, in contrast to the aforementioned studies, wesought the best ways to staff nurses in light of nonsta-tionary patient arrival behavior, including the simula-tion and optimization of nurse shift start times duringthe day. Third, in contrast to Turkcan et al. (2010),we combined simulation and optimization methodsto understand ways to mitigate the impact of uncer-tainty from various sources, including patient servicetimes, nurse availability, and pharmacy procedures.Finally, to the best of our knowledge, we have devel-oped the most comprehensive model of a cancercenter described in the published literature, and webelieve that other cancer centers could adapt it totheir environments. Woodall (2011) provides completedetails of the model formulation, data collection, val-idation, and implementation.

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Model Formulation and ValidationFollowing a detailed assessment of the patient flowthrough the cancer center, the project involved threemodeling phases: (1) development of a discrete-eventsimulation model of the cancer center; (2) develop-ment of a MIP to optimize weekly and monthly nursestaffing decisions in the OTC; and (3) developmentof a simulation-optimization model to determine theoptimal nurse shift start times from the weekly andmonthly nurse staffing decisions. In this section, wedescribe the three models, model parameter estimates,and the model validation activities that led to ourrecommendations.

Discrete-Event Simulation ModelWe built our simulation model using Rockwell’s sim-ulation software, Arena version 11. As part of our pre-liminary work, we collected sample observation timesfor services in all parts of the cancer center. Our datasources included computer information systems, timestudies, and interviews with oncologist, administra-tor, and nurse experts. Collaborative work with thisdiverse group of experts allowed us to make assump-tions about times when data did not exist or were notavailable immediately.

When we finished collecting data, we developed aprototype version of the simulation model. The ini-tial model included the major areas within the can-cer center (e.g., clinics, labs, radiology, pharmacy, andOTC). We estimated patient arrivals by looking at themean number of arrivals during the day using histor-ical data. We chose to use a nonhomogeneous Poissonarrival process because such a stochastic process iscommonly used in similar applications, and becausethis assumption was reasonable based on our modelvalidation. The Poisson arrival process is nonstation-ary because patient arrival rates vary significantlyover the course of a day. From the historical data, wewere able to estimate the average expected arrival rateby each half hour of the day and define the Poissonarrival process. Scheduled resources for each clinicinclude check-in (i.e., patient registration) and check-out (i.e., patient discharge) receptionists, and phle-botomists (i.e., technicians who draw blood samples),nurses, and oncologists; for the OTC, it includes thereceptionist at the OTC check-in desk, charge nurse,nurses by disease-based groups (DBGs), treatment

chairs, and beds. These resources are available accord-ing to predefined schedules that we entered into themodel to define availability over the course of the day.

OTC nurses engage in both direct and indirectpatient care, and work on some activities in paral-lel; to represent this, we assumed that each nurse hassix capacity units available. For a patient infusion tobegin, at least three units of nurse time must be avail-able. Once the nurse finishes the start-up activitiesand begins monitoring the patient, two units of thenurse are freed. As a result of these assumptions, thenumber of patients a nurse can serve at one time islimited to four, which is an upper limit on the num-ber of patients for which a nurse can be responsiblesimultaneously.

We fit probability distributions to historical datausing the Arena 11 input analyzer. Criteria for select-ing distributions were visual inspection and theresults of chi-square and Kolmogorov-Smirnov tests.We also considered the squared error of the fit. If thedata were lacking or unreliable, we fit probability dis-tributions in two ways: time studies and expert opin-ions. We performed three time studies: one for thepharmacy (pharmacist time and drug mixing time),one for the check-out timings at the clinics, and onefor the length of time the OTC charge nurse tookto review patient charts. Table 1 shows the list ofprobability distributions. In the absence of historicaldata and time studies, we solicited expert opinionsfor the minimum, most-frequent (mode), maximum,and average processing times, with which we definedBeta distributions.

Validating the simulation model required that wetake multiple approaches, including requesting expertopinions and statistically validating model outputs.We consulted the following experts: the clinical oper-ations director, the assistant vice president and asso-ciate chief nursing officer of oncology, a managementengineer in oncology, the administrative manager,and healthcare administration staff. Any results iden-tified as potentially invalid were examined further.Accordingly, we made a number of changes to theinitial model to refine our assumptions. Followingthe expert validation, we compared our observationsto the model-generated patient arrival distributions,patient throughput, and the flow times (time from

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Process type Location Probability distribution Source

Charge nurse chart check OTC 005 + 3705 × BETA(1.30, 23.05) Time studyPharmacist processing Pharmacy −005 + LOGNORM(5.46, 6.74) Time studyPharmacy drug mixing Pharmacy 105 + ERLANG(2.94, 2) Time studyOTC nurse chart check OTC 2 + 1505 × BETA(1.84, 4.52) Expert opinionInjection treatment length OTC TRIANGULAR(1, 2.1, 30) Expert opinionOTC nurse IV setup OTC 5 + 25 × BETA(3.31, 4.46) Expert opinionAcuity level 1 treatment time OTC 15 + 75 × BETA(4.46, 3.31) Expert opinionAcuity level 3 treatment time OTC 90 + 120 × BETA(4.6, 2.2) Expert opinionAcuity level 5 treatment time OTC 210 + 210 × BETA(4.36, 3.52) Expert opinionBlood drawn to be processed at labs Labs 905 + GAMMA(12, 1.33) Historical dataProcessing of blood samples Labs 1905 + LOGNORMAL(19.5, 35.7) Historical dataRadiology processing Radiology 30 + GAMMA(62.2, 1.18) Historical data

Table 1: This table provides a list of probability distributions that we included in the simulation model for theOTC and pharmacy (all times are in minutes).

Simulation model(50 replications) Historical data

Cancer Samplecenter area Mean LCL UCL Mean LCL UCL size

Monday arrivals and throughput validationSurgical 146032 143023 149041 147040 130071 164009 21211

oncologyOncology 167004 163012 170096 167087 159047 176026 21518Brain tumor 46018 44003 48033 46053 44006 49001 698Prostate 36008 34027 37089 36040 33019 39061 546Surgery 169056 166044 172068 170087 161043 180030 21563OTC direct 26076 25059 27093 27040 26001 28079 411

arrivalsOTC 100034 97053 103015 103020 98022 108018 11548

throughput

Table 2: This table compares simulation model-estimated arrivals andobserved arrivals to the OTC. OTC direct arrivals are patient arrivalsthat do not originate from a clinic. The 95 percent confidence intervalis defined by the lower confidence limit (LCL) and the upper confidencelimit (UCL).

check in to discharge) to statistically validate our find-ings. Table 2 illustrates results for arrivals and OTCpatient throughput for a Monday. To aid with vali-dation and verification, we created animations, whichwe generally built to troubleshoot behavior that theexperts identified as unusual, for one clinic and forthe OTC.

Mixed-Integer Programming Modelfor Nurse StaffingAs the last step in documenting the patient flow,OTCs are subject to time variability because of up-stream services (i.e., clinics, labs, radiology), and are

also typically resource constrained. In our simulationmodel, patient waiting times showed the OTC as asignificant bottleneck within the cancer center. Furtheranalysis identified OTC nurse availability as the mostsignificant resource constraint. High variations in thetypes of patients, lengths of time required for the infu-sions, and numbers of patients arriving at the OTCthroughout the day, contributed to this problem. Asa result, the project team focused on exploring meth-ods to improve nurse shift schedules, and thus bettermatch nurse supply with patient demand.

Our approach to planning nurse schedules istwofold and hierarchical in nature. Figure 4 illustratesdaily, weekly, and monthly scheduling. Total dailydemandvarying from Monday through Friday, drivesthe weekly and monthly schedules. We used a MIPto solve the monthly and weekly planning prob-lem to allocate a predetermined number of nursesacross the weeks within the month to match aggre-gate nurse supply to historical demand estimates(the appendix has the complete mathematical for-mulation). Although a number of potential choicesfor the objective function are available, we chose tominimize the total shortage of nurse hours relativeto patient demand because (1) project team mem-bers from the cancer center identified it as the mostimportant consideration, and (2) it can be easily inter-preted. The MIP contained many constraints, includ-ing a minimum allocation level across DBGs, fair allo-cation of long weekends (i.e., a Monday or Friday asa day off) among nurses, and others. After looking

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Monthlyschedule

Staffinglevel

Weekly scheduleby disease group

Daily schedule bydisease group

Daily throughputdrives weekly

schedule

Use mixed integerprogram (MIP) to designweekly/monthly schedule

Use simulation-optimization model todesign daily schedule

Figure 4: This diagram illustrates the relationships between major decisions involved in OTC nurse scheduling.The arrows define the information flow.

at the results of the MIP, we developed a simulation-optimization model to optimize daily shift start timesto minimize average patient waiting time. Feedback(i.e., communication) between the monthly, weekly,and daily schedules iteratively improved the nurseschedules.

In our MIP model for weekly and monthly plan-ning, we considered three types of nurses: 10-hournurses who work four days a week (i.e., 40 hours),8-hour nurses who work five days a week (i.e.,40 hours), and part-time nurses who work a vari-able number of days per week, hours per day, andhours per week. Constraints in our MIP define fea-sible schedules for the OTC. First, OTC daily hoursof operation are fixed (e.g., 7:00 am–8:30 pm). At leasttwo nurses must be in the OTC at all times; thus, twoopeners (i.e., nurses who begin their shift at 7:00 am)and two closers (i.e., nurses who end their shift at8:30 pm) must be on duty. In addition, a minimumcoverage level is required in each DBG during peakhours (i.e., 10:00 am–6:30 pm). Furthermore, the OTCwe studied requires each DBG to have a minimumof three nurses scheduled each day, 14 nurses Mon-day–Thursday, and 13 nurses on Friday (reflecting thelower number of patients the OTC sees on Fridays).Constraints also reflect the allocation of days off to10-hour-shift nurses. Each 10-hour-shift nurse worksfour of the five weekdays and receives one day off.Each nurse must receive at least one long weekend

per month (i.e., four-day weekend)—the nurse hasFriday off one week and Monday off the followingweek. For part-time nurses with days off, the loca-tion of the day off is not constrained; thus, part-timenurses tend to be scheduled on the busiest days.

In the monthly and weekly scheduling MIP tests,we sought to determine the best way to allocatenurses across the week to meet variable day-to-daydemand, with a particular focus on constraining themodel to allot one four-day weekend for each 10-hour-shift nurse. The objective was to minimize totalshortage hours in the OTC nurse schedule, withshortage hours determined by the daily schedulingrequirements. Each day, a specific number of nursinghours is required to meet patient demand (assumingthe discrete nature of nurse shifts is relaxed). How-ever, because of the discrete nature of nurse shifts andconstraints on the number of nurses available, theserequirements may not be met each day. We refer tothe nurse shortage as a deficit.

Simulation-Optimization Model forDaily Nurse SchedulingIn our daily schedule-optimization tests, we soughtto determine the best daily shift start times to mini-mize the average time a patient must wait in the OTC.The appendix gives the complete formulation for ourmodel. In this model, each nurse has a series of associ-ated shifts to which he or she could be allocated; theseshifts correspond to start and end times in half-hour

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increments during the day. Binary decision variablesdefine nurse arrival times at these discrete time points(1 represents arrive; 0 represents not arrive). Sets ofopening and closing shifts define the nurses work-ing at the beginning and ending of the day, respec-tively, and off shift defines the sets of nurses whoare working (i.e., on) or not working (i.e., off) on thisday. A constraint determines the shifts for which anurse may be scheduled (i.e., a binary indicator foreach nurse and day combination). Nurse experienceworking in the various DBGs, as defined by the OTCclinical operations director, is the source of this indi-cator. A second constraint enforces the limitation thatnurses do not work on their days off. Additional con-straints require that at least two nurses are availablefor opening and closing the OTC.

In general, models such as the one we describe pre-viously are computationally challenging because ofthe intractable nature of the expectation in the objec-tive function. Closed-form expressions for expectedwaiting time in complex service systems, such as thecancer center we explore, are generally not available.Thus, resorting to heuristics is necessary. As a result,we solved this model using simulation and optimiza-tion by sampling expected patient waiting times viaour discrete-event simulation model, described pre-viously. The Results section provides details of ourimplementation.

ResultsWe conducted our testing on a Dell Optiplex 980 PC,Intel® Core™, 2.93 GHz computer with 8 GB RAM.The preliminary results from the complete simulationmodel for the cancer center and expert opinions aidedus in identifying system bottlenecks. We applied ourdeterministic MIP model to analyze the monthly andweekly nurse-scheduling problem to determine theoptimal allocation of nurses and nurse shift-lengthpolicies, subject to scheduling constraints on variabledaily demand. Finally, our simulation-optimizationmodel helped us to analyze the daily schedulingproblem to find optimal work schedules under vari-ous shift-length policies, with optimal defined as theleast amount of patient waiting time. The remainderof this section illustrates some of the analyses weconducted to obtain the recommendations that DukeCancer Institute implemented.

Min. no. of FT No. of FT No. of FT No. of PT Shortagenurses 10 hours 8 hours nurses (hours)

Staffing level: 21 nurses0 6 1 18 189 7 5 10 17

10 6 5 12 1111 7 4 12 2612 7 5 10 1713 9 4 9 1814 12 2 7 8815 9 6 4 9016 12 4 4 6517 11 6 2 5718 9 9 0 85

Table 3: This table depicts results from the MIP for the optimal allot-ment of full-time (FT) and part-time (PT) nurses for a scenario in which21 nurses are available.

Monthly and Weekly Schedule OptimizationWe chose Premium Solver, the solver add-on, tosolve instances of the MIP for monthly and weeklyplanning using branch and bound, with a tolerance of0.1 percent, because this solver was easy to implementon a standard PC in the clinic environment. Addition-ally, we set the computation time to a maximum ofone hour.

Table 3 shows results for a sample instance ofthe MIP for a 21-nurse scenario. We solved a seriesof model instances in which we varied the mini-mum number of full-time nurses between 9 and 18to control the number of part-time nurses. We alsoinclude the case in which we set this minimum num-ber to zero full-time nurses as a reference point.Table 3 shows the number of full-time nurses for both10-hour and 8-hour shifts, and the number of part-time nurses in the best solution obtained after onehour of computing time. The last column providesthe shortage hours (objective function). The resultssuggest two primary conclusions. First, using part-time nurses could significantly reduce total short-age hours. This is intuitive because shorter nurseshift schedules permit better matching of supply anddemand during the day. Second, as the number ofpart-time nurses increases, the reduction in shortagehours either diminishes or shows no improvement.These results can help decision makers in trading offthe pros and cons of employing part-time nurses tohelp match supply and demand; however, they may

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also require a greater number of handoffs of patientsamong nurses when shifts end.

Daily Schedule Optimization:Simulation-Optimization ResultsWe solved our simulation-optimization model usingOptQuest version 6.4, which applies a combination oftabu search and other heuristics to attempt to find anear-optimal solution (Kelton et al. 2007). We set thestop-time criterion to end the simulation-optimizationrun after 1,000 iterations. We selected this numberas a conservative upper bound when, following oursimulation-optimization runs, we noted that signif-icant objective function improvements did not typ-ically occur after several hundred simulation runs.We set the indifference parameter (a user-definedparameter that defines a threshold at which schedulesare considered indistinguishable) to be 0.1 minutesof patient waiting time. We allowed the optimizer tovary simulation model replications from 10 to 100,under the constraint that confidence interval widthswere within 10 percent of the mean. For each instanceof the model, simulation run times were approxi-mately two to four hours.

The daily schedule simulation-optimization modelused candidate schedules that we developed in aseries of meetings with decision makers; our MIPmodel, combined with expert opinions, gave us thestarting point for the daily schedule. We exam-ined three nurse staffing levels with three promis-ing shift-policy combinations; all 10-hour shifts, all8-hour shifts, and a mix of 10-hour and 8-hourshifts (with the ratio of 10-hour to 8-hour shifts asapproximately 2:1).

Table 4 illustrates each combination of nurse stafflevel and shift policy we considered. Because ofvariations in patient arrivals by day of week, wedefined the simulation-optimization model for eachday of the week in each scheduling scenario.

Table 5 shows the results of the simulation-optimization runs for scenarios 1–3 (the 21-nurse sce-narios). The “Original” column shows the averagewaiting time for the candidate schedules developedmanually; the “Optimal” column shows the averagewaiting time that OptQuest found as the best solutionafter 1,000 iterations. Table 5 also provides the halfwidth of the confidence interval.

Nurse staffScenario level Shift policy Breakdown

1 21 All 10s (21) 10-hour2 21 Mix of 10s and 8s (13) 10-hour, (8) 8-hour3 21 All 8s (21) 8-hour4 18 All 10s (18) 10-hour5 18 Mix of 10s and 8s (12) 10-hour, (6) 8-hour6 18 All 8s (18) 8-hour7 16 All 10s (16) 10-hour8 16 Mix of 10s and 8s (10) 10-hour, (6) 8-hour9 16 All 8s (16) 8-hour

Table 4: This table depicts results for the nurse staff-level and shift-lengthcombinations for daily schedule optimization for various nurse staffinglevels from 16 to 21 nurses.

Original Optimal

Mean HW Mean HW

Mean OTC waiting timeMonday

All 10s 32028 2078 29002 2032(13) 10s 29063 2018 27036 1031All 8s 29033 2029 27069 1084

TuesdayAll 10s 34044 3041 29043 1088(13) 10s 32011 3018 29066 2074All 8s 38048 3024 27073 1045

WednesdayAll 10s 38006 3096 35079 3052(13) 10s 34066 3033 33031 3030All 8s 36006 4002 33047 3033

ThursdayAll 10s 38018 3093 32073 3016(13) 10s 36055 3075 30032 2016All 8s 35017 3065 28086 2078

FridayAll 10s 32081 3016 26095 1063(13) 10s 30075 2097 25088 1058All 8s 32007 3020 25075 1027

Table 5: This table depicts results of the 21-nurse simulation-optimizationby comparing the original and optimal schedules for “All 10s” (i.e., all10-hour nurse shifts), “All 8s” (i.e., all 8-hour nurse shifts), and a mixof shift lengths in which 13 nurses work 10-hour shifts (HW denotes halfwidth).

These results suggest some general conclusions.First, given the complex nature of scheduling, ad-hocscheduling (i.e., scheduling based on expert opin-ion) worked surprisingly well for generating goodsolutions, because many schedules were statisticallyindistinguishable from the simulation-optimization

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Woodall et al.: Improving Patient Access to Chemotherapy Treatment458 Interfaces 43(5), pp. 449–461, © 2013 INFORMS

model-generated schedules. However, in some cases,ad-hoc scheduling did not provide the best solu-tion; in such cases, the simulation-optimization modelwas helpful in improving the schedule. These differ-ences depend on the day of week, suggesting thatthe ad-hoc approach does not consider variationsin patient flow from day to day. Second, the addi-tion of shift starts on the half hour helped, becausethe optimizer chose an 8:30 am–7:00 pm shift for10-hour-shift nurses in most cases. The 8:00 am–6:30 pm and 9:00 am–7:30 pm shifts were not removedcompletely; the optimizer adjusted many of them to8:30 am–7:00 pm.

Another general conclusion we drew is that asnurses become a bottleneck, the optimizer becomesmore advantageous. As we reduced the nurse stafflevel across scenarios, the number of days and shiftpolicies in which the optimizer has a statisticallysignificant improvement on the candidate schedulesincreases. Additionally, as nurses become a scarcerresource, the benefits of shorter shift lengths increasebecause the nurses can be scheduled during periodsof higher demand. However, the impact of changingthe mix of shift types is still significantly less than theimpact of adjusting work schedule times.

Finally, it is important to point out that althoughthe improvement in average waiting time was mod-erate, the benefits were allocated disproportionatelyto patients who were seen during peak hours. Theresults for the maximum waiting times typically aver-aged about 90 minutes, and improvements in wait-ing time at peak times during the day improved byas much as 25 minutes in some cases. Thus, smallchanges in the daily shift schedule can significantlyimpact the waiting time for the patients who are mostaffected by it at the OTC.

ConclusionsIn this section, we describe our general conclusionsthat could apply to other cancer centers.

Our model revealed bottlenecks for phlebotomyand oncologist consultation in the clinics. Patientwaiting times for phlebotomists in the clinics variedby clinic and day of the week, but ranged from 10to 30 minutes on average. Patient wait times for anoncologist in the oncology examination room also var-ied by clinic and day of the week, also ranging from

10 to 30 minutes on average. The model indicated alarge variability in waiting times by clinic and day,which is consistent with what we expected and expertopinions confirmed.

We also identified bottlenecks in the OTC. Patientwait times in the OTC are greatest for chairs, withan average wait time of 2–10 minutes; however, moresignificantly, the maximum average wait time rangesfrom 25 to 40 minutes at peak times during the day.Additionally, patient wait times for OTC nurses rangefrom one to two minutes on average; however, spe-cific DBGs have maximum average wait times ashigh as 30 minutes at peak times of the day. In par-ticular, most of the longer waiting times for nursesare centralized in the hematologic malignancy andoff-service DBG, ranging from five to 30 minutesfor the maximum average waiting time across allreplications.

Our MIP model analyses show that full-time nursesare helpful for covering supply needs during the day,whereas part-time nurses help to meet the variableday-to-day peak demand. Part-time nurses providethe capability to target increased nurse availabilityat peak times during the day. Thus, they can helpin reducing shortage hours in a nurse schedule. Fur-thermore, we found that adding part-time nurses haddiminishing returns in reducing shortage hours; thus,we concluded that a small number of part-time nursescan have a significant impact in reducing shortagehours. As a result, we recommended replacing one ortwo full-time nurses with equivalent levels of part-time nurses.

The simulation-optimization model indicated thatchanging arrival and departure times in nurse sched-ules has the greatest impact on patient waiting time.In particular, the addition of shift starts on thehalf hour helped, because the optimizer frequentlyselected an 8:30 am–7:00 pm shift for 10-hour-shiftnurses. Thus, we recommended changing some ofthe 8:00 am–6:30 pm and 9:00 am–7:30 pm shiftsto 8:30 am–7:00 pm. A combination of 10-hour and8-hour shifts, rather than 10-hour-only or 8-hour-onlyshifts, can also significantly impact average patientwaiting time; however, in our testing, we saw mixedresults on whether this improved patient waitingtimes. Our results indicated that the lower nurse stafflevels are, the more of a bottleneck they become,

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Woodall et al.: Improving Patient Access to Chemotherapy TreatmentInterfaces 43(5), pp. 449–461, © 2013 INFORMS 459

and the larger the improvement optimization meth-ods make in improving candidate schedules.

ImplementationNext, we summarize some of our recommendationsthat the Duke Cancer Institute—Cancer Center inDurham, North Carolina—adopted.

After evaluating the results and conclusionsdescribed previously, Duke Cancer Institute imple-mented the following strategies to optimize its OTCstaffing: First, it hired four part-time nurses to assistin meeting the variable day-to-day peak demand. Sec-ond, it adjusted the start times for both these newhires and some existing nurses to the half-hour mark.Third, it followed our recommendation to hire addi-tional nurses by hiring 1.75 additional full-time equiv-alent (FTE) nurses. The simulation optimizer alsofound that a combination of 10-hour and 8-hour shiftscan impact average patient waiting time; therefore,the cancer center stopped the practice of allowingnurses to work longer shifts.

The changes described previously are some of themost important tangible benefits achieved by apply-ing the operations research methods described in thispaper. We also used our model to explore resourcecapacity planning for a new cancer center that openedin spring 2012. Our model forecast bottlenecks andOTC performance in several scenarios to inform theplanning process for administrators. The results of themodel showed that an expansion of chair capacityfrom current levels removed most of the waiting timefor chairs in the OTC. Thus, the proposed increase inchair levels in the new cancer center was projectedto sufficiently meet the demand needs. Additionally,we identified nurses as the primary bottleneck inthis cancer center under demand increases of 6–12percent. These results helped administrators makestaffing decisions for this cancer center.

AppendixNext, we describe the mathematical formulation of the MIPmodel for the monthly and weekly nurse planning problem:

Indicest = day in the four-week schedule (t = 1121 0 0 0 120).k = disease-based group (DBG) (k = 1121 0 0 0 16).j = index for long weekends during the month

(j = 11 0 0 0 14).

i = index for nurses (i = 1121 0 0 0 13N5;• 4i = 1121 0 0 0 1N corresponds to a part-time nurse);• (i = N + 11N + 21 0 0 0 12N corresponds to a full-time,

8-hour-shift nurse);• (i = 2N +112N +21 0 0 0 13N corresponds to a full-time,

10-hour-shift nurse).

Decision Variablesxit = nurse i scheduled on day t (binary variable).yij = nurse i scheduled on long weekend j (binary variable).zi = nurse i selected on the schedule (binary variable).st = shortage of nurse hours on day t (continuous variable).ot = overage of nurse hours on day t (continuous variable).

Parameters:fi = FTE (full-time equivalent) value for each nurse i.dt = number of nursing hours required for day t.aik = binary indicator defining if a nurse i is associated with

DBG k (aik = 1) or not (aik = 0).N = maximum number of nurses for a particular DBG shift

type.n= minimum nurses in each DBG for the day.rt = minimum number of nurse in OTC on day t.M = maximum number of FTE for OTC nurses.F = minimum number of full-time nurses.wi = number of days worked per week by nurse i;

• wi = 2, 3, 4, or 5 for i = 1121 0 0 0 1N (part-time nursesunder varying policies);

• wi = 5 for i = N + 11N + 21 0 0 0 12N (full-time,8-hour-shift nurses);

• wi = 4 for i = 2N + 112N + 21 0 0 0 13N (full-time,10-hour-shift nurses).

ki = number of hours available per day for nurse i;• ki = 4, 8, or 10 for i = 1121 0 0 0 1N (part-time nurses);• ki = 8 for i =N + 11N + 21 0 0 0 12N (full-time, 8-hour

shift);• ki = 10 for i = 2N + 112N + 21 0 0 0 13N (full-time,

10-hour shift).

MIP Formulation

Minimize20∑

t=1

st

s.t.3N∑

i=1

zifi ≤M Total FTE Constraint1 (1)

3N∑

i=N+1

zi ≥ F Minimum number offull-time nurses used1 (2)

3N∑

i=1

xit ≥ rt ∀ t Minimum nurse requirementfor entire OTC1 (3)

3N∑

i=1

aikxit ≥ n ∀ t1 k Minimum nurse requirementfor each DBG1 (4)

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xit ≤ zi ∀ i1 t Nurse i selected constraint1 (5)3N∑

i=1

xitki + st − ot = dt ∀ t Nursing hoursper day constraint1 (6)

5∑

t=1

xit =wizi ∀ i Days worked perweek 1 constraint1 (7)

10∑

t=6

xit =wizi ∀ i Days worked perweek 2 constraint1 (8)

15∑

t=11

xit =wizi ∀ i Days worked perweek 3 constraint1 (9)

20∑

t=16

xit =wizi ∀ i Days worked perweek 4 constraint1 (10)

xi5 + xi10 + xi15 + xi20 = 3zi i = 2N + 11 0 0 0 13NFT 10-hour nurse no. Fridays worked constraint1 (11)

xi1 + xi6 + xi11 + xi16 = 3zi i = 2N + 11 0 0 0 13NFT 10-hour nurse no. Mondays worked constraint1 (12)

4∑

j=1

yij = zi i = 2N + 11 0 0 0 13N Four-day weekend

constraint for full-time 10-hour shift nurses1 (13)

xi5 + xi6 ≤ 241 − yi15 ∀ i

Long-weekend constraint, weekend 11 (14)

xi10 + xi11 ≤ 241 − yi25 ∀ i

Long-weekend constraint, weekend 21 (15)

xi15 + xi16 ≤ 241 − yi35 ∀ i

Long-weekend constraint, weekend 31 (16)

xi20 + xi1 ≤ 241 − yi45 ∀ i

Long-weekend constraint, weekend 41 (17)

xit1 zi1 yij binary ∀ i1 j1 t

Binary variables constraint1 (18)

st1 ot ≥ 0 ∀ t Nonnegativity constraints0 (19)

Simulation Optimization Model FormulationNext, we describe the mathematical formulation of thesimulation-optimization model for the daily nurse schedul-ing problem. The decision variables xij are binary deci-sion variables that represent whether nurse i, of m nursesis working shift j 4xij = 1), or not (xij = 0). Each nursei has a series of n associated shifts to which he or shecould be allocated that correspond to start and end timesin half-hour increments during the day. The simulation-optimization model can be expressed as follows:

Minimize E6Patient Waiting Time7

s0t0 xij ≤ aij1 ∀ i1 j1n∑

j=1

xij = 01 ∀ i ∈OFFSHIFT1

m∑

i=1

xik ≥ 21

m∑

j=1

xil ≥ 21

xij ∈ 801191 ∀ i1 j1

where j = k and j = l denote the opening and closing shifts,respectively, and OFFSHIFT denotes the sets of nurses whoare working (on) or not working (off) for the day. The firstconstraint determines the shift on which a nurse may bescheduled; indicator aij is 1 if the assignment of nurse ito j is allowed, and 0 otherwise. Nurse experience workingin the various DBGs, as defined by the OTC clinical oper-ations director, determines this indicator. The second con-straint enforces the limitation that nurses do not work ontheir days off. The third and fourth constraints require thattwo nurses are available for opening and closing the OTC.

AcknowledgmentsThe authors are grateful for the help of a number ofproject team members at the Duke Cancer Institute, includ-ing Chad Seastrunk, William T. Fulkerson, Craig Johnson,Nancy Hedrick, Celia Walsh, Steve Power, the pharmacystaff, and the nursing staff. The authors would like to thankBjorn Berg for helpful comments during the preparationof this manuscript. The authors would also like to thankRandy Robinson, Alice Mack, and anonymous reviewers fortheir help in improving the final version of this manuscript.The authors are grateful for funding from Duke UniversityHospital to complete this project. This paper is also basedin part upon work supported by the National Science Foun-dation [Grant CMMI 0844511]. Any opinions, findings, con-clusions, or recommendations expressed in this paper arethose of the authors and do not necessarily reflect the viewsof the National Science Foundation.

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Santibáñez P, Chow VS, French J, Puterman ML, Tyldesley S (2009)Reducing patient wait times and improving resource utiliza-tion at British Coumbia Cancer Agency’s ambulatory care unitthrough simulation. Health Care Management Sci. 12(4):392–407.

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Woodall J (2011) Models for optimizing resource allocation in acancer center. Master’s thesis, North Carolina State University,Raleigh, NC.

Jonathan C. Woodall is a management engineer at DukeMedicine in Durham, NC. He currently provides supportto radiology, labs, and ambulatory clinics for the hospital.He is a member of the Alpha Pi Mu Industrial Engineer-ing Honors Society. Jonathan received his MSc and BS inindustrial and systems engineering from North CarolinaState University, with research interest in discrete-event sim-ulation modeling. While at North Carolina State, Jonathanhad the opportunity to work on several simulation projectsincluding an Endoscopy Suite at UNC Chapel Hill, an ORsuite at the Mayo Clinic, and the Duke Cancer Institute.

Tracy Gosselin is the associate chief nursing officer andassistant vice president at the Duke Cancer Institute inDurham, North Carolina. She has held a variety of adminis-trative and nursing leadership roles at Duke since 1993. Shehas published and presented on a variety of administrative-and clinically-focused oncology topics locally, nationally,and internationally. She received her BSN from Northeast-ern University in Boston, her MSN from Duke Universityin Durham, and her PhD in nursing from the University ofUtah in Salt Lake City.

Amy Boswell, RN, MSN, OCN is the clinical operationsdirector of the Morris Oncology Treatment Center at Duke

Cancer Center in Durham, NC. She is an oncology certi-fied nurse, a member of the Oncology Nursing Society, andcurrent membership chair of the local Triangle chapter. Shecompleted her MSN in nursing and healthcare leadershipat Duke University and her BSN at Middle Tennessee StateUniversity.

Michael Murr is an analyst for the Performance Partnersenterprise within Premier Healthcare Alliance. He receivedhis Master’s in Industrial Engineering from North CarolinaState University in Raleigh, NC. He also received a BS inindustrial and systems engineering from NC State and par-ticipated in the department’s Health Systems EngineeringCertification program.

Brian T. Denton is an associate professor in the Depart-ment of Industrial and Operations Engineering at Universityof Michigan, Ann Arbor, MI. Previously, he was a professorat North Carolina State University, a senior associate consul-tant at Mayo Clinic, and a senior engineer at IBM. He is a fel-low at the Cecil Sheps Center for Health Services Research atUniversity of North Carolina. His primary research interestsare in optimization under uncertainty with applications tohealthcare delivery and medical decision making. He com-pleted his PhD in management science at McMaster Univer-sity, his MSc in physics at York University, and his BSc inchemistry and physics at McMaster University in Hamilton,Ontario, Canada.

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