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Chemotherapy operations planning and schedulingAyten Turkcan a , Bo Zeng b & Mark Lawley ca Northeastern University, Mechanical and Industrial Engineering, 02115, Boston, MA, USAb University of South Florida, Industrial and Management Systems Engineering, Tampa, FL,USAc Purdue University, Weldon School of Biomedical Engineering, West Lafayette, IN, USAAccepted author version posted online: 15 Feb 2012.Published online: 09 May 2012.

To cite this article: Ayten Turkcan , Bo Zeng & Mark Lawley (2012) Chemotherapy operations planning and scheduling, IIETransactions on Healthcare Systems Engineering, 2:1, 31-49, DOI: 10.1080/19488300.2012.665155

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IIE Transactions on Healthcare Systems Engineering (2012) 2, 31–49Copyright C! “IIE”ISSN: 1948-8300 print / 1948-8319 onlineDOI: 10.1080/19488300.2012.665155

Chemotherapy operations planning and scheduling

AYTEN TURKCAN1,", BO ZENG2, and MARK LAWLEY3

1Northeastern University, Mechanical and Industrial Engineering, Boston, MA 02115, USAE-mail: a.turkcan@neu.edu2University of South Florida, Industrial and Management Systems Engineering, Tampa, FL, USA3Purdue University, Weldon School of Biomedical Engineering, West Lafayette, IN, USA

Received November 2011 and accepted February 2012.

Chemotherapy operations planning and scheduling in oncology clinics is a complex problem due to several factors such as thecyclic nature of chemotherapy treatment plans, the high variability in resource requirements (treatment time, nurse time, pharmacytime) and the multiple clinic resources involved. Treatment plans are made by oncologists for each patient according to existingchemotherapy protocols or clinical trials. It is important to strictly adhere to the patient’s optimal treatment plan to achieve the besthealth outcomes. However, it is typically difficult to attain strict adherence for every patient due to side effects of chemotherapy drugsand limited resources in the clinics. In this study, our aim is to develop operations planning and scheduling methods for chemotherapypatients with the objective of minimizing the deviation from optimal treatment plans due to limited availability of clinic resources(beds/chairs, nurses, pharmacists). Mathematical programming models are developed to solve the chemotherapy operations planningand scheduling problems. A two-stage rolling horizon approach is used to solve these problems sequentially. Real-size problems aresolved to demonstrate the effectiveness of the proposed algorithms in terms of solution quality and computational times.

Keywords: chemotherapy, planning, scheduling, oncology, acuity, resource allocation

1. Introduction

Cancer is the second most common cause of death in theUnited States, accounting for 26% of all deaths (Ameri-can Cancer Society, 2010). The National Cancer Instituteestimates that approximately 11.4 million Americans witha history of cancer were alive in January 2006. Some ofthese individuals were cancer-free, while others still hadevidence of cancer and may have been undergoing treat-ment (American Cancer Society, 2010). The demand foroncology services is projected to increase from 41 millionin 2005 to 61 million in 2020 due to the aging population,the age-sensitive nature of cancer, and the increase in can-cer survivors (Erikson et al., 2007; U.S. Census Bureau,2000).

Chemotherapy is one of the most commonly used can-cer treatment therapies, along with surgery and radiother-apy. It is a systemic treatment that uses drugs to kill can-cer cells. Sophisticated treatment methods and improvedmanagement of side effects are increasing the demandfor chemotherapy, and oncology clinics are experiencinghigher workloads that can result in laboratory, pharmacy,and chemotherapy administration delays (McNulty et al.,2001; Grannan et al., 2002; Dobish, 2003; Aboumater et al.,

"Corresponding author

2008). Reducing waiting times for the first visit and wait-ing times in the clinic for chemotherapy administration areamong the highest priorities for quality improvement inoutpatient cancer treatment facilities (Gesell and Gregory,2004).

Studies identified appointment scheduling that does notinclude clinic resources (nurse staffing and chair availabil-ity), and nursing care requirements, as the main causeof delays and unbalanced workload (Gruber et al., 2003;Chabot and Fox, 2005). Most previous studies propose us-ing scheduling templates/rules based on nursing or phar-macy times (Langhorn and Morrison, 2001; Diedrich andPlank, 2003; Hawley and Carter, 2009). Scheduling deci-sions are made in an ad-hoc manner according to physicianand scheduler experiences and patient preferences. To thebest of our knowledge, there is no study that proposes op-timization methods to schedule chemotherapy treatmentsoptimizing several objectives such as minimizing treatmentdelay, patient waiting time and staff overtime, and maxi-mizing staff utilization. In this paper, we develop planningand scheduling methods for chemotherapy patients withthe objective of decreasing patient waiting time and maxi-mizing adherence to treatment plans, while considering thelimited availability of clinic resources (beds/chairs, nurses,pharmacists). This study differs from previous studies inthat it develops and uses optimization methods rather thanscheduling templates and ad-hoc rules.

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We believe the contributions of this study are as follows:

1. We propose an integer programming model to solve theplanning problem for chemotherapy patients in infu-sion clinics. The objective of the planning problem is tominimize unnecessary treatment delays due to limitedresources.

2. We propose a heuristic and an integer programmingmodel to solve the scheduling problem with the objec-tive of minimizing overtime. The proposed schedulingmethod finds appointment times and assigns patients tonurses and chairs simultaneously.

3. To the best of our knowledge, the proposed method is thefirst optimization-based method that considers patientacuities to determine nurse assignments.

The remainder of the article is structured as follows. Thechemotherapy planning and scheduling problem character-istics are explained in detail in Section 2. Section 3 presentsa mathematical programming model, and section 4 presentsa two-stage algorithm to solve planning and schedulingproblems sequentially. Section 5 presents a variety of com-putational studies to illustrate the effectiveness of the pro-posed algorithms, and Section 6 provides some concludingremarks and discusses future work.

2. Problem characteristics

We define the planning and scheduling problems as theallocation of patient treatments to clinic days subject toavailable resource capacities, and setting of appointmenttimes and allocation of chemotherapy patients to nursesand chairs/beds on each day, respectively. Planning andscheduling problems are complicated due to sheer volumeof patients needing treatment, the cyclic nature of theirtreatment plans, the high variability in resources required,

and the complexity of chemotherapy administration(Baldwin, 2006). Chemotherapy regimens, which varywidely in the length of treatment, amount of direct nurs-ing care required, multiple day treatments, and supporttherapies required, are the main inputs to the planningand scheduling problem (Diedrich and Plank, 2003; Bald-win, 2006). Section 2.1 provides a brief explanation ofchemotherapy and the cyclic nature of chemotherapytreatment plans. Section 2.2 presents the oncology clinicenvironment and patient treatment process, which is im-portant in determining appointment durations. Section 2.3discusses the complicating role of patient acuity in planningand scheduling patient treatment, emphasizing the impor-tance of intensity/acuity tools for better allocation of re-sources. Section 2.4 reviews existing planning and schedul-ing studies.

2.1. Chemotherapy treatment plans

Chemotherapy is a systemic treatment that uses drugs totreat cancer patients. The main aims of the chemotherapytreatment are: i) to stop or slow tumor growth, ii) to con-trol or prevent the spread of cancer cells, and iii) to relievecancer symptoms such as pain (palliative chemotherapy).Chemotherapy drugs affect cancer cells by altering cellularactivity during one or more phases of the cell cycle (seeFigure 1a). The treatment decision depends on the stage ofthe disease, expected survival rate, recurrence risk, and thepatient’s health condition. A cancer’s stage is based on theprimary tumor size and whether it has spread to other areasof the body. If cancer cells are present only in the layer ofcells where they developed and have not spread, the canceris referred to as in situ. If cancer cells invade neighbor-ing tissues and spread to other parts of the body throughthe blood and lymph systems, the tumor is said to be in-vasive/malignant (see Figure 1b). Cell-cycle phase specific

Fig. 1. (a) Cell cycle includes five phases (G0, G1, S, G2, M) during which the cell grows, replicates, divides and rests (adapted fromNational Center for Biotechnology Information, 2004; National Institute of General Medical Sciences, 2003); (b) Stages of tumorgrowth (BSCS & Videodiscovery, 1999). (Color figure available online.)

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Planning and scheduling chemotherapy 33

Fig. 2. Treatment plan for treating a cancer patient with Wilms tumor (adapted from IU Simon Cancer Center, 2006). (Color figureavailable online.)

chemotherapy drugs are most effective against cells thatare rapidly dividing (especially when tumor size is small)(Barton-Burke et al., 2001). Cell-cycle phase non-specificchemotherapy drugs affect cells in all phases of the cell cycleand are most effective against slow dividing cells (especiallywhen tumor size is large) (Barton-Burke et al., 2001).

Chemotherapy drugs affect not only cancer cells, butalso rapidly dividing normal cells. Therefore, chemother-apy treatments are given in cycles with intervening periodsof rest that allow the body to recover before the next treat-ment is given (Barton-Burke et al., 2001). Chemotherapyprotocols show the types of drugs, doses, and schedule ofdrugs based on the type of cancer, stage of cancer, andother specifics about the person’s cancer (a comprehensivelist of protocols can be found on the National Comprehen-sive Cancer Network (NCCN) website, 2009). For example,Figure 2 shows the treatment timeline of a chemotherapyprotocol to treat patients with Wilms tumor (a type of kid-ney tumor occurring in children (IU Simon Cancer Center(2006)). The cycle length is 21 days and the cycle is re-peated 7 times. Different drugs are given on different daysof the treatment, and lab tests are performed every week.Typically, the oncologist sees the patient at the beginningof each cycle, checks the lab results, and decides if the pa-tient can start the cycle. If the patient has not recoveredsufficiently from the previous cycle, the treatment might bedelayed or the dosage might be reduced.

It is very important to adhere to the patient’s treatmentplan to achieve the best results, since delaying the treat-ment decreases its effectiveness due to reduced dose inten-sity. Many studies show the correlation between low doseintensity and poor health outcomes (i.e., decreased tumor

growth control, poorer quality of life, and shortened overallsurvival) (Wood et al., 1994; Bonadonna et al., 1995; Cairo,2000; Chang, 2000; Rosenthal, 2007). We consider the min-imization of treatment delays in the planning problem toachieve better health outcomes.

2.2. Clinic environment and patient flow in oncology andinfusion clinics

In the last two decades, chemotherapy administration hasshifted from the inpatient setting to the outpatient settingdue to sophisticated delivery methods, new oral prepara-tions of drugs, and improved management of side-effects,enabling patients to tolerate their treatments without beinghospitalized. Cancer care is provided in a wide range of set-tings from solo practices to large academic medical centers(Buerhaus et al., 2001). Published studies provide informa-tion about the size of the cancer centers and infusion clinics(Chabot and Fox, 2005; Collingwood, 2005; DeLisle, 2009;Hawley and Carter, 2009). Table 1 shows the number ofpatients treated per day in different infusion clinics. Thenumber of patients seen per day shows significant variabil-ity among different settings. Table 2 shows the number ofnew patients seen per week in different settings. Hawleyand Carter (2009) also presented the number of patientsassigned to each nurse per day, which is 6–8 patients.

Figure 3 presents the typical flow of chemotherapy pa-tients in oncology and infusion clinics. All patients arriveto the clinic by appointment. The patient service assistant(PSA) registers the patient. The medical assistant (MA)prepares patient charts and takes vitals. The phlebotomy

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Table 1. Number of patients seen per day in different infusion clinics

Study Setting Number of clinics Number of patients

Collingwood (2005) Mountain States Tumor Institute 5 cancer centers 10–85 patients/dayChabot and Fox (2005) Tower Hematology Oncology Medical Group 1 clinic 50–60 patients/dayDeLisle (2009) A large, private oncology practice 11 clinics 10–70 patients/dayde Raad et al. (2010) Oncology centers in Australia 6 oncology centers 6–26 patients/day

staff draws blood for laboratory tests. If the patient has aport (device inserted under the skin of the patient by sur-gical procedure to facilitate the blood drawing process),these tasks are performed by a nurse. The patient waitsin the waiting room until the laboratory results becomeavailable. Sometimes, the patient has his/her blood drawnand analyzed in another clinic prior to the appointment. Inthat case, there will be no phlebotomy work in the clinic.If the patient has to see the oncologist, all these tasks areperformed in the oncology clinic and the oncologist seesthe patient. If the patient’s health is suitable for the treat-ment, he/she is sent to the infusion clinic for a same-daytreatment or an appointment is scheduled for chemother-apy administration at a later date. If the patient has nottotally recovered, the oncologist may delay the treatmentuntil the patient becomes ready for the treatment. Whenthe patient comes to the infusion clinic, he/she waits foran available chair/bed and nurse. When these resources be-come available, the chemotherapy nurse takes the patient tothe chair/bed. The pharmacy staff prepares the chemother-apy drugs. The portering staff transports the drugs frompharmacy to the clinic. Chemotherapy nurses administerthe chemotherapy to the patient. Sometimes, the patientdoes not have to see the oncologist. In that case, the patientgoes directly to the infusion clinic for treatment. Sadki etal. (2010a,b), Baldwin (2006), van Lent et al. (2006), Sepul-veda et al. (1999) provide similar patient flow diagrams.

Infusion clinics consider one or more of these processesin determining appointment durations for patient schedul-ing. Chemotherapy administration is the main componentof the appointment duration. The lab and pharmacy timesmight be included in the scheduled duration depending onthe current practice in the clinic. For example, if chemother-apy is prepared after the patient is taken to the infusionchair/bed, then the pharmacy time is included in the sched-uled duration. If the patient needs labs and if the clinic isnot using next-day chemotherapy scheduling (where labsare performed and the patient is seen by the oncologist onone day, and chemotherapy is administered on the next day

(Dobish, 2003)), the scheduler might adjust the scheduledappointment duration to give enough time for lab tests. Ifthe patient has to see an oncologist on the same day, thereshould be enough time between oncologist and infusionappointments. If the infusion clinic is small and there isnot enough ancillary service staff, then the vitals are takenby the nurses. If there is no pharmacist, the chemotherapyis prepared by the infusion nurse. Since there is a varietyof clinic practices, we included all processes in the patientflow. In our study, we assume that the scheduled durationincludes all processes that are performed when the patientis in the chair or an infusion nurse is performing the taskeven if the patient is not in the chair.

2.3. Chemotherapy nursing and high variability in resourcerequirements

Chemotherapy chairs/beds and nurses are two keyresources that should be considered while planning andscheduling patients. The availability of these resourcesdetermines the capacity of the clinic. The clinic space is thefixed capacity and nurses determine the flexible capacitysince staffing levels can be adjusted by clinic managers.Even though the clinics might have limited space to handlethe patient volume, nurse staffing is a more crucial problemin oncology settings. According to a survey of chemother-apy infusion clinics (ambulatory, outpatient, and physicianpractices) conducted by the Oncology Nursing Society(ONS), 47% of respondents reported open positions, whichreflects the significance of the nursing shortage (Irelandet al., 2004). In two other surveys, 59% and more than80% of oncology nurses perceived staffing as inadequate(Buerhaus et al., 2001; Lamkin et al., 2001). Differentmethods are used to increase staffing levels such as hiringless experienced nurses (Ireland et al., 2004), asking nursesto work overtime (Buerhaus et al., 2001; Ireland et al.,2004) hiring an increased number of unlicensed ancillarypersonnel (Ireland et al., 2001), and using agency nurses,internal float pools, and nurses from other departments

Table 2. Number of new patients seen per week in different clinics

Study Setting Clinic size Number of new patients

Hawley and Carter (2009) Cleveland Clinic Cancer Center at HillcrestHospital

24 chairs, 3 private rooms 30 new patients/week

Sadki et al. (2010a,b) ICL ambulatory care unit in France 18 beds 21 new patients/week

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Planning and scheduling chemotherapy 35

Fig. 3. Patient flow. (Color figure available online.)

(Buerhaus et al., 2001). Since chemotherapy nurses arespecially trained, pulling nurses from other departments isnot always possible, which makes the resource allocationdecisions even more important.

Chemotherapy nurses perform several tasks such as ad-ministering chemotherapy, managing side-effects, stabiliz-ing patients during an emergency, documenting importantinformation in patient charts, providing counseling to pa-tients and family members, and triaging patient questionsand problems (Oncology Nursing Forum, 2004). Thereis high variability in nurse workflow due to hundreds ofcancer-specific protocols that require different infusionmethods and treatment durations (ranging from 15 min-utes to more than 8 hours). There is also high variability intreatment duration for each regimen due to patient specificfactors such as difficult vein access, risk for side effects, andchange in dosage. The variability in treatment durationsand nurse workflow complicates the planning and schedul-ing process (Delaney et al., 2002; de Raad et al., 2010; Sadkiet al., 2010a,b).

In literature, there are studies that propose patient inten-sity/acuity tools in ambulatory oncology settings (Cusacket al., 2004a; Jones et al., 2004; Chabot and Fox, 2005;Moore and Hastings, 2006; DeLisle, 2009b; Hawley andCarter, 2009) to establish appropriate staffing levels (Gaits,2005; Hawley and Carter, 2009; West and Sherer, 2009) andimprove scheduling with better resource allocation (Cusacket al., 2004b; Jones et al., 2004; Chabot and Fox, 2005).Chabot and Fox (2005) develop a patient-classification sys-tem that represents patient care and staffing needs. Acuitylevels are assigned to each regimen based on the number ofagents, pre-medications, complexity of administration andassessments required. Hawley and Carter (2009) use totaltreatment time, time with patient and/or family members,blood draws and any additional nursing needs assessedby the nurse at the time of the treatment to determinethe acuity level. In this study, we propose acuity-based

planning and scheduling methods for better allocationof nurses.

2.4. Literature on chemotherapy planning and scheduling

Sadki et al. (2010a,b) is the only study that considers thechemotherapy planning problem in literature. They pro-pose an integer programming model to solve the plan-ning problem with the objective of balancing the workloadthroughout the week. We note that the planning problemmay not always be able to generate a feasible daily scheduledue to acuity levels and available nurse capacity. There-fore, the impact of planning can be measured only whenthe scheduling problem is completely solved. We considerboth planning and scheduling problems to generate feasibledaily schedules considering both chair and nursing times.The minimization of treatment delay is the main objectivefor the planning problem in our study.

There are a few studies about chemotherapy appoint-ment scheduling in nursing literature (Langhorn andMorrison, 2001; Diedrich and Plank, 2003; Dobish,2003; Gruber et al., 2003; Chabot and Fox, 2005; Hawleyand Carter, 2009). Langhorn and Morrison (2001),Diedrich and Plank (2003), and Hawley and Carter (2009)propose scheduling templates/rules based on nursing orpharmacy times. These templates show nurses, chairs,and appointment slots on a spreadsheet. Scheduling rulesinclude the maximum number of patients that can bescheduled at any time slot, earliest and latest appointmenttimes that treatments can be scheduled, and calculationof appointment durations based on procedures to beperformed. The scheduling decisions themselves are madein an ad-hoc manner according to physician and schedulerexperiences and patient preferences. Chabot and Fox(2005) discussed the scheduling system changes resultingfrom their previously mentioned patient-classificationsystem. These included representing nurse schedules in

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the schedule log, assigning patients to nurses, and bettercoordination of oncologist appointments and infusionappointments. A three-year implementation of the newacuity system and scheduling guidelines resulted in morepatients treated, more balanced workload throughout theday, reduced overtime, and increased staff and patient satis-faction. Dobish (2003) proposes a next-day chemotherapyschedule, with laboratory and physician appointments onone day and chemotherapy administration on the next day.The implementation of this next–day approach resultedin improved efficiencies for pharmacy and nursing andreduced in-clinic waiting times for patients. After thechange, the pharmacy was able to prepare 95% of theorders on time for the patients seen the day before. Thepercentage was only 44% for the patients seen and treatedon the same day. Gruber et al. (2003) changed nurseworking hours and established scheduling procedures toimprove on-time starts (from 11% to 94%). We note thatnone of these authors use formal methods to optimize anobjective function subject to constraints. The fact that theycould achieve significant improvements without rigorousmodeling reflects the degree of inefficiency in currentpractice.

3. Problem definition

In this section, we first introduce the notation and definethe planning and scheduling problem with its underlyingassumptions. Then, we propose an integer programmingmodel (I P1) to solve the planning problem, which assignsnew patients’ treatments to days without changing the plansof existing patients. The objective is minimizing the treat-ment delays. The planning problem provides an input (theset of patients assigned to each day) to the daily schedulingproblem. An integer programming model (I P2) is proposedto solve the scheduling problem that considers both chairand nurse availabilities. The complexity of the proposedmodel is reduced by considering only the nurse availabil-ities in a revised model, I P3. A heuristic (ALTT) is pro-posed to provide schedules in short computation times. Theproposed scheduling methods find appointment times andassign patients to chairs and/or nurses.

Table 3 provides notation that will be used throughoutthe paper. We assume that the treatment plan (cycle length,Ci , and the number of times a cycle will be repeated, Fi ) isknown for each patient. This is a realistic assumption be-cause chemotherapy treatments are planned by oncologists

Table 3. Notation

PN, PE Set of new and existing patients (i # PN $ PE and PN % PE = &)PR Set of patients who are referred to receive chemotherapy in the last planning period (PR ' PN)PO Set of patients whose treatment cycles have terminated in the last planning period (PO ' PE)PNE Set of patients whose treatments started in the last planning period (PNE ' PN)PRt Total treatment time of the patients assigned to day t in previous planning horizonPBt Total acuity of the patients assigned to day t in previous planning horizonCi Cycle length of the treatment for patient i (in days) (d = 1 · · · Ci )Fi Number of cycles that will be repeated for patient i ( f = 1 · · · Fi )K Number of chairs (k = 1 · · · K)T Length of planning horizon (days) (t = 1 · · · T)! Re-planning frequency (days) (! < T)Nt Number of nurses on day t ( j = 1 · · · Nt)U Target nurse utilizationS Number of slots on each day (s = 1 · · · S)Ht Number of regular working hours on day two Cost of overtimewu Cost of idle timewd

i Effect of treatment delay on patient irid Treatment length on day d of each cycle for patient i (d = 1 · · · Ci )aid Acuity level on day d of each cycle for patient i (d = 1 · · · Ci )esti Earliest treatment start day for patient iAmax Maximum acuity level a nurse can handle at any timeXit Binary variable, 1 if the treatment of patient i starts on day t, 0 otherwiseRit Treatment time required for patient i on day tAit Acuity level of patient i on day t per time slot (Ait = {1, 2, 3, ...})Bit Total acuity of patient i on day t, i.e. Rit AitPt Set of patients who have treatment on day t, i.e., Pt = {i : Rit > 0}Go

t Over utilization on day tGu

t Under utilization on day tCmax

t Completion time of all treatment on day tYi jkst Binary variable, 1 if the treatment of patient i is started by nurse j on chair k at time slot s on day t, 0 otherwiseMjt Completion time of all treatments assigned to nurse j on day t

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Planning and scheduling chemotherapy 37

based on the chemotherapy protocols. If the oncologistchanges the treatment plan for an existing patient, the pa-tient can be considered as a new patient and appointmentscan be scheduled based on the new treatment plan. We as-sume that the length of treatment on each day, rid , is alsoknown. In a real clinic, the appointment durations shouldbe estimated for each treatment regimen based on the in-fusion durations and additional tasks that should be per-formed when the patient is in the clinic (i.e., blood draws,laboratory tests, drug preparations, etc.). Since it would bedifficult for the scheduler to calculate the appointment du-rations, the clinic staff should provide these estimates to thescheduler. The effect of treatment delays changes accordingto cancer type, stage of the disease and the patient’s health.We use priorities (wd

i ) for each patient i to incorporate theeffect of treatment delays. As wd

i increases, the negativeeffect of treatment delay on patient’s health increases.

We assume there are new patients waiting for their treat-ment to start (PN) and existing patients who are already intheir treatment cycles (PE). The planning problem is to as-sign the cycle of treatments to a sequence of days for eachpatient. The scheduling problem is finding appointmenttimes for all patients on their assigned days. The planningproblem is solved for a number of weeks or months. Theplanning horizon (T) is divided into days, and days are di-vided into smaller time slots (S) to find the appointmentdays and times for patients.

The oncology clinic has limited resources. The number ofchairs/beds (K) and other equipment determine the fixedcapacity of the clinics. The clinic staff (nurses, pharmacists,etc.) who provide care determine the flexible capacity. Thus,the capacity can be increased by increasing the number ofnurses (Nt) or working overtime. However, increasing thenumber of nurses and working overtime will increase thecost, since staff are paid more than the regular rate whenthey work after clinic hours. When the clinic resources arenot fully utilized, the idle time can be thought of as lostcapacity and reduced access to care. We assume that thenumber of nurses and their normal working hours (Ht) aregiven for each day in the planning horizon.

As discussed in Section 2.3, the workflow of chemother-apy nurses is complicated. Considering only the total num-ber of patients assigned to each nurse does not reflect the ac-tual workload of the nurses throughout the day. If patientswith high acuity levels are assigned to the same nurse, theremay be delays in treatment and the probability of makingerrors increases because of heavy workload. We consideracuity levels (aid ) for different treatment types to achieve awell-balanced workload for nurses. The acuity levels can bedetermined by nurse input or by performing time studiesand analyzing the time spent for all treatment-related tasks.Even though there are studies that develop patient inten-sity/acuity tools, none of them provide a comprehensivelist of acuity levels for treatment regimens. In this study,we assign acuity levels to the regimens according to the

number of agents, pre-medications and complexity of ad-ministration as in Chabot and Fox (2005). The acuity levelswe assign might not be exact. Our aim is not to develop anacuity system, but to propose an acuity-based schedulingmethod and compare it with current practice that does notconsider acuity levels. Chabot and Fox (2005) propose ascheduling method that assigns patients to nurses accord-ing to the acuity level. In their study, they assume thatall nursing time is required at the beginning of the treat-ment. However, even though the nurse starts the treatmentfor each patient, he/she also has to monitor the patientthroughout the treatment, may need to perform additionaltasks, and must end the treatment. The nursing time maynot be as intense as it is at the beginning of the treatment,but assigning a large number of patients to a nurse at latertimes might increase the risk of errors due to higher work-load. In current practice, some clinics assign a fixed numberof chairs to each nurse to have a balanced workload. How-ever, a simple count of number of patients assigned is nota good representation of the actual workload assigned toa nurse (Delaney et al., 2002). Thus, we use an acuity levelwhich represents the amount of nursing time per slot. Weassume an upper bound (Amax) on the amount of acuitythat can be assigned to a single nurse, thus the maximumnumber of patients that can be assigned depends on theacuity mix. We assume that a nurse can start at most onetreatment per slot since he/she must carefully assess thepatient, check dosages, and start the treatment.

We propose a two-stage algorithm to solve thechemotherapy operations planning and scheduling prob-lems sequentially. At Stage 1, the planning problem issolved to find the treatment start days of the patients (Xit),which is the first day of the first cycle. Since the treat-ment plan for each patient is known, the treatment days(t( = t + ( f ) 1) · Ci + d ) 1), resource requirements (Rit)and acuity levels (Ait) are also determined at this stage.At Stage 2, the daily scheduling problem is solved for thegiven set of patients assigned to the same day. Patients areassigned to chairs, nurses, and appointment times. Figure 4illustrates the two-stage algorithm to solve planning andscheduling problems.

3.1. Planning

The planning problem is as follows:

(IP1) minT!

t=1

(woGot + wuGu

t ) +!

i#PN

T!

t=1

wdi (t ) esti )Xit

(1)

stT!

t=1

Xit * 1 +i # PN (2)

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38 Turkcan et al.

Fig. 4. Chemotherapy operations planning and scheduling.

Rit =Fi!

f =1

Ci!

d=1

rid Xi,t)( f )1)Ci )d+1 +i # PN, t = 1 · · · T

(3)

Ait =Fi!

f =1

Ci!

d=1

aid Xi,t)( f )1)Ci )d+1 +i # PN, t = 1 · · · T

(4)

Bit =Fi!

f =1

Ci!

d=1

ridaid Xi,t)( f )1)Ci )d+1 +i # PN, t = 1 · · · T

(5)

Got ) Gu

t =!

i#PN

Rit + PRt ) K · Ht t = 1 · · · T (6)

!

i#PN

Bit + PBt * U Nt Ht Amax t = 1 · · · T (7)

Xit # {0, 1} +i # PN, t = 1 · · · T (8)

The objective is to minimize total staff overtime andidle time, and total treatment delay. The first term in theobjective function is total overtime and idle time cost ofclinic staff. The second term is the total weighted treat-ment delay, which is calculated according to the earliesttreatment start time (esti ) and planned treatment start time("

t tXit). The earliest treatment start time is determined bythe oncologist and depends on the patient’s health status,the treatment plan and other treatments (surgery, radio-therapy) that need to be coordinated with chemotherapy.The delays are multiplied by wd

i to incorporate the differ-ences among treatments in terms of their effect on healthoutcomes. The patient’s treatment will start on at mostone of the days in the planning horizon, which is guaran-teed by constraint (2). If the planning horizon is not longenough and the number of patients is high, then the treat-

ment may not start in the planning horizon. The resourcerequirement (treatment time, nurse time, pharmacy time)and acuity level per unit time for each patient on a given dayafter the treatment start (a patient’s acuity level is positiveonly on appointment days, zero otherwise) are calculatedin constraints (3) and (4). Total acuity level of patient i on aday is calculated by constraint (5). For example, for patienti" with cycle length of 21 days (Ci" = 21), treatment on days1 and 3 of each cycle (treatment length on day 1 is ri"1 = 90,acuity level on day 1 is ai"1 = 2, treatment length on day3 is ri"3 = 60, acuity level on day 3 is ai"3 = 1, treatmentlength and acuity levels on days 2, 4–21 are zero), and twocycles (Fi" = 2), constraints (3)–(5) would be as follows:

Ri"t = 90Xi"t t = 1, 2Ri"t = 60Xi",t)2 + 90Xi"t t = 3 · · · 21Ri"t = 90Xi"t)21 + 60Xi"t)2 + 90Xi"t t = 22, 23Ri"t = 60Xi",t)23 + 90Xi",t)21 + 60Xi"t)2

+ 90Xi"t t = 24 · · · TAi"t = 2Xi"t t = 1, 2Ai"t = 1Xi",t)2 + 2Xi"t t = 3 · · · 21Ai"t = 2Xi"t)21 + 1Xi"t)2 + 2Xi"t t = 22, 23Ai"t = 1Xi",t)23 + 2Xi",t)21 + 1Xi"t)2

+ 2Xi"t t = 24 · · · TBi"t = 180Xi"t t = 1, 2Bi"t = 60Xi",t)2 + 180Xi"t t = 3 · · · 21Bi"t = 180Xi"t)21 + 60Xi"t)2 + 180Xi"t t = 22, 23Bi"t = 60Xi",t)23 + 180Xi",t)21 + 60Xi"t)2

+ 180Xi"t t = 24 · · · T

Constraint (6) is used to calculate the overtime and idletime. Since the appointment times are not determined inthe planning part, the overtime and idle time are approxi-mate values based on total available capacity (K · Ht). Con-straint (7) is used to control the total acuity of the patients

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Planning and scheduling chemotherapy 39

assigned to nurses on each day. The maximum total acu-ity that can be assigned to nurses during normal workinghours is Nt Ht Amax. However, this is an over-estimate, be-cause it does not consider that a nurse can start at most onetreatment at any slot. Therefore, we multiply the maximumtotal acuity level by U. The value of U should be selectedcarefully in order not to overload nurses. Constraints (6)and (7) are necessary to make a reasonable number of as-signments to days based on available capacity.

3.2. Scheduling

The proposed model finds the treatment days, resourcerequirements, and acuity levels on each treatment day fornew patients. The resource requirements and acuity levelsfor existing patients were already calculated in previoustime periods and are used as inputs in the proposed modelto calculate the overtime and idle time. The resourcerequirements (Rit) and acuity levels (Ait) for all patients(i # PN $ PE) are used as inputs to the scheduling prob-lem. We proposed two integer programming models and aheuristic. The first integer programming model considersboth chair and nurse availabilities. The second integerprogramming model considers just the nurse availabilitiesassuming that nurses are the limited resources. The heuris-tic, which sorts the patients according to treatment durationand assigns them to chairs and nurses one at a time, isproposed to find schedules in small computation times.

3.2.1. Integer programming modelsThe integer programming model considering both chairand nurse availabilities is as follows:

(I P2) minT!

t=1

Cmaxt (9)

stNt!

j=1

K!

k=1

S)Rit+1!

s=1

Yi jkst = 1 +i # Pt, t = 1 · · · T (10)

!

i#Pt

Nt!

j=1

min{S)Rit+1,s}!

u=max{s)Rit+1,1}Yi jkut * 1

k = 1 · · · K, s = 1 · · · S, t = 1 · · · T (11)

!

i#Pt

K!

k=1

min{S)Rit+1,s}!

u=max{s)Rit+1,1}AitYi jkut * Amax

j = 1 · · · Nt, s = 1 · · · S, t = 1 · · · T (12)

!

i#Pt

K!

k=1

Yi jkst * 1 j = 1 · · · Nt, s = 1 · · · S, t = 1 · · · T

(13)

K!

k=1

S)Rit+1!

s=1

Yi jkst(s + Rit ) 1) * Mjt

+i # Pt, j = 1 · · · Nt, t = 1 · · · T (14)

Mjt * Cmaxt j = 1 · · · Nt, t = 1 · · · T (15)

Yi jkst # {0, 1} +i # Pt, j = 1 · · · Nt

k = 1 · · · K, s = 1 · · · S, t = 1 · · · T (16)

The objective is to minimize the total completion timeof all treatments on day t. Each patient who has treatmenton day t (i # Pt) is assigned to a nurse, a chair, and a timeslot by constraint (10). Constraint (11) ensures that at mostone patient is assigned to a chair. The total acuity level as-signed to a nurse cannot exceed the maximum acuity level,which is controlled by constraint (12). Constraint (13) en-sures that a nurse can start at most one treatment per slot.The total completion time of the treatments assigned to anurse on a given day is calculated in constraint (14). Thetotal completion time of all treatments for each day is cal-culated in constraint (15). Constraint (16) is the integralityconstraint. The proposed model I P2 can be decomposedinto smaller subproblems for each day t. The subproblemscan be solved independently to find the appointment times,nurses and chairs on each day.

The proposed integer programming models can besolved optimally. However, the length of the planninghorizon, number of time slots on each day, and num-ber of patients, nurses and chairs affect the computa-tional complexity of the problem. The first model (IP1)has |PN| , T binary variables, 3|PN| , T + T continu-ous variables, and |PN| + 3|PN| , T , maxi {Ci Fi } + 2Tconstraints. The second model (IP2) has |Pt| , Nt , K ,S binary variables, and |Pt| + K , S + 2Nt , S + |Pt| ,Nt + Nt constraints for a given day t. Please note that thesecond model is solved for all days in the planning horizon.If the number of slots, patients, chairs, and nurses are 40,50, 20, and 7, respectively, then there will be 280,000 binaryvariables and 8,687 constraints in IP2 for each day.

One way to reduce the computational complexity ofIP2 is considering a single resource rather than consid-ering both resources (chairs and nurses). For example, ifthe nurses’ time is the limiting resource, as appears tobe the case in most of the clinics, the chair assignmentcan be removed from the model. The index k for chairsand constraint (11) are removed from the model to havea smaller size problem with |Pt| , Nt , S binary variables,and |Pt| + 2Nt , S + |Pt| , Nt + Nt constraints for a givenday t. The patients should be assigned to chairs after thefollowing integer programming model (IP3), which is usedto assign patients to nurses and to find appointment times,is solved.

(IP3) minT!

t=1

Cmaxt (9)

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40 Turkcan et al.

stNt!

j=1

S)Rit+1!

s=1

Yi jst = 1 +i # Pt, t = 1 · · · T (10’)

!

i#Pt

min{S)Rit+1,s}!

u=max{s)Rit+1,1}AitYi jut * Amax

j = 1 · · · Nt, s = 1 · · · S, t = 1 · · · T (12’)!

i#Pt

Yi jst * 1 j = 1 · · · Nt, s = 1 · · · S, t = 1 · · · T (13’)

S)Rit+1!

s=1

Yi jst(s + Rit ) 1) * Mjt

+i # Pt, j = 1 · · · Nt, t = 1 · · · T (14’)

Mjt * Cmaxt j = 1 · · · Nt, t = 1 · · · T (15’)

Yi jst # {0, 1} +i # Pt, j = 1 · · · Nt

s = 1 · · · S, t = 1 · · · T (16’)

3.2.2. HeuristicWe propose a heuristic (ALTTt) to find appointment sched-ules in short computation times. The basic steps of the al-gorithm is as follows:

Algorithm 1 ALTTt: Longest treatment time first rule in-corporating acuity levels

1: NurseAvail jst = 0, ChairAvailkst = 0, TreatStartjst = 0,Mjt = 0

2: Sort all patients according to their treatment times, i.e. Rit -Ri+1,t

3: for all i = 1 to I do4: for all s = 1 to S, j = 1 to J and k = 1 to K do5: Assign patient i to nurse j , chair k and slot s temporarily6: if NurseAvail j,s (,t + Ait * Amax, TreatStart j,s,t < 1 and

ChairAvail j,s (,t < 1 for all k, j , s and s ( where s ( - s ands ( < s + Rit and given t then

7: Go to Step 98: end if9: (s, j, k) satisfies the constraints (6), (7) and (8). Go to

Step 11.10: end for11: (s", j", k") = (s, j, k) and Yi, j",k",s",t = 112: TreatStart j",s",t = 113: for all s ( - s" and s ( * s" + Rit ) 1 do14: NurseAvail j",s (,t = NurseAvail j",s (,t + Ait15: ChairAvail j",s (,t = 116: end for17: if s" + Rit ) 1 > Mj",t then18: Mj",t = s" + Rit ) 119: end if20: if Mj",t > Cmax

t then21: Cmax

t = Mj",t22: end if23: end for

At Step 1, NurseAvail jst, ChairAvailkst, and TreatStart jstare initialized as zero. If NurseAvail jst is zero, that meansnurse j is available at slot s on day t. If ChairAvailkst is zero,that means chair/bed k is available at time slot s on day t.When TreatStart jst is zero, nurse j can start a treatmentat slot s on day t. The completion time of all patientsassigned to nurse j on day t (Mjt) is also initialized tozero at Step 1. At Step 2, the patients are sorted in non-increasing order of their treatment times (Rit). Each patientin the sorted list is assigned to a nurse, a chair/bed and atime slot ( j, k, s) temporarily at step 5. At Step 6, it ischecked whether constraints (6), (7) and (8) are satisfied. Ifall constraints are satisfied, then the patient is assigned tothe corresponding nurse, chair/bed and slot (Step 11). AtSteps 12–16, TreatStart jst, NurseAvail jst, and ChairAvailkstare updated. At Steps 17–19, the completion time of allpatients assigned to a nurse on day t (Mjt) is updated. AtSteps 20–22, the completion time of all treatments on dayt is calculated.

The proposed algorithm is similar to longest processingtime (LPT) rule, which is used to minimize makespan inparallel machine scheduling (Pinedo, 2009). The LPT ruleis modified to control the total acuity level and numberof treatment starts assigned to each nurse. The numberof slots is chosen as large as possible so that all patientscan be assigned to nurses, chairs/beds and time slots. Theproposed algorithm aims to minimize the completion timeof all patients.

The planning and scheduling problems are solved for thepatients who need treatment for several weeks. The sched-ules cannot be fixed for the whole planning horizon. Newpatients with different acuity levels and treatment plans arereferred to receive chemotherapy treatment and they shouldbe added to the existing schedule. The planning problemshould be solved frequently to minimize treatment delaysfor new patients. We use a rolling horizon approach tosolve planning and scheduling problems sequentially. Theproposed method is explained in detail in the followingsection.

4. Rolling horizon methodology

We propose a rolling horizon approach to solve planningand scheduling problems with the objective of minimizingtreatment delays for new patients. We solve the planningproblem every ! days for a planning horizon of T days(where ! * T ). The planning problem finds the treatmentstart days for new patients for the first ! days starting fromcurrent time (tc). Once the first treatment day of a newpatient is identified, the remaining treatment days are fixedin the planning horizon according to the chemotherapycycle. (As noted below, the planning horizon T must besufficiently long to cover the treatment cycles for all patientsreferred up to tc.) No new treatment start is planned fortime periods tc + ! + 1, · · · , T.

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Planning and scheduling chemotherapy 41

The detailed appointment schedule is generated only forthe first ! days. After the schedule for ! days is executed,the planning and scheduling problems are solved for timeperiods tc + ! + 1, · · · , tc + ! + T. The planning horizonT must be updated every time the planning problem issolved since the set of new patients will change. The patientswhose treatments have started in the last ! days, PNE, areremoved from the set of new patients, PE, and added tothe set of existing patients, PE. Patients referred to receivechemotherapy in the last ! days, PR, are added to thenew patient set, PE, and patients whose treatment cyclesfinished in the last ! days, PO, are removed from the setthe existing patient set, PE.

The planning horizon T is calculated according to thetotal treatment length of new patients and the treatmentcompletion time of existing patients, i.e. T = max{tc + ! +maxi#PN{Ci Fi }, max{t : Rit > 0, i # PE}}. The first term isthe maximum treatment completion time for the new pa-tients. The second term is the treatment completion timefor all existing patients. The treatment length of a new pa-tient is calculated by multiplying the cycle length (Ci ) withnumber of cycles (Fi ). The treatment completion time ofan existing patient i is the last time period that the patientis treated and is calculated as max{t : Rit > 0}.

The following algorithm gives the basic steps of a rollinghorizon approach:

Algorithm 2 Rolling horizon algorithm

1: Initialize tc = 0.2: Calculate the planning horizon for the given set of existing

(PE) and new patients (PN).T = max{tc + ! + maxi#PN {Ci Fi }, max{t : Rit >0, i # PE}}.

3: Solve I P1 for all patients in PE $ PN and the planning hori-zon of [tc + 1, T].

4: for all t = tc + 1, tc + 2, · · · , tc + ! do5: Determine the set of patients who have treatment on day t.

Pt = {i : Rit > 0 and i # PE $ PN}.6: Solve I P2 (or I P3) for day t.7: end for8: After ! days, find sets PNE, PO and PR.

PNE = {i : Xit = 1, t # [tc + 1, tc + !], i # PN}PO = {i : max{t : Rit > 0} # [tc + 1, tc + !]}PR = {i : esti # [tc + 1, tc + !]}

9: Update PE and PN.PE = PE $ PNE \ PO

PN = PN $ PR \ PNE

10: Increase tc by ! and go to Step 2.

At Step 1, the current time is initialized to zero. The set ofexisting patients (PE) and new patients (PN) are assumedto be known at time zero. At Step 2, the planning hori-zon T is calculated according to the treatment completiontimes of existing patients and maximum completion timeof new patients. The planning problem I P1 is solved at Step3. The treatment start times are found for the time interval

[tc + 1, tc + !]. The scheduling problem is solved for eachday t # [tc + 1, tc + !] at Step 6. According to the solu-tion of the planning problem, patients whose treatmentshave started (PNE) and whose treatment cycles have termi-nated (PO) during time interval [tc + 1, tc + !] are foundat Step 8. The set of patients who are referred to receivechemotherapy are also found at this step. At Step 9, thesets of existing and new patients are updated. The sched-ules for time periods tc + 1 · · · tc + ! are executed and thecurrent time is updated at Step 10. Steps 2–10 are repeatedevery ! days.

We present a numerical example to clarify the basic stepsof the rolling horizon approach. Assume that there are twonurses (Nt = 2) and five chairs (K = 5) in an infusion clinic.Four hours is used as the normal working hours. The to-tal working hours is 20 hours (KHt = 5 , 4 = 20 hours =1200 minutes). ! is chosen as seven days, which corre-sponds to a week. At the beginning of week 11, the setof existing patients is PE = {1, 2, · · ·, 80} and the set ofnew patients is PN = {81, 82, · · ·, 96}. The planning prob-lem is solved to assign new patients to days 71–75 (Mon-day through Friday) at week 11. The clinic is closed onweekends, therefore no patient is scheduled on days 76–77.Table 4 shows existing and new patients assigned to eachday after the planning problem is solved. The bold numbersrepresent the new patients whose treatment started at week11. Patients who have multiple day treatments can easily beseen in the table. For example, patient 14 has treatments ondays 71, 72, and 73. The total workload (total treatmentduration) on each day is also shown in the same table.

During week 11, the treatments of patients 84, 85, 86,87, 89, 90, 91, 92, 93, 95, 96 (PNE) are started, and treat-ment of patients 4, 8, 10, 35, 62 (PO) are terminated. Tennew patients (PR = {97, 98, ..., 106}) are referred to receivetreatment. At the beginning of week 12, sets of existing andnew patients are updated as follows:

PE = PE $ PNE \ PO = {1, 2, · · · , 80} $ {84, 85, 86, 87, 89,

90, 91, 92, 93, 95, 96} \ {4, 8, 10, 35, 62}PN = PN $ PR \ PNE = {81, 82, · · · , 96} $ {97, 98, . . . , 106}

\{84, 85, 86, 87, 89, 90, 91, 92, 93, 96, 96}

After the patients who have treatments on days 71–75are determined, appointment scheduling problem is solvedfor each day. An example schedule for day 74 can be seenin Figure 5. Different colors show different nurses; patients10, 59, 66, 74, 84, 92 are assigned to nurse 1, and patients57, 85, 89, 91, 95, 96 are assigned to nurse 2. The num-bers in the boxes show the patient number and the acuitylevel. For example, patient 96 with acuity level 2 (96, 2) isassigned to chair 1. Patients 92 and 96 are scheduled toarrive at time zero. Patients 74 and 89 are scheduled to ar-rive at time 15. They cannot be scheduled to arrive at timezero, because a nurse can start the treatment of only onepatient. The completion time is 270 minutes, which is cor-responds to 30 minutes of overtime. Even though the total

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42 Turkcan et al.

Table 4. Numerical example - planning

Total workload (minutes)Day, t Pt = {i : Rit > 0, i # PE} $ {i : Xit = 1, i # PN} (Existing + New)

During week 11 71 1 4 5 7 8 10 11 12 13 14 15 16 17 20(Steps 3 and 5) 37 59 60 62 1185 (1155 + 30)

93

72 10 13 1423 34 35 36 38 40 43 45 59 60 62 1185 (1185 + 0)

70

73 10 1457 59 61 1155 (675 + 480)

85 95 96

74 1057 59 1155 (555 + 600)

66 74 84 85 89 91 92 95 96

75 1057 59 1005 (300 + 705)

85 86 87 90 95 96

treatment duration assigned to day 74 is less than the totalavailable nursing time (1155 < 1200), the actual schedulehas overtime because of nurse capacity.

5. Comparison of proposed method with current practice

The aim of this section is to compare the proposed rollinghorizon approach with current practice. In the followingsections, we first explain the planning/scheduling methodused in current practice and then use randomly generated

Fig. 5. Example schedule for day 74. (Color figure available on-line.)

patient mixes to compare the proposed rolling horizon ap-proach with current practice.

5.1. Current practice

In current practice, after the oncologist sees the patient, theoncology nurse sends the scheduling request to the sched-uler. The scheduler selects the patient whose appointmentrequest is made first and schedules all treatments in onetreatment cycle. The patient’s treatments are scheduled tothe first available time slot according to patient and nursepreferences. Even though the chair availability is consid-ered, the acuity levels and nurse availabilities are not con-sidered by the scheduler due to either no expertise in in-fusion nursing or no software capability to consider chairand nurse availabilities and treatment acuity levels at thesame time. Figure 6 shows the planning/scheduling processperformed by the scheduler.

After the schedules are made, the charge nurse in theinfusion clinic assigns nurses to patients based on acuitylevels, nurse skills, nurse working hours, and patient pref-erences. Nurse assignment is performed the day before theclinic session starts.

5.2. Experimental settings

The main input to the planning and scheduling problem ispatient mix, that is, the number of patients, their prescribedtreatment regimens, and their acuity levels. The cancer typeand the treatment regimen are determined randomly. Thecancer type is generated according to the expected numberof new cases (American Cancer Society, 2010), and per-centage of patients receiving chemotherapy (Warren et al.,

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Planning and scheduling chemotherapy 43

Fig. 6. Planning and scheduling algorithm in current practice. (Color figure available online.)

2008). We considered four cancer types with highest inci-dence rates (lung, breast, colorectal, and prostate). Table 5shows the percentage of expected new cases over all newcases for each cancer type. The percentage of the patientsreceiving chemotherapy is based on the values for 2002 pre-sented by Warren et al. (2008), who used SEER-Medicaredata for 306,709 persons aged 65 and older and diagnosedwith breast, lung, colorectal, or prostate cancer.

When new patients are generated, the percentages in thelast column are used. Thus, the probability that the patientwill have lung cancer is 0.4184. The probabilities for breast,prostate and colorectal are 0.2540, 0.0717 and 0.2560, re-spectively. The treatment protocol of the patient is deter-mined according to the cancer type. Fifty-nine treatmentprotocols obtained from NCCN website (2009) are usedin our computations. Each protocol has equal probabil-ity of being selected. The cycle length, number of cycles,treatment days, and treatment times for each treatment daychange between 7–28 days, 1–6 cycles, 1–5 days, and 30–480minutes, respectively.

The second important parameter is the acuity level ofeach chemotherapy regimen. None of the existing studiesprovide a comprehensive list of acuity levels for treat-ment regimens. In our experiment, we used three acuitylevels as in Chabot and Fox (2005). According to theirpatient classification system, we assigned acuity levelsto the regimens according to the number of agents,

pre-medications and complexity of administration. Asthe number of drugs and treatment duration increase, theacuity level increases.

The other parameters are related to the clinic environ-ment. We consider an infusion clinic with 20 chairs and7 nurses. The Poisson distribution with mean five is usedto generate the number of new patients per day. The plan-ning model assigns approximately 53.5 patients to each day(as explained in Section 5.3). A nurse sees approximately7–8 patients per day. The problem size is comparable tothe ones reported in the literature (see Table 1 in Section2.2). The normal working hours are 8:00 am – 4:00 pm (Ht= 480 minutes). If the total completion time of all treat-ments exceeds eight hours, overtime cost is incurred. Theslot length is chosen as 30 minutes and number of slots fornormal working hours is 16. The clinic is assumed closedon weekends.

5.3. Planning

The proposed planning and scheduling methods and thecurrent practice are coded in C++. The callable librariesof Cplex 12.0 are used to solve the integer programmingmodels. All computations are performed in a personalcomputer with 2 GHz CPU and 2 GB memory. In ourcomputations, we solve the planning problem for oneyear with a non-empty schedule at the beginning. The

Table 5. Patient mix

Percentage of Percentage of Percentage of patientsnew cases patients receiving Average treatment in the patient mix

Cancer (American Cancer chemotherapy Number of duration per day (14.83 , 35.3 + 13.13 , 24.2 +type Society, 2010) (Warren et al., 2008) regimens (min, max) 13.00 , 6.9 + 10.93 , 29.3 = 1250)

Lung 14.83% 35.3% 11 74 (15,240) 14.83 , 35.3/1250 = 41.84%Breast 13.13% 24.2% 39 63 (15,240) 13.13 , 24.2/1250 = 25.40%Prostate 13.00% 6.9% 4 98 (60,180) 13.00 , 6.9/1250 = 7.17%Colorectal 10.93% 29.3% 5 137 (75,255) 10.93 , 29.3/1250 = 25.60%

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44 Turkcan et al.

Fig. 7. Distribution of daily workload (total treatment duration). (Color figure available online.)

performance measures are number of patients assignedto each day, daily workload in terms of total treatmentduration, total acuity violation, and total treatment delay.

The average number of patients assigned to each dayis 53.5 for both current practice and proposed planningmethod. The standard deviation is 7.3 for current practiceand 8.6 for the proposed method. The F-test to test theequality of variances shows that variances are not equal(p-value = 0.01). The number of patients assigned to eachday can be a misleading performance measure due to highvariability in treatment durations. Figure 7 shows the distri-bution of daily workload (total treatment duration in termsof number of slots) for both methods. The average work-load per day is 185 slots (92.5 hours) for both methods.Since the fixed capacity is 320 slots (8 hours *20 chairs =160 chair hours = 320 slots), the chairs are not fully uti-lized. This is due to patient mix with high acuities. The

standard deviations of daily workload are 26.4 and 35.8for current practice and the proposed method, respectively.F-test with a p-value of less than 0.001 shows that theproposed method gives daily plans with higher variances.

The next performance measure is total acuity viola-tion. Current practice considers only chair availabilitywhile scheduling patients. Therefore, the generated sched-ules might not be feasible due to limited nurse availabil-ity. If the total acuity level assigned to a slot exceeds theavailable nurse capacity, either additional nurses are re-quired to treat the patients on time or the patients haveto wait until the nurses become available. Figure 8(a)shows the distribution of total acuity violation per dayfor current practice. The acuity violation is calculatedas

"s max(0, TotalAcuityst ) Nt Amax). The average acuity

violation per day is 94.6. Since a nurse can handle a totalacuity of at most 64 (Amax, Number of slots = 4 , 16)

Fig. 8. (a) Distribution of total acuity violation, (b) Distribution of difference between total acuity violation and under-utilizationper day. (Color figure available online.)

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Planning and scheduling chemotherapy 45

Table 6. Comparison of current practice and the proposed planning method: summary of the results

Performance measure Current practice Proposed method F-test (or t-test) Significant difference?

Number of patients seen per day Mean 53.5 53.5Std. dev. 7.3 8.6 p < 0.01 Yes

Total workload per day (hours) Mean 92.5 92.5Std. dev. 26.4 35.8 p < 0.001 Yes

Acuity violation per day Mean 94.6 No violationAcuity violation > underutilization Mean 76 days No violationTreatment delay (days) Mean 4.0 3.1 p = 0.14 No

Range 0–45 0–19Std. dev. 7.7 4.1 p < 0.001 Yes

in 8 hours, 1.5 additional nurse FTE is required to handlethe workload. If the salary of an oncology nurse is $40,000per year, then the annual cost of additional nurses would be$60,000 (1.5FTE , 40, 000/FTE). Even though total treat-ment time is much less than the available chair capacity, theacuity violation shows that the nurse capacity is exceededby current practice. In contrast, the proposed method doesnot lead to acuity violation, since this is stated explicitly asa constraint.

Another important point that should be noted is thatthe workload throughout the day generated by the currentpractice shows high variability. While there are time slotswith high acuity, there are underutilized time slots as well.In order to see if the underutilized time slots can be enoughto accommodate the workload of overloaded slots, weshould look at the difference between the total acuity viola-tion and underutilization. Figure 8(b) shows the histogramof these differences. In 76 days out of 260 days (29%), theunderutilized time slots are not enough to accommodatethe workload in overloaded time slots. This means currentpractice cannot generate feasible schedules (schedules free

of acuity violation) unless additional resources and/orovertime are used. If no additional resources are used,the schedules generated by current practice will causeadditional patient waiting times or safety problems due toheavy workload assigned to nurses. Since nurses have toperform several tasks such as assessment, education, IVaccess, and monitoring, the patients will have to wait whilethe nurse is completing the tasks for other patients. Theheavy workload will increase the nurse burnout, whichmay increase the risk of making safety errors.

The objective of the planning model is to minimize thetotal treatment delay. Figure 9 shows the histograms oftreatment delays per day. The treatment delay for a day iscalculated for all patients planned on that day. The treat-ment delay for new patients is calculated as (esti ) tXit),where esti is the earliest start time for patient i and tXit is theassigned treatment start time. The treatment delay for theexisting patients is calculated as (tXit + ( f ) 1)Ci ) tX f

it),where (tXit + ( f ) 1)Ci ) is the desired treatment start timeof cycle f and tX f

it is the actual start time of cycle f . Thehistograms show that proposed method can start patient

Fig. 9. Distribution of treatment delay per day for (a) current practice and (b) the proposed method. (Color figure available online.)

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Fig. 10. Chair and nurse utilization per slot (average of 50 schedules). (Color figure available online.)

treatments at their earliest start time on 133 days (51% ofdays). The current practice can do that for only 103 days(40% of days). The treatment delays range between 0 and19 days for the proposed method, and between 0 and 45days for the current practice. The average treatment de-lay is 4.0 days with current practice and 3.1 days for theproposed method. The standard deviation is 7.7 and 4.1for current practice and the proposed method, respectively.Although the difference between the mean treatment delaysis not statistically significant (the p-value of paired t-test is0.14), the variance of treatment delays is higher for thecurrent practice (the p-value of the F-test is < 0.001). Webelieve the main reason for high variability in treatmentdelays among days is that the proposed method plans thetreatments for all cycles at the beginning, which does notcause any other delay during the treatment. The proposedmethod also plans several patients simultaneously, whichgives a smaller treatment delay compared to planning ofpatient treatments one by one.

To summarize these findings, the proposed method gen-erates plans with greater variance in total daily treatmenttime than does current practice. This is due to the fact thatcurrent practice does not consider limits on daily acuity,whereas the proposed method generates plans with no acu-ity violation. Thus, while total treatment time from day today is more consistent under current practice, nurses aremore often overloaded with too much acuity assignment.That is, they are handling too many complicated tasks atthe same time, which has implications for patient safety.In fact, protecting nurses and patients from acuity over-load seems to come at the price of more variation in totaldaily treatment time. On a final note, the proposed methodprovides smaller average treatment delay as well as smallervariance and maximum value of treatment delay, whichbodes well for healthier patient outcomes.

5.4. Scheduling

It is important to verify that the generated plans canbe used to find feasible daily schedules. We solve thescheduling problem using the proposed heuristic andinteger programming model, I P3. We compare the dailyschedules generated by the proposed integer programmingmodel and current practice. The performance measuresused for comparison are chair utilization, total acuity perslot, completion times (Cmax), and computation times.

The scheduling problem is solved for 50 days. Figure 10shows the number of chairs occupied and total acuity perslot for current practice and the proposed integer program-ming method, IP3. Since both methods consider chair avail-ability, the number of patients at each slot does not ex-ceed the total number of chairs. However, as is clear, theproposed method provides much better control over nurseworkload throughout the day by guaranteeing that assignedacuity level does not exceed the required threshold. Undercurrent practice, the charge nurse must either request ad-ditional nurses to treat all patients on time, or the patientsshould wait until the nurses become available to safely de-liver the chemotherapy.

We compare the proposed integer programming modeland the heuristic in terms of treatment completion timesfor each day. Figure 11 shows the histogram of thedifference between completion times found by ALTT andI P3. The proposed integer programming model gives thesame results as the proposed heuristic for 14 days (28%),and gives better results for 36 days (72%). Paired t-test(with a p-value <0.001) shows that the difference betweenthe methods is significant. The average difference betweenthe completion times is 0.97 slots, which corresponds to29.1 minutes. Since there are seven nurses in the clinic, theimprovement in total nurse time is 3.4 hours. If the demandfor chemotherapy treatment is high, more patients can be

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Planning and scheduling chemotherapy 47

Fig. 11. Difference between the completion times found by ALTTand IP3. (Color figure available online.)

treated in that 3.4 hours. If one more treatment (with atreatment duration of less than 3.4 hours) can be added toeach day, 260 more treatments can be performed in a year,which improves the access to care.

Since IP3 considers just nurse availability, we check theschedules in terms of chair feasibility. The maximum num-ber of chairs is 18, which means all schedules are chairfeasible. This may not be the case in other infusion clin-ics where there is limited space for the chairs. In that case,IP2 that considers both chair and nurse availability shouldbe solved or clinic managers should reduce the number ofnurses and solve IP3 again.

When we look at the computation times, IP3 finds anoptimal schedule in 13 minutes on the average. The pro-posed heuristic finds solutions in less than one second. Togive an idea about the computation times of IP2, we solvedfive problems with a time limit of one hour. IP2 could notfind any optimal solution within the one-hour time limit.The percentage gap between the lower bound and the bestfeasible solution was 40% at the end of one hour. All theseproblems were solved optimally by IP3 within 20 minutes,and the average computation time was 7.5 minutes. If thecomputation times are important or an optimization soft-ware is not available, then proposed heuristic that considersboth chair and nurse availabilities can be used to find betterschedules than current practice.

5.5. Recommendations for implementing optimization andheuristic methods

Due to high variability in clinic practices, we would liketo make some recommendations to clinic managers aboutwhen and how the proposed methods can be used.

1. If there is a patient waiting list and reducing the wait-ing times for treatment is important for the clinic, thenthe proposed optimization-based planning method canreduce the treatment delays by planning several patienttreatments simultaneously rather than planning each pa-tient’s treatment one by one. (In our experiments, thereduction in treatment delay was on average one day.This improvement was achieved by just considering onthe average five patients (which is equal to the numberof new patient arrivals per day) simultaneously insteadof one patient at a time, and planning all treatmentsinstead of only one cycle at a time.)

2. The proposed rolling horizon approach is flexible in thesense that it can be solved as frequently as needed (ev-ery day or every week). That means, if an urgent patientarrives, the treatment plan can be prepared without wait-ing for the next planning horizon.

3. The importance of treatment delays (wid ) can be deter-mined by oncologists. The stage of the disease, invasive-ness of the tumor, and patients health condition are thefactors that might be important in determining wid .

4. The decision to use one of the scheduling methods (IP2,IP3, or ALTT) should be based on the patient mix andavailable computational resources. If the patients havehigh acuities, IP3 should be used because nurses wouldbe the limiting resource. If the patients have low acuities,then IP2 that considers both chair and nurse availabili-ties should be used. If the computation time is importantand/or an optimization package is not available, thenALTT can be used.

5. Time studies or nurse experience can be used to deter-mine acuity levels for each treatment. Treatment acuitylevels can be adjusted after the first treatment accordingto difficulty and workload the nurse experiences. Thecharge nurses who do the nurse assignments can deter-mine the maximum acuity level a nurse can handle safelyat any given time.

6. The computational results show that there is high vari-ability in treatment completion times among days. Inorder to reduce the variability in completion times, werecommend the clinic managers to adjust the numberof nurses by solving the scheduling problem with differ-ent number of nurses to have similar completion timesamong different days. Another way is to add workloadbalancing as an another objective to the planning model.We would like to refer the reader to the studies by Sadkiet al. (2010a,b) how workload balancing can reduce thevariability between days. We are currently developingplanning models to find a more balanced workload (interms of treatment duration and total acuity) amongdays while keeping the treatment delays low.

7. If staggered nurse schedules are used, the proposedscheduling methods can easily handle nurse workinghours with additional constraints.

8. The proposed scheduling methods assume that the ap-pointment times and nurse/chair assignments are made

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once all patients assigned to a day are known. However,it is not practical for the clinic to wait until the last day toschedule all the patients. Therefore, a cut-off point likeone week can be determined by the clinic. All patientsthat are assigned to come on a day can be scheduled oneweek before the appointment day. If new patients arriveafter the schedule is generated, they can be added tothe existing schedule by using the proposed schedulingmethods.

6. Concluding remarks and future research

In oncology clinics, the scheduling of treatments (surgery,radiotherapy, chemotherapy, etc.) for cancer patients is veryimportant in delivering the right care at the right time. Weconsider planning and scheduling problems for chemother-apy patients. Chemotherapy drugs are given to patients onseveral days with the objective of minimizing the numberof cancer cells while sparing the normal cells. Adherenceto chemotherapy protocols is crucial in achieving the besthealth outcomes. Our aim is to achieve the best health out-comes by minimizing the treatment delays due to limitedresources.

Most previous studies propose using scheduling tem-plates/rules based on nursing or pharmacy times. Thescheduling decisions are made in an ad-hoc manner accord-ing to physician and scheduler experiences and patient pref-erences. To the best of our knowledge, there is no study thatproposes optimization methods to schedule chemotherapytreatments considering acuity levels and optimizing sev-eral objectives such as minimization of treatment delay,minimization of staff overtime and under-utilization, andmaximization of staff utilization.

We proposed a two-stage approach to solve the plan-ning and scheduling problems sequentially. Integer pro-gramming models are proposed to solve these problems. Arolling horizon approach is used to schedule new patientswho are referred to receive chemotherapy. A heuristic andan integer programming model are proposed to reduce thecomputation times for the scheduling problem. The com-putational results show that the planning and schedulingproblems can be solved in reasonable times. The proposedplanning and scheduling models can be used as a decisionmaking tool in determining the optimal staffing levels.

In a real clinic environment, there are many uncertaintiessuch as delays in getting lab results, cancellations, add-onpatients, and variability in treatment durations, that affectthe daily performance. Delays increase the patient waitingtimes and affect the clinic flow. Most of the cancellationsoccur on the same day after the lab results are performedand the patient is seen by the oncologist. Add-ons mightoccur within the last few days. One future research areais developing stochastic planning and scheduling methodsthat consider uncertain treatment durations, cancellationsand add-ons. One drawback of the proposed deterministic

planning model is the high variability in workload amongdays. As a future research, the workload balancing objectivecan also be added to the stochastic planning model to findplans with a more balanced workload among days whilekeeping the treatment delays low.

There are several clinics where patients are scheduledaccording to chair availability and nurse staffing/nurse as-signments are performed after the schedule is generated.Nurse assignment is either performed by the charge nurseaccording to the patient acuities, nurse skills, nurse work-ing hours, and patient and nurse preferences, or the nurseassignment is done in real time where the arriving patient isassigned to the nurse who has the least number of patients.Even though we mentioned about nurse assignment whenwe explained the current practice in Section 5.1, we didnot use any staffing method to find feasible nurse assign-ments for a given schedule. Another future research areais developing nurse assignment/staffing models that incor-porate patient acuities, nurse skills, and patient and nursepreferences.

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