An optimal decision-making approach for the management of radiotherapy patients

Transcription

1 OR Spectrum DOI /s y REGULAR ARTICLE An optimal decision-making approach for the management of radiotherapy patients D. Conforti F. Guerriero Rosita Guido M. Veltri Springer-Verlag 2009 Abstract In this paper, novel integer programming formulations are developed for solving the optimal scheduling of patients waiting for radiotherapy treatment. In this specific clinical domain, the suitable management and control of a patients waiting list strongly affect both the quality of the therapeutical outcome, in terms of effectiveness, and the cost-saving use of the overall therapeutical resources, in terms of efficiency. The proposed models allow the best scheduling strategy to be devised by taking into account the quality of the health care service offered to the patient as well as the status and the preferences of the patient. The computational experiments, carried out on realistic scenarios and considering real data, are very promising and show the efficiency and robustness of the proposed models to address the problem under consideration. Keywords Radiotherapy patient scheduling Mathematical programming 1 Introduction Government health care policy makers, health insurance companies, service providers and user s organizations are changing the face of health care delivery, especially across developed western countries, requiring that quality and cost efficiency of health care, D. Conforti F. Guerriero R. Guido (B) Laboratory of Decisions Engineering for Health Care Delivery, Dipartimento di Elettronica, Informatica e Sistemistica (DEIS), Università della Calabria, Via P. Bucci, cubo 41 C, Arcavacata di Rende (CS), Italy rosita.guido@deis.unical.it M. Veltri Radiotherapy Division, General Hospital of Cosenza, Cosenza, Italy

2 D. Conforti et al. safety and empowerment of patients play an even more crucial role in the management of the national and regional health care systems. Improving health care quality, while reducing costs, requires the elimination of unintended and unnecessary overheads in the entire care process and the application of new and more accurate quantitative procedures for the organization and management of health care delivery. To this end, advanced technologies and innovative quantitative approaches can play a strategic role. In fact, during the last few years, there has been an increasing development of high technologically and methodologically effective solutions to foster evidence-based medicine and best health care practices. By integrating these advanced technologies and methodologies and exploiting the high potentiality of problem representation and solution, which characterizes models and methodologies of decision science, it is possible to improve and make more efficient, effective and accurate all the health care processes within several health care domains. Within this broad context, a very challenging area is that related to the development of optimal scheduling procedures, which allows not only to reduce to a minimum of staff idle time, but also to improve the patient flow, providing timely treatment and maximum utilization of the available medical resources. The majority of the literature on the scheduling in health care systems is centred around nurse scheduling (Cheang et al. 2003; Burke et al. 2004a; Ernst et al. 2004a,b). Over the years, the problem of determining a high quality solution for the staff scheduling problem in hospitals has been addressed by many scientists and a wide range of solution techniques has been investigated. Indeed, in Ernstetal.(2004a,b), the methods proposed in the scientific literature were grouped into 28 different categories. They range from mathematical programming methods (see, e.g. Millar and Kiragu 1998; Miller et al. 1976; Jaumard et al. 1998) to (meta)heuristic and hybrid approaches (see, e.g. Bellanti et al. 2004; Dowsland 1998; Dowsland and Thompson 2000; Valouxis and Housos 2000; Burke et al. 1998, 2004b); from expert systems to simulation techniques (see, e.g. Petrovic et al. 2003; Chen and Yeung 1993; Valouxis and Housos 2000). Of all the proposed methods, metaheuristic methods seem to be the most suitable to solve real problems (Burke et al. 2004b). Another very challenging problem in health care systems is related to the optimal assignment of patients to medical resources. Solving this problem relies on the development of procedures that allow the determination of how patient appointments are scheduled, their time length and the time between appointments. The ultimate goal is to guarantee the delivery of the right treatment at the right time, by ensuring the effective use of all resources involved. The problem is quite complex, because several goals are pursued and a large set of constraints have to be taken into account (Harper and Gamlin 2003). Patient scheduling has been the subject of scientific investigation since the beginning of the 1950s. Indeed, the first work dates from 1952, with the prominent contribution of Bailey and Welch (1952), where the first advanced outpatient appointment scheduling rule is proposed and tested, through a simulation approach. Since then the research has expanded considerably and particular emphasis has been given to the outpatient scheduling problem. The scientific work on appointment scheduling in outpatient services was surveyed in two excellent reviews by Cayirli and Veral (2003), Cayirli et al.(2006), whereas a bibliography of the application of queuing theory to the problem under consideration was presented in Preater (2001).

3 An optimal decision approach for management of radiotherapy patients In the context of patient scheduling, a new and specific area of concern is represented by medical treatments involving radiation (i.e., radiotherapy treatments). Basically, these treatments involve the effective clinical use of ionizing radiation delivered by a linear accelerator (linac) for cancer treatment (Washington and Leaver 2003; Perez et al. 2004). Within this clinical domain, a crucial and complex problem to be faced is the effective and efficient scheduling of the patients waiting to start a treatment plan. In particular, the management of the patient s waiting list strongly affects both the quality of the therapeutical treatment, in terms of effectiveness, and the cost-saving use of the linac, in terms of efficiency (Han et al. 2005). Indeed, improper scheduling procedures can have a severe impact on the success of treatments and, above all, could potentially affect the survival rate of the patients involved (Ragaz et al. 2004). The radiotherapy patient scheduling problem can be classified into two main categories: block system and non-block system. In the block system, the workday is split into a fixed number of time blocks/slots, usually with the same duration (10/15 min), during which one radiotherapy session can be delivered. In the non-block system, a different treatment time is assigned to each patient (Burnet et al. 2001). Even though the use of uniform appointment blocks leads to a poor representation of the real workload (in fact, the real treatments can take either more or less time than the assigned time block), this booking strategy is adopted by the majority of radiotherapy centers, for its easy applicability. Given its practical importance, the block-system is considered in this paper. It is worth observing that, although the use of quantitative approaches can improve the performance of an appointment-based system, the development of optimization models and methods to address the radiotherapy patient scheduling problem has not attracted much attention in the scientific community. In particular, the contribution of Kapamara et al. (2006) is quite remarkable, since it presents a review of scheduling problems in radiotherapy and proposes to formulate the scheduling problem as a dynamic job-shop problem. In Petrovic et al. (2006), two algorithms to schedule radiotherapy treatments for patients of different categories on a daily basis, by taking into account due-date and release date constraints were proposed. The first approach schedules the patients forward from the release date, whereas the second method books the patients backward from the due date. The main aim is to find a feasible schedule, by which the number of patients, not matching a given set of time constraints, is minimized. An efficient block scheduling strategy was proposed by three of the authors in Conforti et al. (2008), where innovative mathematical models for booking patients, in a prioritized waiting list, are presented. These models allow patients having treatment in course to be rescheduled, if it results in an increase in the total number of scheduled patients (i.e., the number of new patients that can begin their treatment session during the planning horizon is maximized). The main contribution of the present paper is twofold: (i) the models proposed in Conforti et al. (2008) are extended, by taking into account other important requirements, such as patient availability, and (ii) the performance of the proposed mathematical models on a set of randomly generated instances and on a real case study is validated.

4 D. Conforti et al. The paper is structured as follows. In Sect. 2, the proposed approaches to schedule radiotherapy treatment sessions are illustrated in detail. In Sect. 3, the results obtained by some numerical experiments are compared and discussed, underlining the possible improvement of working lives, reduction of waiting time and delays in the start of radiotherapy planning for the patients. Finally, some concluding remarks complete the paper. 2 Problem statement and mathematical formulation Radiotherapy is a quite effective way for treating many kinds of cancer, allowing several therapeutical goals (Washington and Leaver 2003; Perez et al. 2004) tobe achieved. It is often given on its own, in order to destroy a tumor and to cure the cancer. In this respect, it is described as radical/curative radiotherapy aiming at giving long-term benefits to the patient. On the other hand, the radiotherapy may be given before surgery to shrink a tumor or just after surgery to stop the growth of cancer cells that still remain in situ. It can also be given before, during, or after chemotherapy to improve treatment outcomes. Sometimes, when it is not possible to effectively cure a cancer, the radiotherapy is used as palliative treatment to reduce pain or relieve other severe symptoms. The amount of radiotherapy to be delivered depends on different factors, such as site, size and type of cancer, and overall pathological conditions of the patient. The total amount of radiation is computed by a radiotherapist, who also determines the dose fractions which will be delivered during treatment sessions, along a planned time horizon (Barendsen 1982). Typically, a radiotherapy treatment plan is devised so that it is well tailored to the patient; consequently, several and different radiotherapy treatment plans are possible. Some patients have long treatment plans, and their treatment sessions everyday for a predetermined period of time or in some cases once or twice a week (i.e., for palliative treatments (DeVita et al. 2007). The main requirements that should be taken into account in developing the radiotherapy treatment plan, are: a fixed number of treatment sessions has to be carried out on consecutive days and, appropriately spaced out, on consecutive weeks, as prescribed by the radiation oncologist; only one treatment session per day can be delivered to each scheduled patient; the same linac must be used during the entire treatment plan, since technical characteristics could vary among machines. After completing all phases that precede the treatment planning, the radiation oncologist assigns to the patient a priority value on the basis of the severity of pathological conditions (Lim et al. 2005). If it is not possible to make a booking during the planning horizon, the patient is inserted in a waiting list. As a consequence, the waiting list is partitioned into ordered sublists, such that each of them is a collection of patients with the same assigned priority. In this respect, it is important to point out that the priority value has to be frequently updated, according to the possible variations in patient conditions.

5 An optimal decision approach for management of radiotherapy patients Table 1 Example of booked treatment sessions Time slots Monday P 1 P 3 P 4 Tuesday P 1 P 3 P 4 4 patients (P 1, P 2, P 3, P 4 )have already begun treatment plans in the past. The appointments have been scheduled by respecting their known availability Wednesday P 1 P 2 P 3 P 4 Thursday P 1 P 2 P 3 P 4 Friday P 1 P 2 P 3 Saturday P 2 In this paper, the patient radiotherapy scheduling problem is considered over a welldefined planning horizon, by which the radiotherapist determines, for each patient, the total number of due therapy sessions. It is assumed that the planning horizon is a week (i.e., 6 days). This is reasonable, because the patients waiting for a radiotherapy treatment are generally scheduled on a week-to-week basis. In addition, it is assumed that each weekday is partitioned into a given number of time slots with the same fixed duration, during which the therapy session is carried out. It can be observed that the entire system usually is partially booked since there are some time slots already assigned to booked patients. An example of a partially booked system is reported in Table 1 where, for simplicity, only seven time slots are reported and those already assigned to patients are highlighted. In what follows, two optimization models, which allow radiotherapy staff to schedule patients aiming at reducing the size of the waiting list and efficiently using the linac, are proposed and described. The models are based on a block scheduling strategy, since most of the radiotherapy appointment booking systems, typically performed by hand, are based on this approach. The two proposed optimization models represent two different scenarios: in the first model, appointments of patients already booked cannot be modified, whilst in the second model, it is possible (if necessary) to reschedule some patients on the basis of availability and taking into account the maximum interval between each pair of consecutive week-sessions. 2.1 Notation and assumption We now describe our notation and present the specific conditions, on the basis of which the proposed models have been developed. Let K (indexed by k) be the set of workdays in the planning horizon (i.e., a week), and W (indexed by w) be the set of time slots of each workday. Let, also, WP (indexed by j) be the waiting list of unscheduled patients, i.e., the list of patients ready to start their own treatment plan, and BP (indexed by p) be the set of booked patients that have already begun the treatment plan in the past. The data used for both models, for each patient j WP, are the following: pr j, priority value assigned by the radiation oncologist on the basis of the severity of patient conditions;

6 D. Conforti et al. t j, number of treatment sessions per planning horizon (i.e., the number of consecutive days, since only one treatment session per day can be delivered); mts j, minimum number of treatment sessions per planning horizon; ld j, the latest first day session if all the prescribed treatment sessions t j are booked; ld j, the latest first day session if mts j treatment sessions are booked; RTS j and DTS j, dimensional vectors, used to take into account the release time slot and the due time slot, respectively, during each working day Scheduling of the treatment sessions A new patient j WP is scheduled in the planning horizon if it is possible to deliver all t j treatment sessions on consecutive t j days, as prescribed by the radiation oncologist, without interruption. In some cases, it is convenient to relax this hard constraint in order to maximize the number of patients starting their treatment plan. In fact, some specific situations can happen when booking a new patient j in the planning horizon: 1. the number of free time slots on consecutive days is less than the required t j sessions; 2. the patient j is not available in some free time slots, and it is impossible to book all t j sessions. In these particular situations, the number of possible sessions is less than the prescribed t j. To make the optimization models more flexible, the parameter mts j has been introduced for each waiting patient j; it defines the minimum number of consecutive treatment sessions that every newly booked patient must carry out during the planning week. In particular, mts j = t j means that all prescribed treatment sessions must be carried out, and the first treatment session can take place, at the latest, on the day ld j = t j +1. However, mts j < t j means that at least mts j sessions have to be booked and the latest starting day is computed, in this case, as ld j = mts j + 1. In the best case, the number of assigned treatment sessions to each new scheduled patient j is equal to the prescribed number t j First radiotherapy session Another important issue to be considered concerns the first treatment session, during which several setting operations have to be carried out (Turner and Qian 2002). To this end, an auxiliary time slot must be assigned to each new booking patient such that the first treatment session covers two consecutive time slots Patient availability In case, the radiotherapy treatment is combined with chemotherapy or it is prescribed before surgical intervention, the same treatment should be given in a predefined period of the time. For this reason, it is important to take into account the availability of a patient each weekday of the considered planning horizon. To this end, we define for each patient i the release time slot r ik and the due time slot d ik as the start and the final time slots on day k, respectively; in this way, the period during which the patient is

7 An optimal decision approach for management of radiotherapy patients available for treatment sessions, is determined as [r ik, d ik ]. Obviously, the following conditions have to be satisfied: d ik r ik ; 0 r ik W ; 0 d ik W, i WP BP, k. In addition, if patient i is not available on day k K, wehaver i k = 0 and d i k = Linear accelerator availability We observe that the availability of only one linac has been assumed, even though the proposed models can be easily extended to address the case of more than one linac. In the sequel, the first and second models are referred to as basic and enhanced model, respectively. 2.2 Basic model The basic model has been developed assuming that the appointments of patients already booked cannot be modified. To keep track of the already assigned time slots, the matrix sched W has been used. Its generic element is defined as: { 1 if the time slot w on day k is already assigned; sched kw = 0 otherwise. The basic decisions to be taken concern which patient to schedule among those waiting, and consequently when to start the first therapy session and when to schedule the subsequent sessions. These decisions can be mathematically formulated using the following binary variables: { 1 if the first appointment for patient j is assigned to time slot w on day k; z jkw = 0 otherwise. { 1 if the patient j is assigned to time slot w on day k; y jkw = 0 otherwise. Since the first session covers two consecutive time slots, the binary variable f jkw, with the same meaning of variable z, has been introduced. In case, the first appointment of patient j is assigned to the time slot w on day k (z jkw = 1), then also the successive time slot w + 1 is assigned to the same patient j, hence f jk(w+1) = 1. The condition that guarantees two consecutive time slots are assigned to each new booked patient for only the first session of the planned course, is imposed by the constraints (1); constraints (2) can be interpreted as boundary conditions: z jkw = f jk(w+1) j, k, w < W (1) f jk1 = 0, z jk W = 0 j, k. (2) Since the first session is one booked appointment, we have: z jkw y jkw j, k, w. (3)

8 D. Conforti et al. The following set of constraints assures that it is possible to deliver the first session only once during the planning week, such as to guarantee at least mts booked sessions, and only when the patient is available [constraints (4)]. In fact, constraints (5) and (6) avoid fixing any appointment when at least mts sessions are not bookable or when the patient is unavailable: ld j d jk z jkw 1 j (4) W k>ld w=1 j r jk 1 w=1 z jkw = 0 j (5) y jkw = 0, W w>d jk y jkw = 0 j, k. (6) For example, z j k w = 1 means that patient j begins the treatment plan in the time slot w on day k, and it is also assured that at most mts j treatment sessions can take place on consecutive days during the planning horizon. It is well known that positive therapeutical effects are obtained when all the prescribed treatment sessions t j are booked. With the aim to book the remaining t j mts j sessions when it is possible (if mts j < t j ), the auxiliary binary variables ȳ jk = {0, 1}, j and k are introduced. More specifically, ȳ jk = 1 means that another session of patient j could be carried out during day k, but it is impossible to settle it, since either the patient is not available or all the time slots during this day have already been assigned to some other patient. Hence: d jk y jkw +ȳ jk d jk k s=1 d jk z jsw j, k (7) ȳ jk 1 y j (k+1)w j, mts j < k <. (8) Constraints (7) impose that, on each weekday k, a new appointment can be booked only if a first appointment has already been booked. Constraints (8) ensure that the booking of treatment sessions are consecutive and therefore without interruption. In order to ensure that each new scheduled patient j has t j booked sessions in t j consecutive days during the planning horizon, the following constraints (9) (11) j WP, are defined: d jk k+t j 1 s=k d jk y jsw + ȳ js1 t j z jkw k ld j (9) s 1 =k+mts j

9 An optimal decision approach for management of radiotherapy patients d jk s=k y jsw + d jk y jkw + d jk ȳ js1 start j z jkw k > ld j (10) s 1 =k+mts j ld j ȳ jk = t j d jk z jkw + ld j k>ld j d jk start j z jkw (11) where start j = ( t j k + ld j ) is zero only if the start day of the weekly booking of sessions precedes the ld j weekday; otherwise, t j sessions will not be booked. Constraints (9) guarantee that the scheduled treatment sessions will be at least equal to the number of the prescribed treatment sessions. Thus, if exactly the prescribed t j sessions are booked to patient j, constraints (9) will be active. The same applies to constraints (10) if the booked sessions are at least mts j. In this case, also constraints (10) will be active. Finally, constraints (11) represent boundary conditions, imposing that the total number of booked sessions in the planning horizon is equal to the total number of prescribed treatment sessions. In the best case, exactly t j treatment sessions are reserved to patient j. Finally, taking into account that there are already booked patients, and that only one patient at a time is treated in a given time slot for each day, the following constraints are formulated: WP j=1 ( y jkw + f jkw ) + schedkw 1 k,w. (12) The goal of the model is to maximize the number of new scheduled patients taking into account the priority values and the position in the waiting list. It is also important to maximize the number of booked sessions during the week. Then, the objective function is given as a sum of two terms: the first one is the total number of newly scheduled patients (i.e., WP j=1 the booked appointments (i.e., WP assumes the following form: max WP j=1 j=1 W w=1 W pr j w=1 j z jkw ); the second term is the sum of pr j j y jkw ). The objective function W w=1 pr j (z jkw + y jkw ). (13) j The factor 1/j has been introduced in order to avoid the generation of equivalent solutions, i.e., schedules with the same number of scheduled patients, but with a violated precedence rule among patients belonging to the same priority sub-queue. Indeed, this factor allows discrimination among patients with the same priority value and the same number of treatments per week, on the basis of their access to the waiting list WP. In the Appendix, the overall mathematical formulation of the basic model is summarized.

10 D. Conforti et al. Table 2 Used data of the booked patient P 1 for planning week t lps r d break sd delay lsd P Enhanced model With the aim to further improve the booking of treatment sessions of patients on W P list, an enhanced version of the basic model is now described. The enhanced model is based on rescheduling the already booked patients with a treatment plan in progress (i.e., each patient p belonging to BP). This will be achieved by exploiting patient availability since for each of them the release time and due time slots on each weekday are known. Rescheduling has a double meaning: only the time slots could change with respect to the last planned week or it is possible to delay the day of the first weekly session. We remark that the enhanced model reschedules some patients, who have a treatment plan in progress, only when it is necessary. This aspect will be better explained in Sect The rescheduling is done by considering for each patient p BP the following additional information: t p, number of treatment sessions per planning horizon (i.e., from 1 to 6 treatment sessions); lps p, weekday of the last planned treatment session in the last week; r pk and d pk, release and due time slot on each day, respectively. The patient availability interval time is [r pk, d pk ], k K; break p, feasible interval (in days) between two consecutive week sessions; sd p, starting day in the planning horizon; delay p, feasible delay (in days) in starting the first weekly session; lsd p, latest starting day in the planning horizon. For the sake of clarity, let us consider the scenario referring to the last planned week and reported in Table 1, as well as patient P 1, whose data, relevant to the planning week, are summarized in Table 2. As reported in Table 1, his/her last booked weekly treatments were t = 5, from Monday to Friday, with lps = 5. The interval between two consecutive week sessions is set to break = 2, thus the first treatment session within the planning week will be on Monday, that is sd = 1. Generally, the weekly start day is computed as follows: sd = { break +lps, if break > lps 1 otherwise. In some cases, this patient could delay the weekly start day by at most delay 0 days, if this leads to the maximization of the number of new booked patients. Since the completion of t sessions must be assured in the planning week, the latest start day is computed as lsd = min {sd + delay, t + 1}. Thus, the start day will be in the range [sd; lsd]. For instance, delay P1 = 3 means that patient P 1 could start the weekly treatments on Monday, Tuesday, Wednesday or Thursday, but since all the

11 An optimal decision approach for management of radiotherapy patients t = 5 treatments must take place, he/she can start only on either Monday or Tuesday which is thus the latest feasible day (given as t + 1 = = 2) for starting the 5 treatment sessions. The constraints relevant to the rescheduling of patients p BP are formulated in what follows, from (14) to(21), taking into account the above mentioned conditions. The constraints (14) (16) have to be imposed to fix the first treatment in the planning horizon, where both patient availability and the break between two consecutive planned weeks are considered. Then, p BP, wehave: lsd p d pk k=sd p r pk 1 w= r pk z pkw = 1 (14) z pkw + w=1 W k>lsd p w=1 W w=d pk +1 z pkw = 0 (15) z pkw = 0. (16) It can be observed that every patient p BP must be booked, hence the constraints (14). Each patient p can be assigned only to one time slot per weekday, and obviously it is not possible to treat when the patient is not available: z pkw y pkw p, k,w (17) d pk w=r pk y pkw 1 p, k (18) r pk 1 w=1 y pkw + W w=d pk +1 y pkw = 0 p, k. (19) Further, it must be asserted that t p treatment sessions are fixed in t p consecutive days, that is p BP: d pk w=r pk k=sd p k+t p 1 s=k d pk d pk y psw t p z pkw k = sd p,...,lsd p (20) w=r pk w=r pk y pkw = t p lsd p k=sd p d pk w=r pk z pkw. (21) The constraints, imposed in order to schedule the patients belonging to WP list, remain the same as in the basic model [see constraints (1) (11)].

12 D. Conforti et al. Finally, since it is possible to treat only one patient belonging either to BP or to WP list, at a time, the constraints (12) are replaced by the following constraints: j WP ( y jkw + f jkw ) + p BP y pkw 1 k, w. (22) The goal of the enhanced model is still the same as the basic model, but since there are some rescheduled patients, with the aim to minimize their delay in starting the weekly sessions, a second term in the objective function is added. The objective function thus assumes the following form: max WP j=1 W w=1 pr j (z jkw + y jkw ) j BP (k sd p ) z pkw (23) p=1 The overall mathematical formulation of the enhanced model is given in the Appendix. W w=1 3 Computational experiments This section reports the results obtained on the basis of an extensive series of computational experiments. First, the basic properties and the main advantages of the proposed models are illustrated by some toy examples. Then, the efficiency of the models, in terms of computational workload, are evaluated by considering quite a large set of randomly generated instances, obtained by varying the number of patients on the waiting lists, the number of time slots per day, the number of patients already scheduled and by considering different operational conditions. Finally, to evaluate the usability of the models within a real hospital setting, a real case study is considered. The computational experiments were carried out by using the MIP solver CPLEX 10.1 ( on a PC Pentium IV, with 2.8 GHz and 2 GB of RAM. The proposed models provide more than one feasible sequence, since several permutations of the treated patients, on the same day, according to the availability of each patient, are possible. Note that both the basic and the enhanced model applied to all instances herein reported always find a feasible solution. It is assumed that the availability of patients p BP is given almost the same appointments as the previous week; this means that the same appointments as the previous week can be assigned to the patient in the worst case. 3.1 Toy examples The toy examples considered in this section are aimed at assessing the properties and the advantages of the proposed models. These simple examples are characterized by only four time slots per day. The scenario of the last planned week is characterized by

13 An optimal decision approach for management of radiotherapy patients Table 3 The booked treatment sessions for the toy examples Time slots Monday P 1 P 5 P 3 Tuesday P 1 P 2 P 3 P 4 Wednesday P 1 P 2 P 3 P 4 Thursday P 1 P 2 P 3 P 4 Friday P 1 P 2 P 3 P 4 Saturday P 4 Table 4 Data of patients on WP list of toy examples Data Patients J 1 J 2 J 3 J 4 J 5 J 6 J 7 J 8 pr t mts (scen 1 2) r d mts (scen 3) five patients, with a treatment in course; the time slots assigned to them are highlighted in Table 3. The patients (P 3, P 4, P 5 ) will complete their treatment plan in the current week; consequently, the corresponding time slots can be used to schedule new patients. Three different toy examples are considered, which share the following conditions: BP = 2, that is 2 patients (P 1, P 2 ) with treatment plans in progress; WP = 8, that is 8 patients waiting to start their treatment plan. The list WP is partitioned into two subqueues, according to the priority value assigned to each patient. The data of patients on WP list (i.e., assigned priority, prescribed number of treatment sessions, and minimum number of treatment sessions to deliver) are summarized in Table 4, where the row labeled mts(scen 1 2) shows the minimum number of treatments per planning horizon under the first and the second scenario (i.e., the same values are considered in both scenarios), whereas the last row, labeled mts(scen 3), gives the same information for the third scenario. For the sake of simplicity, it is assumed that the release time slot and the due time slot, on each day, are the same for all patients, that is r jk = 1 and d jk = 4, j WP, k. These toy examples were built with the goal to represent the different solutions that it is possible to obtain when the system is partially booked. First scenario It is assumed that the patients P 1 and P 2 are available only in the first and second slot of each day, respectively, thus: r 1k = d 1k = 1

14 D. Conforti et al. Table 5 Data of patients on BP list in the first toy example Patients Data t lps r d break sd delay lsd P P Table 6 Scheduled patients with the basic and the enhanced model in the first toy example Time slots Monday P 1 J 1 J 1 Tuesday P 1 P 2 J 1 Wednesday P 1 P 2 J 1 Thursday P 1 P 2 J 1 Friday P 1 P 2 Saturday and r 2k = d 2k = 2, k K. The number of treatment fractions per planning horizon is equal to five for patient P 1 (i.e., t 1 = 5) and four for patient P 2 (i.e., t 2 = 4). Delay in starting the weekly treatments is not allowed (i.e., delay 1 = delay 2 = 0); break between each consecutive weekly treatment sessions is 2 days; thus, the start day (sd) is Monday and Tuesday for P 1 and P 2, respectively. In this first example, it is not possible to delay the start of the weekly treatment sessions; this implies that the latest starting day coincides with the start day for both patients (i.e., sd = lsd). Information, related to the first scenario, is summarized in Table 5. The optimal schedule, obtained by applying the basic model, is reported in Table 6, where it can be observed that only patient J 1 is scheduled; he/she is the first patient into the sublist with maximum priority. It is important to observe that the same schedule is obtained by applying the enhanced model. This behavior can be easily explained noting that it is not possible to reschedule the patients belonging to BP, in the considered scenario, given their limited availability and since delay p = 0, p BP. Only in similar cases, the two models give the same solution. Second scenario The data characterizing this example is the same as the previous scenario, but now it is possible to delay the first treatment up to two days for both patients P 1 and P 2 (delay 1 = 2 and delay 2 = 2); this means that each patient can start the weekly treatments on Monday, Tuesday or Wednesday. By applying the enhanced model, a different optimal schedule is obtained, as reported in Table 7; the result clearly highlights that, by delaying the start of weekly

15 An optimal decision approach for management of radiotherapy patients Table 7 Scheduled patients with enhanced model in the second toy example Time slots Monday P 1 J 2 J 2 Tuesday P 1 J 2 J 1 J 1 Wednesday P 1 P 2 J 1 J 2 Thursday P 1 P 2 J 1 J 2 Friday P 1 P 2 J 1 J 2 Saturday P 2 J 2 Table 8 Scheduled patients with enhanced model Time slots Monday J 1 J 1 J 2 J 2 Tuesday P 1 P 2 J 1 J 2 Wednesday P 1 P 2 J 1 J 2 Thursday P 1 P 2 J 1 J 2 Friday P 1 P 2 J 3 J 3 Saturday P 1 J 3 treatment sessions of patient P 2 by only one day (Wednesday instead of Tuesday), it is possible to schedule both patients J 1, J 2. This simple toy example underlines a specific feature of the defined objective function (23): its second term avoids every feasible delay that does not lead to an increase in the total number of new scheduled patients. Indeed, for example, a feasible schedule has the patient P 1 starting on Tuesday and the patient P 2 starting on Wednesday: since this delay does not improve the final result, this feasible schedule is not taken into account. Third scenario The third toy example has been obtained by using the second one and by changing only the mts value of patients on WP list, as reported in last row of Table 4. From the optimal schedule, reported in Table 8, it is evident that: it is possible to schedule 3 new patients, ensuring that the minimum number of needed treatments (mts) are delivered, the weekly start day for patient P 1 is delayed by one day, the patient J 1 completes the weekly treatment sessions, 4 out of 5 treatment sessions are scheduled for patient J 2, and only 2 sessions for patient J 3. It is important to observe that the scheduled treatment sessions for patient J 2 are greater than the minimum required number (mts 2 = 3). The obtained optimal schedule underlines that the maximization of the value of the objective function leads to the maximization of the

16 D. Conforti et al. number of new scheduled patients. As a matter of fact, given the original unbooked time slots configuration, patient J 2 could complete all the treatment sessions. However, under the optimal schedule, only 4 scheduled sessions are assigned to patient J 2, by making possible the scheduling of patient J 3. The scheduling of patient J 4 is feasible, but J 4 is not scheduled, since the objective function is defined in such a way that the precedence relations among the scheduling patients are fulfilled. The results reported above underline that the enhanced model allows the scheduling of the maximum number of patients. Consequently, the waiting list is minimized as much as possible and the constraints, related to the prescribed treatment plan and patient availability, are satisfied. 3.2 Randomly generated instances In this section, the effectiveness of the proposed enhanced model is evaluated on quite a large number of randomly generated instances, obtained by varying the number of patients of the waiting list, the number of time slots per day, and the number of patients belonging to BP. Different operational conditions are also taken into account. In particular, two scenarios, characterized by different system load conditions, that is unbooked (BP list is empty) and partially booked systems, are considered. The influence of the patients preferences of the minimum number of requested treatments and of the number of feasible delay days on the computational time has also been investigated. The characteristics of the test problems are reported in Table 9, in which for each instance, the number of patients on the waiting list, the number of time slots per day and the number of patients belonging to BP are reported. The numerical results obtained under the unbooked system scenario, when mts j = 5, j WP and there are no restrictions on patient availability, are reported in Table 10, where, for each instance, in addition to the execution time (in seconds), the number of variables and the number of constraints are also given. Table 11 collects the computational results obtained, when the patients are available only in half of the time slots. It is worth observing that, in this first set of experiments, the basic model is applied. The results of Table 10 clearly underline that, if the number of time slots is kept fixed, the computational overhead increases substantially as the number of patients belonging to WP is increased. Indeed, the average execution time when WP =40 is equal to s, whereas the computational effort is equal to and s, for WP = 50 and WP = 100, respectively. A similar behavior is observed when the patients are available for treatments only in half of the time slots (see, Table 11). More specifically, the average execution time is equal to 45.87, and s, for WP = 40, WP = 50 and WP = 100, respectively.

17 An optimal decision approach for management of radiotherapy patients Table 9 Characteristics of the randomly generated instances Test WP W BP T T T T T T T T T T T T T T T T T T Table 10 Execution time for the unbooked system scenario, mts j = 5, j WP,thereare no restrictions on patient availability Test Number of variables Number of constraints Execution time T 1 21,960 15, T 2 36,040 25, T 3 71,500 50, T 4 35,800 25, T 5 45,400 31, T 6 89,500 62,700 1, T 7 43,000 30, T 8 53,750 37, T 9 107,500 74,760 2, On the other hand, if the number of patients on the waiting list is kept fixed and the number of time slots is increased, the computational effort does not always increase (see, for example instances T 7 and T 8 in Table 10). This behavior can be explained by observing that, the larger the number of slots, the higher the probability that a feasible solution found in the early stage of the search process will include all patients belonging to WP. Given the specific form of the defined objective function, such a solution is optimal. By comparing Tables 10 and 11, it can be seen that when some restrictions on patient availability are imposed, the computational effort required to solve the problem decreases (see, for example the instance T 9 ). Indeed, the average execution time

18 D. Conforti et al. Table 11 Execution time for the unbooked system scenario, mts j = 5, j WP,the patients of WP are available only in half of the time slots Test Number of variables Number of constraints Execution time T 1 28,600 20, T 2 35,990 25, T 3 71,500 50, T 4 36,100 25, T 5 45,050 31, T 6 89,500 62, T 7 43,000 30, T 8 53,750 37, T 9 107,500 74, Table 12 Execution time for the unbooked system scenario, mts j = 4, j WP,the patients of WP are available in only half of the time slots Test Number of variables Number of constraints Execution time T 1 28,520 20, T 2 35,720 25, T 3 42,920 30, T 4 35,650 25, T 5 44,650 31, T 6 53,650 37,660 1, T 7 71,300 50, T 8 89,300 62, T 9 107,300 74, when the patients are available in all the time slots is equal to s, whereas the computational overhead decreases to s when the number of available slots for each patient is halved. The results obtained under the unbooked system scenario, when mts j = 4, j WP, and the patients are available only in half of the time slots, are reported in Table 12. As expected, the comparison among the computational results given in Tables 11 and 12, underlines that generally the execution time is increased when the minimum number of requested treatments is decreased (i.e., a less constrained situation is considered). Indeed, in this last case, the average computational workload increases by a factor of about 2.0 (i.e, the average execution time is equal to s). It is worth observing that, an increase in the number of constraints and/or the variables cannot necessarily lead to a worsening in the computational performance. A similar behavior is also observed when a partially booked scenario is considered. Tables 13, 14 and 15 collect the test results obtained under the partially booked system scenario. In particular, Table 13 reports the execution time required to solve the problem when the appointments of the patients already booked in the past cannot be modified and the minimum number of treatment sessions is set equal to 5 for all the patients belonging to WP; the results of Table 14 refer to the case in which mts j = 4, j WP, whereas Table 15 shows the computational effort when

19 An optimal decision approach for management of radiotherapy patients Table 13 Execution time for the partially booked system scenario: mts j = 5, j WP, and the appointments of the patients belonging to BP cannot be modified Test Number of variables Number of constraints Execution time T 10 23,858 15, T 11 30,988 20, T 12 52,378 35, T 13 29,880 19, T 14 38,830 25, T 15 65,680 44, T 16 35,880 22, T 17 46,630 30, T 18 78,880 52, Table 14 Execution time for the partially booked system scenario, mts j = 4, j WP, the appointments of the patients belonging to BP cannot be modified Test Number of variables Number of constraints Execution time T 10 23,840 15, T 11 39,700 20, T 12 52,360 39, T 13 29,840 19, T 14 38,770 25, T 15 65,560 44, T 16 35,840 22, T 17 46,570 30, T 18 78,760 52, Table 15 Execution time for the partially booked system scenario, mts j = 4, j WP, a delay of two days in starting the weekly sessions is allowed for the patients belonging to BP Test Number of variables Number of constraints Execution time T 10 23,858 15, T 11 30,988 20, T 12 52,378 35, T 13 29,858 19, T 14 38,788 25, T 15 65,578 44, T 16 35,858 22, T 17 46,588 30, T 18 78,778 52, mts j = 4, j WP and a delay of two days in starting the first weekly session is allowed. The results collected in Table 13 underline a behavior similar to the one observed in the case of the unbooked system scenario (see Table 10); indeed, if the number of time slots is kept fixed, the larger the number of patients belonging to WP, the higher the computational time; on the other hand, if the number of patients of WP is kept fixed, an increase in the number of time slots cannot necessarily lead to a worsening

20 D. Conforti et al. in the computational performance. This last behavior can be explained by the same considerations introduced for the unbooked system scenario. As expected, from Tables 13, 14 and 15, it can be observed that the computational effort increases when less constrained operational conditions are considered. On the basis of the collected computational results and their comparison, we remark that the maximum computational time employed to solve the considered instances is equal to 2, s (see, test T 9 of Table 10). This computational effort can be considered acceptable in real settings, mainly because the scheduling of patients is done on the basis of an off-line procedure that need only be applied once a week, generally on Saturday. In addition, the worst performance is obtained under the unbooked system scenario, which rarely happens in a real life situation. Consequently, the use of a general purpose solver like CPLEX is a viable tool to address the problem under consideration. 3.3 Use-case study In what follows, we present the computational results obtained by applying the enhanced model on real data collected at the Radiotherapy Division of the General Hospital of Cosenza (Italy), which has also provided expertise in the domain of radiotherapy. The data were collected from September to November The strategy adopted by the relevant clinical setting is quite similar to the scheduling policy followed by medium-size Italian radiotherapy oncology divisions. Thus, the conclusions drawn in this study can be considered of general validity. Within the considered real context, radiotherapy treatments are given by using only one linac, which works continuously from 7.30 a.m. to 8.30 p.m. Each treatment session is delivered during one time slot of 15 min. Thus, the number of available slots per day is 52. Currently, the Hospital does not use any software tool for the radiotherapy patients scheduling, that is, the scheduling procedures are handled manually. This means that the used scheduling strategy is rather rigid. Indeed, the patients, waiting to start their treatment plan, are scheduled without taking into account their preferences and no priority policy is applied (i.e., the patients are simply scheduled based on the order of arrival). The first appointment for these patients is given in a single time slot, during which the relevant parameter values, already set, should be verified, before delivering the radiotherapy fraction. It is important to note that during the first appointment no radiotherapy treatment is delivered to the patient. Obviously, it can happen that the time needed to carry out all the required operations is greater than the allowed amount (i.e., 15 min), resulting in overtime. On Saturday morning, the Hospital performs only palliative treatments. Moreover, the possibility to reschedule the appointments of patients belonging to BP is not considered. In what follows, the collected real data relevant to the first week of October 2007 are reported. For the sake of simplicity, only the first 26 slots and the subset of patients belonging to WP, available only in the considered time slots, are considered for each day. In particular, the main features of the real scenario under consideration are: W = 26: there are 26 time slots per day.

21 An optimal decision approach for management of radiotherapy patients Table 16 Data collected at General Hospital (a) Data of patients on BP list P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 P 12 t lps r d P 13 P 14 P 15 P 16 P 17 P 18 P 19 P 20 P 21 P 22 P 23 t lps r d (b) Data of patients on waiting list WP J 1 J 2 J 3 J 4 J 5 J 6 J 7 J 8 J 9 J 10 J 11 J 12 t pr BP = 23: there are 23 patients, who have started their treatment plan in the past weeks and thus they must be scheduled in the considered planning horizon. The related information is reported in Table 16a, where a value of t less than 4 means that the corresponding patient will end the treatment plan. WP = 12: there are 12 patients, waiting to start the radiotherapy treatment plan. The assigned priority value and the number of their treatment sessions are summarized in Table 16b. For simplicity, it is assumed that all patients are available in each time slot (i.e., r jk = 1 and d jk = 26, j WP, k), and mts j = 2, j WP. The way in which the patients belonging to WP are actually scheduled by the radiotherapy division of the General Hospital of Cosenza is reported in Fig. 1a. It is important to notice that the first time slot assigned to a given patient is not used to deliver the radiotherapy dose, but it is spent on the radiotherapy parameters verification; consequently, only 4 treatments are booked for the patients J 1, J 2, J 3, 2 treatments for J 4, whereas only one radiotherapy treatment is booked for the remaining five scheduled patients (i.e., J 5, J 6, J 7, J 8, J 9 ). The schedule obtained by using the enhanced model, if delay = 1 is assigned to all patients in the waiting list WP and the priority values are those reported in Table 16,is depicted in Fig. 1b. Figure 1b clearly highlights that 9 out of the 12 patients belonging to the waiting list WP are scheduled and, thus, can start their treatment plan. For the patients J 1, J 2, J 3, J 7, having high priority value, all prescribed treatments are scheduled, whilst for the 5 remaining patients only 2 treatments are booked. It is important to observe that patient J 6 has the same priority value as patient J 7, but his/her number of treatments per week is less than that prescribed to patient J 7 (i.e., t 6 = 4 < t 7 = 5). Since the enhanced model maximizes the number of booked