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

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1 Health Care Manag Sci DOI /s Stochastic online appointment scheduling of multi-step sequential procedures in nuclear medicine Eduardo Pérez Lewis Ntaimo César O. Malavé 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 procedures has been recognized as one of the contributors to the rise of health care costs in the U.S. in the last few years. Nuclear medicine is a subspecialty of radiology that uses advanced technology and radiopharmaceuticals for the diagnosis and treatment of medical conditions. Procedures in nuclear medicine require the use of radiopharmaceuticals, are multi-step, and have to be performed under strict time window constraints. These characteristics make the scheduling of patients and resources in nuclear medicine challenging. In this work, we derive a stochastic online scheduling algorithm for patient and resource scheduling in nuclear medicine departments which take into account the time constraints imposed by the decay of the radiopharmaceuticals and the stochastic nature of the system when scheduling patients. We report on a computational study of the new methodology applied to a real clinic. We use both patient and clinic performance measures in our study. The results show that the new method schedules about 600 more patients per year on average than a scheduling policy that was used in practice by improving the way limited resources are managed at the clinic. The new E. Pérez ( ) Ingram School of Engineering, Texas State University, 601 University Drive, San Marcos, TX 78666, USA eduardopr@txstate.edu L. Ntaimo C. O. Malavé Department of Industrial and Systems Engineering, Texas A&M University, 3131 TAMU, College Station, TX 77843, USA C. Bailey P. McCormack Scott and White Clinic, 2401 S. 31st Street, Temple TX 76508, USA methodology finds the best start time and resources to be used for each appointment. Furthermore, the new method decreases patient waiting time for an appointment by about two days on average. Keywords Health care Nuclear medicine Patient service Online scheduling Stochastic programming Mathematics Subject Classifications (2010) 90B99 68M20 68W27 1 Introduction In this paper, we derive a stochastic online scheduling methodology for patient and resource management in nuclear medicine clinics. Nuclear medicine is a branch of medical imaging that uses small amounts of radioactive materials to diagnose and treat a variety of diseases, including, many types of cancers, heart diseases and certain other abnormalities within the body. The cost of advance imaging procedures has grown disproportionately compared with the overall cost of health care and has become a major factor in the U.S. government expenses [8]. According to [28], nuclear procedures were used most extensively in the United States in 2007, with 1,000 or more procedures performed per 100,000 people. Suthummanon et al. [25] showed that machine time, direct labor time, and radiopharmaceuticals account for most of the cost per procedure in nuclear medicine. Also, the way resources are managed has a direct impact in the quality of service. The increased demand and the complexity in the procedure protocols make the scheduling of patients and resources in nuclear medicine a challenging problem.

2 E. Pérez et al. Nuclear medicine procedures require the administration of a radiopharmaceutical to the patient and involve multiple sequential steps. Each procedure step requires several resources such as technologists, nurses, gamma cameras, and sometimes a treadmill. Radiopharmaceuticals allow for the imaging (scans) or treatment of a specific organ of the human body. Radiopharmaceuticals are prepared in radiopharmacies outside nuclear medicine clinics. They have to bemanaged carefullyand theirdelivery requirewell-planned lead time. The short half-life of the radiopharmaceuticals imposes strict time constraints on scheduling patients and resources. To successfully complete a procedure, every step has to be initiated and completed within a specific time window. If the procedure protocol is not followed, a poor scan will result causing poor utilization of resources and patient rescheduling. A scan could last from minutes to hours and a procedure may require multiple scans in a day. Some of the resources needed to perform nuclear medicine procedures include technologists, radiopharmaceuticals, gamma cameras, and sometimes a treadmill, a nurse or an EKG technician. Gamma cameras take advantage of radiopharmaceutical properties to capture images from within the human body. The cost of these cameras can be as high as a $ 1 million. Since the gamma cameras can be very expensive, they have to be utilized most of the time. Resources required to serve a patient must be available at the time they are scheduled. Each procedure step requires the scheduling of a pair of resources, a human resource and a clinic room or station. A member of the staff is always required to operate equipment in a station and to take care of patients. Patient requests in nuclear medicine arrive during the day as the scheduling proceeds. Requests are handled as they arrive and appointments are provided in an online fashion without taking into consideration future requests. Finding ways to improve patient service is of great interest for nuclear medicine managers. The characteristics of this problem make it unique with very limited research reported in the literature. Stochastic planning techniques are an alternative to address this problem. In this paper, a stochastic online scheduling algorithm for patient and resource management in nuclear medicine clinics is derived. The algorithm uses clinic historical information to make more informed decisions when selecting an appointment for the patient request on hand. Conceptually, historical data regarding patient appointment requests and arrivals to the clinic can be incorporated into a multistage mathematical program for optimal scheduling plans. However, such optimization problems tend to be very large even for a medium size clinic and very challenging to solve. In contrast, stochastic online algorithms are suboptimal, but scalable ways of solving stochastic integer programs. The contributions of this paper include new mathematical models for scheduling nuclear medicine procedures, a stochastic online algorithm for patient and resource scheduling, and a new methodology for scheduling medical multi-step sequential procedures that consider both patient and clinic perspectives. The new methodology takes into account the time constraints imposed by the decay of radiopharmaceuticals and the stochastic nature of the system in terms of the future procedure requests happening after the current patient. These contributions will help the practice of nuclear medicine by providing shorter patient waiting times, increased patient throughput, minimal delays in radiopharmaceutical delivery, higher utilization of resources, and ultimately lower health care costs. The rest of the paper is organized as follows: Section 2 provides a review of the existing literature for this problem. The nuclear medicine scheduling problem is described in Section 3. Section 4 provides a description of the scheduling problem and presents the stochastic online algorithm. In Section 5 we describe the setting where the algorithm is applied and the experimental frame. A computational study and discussion of the results is given in Section 6. The paper ends with some concluding remarks and directions for future research in Section 7. 2 Related work Patient scheduling has been studied widely in the past thirty years. The major problem is to find the best appointment rule, which is the best algorithm that specifies the appointment interval lengths and resources to be used [6]. Medical facilities dedicated to the diagnosis and treatment of patients are classified as specialty clinics. Radiology, CT scan, and magnetic resonance imaging (MRI) clinics are within this group. Nuclear medicine is a sub-specialty of radiology. Procedures in nuclear medicine differ from most other imaging modalities in that they show the physiological function of the part of the body being investigated. In addition, these procedures rely on the process of radioactive decay, involve several steps, and multiple resources making them difficult to manage. The existing literature on the nuclear medicine scheduling problem is limited. Most of the existing published work for specialty clinics focuses in scheduling CT scan and MRI procedures, which are relatively less difficult to manage since they do not consider multi-step sequential procedures. Also, most of the existing literature assumes that a pool of patients is always available at the beginning of the day. However, in nuclear medicine patients arrive in an online fashion and are scheduled one at a time. Green et al. [10] address the problem of scheduling randomly arriving patients of different types in an MRI facility.

3 Stochastic online appointment scheduling of multi-step sequential procedures in nuclear medicine They formulate the problem as a finite-horizon dynamic program for a clinic schedule that allows at most one patient per period and a single server, where only one patient can be served at a time. The authors derive properties of the optimal scheduling policies and identify a service sequence that minimize the expected total cost of serving patients. Patrick et al. [19] study a similar problem but they characterize patients with different priorities. Kolisch and Sickinger [15] consider a similar problem but with two servers. The authors compare the linear capacity allocation (LCA), first-come first-served (FCFS), and random selection (RS) decision and conclude that the FCFS provides a better performance. Patrick and Puterman [18] consider the problem of scheduling patients in a CT scan clinic. They formulate an optimization problem that returns a reservation policy that minimizes the non-utilization of resources subject to an overtime constraint. Their approach assumes the availability of a pool of patients that can be called to occupy unused time slots. The authors use simulation to demonstrate a reduction in outpatient waiting time. Sickinger and Kolisch [23] propose a generalization of the Bailey-Welch rule as well as a neighborhood search heuristic for a medical service facility with two servers. The Bailey-Welch rule states that for one server clinic the best performance in terms of patient waiting and server idle time is to schedule two patients for the first appointment space and one patient on the ones that follow. The authors analyze the impact of different problem parameters on the total reward. Standridge and Steward [24] propose a simulation model that includes a control logic for patient scheduling. The system presented by the authors schedules patients within a simulation framework. Vermeulen et al. [27] devisean adaptive approach to automatic optimization of resource calendars in a CT scan clinic. They implement a simulation model for a case study analysis to demonstrate that their approach makes efficient use of resources capacity. Pérez et al. [20] develop a discrete event specification (DEVS) simulation model for a nuclear medicine facilities. The authors propose several scheduling algorithms that suggest different system configurations for serving patients. The benefits of taking into account future events when optimizing decision processes are reported in the literature. It has been shown that using additional stochastic information can improve the quality of solutions in scheduling 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 different ways; however, the unifying theme seen throughout the research is that there are considerable advantages when stochastic information is taken into account. The common strategy is to predict the future requests using a statistical model by sampling the observations on the history. Chang et al. [7] study the multiclass packetnetwork scheduling problem. The authors use a Hidden Markov Model (HMM) to generate the tasks arrivals for each class with a particular weight and develop the expectation algorithm which adapts an optimal offline algorithm into an online algorithm by sampling possible future tasks sequences from the HMM. The expectation algorithm has some resemblances to the sample average approximation method for non-dynamic stochastic programming [14, 22] where the solution depends of a deterministic part and a stochastic part. The deterministic part gives the immediate plan and the stochastic part gives a penalty for changing the plan to accommodate the best as possible the scenario that has become reality. One must average over many scenarios to find the best expected solution. Unfortunately, the expectation algorithm does not perform well under time constraints, since it must distribute its available optimizations across all requests. This issue was recognized and addressed in [3] where a consensus algorithm was proposed. The consensus algorithm solves as many samples as possible to select the request which is chosen most often in the sample solutions at time t. This algorithm was shown to outperform the expectation algorithm for the online packet scheduling problem under time constraints. However, as decision time increases, the quality of the consensus algorithm levels off and is eventually outperformed by the expectation algorithm. The regret algorithm proposed in [2] combines the features of both expectation and consensus algorithms. The regret algorithm assess every decision on all samples but has the ability to avoid distributing the samples among decisions. Stochastic online optimization assumes a known distribution of future requests, or an approximation thereof, is available for sampling [3]. The typical case is the existence of either historical data or predictive models. Decisions made using stochastic online optimization are usually constrained by time, meaning that there is a limited time to report a solution for each request to the decision maker. Awasthi and Sandholm [1] consider the scheduling of human kidney transplants using a stochastic online framework. They propose an adaptation of the regret algorithm. Van Hentenryck et al. [26] consider the online stochastic reservation problem where the goal is to allocate requests that are received online to a limited group of resources in order to maximize profit. The authors adapted the consensus and regret algorithms for their problem. Their modification of the regret algorithm is based on a constant sub-optimality approximation of multi-knapsack problem. The authors used two black-boxes to handle the stochastic arrivals of reservation requests for hotel rooms. One black-box is the

4 E. Pérez et al. sub-optimal approximation module and the other is the sampling module, which relies on the observations of the past arrivals. This research differs from earlier studies in multiple ways. First, appointment schedules deal with radiopharmaceutical lead times and require multiple resources at each step of the procedure. Second, nuclear medicine procedures follow sequential protocols that impose strict time-window constraintsonthe starting ofeach step. These characteristics make scheduling patients in nuclear medicine clinic a new challenging problem. In fact Gupta and Denton [11] identified the problem of scheduling patients in constrained health care clinics as an open research challenge. 3 Problem definition and notation To define the problem, we first state some assumptions. The operation time in a day is divided into time slots of equal length. Patients arriving to the clinic have an appointment and that patient appointment times coincide with the beginning of a time slot. It is also assumed that patients show for their appointment most of the time. Let S denote the set of stations inside the clinic and R be the set of human resources. Stations contain at least one type of equipment and are classified based on the type of equipment they have. Equipment include gamma cameras and treadmills. Human resources include physicians, technologists, nurses, and a manager. Each human resource is responsible for a specific set of tasks. The number of tasks and the time required to complete them depend on the human resource experience and expertise. Table 1 lists some of the tasks performed by the human resources. In nuclear medicine, patient appointments are scheduled by the patient s primary physician. A procedure request involves two pieces of information: a procedure type from set P and a patient preferred day for the appointment q. Patient appointments are given by the clinic at the time the request is received without taking into account requests arriving 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 nuclear medicine with their current procedural terminology (CPT) codes. Let A denote the set of radiopharmaceuticals and at least one radiopharmaceutical from this set is required per nuclear medicine procedure. Radiopharmaceuticals need to be at the clinic by the time of the patient appointment. Therefore, they are requested in advance allowing a lead time for preparation and delivery. Each procedure p P has a finite number of tasks/steps denoted by n p.letβ kp define the duration of step k for procedure p. Then the total duration of the procedure is n p k=1 β kp. The rest of the notation required for our problem is listed in Table 3. Table 4 lists the steps for the MSC-bone imaging procedure (CPT 78315). It has an average total duration of 245 minutes. Observe that the procedure has four steps and each step has different requirements in terms of time duration, stations, and human resources. For instance, step 1 of the procedure takes about 20 minutes on average and requires one station and one human resource from those listed in the table. Let R kp be the set of human resources qualified to perform step k of procedure p. Similarly, let S kp be the set of stations where step k of procedure p is performed. Figure 1 illustrates a typical nuclear medicine schedule for two different nuclear medicine procedures. Two procedure protocols are used in the example. For illustration purposes only a limited number of resources are used in the example. In addition, procedures are scheduled as they arrive using an as soon as possible strategy. The first procedure listed is the MSC-bone imaging (CPT 78815) presented in Table 4. This procedure is scheduled at the beginning of the day and is represented by the white Table 2 Examples of nuclear medicine procedures CPT Code Name Cardiovascular Event (CVE) Myocardial Imaging (SP-M) Positron Emission Tomography (PET)/ Computed Tomography (CT) MSB-Bone Imaging (Whole Body) MSC-Bone Imaging (Three Phase) GIC-Hepatobiliary Imaging CVJ-Cardiac Blood Pool REB-Pulm Perfusion / Ventilation ENC-Thyroid Imaging HEE-Lymphatic Imaging CVD-Myocardial Imaging (SP-R ORS)

5 Stochastic online appointment scheduling of multi-step sequential procedures in nuclear medicine Table 3 Scheduling problem sets and parameters Sets J : Set of patients, indexed j P : Set if procedures, indexed p J p : 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 S kp : Set of stations where step k of procedure p can be performed R kp : Set of human resources qualified to perform step k of procedure p I kp : Set of resources that can be used to perform step k of procedure p, I kp ={R kp S kp } L itj : Set of appointment star times that require the use of resource i at time-slot t for patient j K itj : Set of procedure steps that require the use of resource i at time-slot t for patient j T ij : Set of time-slots where resource i could be used to serve patient j T aj : Set of time-slots where radiopharmaceutical a could be used to serve patient j L j : Set of day and time slot pairs, (d, t), for patient j. U r : Set of day and time slot pairs, (d, t), for human resource r schedule. V s : Set of day and time slot pairs, (d, t), for station s schedule. i : j : a : p : k : l : t : τ : β kp : n p : ρ : δ p : ω : μ : m : q : Parameters Subscript, for the i resource Subscript, for the j patient Subscript, for the a radiopharmaceutical Subscript, for the p procedure Subscript, for the k step of a procedure Subscript, for the l starting time-slot for a patient appointment Subscript, for the t time-slot, incremental time Total number of time-slots in a day, indexed t,...,τ Number of time-slots required to complete step k of procedure p Total number of steps for procedure p, indexed k,...,n p Parameter representing resource r or station s total duration of procedure p Number of days in a week Number of days in a month Number of days before arrival of radiopharmaceutical after placing order 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.) Station Axis; P2000; Meridian; TRT Axis; P2000; Meridian Waiting Axis; P3000; Meridian Human Resource Technologists; Nurse; Manager Technologists; Manager None Technologists; Manager

6 E. Pérez et al. time = 20 s = TRT r = nurse Procedure CPT time = 150 waiting time = 15 s = axis r = technologist Procedure CPT time = 45 s = axis r = technologist Technologist EKG Technologist Nurse 1 TRT 1 1 Treadmill 2 2 Axis time = 5 s = TRT r = technologist time = 30 s = treadmill r = EKG tech time =60 waiting time = 30 s = axis r = technologist Waiting 3 Waiting 3 schedule for the first request (CPT 78815) schedule for the second request (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 second procedure are already occupied at the beginning of the day, the second procedure, in gray, is scheduled later in the day. As more procedure requests arrive, managing resources in nuclear medicine clinics become very challenging. For instance, in this example no other procedure can be added to the schedule without overlapping. It is important to notice that most of the resources are not fully utilized. A set of measures are used to quantify the performance of the scheduling algorithms. These performance measures are defined using both patient and clinic perspectives. The fiveperformancemeasures selected were identified as those commonly used in literature and they are summarized in Table 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 scheduling patients and resources in nuclear medicine clinics using three approaches; namely, offline, online, andstochastic online. We start with the offline scheduling approach in Section 4.1. In offline scheduling, the problem is solved on a day by day basis and it is assumed that all requests for the day are known in advance. The problem is modeled using integer programming (IP). We present the online version of the problem in Section 4.2. Unlike offline scheduling, in online scheduling requests arrive sequentially one at a time and scheduling decisions are made when the request is received. For the online version of the problem, we devise an online framework for scheduling patients and resources. Lastly, in Section 4.3, we discuss the stochastic online version of the problem. Similar to online scheduling requests arrive sequentially one at a time. However, possible future requests are now taken into account when making scheduling decisions. Consequently, we formulate the problem using stochastic integer programming (SIP) to deal with the uncertainty within the online framework. 4.1 The offline problem In the offline scheduling problem we assume a finite number of days in the scheduling horizon and require each resource i I to have a schedule that contains τ number of time-slots per day. In the offline version of the problem, it is assumed that all patient requests for the day are known in advance. Thus, patient appointments are provided by taking into account all the requests for the day. A set of patient requests (J p ) is used as input to the model. Each

7 Stochastic online appointment scheduling of multi-step sequential procedures in nuclear medicine patient requests a procedure type p. The goal is to schedule a subset B J p of patients that satisfies the problemspecific constraints and maximize the objective function. Let xjptl ik =1 if patient j requesting procedure p is scheduled to use resource i at time-slot t when procedure is started at time l for the k step of the procedure, otherwise xjptl ik =0. Similarly, let wjl ik =1 if resource i is selected to serve patient j in step k when procedure is started at time l, otherwise wjl ik =0. The offline problem can be formulated as an IP1 as follows: IP1 : Max : p P s.t. p P j J p k K itj i I, t T i R kp i S kp t T j J p i S 1p t T l L itj l L itj x ik jl 1, jptl 1, l L itj w i1 jl p P, j J p,k= 1,...,n p t T l L itj jl 1, p P, j J p,k= 1,...,n p xjptl ik wik jl = 0, p P, j J p, i I kp,t T, l L itj,k= 1,...,n p jl jl = 0, p P, i R kp i R (k 1)p w i(k 1) j J p,t T, l L itj,k= 1,...,n p jl w i(k 1) i S kp i S (k 1)p i R 1p w i1 jl = 0, p P, j J p,t T, l L itj,k= 1,...,n p jl i S 1p w i1 jl = 0, p P, j J p,t T, l L itj xjptl ik,wik jl {0, 1} (1i) (1a) (1b) (1c) (1d) (1e) (1f) (1g) (1h) Model IP1 allocates a subset B of patients to available resources so that their capacities are not exceeded. The objective function (1a) maximizes the number of patients scheduled on a given day. Constraint (1b) enforces for each patient the time-slot by time-slot requirements for procedure completion, that is, at most one patient is assigned to each resource each time period. Constraint (1c) and constraint (1d) select the staff and station, respectively, per procedure step and decide the start-time of the appointment for each patient. Constraint (1e) makes sure that the same resource is scheduled for the duration of each procedure step. Constraint (1f) and constraint (1g) verify that the staff and stations, respectively, selected to serve a patient follow the procedure sequence protocol. Constraint (1h) matches a station to a staff member for each step of the procedure requested by the patient. Finally, constraint (1i) imposes binary restrictions on the decision variables. Given the patient request demand, IP1 can be solved using a direct IP solver. 4.2 The online problem In nuclear medicine, patient requests are not known in advance, rather, they arrive sequentially one at a time and are scheduled as they arrive. Let ξ t be a patient appointment request at time t, if a sequence of requests ξ = ξ 1,...,ξ t 1,ξ t is revealed at different times of the day, the requests ξ 1,...,ξ t 1 are already scheduled at time t when request ξ t is received. At time t the problem is to decide how to schedule request ξ t by keeping all the other patients already scheduled fixed. Model IP1 in Section 4.1 can be modified to schedule patients and resources in an online fashion. Instead of scheduling a group of requests simultaneously, the modified version of the model will be used to schedule procedure requests one at a time. The online problem can be formulated as an IP as follows: IP2 : Min : (2a) s.t. i S 1p t T x ik k K itj l L itj i R kp i S kp l L itj lw i1 jl jptl 1, i I, t T jl 1, k = 1,...,n p t T l L itj jl 1, k = 1,...,n p t T l L itj xjptl ik wik jl = 0, i I kp,t T, l L itj,k= 1,...,n p jl jl = 0, i R kp i R (k 1)p w i(k 1) t T, l L itj,k= 1,...,n p jl jl = 0, i S kp i S (k 1)p w i(k 1) t T, l L itj,k= 1,...,n p (2b) (2c) (2d) (2e) (2f) (2g)

8 E. Pérez et al. i R 1p w i1 jl i S 1p w i1 xjptl ik,wik jl {0, 1} (2i) jl = 0, t T, l L itj (2h) Model IP2 share the same decision variables and constraints as model IP1. The only difference between the two models is the objective function. The objective function (2a) minimizes the waiting time Type 1 for the patient. Since the problem now is to schedule one patient at a time, we do not need to maximize the throughput. Instead, we find the best schedule for each new patient request by minimizing the waiting time using IP2. Model IP2 requires an online scheduling framework to provide schedules as requests are received. Therefore, we devise an online framework to schedule patients every time a request is received using IP2. Some notation is defined before describing the framework. Let U r = {(d, t) 1 d h, 1 t τ} define a set of day and time slot pairs (d, t) for human resource r. Likewise, we define a set V s = {(d, t) 1 d h, 1 t τ} for station s. Both sets, U r and V s include all the time slots that are already occupied by other patients. Parameter h is used to define the total number of days in the scheduling horizon. L j is the set of the appointment schedule found for the patient. In addition, in what follows is used to denote an assignment. The online framework is presented next and is named the Nuclear Medicine Online Scheduling (NMOS) algorithm. We state the steps of the algorithm in Algorithm 1. Algorithm 1 NMOS Algorithm The first step of NMOS algorithm (line 1) initializes the patient set J and the patient ID. Line 2 defines the scheduling horizon in terms of days and line 3 defines the number of time periods per day. The maximum number of time periods is given by τ. Method GetPatientRequest() is invoked when a request is received at the clinic. The method gets the patient information required to find an appointment. The information includes the nuclear medicine procedure p and the preferred day for an appointment q (line 4). A counter λ is initialized to zero and the time and day of the request is assigned to the patient parameters in line 5. Line 6 finds the earliest day to begin the search for an appointment α by taking into account the radiopharmaceutical lead time and the patient preferred day. The method ServeRequest() uses parameters j, p j,and α to find an appointment for the patient in line 7. The method builds and solves the IP2 model for day α. Those time slots that are already taken to serve other patients are taken into account as follows: 1. For R I, set right hand side of constraint (1b) = 0if (α, t) U r. 2. For S I, set right hand side of constraint (1b) = 0if (α, t) V s. IP2 has about 2,500 variables and 500 constraints, on average. The NMOS algorithm creates an object of type IloCplex and use the Concert Technology modeling interface implemented by ILOG CPLEX to solve IP2. It takes less than a minute to solve IP2 using ILOG CPLEX. If no appointment is found (line 16), the algorithm considers the following week in the scheduling horizon, α = α + ω. If an appointment is found, the algorithm checks if the waiting time is shorter than a month (lines 8-9). If the waiting time is longer than a month, the algorithm searches for an alternative appointment on a different date. This new search starts on day α d j + m + λ. If the waiting time is less than a month, the appointment information is passed back to the system and to the patient (line 13). 4.3 The stochastic online problem The stochastic online problem is an extension of the online problem described in Section 4.2. In this version of the problem we are interested in deciding how and when to serve a procedure request by keeping previous scheduled patient appointments fixed. However, the stochastic online problem takes into account additional information when making those decisions. Specifically, we take into consideration possible patients requests that could arrive after the patient request we have on hand. Taking into account possible future patient requests allows us to make more

9 Stochastic online appointment scheduling of multi-step sequential procedures in nuclear medicine informed decisions when scheduling patients and resources under uncertainty. The goal is to find the best day and time to accommodate the current request such that we can improve the system s performance measures. Consequently, the stochastic online problem can be formulated as a Two- Stage Stochastic Integer Programming (SIP) model. In this formulation, here and now decisions are made at the first stage before the realization of uncertain data. The uncertain data is represented by ω, which is a scenario realization of possible future patient requests. In the second stage, after a realization of ω becomes known, a patient schedule for the current request is generated by solving the appropriate optimization problem. Two new binary variables are defined for the second stage of the problem. For each scenario ω, variable yj ikω tl =1 if patient j is scheduled to use resource i at time-slot t and when the procedure is started at time l for the k step of the procedure. Otherwise yj ikω tl =0. Variable zikω jl =1 if resource i is selected to serve patient j in step k and when the procedure is started at time l. Otherwise zj ikω l =0. SIP is formulated as follows: SIP : Min : s.t. x ik k K itj l L itj i R kp i S kp lw i1 i S 1p t T l L itj jptl 1, i I, t T jl E[Q(x, ω)] jl 1, k = 1,...,n p t T l L itj jl 1, k = 1,...,n p t T l L itj x ik jptl wik jl = 0, i I kp,t T, l L itj,k= 1,...,n p jl w i(k 1) i R kp i R (k 1)p jl = 0, t T, l L itj,k= 1,...,n p jl w i(k 1) i S kp i S (k 1)p jl = 0, t T, l L itj,k= 1,...,n p i R 1p w i1 jl i S 1p w i1 xjptl ik,wik jl {0, 1} (3i) jl = 0, t T, l L itj (3a) (3b) (3c) (3d) (3e) (3f) (3g) (3h) for each outcome ω of ω, Q(x, ω) = Max : p P s.t. p P j Jp ω i I, t T i R kp t T k K itj l L itj y ikω j Jp ω i S 1p t T zj i1ω l l L itj yj ikω ptl + xik jptl 1, l L itj j ptl 1, p P, j Jp ω,k= 1,...,n p j ptl 1, i S kp t T l L itj y ikω p P, j J ω p,k= 1,...,n p yj ikω ptl zi1ω j l = 0, p P, j J p ω, i I kp,t T, l L itj,k= 1,...,n p j l j l = 0, p P, i R kp z ikω i R (k 1)p z i(k 1)ω j Jp ω,t T, l L itj, k= 1,...,n p j l j l = 0,p P, i S kp z ikω i S (k 1)p z i(k 1)ω j Jp ω,t T, l L itj, k= 1,...,n p j l j l = 0, i R 1p z i1ω i S 1p z i1ω p P, j J ω p,t T, l L itj yj ikω ptl,zikω j l {0, 1} (4i) (4a) (4b) (4c) (4d) (4e) (4f) (4g) (4h) The first stage of SIP (3a 3i) decides when to schedule the request on hand and which resources to use. The first stage of SIP model is similar to the IP2 formulation presented in Section 4.2 but differs in the objective function (3a). In addition to minimize the waiting time for the current patient, the objective function has an additional coefficient that accounts for the expected value of the model second stage objective function. This additional coefficient allows for recourse/corrective actions on the schedule based on 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 a group of possible requests that could arrive after our current patient request and that also share the same preferred day for an appointment. Hence, the evaluation of the appointment for our current patient will be an aggregation of many

10 E. Pérez et al. scenarios. These scenariosare generatedusing Monte-Carlo simulations with empirical distributions derived from historical data. The number of patients generated per scenario is conditioned on the number of patients currently scheduled on the earliest day we can schedule our current request. The second-stage problem is similar to the offline problem presented in Section 4.1. The only difference is in constraint (4b) where the value of xjtl ik is used as a parameter. The solution (patient schedule) obtained in the first-stage of the problem is now considered a parameter in this constraint. Since patients are assumed to arrive one at a time, we need to solve SIP every time a request is received at the clinic. Therefore, we derive a stochastic online framework that uses the SIP model to schedule patients every time a request is received. The stochastic online framework is presented in Algorithm 2 and is named the Nuclear Medicine Stochastic Online Scheduling (NMSOS) algorithm. Algorithm 2 NMSOS Algorithm to the patient parameters in line 5. Line 6 finds the earliest day that can be used to begin our search for an appointment α. TheGetSample() method is used to generate future possible requests occurring after time t. The method uses empirical distributions that are based on historical demand data from the clinic. From this group of requests, we select those possible requests that ask for the same day of the week (q) that our current patient (line 8). The selected requests for scenario η arestoredinthesetg η and each scenario set is added to the set G. Once all the scenarios required for the model are generated we proceed to step 11. All the parameters and sets are passed to the ServeRequest() function in line 11. The ServeRequest() function uses the information provided to build and solve SIP. The method ServeRequest() builds SIP model for day α. Those time slots that are already taken to serve other patients are blocked off as follows: 1. For R I, set right hand side of constraint (1b) = 0if (α, t) U r. 2. For S I, set right hand side of constraint (1b) = 0if (α, t) V s. If no appointment is found (line 20), we search the following week in the scheduling horizon, α = α + ω. If an appointment is found, we check if the waiting time is shorter than a month (lines 11-12). If the waiting time is longer than a month, the algorithm searches for an alternative appointment on a different date. This new search starts on day α d j + m + λ. If the waiting time is less than a month for the appointment found, the information is passed back to the scheduler and to the patient (line 17). The intuition behind the NMSOS algorithm is explained with an example in the Appendix for the interested reader. 5 Application The NMSOS algorithm first initializes both the patient set J and the patient identification number. The scheduling horizon in terms of days is defined in line 2. The number of time periods in a day are defined in line 3. Parameter τ defines the maximum number of time periods in a day. Method GetPatientRequest() is invoked by the algorithms when a request is received at the clinic. The method gets the required information from the patient which includes the nuclear medicine procedure p and the preferred day for an appointment q (line 4). A counter λ is initialized to zero and the time and day of the request is assigned The methodology derived in the previous sections was implemented and applied to the Scott & White Clinic Nuclear Medicine Clinic at Temple, TX. We conducted a series of simulation experiments to test the performance of our algorithms in a real environment. The NMOS and NMSOS scheduling algorithms were compared against a benchmark scheduling algorithm called Fixed Resource (FR). The FR algorithm was validated in [21] and represents an example of current practice. The discrete event specification (DEVS) simulation developed by [20] was used to assess the performance of the algorithms based on historical data of the Scott & White Nuclear Medicine. The NMOS and NMSOS algorithms were implemented in JAVA and ILOG CPLEX.

11 Stochastic online appointment scheduling of multi-step sequential procedures in nuclear medicine 5.1 Real nuclear medicine setting The Scott & White Health System in Temple, Texas has one of the largest fully accredited nuclear laboratories for general nuclear imaging and non-imaging in the U.S. This facility operates from Monday to Friday from 8:00 am to 5:00 pm. At the time of the study, the clinic had eight technologists, 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 procedures. Nurses provide support to the other members of the staff. The division manager can assist in most of the activities in the absence of one of the regular staff. The clinic also has nuclear medicine physicians, radiology residents, and a cardiologist. There are twelve stations within this facility. Equipment such as gamma cameras are located inside the stations. The facility has seven gamma camera stations. Five of these cameras are planar, and capable of doing 2D whole-body imaging and 3D Single Photon Emission Computed Tomography (SPECT). The other two cameras are planar as well, but one is SPECT capable and the other is for imaging only. Two of the stations are equipped with EKG capability. The other stations are treatment (TRT) rooms for patient hydration and waiting. In this clinic around sixty different procedures are performed. Table 2 in Section 3 depicts the procedures that were performed more frequently at the clinic during the year. These are the procedures used in our study because they account for more than 90 % of the procedures requested. 5.2 Experimental setup The optimization models used in the scheduling algorithms were formulated based on the data provided by the Scott & White Healthcare Clinic Nuclear Medicine Department. The models involve the following stations: 7 rooms with gamma cameras, 2 stress rooms with treadmills, and 3 TRT 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 resource names at the clinic with the following names: Technologist(1), Technologist(2), Technologist(3), Technologist(4), Technologist(5), Technologist(6), Technologist(7), Technologist(8), Technologist(9), Technologist(10), Nurse(1), and Manager(1). The scenarios for the SIP model are generated using empirical distributions that are based on historical demand data from the clinic. A scenario comprises several procedures with a high chance of been requested after serving the current patient. Monte Carlo simulations were used to obtain the scenarios required for each SIP model. The number of procedures per scenario range from 1 to 20. As explained in Section 4.3, we condition the number of patients per scenario to the current number of patients already schedule on the earliest day we can schedule our current request. This strategy is used to reduce the size of the SIP model. We conducted a series of simulation experiments to compare the performance of the NMOS and NMSOS scheduling algorithms against a benchmark algorithm called Fixed Resource (FR). The FR algorithm details are explained in [21]. The FR algorithm schedules patients and resources as early in the schedule as possible by taking into account the patient preferred day for an appointment. In this algorithm some of the human resources are assigned to always serve patients on specific stations. For example, two of the technologists can always be assigned to serve patients in two of the gamma camera stations. The algorithm also takes into account patient waiting times for an appointment. In case the appointment found for the patient results in a waiting greater than a month, the algorithm searches for another appointment on a different day of the week. Two empirical distributions were derived to generate the procedure type requested and preferred day of the appointment for each patient. A list of the nuclear medicine procedure types considered is presented in Table 2. Intermsof the preferred day of the appointment, Monday and Friday were identified as the days of the week requested the most by patients. Since procedure request arrivals are independent, a Poisson process was assumed for patient procedure requests. The monthly call interarrival times in minutes followed an exponentialdistribution 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, In our experiments, we considered the impact of having different demand levels. We defined three patient demand levels: low demand, base demand,andhigh demand.thebase demand level uses interarrival times that are based on the historical data provided by the clinic. For the low demand and high demand levels we decreased and increased the demand rate by 10 %, respectively. The performance measures listed in Table 5 were used to measure the system performance under the different scheduling algorithms. These performance measures provide an assessment of the system in terms of both patient and management perspectives. We conducted a total of nine experiments each involving 20 replications with different seeds for the random number generator to allow for independence among the replications. The experiments involved running the following algorithms: Fixed Resource, Nuclear Medicine Online Scheduling (NMOS), and Nuclear

12 E. Pérez et al. Medicine Stochastic Online Scheduling (NMSOS). For each simulation run we used a scheduling horizon of 12 months. We compute several statistics for each performance measure. The experiments were conducted on a Dell X5355 computer with 2 Intel(R) Xeon(R) X processors at 2.66 GHz each with 12.0 GB of RAM. 6 Computational experiments We now report computational results to evaluate the robustness of the NMOS and NMSOS algorithms. The results were compared against the FR algorithm, for each of the three patient demand level scenarios: (a) low patient demand, (b) base patient demand, and(c)high patient demand. The FR algorithm schedules patients as early as possible, which may not be appropriate when your resources are limited in terms of capacity and expertise. The NMOS and NMSOS algorithms use optimization models to improve patient schedules. Finally, we present a sensitivity analysis to evaluate the performance of the algorithms when the procedure type demand is not based on empirical distributions. 6.1 Computational results We report the average number of patients served using the FR, NMOS, and NMSOS algorithms under three demand levels in Fig. 2. Under the low patient demand level, the FR and NMOS algorithms provide similar results. The NMSOS algorithm provides over 1 % improvement compared to the FR algorithm. Under the base patient demand level, the results show a pattern similar to the one observed under the low patient demand. The FR and NMOS algorithms provide similar results and the NMSOS algorithm provides an improvement of over 1 %. For the high patient demand level the NMSOS algorithm provides an over 4 % improvement compared to the FR algorithm. A 4 % improvement is significant because it translates into having about 600 more patients served on average per year. Tables 6, 7,and8 provide the average number of patients served per month with a 95 % confidence interval under the low, base, and high patient demand levels respectively. The results show that NMSOS algorithm is able to accommodate more patients early in the year for the low patient demand and base patient demand levels. The FR and NMOS algorithms start accommodating more patients later in the year which is simply the early demand pushed into the future. This has an impact on the patient waiting time which is discussed later in this section. Under the high patient demand level, the NMSOS algorithm outperform FR and NMOS accommodating more patients per month. Average number of patients served Low Base High FR 14, , , NMOS 14, , , NMSOS 14, , , Fig. 2 Average number of patients served per year under three demand 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 utilization is shown on the left side of the figure and the average station utilization on the right side of the figure. The graphs display the average utilization of each resource for a year of clinic operation. Figure 3 shows that the FR algorithm provides a higher utilization for most of the technologists at the clinic and a lower utilization for the nurse and the manager. On the other hand, the NMOS and NMSOS algorithms provide a more balanced utilization of the human resources. In terms of station utilization, the NMSOS algorithm was able to increase the utilization of the gamma camera stations of type Axis and also of the Treadmill stations. The FR algorithm provides a more balanced utilization of the gamma cameras. The overall average utilization of both human resources and stations was about the same for the FR and NMOS algorithms under the patient low demand, but it was increased by about 1 % with the NMSOS algorithm. Figure 4 shows a plot of the resources utilization under base patient demand for each scheduling algorithm. The graph depicts a similar pattern when compared to the low demand case. Again, the overall average utilization of both human resources and stations was about the same for the FR and NMOS algorithms, but it was increased by over 1 % with the NMSOS algorithm. Figure 5 show that the NMSOS algorithm provides a higher utilization for most of the technologists at the clinic and a higher utilization of the nurse. NMSOS also provides a more balanced utilization of the human resources. In terms of station utilization, the NMSOS algorithm was able to

13 Stochastic online appointment scheduling of multi-step sequential procedures in nuclear medicine Table 6 Number of patients served per month under low demand Month FR NMOS NMSOS Jan ± ± ± 3.67 Feb 1, ± , ± , ± 3.85 Mar 1, ± , ± , ± 3.68 Apr 1, ± , ± , ± 3.91 May 1, ± , ± , ± 3.17 Jun 1, ± , ± , ± 3.56 Jul 1, ± , ± , ± 4.53 Aug 1, ± , ± , ± 4.14 Sep 1, ± , ± , ± 4.37 Oct 1, ± , ± , ± 3.80 Nov 1, ± , ± , ± 3.73 Dec 1, ± , ± , ± 5.93 Jan 1, ± , ± , ± 3.46 Table 7 Number of patients served per month under base demand Month FR NMOS NMSOS Jan 1, ± , ± , ± 4.07 Feb 1, ± , ± , ± 2.60 Mar 1, ± , ± , ± 3.91 Apr 1, ± , ± , ± 4.99 May 1, ± , ± , ± 3.51 Jun 1, ± , ± , ± 4.71 Jul 1, ± , ± , ± 2.49 Aug 1, ± , ± , ± 4.34 Sep 1, ± , ± , ± 3.42 Oct 1, ± , ± , ± 3.14 Nov 1, ± , ± , ± 2.04 Dec 1, ± , ± , ± 2.43 Jan 1, ± , ± , ± 3.00 Table 8 Number of patients served per month under high demand Month FR NMOS NMSOS Jan 1, ± , ± , ± 2.65 Feb 1, ± , ± , ± 1.38 Mar 1, ± , ± , ± 2.23 Apr 1, ± , ± , ± 1.62 May 1, ± , ± , ± 2.21 Jun 1, ± , ± , ± 2.78 Jul 1, ± , ± , ± 2.10 Aug 1, ± , ± , ± 2.79 Sep 1, ± , ± , ± 2.36 Oct 1, ± , ± , ± 3.15 Nov 1, ± , ± , ± 3.56 Dec 1, ± , ± , ± 3.51 Jan 1, ± , ± , ± 3.07

14 E. Pérez et al. Average human resource utilization (%) Average station utilization (%) FR NMOS NMSOS FR NMOS NMSOS Fig. 3 Resources utilization under low demand Average human resource utilization (%) Average station utilization (%) FR NMOS NMSOS FR NMOS NMSOS Fig. 4 Resources utilization under base demand Average human resource utilization (%) Average station utilization (%) FR NMOS NMSOS FR NMOS NMSOS Fig. 5 Resources utilization under high demand

15 Stochastic online appointment scheduling of multi-step sequential procedures in nuclear medicine increase the utilization of the gamma camera stations and the treadmill stations. NMSOS algorithm also provides a more balanced utilization of the gamma cameras. The overall average utilization of both human resources and stations was about the same for the FR and NMOS algorithms under patient high demand, but it was increased by over 4 % with the NMSOS algorithm. The results related to patient perspective are reported in Figs. 6 and 7. Figure 6 reports the patient average waiting time Type 1 for the three algorithms under the three patient service demands. Recall that the patient waiting time Type 1 is the time a patient has to wait from the time they make a request to the actual appointment time. The NMSOS algorithm provides a lower waiting time for the patients under the three demand levels. NMSOS reduces the waiting time compared to the FR algorithm by about 10 % for the patient low and base demand cases, and by about 25 % for the high demand case. This performance can be attributed to the fact that NMSOS is able to accommodate more patients early in the year. Thus most patients end up waiting less time. Figure 7 shows the patient preference ratio for the three algorithms under the three different patient service demands: low, base, and high. Recall that the patient preference ratio is the portion of patients scheduled on their requested day. The NMSOS algorithm provides a higher preference ratio for the patients under the three demand levels when compared to the FR algorithm. NMSOS increases the patient preference ratio by about 5 % for the patient low and base demand cases, and by about 80 % for the high demand case. In terms of computational time, it takes about two minutes to get an appointment for a patient using the NMSOS Average patient waiting Type1 (days) Low Base High FR NMOS NMSOS Fig. 6 Patient waiting type 1 Average patient preference satisfaction ratio (%) Low Base High FR NMOS NMSOS Fig. 7 Patient preference satisfaction ratio algorithm. The FR and NMOS algorithms take only a few seconds. The NMSOS algorithm takes longer because it formulates and solves an SIP model for each patient request. 6.2 Sensitivity analysis We now present sensitivity analysis for the performance of the algorithms under different demand patterns. So instead of using an empirical distribution, we considered a uniform distribution to generate the type of procedure requested by the patient. A uniform distribution was chosen to give each procedure type the same probability of being requested. This same distribution is used to generate the scenarios in the SIP model. We now report the computational results to evaluate the robustness of the NMOS and NMSOS algorithms. The results were compared against the FR algorithm, for each of the three demand levels: (a) low patient demand, (b)base patient demand, and(c)high patient demand. The performance of the algorithms is similar to what is reported in the previous section. Hence, we report results for only three of the performance measures, the number of patients served per year, patient waiting time Type 1, and patient preference satisfaction ratio. These three performance measures represent both patient and management perspectives. Figure 8 reports the average number of patients served using the FR, NMOS, and NMSOS algorithms under the three demand levels. For the low demand case, FR and NMOS report similar results and the NMSOS algorithm reports about 1 % improvement over the FR algorithm. The same behavior is observed under the base demand case. Lastly, under high demand, NMSOS outperforms the FR and NMOS

16 E. Pérez et al. Average number of patients served Low Base High FR 12, , , NMOS 12, , , NMSOS 12, , , Fig. 8 Sensitivity analysis average number of patients served per year under three demand levels: low, base, and high algorithms by 2 %. This improvement represents about 300 more patients served per year. As reported earlier, NMSOS is able to accommodate more patients into the schedule with the same number of resources when the demand at the clinic is increased. Next we report the results for the patient perspective performance measures. Figure 9 reports the average waiting time Type 1 for the three algorithms under the three patient demand levels. The NMSOS algorithm provides a lower waiting time for the patients under the three demand levels. When compared to the FR algorithm, NMSOS decreases waiting time by 11 % for low and base demand cases, and by 26 % for the high Average patient waiting Type Low Base High FR NMOS NMSOS Fig. 9 Sensitivity analysis patient waiting type 1 Average patient preference satisfaction ratio (%) Low Base High FR NMOS NMSOS Fig. 10 Sensitivity analysis patient preference satisfaction ratio demand case. Figure 10 shows the patient preference ratio for the algorithms under the three demand levels. When compared to the FR algorithm, NMSOS provides a higher preference ratio under the three demand cases. NMSOS increases patient preference ratio by about 3 % for the low and base demand cases, and by about 73 % for the high demand case. The results presented in this section confirm the robustness of the performance of the NMSOS algorithm. 7 Summary and conclusions Appointment scheduling in specialized clinics such as nuclear medicine departments is a very challenging problem. Radiopharmaceutical properties require nuclear medicine procedures to be performed following strict protocols that must be adhered to by the human resources. In this paper we derive an online scheduling (NMOS) algorithm and a stochastic online scheduling (NMSOS) algorithm for patient and resource scheduling in nuclear medicine departments. The scheduling algorithms take into account the time constraints imposed by the decay of the radiopharmaceuticals. Both algorithms were implemented within a simulation framework and experiments are based on historical data from an actual clinic. We compared the results of our algorithms against the Fixed Resource (FR) algorithm, which is based on the operation of an actual clinic. We obtain computational results that provide evidence of the benefits of considering stochastic future arrivals information when scheduling patients in health care clinics. We found that the number of patients served was significatively larger under the NMSOS scheduling algorithm when compared to the FR scheduling algorithm, especially under high