stochastic online appointment scheduling of multi...

Health Care Manag SciDOI 10.1007/s10729-013-9224-4

Stochastic online appointment scheduling of multi-stepsequential procedures in nuclear medicine

Eduardo Perez · Lewis Ntaimo · Cesar O. Malave ·Carla Bailey · Peter McCormack

Received: 29 September 2012 / Accepted: 14 February 2013© Springer Science+Business Media New York 2013

Abstract The increased demand for medical diagnosis pro-cedures has been recognized as one of the contributors tothe rise of health care costs in the U.S. in the last fewyears. Nuclear medicine is a subspecialty of radiology thatuses advanced technology and radiopharmaceuticals for thediagnosis and treatment of medical conditions. Proceduresin nuclear medicine require the use of radiopharmaceuti-cals, are multi-step, and have to be performed under stricttime window constraints. These characteristics make thescheduling of patients and resources in nuclear medicinechallenging. In this work, we derive a stochastic onlinescheduling algorithm for patient and resource schedulingin nuclear medicine departments which take into accountthe time constraints imposed by the decay of the radio-pharmaceuticals and the stochastic nature of the systemwhen scheduling patients. We report on a computationalstudy of the new methodology applied to a real clinic.We use both patient and clinic performance measures inour study. The results show that the new method sched-ules about 600 more patients per year on average than ascheduling policy that was used in practice by improving theway limited resources are managed at the clinic. The new

E. Perez (�)Ingram School of Engineering, Texas State University,601 University Drive, San Marcos, TX 78666, USAe-mail: [email protected]

L. Ntaimo · C. O. MalaveDepartment of Industrial and Systems Engineering,Texas A&M University,3131 TAMU, College Station, TX 77843, USA

C. Bailey · P. McCormackScott and White Clinic,2401 S. 31st Street, Temple TX 76508, USA

methodology finds the best start time and resources to beused for each appointment. Furthermore, the new methoddecreases patient waiting time for an appointment by abouttwo days on average.

Keywords Health care · Nuclear medicine · Patientservice · Online scheduling · Stochastic programming

Mathematics Subject Classifications (2010) 90B99 ·68M20 · 68W27

1 Introduction

In this paper, we derive a stochastic online schedul-ing methodology for patient and resource management innuclear medicine clinics. Nuclear medicine is a branchof medical imaging that uses small amounts of radioac-tive materials to diagnose and treat a variety of dis-eases, including, many types of cancers, heart diseasesand certain other abnormalities within the body. Thecost of advance imaging procedures has grown dispropor-tionately compared with the overall cost of health careand has become a major factor in the U.S. governmentexpenses [8]. According to [28], nuclear procedures wereused most extensively in the United States in 2007, with1,000 or more procedures performed per 100,000 people.Suthummanon et al. [25] showed that machine time, directlabor time, and radiopharmaceuticals account for mostof the cost per procedure in nuclear medicine. Also, theway resources are managed has a direct impact in thequality of service. The increased demand and the com-plexity in the procedure protocols make the scheduling ofpatients and resources in nuclear medicine a challengingproblem.

mailto:[email protected]

E. Perez et al.

Nuclear medicine procedures require the administrationof a radiopharmaceutical to the patient and involve multi-ple sequential steps. Each procedure step requires severalresources such as technologists, nurses, gamma cameras,and sometimes a treadmill. Radiopharmaceuticals allow forthe imaging (scans) or treatment of a specific organ of thehuman body. Radiopharmaceuticals are prepared in radio-pharmacies outside nuclear medicine clinics. They have tobemanaged carefullyand theirdelivery requirewell-plannedlead time. The short half-life of the radiopharmaceuticalsimposes strict time constraints on scheduling patients andresources. To successfully complete a procedure, every stephas to be initiated and completed within a specific time win-dow. If the procedure protocol is not followed, a poor scanwill result causing poor utilization of resources and patientrescheduling. A scan could last from minutes to hours and aprocedure may require multiple scans in a day.

Some of the resources needed to perform nuclearmedicine procedures include technologists, radiopharma-ceuticals, gamma cameras, and sometimes a treadmill, anurse or an EKG technician. Gamma cameras take advan-tage of radiopharmaceutical properties to capture imagesfrom within the human body. The cost of these cameras canbe as high as a $ 1 million. Since the gamma cameras canbe very expensive, they have to be utilized most of the time.Resources required to serve a patient must be available atthe time they are scheduled. Each procedure step requiresthe scheduling of a pair of resources, a human resource anda clinic room or station. A member of the staff is alwaysrequired to operate equipment in a station and to take careof patients.

Patient requests in nuclear medicine arrive during the dayas the scheduling proceeds. Requests are handled as theyarrive and appointments are provided in an online fashionwithout taking into consideration future requests. Findingways to improve patient service is of great interest fornuclear medicine managers. The characteristics of this prob-lem make it unique with very limited research reported inthe literature. Stochastic planning techniques are an alterna-tive to address this problem. In this paper, a stochastic onlinescheduling algorithm for patient and resource managementin nuclear medicine clinics is derived. The algorithm usesclinic historical information to make more informed deci-sions when selecting an appointment for the patient requeston hand.

Conceptually, historical data regarding patient appoint-ment requests and arrivals to the clinic can be incorpo-rated into a multistage mathematical program for optimalscheduling plans. However, such optimization problemstend to be very large even for a medium size clinic andvery challenging to solve. In contrast, stochastic onlinealgorithms are suboptimal, but scalable ways of solvingstochastic integer programs.

The contributions of this paper include new mathemat-ical models for scheduling nuclear medicine procedures,a stochastic online algorithm for patient and resourcescheduling, and a new methodology for scheduling medicalmulti-step sequential procedures that consider both patientand clinic perspectives. The new methodology takes intoaccount the time constraints imposed by the decay of radio-pharmaceuticals and the stochastic nature of the system interms of the future procedure requests happening after thecurrent patient. These contributions will help the practice ofnuclear medicine by providing shorter patient waiting times,increased patient throughput, minimal delays in radiophar-maceutical delivery, higher utilization of resources, andultimately lower health care costs.

The rest of the paper is organized as follows: Section 2provides a review of the existing literature for this problem.The nuclear medicine scheduling problem is described inSection 3. Section 4 provides a description of the schedul-ing problem and presents the stochastic online algorithm.In Section 5 we describe the setting where the algorithm isapplied and the experimental frame. A computational studyand discussion of the results is given in Section 6. Thepaper ends with some concluding remarks and directions forfuture research in Section 7.

2 Related work

Patient scheduling has been studied widely in the pastthirty years. The major problem is to find the best appoint-ment rule, which is the best algorithm that specifies theappointment interval lengths and resources to be used [6].Medical facilities dedicated to the diagnosis and treatmentof patients are classified as specialty clinics. Radiology, CTscan, and magnetic resonance imaging (MRI) clinics arewithin this group. Nuclear medicine is a sub-specialty ofradiology. Procedures in nuclear medicine differ from mostother imaging modalities in that they show the physiolog-ical function of the part of the body being investigated. Inaddition, these procedures rely on the process of radioactivedecay, involve several steps, and multiple resources makingthem difficult to manage.

The existing literature on the nuclear medicine schedul-ing problem is limited. Most of the existing published workfor specialty clinics focuses in scheduling CT scan and MRIprocedures, which are relatively less difficult to managesince they do not consider multi-step sequential procedures.Also, most of the existing literature assumes that a poolof patients is always available at the beginning of the day.However, in nuclear medicine patients arrive in an onlinefashion and are scheduled one at a time.

Green et al. [10] address the problem of scheduling ran-domly arriving patients of different types in an MRI facility.

Stochastic online appointment scheduling of multi-step sequential procedures in nuclear medicine

They formulate the problem as a finite-horizon dynamicprogram for a clinic schedule that allows at most one patientper period and a single server, where only one patient can beserved at a time. The authors derive properties of the opti-mal scheduling policies and identify a service sequence thatminimize the expected total cost of serving patients. Patricket al. [19] study a similar problem but they characterizepatients with different priorities. Kolisch and Sickinger [15]consider a similar problem but with two servers. The authorscompare the linear capacity allocation (LCA), first-comefirst-served (FCFS), and random selection (RS) decisionand conclude that the FCFS provides a better performance.

Patrick and Puterman [18] consider the problem ofscheduling patients in a CT scan clinic. They formulatean optimization problem that returns a reservation policythat minimizes the non-utilization of resources subject to anovertime constraint. Their approach assumes the availabilityof a pool of patients that can be called to occupy unused timeslots. The authors use simulation to demonstrate a reductionin outpatient waiting time.

Sickinger and Kolisch [23] propose a generalization ofthe Bailey-Welch rule as well as a neighborhood searchheuristic for a medical service facility with two servers. TheBailey-Welch rule states that for one server clinic the bestperformance in terms of patient waiting and server idle timeis to schedule two patients for the first appointment spaceand one patient on the ones that follow. The authors ana-lyze the impact of different problem parameters on the totalreward.

Standridge and Steward [24] propose a simulation modelthat includes a control logic for patient scheduling. Thesystem presented by the authors schedules patients withina simulation framework. Vermeulen et al. [27] devise anadaptive approach to automatic optimization of resourcecalendars in a CT scan clinic. They implement a simulationmodel for a case study analysis to demonstrate that theirapproach makes efficient use of resources’ capacity. Perez etal. [20] develop a discrete event specification (DEVS) sim-ulation model for a nuclear medicine facilities. The authorspropose several scheduling algorithms that suggest differentsystem configurations for serving patients.

The benefits of taking into account future events whenoptimizing decision processes are reported in the litera-ture. It has been shown that using additional stochasticinformation can improve the quality of solutions in schedul-ing applications such as: dynamic vehicle routing [2, 4,5], packet scheduling [2, 3, 7, 12], reservation systems[26], inventory management [5, 13], organ transplants [1],surgery scheduling [9, 16], and elevator dispatching [17].In these applications, stochastic information is used in dif-ferent ways; however, the unifying theme seen throughoutthe research is that there are considerable advantages whenstochastic information is taken into account.

The common strategy is to predict the future requestsusing a statistical model by sampling the observations onthe history. Chang et al. [7] study the multiclass packet-network scheduling problem. The authors use a HiddenMarkov Model (HMM) to generate the tasks arrivals foreach class with a particular weight and develop the expec-tation algorithm which adapts an optimal offline algorithminto an online algorithm by sampling possible future taskssequences from the HMM. The expectation algorithm hassome resemblances to the sample average approximationmethod for non-dynamic stochastic programming [14, 22]where the solution depends of a deterministic part and a sto-chastic part. The deterministic part gives the immediate planand the stochastic part gives a penalty for changing the planto accommodate the best as possible the scenario that hasbecome reality. One must average over many scenarios tofind the best expected solution. Unfortunately, the expecta-tion algorithm does not perform well under time constraints,since it must distribute its available optimizations across allrequests.

This issue was recognized and addressed in [3] wherea consensus algorithm was proposed. The consensusalgorithm solves as many samples as possible to select therequest which is chosen most often in the sample solu-tions at time t. This algorithm was shown to outperformthe expectation algorithm for the online packet schedul-ing problem under time constraints. However, as decisiontime increases, the quality of the consensus algorithm lev-els off and is eventually outperformed by the expectationalgorithm. The regret algorithm proposed in [2] combinesthe features of both expectation and consensus algorithms.The regret algorithm assess every decision on all sam-ples but has the ability to avoid distributing the samplesamong decisions. Stochastic online optimization assumesa known distribution of future requests, or an approxima-tion thereof, is available for sampling [3]. The typical caseis the existence of either historical data or predictive mod-els. Decisions made using stochastic online optimization areusually constrained by time, meaning that there is a limitedtime to report a solution for each request to the decisionmaker.

Awasthi and Sandholm [1] consider the scheduling ofhuman kidney transplants using a stochastic online frame-work. They propose an adaptation of the regret algorithm.Van Hentenryck et al. [26] consider the online stochasticreservation problem where the goal is to allocate requeststhat are received online to a limited group of resources inorder to maximize profit. The authors adapted the consensusand regret algorithms for their problem. Their modificationof the regret algorithm is based on a constant sub-optimalityapproximation of multi-knapsack problem. The authorsused two black-boxes to handle the stochastic arrivals ofreservation requests for hotel rooms. One black-box is the

E. Perez et al.

sub-optimal approximation module and the other is the sam-pling module, which relies on the observations of the pastarrivals.

This research differs from earlier studies in multipleways. First, appointment schedules deal with radiopharma-ceutical lead times and require multiple resources at eachstep of the procedure. Second, nuclear medicine proceduresfollow sequential protocols that impose strict time-windowconstraints on the starting of each step. These characteristicsmake scheduling patients in nuclear medicine clinic a newchallenging problem. In fact Gupta and Denton [11] identi-fied the problem of scheduling patients in constrained healthcare clinics as an open research challenge.

3 Problem definition and notation

To define the problem, we first state some assumptions. Theoperation time in a day is divided into time slots of equallength. Patients arriving to the clinic have an appointmentand that patient appointment times coincide with the begin-ning of a time slot. It is also assumed that patients show fortheir appointment most of the time.

Let S denote the set of stations inside the clinic andR be the set of human resources. Stations contain at leastone type of equipment and are classified based on the typeof equipment they have. Equipment include gamma cam-eras and treadmills. Human resources include physicians,technologists, nurses, and a manager. Each human resourceis responsible for a specific set of tasks. The number oftasks and the time required to complete them depend on thehuman resource experience and expertise. Table 1 lists someof the tasks performed by the human resources.

In nuclear medicine, patient appointments are scheduledby the patient’s primary physician. A procedure requestinvolves two pieces of information: a procedure type fromset P and a patient preferred day for the appointment q.Patient appointments are given by the clinic at the timethe request is received without taking into account requestsarriving later in the day.

Table 1 Human resources tasks

Task Technologist Nurse Physician Manager

Hydrate patient X X X X

Radiopharmaceutical X X X

preparation

Imaging X

Draw doses X X

Radiopharmaceutical X X

administration

There are more than 60 procedures in nuclear medicine,each of them having a protocol with a finite number of steps.Table 2 list some of the procedures performed in nuclearmedicine with their current procedural terminology (CPT)codes. Let A denote the set of radiopharmaceuticals and atleast one radiopharmaceutical from this set is required pernuclear medicine procedure. Radiopharmaceuticals needto be at the clinic by the time of the patient appoint-ment. Therefore, they are requested in advance allowinga lead time for preparation and delivery. Each procedurep ∈ P has a finite number of tasks/steps denoted bynp . Let βkp define the duration of step k for procedurep. Then the total duration of the procedure is

∑np

k=1 βkp .The rest of the notation required for our problem is listedin Table 3.

Table 4 lists the steps for the MSC-bone imaging proce-dure (CPT 78315). It has an average total duration of 245minutes. Observe that the procedure has four steps and eachstep has different requirements in terms of time duration,stations, and human resources. For instance, step 1 of theprocedure takes about 20 minutes on average and requiresone station and one human resource from those listed in thetable. Let Rkp be the set of human resources qualified toperform step k of procedure p. Similarly, let Skp be the setof stations where step k of procedure p is performed.

Figure 1 illustrates a typical nuclear medicine sched-ule for two different nuclear medicine procedures. Twoprocedure protocols are used in the example. For illus-tration purposes only a limited number of resources areused in the example. In addition, procedures are sched-uled as they arrive using an ‘as soon as possible’ strategy.The first procedure listed is the MSC-bone imaging (CPT78815) presented in Table 4. This procedure is scheduledat the beginning of the day and is represented by the white

Table 2 Examples of nuclear medicine procedures

CPT Code Name

78465 Cardiovascular Event (CVE)

Myocardial Imaging (SP-M)

78815 Positron Emission Tomography (PET)/

Computed Tomography (CT)

78306 MSB-Bone Imaging (Whole Body)

78315 MSC-Bone Imaging (Three Phase)

78223 GIC-Hepatobiliary Imaging

78472 CVJ-Cardiac Blood Pool

78585 REB-Pulm Perfusion / Ventilation

78006 ENC-Thyroid Imaging

78195 HEE-Lymphatic Imaging

78464 CVD-Myocardial Imaging (SP-R ORS)


Table 3 Scheduling problemsets and parameters Sets

J : Set of patients, indexed j

P : Set if procedures, indexed p

Jp: Set of patients requesting procedure p, indexed j

T : Set of time periods t

H : Set of days, indexed h

I : Set of resources, indexed i

S : Set of stations, indexed s

R : Set of human resources, indexed r

A : Set of radiopharmaceuticals, indexed a

Skp : Set of stations where step k of procedure p can be performed

Rkp : Set of human resources qualified to perform step k of procedure p

Ikp : Set of resources that can be used to perform step k of procedure p, Ikp = {Rkp ∪ Skp}Litj : Set of appointment star times that require the use of resource i at time-slot t for patient j

Kitj : Set of procedure steps that require the use of resource i at time-slot t for patient j

Tij : Set of time-slots where resource i could be used to serve patient j

Taj : Set of time-slots where radiopharmaceutical a could be used to serve patient j

Lj : Set of day and time slot pairs, (d, t), for patient j.

Ur : Set of day and time slot pairs, (d, t), for human resource r schedule.

Vs : Set of day and time slot pairs, (d, t), for station s schedule.

Parameters

i : Subscript, for the i resource

j : Subscript, for the j patient

a : Subscript, for the a radiopharmaceutical

p : Subscript, for the p procedure

k : Subscript, for the k step of a procedure

� : Subscript, for the � starting time-slot for a patient appointment

t : Subscript, for the t time-slot, incremental time

τ : Total number of time-slots in a day, indexed t, . . . , τ

βkp : Number of time-slots required to complete step k of procedure p

np : Total number of steps for procedure p, indexed k, . . . , np

ρ : Parameter representing resource r or station s

δp : total duration of procedure p

ω : Number of days in a week

μ : Number of days in a month

m : Number of days before arrival of radiopharmaceutical after placing order

q : Day of the week requested by patient, indexed q = 1, . . . ,5, where

1=Monday, 2=Tuesday, 3=Wednesday, 4= Thursday, 5=Friday

Table 4 Procedure 78315: MSC-bone imaging (three phase)

Step 1 Step 2 Step 3 Step 4

Activity Radiopharmaceutical Administration Scan Acquisition Patient Wait Scan Acquisition

Average Time (mins.) 20 15 165 45

Station Axis; P2000; Meridian; TRT Axis; P2000; Meridian Waiting Axis; P3000; Meridian

Human Resource Technologists; Nurse; Manager Technologists; Manager None Technologists; Manager

E. Perez et al.

4

4

250190

200210

220230

130140

150160

170180

70 80 90 100

Treadmill 2

Waiting 3

110120

10 20 30 40 50

Waiting 3

60

Technologist 2 1 4

EKG Technologist

2

Nurse 1

TRT 1 1

Axis 2 4

schedule for the firstrequest (CPT 78815)

schedule for the secondrequest (CPT 78465)

time = 20s = TRTr = nurse

1 2 3 4time = 15s = axisr = technologist

time = 150waiting

time = 45s = axisr = technologist

time = 5s = TRTr = technologist

1 2 3 4time = 30s = treadmillr = EKG tech

time =60waiting

time = 30s = axisr = technologist

Procedure CPT 78815

Procedure CPT 78465

Fig. 1 Example schedule for two nuclear medicine procedures scheduled as early as possible

bars. Since some of the resources required for the sec-ond procedure are already occupied at the beginning ofthe day, the second procedure, in gray, is scheduled laterin the day. As more procedure requests arrive, managingresources in nuclear medicine clinics become very chal-lenging. For instance, in this example no other procedurecan be added to the schedule without overlapping. It isimportant to notice that most of the resources are not fullyutilized.

A set of measures are used to quantify the performanceof the scheduling algorithms. These performance measuresare defined using both patient and clinic perspectives. Thefive performance measures selected were identified as thosecommonly used in literature and they are summarized inTable 5.

Table 5 Performance measures

Name Description Viewpoint

Waiting Waiting time from the time of Patient

time type 1 the request until the time of

the appointment

Preference Number of times patients are Patient

ratio scheduled on the date requested

above all requests

Equipment The amount of time an Clinic

utilization equipment is used during

operating hours

Human resource The amount of time a human Clinic

utilization resource is used during

operating hours

Patient Number of patients Clinic

throughput served per day

4 Scheduling problem

We will now describe and formulate the problem of schedul-ing patients and resources in nuclear medicine clinics usingthree approaches; namely, offline, online, and stochasticonline. We start with the offline scheduling approach inSection 4.1. In offline scheduling, the problem is solvedon a day by day basis and it is assumed that all requestsfor the day are known in advance. The problem is mod-eled using integer programming (IP). We present theonline version of the problem in Section 4.2. Unlikeoffline scheduling, in online scheduling requests arrivesequentially one at a time and scheduling decisions aremade when the request is received. For the online ver-sion of the problem, we devise an online framework forscheduling patients and resources. Lastly, in Section 4.3,we discuss the stochastic online version of the problem.Similar to online scheduling requests arrive sequentiallyone at a time. However, possible future requests are nowtaken into account when making scheduling decisions.Consequently, we formulate the problem using stochasticinteger programming (SIP) to deal with the uncertaintywithin the online framework.

4.1 The offline problem

In the offline scheduling problem we assume a finite num-ber of days in the scheduling horizon and require eachresource i ∈ I to have a schedule that contains τ numberof time-slots per day. In the offline version of the problem,it is assumed that all patient requests for the day are knownin advance. Thus, patient appointments are provided bytaking into account all the requests for the day. A set ofpatient requests (Jp) is used as input to the model. Each


patient requests a procedure type p. The goal is to schedulea subset B ⊆ Jp of patients that satisfies the problem-specific constraints and maximize the objective function.Let xik

jpt� =1 if patient j requesting procedure p is scheduledto use resource i at time-slot t when procedure is started attime � for the k step of the procedure, otherwise xik

jpt�=0.

Similarly, let wikj�=1 if resource i is selected to serve patient

j in step k when procedure is started at time �, otherwisewik

j�=0. The offline problem can be formulated as an IP1 asfollows:

IP 1 : Max :∑

p∈P

∑

j∈Jp

∑

i∈S1p

∑

t∈T

∑

�∈Litj

wi1j� (1a)

s.t.∑

p∈P

∑

j∈Jp

∑

k∈Kitj

∑

�∈Litj

xikjpt� ≤ 1,

i ∈ I, t ∈ T (1b)

∑

i∈Rkp

∑

t∈T

∑

�∈Litj

wikj� ≤ 1,

p ∈ P, j ∈ Jp, k = 1, . . . , np (1c)

∑

i∈Skp

∑

t∈T

∑

�∈Litj

wikj� ≤ 1,

p ∈ P, j ∈ Jp, k = 1, . . . , np (1d)

xikjpt� − wik

j� = 0, p ∈ P, j ∈ Jp,

i ∈ Ikp, t ∈ T , � ∈ Litj , k = 1, . . . , np (1e)

∑

i∈Rkp

wikj� −

∑

i∈R(k−1)p

wi(k−1)j� = 0, p ∈ P,

j ∈ Jp, t ∈ T , � ∈ Litj , k = 1, . . . , np (1f)

∑

i∈Skp

wikj� −

∑

i∈S(k−1)p

wi(k−1)j� = 0, p ∈ P,

j ∈ Jp, t ∈ T , � ∈ Litj , k = 1, . . . , np (1g)

∑

i∈R1p

wi1j� −

∑

i∈S1p

wi1j� = 0,

p ∈ P, j ∈ Jp, t ∈ T , � ∈ Litj (1h)

xikjpt�, wik

j� ∈ {0, 1} (1i)

Model IP1 allocates a subset B of patients to availableresources so that their capacities are not exceeded. Theobjective function (1a) maximizes the number of patientsscheduled on a given day. Constraint (1b) enforces for eachpatient the time-slot by time-slot requirements for proce-dure completion, that is, at most one patient is assignedto each resource each time period. Constraint (1c) andconstraint (1d) select the staff and station, respectively, per

procedure step and decide the start-time of the appointmentfor each patient. Constraint (1e) makes sure that the sameresource is scheduled for the duration of each procedurestep. Constraint (1f) and constraint (1g) verify that the staffand stations, respectively, selected to serve a patient followthe procedure sequence protocol. Constraint (1h) matchesa station to a staff member for each step of the procedurerequested by the patient. Finally, constraint (1i) imposesbinary restrictions on the decision variables. Given thepatient request demand, IP1 can be solved using a direct IPsolver.

4.2 The online problem

In nuclear medicine, patient requests are not known inadvance, rather, they arrive sequentially one at a timeand are scheduled as they arrive. Let ξt be a patientappointment request at time t, if a sequence of requestsξ = 〈ξ1, . . . , ξt−1, ξt 〉 is revealed at different times of theday, the requests ξ1, . . . , ξt−1 are already scheduled at timet when request ξt is received. At time t the problem is todecide how to schedule request ξt by keeping all the otherpatients already scheduled fixed.

Model IP1 in Section 4.1 can be modified to sched-ule patients and resources in an online fashion. Insteadof scheduling a group of requests simultaneously, themodified version of the model will be used to schedule pro-cedure requests one at a time. The online problem can beformulated as an IP as follows:

IP 2 : Min :∑

i∈S1p

∑

t∈T

∑

�∈Litj

�wi1j� (2a)

s.t.∑

k∈Kitj

∑

�∈Litj

xikjpt� ≤ 1, i ∈ I, t ∈ T (2b)

∑

i∈Rkp

∑

t∈T

∑

�∈Litj

wikj� ≤ 1, k = 1, . . . , np (2c)

∑

i∈Skp

∑

t∈T

∑

�∈Litj

wikj� ≤ 1, k = 1, . . . , np (2d)

xikjpt� − wik

j� = 0,

i ∈ Ikp, t ∈ T , � ∈ Litj , k = 1, . . . , np (2e)

∑

i∈Rkp

wikj� −

∑

i∈R(k−1)p

wi(k−1)j� = 0,

t ∈ T , � ∈ Litj , k = 1, . . . , np (2f)

∑

i∈Skp

wikj� −

∑

i∈S(k−1)p

wi(k−1)j� = 0,

t ∈ T , � ∈ Litj , k = 1, . . . , np (2g)

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∑

i∈R1p

wi1j� −

∑

i∈S1p

wi1j� = 0, t ∈ T , � ∈ Litj (2h)

xikjpt�, wik

j� ∈ {0, 1} (2i)

Model IP2 share the same decision variables and con-straints as model IP1. The only difference between the twomodels is the objective function. The objective function (2a)minimizes the waiting time Type 1 for the patient. Since theproblem now is to schedule one patient at a time, we do notneed to maximize the throughput. Instead, we find the bestschedule for each new patient request by minimizing thewaiting time using IP2.

Model IP2 requires an online scheduling framework toprovide schedules as requests are received. Therefore, wedevise an online framework to schedule patients every timea request is received using IP2. Some notation is definedbefore describing the framework. Let Ur = {(d, t)| 1 ≤d ≤ h, 1 ≤ t ≤ τ } define a set of day and time slotpairs (d, t) for human resource r. Likewise, we define aset Vs = {(d, t)| 1 ≤ d ≤ h, 1 ≤ t ≤ τ } for sta-tion s. Both sets, Ur and Vs include all the time slotsthat are already occupied by other patients. Parameter h isused to define the total number of days in the schedulinghorizon. Lj is the set of the appointment schedule foundfor the patient. In addition, in what follows ← is used todenote an assignment. The online framework is presentednext and is named the Nuclear Medicine Online Scheduling(NMOS) algorithm. We state the steps of the algorithm inAlgorithm 1.

Algorithm 1 NMOS Algorithm

The first step of NMOS algorithm (line 1) initializes thepatient set J and the patient ID. Line 2 defines the schedul-ing horizon in terms of days and line 3 defines the numberof time periods per day. The maximum number of time peri-ods is given by τ . Method GetPatientRequest() isinvoked when a request is received at the clinic. The methodgets the patient information required to find an appoint-ment. The information includes the nuclear medicine pro-cedure p and the preferred day for an appointment q(line 4). A counter λ is initialized to zero and the timeand day of the request is assigned to the patient param-eters in line 5. Line 6 finds the earliest day to begin thesearch for an appointment α by taking into account theradiopharmaceutical lead time and the patient preferred day.The method ServeRequest() uses parameters j, pj , andα to find an appointment for the patient in line 7. Themethod builds and solves the IP2 model for day α. Thosetime slots that are already taken to serve other patients aretaken into account as follows:

1. For R ∈ I , set right hand side of constraint (1b) = 0 if(α, t) ∈ Ur .

2. For S ∈ I , set right hand side of constraint (1b) = 0 if(α, t) ∈ Vs .

IP2 has about 2,500 variables and 500 constraints, onaverage. The NMOS algorithm creates an object of typeIloCplex and use the Concert Technology modeling inter-face implemented by ILOG CPLEX to solve IP2. It takesless than a minute to solve IP2 using ILOG CPLEX. Ifno appointment is found (line 16), the algorithm consid-ers the following week in the scheduling horizon, α =α + ω. If an appointment is found, the algorithm checksif the waiting time is shorter than a month (lines 8-9).If the waiting time is longer than a month, the algo-rithm searches for an alternative appointment on a differentdate. This new search starts on day α ← dj + m + λ.If the waiting time is less than a month, the appoint-ment information is passed back to the system and to thepatient (line 13).

4.3 The stochastic online problem

The stochastic online problem is an extension of the onlineproblem described in Section 4.2. In this version of theproblem we are interested in deciding how and when toserve a procedure request by keeping previous scheduledpatient appointments fixed. However, the stochastic onlineproblem takes into account additional information whenmaking those decisions. Specifically, we take into consid-eration possible patients requests that could arrive afterthe patient request we have on hand. Taking into accountpossible future patient requests allows us to make more


informed decisions when scheduling patients and resourcesunder uncertainty. The goal is to find the best day andtime to accommodate the current request such that we canimprove the system’s performance measures. Consequently,the stochastic online problem can be formulated as a Two-Stage Stochastic Integer Programming (SIP) model. In thisformulation, “here and now” decisions are made at the firststage before the realization of uncertain data. The uncertaindata is represented by ω, which is a scenario realization ofpossible future patient requests. In the second stage, aftera realization of ω becomes known, a patient schedule forthe current request is generated by solving the appropriateoptimization problem.

Two new binary variables are defined for the second stageof the problem. For each scenario ω ∈ �, variable yikω

j ′t�=1if patient j’ is scheduled to use resource i at time-slot t andwhen the procedure is started at time � for the k step of theprocedure. Otherwise yikω

j ′t�=0. Variable zikωj� =1 if resource i

is selected to serve patient j in step k and when the procedureis started at time �. Otherwise zikω

j ′� =0. SIP is formulated asfollows:

SIP : Min :∑

i∈S1p

∑

t∈T

∑

�∈Litj

� wi1j� −E[Q(x, ω)] (3a)

s.t.∑

k∈Kitj

∑

�∈Litj

xikjpt� ≤ 1, i ∈ I, t ∈ T (3b)

∑

i∈Rkp

∑

t∈T

∑

�∈Litj

wikj� ≤ 1, k = 1, . . . , np (3c)

∑

i∈Skp

∑

t∈T

∑

�∈Litj

wikj� ≤ 1, k = 1, . . . , np (3d)

xikjpt� − wik

j� = 0,

i ∈ Ikp, t ∈ T , � ∈ Litj , k = 1, . . . , np (3e)

∑

i∈Rkp

wikj� −

∑

i∈R(k−1)p

wi(k−1)j� = 0,

t ∈ T , � ∈ Litj , k = 1, . . . , np (3f)

∑

i∈Skp

wikj� −

∑

i∈S(k−1)p

wi(k−1)j� = 0,

t ∈ T , � ∈ Litj , k = 1, . . . , np (3g)

∑

i∈R1p

wi1j� −

∑

i∈S1p

wi1j� = 0, t ∈ T , � ∈ Litj (3h)

xikjpt�, wik

j� ∈ {0, 1} (3i)

for each outcome ω ∈ � of ω,

Q(x, ω) = Max :∑

p∈P

∑

j ′∈Jωp

∑

i∈S1p

∑

t∈T

∑

�∈Litj

zi1ωj ′� (4a)

s.t.∑

p∈P

∑

j ′∈Jωp

∑

k∈Kitj

∑

�∈Litj

yikωj ′pt� + xik

jpt� ≤ 1,

i ∈ I, t ∈ T (4b)

∑

i∈Rkp

∑

t∈T

∑

�∈Litj

yikωj ′pt� ≤ 1,

p ∈ P, j ∈ Jωp , k = 1, . . . , np (4c)

∑

i∈Skp

∑

t∈T

∑

�∈Litj

yikωj ′pt� ≤ 1,

p ∈ P, j ∈ Jωp , k = 1, . . . , np (4d)

yikωj ′pt� − zi1ω

j ′� = 0, p ∈ P, j ∈ Jωp ,

i ∈ Ikp, t ∈ T , � ∈ Litj , k = 1, . . . , np (4e)

∑

i∈Rkp

zikωj ′� −

∑

i∈R(k−1)p

zi(k−1)ω

j ′� = 0, p ∈ P,

j ′ ∈ Jωp , t ∈ T , � ∈ Litj , k = 1, . . . , np (4f)

∑

i∈Skp

zikωj ′� −

∑

i∈S(k−1)p

zi(k−1)ω

j ′� = 0, p ∈ P,

j ′ ∈ Jωp , t ∈ T , � ∈ Litj , k = 1, . . . , np (4g)

∑

i∈R1p

zi1ωj ′� −

∑

i∈S1p

zi1ωj ′� = 0,

p ∈ P, j ′ ∈ Jωp , t ∈ T , � ∈ Litj (4h)

yikωj ′pt�, zikω

j ′� ∈ {0, 1} (4i)

The first stage of SIP (3a–3i) decides when to sched-ule the request on hand and which resources to use. Thefirst stage of SIP model is similar to the IP2 formula-tion presented in Section 4.2 but differs in the objectivefunction (3a). In addition to minimize the waiting time forthe current patient, the objective function has an additionalcoefficient that accounts for the expected value of the modelsecond stage objective function. This additional coefficientallows for recourse/corrective actions on the schedule basedon what it is observed on the second stage of SIP.

The second stage of the model (4a–4i) depicts a“scenario” for the problem. A scenario is defined as agroup of possible requests that could arrive after our currentpatient request and that also share the same preferred dayfor an appointment. Hence, the evaluation of the appoint-ment for our current patient will be an aggregation of many

E. Perez et al.

scenarios. These scenarios are generated using Monte-Carlosimulations with empirical distributions derived from his-torical data. The number of patients generated per scenariois conditioned on the number of patients currently sched-uled on the earliest day we can schedule our current request.The second-stage problem is similar to the offline problempresented in Section 4.1. The only difference is in constraint(4b) where the value of xik

jt� is used as a parameter. Thesolution (patient schedule) obtained in the first-stage of theproblem is now considered a parameter in this constraint.

Since patients are assumed to arrive one at a time, weneed to solve SIP every time a request is received at theclinic. Therefore, we derive a stochastic online frameworkthat uses the SIP model to schedule patients every time arequest is received. The stochastic online framework is pre-sented in Algorithm 2 and is named the Nuclear MedicineStochastic Online Scheduling (NMSOS) algorithm.

Algorithm 2 NMSOS Algorithm

The NMSOS algorithm first initializes both the patientset J and the patient identification number. The schedulinghorizon in terms of days is defined in line 2. The numberof time periods in a day are defined in line 3. Parame-ter τ defines the maximum number of time periods in aday. Method GetPatientRequest() is invoked by thealgorithms when a request is received at the clinic. Themethod gets the required information from the patient whichincludes the nuclear medicine procedure p and the preferredday for an appointment q (line 4). A counter λ is initial-ized to zero and the time and day of the request is assigned

to the patient parameters in line 5. Line 6 finds the earliestday that can be used to begin our search for an appointmentα. The GetSample() method is used to generate futurepossible requests occurring after time t. The method usesempirical distributions that are based on historical demanddata from the clinic. From this group of requests, we selectthose possible requests that ask for the same day of the week(q) that our current patient (line 8). The selected requestsfor scenario η are stored in the set Gη and each scenarioset is added to the set G. Once all the scenarios requiredfor the model are generated we proceed to step 11. All theparameters and sets are passed to the ServeRequest()function in line 11. The ServeRequest() function usesthe information provided to build and solve SIP.

The method ServeRequest() builds SIP model forday α. Those time slots that are already taken to serve otherpatients are blocked off as follows:

1. For R ∈ I , set right hand side of constraint (1b) = 0 if(α, t) ∈ Ur .

2. For S ∈ I , set right hand side of constraint (1b) = 0 if(α, t) ∈ Vs .

If no appointment is found (line 20), we search thefollowing week in the scheduling horizon, α = α +ω. If an appointment is found, we check if the waitingtime is shorter than a month (lines 11-12). If the wait-ing time is longer than a month, the algorithm searchesfor an alternative appointment on a different date. Thisnew search starts on day α ← dj + m + λ. If the wait-ing time is less than a month for the appointment found,the information is passed back to the scheduler and to thepatient (line 17). The intuition behind the NMSOS algo-rithm is explained with an example in the Appendix for theinterested reader.

5 Application

The methodology derived in the previous sections wasimplemented and applied to the Scott & White ClinicNuclear Medicine Clinic at Temple, TX. We conducted aseries of simulation experiments to test the performanceof our algorithms in a real environment. The NMOS andNMSOS scheduling algorithms were compared against abenchmark scheduling algorithm called Fixed Resource(FR). The FR algorithm was validated in [21] and representsan example of current practice. The discrete event specifi-cation (DEVS) simulation developed by [20] was used toassess the performance of the algorithms based on historicaldata of the Scott & White Nuclear Medicine. The NMOSand NMSOS algorithms were implemented in JAVA andILOG CPLEX.


5.1 Real nuclear medicine setting

The Scott & White Health System in Temple, Texas has oneof the largest fully accredited nuclear laboratories for gen-eral nuclear imaging and non-imaging in the U.S. This facil-ity operates from Monday to Friday from 8:00 am to 5:00pm. At the time of the study, the clinic had eight technolo-gists, two EKG technologists, one nurse, and one manager.Technologists have several responsibilities that include:radiopharmaceutical administration and image acquisitions.EKG technicians perform stress exams for cardiac proce-dures. Nurses provide support to the other members of thestaff. The division manager can assist in most of the activi-ties in the absence of one of the regular staff. The clinic alsohas nuclear medicine physicians, radiology residents, and acardiologist.

There are twelve stations within this facility. Equipmentsuch as gamma cameras are located inside the stations. Thefacility has seven gamma camera stations. Five of thesecameras are planar, and capable of doing 2D whole-bodyimaging and 3D Single Photon Emission Computed Tomog-raphy (SPECT). The other two cameras are planar as well,but one is SPECT capable and the other is for imagingonly. Two of the stations are equipped with EKG capabil-ity. The other stations are treatment (TRT) rooms for patienthydration and waiting. In this clinic around sixty differ-ent procedures are performed. Table 2 in Section 3 depictsthe procedures that were performed more frequently at theclinic during the year. These are the procedures used inour study because they account for more than 90 % of theprocedures requested.

5.2 Experimental setup

The optimization models used in the scheduling algorithmswere formulated based on the data provided by the Scott& White Healthcare Clinic Nuclear Medicine Department.The models involve the following stations: 7 rooms withgamma cameras, 2 stress rooms with treadmills, and 3TRT rooms. The names of the gamma camera rooms are:Axis(1), Axis(2), Axis(3), P2000(1), P2000(2), P2000(3),and Meridian(1). The stress rooms are named Treadmill(1)and Treadmill(2) whereas the treatment rooms are TRT(1),TRT(2), and TRT(3). We identify the human resourcenames at the clinic with the following names: Technolo-gist(1), Technologist(2), Technologist(3), Technologist(4),Technologist(5), Technologist(6), Technologist(7), Tech-nologist(8), Technologist(9), Technologist(10), Nurse(1),and Manager(1).

The scenarios for the SIP model are generated usingempirical distributions that are based on historical demanddata from the clinic. A scenario comprises several proce-dures with a high chance of been requested after serving

the current patient. Monte Carlo simulations were usedto obtain the scenarios required for each SIP model. Thenumber of procedures per scenario range from 1 to 20.As explained in Section 4.3, we condition the number ofpatients per scenario to the current number of patientsalready schedule on the earliest day we can schedule ourcurrent request. This strategy is used to reduce the size ofthe SIP model.

We conducted a series of simulation experiments to com-pare the performance of the NMOS and NMSOS schedulingalgorithms against a benchmark algorithm called FixedResource (FR). The FR algorithm details are explained in[21]. The FR algorithm schedules patients and resources asearly in the schedule as possible by taking into account thepatient preferred day for an appointment. In this algorithmsome of the human resources are assigned to always servepatients on specific stations. For example, two of the tech-nologists can always be assigned to serve patients in two ofthe gamma camera stations. The algorithm also takes intoaccount patient waiting times for an appointment. In casethe appointment found for the patient results in a waitinggreater than a month, the algorithm searches for anotherappointment on a different day of the week.

Two empirical distributions were derived to generate theprocedure type requested and preferred day of the appoint-ment for each patient. A list of the nuclear medicine proce-dure types considered is presented in Table 2. In terms ofthe preferred day of the appointment, Monday and Fridaywere identified as the days of the week requested the mostby patients. Since procedure request arrivals are indepen-dent, a Poisson process was assumed for patient procedurerequests. The monthly call interarrival times in minutes fol-lowed an exponential distribution with the following means:January, 6.00; February, 6.25; March 6.58; April, 6.67; May,6.75; June, 6.88; July, 6.96; August, 7.04; September, 7.10;October, 7.29; November, 7.34; and December, 7.44. In ourexperiments, we considered the impact of having differentdemand levels. We defined three patient demand levels: lowdemand, base demand, and high demand. The base demandlevel uses interarrival times that are based on the historicaldata provided by the clinic. For the low demand and highdemand levels we decreased and increased the demand rateby 10 %, respectively.

The performance measures listed in Table 5 were usedto measure the system performance under the differentscheduling algorithms. These performance measures pro-vide an assessment of the system in terms of both patientand management perspectives. We conducted a total ofnine experiments each involving 20 replications with dif-ferent seeds for the random number generator to allowfor independence among the replications. The experimentsinvolved running the following algorithms: Fixed Resource,Nuclear Medicine Online Scheduling (NMOS), and Nuclear

E. Perez et al.

Medicine Stochastic Online Scheduling (NMSOS). Foreach simulation run we used a scheduling horizon of 12months. We compute several statistics for each performancemeasure. The experiments were conducted on a Dell X5355computer with 2 Intel(R) Xeon(R) X processors at 2.66 GHzeach with 12.0 GB of RAM.

6 Computational experiments

We now report computational results to evaluate the robust-ness of the NMOS and NMSOS algorithms. The resultswere compared against the FR algorithm, for each ofthe three patient demand level scenarios: (a) low patientdemand, (b) base patient demand, and (c) high patientdemand. The FR algorithm schedules patients as earlyas possible, which may not be appropriate when yourresources are limited in terms of capacity and expertise. TheNMOS and NMSOS algorithms use optimization models toimprove patient schedules. Finally, we present a sensitiv-ity analysis to evaluate the performance of the algorithmswhen the procedure type demand is not based on empiricaldistributions.

6.1 Computational results

We report the average number of patients served using theFR, NMOS, and NMSOS algorithms under three demandlevels in Fig. 2. Under the low patient demand level, the FRand NMOS algorithms provide similar results. The NMSOSalgorithm provides over 1 % improvement compared to theFR algorithm. Under the base patient demand level, theresults show a pattern similar to the one observed under thelow patient demand. The FR and NMOS algorithms pro-vide similar results and the NMSOS algorithm provides animprovement of over 1 %. For the high patient demand levelthe NMSOS algorithm provides an over 4 % improvementcompared to the FR algorithm. A 4 % improvement is sig-nificant because it translates into having about 600 morepatients served on average per year.

Tables 6, 7, and 8 provide the average number of patientsserved per month with a 95 % confidence interval underthe low, base, and high patient demand levels respec-tively. The results show that NMSOS algorithm is able toaccommodate more patients early in the year for the lowpatient demand and base patient demand levels. The FRand NMOS algorithms start accommodating more patientslater in the year which is simply the early demand pushedinto the future. This has an impact on the patient wait-ing time which is discussed later in this section. Underthe high patient demand level, the NMSOS algorithm out-perform FR and NMOS accommodating more patientsper month.

Low Base High

FR 14,470.83 15,924.77 16,929.80

NMOS 14,467.30 15,928.44 16,933.51

NMSOS 14,544.70 16,008.33 17,587.22

14000

14500

15000

15500

16000

16500

17000

17500

18000

Ave

rag

e n

um

ber

of

pat

ien

ts s

erve

d

Fig. 2 Average number of patients served per year under threedemand scenarios: low, base, and high

The results for resource utilization for the low, base,and high patient demand are presented in Figs. 3, 4,and 5 respectively. The average human resource utilizationis shown on the left side of the figure and the average stationutilization on the right side of the figure. The graphs displaythe average utilization of each resource for a year of clinicoperation.

Figure 3 shows that the FR algorithm provides a higherutilization for most of the technologists at the clinic and alower utilization for the nurse and the manager. On the otherhand, the NMOS and NMSOS algorithms provide a morebalanced utilization of the human resources. In terms of sta-tion utilization, the NMSOS algorithm was able to increasethe utilization of the gamma camera stations of type Axisand also of the Treadmill stations. The FR algorithm pro-vides a more balanced utilization of the gamma cameras.The overall average utilization of both human resourcesand stations was about the same for the FR and NMOSalgorithms under the patient low demand, but it wasincreased by about 1 % with the NMSOS algorithm.

Figure 4 shows a plot of the resources’ utilization underbase patient demand for each scheduling algorithm. Thegraph depicts a similar pattern when compared to the lowdemand case. Again, the overall average utilization of bothhuman resources and stations was about the same for theFR and NMOS algorithms, but it was increased by over 1 %with the NMSOS algorithm.

Figure 5 show that the NMSOS algorithm provides ahigher utilization for most of the technologists at the clinicand a higher utilization of the nurse. NMSOS also providesa more balanced utilization of the human resources. In termsof station utilization, the NMSOS algorithm was able to


Table 6 Number of patientsserved per month under lowdemand

Month FR NMOS NMSOS

Jan 931.86 ± 3.26 937.10 ± 3.48 995.00 ± 3.67

Feb 1, 179.48 ± 3.91 1, 172.20 ± 3.86 1, 167.50 ± 3.85

Mar 1, 179.66 ± 4.34 1, 186.50 ± 3.31 1, 174.00 ± 3.68

Apr 1, 163.90 ± 4.26 1, 156.40 ± 4.31 1, 150.00 ± 3.91

May 1, 148.14 ± 3.60 1, 149.10 ± 3.35 1, 145.90 ± 3.17

Jun 1, 143.48 ± 4.00 1, 134.70 ± 3.88 1, 139.40 ± 3.56

Jul 1, 134.79 ± 4.70 1, 137.90 ± 3.83 1, 141.10 ± 4.53

Aug 1, 129.38 ± 4.66 1, 139.90 ± 3.52 1, 141.50 ± 4.14

Sep 1, 114.24 ± 4.50 1, 118.90 ± 3.73 1, 121.50 ± 4.37

Oct 1, 095.28 ± 3.91 1, 090.50 ± 3.11 1, 103.30 ± 3.80

Nov 1, 093.59 ± 3.81 1, 094.70 ± 3.00 1, 111.90 ± 3.73

Dec 1, 085.34 ± 4.51 1, 081.20 ± 4.31 1, 086.60 ± 5.93

Jan 1, 071.69 ± 4.10 1, 068.20 ± 3.89 1, 067.00 ± 3.46

Table 7 Number of patientsserved per month under basedemand

Month FR NMOS NMSOS

Jan 1, 005.90 ± 2.30 1, 009.78 ± 2.35 1, 056.83 ± 4.07

Feb 1, 297.27 ± 2.48 1, 309.33 ± 2.53 1, 293.50 ± 2.60

Mar 1, 301.87 ± 2.98 1, 305.44 ± 3.64 1, 310.83 ± 3.91

Apr 1, 289.90 ± 3.66 1, 288.67 ± 4.89 1, 296.67 ± 4.99

May 1, 274.10 ± 3.66 1, 259.89 ± 4.09 1, 255.00 ± 3.51

Jun 1, 261.47 ± 3.41 1, 259.78 ± 3.78 1, 260.33 ± 4.71

Jul 1, 253.23 ± 3.99 1, 257.78 ± 2.94 1, 267.00 ± 2.49

Aug 1, 238.27 ± 4.24 1, 242.00 ± 3.18 1, 255.00 ± 4.34

Sep 1, 223.73 ± 4.13 1, 219.00 ± 4.71 1, 224.33 ± 3.42

Oct 1, 214.70 ± 3.18 1, 211.11 ± 3.05 1, 214.67 ± 3.14

Nov 1, 197.83 ± 3.34 1, 193.89 ± 3.68 1, 203.83 ± 2.04

Dec 1, 178.03 ± 4.45 1, 175.22 ± 6.43 1, 190.83 ± 2.43

Jan 1, 188.47 ± 4.41 1, 196.56 ± 5.27 1, 179.50 ± 3.00

Table 8 Number of patientsserved per month under highdemand

Month FR NMOS NMSOS

Jan 1, 006.03 ± 2.34 1, 009.91 ± 1.07 1, 123.78 ± 2.65

Feb 1, 308.43 ± 2.35 1, 320.50 ± 2.26 1, 377.11 ± 1.38

Mar 1, 322.43 ± 2.51 1, 326.01 ± 2.20 1, 373.56 ± 2.23

Apr 1, 329.40 ± 2.34 1, 328.17 ± 1.98 1, 376.67 ± 1.62

May 1, 327.27 ± 2.37 1, 313.06 ± 2.40 1, 359.11 ± 2.21

Jun 1, 322.53 ± 2.99 1, 320.84 ± 2.99 1, 372.89 ± 2.78

Jul 1, 324.47 ± 2.52 1, 329.01 ± 2.34 1, 384.44 ± 2.10

Aug 1, 328.27 ± 2.29 1, 332.00 ± 2.27 1, 383.33 ± 2.79

Sep 1, 329.77 ± 2.52 1, 325.03 ± 1.83 1, 375.00 ± 2.36

Oct 1, 334.70 ± 1.88 1, 331.11 ± 2.53 1, 377.44 ± 3.15

Nov 1, 332.97 ± 2.12 1, 329.02 ± 3.14 1, 359.67 ± 3.56

Dec 1, 335.53 ± 2.74 1, 332.72 ± 3.38 1, 360.11 ± 3.51

Jan 1, 328.03 ± 2.44 1, 336.12 ± 3.16 1, 364.11 ± 3.07

E. Perez et al.

20

30

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80

Ave

rag

e st

atio

n u

tiliz

atio

n (

%)

Ave

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e h

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eso

urc

e u

tiliz

atio

n (

%)

FR NMOS NMSOS

25

35

45

55

65

75

85

FR NMOS NMSOS

Fig. 3 Resources utilization under low demand

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55

65

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85

Ave

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%)

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25

35

45

55

65

75

85A

vera

ge

stat

ion

uti

lizat

ion

(%

)

FR NMOS NMSOS

Fig. 4 Resources utilization under base demand

30

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30

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e st

atio

n u

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atio

n (

%)

FR NMOS NMSOS

Fig. 5 Resources utilization under high demand


increase the utilization of the gamma camera stations andthe treadmill stations. NMSOS algorithm also provides amore balanced utilization of the gamma cameras. The over-all average utilization of both human resources and stationswas about the same for the FR and NMOS algorithms underpatient high demand, but it was increased by over 4 % withthe NMSOS algorithm.

The results related to patient perspective are reported inFigs. 6 and 7. Figure 6 reports the patient average wait-ing time Type 1 for the three algorithms under the threepatient service demands. Recall that the patient waiting timeType 1 is the time a patient has to wait from the time theymake a request to the actual appointment time. The NMSOSalgorithm provides a lower waiting time for the patientsunder the three demand levels. NMSOS reduces the waitingtime compared to the FR algorithm by about 10 % for thepatient low and base demand cases, and by about 25 % forthe high demand case. This performance can be attributed tothe fact that NMSOS is able to accommodate more patientsearly in the year. Thus most patients end up waiting lesstime.

Figure 7 shows the patient preference ratio for thethree algorithms under the three different patient servicedemands: low, base, and high. Recall that the patient pref-erence ratio is the portion of patients scheduled on theirrequested day. The NMSOS algorithm provides a higherpreference ratio for the patients under the three demand lev-els when compared to the FR algorithm. NMSOS increasesthe patient preference ratio by about 5 % for the patient lowand base demand cases, and by about 80 % for the highdemand case.

In terms of computational time, it takes about two min-utes to get an appointment for a patient using the NMSOS

Low Base High

FR 4.99 5.15 9.48

NMOS 4.92 5.06 9.39

NMSOS 4.50 4.63 7.18

4

5

6

7

8

9

10

Ave

rag

e p

atie

nt

wai

tin

g T

ype1

(day

s)

Fig. 6 Patient waiting type 1

Low Base High

FR 92.35 90.12 35.88

NMOS 94.25 89.85 35.61

NMSOS 96.38 93.60 66.26

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100

Ave

rag

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nt

pre

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sa

tisf

acti

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rat

io (

%)

Fig. 7 Patient preference satisfaction ratio

algorithm. The FR and NMOS algorithms take only a fewseconds. The NMSOS algorithm takes longer because itformulates and solves an SIP model for each patient request.

6.2 Sensitivity analysis

We now present sensitivity analysis for the performance ofthe algorithms under different demand patterns. So insteadof using an empirical distribution, we considered a uniformdistribution to generate the type of procedure requested bythe patient. A uniform distribution was chosen to give eachprocedure type the same probability of being requested.This same distribution is used to generate the scenarios inthe SIP model. We now report the computational resultsto evaluate the robustness of the NMOS and NMSOSalgorithms. The results were compared against the FR algo-rithm, for each of the three demand levels: (a) low patientdemand, (b) base patient demand, and (c) high patientdemand.

The performance of the algorithms is similar to what isreported in the previous section. Hence, we report resultsfor only three of the performance measures, the num-ber of patients served per year, patient waiting time Type1, and patient preference satisfaction ratio. These threeperformance measures represent both patient and manage-ment perspectives. Figure 8 reports the average numberof patients served using the FR, NMOS, and NMSOSalgorithms under the three demand levels.

For the low demand case, FR and NMOS report sim-ilar results and the NMSOS algorithm reports about 1 %improvement over the FR algorithm. The same behavioris observed under the base demand case. Lastly, underhigh demand, NMSOS outperforms the FR and NMOS

E. Perez et al.

Low Base High

FR 12,419.80 13,520.43 14,309.50

NMOS 12,433.63 13,527.40 14,330.50

NMSOS 12,497.50 13,642.80 14,606.50

12000

12500

13000

13500

14000

14500

15000A

vera

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nu

mb

er o

f p

atie

nts

ser

ved

Fig. 8 Sensitivity analysis average number of patients served per yearunder three demand levels: low, base, and high

algorithms by 2 %. This improvement represents about 300more patients served per year. As reported earlier, NMSOSis able to accommodate more patients into the schedulewith the same number of resources when the demand at theclinic is increased. Next we report the results for the patientperspective performance measures.

Figure 9 reports the average waiting time Type 1 for thethree algorithms under the three patient demand levels. TheNMSOS algorithm provides a lower waiting time for thepatients under the three demand levels. When compared tothe FR algorithm, NMSOS decreases waiting time by 11 %for low and base demand cases, and by 26 % for the high

Low Base High

FR 9.01 10.59 20.42

NMOS 8.91 10.34 19.50

NMSOS 8.04 9.57 15.16

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Fig. 10 Sensitivity analysis patient preference satisfaction ratio

demand case. Figure 10 shows the patient preference ratiofor the algorithms under the three demand levels. Whencompared to the FR algorithm, NMSOS provides a higherpreference ratio under the three demand cases. NMSOSincreases patient preference ratio by about 3 % for the lowand base demand cases, and by about 73 % for the highdemand case. The results presented in this section confirmthe robustness of the performance of the NMSOS algorithm.

7 Summary and conclusions

Appointment scheduling in specialized clinics such asnuclear medicine departments is a very challenging prob-lem. Radiopharmaceutical properties require nuclear me-dicine procedures to be performed following strict proto-cols that must be adhered to by the human resources. In thispaper we derive an online scheduling (NMOS) algorithmand a stochastic online scheduling (NMSOS) algorithm forpatient and resource scheduling in nuclear medicine depart-ments. The scheduling algorithms take into account the timeconstraints imposed by the decay of the radiopharmaceuti-cals. Both algorithms were implemented within a simulationframework and experiments are based on historical datafrom an actual clinic. We compared the results of ouralgorithms against the Fixed Resource (FR) algorithm,which is based on the operation of an actual clinic.

We obtain computational results that provide evidence ofthe benefits of considering stochastic future arrivals infor-mation when scheduling patients in health care clinics. Wefound that the number of patients served was significativelylarger under the NMSOS scheduling algorithm when com-pared to the FR scheduling algorithm, especially under high


Fig. 11 Example schedule for anuclear medicine proceduresscheduled using NMSOS

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demand, where an improvement of 4 % was achieved onthe overall throughput for the year. This improvement trans-lates into 600 more patients served on average per year.In terms of patient perspective, the NMSOS algorithm pro-vides appointment with less waiting time for the patientswhen high demand is expected. Specifically, a 25 % reduc-tion on the waiting time was achieved when compared to theFR algorithm which translates on patients having to wait onaverage two days less for their appointments.

From a practical perspective, nuclear medicine man-agers with limited resource capacity and increasing patientdemand should consider the use of the NMSOS algorithm.The NMSOS algorithm showed that managers can increasethe number of patients served for a year with the samenumber of resources. In addition, the NMSOS algorithm

provides a better service to their patient by making themwait a shorter amount of time.

Even though this work focus in nuclear medicine webelieve it can also benefit other health care settings. Forexample, algorithms can be extended to other health caresettings with multi-step medical procedures such as oncol-ogy clinics and operating rooms. Further research alsoincludes the extension of the algorithms to account forother sources of uncertainty such as patient no-shows,patient re-scheduling, human resource fatigue, and equip-ment failures. Patient no-show is not critical in nuclearmedicine but it has a significant impact when schedul-ing patients in oncology clinics and operating rooms.Constraints will have to be added to the SIP model toaccount for no-shows in the methodology presented in

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schedule for the firstrequest (CPT 78815)

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Fig. 12 Example schedule for three nuclear medicine procedures scheduled using NMSOS

E. Perez et al.

this paper. Likewise, new constraints will have to be addedto the SIP model to account for patient fatigue and equip-ment failures.

Acknowledgments The author wish to thank Wayne Stockburger ofthe Scott and White Hospital for providing access to the clinic and his-torical data. The first author wishes to acknowledge research supportfrom the Sloan Foundation and the National GEM Consortium.

Appendix

The intuition behind the NMSOS algorithm is explained inthis section. The example presented in Section 3 is used asa reference. Figure 1 illustrates a typical nuclear medicineschedule for two different nuclear medicine procedures. Theprocedures are scheduled as they arrive using an ‘as soonas possible’ strategy. The first procedure (CPT 78815) isscheduled at the beginning of the day and it is representedby white bars. Since some of the resources required forthe second procedure (CPT 78465) are already occupiedat the beginning of the day, the second procedure, in graybars, is scheduled later in the day. In this example no otherprocedure can be added to the schedule without overlapping.

The NMSOS algorithm takes into account possible futureprocedure requests which allows for making more informeddecisions when scheduling patients and resources underuncertainty. For instance, a different appointment schedulewill be provided to the first procedure request (CPT 78815)if the uncertainty about future procedure requests is takeninto account. Figure 11 shows the schedule for the first pro-cedure request when the scenarios generated for the SIPmodel show a high probability of having multiple procedurerequests of type CPT 78465 later in the day.

The schedule has the first patient request arriving laterin the day when compared to the appointment presentedin Fig. 1. The change in the appointment time providesmore flexibility to the clinic in terms of scheduling morepatients. In fact, the new schedule will allow the system toschedule multiple procedures requests of type CPT 78465as presented in Fig. 12. The NMSOS algorithm providesappointment times and resource assignments that allows theclinic to schedule more patients in the long run.

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