Health Care Manag Sci (2011) 14:74–88
DOI 10.1007/s10729-010-9144-5
An optimal decision making model for supporting
week hospital management
Domenico Conforti · Francesca Guerriero ·
Rosita Guido · Marco Matucci Cerinic ·
Maria Letizia Conforti
Received: 28 May 2010 / Accepted: 3 November 2010 / Published online: 18 November 2010
© Springer Science+Business Media, LLC 2010
Abstract Week Hospital is an innovative inpatient
health care organization and management, by which
hospital stay services are planned in advance and delivered on week-time basis to elective patients. In
this context, a strategic decision is the optimal clinical
management of patients, and, in particular, devising
efficient and effective admission and scheduling procedures, by tackling different requirements such as beds’
availability, diagnostic resources, and treatment capabilities. The main aim is to maximize the patient flow,
by ensuring the delivery of all clinical services during
the week. In this paper, the optimal management of
Week Hospital patients is considered. We have developed and validated an innovative integer programming
model, based on clinical resources allocation and beds
utilization. In particular, the model aims at scheduling
Week Hospital patients’ admission/discharge, possibly
reducing the length of stay on the basis of an available
timetable of clinical services. The performance of the
model has been evaluated, in terms of efficiency and
robustness, by considering real data coming from a
Week Hospital Rheumatology Division. The experimental results have been satisfactory and demonstrate
the effectiveness of the proposed approach.
Keywords Health care services · Week hospital ·
Patient scheduling · Decision making problem ·
Mathematical programming
Mathematics Subject Classifications (2010) 90B35 ·
90C10
1 Introduction
D. Conforti · F. Guerriero · R. Guido (B)
Laboratory of Decisions Engineering for Health Care
Delivery, Department of Electronics, Computer Science
and Systems, University of Calabria, Ponte Pietro
Bucci 41C, 87036 Rende Cosenza, Italy
e-mail: rguido@deis.unical.it
D. Conforti
e-mail: mimmo.conforti@unical.it
F. Guerriero
e-mail: guerriero@deis.unical.it
M. Matucci Cerinic · M. L. Conforti
Rheumatology Division, Department of Internal Medicine,
University of Florence, Viale Pieraccini 18,
50139 Firenze, Italy
M. Matucci Cerinic
e-mail: marco.matuccicerinic@unifi.it
Scientific and technological innovations, aging population, and limited economic resources are factors that,
among the others, have generally affected the health
care scenarios of recent years since their remarkable
impact to services planning, organization, and management. The same factors raised also need for a substantial improvement of hospital services organization
and management. In fact, some important issues have
been related to a greater integration with primary care
and continuous assistance services, internal reorganization matching criteria of efficiency and effectiveness,
stronger involvement of health professionals in change
processes.
Under this respect, the basic orientation of health
care policies at national and regional level moves to
define and test new organizational models that may
Health Care Manag Sci (2011) 14:74–88
effectively impact on care workflows and, on the other
side, optimise the use of resources. For instance, several clinical domains (e.g., gastroenterology, diabetes,
endocrinology, rheumatology) periodically require that
a patient carries out diagnostic tests and therapeutic
treatments, usually performed during a hospitalization
time with night stay. Following these issues, a novel
organizational model of inpatient hospital care services
planned and delivered on a weekly time horizon, seems
to fully respond to these requirements, mainly allowing
a reduction of inpatient length of stay with a better
benefit for patient health. We refer to this model as
Week Hospital (WH for short) system.
A WH system is a planned hospital stay whose maximum duration is five weekdays. It basically aims at
performing inpatient diagnostic tests and clinical care
in a more effective and efficient way with respect to ordinary hospitalization, by exploiting the available time
horizon of one week. Each elective patient, waiting
to be admitted to WH, is characterized by a clinical
pathway, defined by physician and detailing the set of
diagnostic tests and therapeutic treatments. Moreover,
effective WH patient management implies also the assignment of a suitable clinical priority.
It is our aim, by this paper, to introduce and propose
WH as a new standard, which extends the taxonomy
of hospitalization: day hospital, week hospital, ordinary/
long stay hospital.
The main issues of WH organization concern the
admission policies and the patient scheduling. The
admission planning problem aims at defining the set
of admitting patients, whereas the patients scheduling
problem deals with the management of diagnostic tests
and clinical care. In the following, we briefly refer
to diagnostic tests, therapeutic treatments, and related
resources as “clinical services”.
In order to allow an efficient use of clinical services
and increase, if possible, the admission rate to WH, we
have developed an optimization model for supporting
the area manager in planning and scheduling patients
and clinical services into a WH division. Furthermore,
our contribution lies also on defining a novel rule able
to manage several dynamic aspects characterizing this
context.
Even though, to the best of our knowledge, this
specific problem has not been yet addressed in the
scientific literature, it shares some common features
with the outpatients appointment scheduling problem.
An extensive review on this topic is due to Cayirli and
Veral [1]. Interesting contribution has been given by
the paper of Gupta and Denton [2], which describes
the most common types of health care delivery systems
with focusing on the factors that make rather challeng-
75
ing appointment scheduling problems. Several authors
[3–7] recognized that demand and capacity have to be
balanced otherwise patient dissatisfaction is high and
wasting of resources is unavoidable; in addition, also
timely access and patient no-show need to be considered [3, 8, 9]. In most of the cases, queueing models have been proposed for appointment-driven health
care systems [10].
It is worth observing that these existing approaches
cannot be applied to a WH system, since the main
requirements are rather different. In fact, the WH management aims at satisfying time constraints (i.e., patient
gets admission care if and only if all prescribed clinical
services are booked during the week) and specific resource constraints (given by several clinical services and
bed availability), which are not considered in outpatient
appointment scheduling.
On the other hand, it is worth observing that bed
availability has been taken into account to address
patients admission planning problem in a general hospital [6]. Usually, the main decision is to select waiting patients by effective and efficient utilization of
nursing staff and resource and bed availability; however, time constraints are generally not considered.
Recently, a hospital appointment scheduling model,
characterized by time and resource constraints, has
been proposed by Chern et al. [11], by assuming that
every resource is always available. This assumption is
rather unrealistic since hospital resources are limited
and makes the problem rather difficult to solve. In fact,
the proposed model has been solved with a heuristic
approach. The appointments are scheduled for checkups without overnight stay (thus bed availability is not
needed); consequently, a patient does not have admission/discharge date since all procedures are performed
in a “day hospital” outpatient manner. This is substantially the main difference with the WH management
problem addressed in this paper.
The paper is structured as follows. In Section 2, we
give a general description of the relevant characteristics of WH, with an analysis of the main management
problems. In Section 3, we present the optimization
model, which is tested in Section 4 within a pilot setting
in the Rheumatology Division of “Careggi” University
Hospital (Florence, Italy). A discussion on the results
and the conclusion section complete the paper.
2 Week hospital: definition and problem statement
Generally, hospital admission classification is related to
length of stay (LOS). As a matter of fact, we typically
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Health Care Manag Sci (2011) 14:74–88
refer to “day hospital”, “ordinary hospital”, and “long
stay” according to the relevant inpatient care.
We now remark that several clinical domains, typically characterized by chronic diseases, very often
require that patients periodically carry out specialized diagnostic tests and personalized therapeutic treatments. To this end, the physician prescribes inpatient
care based on hospital stay with a given minimum
LOS, typically independent from the required diagnostic tests, but aimed at obtaining, for example, a detailed
monitoring of overall patient’s conditions, an effective
tuning of drug dosage, and an assessment of therapy
side effects.
In order to meet these requirements, WH is a new
way of planned hospital stay allowing an effective
and efficient inpatient clinical management along a
time horizon of five consecutive weekdays. Since maximum hospital stay is a week, all decisions about
appointment time have to be made before the admission phase, in order to efficiently schedule clinical
services.
It is worth while to observe that one of the aims of
WH system is to avoid frequent hospital re-admissions
(following a discharge) since they have high cost implications for a general health care system. As we already pointed out, from a clinical point of view, WH
is typically motivated for the management of chronic
diseases. In this case, periodically planned WH stays are
also aimed at avoiding or anticipating possibly worsening events, which very often imply frequent acute
hospital admissions. During a WH stay, prescribed
diagnostic tests, specific therapeutic treatments and
clinical assessment of the patient conditions can be
efficiently carried out. Hence, an optimal management
can beneficially impact on the overall economic and
social costs associated to inpatient care.
Generally, the organizational layout of WH is based
on a dedicated theater composed of a specific ward with
a dedicated nurse staff, a given number of available
beds and commonly sharing the overall clinical services
available in the hospital. WH ward typically works from
Monday morning to Friday midday and WH patients
usually stay more than 24 h.
Several issues in developing a WH affect the internal organizational dimension of each involved division,
also raising some logistic problems. The main aspects
are the following:
–
–
a detailed planning and standardization of admission/discharge procedures;
the respect of the operative times, also through an
efficient management of the internal connections;
–
–
–
–
–
the organization of the activities by a logic of division clinical services aiming at improving welfare
levels;
the enlargement of the hospital pre-admission;
the prevision of the possibility to be not discharged
by Friday morning and then the singling out of beds
that are available at a continuous cycle;
the guarantee of the availability during weekends
of emergency services and assistance for the discharged patients;
the definition of suitable care protocols, shared
with the primary care, for patients needing continuous assistance.
A typical WH process consists of the following main
phases
1. pre-admission: usually a pre-admission phase includes a review of patient’s health care history.
During a baseline visit, the health care team, by following specific guidelines, prescribes a complete list
of diagnostic tests and treatments to be performed
assigning also a clinical priority. A set of clinical
services is then allotted to evaluate patient’s condition, performing also appropriate care. In case
of not immediate hospitalization, the patient is inserted into a waiting list;
2. patient hospitalization, during which all prescribed
clinical services are delivered. The admission date
is planned in such a way the prescribed clinical
services can be delivered by (at most) five consecutive weekdays. Worsening of conditions or acute
events may require that the planned hospital stay
be rescheduled. In this case, the patient is immediately transfered to a dedicated care unit;
3. follows up: a first check follows patient discharge,
which can be followed by next checks if necessary;
4. continuous assistance: appropriate integration of
primary, secondary, and tertiary care for selected
groups of patients after discharge.
The date and time of patient’s discharge is defined
in advance so that the activity of responsible personnel
(i.e., planning admission/discharge and scheduling of
the several clinical services) is organized accordingly.
It is quite evident that this complex problem requires a
centralized planning and scheduling function.
The number of weekdays between the admission and
the discharge dates of a patient defines the related
LOS. Obviously, shorter hospital stays, by making a
more efficient use of hospital resources, are preferred
in terms of overall system efficiency, since more patients can get care treatment during the same period,
Health Care Manag Sci (2011) 14:74–88
improving hospital service, and allowing cost reduction
[12]. In this context, an important issue is to ensure,
by a more efficient planning of clinical services during
the hospitalization period, that patients recover more
quickly, leaving hospital as soon as possible. Under this
respect, we remark that the overall quality of care from
the patient’s point of view is guaranteed by the required
minimum LOS prescribed by the physician during the
baseline visit.
WH management typically involves the following
three hierarchical decision levels:
Strategic level
Tactical level
Operational level
is devoted both to clinical service
capacity sizing and allocation of the
clinical resources;
concerns the assignment of the required clinical resources to a specific clinical specialty, typically
according to a block time strategy
(clinical services timetable);
is devoted to WH patients planning
and scheduling. The main goal is
to decide when a waiting patient is
hospitalized, that is the definition of
the admission/discharge date.
The planning problem considered in this paper is
to generate an admission profile for a specified clinical specialty when target patient throughput and hospital resource availability are known. Moreover, a
planned admission requires an appropriate coordination of involved staff members and resources. In detail,
two different issues affect patients scheduling: (1) a
timetable related to the availability of clinical personnel
and equipments, in order to carry out the prescribed
clinical services, and (2) the number of available beds.
Both issues could potentially represent a bottleneck for
patients flow within a WH organization. The first is
solved at the tactical decision level: a five weekdays
timetable of clinical services is developed, typically on
the basis of retrospective data and estimated demand.
The resulting timetable is obtained at the end of a
suitable negotiation phase between clinical division and
hospital health care manager.
In this paper, we focus our attention on a simplified
case study setting, since we take into account a WH
organization based on only one specialty. The specific
objective of this study is to design an effective and
efficient WH admission/discharge scheduling system.
Furthermore, we will show that both tactical and operational decisions affect the efficiency level of WH
with some important consequences on patients waiting
list and clinical management. In particular, the proposed optimization model allows to schedule patients
77
by efficiently booking all the resources for performing
the several clinical services during his/her LOS. Available beds and clinical service capacity are taken into
account for defining the “optimal” schedule.
As final remark, we observe that WH management
resembles, in terms of structure, the operating rooms
management [13] even though it is more complex. In
general, it requires a multiple assignment of several
clinical resources shared among several divisions.
3 The optimization model
In order to effectively address the described problem,
we have developed an integer linear programming
model allowing suitable management of the workload
of inpatients flow. The decisions are related to which
waiting patient is hospitalized and when, on the basis
of several patient’s clinical paths and hospital resource
availability, including beds. Moreover, only one admission date is possible for every hospitalized patient
during the planning week. A bed is assigned to a patient
only if he/she can undergo all the prescribed clinical
services in the planning week (i.e., the assignment of a
bed to a patient and then his/her hospitalization implies
the booking of all the prescribed services); if this is not
the case, patient is not hospitalized.
A crucial aspect that affects a schedule is the planning horizon definition. Herein, we consider a week
as planning horizon so that the waiting list is weekly
updated. This may allow the respect of equity criteria in
the health care system. Furthermore, future analysis of
various situations are useful for providing recommendations (such as a reallocation of some resources).
To simplify the mathematical formulation, we introduce the so-called “blocks”. We assume that each
weekday is divided in two blocks, i.e., morning and
afternoon. Since the services of a WH end on Friday
midday, there are nine ordered blocks (from Monday
morning to Friday morning) in a planning week. A
schematic representation of a planning week is reported in Fig. 1. For example, the morning block could
start at 8.00 a.m. and end at 2 p.m., whereas the afternoon block starts at 2 p.m. and includes overnight
stay. Moreover, every block is split in a fixed number of time slots (TSs) during which specific clinical
Fig. 1 Representation of a planning week with 9 blocks
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Health Care Manag Sci (2011) 14:74–88
services can be delivered (for the sake of simplicity,
it is assumed that the time slots have the same duration). However, this assumption does not affect the
validity of the approach in a different situation (e.g.,
a different number of blocks and different duration of
TSs). In each “cell” of such a timetable the clinical
service availability and the related capacity per time
slot are settled. In particular, we observe that, in most
of the cases, the resources for the WH clinical services
are shared among several hospital divisions/wards. This
implies that the clinical services timetable, which has
been defined at the tactical level for a given WH division, denotes the clinical service availability in each
time slot of a related block. Moreover, even though
for most of the clinical domains, a specific order for
services is not prescribed, a timetable implicitly could
define a sequence for performing clinical services. This
occurs in presence of limited resources. The admissiondischarge planning problem is thus represented as a
block-scheduling resource-constrained problem. In this
way, the number of admitted/discharged patients during each weekday is known per each block.
Before going into the details of the mathematical
model formulation, we introduce the used notation and
suitable assumptions. More specifically, we have:
–
–
–
–
–
blocks set B = {b : b = 1, . . . , 9};
time slots set K = {k : k = 1, . . . , l};
clinical services set S = {i : i = 1, . . . , m};
patients’ ordered waiting list P = { p : p =
1, . . . , n};
number of available beds D.
As already mentioned, we basically assume that both
the timetable and the capacity of clinical services are
defined and known in advance, as well as the number
of beds into the hospital division. We thus know the
following data:
–
–
for each time slot k ∈ K of a block b ∈ B , the capacity µib k ≥ 0 of the clinical service i ∈ S is given: for
the sake of simplicity, µib k is defined as the number
of patients that can undergo the clinical service i in
the slot k of the block b ;
for each waiting patient p ∈ P : the date of the basep
line visit D0 , the assigned clinical priority prp , the
minimum LOS nb p (in terms of number of blocks)
required by the physician, and the set of prescribed
clinical
services expressed as
1 if the service i is prescribed to patient p
i
sp =
0 otherwise.
i
We denote as nsp =
s p the number of prescribed
i∈S
clinical services to patient p.
Decision variables
⎧
⎨ 1 patient p undergoes the clinical service
xipbk =
i during slot k of block b
⎩
0 otherwise
1 if patient p occupies a bed during block b
ypb =
0 otherwise
1 if patient p is admitted during block b
adm pb =
0 otherwise
rspb = number of the remaining clinical services to
which patient p has to be undergone during
block b .
rb pb = number of the remaining blocks with respect to
the required minimum LOS (i.e., nbp ) for
patient p
Constraints For each patient p, every prescribed clinical service has to be booked and performed only once
during the planning week:
xipbk ≤ sip , ∀i ∈ S , ∀p ∈ P .
(1)
b ∈B k∈K
Obviously, a patient p can undergo at most only one
clinical service during slot k of block b :
xipbk ≤ 1, ∀p ∈ P , ∀b ∈ B , ∀k ∈ K.
(2)
i∈S
The number of patients undergoing the clinical service i ∈ S during slot k of block b is upper bounded by
the related capacity µib k , that is:
xipbk ≤ µibk , ∀i ∈ S , ∀b ∈ B , ∀k ∈ K.
(3)
p∈P
Furthermore, only one admission is possible for each
hospitalized patient. The first booked clinical service
defines the patient admission block:
adm pb ≤ 1 ∀p ∈ P ,
(4)
b ∈B
and no clinical service can be performed before the
admission:
b
j=1
adm pj ≤
b
xipjk ∀ p ∈ P , ∀b ∈ B .
(5)
j=1 k∈K
As already stated, each clinical service prescribed to
a patient is booked only during his/her hospitalization
time. Consequently, by constraints 6–8, we impose that
Health Care Manag Sci (2011) 14:74–88
79
each patient occupies a bed, even though some services
cannot be performed during a defined block:
xipbk ≤ ypb
(6)
∀i ∈ S , ∀ p ∈ P , ∀b ∈ B , ∀k ∈ K
rsp1 = nsp adm p1
(7)
∀p ∈ P
Objective function The objective function aims at
maximizing the number of admitted patients during
the planning week and at scheduling the patients with
higher value of weight w:
max
wp adm pb .
(17)
p∈P b ∈B
rspb = nsp
b
admpj −
b −1
xipjk
i∈S j=1 k∈K
j=1
(8)
∀ p ∈ P , b = 2, . . . , |B |.
Moreover, by the following constraint, a bed can be
occupied not before the relevant patient’s admission
block:
b
y pj ≤ b
b
(9)
adm pj, ∀ p ∈ P , ∀b ∈ B .
j=1
j=1
We now remind that LOS of each hospitalized patient has to be at least equal to the number of blocks
required by the physician, and obviously a bed is occupied during the related period:
rbp1 = nbp adm p1
rbpb ≤ nbp
b
(10)
∀p ∈ P
admpj −
b −1
ypj
∀ p, b = 2, . . . , |B |
j=1
j=1
(11)
The weight w is a value assigned to each waiting
patient and, for the specific clinical specialty considered
in this paper, represents a score computed by taking
into account priority to clinical services, elapsed waiting
time, and maximum allowed waiting time. As we will
see in the following section, this score is weekly updated
for those patients that remain on the waiting list for
over one week.
The complete mathematical formulation is reported
in the following (where it is not explicitly specified, i ∈
S , p ∈ P , b ∈ B , k ∈ K):
wp adm pb
(18)
max
p∈P b ∈B
s.to
p∈P
xipb k ≤ µib k
b ∈B k∈K
i
x pb k
i∈S
∀i, ∀b , ∀k
xipb k ≤ sip
∀i, ∀ p
≤1
∀ p, ∀b , ∀k
adm pb ≤ 1
∀p
b ∈B
nbp ypb ≥ rbpb
∀ p ∈ P , ∀b ∈ B
(12)
b
adm pj ≤
j=1
i∈S j=1 k∈K
xipb k ≤ ypb
b +nbp−1
ypj ≥ nbp admpb
∀ p ∈ P , b > |B | − nbp + 1
(13)
rspb = nsp
b
adm pj −
b ∈B
Finally, the number of beds D is the upper bound for
the number of hospitalized patients in each block:
ypb ≤ D, ∀b ∈ B .
(16)
p∈P
∀p
−1
b
i∈S j=1 k∈K
(14)
We remark that a patient p has admission care only
and only if all clinical services are booked. If a patient
is admitted during the planning week, the corresponding binary decision variable adm p = 1 else adm p = 0,
that is:
xipb k = nsp
adm pb ∀ p ∈ P .
(15)
i∈S b ∈B k∈K
∀ p, ∀b
∀i, ∀ p, ∀b , ∀k
j=1
y pb = 0
xipjk
rsp1 = nsp adm p1
j=b
∀ p ∈ P , b = 1, . . . , |B | − nbp + 1
b
b
y pj ≤ b
j=1
b
adm pj
b
adm pj −
∀ p, b = 2, . . . , |B |
∀ p, ∀b
j=1
rbp1 = nbp adm p1
rbpb ≤ nbp
j=1
nbp ypb ≥ rbpb
b +nb
p−1
xipjk
∀p
b
−1
y pj
∀ p, b = 2, . . . , |B |
j=1
∀ p, ∀b
y pj ≥ nbp adm pb
j=b
∀ p, b = 1, . . . , |B |
−nb p + 1
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Health Care Manag Sci (2011) 14:74–88
ypb = 0
i∈S b ∈B k∈K
∀p, b > |B | − nb p + 1
xipb k = nsp
ypb ≤ D
adm pb ∀p
b ∈B
∀b
p∈P
xipb k ∈ {0, 1}
∀i, ∀ p, ∀b , ∀k
ypb , admpb ∈ {0, 1}
∀ p, ∀b
We observe that the proposed formulation represents an “application driven” discrete (i.e., integer linear) optimization model, specifically developed for the
robust and efficient solution of the considered problem.
The computational complexity of this class of optimization problems is generally exponentially affected by the
size of the instance to be solved, and, in our case, this
is mainly related to the cardinalities of the sets B , K,
P and S . Under this respect, we remark that we are
mainly interested to obtain a quite reliable solution
process, in order to have an effective optimal solution,
and we do not need to determine the solution of the
model in “real time”. As we will see in the next section,
the typical size of the above mentioned sets is of the
order of tens. Hence, the choice to use theoretically
well sound general purpose methods based on exact
solution approaches is well-founded.
4 Pilot study within a rheumatology division
The Rheumatology division of “Careggi” University
Hospital (Florence, Italy) is one of the most important national clinical site within its own field. During
2009, the division performed 5,400 outpatient visits and
11,786 clinical treatments on day-hospital basis. Sclerosis systemic (i.e., a systemic disorder that affects the
connective tissue of the skin, internal organs, and the
walls of blood vessels), rheumatoid arthritis, psoriatic
arthritis, and spondylitis rheumatology are the main
treated pathologies. Patients can be admitted to the
WH for many reasons, such as advanced diagnostic
tests, drug dosage tuning, stabilization and monitoring
of current health conditions.
Rather recently, the division started to experiment
WH services, devoted to both diagnostic tests and
therapeutic treatments (i.e., clinical services); these are
prescribed by physician by considering patient’s health
history and current pathological conditions. Data collected during the pilot study concerns patients affected
by different rheumatic diseases; each patient has a
proper set of prescribed clinical services to be performed. Currently, WH organization is based on a
timetable of available clinical resources, exploiting the
availabilities of only four dedicated beds. Patient flow
management is carried out manually, with a considerable effort of area manager, and so far this has implied
that trivially only four patients per week have been
admitted to the WH.
Generally speaking, it is well known that variability
inherently characterizes different health care processes
for several reasons. In order to suitable assess the proposed model, we have tackled both clinical variability
and flow variability, through simple but rather realistic
test cases. A clinical variability is defined by various
patient’s pathological conditions; this means that the
clinical pathway (represented here as a set of prescribed clinical services and minimum required LOS)
is typically different for every waiting patient. A flow
variability is given by patients arriving during a given
period. Moreover, the rheumatologist has typically a
close working relationship with other medical doctors
(e.g., orthopedists, ophthalmologists, physical and occupational therapists, child psychologists, and nephrologists); this particular aspect increases the difficulty in
efficiently tackling the considered problem. As we will
show, we have tackled also this variability in the third
test case by changing the capacity, defined per time slot,
of some clinical services.
To capture some dynamic aspects (which are typical
in this context, as already stated) and to manage the
waiting list P , we have designed the following rule that
assigns a score to each waiting patient:
p
wp = prp D1 − D0 W̄ − MaxWait p , ∀ p ∈ P ,
(19)
where D1 is the date of the planning-scheduling, D1 −
p
D0 denotes the number of elapsed waiting days for
patient p until D1 , pr p is the clinical priority and
MaxWait p denote the maximum waiting time (in days)
allowed for patient p, typically fixed on the basis of
the clinical protocols of the relevant specialty. W̄ is an
appropriate upper bound on the value of MaxWait p
(W̄ ≥ max p∈P {MaxWait p } + 1).
The choice of the multiplicative form of Eq. 19 is motivated by its capability to sharper represent (with respect, for example, to an additive form) the differences
among the patients in terms of the related values of pr p ,
p
D1 − D0 , and MaxWait p .
To differentiate the several rheumatic pathologies
that affect patients, three priority classes have been
defined by taking into account illness severity. In
Table 1, we report these classes with the corresponding
priority value and the maximum waiting time for a
given patient. It is worth noting, that the maximum
waiting time for a patient of class A is smaller than the
Health Care Manag Sci (2011) 14:74–88
Table 1 Priority classes and
maximum waiting time (in
days) of patients into a
Rheumatology division
81
Class
Priority
MaxWait
(days)
A
B
C
3
2
1
10
30
60
waiting time of a patient of class B and class C, due to
the higher priority.
In our proposed approach, patient’s weight is computed by considering clinical priority, elapsed waiting
time (defined as the elapsed time between the date
of baseline visit and the date of planning-scheduling),
and the maximum number of waiting days. Moreover,
patient’s waiting time depends on several factors, as we
will see in the following. Specifically, we have managed
patient’s variability and flow variability by considering
scenarios concerning the following aspects:
1. different diseases/pathologies with different severity;
2. different minimum required LOS of hospitalized
patients;
3. variable patients arrival rate along the planning
period;
4. variable capacity of some clinical services.
We suppose all care providers are equal in their
ability to provide quality health care.
4.1 Experimental setting and computational results
In this section, we assess the efficiency of the proposed
optimisation model 18. To this end, computational experiments have been carried out on realistic instances,
by taking as use case the Rheumatology division of
“Careggi” University Hospital. The typical set of clinical services, mainly based on diagnostic tests, is reported in Table 2.
The overall health care organization of the hospital
is such that the health care manager defines a specific
Table 2 Clinical services of rheumatology division
1) Computed tomography scan
2) X-rays
3) Nuclear magnetic resonance
4) Biopsia
5) Ecography
6) Eco-doppler
7) Hematochemical analysis
8) Human leucocyte antigens
9) Pulmonary function test
10) Scintigraphy
11) Eco heart
12) Holter heart
13) Esophageal manometry
14) Gastroscopy
15) Eyes exams
16) Mammography
17) Angiography
18) Colonoscopy
timetable for every division. The relevant clinical services timetable of the Rheumatology division, fixed at
the tactical level, is reported in Table 3, where the
clinical services are numbered according to the notation of Table 2. Note that, for each slot of every
block, the cell entries of Table 3 show the available
clinical services, where the reported enumeration does
not indicate a prescribed order for service. In fact,
as we have already stated, in the case of rheumatic
diseases clinical domain, a prescribed order for services
is not strictly requested. On the other hand, since we
assume that the clinical services timetable is already
provided by the hospital health care management, an
order for services is implicitly defined by the same
timetable.
As mentioned in the previous section, for the sake
of simplicity, every time slot (TS) of the timetable has
the same duration. However, this does not impair the
applicability of the optimisation model to other settings
because an important requirement for patients scheduling problem is the defined clinical resource capacity
per TS. Indeed, commonly the hospital manager has
previously computed a resource capacity per TS, either
by taking into account if there are more than one
resource for a specific clinical service per TS or if it is
possible to execute more than one in the same time. In
this way, the capacity of a given resource per TS defines
the maximum number of patients that can carry out the
relevant clinical service during the defined TS; thus, the
time to be spent for executing a specific clinical service
is not specified here.
In the following, we have considered three realistic and different test cases (coming out from data
of “Careggi” Rheumatology division) representing the
decision making process during a planning horizon of
two consecutive weeks. A schematic representation of
the patient flow and related decision making process,
along two consecutive weeks, is reported in Fig. 2.
The number of available beds is 4 in each block.
If not specified, the capacity of each clinical service,
available into a defined TS, is assumed equal to 2
(i.e., at most two patients can carried out the related
clinical service during this TS) and the minimum LOS
required by physician equal to 2 blocks for each patient
(i.e., nb p = 2, ∀ p ∈ P ). The main aim of WH, for this
specific division, is to monitor the course of diseases
through suitable clinical tests and assess signs and
symptoms for a prompt titration of drug treatments. We
remark that the minimum LOS imposed by physician is
mainly due to specific medical treatments/procedures
that a patient needs to undergo. As we will see, this
value typically affects the optimal schedule.
82
Health Care Manag Sci (2011) 14:74–88
Table 3 Clinical services
timetable for the
Rheumatology division (m =
morning, a = afternoon).
Note that the enumeration
does not imply a defined
order for service
4.2 Test case 1
Within this test case (which consists of two consecutive
planning weeks), we assume a waiting list of 20 patients
at the beginning of the first week. The related information (i.e., prescribed clinical services, priority value, and
computed score) are reported in Table 4. We remark
that the score of patient p is computed by considering
the elapsed time (measured in days) between current
p
date D1 of planning and scheduling process and D0
(date of the baseline visit).
Even though the small size of this instance, it is quite
evident the strong difficulty in manually determining
an efficient patients scheduling with only “paper and
pencil”, basically due to the complex combinatorial
nature of the WH management problem. The process is
time-consuming due to the lack of an efficient software
support system and the schedule is, in some cases, far
to being optimal. On the contrary, both patients admis-
sion/discharge planning problem, and thus the scheduling of clinical services problem, is solved to the optimality, in a few seconds by implementing and solving
the proposed optimisation model 18 with the standard
MIP solver of ILOG CPLEX 10 (www.ilog.com), on a
PC MS Windows XP Professional SP3, Intel Core duo
Processor T2400, 1.83 GHz CPU with 1GB RAM. The
model is able to select the waiting patients with high
score by matching efficiently both the requirements of
availability and capacity of the clinical services.
The obtained solution defines the set of patients
to hospitalize during the considered week, the related
admission and discharge blocks, and the related appointments for performing the required clinical services. In this specific week, eight waiting patients in
P (highlighted in Table 4 with grey color) are selected. The optimal schedule is reported in Fig. 3: for
each hospitalized patient both admission and discharge
blocks, and thus also his/her LOS, are defined. For
Fig. 2 Patient flow and decision making process for two consecutive weeks. Given a waiting list, some patients are scheduled in the
first week. The unscheduled patients with new arrivals form a new waiting list, used to plan the second week
Health Care Manag Sci (2011) 14:74–88
83
Table 4 Waiting list of the
first week of test case 1:
waiting time, priority, score,
and set of prescribed clinical
services
the sake of brevity, the booked appointments (for this
and the other instances) are not reported. We observe
that some patients (e.g., P4 , P5 , P7 , P9 ) have LOS of
only two blocks: these patients undergo all prescribed
clinical tests during the minimum required LOS (which
is 2 blocks in this specific instance). On the other hand,
the LOS of other patients (e.g., P11 that has to undergo
only 2 clinical services) is longer than 2 blocks. These
particular cases can be explained by remarking that
not only the specific waiting list and the prescribed
clinical services affect the optimal schedule, but also
the availability and capacity of the required resources
to perform such services. Furthermore, we observe
that patients P1 , P2 , P3 have the same score and P3
is scheduled. As it has been verified, the schedule still
remains optimal if P2 is hospitalized instead of patient
P3 . On the other hand, if P1 is scheduled, a worse
solution (measured in terms of number of hospitalized
patients, resources utilization, and total score) is ob-
Fig. 3 Test case 1: scheduled
patients and LOS of the first
planned week (m = morning,
a = afternoon)
tained. Finally, we notice that the number of available
beds is the upper bound on the number of the scheduled
patients. Of course, all these considered issues could
not be drawn when a manually scheduling approach is
used.
The next step of a good WH management concerns
the updating phase of data used to plan the subsequent
week. The dynamic aspects of the problem are captured
by using the rule 19 and by updating the waiting list
with the new arrivals: the set of not currently scheduled
patients (e.g., P1 , P2 , and so on) have an updated score
due to a further wait (D1 is now different), whereas the
score of new arrivals is defined by their priority class.
The value of parameter W̄, used to update the score of
patients is set to 360. Obviously, a new arrived patient
could be scheduled while some patients could remain
still waiting. This specific case is provided by the second
week, which is characterized by 11 new arrivals. These
patients have been visited during the previous week by
84
Table 5 Waiting list of the
second week of test case 1:
waiting time, priority, score,
and prescribed clinical
services. The scheduled
patients are in grey color
Fig. 4 Test case 1: scheduled
patients and LOS of the
second planned week (m =
morning, a = afternoon)
Table 6 Waiting list of the
first week of test case 2:
waiting time, priority, score,
minimum LOS (LOSmin )
required by physician, and set
of the prescribed clinical
services. The scheduled
patients are in grey color
Health Care Manag Sci (2011) 14:74–88
Health Care Manag Sci (2011) 14:74–88
Fig. 5 Test case 2: scheduled
patients and LOS of the first
planned week (m = morning,
a = afternoon)
Table 7 Waiting list of the
second week of test case 2:
waiting time, priority, score,
minimum LOS (LOSmin )
required by physician, and
prescribed clinical services.
The scheduled patients are in
grey color
Fig. 6 Test case 2: scheduled
patients and LOS of the
second planned week (m =
morning, a = afternoon)
Fig. 7 Test case 3: scheduled
patients and LOS (m =
morning, a = afternoon)
85
86
the rheumatology staff. The corresponding waiting list
is thus constructed by updating the score of patients
not scheduled during the first week, according to the
described rule 19 and by inserting the 11 new patients.
Hence, the new waiting list has 12 patients (from P1 to
P12 ) with the updated score, and 11 new patients (from
P13 to P23 ). The related data are reported in Table 5.
By solving the optimisation model 18, the new set
of patients to hospitalize is determined (patients are
highlighted in grey color in Table 5). The optimal
schedule is reported in Fig. 4. We observe that also
in this case there are patients with LOS of only two
blocks (e.g., P1 , P4 ); some of the scheduled patients
have a high score since the elapsed waiting time is
higher than that of patients arrived during the previous
week. Moreover, it may happen that patients belonging
to class A are not scheduled despite the highest priority
class (e.g., P10 and P18 ): this is due both to the score
of the other patients and the availability of clinical
resources (i.e., timetable structure).
4.3 Test case 2
This second instance, consisting of two consecutive
planning weeks, has been defined with the aim of showing as a small variation in the data may result in a significant change in the optimal schedule. In particular,
the same data of the first week of test case 1 are used,
with the only exception of the minimum required LOS,
which depends now on patient, as reported in Table 6.
By solving the optimisation model 18, a set of hospitalized patients different from that determined for the
test case 1 is obtained (see Fig. 5).
The second week is characterized by the same 11 new
arrivals of test case 1. Now, the waiting list consists of
10 patients (from P1 to P10 ), with updated score, and
11 new patients (from P11 to P21 ). The corresponding
data are reported in Table 7. The optimal schedule
determined in this case is given in Fig. 6, from which
it is evident that the number of scheduled patients
(i.e., 10) is higher than that obtained in the test case
1, and this is mainly due to a different value of the
minimum required LOS. It is worth observing that, also
in this simple case, a manually drawn schedule requires
a strong human effort and, in any case, it is difficult to
define an optimal schedule.
4.4 Test case 3
The third test case has been defined with the aim
of investigating the influence of the clinical resources
availability on the patient scheduling. This instance has
been defined by considering the second week of the test
Health Care Manag Sci (2011) 14:74–88
case 2 and modifying the capacity of clinical services 3,
7, and 8, which has been increased to 3.
The corresponding optimal schedule is reported in
Fig. 7. We exploit this schedule to highlight a very
important aspect, that is just the number of scheduled
patients does not allow to have a clear indication of how
the system is performing.
By comparing the optimal schedule of the test case
3 with that obtained on the second week of the test
case 2 (see Fig. 6), we observe that a lower number of
patients is scheduled (i.e., 9 scheduled patients instead
of 10), but the optimal value of the objective function is
greater. This improvement, in terms of objective function is due to the increased capacity of some clinical services allowing to schedule patients with higher weights:
the patients P8 , P18 , and P19 (that are scheduled in
the second week of the test case 2 and not in the
test case 3) have an overall weight 13.5, whereas the
patients P2 and P16 (scheduled in test case 3 and not
in test case 2) have an overall weight 21.15. Since the
optimal decision criterion of the optimization model 18
is the maximization of the total weighted sum of all
scheduled patients, the increased availability of some
clinical resources (test case 3) allows to further increase
the optimal value of the objective function.
Hence, the efficiency of a schedule cannot be measured by only counting the scheduled patients; as it has
showed in this simple test case, both the total score of
scheduled patients and the clinical resources’ utilization
rate need to be taken into account.
5 Discussion and conclusions
In this paper, a novel WH management has been
considered by exploiting quantitative optimisation approaches. In particular, the proposed optimal decision
making model may represent an effective tool for supporting the hospital health care manager aiming at improving the quality of health care delivered to patients.
By comparing the real schedules of the Rheumatology division of “Careggi” University Hospital and the
optimal schedules obtained by solving the optimization
model 18 on the depicted test cases, there is a clear
improvement in terms of number of scheduled patients.
Due to the high complexity of the problem, actually
the manual schedule is able to hospitalize only four
patients per week. Furthermore, we have shown that it
is possible to provide a more efficient use of the clinical
services with respect to a manually defined schedule.
When the values of some parameters change (e.g., the
minimum required LOS of patients and capacity of clinical services) also the optimal schedule could change
Health Care Manag Sci (2011) 14:74–88
but the use of clinical service is always optimised as
possible.
We remark that the number of available beds is
the crucial determinant affecting patients scheduling.
Indeed, the beds availability represents potentially a
bottleneck for patients flow and is a very tight constraint in case of manual scheduling, even if resource
capacity increases.
Thus the proposed test cases give an idea about
the complexity of the WH management and motivate
the need of effective and efficient tools for optimally
supporting the decision making process.
On the other hand, WH organization can be
beneficial also in terms of remarkable reduction of
economic costs. From a recent study made in Italy
considering cost analysis in sclerosis systemic [14], it is
possible to estimate a reduction of direct and indirect
costs of about four times with respect to the ordinary
inpatient hospitalization.
On the basis of the reported experimental validation
phase and its results, the following general conclusions
can be drawn.
–
–
–
–
The proposed model can be also applied to verify if
a patient could be admitted in WH. In fact, during
the baseline visit, it is possible to confirm whether a
patient may undergo all the clinical services during
a week on the basis of the given timetable. If this is
not possible, patient will not be added into the WH
waiting list.
As far as the structure of the objective function of
our model is concerned, we remark that the maximization of the sum of the weight of scheduled patients allows a LOS reduction for some patients. On
the other hand, it could be a conflicting goal in some
cases with LOS minimization; this specific problem,
can be handled by considering a bi-objective integer programming model whose objective function
is a linear combination (with appropriate weights)
of the two mentioned goals. It is our intention
to further investigate this issue in a forthcoming
paper.
Finally, a waiting time reduction could be possible
by allocating resources in a different manner or
changing their capacity. Since a timetable of resources is a factor that strongly affects the optimal
schedule, our future work will be an integration of
tactical decision level.
We remark that the optimal schedule obtained by
the optimisation model can be interpreted from
different point of view:
–
–
–
–
87
from a patient perspective, since he/she knows
in advance admission and discharge date (if
no complication occurs) and the clinical services appointments. Any repeated admission is
avoided with a consequence reduction of stress
and social costs;
from a health care operator since he/she knows
in advance the clinical procedures per TS to
perform;
from a manager perspective, since costs and
utilization of all resources are known in advance. He/she analyzes if current utilization is
in line with best practice and is able to identify
critical resources. A timetable is modified when
changes in volume of clinical service demands
or other requirements occur.
The application of the optimisation model may
support the management of health care personnel.
Since every clinical service requires a specialized
medical staff, activities of personnel/staff can be
planned in advance (a medical doctor/staff constraint is solved by manager when he/she defines
the timetable).
References
1. Cayirli T, Veral E (2003) Outpatient scheduling in health
care: a review of literature. Prod Oper Manag 12(4):519–549
2. Gupta D, Denton B (2008) Appointment scheduling in health
care: challenges and opportunities. IIE Trans 40(9):800–
819
3. Kopach R, De Laurentis PC, Lawley M, Muthuraman K,
Ozsen L, Rardin R, Wan H, Intrevado P, Qu X, Willis D
(2007) Effects of clinical characteristics on successful open
access scheduling. Health Care Manag Sci 10:111–124
4. de Bruin A, Bekker R, van Zanten L, Koole G (2010) Dimensioning hospital wards using the Erlang loss model. Ann
Oper Res 178:23–43
5. Williams P, Tai G, Lei Y (2010) Simulation based analysis
of patient arrival to health care systems and evaluation of
an operations improvement scheme. Ann Oper Res 178:263–
279
6. Kusters R, Groot P (1996) Modeling resources availability
in general hospitals design and implementation of a decision
support model. Eur J Oper Res 88:428–45
7. Vanden Bosch P, Dietz D (2000) Minimizing expected waiting in a medical appointment system. IIE Trans 32(9):841–8
8. Kaandorp GC, Koole G (2007) Optimal outpatient appointment scheduling. Health Care Manag Sci 10:217–229
9. Zeng B, Turkcan A, Lin J, Lawley M (2010) Clinic scheduling
models with overbooking for patients with heterogeneous noshow probabilities. Ann Oper Res 178(1):121–144
88
10. Creemers Sand Lambrecht M (2010) Queueing models for
appointment-driven systems. Ann Oper Res 178:155–172
11. Chern CC, Chien PS, Chen SY (2008) A heuristic algorithm
for the hospital health examination scheduling problem. Eur
J Oper Res 186(3):1137–1157
12. Lambert CM Hurst N (1995) Health economics as an aspect of health outcome: basic principles and application in
rheumatoid arthritis. Br J Rheumatol 34(8):774–80
Health Care Manag Sci (2011) 14:74–88
13. Santibanez P, Begen M, Atkins D (2007) Surgical block
scheduling in a system of hospitals: an application to resource
and wait list management in a British Columbia health authority. Health Care Manag Sci 10:269–282
14. Belotti Masserini A, Zeni S, Cossutta R, Soldi A, Fantini
F (2003) Cost-of-illness in systemic sclerosis: a retrospective study of an italian cohort of 106 patients. Reumatismo
55(4):245–55