OR Spectrum
DOI 10.1007/s00291-009-0170-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,
87036 Arcavacata di Rende (CS), Italy
e-mail: [email protected]
M. Veltri
Radiotherapy Division, General Hospital of Cosenza, Cosenza, Italy
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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 Ernst et al. (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).
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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.
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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) to be
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.
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An optimal decision approach for management of radiotherapy patients
Table 1 Example of booked
treatment sessions
Time slots
1
4 patients (P1 , P2 , P3 , P4 ) have
already begun treatment plans in
the past. The appointments have
been scheduled by respecting
their known availability
2
3
4
5
Monday
P1
P3
P4
Tuesday
P1
P3
P4
Wednesday
P1
P2
P3
Thursday
P1
P2
P3
Friday
P1
P2
P3
Saturday
P2
6
7
P4
P4
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;
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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 , |K| dimensional vectors, used to take into account the release
time slot and the due time slot, respectively, during each working day.
2.1.1 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 = |K|−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 = |K| − 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 .
2.1.2 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.
2.1.3 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 rik and the due time slot dik as the start and the final
time slots on day k, respectively; in this way, the period during which the patient is
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An optimal decision approach for management of radiotherapy patients
available for treatment sessions, is determined as [rik , dik ]. Obviously, the following
conditions have to be satisfied:
dik ≥ rik ; 0 ≤ rik ≤ |W|; 0 ≤ dik ≤ |W|, ∀i ∈ WP ∪ BP, ∀k.
In addition, if patient i is not available on day k̄ ∈ K, we have ri k̄ = 0 and di k̄ = 0.
2.1.4 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|K| × |W| has been used. Its generic element is defined as:
schedkw =
1
0
if the time slot w on day k is already assigned;
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|
f jk1 = 0, z jk|W | = 0 ∀ j, ∀k.
(1)
(2)
Since the first session is one booked appointment, we have:
z jkw ≤ y jkw ∀ j, ∀k, ∀w.
(3)
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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
d jk
j z jkw ≤ 1 ∀ j
(4)
z jkw = 0 ∀ j
(5)
k=1 w=r jk
|K | |W |
k>ld j
w=1
r jk −1
y jkw = 0,
w=1
|W |
y jkw = 0 ∀ j, ∀k.
(6)
w>d jk
For example, z j k̄ w̄ = 1 means that patient j begins the treatment plan in the time slot
w̄ on day k̄, 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 ≤
w=r jk
ȳ jk ≤ 1 −
d jk
k z jsw ∀ j, ∀k
(7)
∀ j, mts j < k < |K|.
(8)
s=1 w=r jk
d jk
y j (k+1)w
w=r jk
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
w=r jk
s=k
123
y jsw +
|K |
s1 =k+mts j
ȳ js1 ≥ t j
d jk
w=r jk
z jkw
∀k ≤ ld j
(9)
An optimal decision approach for management of radiotherapy patients
d jk |K|
|K |
y jsw +
w=r jk s=k
s1 =k+mts j
d jk
|K | |K |
y jkw +
k=1 w=r jk
k=1
ȳ js1 ≥ star t j
ȳ jk = t j
d jk
z jkw ∀k > ld j
(10)
w=r jk
ld j d jk
ld
z jkw +
k=1 w=r jk
d jk
j star t j z jkw
(11)
k>ld j w=r jk
where star t 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
|
y jkw + f jkw + schedkw ≤ 1
∀k, w.
(12)
j=1
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
|WP | |K| |W | pr j
scheduled patients (i.e., j=1
w=1 j z jkw ); the second term is the sum of
k=1
|WP | |K| |W | pr j
the booked appointments (i.e., j=1
w=1 j y jkw ). The objective function
k=1
assumes the following form:
max
|K | |W |
|WP
| j=1 k=1 w=1
pr j
j
(z jkw + y jkw ).
(13)
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.
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D. Conforti et al.
Table 2 Used data of the
booked patient P1 for planning
week
P1
t
lps
r
d
break
sd
delay
lsd
5
5
1
1
2
1
3
2
2.3 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. 3.1.
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 dpk , 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 P1 , 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 − |K| + lps,
1
if break > |K| − lps
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, |K| − t + 1}. Thus, the start day will be
in the range [sd; lsd]. For instance, delay P1 = 3 means that patient P1 could start
the weekly treatments on Monday, Tuesday, Wednesday or Thursday, but since all the
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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 |K| − t + 1 = 6 − 5 + 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, we have:
d pk
lsd p
z pkw = 1
(14)
k=sd p w= r pk
|K |
⎛
r pk −1
⎝
z pkw +
w=1
k=1
|K | |W |
⎞
|W |
z pkw ⎠ = 0
(15)
w=d pk +1
z pkw = 0.
(16)
k>lsd p w=1
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
y pkw ≤ 1 ∀ p, k
(18)
w=r pk
r pk −1
|W |
y pkw +
w=1
y pkw = 0 ∀ p, k.
(19)
w=d pk +1
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 |
k+t p −1
y psw ≥ t p
s=k
k=sd p w=r pk
z pkw k = sd p , . . . , lsd p
(20)
w=r pk
d pk
d pk
y pkw = t p
d pk
lsd p
z pkw .
(21)
k=sd p w=r pk
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)].
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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:
y pkw ≤ 1 ∀k, ∀w.
y jkw + f jkw +
j∈WP
(22)
p∈BP
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
|K | |W |
|WP
| j=1 k=1 w=1
pr j
j
(z jkw + y jkw ) −
|K |
|
BP | |W |
p=1 k=1
w=1
(k − sd p )
z pkw
(23)
The overall mathematical formulation of the enhanced model is given in the
Appendix.
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 (http://www.ilog.com) 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
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An optimal decision approach for management of radiotherapy patients
Table 3 The booked treatment
sessions for the toy examples
Time slots
1
2
3
4
Monday
P1
P5
P3
Tuesday
P1
P2
P3
P4
Wednesday
P1
P2
P3
P4
Thursday
P1
P2
P3
P4
Friday
P1
P2
P3
P4
Saturday
Table 4 Data of patients on
WP list of toy examples
P4
Data
Patients
J1
J2
J3
J4
J5
J6
J7
J8
1
pr
10
10
10
10
1
1
1
t
4
5
4
4
5
4
5
5
mts (scen 1–2)
4
4
3
3
3
3
5
5
r
1
1
1
1
1
1
1
1
d
4
4
4
4
4
4
4
4
mts (scen 3)
4
3
2
1
2
3
5
5
five patients, with a treatment in course; the time slots assigned to them are highlighted
in Table 3.
The patients (P3 , P4 , P5 ) 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 (P1 , P2 ) 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 P1 and P2 are available only in the
first and second slot of each day, respectively, thus: r1k = d1k = 1
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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
P1
5
5
1
1
2
1
0
1
P2
4
6
2
2
2
2
0
2
Table 6 Scheduled patients
with the basic and the enhanced
model in the first toy example
Time slots
1
2
3
Monday
P1
J1
J1
Tuesday
P1
P2
J1
Wednesday
P1
P2
J1
Thursday
P1
P2
J1
Friday
P1
P2
4
Saturday
and r2k = d2k = 2, ∀k ∈ K. The number of treatment fractions per
planning horizon is equal to five for patient P1 (i.e., t1 = 5) and four for
patient P2 (i.e., t2 = 4). Delay in starting the weekly treatments is not
allowed (i.e., delay1 = delay2 = 0); break between each consecutive
weekly treatment sessions is 2 days; thus, the start day (sd) is Monday
and Tuesday for P1 and P2 , 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 J1
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 P1 and P2 (delay1 = 2 and delay2 = 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
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An optimal decision approach for management of radiotherapy patients
Table 7 Scheduled patients
with enhanced model in the
second toy example
Time slots
1
2
3
4
Monday
P1
J2
J2
Tuesday
P1
J2
J1
J1
Wednesday
P1
P2
J1
J2
Thursday
P1
P2
J1
J2
Friday
P1
P2
J1
J2
Saturday
Table 8 Scheduled patients
with enhanced model
P2
J2
Time slots
1
2
3
4
Monday
J1
J1
J2
J2
Tuesday
P1
P2
J1
J2
Wednesday
P1
P2
J1
J2
Thursday
P1
P2
J1
J2
Friday
P1
P2
J3
J3
Saturday
P1
J3
treatment sessions of patient P2 by only one day (Wednesday instead
of Tuesday), it is possible to schedule both patients J1 , J2 .
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 P1
starting on Tuesday and the patient P2 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 P1 is delayed by one day,
– the patient J1 completes the weekly treatment sessions,
– 4 out of 5 treatment sessions are scheduled for patient J2 , and
only 2 sessions for patient J3 .
It is important to observe that the scheduled treatment sessions for
patient J2 are greater than the minimum required number (mts2 = 3).
The obtained optimal schedule underlines that the maximization of
the value of the objective function leads to the maximization of the
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D. Conforti et al.
number of new scheduled patients. As a matter of fact, given the
original unbooked time slots configuration, patient J2 could complete
all the treatment sessions. However, under the optimal schedule, only
4 scheduled sessions are assigned to patient J2 , by making possible
the scheduling of patient J3 .
The scheduling of patient J4 is feasible, but J4 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 114.61 s, whereas the computational effort is equal to 264.32 and 1422.09 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, 70.40 and 483.16 s, for |WP| = 40, |WP| = 50 and |WP| = 100,
respectively.
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An optimal decision approach for management of radiotherapy patients
Table 9 Characteristics of the
randomly generated instances
Table 10 Execution time for
the unbooked system scenario,
mts j = 5, ∀ j ∈ WP, there are
no restrictions on patient
availability
|WP|
Test
|W|
|BP|
T1
40
40
T2
50
40
0
0
T3
100
40
0
0
T4
40
50
T5
50
50
0
T6
100
50
0
T7
40
60
0
T8
50
60
0
T9
100
60
0
T10
20
40
20
T11
30
40
20
T12
60
40
20
T13
20
50
20
T14
30
50
20
T15
60
50
20
T16
20
60
20
T17
30
60
20
T18
60
60
20
Test Number of variables Number of constraints Execution time
T1
21,960
15,390
108.23
T2
36,040
25,730
265.19
T3
71,500
50,640
847.33
T4
35,800
25,260
224.28
T5
45,400
31,550
393.70
T6
89,500
62,700
1,337.63
T7
43,000
30,120
11.31
T8
53,750
37,560
303.48
T9
107,500
74,760
2,081.33
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 T7 and T8 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 T9 ). Indeed, the average execution time
123
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
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
T1
28,600
20,400
T2
35,990
25,440
32.42
T3
71,500
50,640
T4
36,100
25,260
T5
45,050
31,500
66.56
T6
89,500
62,700
419.11
T7
43,000
30,120
4.52
T8
53,750
37,560
94.47
T9
107,500
74,760
581.17
51.94
449.2
64.78
Test Number of variables Number of constraints Execution time
T1
28,520
20,480
68.69
T2
35,720
25,340
124.95
T3
42,920
30,200
483.23
T4
35,650
25,540
31.83
T5
44,650
31,600
94.36
T6
53,650
37,660
1,237.65
T7
71,300
50,840
5.94
T8
89,300
62,900
38.23
T9
107,300
74,960
979.53
when the patients are available in all the time slots is equal to 68.80 s, whereas the
computational overhead decreases to 21.78 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 42.56 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
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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
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
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
T10
23,858
15,508
6.72
T11
30,988
20,568
50.25
T12
52,378
35,748
136.50
T13
29,880
19,080
1.64
T14
38,830
25,320
35.30
T15
65,680
44,040
156.47
T16
35,880
22,740
1.23
T17
46,630
30,180
5.56
T18
78,880
52,500
259.94
Test Number of variables Number of constraints Execution time
T10
23,840
15,460
4.25
T11
39,700
20,520
27.14
T12
52,360
39,700
131.49
T13
29,840
19,120
1.69
T14
38,770
25,380
12.62
T15
65,560
44,160
269.19
T16
35,840
22,780
1.19
T17
46,570
30,240
8.11
T18
78,760
52,620
417.13
Test Number of variables Number of constraints Execution time
T10
23,858
15,508
6.66
T11
30,988
20,568
50.13
T12
52,378
35,748
136.42
T13
29,858
19,168
2.56
T14
38,788
25,428
17.48
T15
65,578
44,208
313.27
T16
35,858
22,828
1.76
T17
46,588
30,288
10.39
T18
78,778
52,668
512.16
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
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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 2,081.33 s (see, test T9 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 2007.
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.
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An optimal decision approach for management of radiotherapy patients
Table 16 Data collected at General Hospital
(a) Data of patients on BP list
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
t
5
4
3
3
4
4
4
5
5
3
5
5
lps
5
5
5
5
5
5
5
5
5
5
5
5
r
1
3
4
5
7
8
9
10
11
12
13
15
d
1
3
4
5
7
8
9
10
11
12
13
15
P13
P14
P15
P16
P17
P18
P19
P20
P21
P22
P23
5
4
2
3
3
4
5
5
5
5
5
t
5
5
5
5
5
5
5
5
5
5
5
r
lps
16
17
18
19
21
23
24
25
22
2
26
d
16
17
18
19
21
23
24
25
22
2
26
J6
J7
J8
J9
J10
J11
P12
(b) Data of patients on waiting list WP
J1
J2
J3
J4
J5
J12
t
5
5
5
4
5
4
5
5
5
4
4
4
pr
10
10
10
1
1
10
10
10
10
1
1
10
– |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 J1 , J2 , J3 , 2 treatments for J4 ,
whereas only one radiotherapy treatment is booked for the remaining five scheduled
patients (i.e., J5 , J6 , J7 , J8 , J9 ).
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 J1 , J2 , J3 , J7 , 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 J6 has the same priority value as patient J7 ,
but his/her number of treatments per week is less than that prescribed to patient J7
(i.e., t6 = 4 < t7 = 5). Since the enhanced model maximizes the number of booked
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D. Conforti et al.
Fig. 1 a “Manual” schedule of radiotherapy division. b “Optimal” schedule with delay = 1
treatments, patient J7 can complete the treatments in the planned week, whilst patient
J6 has only 2 booked treatments.
Consequently, even though by using the enhanced model it is possible to schedule
the same number of patients as those booked manually (i.e., nine), the solution obtained
by the enhanced model delivers a higher number of treatment sessions. Indeed, the
total number of sessions booked by the manager of the radiotherapy division is equal
to 28, whereas by applying the enhanced model the number of scheduled sessions
increases to 37. In addition, since the strategy adopted by the considered radiotherapy
division assumes that the first appointment for new scheduled patients is given in a
single slot to verify only the values of the relevant parameters for the treatment plan,
the number of treatment sessions reduces to 19. Thus, by using the proposed model,
we have registered an increase of 47% the number of booked treatment sessions.
On the basis of the computational results collected and their comparison, it is evident
that by using the enhanced model, the wait time of each patient is reduced. In addition, more treatment sessions than those scheduled manually by the manager of the
radiotherapy division are booked.
4 Conclusions
In this paper, the optimal management of patients waiting for radiotherapy treatment
has been considered by devising, implementing and testing well-tailored integer programming models.
The robustness and efficiency of the proposed models were evaluated on the basis
of an experimental study, carried out on dummy scenarios and real data.
The computational results collected show the effectiveness and efficiency of the
proposed models, which can help to optimally manage the appointments service, by
suiting different patient requests and minimizing the total wait time.
It is worth observing that, using a radiotherapy appointment booking system managed by “hand”, it is very difficult to book the appointments by taking into account all
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An optimal decision approach for management of radiotherapy patients
the aforementioned requirements. More specifically, the benefits gained by the impact
of the proposed methodologies, are generally determined by the relevant flexibility
and suitability features, which allow the system to meet the various conditions that
may be encountered in real cases. In fact:
– conditions of “availability” of the patient, to carry out the treatment session, are
taken into account, but at the same time limiting the variability of that availability,
so that it does not negatively impact other patients;
– the pathological conditions of the patient are considered, by exploiting the assessments of cancer staging and prognosis, through the definition of the level of intervention priority, determined by the medical doctor;
– a continuous monitoring of the patient waiting list is made possible, allowing for
appropriate control of the maximum size of the same list on a given time horizon;
– conditions of management and economic type of the health care system are considered, through a more efficient use of equipment resources (linacs) and a more
rational organization of health care personnel (ability to predefine with some accuracy work shifts and reduce periods of inactivity).
Acknowledgment We would like to thank the anonymous reviewers for their comments that have
improved the paper.
5 Appendix
Complete mathematical formulation of the basic model.
max
|K | |W |
|WP
| j=1 k=1 w=1
pr j
(z jkw + y jkw )
j
s.t.
z jkw = f jk(w+1)
∀ j, ∀k, ∀w < |W |
f jk1 = 0, z jk|W | = 0,
∀ j, ∀k
z jkw ≤ y jkw
∀ j, ∀k, ∀w
ld j
d jk
k=1 w=r jk
|K| |W |
k>ld j
z jkw ≤ 1
∀j
z jkw = 0
∀j
w=1
r jk −1
w=1
d jk
|W |
y jkw = 0,
y jkw = 0
∀ j, ∀k
w>d jk
y jkw + ȳ jk ≤
w=r jk
ȳ jk ≤ 1 −
d jk
k z jsw ∀ j, ∀k
s=1 w=r jk
d jk
y j (k+1)w
∀ j, mts j < k < |K|
w=r jk
123
D. Conforti et al.
d jk
k+t j −1
|K|
y jsw +
w=r jk s=k
d jk |K|
ȳ js1 ≥ star t j
w=r jk s=k
s1 =k+mts j
d jk
|K| |K|
y jkw +
k=1 w=r jk
|WP
|
∀ j, k ≤ ld j
z jkw
w=r jk
d jk
s1 =k+mts j
|K|
y jsw +
d jk
ȳ js1 ≥ t j
∀ j, k > ld j
z jkw
w=r jk
ȳ jk = t j
ld
ld j d jk
z jkw +
k=1 w=r jk
k=1
d jk
j star t j z jkw ∀ j, ∀k
k>ld j w=r jk
y jkw + f jkw + schedkw ≤ 1
∀k, ∀w
j=1
z jkw , f jkw , y jkw ∈ {0, 1}
∀ j, ∀k, ∀w
ȳ jk ∈ {0, 1}
∀ j, ∀k
Complete mathematical formulation of the enhanced model.
max
|K | |W |
|WP
| pr j
(z jkw + y jkw ) −
j
j=1 k=1 w=1
|
BP | |K|
|W |
p=1 k=1
w=1
(k − sd p )
z pkw
s.t.
z jkw = f jk(w+1)
∀ j, ∀k, ∀w < |W |
f jk1 = 0, z jk|W | = 0,
∀ j, ∀k
z jkw ≤ y jkw
∀ j, ∀k, ∀w
ld j
d jk
k=1 w=r jk
|K| |W |
k>ld j
z jkw ≤ 1
∀j
z jkw = 0
∀j
w=1
r jk −1
w=1
d jk
|W |
y jkw = 0,
y jkw + ȳ jk ≤
w=r jk
∀ j, ∀k
d jk
k ∀ j, ∀k
z jsw
s=1 w=r jk
ȳ jk ≤ 1 −
d jk
y jkw = 0
w>d jk
d jk
w=r jk s=k
d jk |K|
y jsw +
y jsw +
w=r jk s=k
123
∀ j, mts j < k < |K|
y j (k+1)w
w=r jk
k+t j −1
|K|
ȳ js1 ≥ t j
s1 =k+mts j
|K|
d jk
ȳ js1 ≥ star t j
s1 =k+mts j
z jkw
∀ j, k ≤ ld j
w=r jk
d jk
w=r jk
z jkw
∀ j, k > ld j
An optimal decision approach for management of radiotherapy patients
d jk
|K | y jkw +
k=1 w=r jk
|K | |W |
|K |
ȳ jk = t j
ld j d jk
k=1 w=r jk
k=1
z pkw = 0
z pkw ≤ y pkw
y pkw +
∀ p, ∀k
|W |
y pkw = 0
w=1
w=d pk +1
d pk k+t p −1
w=r pk s=k
d pk
| K| y psw ≥ t p
z pkw
w=r pk
d pk
lsd p
∀ p, k = sd p , . . . , lsd p
z pkw
∀ p, ∀k
y pkw ≤ 1
∀k, ∀w
k=sd p w=r pk
(y jkw + t jkw ) +
j∈WP
∀ p, ∀k
d pk
y pkw = t p
k=sd p w=r pk
star t j z jkw ∀ j, ∀k
k>ld j w=r jk
∀ p, ∀k, ∀w
y pkw ≤ 1
w=r pk
r pk −1
z jkw +
∀p
k>lsd p w=1
d pk
ld
d jk
j p∈BP
z jkw , f jkw , y jkw ∈ {0, 1}
∀ j, ∀k, ∀w
ȳ jk ∈ {0, 1}
∀ j, ∀k
z pkw , y pkw ∈ {0, 1}
∀ p, ∀k, ∀w
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