OR Spectrum (2012) 34:319–348
DOI 10.1007/s00291-011-0272-1
REGULAR ARTICLE
Organizing hospitals into networks: a hierarchical
and multiservice model to define location, supply
and referrals in planned hospital systems
Ana Maria Mestre · Mónica Duarte Oliveira ·
Ana Barbosa-Póvoa
Published online: 28 October 2011
© Springer-Verlag 2011
Abstract Health care planners in countries with a system based on a National Health
Service (NHS) have to make decisions on where to locate and how to organize hospital services, so as to improve the geographic equity of access in the delivery of care
while accounting for efficiency and cost issues. This study proposes a hierarchical
multiservice mathematical programming model to inform decisions on the location
and supply of hospital services, when the decision maker wants to maximize patients’
geographical access to a hospital network. The model considers the multiservice structure of hospital production (with hospitals producing inpatient care, emergency care
and external consultations) and the costs associated with reorganizing the hospital
network. Moreover, it considers the articulation between different hospital services
and between hospital units, and the ascendant and descendent flows related to twoway referrals of patients in the hospital hierarchy. The proposed approach differs from
previous literature by accounting simultaneously for these issues and provides crucial information for health care planners on referral networks, on hospital catchment
areas, on the location and structure of hospital supply as well as on the costs required
to improve access. The results from applying the model are illustrated in an application
to the South region of the Portuguese NHS. Three scenarios are portrayed to describe
how the model can be used in distinct institutional settings and policy contexts and
when there is uncertainty concerning the key parameters of the model.
Keywords Public facility planning · Location models · Network design ·
Health care management
A. M. Mestre · M. D. Oliveira (B) · A. Barbosa-Póvoa
Centre for Management Studies of Instituto Superior Técnico, Technical University of Lisbon,
Avenida Rovisco Pais, 1049-001 Lisbon, Portugal
e-mail: [email protected]
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List of symbols
Abbreviations
DH District hospital
CH Central hospital
Sets
i∈I
j, l ∈ J
k∈K
w, v, a ∈ W
Demand points
Potential locations for DH
Potential locations for CH
Hospital services (w = 1 for inpatient care, w = 2 for emergency
service and w = 3 for external consultation)
Parameters
2
di1j dik
Travel time between a demand point iand a DH j/CH k
d 3jk
Travel time between DH j and CH k
Diw
DCwv
Demand for service w in demand point i
Share of demand transferred from service w in a DH to service v
in a CH
Share of demand transferred from service v in a CH to service w
CDwv
in a DH
Share of demand transferred from service w to v within the same
p wv
hospital
Length of stay spent in service w in a DH/CH
adw acw
Maximum capacity allowed for service w operating in a DH and
DHw CHw
in a CH
Minimum capacity required for service w operating in a DH and
dhw chw
in a CH
α (0 ≤ α ≤ 1)
Weighting factor to differentiate first from second attendances
(expressing planners’ preferences)
β w (0 ≤ β w ≤ 1) Weighting factor to differentiate hospital services
(expressing planners’ preferences)
Population located in demand point j
pop j
popmin
n
stnd
DHcwj CHcw
k
DHec j DHic j
Minimum population located in a demand point required to open
a new hospital
Number of facilities to be opened
Maximum travelling time allowed for a population to access a
hospital
Total unit costs for delivering care in a DH j and in a CH k
icap_X wj
Cost of expanding/investing one bed in an existing/new DH
located in j
Current capacity of DH j
TOC
TIC
Total annual operating costs for all DHs
Total investment costs for all DHs
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Variables
Determines the opening (=1) or closure (=0) of a DH in location j that
X wj
provides service w
Ykw
Determining the opening (=1) or closure (=0) of a CH in location k
that provides service w
Flow from population point i to DH j for service w
fdiwj
w
fcik
Flow from population point i to CH k for service w
zdcwv
jk
zcdwv
kj
tdwv
j
tcwv
k
Ascendant flow from service w in DH j to service v in CH k
cap_X wj
cap_Ykw
Capacity for service w in a DH j
Descendent flow from service w in CH k to service v in DH j
Flow between service w and v inside DH j (auxiliary variable)
Flow between service w and v inside CH k (auxiliary variable)
Capacity for service w in a CH k
1 Introduction
Health care systems in most countries attempt primarily to maximize the populations’
health, equity, efficiency and quality, and, at a second level, to control and/or minimize
health care spending. In order to pursue these objectives, health care systems based on
a National Health Service (NHS) need to plan hospital resources. Accordingly, several
related decisions as follows need to be made: where should hospitals be located so
as to improve the geographic equity of access? What is the optimal structure of the
hospital network? How should the catchment area of each hospital be defined? What
are the costs required to improve access, and are these acceptable?
Such strategic decisions about locating hospitals and organizing hospital networks
have well-known trade-offs with respect to the pursuit of some of the above objectives, such as the trade-off between equity, efficiency and costs (Current et al. 1990).
For example, increasing the geographic equity of access might imply building small
hospitals close to populations, which leads to inefficiencies in scale and higher costs.
On the other hand, the high cost of some medical equipment and the low availability of highly skilled human resources (such as specialized doctors) might imply that
the supply of services is delivered to large populations, which might have a negative
impact on geographic access. Subsequently, planning models defining the supply and
location of hospital services need to account for these aspects.
The present study aims at developing a model that can act as a support tool to help
health care planners to improve the geographic equity of access through the reorganization of hospital networks, while accounting for scale-related efficiency issues (concerning the minimum and maximum capacity of hospital providers) and informing on
the implications of improvements in access for costs. In line with this, a mathematical
programming model is proposed for analysing location allocation. The model simultaneously considers the generic characteristics of hospital systems such as the hierarchical and multiservice structure of hospitals’ production; the connection between
different hospital services and hospital units by considering the flows of patients that
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move through the system; and the ascendant and descendent flows of patients in the
hospital hierarchy. It also computes the costs associated with investments and the
delivery of services in a hospital network. According to the authors’ best knowledge,
two-directional referrals of patients in a hierarchical and multiservice model have not
been considered in the location literature.
The proposed model takes into account the institutional context of a health system
based on an NHS. For instance, it is built so as to maximize the access of patients to
hospital services and to plan public hospital supply strategically. It produces as outputs crucial information on the location and structure of hospital supply desegregated
by service, on hospital catchment population areas, on the referral network between
hospitals and on hospital costs. The constraints used in the model encapsulate several
policy concerns: organization and policy issues, scale-related efficiency issues and
implications of changes in the network for costs. The model is generic and might be
adapted to distinct planned and hierarchical health systems. The applicability of the
model is shown through the solution of a real case study of the Portuguese health
care system (which is similar to other systems, e.g. to the English NHS). The results
illustrate how the model can be applied to distinct institutional settings and policy
contexts, with three scenarios exploring different features of the model being studied.
The first two scenarios examine, which changes in hospital size of existing hospitals
can improve patients’ access to hospital services. In view of that, facility closures are
not allowed but new facilities might be opened. Each one of these two scenarios correspond to different assumptions regarding the provision of care in remote areas—either
to be provided by very small hospitals or by specialized primary care and emergency
services. For the second of these scenarios, a sensitivity analysis is performed showing
how robust the model results are and the extent to which there is a trade-off between
improved access and costs is analysed. The third scenario is informed by directives
from the Portuguese government that indicate the need for building new replacement
hospitals. The best locations for these replacement hospitals are analysed, with hospital
closures being allowed for selected sites.
The remainder of the paper is organized as follows: Section 2 makes a brief review
of the location models relevant to planning health care facilities. In Sect. 3 we present
some background information on planned health care systems that is important to
frame the proposed model. In Sect. 4 we explain the developed multiservice hierarchical model. The model is then applied to the Portuguese health care system, with
some relevant scenarios being explored in Sect. 5. Finally, in Sect. 6 we report the
experience from the use of the model by planners of the North Administrative Health
Region in Portugal and present some concluding remarks.
2 Literature review
2.1 Location models
Location models have been widely studied as decision support tools, due to their
usefulness and due to the importance of facilities’ location to the performance of
systems, as shown in the reviews by Owen and Daskin (1998), Marianov and Serra
(2004), ReVelle and Eiselt (2005), more recently by Smith et al. (2009b) and by Melo
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et al. (2009). Such reviews present the main classes of location problems and their
applications and strategies to overcome the computational difficulties that typically
arise in real case studies. The review by Marianov and Serra (2004) focuses on public
sector facility location problems.
Daskin (2008) identified two large groups of discrete location models: covering
and median-based models. In covering models, a critical distance is established and
cannot be exceeded for demand to be considered satisfied. Given the critical distance,
the objective may be the maximization of the number of demand points that are within
the covering distance (maximum covering model) (Marianov and Serra 2004) or the
minimization of the total number of facilities (set covering model) (Daskin 1995).
Otherwise, the objective of covering models can be the minimization of the covering
distance (centre model). In the health sector, covering models are particularly appropriate for analysing emergency services where the time response is critical. Conversely,
median-based models have been used to maximize patients’ access to services, which
is achieved through an objective function that minimizes the demand-weighted travel
distance.
Incorporating equity into facility location models has become an important subject
since the former works by McAllister (1976) and Savas (1978). McAllister (1976)
studied the effects of efficiency and equity and concluded that equity is more sensitive
and should play an important rule in the design of public services. Following these
earlier works, several studies on the public location of facilities in general, and on the
location of health care facilities in particular, have considered the pursuit of multiple
and conflicting objectives, embodied by alternative definitions of equity, efficiency
and access (Mayhew and Leonardi 1982; McAllister 1976; Smith et al. 2010). The
measures developed for equity were reviewed by Marsh and Schilling (1994), who
concluded that there is little consensus on how equity should be measured. Nonetheless, the definition of equity seems to be problem-related and to depend on the
planners’ preference. Moreover, there is no unique overarching equity principle to
guide the distribution of resources in a health care system (Culyer 2001) and public
services encounter difficulties when attempting to satisfy different aspects of equity
(Truelove 1993), as shown in several studies analysing location and related resourceallocation aspects. For instance, Mayhew and Leonardi (1982) analysed alternative
definitions of equity, efficiency and access for resource allocation in the health-sector context. Smith et al. (2010) combined efficiency and equity objectives in location
models where equity is introduced with a service standard distance, from which absolute deviation is minimized (note that the definition of efficiency in these studies is
defined as alternative definitions of equity in other studies Oliveira and Bevan 2006).
Eiselt and Laporte (1995) presented a comprehensive review of the objectives in location problems where the minimization of the total travel time (minisum objective) is
associated with the private sector and the minimization of the maximum travel time
(minimax objective) with the public sector.
The applicability of the minimax models to problems where there are significant
differences in population access may be restrictive since they tend to focus on the case
of the most disadvantaged populations (in terms of access), meaning that a minisum
objective constrained by upper bounds to travel time might be more suitable in many
systems. This means that median-based models with a minisum objective allow for the
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analysis of improvements in demand-weighted travel distance while ensuring that a
maximum travel time is ensured for all patients (Eiselt and Laporte 1995). This latter
feature is consistent with health-policy objectives in Portugal and in other NHS-based
systems. Thus, when comparing with the previous types of models referred above,
median-based models appear as adequate and flexible to consider an existing network
of hospitals and to represent and model patient flows through a network of hospitals (which are key features of hospital systems).Following this, an extension of a
median-based model with a minisum objective has been developed in this article, as
it is compatible with the institutional characteristics and the policy objectives of the
health system to be modelled.
Planned networks of hospitals are composed of different types of providers organized in a hierarchy, as will later be described in greater detail. Therefore, hospital
location analysis needs to consider the potential flows of patients and the interactions
between providers. Hierarchical models are an extension of median-based models that
specifically deal with these issues, and although we acknowledge the usefulness of
other location models in health care planning (Rahman and Smith 2000), we focus our
review on hierarchical models that specifically relate to the objectives of our study
and that are meaningful for the context of NHS systems.
2.2 Hierarchical location models
Narula (1986) presented the first review on the use of hierarchical location models, and
a more recent survey has been published by Sahin and Süral (2007). In a hierarchical
model, the facilities can be classified as successively inclusive when higher-level facilities offer both higher- and lower-level services, and lower-level facilities only deliver
lower-level services, or as successively exclusive, when each facility only ensures its
own level of services. These types of hierarchies allow for the modelling of different
ways for users to access services and depict the distinct underlying characteristics of
health systems. For planned systems where there is a gatekeeping referral system—in
which users must enter through lower-level facilities and afterwards can be referred
to higher levels, and higher-level facilities provide both higher- and lower-level services—the use of a successfully inclusive facility hierarchy is appropriate to represent
the hospital network problem. In health systems where users have direct access to services at any level of the network, the use of a successively exclusive hierarchy might
be more adequate. The use of a successfully inclusive facility hierarchy is consistent
with the structure of the Portuguese health care system (and of other NHS plan-based
systems such as the English system). In these systems, the access of patients to hospital
services is achieved according to a gatekeeping system, with patients assigned to a specific facility (lower or higher level) and being transferred from lower- to higher-level
hospitals when they require higher-level services. Lower-level services are provided
in smaller hospitals closer to populations, and higher-level services are provided in
larger hospitals concentrated in urban centres; higher-level hospitals commonly provide both lower-level services to their local populations and higher-level services to a
larger catchment area.
The first hierarchical model analysed in the scope of this work belongs to Ruth
(1981). This study presented a model for planning hospital inpatient services that
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considers three levels of services working in a successively inclusive hierarchy. This
model includes minimum capacities that take into account indirectly economies of
scale and efficiency issues. This threshold on capacity breaks the assumption of assigning each demand point to a single facility and to the closest facility. This is because
in the presence of minimum capacities, the demand might not be met in the closest
facility and/or one demand point might be assigned to more than one facility (Teixeira
et al. 2007). Although the use of minimum capacities potentiates economies of scale in
health care delivery, it might worsen the access to services for part of the population.
Nevertheless, while planners pursue economies of scale, splitting demand points or
making demand points not to use the closest facility is hardly accepted in public facility location, meaning that models should contain explicit restrictions for the single
and closest assignment (Gerrard and Church 1996).
Recent studies have developed several features of hierarchical models. Galvão et al.
(2002) applied a three-level hierarchical model for the delivery of perinatal care in the
municipality of Rio de Janeiro and considered three main types of facilities that cooperated in a successively inclusive hierarchy with service referrals. Galvão et al. (2006)
introduced capacity constraints that increased the model complexity, requiring the use
of Lagrangian heuristics to solve the problem. More recently, Smith et al. (2009a)
proposed a set of hierarchical models that range from covering type to p-median.
These models were developed in the context of planning sustainable community health
schemes and applied to a rural Indian region. Also, Smith et al. (2010) built a range of
discrete hierarchical location models with bicriteria efficiency/equity objectives using
multiobjective mathematical programming formulations.
In the context of the Portuguese NHS, Oliveira and Bevan (2006) analysed the redistribution of hospital supply through the definition and comparison of three alternative
models. These models used different objective functions representing alternative definitions of equity of access and utilization. Also, different constraints in the model
captured different institutional characteristics of the hospital system and alternative
assumptions on the behaviour of patients when using hospital services. Nevertheless,
none of these three models explicitly accounted for the administrative hierarchy of
hospitals or for the flows of patients between hospitals.
To the best of our knowledge, hospitals have always been taken as single-service facilities, with inpatient care being the main service, and hospital flows have
only accounted for ascendant ways in the hierarchy. Health policy-makers have been
increasingly recognizing the need to reorganize and design networks of providers,
where the hierarchical and multiservice nature of hospital services is explicitly stated,
and any health care reform needs to be informed by its cost implications. In Portugal,
the recent reforms are in line with these research needs: the emergency care network
is currently being reorganized so as to consider population density and accessibility
criteria adequately (Health Ministry 2007); the redefinition of patients’ pathways in
the hospital system is important for coordinating services delivered at different levels of the hierarchy, it being critical to account for descendent referral flows (Health
Ministry 2002).
The present study differs from previous research by proposing a hierarchical model
that accounts for the multiservice nature of health care delivery, for two-directional
referral of patients between higher- and lower-level units and for the cost implications
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of redesigning the network of providers. The model produces information on location,
supply and referral and allows for the examination of the trade-off between improvements in access and cost implications. Although these features are new in the location
literature for the health sector, some of them have already been studied in other contexts and are shown to be useful when analysing hospital networks—for example,
modelling reverse flows is a feature that has been used in the development of green
logistics in the area of supply chain management (Salema et al. 2009) (one should note
that this branch of the literature does not consider other features, such as the hierarchical structure and two-directional referral of patients between higher- and lower-level
units).
3 Health-system background information
We consider the context of a health system based on an NHS with universal coverage,
nearly free access at the point of use, funded by public taxation and with a role of
the state in planning health care supply. This is the case of the Portuguese NHS, the
English NHS and the health care systems of several Spanish autonomous communities. A key objective of the political system in countries with an NHS-based system
is to achieve equity among the citizens, independently of their economic condition
or geographic distribution (Health Ministry 1990; Rice and Smith 2001). Efficiency,
quality and cost containment in health care delivery are other important policy objectives. The supply of health care services in these systems is dominated by a set of
public providers that cooperate in an integrated network of care. Planned hospital systems are commonly organized in several administrative types of hospitals (from lower
to higher technological complexity, from small to large catchment populations and
from basic to specialized care), which indirectly capture issues related to economies
of scale, efficiency and the availability of human resources in health. Typically, lowerlevel hospitals, often named district hospitals, refer patients to higher-level hospitals,
named central hospitals, which afterwards might refer patients back to lower-level
hospitals (reverse flow). Each hospital has a catchment area that comprises a set of
populations from adjacent municipalities (Health Ministry 2002).
Hospitals provide three main types of services: external consultations, emergency
care and inpatient care. Within a gatekeeping system—where patients enter the system via a primary care consultation or an emergency episode—access to external
consultations might follow a first entry into the system. Portugal has a level of hospital supply (as measured by per capita indicators) close to the average of countries of the Organisation for Economic Co-operation and Development (Oliveira and
Pinto 2005). This means that changes in hospital supply might be achieved mainly
through the redistribution of current supply, or by building replacement hospitals
(Oliveira and Bevan 2006). Hence, the chosen scenarios will take this context into
account.
To sum up, building a decision support tool to inform simultaneously on hospital
location, supply, referral and related costs meets the needs of health care planners in
NHS-based countries in general, and of Portuguese planners in particular.
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4 The hierarchical and multiservice model
We propose a multiservice hierarchical mathematical programming model that can be
used as a tool to support the decisions of planners who want to maximize patients’
geographic access to hospital services, while taking into account the population needs,
the characteristics of the hospital system, scale-related efficiency issues, political constraints and the implications of changes for costs. The model is formulated as a mixed
integer linear programming (MILP) model, where the integer variables are associated
with the locations of the hospitals and the continuous variables are related to the flows
of patients within the network.
We start by defining the conceptual representation of the system. We consider that
in most countries based on a NHS structure, the hospital hierarchy can be represented
by two main levels: central hospitals (CH) and smaller-scale district hospitals (DH).
The hierarchical relationships and the main flows of patients in hospital systems may
be generally represented as in Fig. 1. Thus, Fig. 1 conceptualizes the flows for the
hierarchical and multiservice model proposed in this study, including only the most
significant flows between hospital services and between hospital units, and for which
there is routinely collected data in most health systems. The model can be easily
adapted to account for other flows that might exist and be meaningful for a specific
health care system.
Figure 1 should be read as follows: the demand for three health care services
(inpatient care, emergency care and external consultation) is set for each small area,
according to the population need for hospital services (which might be influenced by
population numbers, the age and sex structure of the population, and by admission
rates). Those services can be provided in DH and/or in CH hospitals. When entering
the hospital system, a patient has a predefined probability of changing to a different
service—for instance, see in Fig. 1 the flow from emergency to inpatient service. If the
population demand point is allocated to a lower-level unit (DH), it has a predefined
probability of being referred to another service within the same hospital and/or of
Fig. 1 Patient flow scheme (within and between hospitals)
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being referred to a higher-level unit (CH) with more specialized services. Moreover,
after inpatient treatment in a CH, there is a probability of the patient being referred
to a lower-level hospital, e.g. a proportion of patients will follow a reverse flow (from
CH to DH). Based on this we find it appropriate to assume that hospitals are organized
in a successively inclusive hierarchy with two types of hospital services:
– Lower-level services are provided in DHs and in CHs for their local catchment
areas;
– higher-level services are specialized services provided only in CHs to local populations and to non-local populations located within the DH catchment population
areas.
Descendent flows in the hierarchy represent patients who have already been treated
and who no longer need specialized services. An optimization model based on the
described hierarchical structure was developed using the notation displayed in “List
of symbols”.
The model is formulated as
⎡
2
w
βw × ⎣
di1j × fdiwj +
dik
× fcik
Min z =
w∈W
⎛
+α× ⎝
i∈I j∈J
i∈I k∈K
d 3jk ×zdcwv
jk +
j∈J k∈K v∈W
⎞⎤
⎠⎦
d 3jk ×zcdvw
kj
(1)
j∈J k∈K v∈W
Subject to
fdiwj +
j∈J
w
fcik
= Diw ∀i∈I, w∈W
(2)
k∈K
fdiwj × p wv = tdwv
∀ j∈J, w,v∈W
j
(3)
w
fcik
× p wv = tcwv
∀k∈K , w,v∈W
k
(4)
i∈I
i∈I
fdiwj +
i∈I
zcdvw
kj
=
tdaw
× DCwv =
j
a∈W
wv
zdcwv
jk × CD
cap_X wj =
cap_Ykw
fdiwj +
zdcwv
jk ∀ j∈J, w,v∈W
(5)
k∈K
∀ j∈J, k∈K , w,v∈W
tdvw
j +
(6)
zcdvw
× adw ∀ j∈J, w∈W
kj
(7)
⎛
⎞
w
⎠ × acw ∀k∈K , w∈W
=⎝
fcik
+
tcvw
zdcvw
k +
jk
(8)
i∈I
v∈W
k∈K v∈W
i∈I
v∈W
j∈J v∈W
cap_X wj ≥ dhw × X wj ∀ j∈J, w∈W
123
(9)
Organizing hospitals into networks
329
cap_Ykw ≥ chw × Ykw ∀k∈K , w∈W
(10)
cap_X wj
cap_Ykw
(11)
X wj +
X wj +
w
≤ DH × X wj ∀ j∈J, w∈W
≤ CHw × Ykw ∀k∈K w∈W
fdilw
≤ 1 ∀i∈I, j,l∈J, w∈W j∈Cil ={ j|d 1 <d 1 }
ij
il
Diw
w
fcik
≤ 1 ∀i∈I, j∈J, k∈K , w∈W j∈Cik ={ j|d 1 <d 2 }
ij
ik
Diw
fdiwj = 0 ∀i∈I, j∈J, w∈W
j∈{ j|di1j ≥stnd}
(12)
(13)
(14)
(15)
fdiwj ≤ Diw X wj ∀i∈I, j∈J, w∈W
(16)
w
fcik
(17)
≤
Diw Ykw
zdcwv
jk ≤
∀i∈I, k∈K , w∈W
Diw Ykw ∀ j∈J, k∈K , w,v∈W
(18)
i∈I
The model minimizes the total travel time for patients to access hospital services
through the objective function expressed in Eq. (1). This objective function encapsulates a principle of geographical access since one unit of travelling time to
access services for each patient is equally valued. This definition is equivalent to
the minimization of total travelling times, to the minimization of average travelling times and to the minimization of utilization-weighted travelling times. When
interpreting the value of the objective function, one should also take into account
that Eq. (15) ensures that no member of the population should take more than
a maximum (acceptable) amount of time to access hospital services at a DH—
i.e. it is ensured that the minimum acceptable levels are respected for the populations with lower geographic access to services. This concept of maximum critical
time (stnd) has been used in previous studies to indicate that a population has to
access services within a maximal travelling distance (Marianov and Serra 2004). As
explained in the literature review section, using Eq. (15) together with the objective
function guarantees that one specific definition of geographic equity of access is pursued.
Equation (1) includes four terms representing, respectively, the demand-weighted
travel times for patients to reach all DHs, to reach all CHs, to be transferred from
DHs to CHs (ascendant flow) and to be transferred from CHs to DHs (descendent
flow). The β w factors allow the decision-maker to differentiate the travelling time of
patients accessing different hospital services. The α factor allows the differentiation of
the travelling times for patients who are transferred between hospitals. These weights
depend on the preferences of the decision-maker, who might, for example, give a
higher weight to the travelling time of patients accessing emergency care and a lower
weight to the travelling time of patients transferred between hospital units.
Equation (2) ensures that the demand for all hospital services is satisfied. The
demand for hospital service w and for a population point i is measured by the parameter Diw . This is the result of converting population numbers to hospital admissions
through estimates of population needs for hospital services. These estimates should
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ideally consider specific adjustment factors to population numbers such as the age and
gender structure and the level of morbidity of populations.
Equations (3)–(4) assist in the definition of flows within DHs and CHs, respectively.
Equations (5)–(6) define the two-directional flows between DHs and CHs. Equation (5)
establishes the ascendant flows in the hierarchy, by defining the patients who are transferred from DHs to CHs. Equation (6) defines the descendent flows of patients who
are transferred from CHs to DHs.
Equations (7)–(8) are auxiliary constraints (with reference to DHs and CHs, respectively) that convert hospital utilization for each service into a measure of capacity for
that service. These are particularly relevant for inpatient care where capacity is limited
as measured by the number of beds. For each service, these equations are composed
of three parts. These represent the following: new entries of patients into the hospital
system; patients who are within the same hospital unit and who are transferred to that
service; and patients who are in another hospital unit and who are transferred to that
service. The parameters adw and acw convert the flows into a capacity measure: for
inpatient care, this represents the average length of stay (ALOS) measured in days; for
other services, there is no need to convert flows into a different capacity measure—the
capacity is simply the number of attendances.
Equations (9)–(12) ensure that the service offer is constrained by the hospital capacity, incorporating information from health care planners and evidence from the literature in that very small hospitals and very large hospitals operate under diseconomies
of scale (Rosko and Broyles 1988).
The allocation of patients from a population point to the nearest existing facility and
the uniqueness of that allocation is obtained through Eqs. (13)–(14). These constraints
define that within the model, and for each demand point, if a closer hospital is opened,
then the population cannot be allocated to another hospital. Equation (13) applies this
rule for a DH, while Eq. (14) applies this to the case of a CH. Similar equations were
defined for opening CHs (for the sake of simplicity, we exclude these equations from
the text). These are examples of the Dobson–Karmarkar closest assignment constraints
(Gerrard and Church 1996) formulated for continuous variables where a standardization is needed (i.e. conversion into a zero and one scale).
Equations (16)–(18) enhance the model performance. Equations (16)–(17) ensure
that the patients are only assigned to open facilities, in the case of a DH and a CH,
respectively. Equation (18) guarantees that transfers are only made to open facilities.
Summing up, Eqs. (1)–(18) describe the basic hierarchical and multiservice model.
However, depending on the hospital system under analysis, additional constraints
might be used. For example, providing one service might depend on the provision
of another service, which indirectly models the existence of economies of scope. For
example, for the case of Eq. (19), emergency services (i.e. w = 2) and external consultations (i.e. w = 3) can only be served in a DH where inpatient care (i.e. w = 1)
is provided. Therefore, we can write
X 1j ≥ X 2j and X 1j ≥ X 3j
∀ j∈J
(19)
Furthermore, given the use of an objective function that minimizes the time travelled
by patients to access services, it is likely that the chosen locations will be the ones
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Organizing hospitals into networks
331
with higher levels of demand (and of populations). Nevertheless, the model might
also locate hospitals in areas with low population numbers (e.g. this might happen in
remote areas with greater accessibilities and with surrounding large areas with small
populations), and this might not be accepted by health care planners. For instance,
locating public facilities like hospitals in locations with a small population might
be less attractive because of a lack of surrounding public infrastructures and human
resources. Within this context, we can add an extra constraint, Eq. (20), which models
the case when DHs can only be opened in locations that have a local population higher
than a predefined value (popmin ).
X wj = 0 ∀w∈W
j∈J, j∈{ j|pop j <popmin }
(20)
If the planner wishes to indicate the number of units to open, this can also be
included in the model through Eq. (21), where n represents the number of DHs to
open.
X wj = n
∀w∈W
(21)
j∈J
In applying the multiservice hierarchical location model, one should be aware that
combining minimum and maximum capacities (Eqs. (9)–(12)) with an existing network, with closest assignment constraints (Eqs. (13)–(14)), with minimum population
requirements (Eq. (20)), and/or with maximum critical time (Eq. (15)), can lead to
infeasible solutions (as will be discussed in detail Sect. 5.3). In these cases, the maximum capacity might be relaxed and replaced by, for example, the sum of all utilization.
Furthermore, the number of facilities opened in each potential location may be iteratively defined after optimization.
Additionally, the model comprises a set of constraints to compute the financial costs
associated with the reorganization of the network including investment costs related to
the construction of new hospitals, investment costs related to expanding the capacity
of existing hospitals and the annual operating costs that result from hospitals delivering health care services. The definitions of variables related to costs are presented
separately, as the formulation of cost-related equations might depend on the structure
of the available data or on alternative assumptions on costs. We have adopted a formulation for investment costs compatible with the format of data available for Portuguese
hospitals, in which costs vary according to the number of beds of the hospital. Nevertheless, a different cost structure can be easily used within the model proposed (for
example, accounting not only for variable costs, but also for fixed costs). The cost
parameters are synthesized in Table 1. In the selected formulation, the unit costs differ
for DHs and CHs. The costs for building new hospitals and for extending existing
hospitals are defined per bed unit and depend on the capacity of each new hospital and
on the expansion of existing hospitals. The annual operating costs are defined per unit
of emergency entry, per external consultation and per inpatient admission.
Equation (22) defines the operating costs for delivering services in all hospitals,
taking into account the differences in unit costs from providing care in DH and in
CH settings. Equation (23) computes the investment costs associated with investment
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A. M. Mestre et al.
in new DHs and with expanding capacity in existing DHs. A similar equation to
Eq. (23) is defined for CHs. It would also be possible to consider in the model the
costs associated with hospital closures or with the reduction of the existing capacity.
Yet, we have chosen not to introduce these components of cost into the model given
that in many health care systems (such as the Portuguese case), policy-makers prefer
not to close hospitals, the common policy being the conversion of hospitals into other
types of health care facilities.
TOC =
⎛
⎞
⎝
⎠
cap_X wj × DHcwj +
cap_Ykw × CHcw
k
w∈W
TIC =
j∈
+
j
(cap_X 1j − icap_X 1j ) × DHec j
J |cap_x 1j −icap_X 1j >0
∧icap_X 1j =0
j∈
J |cap_X 1j >0
∧icap_X 1j =0
(22)
k
cap_X 1j × DHic j
(23)
Other financial costs for the NHS could also potentially be added to the model,
such as the costs incurred by hospitals for patients’ transportation for emergencies
and for hospital transfers. In addition, other constraints could be added to the model
so as to represent the availability of funding to reorganize the network. For example,
the health care planner could limit the amount available to invest in the reorganization
of the network. The model could easily consider these situations, amongst others.
5 Case study
In this section the model applicability is shown through the solution of a real case
study. The model is applied to the South region of the Portuguese NHS. This section
starts by presenting specific information on the South region, followed by a description
of the data in use. The results for some illustrative scenarios relevant to policy are then
presented and analysed.
5.1 The South region
Portugal has an administrative division in the health sector that allows for the delimitation of three independent and self-sufficient geographic areas: the North, the Centre
and the South regions. The present study focuses on the South region, which contains
a population of 4,572,202 inhabitants and includes 3 Administrative Health Regions
with 7 health sub-regions as depicted in Fig. 2: the Lisbon and Tagus Valley administrative region (Setúbal, Santarém and Lisbon sub-regions), the Alentejo region (Beja,
Portalegre and Évora sub-regions) and the Algarve region (Faro sub-region). The
South region is also divided into 109 smaller area population units (“Concelhos”),
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Organizing hospitals into networks
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333
(b)
District Hospitals
Inpatient Emergency
Service
service
External
Consultation
Faro
491
126,554
Lagos
55
28,781
151,937
4,072
Portimao
244
84,769
91,419
Beja
251
49,641
72,326
Serpa
34
---
4,509
Elvas
108
36,765
28,931
Portalegre
250
45,363
47,543
Évora
384
60,361
129,198
Almada
482
154,638
198,713
Barreiro
379
113,722
144,531
Montijo
95
48,560
15,662
Santiago do Cacem
38
38,394
11,109
Setubal
328
113,273
137,556
Abrantes
151
61,710
48,023
Santarem
393
100,415
125,962
Tomar
151
61,710
48,023
Torres Novas
151
61,710
48,023
Amadora
664
214,806
225,438
Cascais
248
117,643
72,955
Torres Vedras
277
84,951
61,860
Vila Franca de Xira
CH Lisboa
211
107,667
60,747
5,758
694,640
1,883,292
Fig. 2 Geographical areas and the current location of hospitals in the South region in Portugal (hospitals
classified by size, as measured by the number of beds). a Current hospital locations and administrative
sub-regions. b Hospital current capacity in terms of beds, emergency admissions and external consultations
appointments
which are small wards that correspond to administrative areas with political elected
bodies—we consider these small wards to be our geographic unit of analysis. Figure 2
also displays the small wards, as well as the configuration of the current hospital network (that from now on will be referred as current network), which is composed of
12 general CHs and 6 very small specialized CHs, all of them located in the Lisbon
sub-region (17 of these being located in the Lisbon small ward), and of 19 DHs that
are more evenly spread across the region. The South region includes a diversity of
urban and rural areas. Urban areas benefit from improved physical accessibility and
higher geographic proximity to hospital services, while rural areas have seen their
populations decreasing and ageing at a fast rate in the past decades with no hospitals
nearby. The most rural and remote areas are located within the Alentejo region.
5.2 Data in use
The application of the hierarchical and multiservice model requires the use of the best
available information and the computation of estimates for some of the parameters
defined in the mathematical programming model. To compute estimates of population needs for inpatient care, we used the 2003 data from the diagnostic-related group
(DRG) database system (which contains records for all the patients entering Portuguese
NHS hospitals for inpatient care). We calculated the rates of hospital utilization by age
and sex, and through the use of demographic data, projected future utilization. Figure 3
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A. M. Mestre et al.
400
Male
350
Female
Average of 10%
300
250
200
150
100
50
85 +
80 to 84
75 to 79
70 to 74
65 to 69
60 to 64
55 to 59
50 to 54
45 to 49
40 to 44
35 to 39
30 to 34
25 to 29
20 to 24
15 to 19
10 to 14
5 to 9
0 to 4
0
Fig. 3 Estimates on the utilization of inpatient services per 1,000 inhabitants (national average and rates
by sex and age groups)
shows those estimates and should be read as follows: for each 1,000 inhabitants from a
population area, one expects on average 100 hospital admissions per year. These values
will be higher for older populations (in particular for males). These estimates of need
imply that two areas with the same total population might have different utilizations
of inpatient services due to differences in their demographic structure.
Table 1 provides the key data used to run the hierarchical and multiservice model.
Estimates of the need for external consultations and for emergency services were
computed with 2003 data from the General Directorate of Health (Health Ministry
2003). For example, one should interpret 579 in Table 1 as follows: for each 1,000
inhabitants one expects 579 external consultation appointments per year to be provided
in lower-level services. Demographic data for small wards were obtained from the
National Institute of Statistics (2006). The DRG database (described earlier) was used
to estimate transfer rates. Travelling time was calculated between centroids of small
wards using the ViaMichelin Internet website (2006)—these travelling time estimates
take into account road accessibility and road conditions. Two estimates of inpatient
services’ ALOS are used for all DHs and for all CHs (i.e., the ALOS does not vary by
hospital provider). The ALOS data figures were taken from the 2003 DRG database.
The standard limits for hospital capacities for different services made use of information reported by the Portuguese Health Ministry (2006a) and evidence taken from
the literature on the existence of economies of scale. With regard to operational costs,
data were used on the total unit costs per each existing hospital and per service from
the Ministry of Health (2006b). For new hospitals and for a few hospitals with missing data, we assumed average total unit costs available in Health Ministry (2006b).
For estimating the costs associated with investment in new hospital facilities or with
extending capacity in existing facilities, estimates from a study carried out for the
Algarve health region (2004) were taken.
Running the model demanded for the use of other assumptions drawing on knowledge about the system under analysis. For example, assumptions that capture the preferences of decision-makers were used to select the weighting factors. Each hospital was
considered to provide the three services simultaneously, the assumption of economies
of scope being implicit [captured by Eq. (19)]. Patients were required to access a DH
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Table 1 Key data used in the hierarchical and multiservice model
Notation
Value
Description
Diw
579
Number of external consultation appointments per 1,000
inhabitants
Diw
584
Number of entries in the emergency service per 1,000
inhabitants
13%
Percentage of patients transferred from inpatient service
w in DH to inpatient service v in CH
Percentage of patients transferred from external
consultation service w in DH to external
consultation service v in CH
DCwv
19.6%
CDwv
1.24%
Percentage of patients transferred from inpatient service
w in CH to inpatient service v in DH
p wv
9%
Percentage of emergencies in service w that will be
admitted to inpatient service v
adw acw
7 and 8.9
Average length of stay for inpatient service w in a DH
and in a CH
dhw chw
150 and 300
Minimum capacity for inpatient service w in a DH and in
a CH in terms of beds, which are afterwards converted
into inpatient days
=C 380 and =C 544
Average cost of an inpatient day w in a DH j and in a
CH k
Average cost of a hospital entry to emergency service w
in a DH j and in a CH k
Average cost of an external consultation w in a DH jand
in a CH k
DHcwj CHcw
k
=C 143 and =C 119
=C 94 and =C 118
DHic j CHick
=C 200,000 and =C 224,500
Cost of a new bed in a new DH j and CH k
DHec j CHeck
=C 100,000 and =C 124,750
Cost of expanding a bed in an existing DH j or in a CH k
α
0.5
Weighting factor to differentiate first and second
attendances
βw
1
Weighting factor to differentiate between hospital services
popmin
15,000 and 50,000
Minimum number of inhabitants for a DH and a CH
located in the administrative region where they live. Given the Portuguese policy context, a constraint on the number of hospitals that might be opened was not used in the
case study [i.e. for this case we did not apply Eq. (21)]. Information on past utilization
was used to estimate many parameters, and one should bear in mind that figures on past
utilization are possibly affected by the phenomenon of supply-induced demand and do
not account for the unmet needs of some population groups (Folland et al. 1997). Only
the flows between hospitals and within services depicted in Fig. 1 were considered,
although the remaining flows assume a low magnitude in the Portuguese system.
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While analysing the results one should bear in mind the data limitations and that
accurate estimates of normative utilization are strictly required for obtaining reliable
results. The cost figures in use are crude estimates that do not consider either the
geographic location of the hospital or other factors affecting investment costs. Given
the lack of information to compute some parameters, the limitations of the data in use
and the use of assumptions, a sensitivity analysis was carried out to test the impact of
changing key parameters on the model results (see Sect. 5.3.3).
5.3 Studied scenarios
To exemplify the usefulness of the model, three scenarios (scenarios I, II and III) that
correspond to different policy options and reforming contexts were selected. These
scenarios do not aim to provide a unique answer to the optimal hospital location and
structure of hospital supply, but are illustrative of how the model can assist the planning
of hospital networks.
The first results from running the model for the South region have shown that when
the current network was considered in combination with minimum capacities, with
closest assignment constraints and with maximum critical time, infeasible solutions
were obtained. This happens because, in order to ensure that the maximum critical
time was respected, the model required building very small hospitals in rural areas that
did not respect minimal capacity requirements. Given that, we have built scenarios
I and II that study incremental changes to the current network to improve patients’
access to the hospital network under different contexts. The results for these scenarios
might be seen as a second best result, given that they depart from the current hospital
network, which is maintained (i.e. facility closures are not allowed), and improvements are sought through marginal changes in the capacity of existing hospitals and
through the opening of new hospitals. For these scenarios we have as key differences
– In scenario I, a maximum critical time of 60 min is applied and the minimum
capacity requirement of 150 beds for new hospitals is relaxed;
– Conversely, in scenario II the minimum capacity requirement of 150 beds is considered and the maximum critical time of 60 min is relaxed. In this scenario, the
following assumptions apply: populations located at remote areas incur in higher
travelling times to access hospital services; these populations are additionally provided at the local level with specialized primary care (e.g., beds) and emergency
services (e.g., helicopters), so that they have improved access to some services to
be delivered in primary care settings and their real travel time to access emergency
services is decreased through special transportation modes.
When running scenarios I and II, further assumptions were used. For example, it was
assumed for all the hospitals of the Lisbon ward a minimum capacity of 2,000 beds.
This was necessary because it was conflicting to use closest assignment constraints and
higher values for hospital capacity together in Lisbon, as the Lisbon ward currently
has a large supply of hospital services.
After analysing the results for scenarios I and II (see Sects. 5.3.1, 5.3.2), a sensitivity analysis of key parameters of the model was also developed (see Sect. 5.3.3).
This was performed taking as basis the conditions of scenario II since this represents
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Organizing hospitals into networks
337
a much more realistic solution for the network under study. This study shows how the
model can be used to analyse the equity/cost trade-off from introducing changes to
the network (see Sect. 5.3.4). Taking scenario II as the basis, an analysis on costs of
improved access was performed.
Finally, a third scenario, scenario III, was studied (see Sect. 5.3.5). This scenario
considers the Portuguese context where government directives have announced the
intention of building replacement hospitals. It analyses which are the best locations
for these replacement hospitals (i.e. hospital closures are hence allowed at selected
hospital sites). This scenario also considers seasonal variations in the demand for
services of populations located in the Algarve region, a region of tourism in Portugal.
In all the scenarios, allocation to lower level services is only possible within the
administrative health regions, with exception for some bordering locations where
cooperation could bring benefit in access, and where this is currently observed.
The model was implemented in the general algebraic modelling system (GAMS
2010) and solved through the branch and cut method of the commercial solver CPLEX
11. The results were obtained with an Intel® CoreTM i3 CPU [email protected] GHz.
GAMS default options were used to run the studied scenarios and a zero per cent gap
was used as a stopping criterion.
5.3.1 Scenario I
As referred earlier, scenario I considers marginal changes to the current hospital network presented in Fig. 2. Hospital closures are not allowed, but new hospitals might
be constructed. A maximum travel time of 60 min to access hospital services—a value
that has been often mentioned as a reference in Portugal (Health Ministry 2006a,
2007)—is applied. As explained earlier, the minimum capacity constraint of 150 beds
for new hospitals is not applied.
The results for the hierarchical multiservice model are illustrated in Fig. 4, where
with reference to scenario I we can observe in Fig. 4a the location of the current and
new hospitals, the hospitals’ catchment areas (defined by colours) and the flow of
transfers between DHs and CHs (defined by arrows). Figure 4b allows for the comparison of the model results with the values of the current network, as well as it provides
additional information regarding model performance and total costs.
These results suggest changes to both the location and the structure of hospital supply. Therefore, if the decision-maker wants to maximize patients’ access through the
minimization of travelling times and simultaneously ensure that the hospital supply
meets the need for hospital care, he/she should consider (1) building three new CHs
in the Lisbon metropolitan area, while current hospitals located in the Lisbon small
ward should have their capacity decreased; (2) building several new very small DH
facilities in rural areas of Alentejo and Algarve (namely, in Castro Marim, 111 beds,
Moura, 44 beds, Odemira, 68 beds and Estremoz, 143 beds). This result for Lisbon
might be explained by the large populations living in wards contiguous to the Lisbon
ward (which corresponds to the Lisbon city centre). Results for building small hospitals illustrate that gains in equity can only be obtained with very small hospitals closer
to populations and with a level of supply for services that might not respect minimal
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A. M. Mestre et al.
(b)
Model results
Current
Inpatient Emergency
External
Inpatient
Service
service
Consultation Service
Castro Marim
111
31,178
30,911
Faro
353
103,829
102,940
Lagos
80
22,659
22,465
55
Portimao
303
88,513
87,756
244
Beja
177
50,148
49,717
251
12,518
12,412
District Hospitals
Moura
Odemira
68
18,282
18,126
Serpa
50
13,720
13,602
Elvas
62
17,999
17,846
108
Portalegre
111
29,317
29,068
250
Estremoz
143
38,725
34
38,393
Évora
210
59,130
58,622
Almada
633
196,598
194,915
482
Barreiro
284
87,385
86,637
379
114
Santiago do Cacém
117
33,760
33,472
38
Setubal
502
149,974
148,689
328
Abrantes
168
45,204
44,817
151
Santarem
330
94,532
93,723
Tomar
204
59,564
59,053
151
Torres Novas
195
56,985
56,497
151
Amadora
669
201,591
199,864
664
362
33,374
108,203
33,089
384
Montijo
Cascais
CH
44
486
107,277
95
393
248
Torres Vedras
249
73,915
73,281
277
Vila Franca de Xira
506
154,536
153,213
211
Total
6,045
1,781,639
1,766,385
5,381
Lisboa
2,393
297,695
506,578
5,758
Loures
843
115,380
190,718
Odivelas
829
86,962
213,236
Sintra
1,404
289,036
313,381
Total
5,469
789,073
1,223,913
5,758
Fig. 4 Hospital locations, referral network, hospital capacities, catchment populations and hospital costs
for scenario I. a Hospital locations, referral networks and catchment populations. b Hospital capacity in
terms of beds, emergency admissions and external consultation appointments and costs (selected indicators
are presented below)
size conditions. Such case might even be considered impractical given the difficulties
associated with the recruitment of health personnel in rural areas in Portugal.
Improving access as set in scenario I is accompanied by an expansion of the network that demands additional capacity in existing hospitals and building new hospitals.
These figures are shown in Fig. 4b).
5.3.2 Scenario II
Scenario II differs from scenario I by relaxing the constraint imposing that maximum
critical time is 60 min and by imposing that new hospitals should have a capacity
higher than 150 beds. The existing rural hospitals might see their capacity decreased
by a maximum of 30% in rural areas and 20% in urban areas (as remarked above, this
constraint is not applied to Lisbon). In comparison with scenario I, scenario II informs
about the hospital network that should be built if the decision-maker does not want
to install very small hospitals and the choice is to provide specialized primary care
and emergency services complementary to hospital care for the populations located in
remote areas (i.e., with travel time higher than 60 min). The results for the hierarchical
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Organizing hospitals into networks
(a)
339
(b)
Model results
Inpatient Emergency
External
Service
service
Consultation
CH
District Hospitals
Faro
464
135,007
133,851
Lagos
80
22,659
22,465
Portimao
303
88,513
87,756
Beja
190
53,464
53,005
Serpa
87
24,269
24,062
Elvas
153
42,905
42,538
Portalegre
136
36,025
35,719
Évora
269
75,314
74,667
Almada
633
196,598
194,915
Barreiro
284
87,385
86,637
Montijo
114
33,374
Santiago do Cacem
172
48,726
48,310
Setubal
476
142,870
141,646
33,089
Abrantes
168
45,204
44,817
Santarem
330
94,532
93,723
Tomar
204
59,564
Torres Novas
195
56,985
56,497
Amadora
669
201,591
199,864
Cascais
362
108,203
107,277
Torres Vedras
249
73,915
73,281
Vila Franca de Xira
506
154,536
153,213
Total
6,046
1,781,639
1,766,385
Lisboa
2,515
297,695
531,539
Loures
813
115,380
186,187
Odivelas
738
86,962
192,807
Sintra
1,404
289,036
313,381
Total
5,469
789,073
1,223,913
59,053
Fig. 5 Hospital locations, referral network, hospital capacities, catchment populations and hospital costs
for scenario II. a Hospital locations, referral networks and catchment populations. b Hospital capacity in
terms of beds, emergency admissions and external consultation appointments and costs (selected indicators
are presented below)
multiservice model in scenario II are illustrated in Fig. 5 (results compare to the current
network displayed in Fig. 2).
Similarly to scenario I, the results displayed in Fig. 5 suggest changes to both the
location and the structure of hospital supply. Three new CHs in the Lisbon metropolitan area are to be built, with decreases in capacity of existent hospitals in the Lisbon
small ward. In a different way, in scenario II there is no building of new DHs, but an
adjustment to the capacity of most DHs in rural areas. The results for DHs show that
the target of 150 attendees per day for emergency services (referred to in some policy
documents) is not achieved in seven hospitals, yet all of them are small-sized units,
and some of them are located in rural areas (for instance, Elvas and Serpa). This result
provides (again) evidence of a trade-off between equity and efficiency in designing the
hospital network. So, in order to have improved access for remote populations living
in rural areas, some services need to be operating with low levels of production.
Figure 6 shows some indicators of variations in travelling times for populations
living in different areas that should be used to complement the information provided by
the model to the decision-maker. Although the average travel time is always <45 min,
it is observed that the maximum travel time exceeds 60 min for patients living in some
wards in Algarve and Alentejo. Despite the chosen objective function minimizing the
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(a)
(b)
90
84
80
70
70
68
Time (minutes)
61
60
50
45
40
35
30
31
30
24
20
10
17
12
21
11
8
0
Average travel time
Maximum travel time
Fig. 6 Travel time indicators for scenario II. a Average and maximum travel time (in minutes) for the
population living in each health sub-region. b Maximum travel time (min) for individuals living in each
ward
travelling times for patients to access hospital services, the proposed network of hospitals still hides inequalities in access for populations living in different areas. This is
an expected result given the previous literature (Marsh and Schilling 1994; Oliveira
and Bevan 2006) and the discussion provided in the literature review section, as well
as the features selected for scenario II [namely, relaxation of the use of Eq. (15)].
Nonetheless, these inequalities can only be accepted if populations in those points
are also provided with specialized primary care resources and improved pre-hospital
emergency service. Finally, improving access as set in scenario II implies investments
in capacity expansion of existing units and in new hospitals installation. These figures are shown in Fig. 5b), with investment costs being lower in scenario II (when
compared to scenario I).
5.3.3 Sensitivity analysis to scenario II
Since there is uncertainty associated with the value of some parameters (mainly due
to limitations of the data in use and to the employment of past data to analyse future
changes), sensitivity analyses were performed to allow the decision-maker to understand how changes in key parameters impact on model outputs (Morgan and Henrion
1990). Sensitivity analysis was performed on the following parameters:
1. transfer rates, because there was evidence of underreporting;
2. ALOS given a downward trend in the ALOS and the commitment of the Portuguese
Ministry of Health to promote a decrease in the ALOS in the hospital system;
3. demand for hospital services, because of uncertainty associated with forecasts.
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Sensitivity analysis on transfer rates Increases in transfer rates between DH and CH
were tested by increasing ascendant transfer rates between DH and CH by 2 and 4%
(leading to sub-scenarios ‘DH-CH 1’ and ‘DH-CH 2’ in Table 2) and by increasing
descendent transfer rates between DH and CH by 2 and 4% (leading to sub-scenarios
‘DH-CH 1’ and ‘DH-CH 2’ in Table 3). Results in Table 2 suggest that hospital locations are robust for all those sub-scenarios, i.e. changes in those input parameter values
do not impact on the model’s recommendations of where to build new hospitals—only
results for hospital beds are reported in Table 2. Yet, as expected, hospital capacities
vary for each sub-scenario. In comparison with the base scenario II, increases in ascendant transfer rates have a direct impact on CHs capacity and a very small impact on a
few DHs; and increases in descendant transfer rates only affect the capacities of DHs.
All these sub-scenarios lead to an increased level of global hospital activity and to a
raise in the objective function and costs.
Sensitivity analysis on ALOS Table 2 also shows the impact of decreasing ALOS by
half day and by one day, leading to sub-scenarios ‘ALOS 1’ and ‘ALOS 2’, respectively. Results also suggest that locations are robust to changes in ALOS. Yet, when
compared with the impact of varying transfer rates, decreases in ALOS have a stronger
impact in the choice of hospital capacities and might justify a substantial reduction in
global hospital capacity. For instance, 1 day reduction in ALOS justifies a reduction of
13% in the total number of hospital beds. The objective function remains unchanged in
the tested ALOS sub-scenarios, while investment and operating costs comparatively
decrease, as a result of decreases in capacity.
Sensitivity analysis on demand for hospital services Given the difficulties associated
with forecasting, three sub-scenarios for demand were considered: ‘Dem 1’, ‘Dem 2’
and ‘Dem 3’, which correspond to variations of demand in the base scenario of −2,
2 and 4%, respectively. Results in terms of inpatient beds for these sub-scenarios are
presented in Table 2. Although locations are robust to smaller changes in demand,
this does not hold for larger variations—for example, an increase of 4% in demand
justifies a new DH in Palmela. It is also observed that although a reduction in demand
leads to a reduction in the objective function, increases in demand do not always lead
to significant increases in the objective function. This can be explained by a balance
obtained between the travel time reduction due to the installation of a new facility
and the effect of more patients travelling due to a demand increase. Investment and
operational costs show to be sensitive to changes in demand.
5.3.4 Evaluating the costs of improved access in scenario II
The multiservice hierarchical model can be used to examine the equity of access/cost
trade-off, i.e. to analyse the costs associated with improved access. In order to illustrate
this, the minimum capacity requirements were relaxed so that smaller hospitals could
be opened to improve access to services. Also Eq. (21) was added to the model and in
each sub-scenario (represented in Table 3; Fig. 7) the number of DHs was increased by
1 facility, up to 5 facilities (from 21 to 26 DH hospitals). With this procedure, access
was improved and the costs of changing the network were quantified.
123
123
7.80E+08
457
Operational costs (=
C)
Investments costs (=
C)
CPU (s)
633
114
–
172
476
168
330
204
195
Montijo
Palmela
Santiago do Cacém
Setubal
Abrantes
Santarem
Tomar
Torres Novas
269
Évora
284
136
Portalegre
Barreiro
153
Elvas
Almada
190
87
303
Portimao
Serpa
80
Lagos
Beja
464
Faro
District hospitals-capacity
6.78E+07
2.60E+09
Objective function
13.0%
Parameters’ value
196
204
330
168
476
172
–
114
284
633
269
136
153
87
190
303
80
465
407
7.96E+08
2.63E+09
6.81E+07
15.0%
196
204
330
168
476
172
–
114
284
634
269
136
153
87
190
303
80
465
462
8.12E+08
2.66E+09
6.83E+07
17.0%
195
204
330
168
476
172
–
114
284
633
269
136
153
87
190
303
80
464
457
7.80E+08
2.60E+09
6.78E+07
1.2 %
196
205
331
168
477
173
–
114
285
635
269
137
153
88
191
304
81
466
478
7.81E+08
2.61E+09
6.79E+07
3.2%
CH–DH 1
Base
DH–CH 2
Base
DH–CH 1
Transfer rates CH–DH
Transfer rates DH–CH
196
205
332
169
478
173
–
114
286
637
270
137
154
88
191
305
81
467
430
7.82E+08
2.61E+09
6.79E+07
5.2%
CH–DH 2
195
204
330
168
476
172
–
114
284
633
269
136
153
87
190
303
80
464
457
7.80E+08
2.60E+09
6.78E+07
182
190
306
156
442
160
–
106
264
588
250
127
142
81
177
282
75
431
412
7.17E+08
2.48E+09
6.78E+07
6.50
8.40
7.00
ALOS 1
8.90
Base
Average length of stay
168
175
283
144
408
147
–
98
244
543
230
117
131
75
163
260
69
398
389
6.56E+08
2.36E+09
6.78E+07
7.90
6.00
ALOS 2
192
200
323
164
467
169
–
112
279
621
263
134
150
86
186
297
79
455
447
7.58E+08
2.55E+09
195
204
330
168
476
172
–
114
284
633
269
136
153
87
190
303
80
464
457
7.80E+08
2.60E+09
6.78E+07
–
−2%
6.65E+07
Base
Dem 1
Demand
Table 2 Results from the sensitivity analyses to the transfer rates, ALOS and demand (hospital capacity measured by the number beds)
199
208
337
171
486
176
–
116
290
646
274
139
156
89
194
309
82
474
422
8.01E+08
2.66E+09
6.92E+07
+2%
Dem 2
203
212
343
174
344
179
151
118
296
659
279
142
159
91
198
315
84
483
335
8.49E+08
2.67E+09
6.92E+07
+4%
Dem 3
342
A. M. Mestre et al.
362
249
506
6,046
Torres Vedras
Vila Franca de Xira
Total
2,515
813
738
1,404
5,469
Lisboa
Loures
Odivelas
Sintra
Total
CH-capacity
669
Cascais
5,622
1,430
757
838
2,597
6,048
506
249
362
669
5,776
1,456
776
864
2,680
6,049
506
249
363
669
5,469
1,404
738
813
2,515
6,046
506
249
362
669
5,469
1,527
615
813
2,515
6,062
507
250
363
671
CH–DH 1
Transfer rates CH–DH
Base
DH–CH 2
Base
DH–CH 1
Transfer rates DH–CH
Amadora
Table 2 continued
5,469
1,404
738
813
2,515
6,077
508
250
364
672
CH–DH 2
5,469
1,293
848
813
2,515
6,046
506
249
362
669
5,162
1,337
684
767
2,374
5,614
470
231
336
621
ALOS 1
Average length of stay
Base
4,854
1,257
644
721
2,232
5,183
433
213
311
573
ALOS 2
Demand
5,360
1,388
711
796
2,465
5,925
496
244
355
656
Dem 1
5,469
1,404
738
813
2,515
6,046
506
249
362
669
Base
5,578
1,445
740
829
2,565
6,167
516
254
370
682
Dem 2
5,688
1,460
767
898
2,563
6,288
526
259
377
696
Dem 3
Organizing hospitals into networks
343
123
344
A. M. Mestre et al.
Table 3 Trade-off between access and cost for selected sub-scenarios
Parameter No. of DH
21
22
23
24
25
26
Objective function 6.78E+07 6.56E+07 6.35E+07 6.15E+07 6.02E+07 5.90E+07
Results
Total cost
3.383E+09 3.377E+09 3.384E+09 3.394E+09 3.394E+09 3.412E+09
CPU (s)
457
349
360
15
15
14
14
13
13
12
12
11
11
10
21
444
387
(b)
22
23
24
25
Number of DH facilities
Average travel time
Total Cost
3,420E+09
3,410E+09
3,400E+09
3,390E+09
3,380E+09
3,370E+09
3,360E+09
3,350E+09
3,340E+09
26
Nº of DH
Faro
Costs
Average travel time
(a)
368
Be ja
He alth Portale gre
Sub- Évora
re gion Se túbal
Santaré m
Lisboa
21
16.7
22
16.7
23
16.7
31.0
21.5
24.3
10.6
12.1
8.3
31.0
21.5
24.3
10.6
12.1
6.4
31.0 19.9 19.9 19.9
21.5 21.5 21.5 21.5
24.3 24.3 24.3 24.3
8.5
8.5
8.5
8.5
12.1 12.1 12.1 10.0
6.4
6.4
6.4
6.4
24
25
16.7 14.2
26
14.2
Fig. 7 Cost impact of improved access for selected sub-scenarios. a Total costs versus average travel time.
b Average travel time by health sub-region
The results show that if more capital is invested, access improves as captured by
lower weighted average travel time. In Fig. 7a it is possible to observe that the total
costs do not always grow, as the increase in costs depends on the size and type of
the hospital facilities to be opened. In Fig. 7b it is also possible to identify the health
sub-regions that mostly benefit from improvements in the average travel time. For
example, moving from 21 to 22 hospitals will improve access for populations located
in the Lisbon sub-region.
5.3.5 Scenario III
Scenario III tests specifically where to locate some replacement hospitals. It departs
from the assumptions used in scenario II, but considers additional information from
the Portuguese health system. First, the Portuguese government is currently analysing
the construction of replacement hospitals in Évora (in the Alentejo Health Region) and
a joint hospital in Faro-Loulé (in the Algarve Health Region). Second, there has been
a high fluctuation in hospital utilization in the Algarve sub-region due to seasonal
variations in populations (mainly explained by variations in the number of tourists
throughout the year). Given this, we have chosen to analyse the results of the model
when the demand for the Algarve health sub-region is increased by 10% (to model the
population fluctuation that occurs every year during the summer). In scenario III we
allow for high changes in the capacities of the hospitals currently located in Évora and
Faro, so as to test whether the model suggests replacement hospitals or other adjustments to the current network in these locations or in surrounding areas. The results
are presented in Fig. 8.
The results show that two new CHs should be opened in Évora and Loulé and a DH
should be maintained in Faro. The new CH hospitals supply services to populations
located in the Alentejo and in the Algarve regions, respectively. We observe that the
123
Organizing hospitals into networks
(a)
345
(b)
District Hospitals
Current
Inpatient
Service
Faro
219
65,529
64,969
Lagos
88
24,925
24,712
491
55
Portimao
256
73,439
72,811
244
Beja
190
53,464
53,005
251
Serpa
87
24,269
24,062
34
Elvas
153
42,905
42,538
108
Portalegre
136
36,025
35,719
250
633
196,598
Évora
194,915
384
Almada
284
87,385
86,637
482
Barreiro
114
33,374
33,089
379
Montijo
172
48,726
48,310
Santiago do Cacém
476
142,870
141,646
38
Setubal
168
45,204
44,817
328
Abrantes
330
94,532
93,723
151
95
Santarem
204
59,564
59,053
393
Tomar
195
56,985
56,497
151
Torres Novas
669
201,591
199,864
151
Amadora
362
108,203
107,277
664
Cascais
249
73,915
73,281
248
Torres Vedras
249
73,915
73,281
277
Vila Franca de Xira
506
154,536
153,213
211
5,742
1,697,954
1,683,419
5,385
662
Total
Loulé
CH
Model results
Inpatient Emergency External
Service
service
Consultation
106,904
146,614
Évora
530
75,314
113,498
Lisboa
1,946
297,695
Loures
813
115,380
186,187
Odivelas
738
86,962
192,807
Sintra
1,404
289,036
313,381
Total
6,092
971,291
1,365,510
413,023
5,758
5,758
Fig. 8 Hospital location, referral network, hospital capacities, catchment populations and costs for scenario III. a Hospital location, referral networks and catchment populations. b Hospitals’ capacity in terms
of beds, emergency admissions and external consultation appointments and costs (selected indicators are
presented below)
results obtained in the context of redesigning the current network (scenario II) are
quite different from the results when replacement hospitals are allowed (see Figs. 5,
8, respectively). The locations appear to be sensitive to the increase in demand in the
Algarve region.
While there are improvements in access in scenario III in comparison with scenario
II (as observed by a decrease in the objective function), the costs in scenario III are
larger than those in scenario II (both investment costs and annual operating costs), as
can be read by comparing Figs. 5 and 8. The improvements in access in scenario III
are mostly obtained by a reduction in average travel times for populations living in
the Faro sub-region (with a reduction from 17 to 13 min).
Further scenarios could be created to test different configurations relevant for the
decision-maker.
6 Conclusions
We have proposed a hierarchical and multiservice model to help health care planners to
decide upon the location and structure of hospital supply when their main objective is
123
346
A. M. Mestre et al.
to improve access to hospital services. The hierarchical structure of a hospital system,
the multiservice nature of hospitals’ activity and the referral system between hospitals,
as well as other constraints that capture the institutional characteristics of the system
were considered. The present work adds to the previous literature by considering the
multiservice nature of hospital activity, the interaction between various services and
hospital levels, allowing for two-way flows of patients in the hospital hierarchy, and
by looking simultaneously at the cost implications of improving access.
The application to a real case study provided a framework to test and illustrate the
usefulness of the hierarchical and multiservice location model as a flexible decision
support tool to assist the planning of hospital networks. The proposed model was also
adapted and used by planners of the North Administrative Health Region in Portugal to
analyse where to locate new replacement hospitals (Mestre et al. 2009). The planners
of the North region valued certain features of the model. The first one was concerned
with the fact that this is able to capture key aspects that policy-makers consider in
the reorganization of hospital services. Also, it generates simultaneously information
on location, size, referral and hospital catchment areas, allowing for an integrated
analysis. Furthermore, it can easily be adapted to capture specific circumstances of
the regional health system, such as the direct referral of patients from primary care
centres to central hospitals in some specialties. Finally, the model could easily be run
under alternative policy scenarios. When adapting and applying it to the North health
region, it was observed that it would be useful if the model could differentiate between
two groups of DH hospitals, representing smaller and larger DHs (as those hospitals
might still provide a different range of services); concerns were raised with regard to
sensitivity analysis issues (in some scenarios, the location was particularly sensitive
to some parameters); and the model should provide estimates of the financial costs
required to implement those changes. We took into account the latter comment to
develop the model further and present that development in this paper.
In this paper, we have chosen three scenarios to illustrate the usefulness and the
behaviour of the proposed model. These scenarios inform different policy contexts and
depart from the status quo to analyse potential improvements to the hospital network.
The model was shown to produce key information on referral networks, on hospital
catchment populations and on the location and structure of hospital supply for the
health care planner in an organized and concise format. We observed that the proposed model is data demanding (although it mostly makes use of routinely collected
data) and cannot be used without specific knowledge about the system under analysis. Introducing costs allowed for the analysis of the cost implications of improving
geographic access and thus for analysing the trade-off between access and costs.
Regarding future developments, we consider important to develop multi-objective formulations that explore the reorganization of hospital networks and a balanced
impact on equity of access and on costs (e.g. using multi-objective mathematical programming). Also, it will be interesting to extend the present work so as to consider
the long-term care network of hospitals and the local supply of primary care services
that might substitute or complement the provision of hospital services in different
geographic areas (networks of long-term care services are being created in several
countries and their activity interacts with the configuration and size of a hospital network). Furthermore, it will be important to develop the model with information on
123
Organizing hospitals into networks
347
future forecasts of the population demand for services and for the future structure
of hospital supply (which is evolving throughout time and being changed by new
technologies). Accounting for these improvements is expected to increase the model
complexity, which means that exploring alternative solution techniques (e.g. hybrid
solutions where exact methods might be combined with heuristics) might be required
to improve the model performance.
Acknowledgments The authors acknowledge financing from the Fundação para a Ciência e Tecnologia
(Portugal) (SFRH/BD/36895/2007). The authors are grateful to two anonymous referees for their thorough
and insightful comments that helped improving this paper.
References
Algarve Health Region (2004) Plano Director Regional da Saúde do Algarve (Regional plan for health in
Algarve). Faro
Culyer A (2001) Equity—some theory and its policy implications. J Med Ethics 27:275–283
Current J, Min H, Schilling D (1990) Multiobjective analysis of facility location decisions. Eur J Oper Res
49(3):295–307
Daskin M (1995) Network and discrete location. Wiley, New York
Daskin M (2008) What you should know about location modelling. Nav Res Logist 55:283–294
Eiselt HA, Laporte G (1995) Objectives in location problems. Drezner Z (ed) Facility location—a survey
of applications and methods Springer, New York, pp 151–180
Folland S, Goodman A, Stano M (1997) The economics of health and health care, 2nd edn. Prentice-Hall,
Upper Saddle River
Galvão RD, Espejo A, Gonzalo L, Boffey B (2002) A hierarchical model for the location of perinatal
facilities in the municipality of Rio de Janeiro. Eur J Oper Res 138(3):495–517
Galvão RD, Acosta E, Luis G, Boffey B, Yates D (2006) Load balancing and capacity constraints in a
hierarchical location model. Eur J Oper Res 172(2):631–646
GAMS (2010) General algebraic modeling system version 23.6.5. Washington, DC, USA
Gerrard R, Church R (1996) Closest assignment constraints and location models: properties and structure.
Locat Sci 4(4):251–270
Health Ministry (1990) Lei no 48/90: Lei de Bases da Saúde (fundamental principles of health) Diário da
Républica: I série, no 195 de 24 de Agosto
Health Ministry (2002) Rede de referenciação hospitalar em oncologia (hospital network in oncology).
Lisbon
Health Ministry (2003) Centros de saúde e hospitais: recursos e produção do SNS (primary care centre and
hospitals: production and resources of the NHS). Lisbon
Health Ministry (2006a) Reordenamento das capacidades hospitalares da cidade de Lisboa (redevelopment
of hospital capacity in the city of Lisbon). Lisbon
Health Ministry (2006b) Contabilidade Analítica—Hospitais do SNS (cost accounting). Lisbon
Health Ministry (2007) Proposta da rede de urgências—Relatório Final (proposal of the emergency network—final report). Lisbon
Marianov V, Serra D (2004) Location problems in the public sector In: Drezner Z, Hamacher H (eds) Facility
location: applications and theory. Springer, Berlin, pp 119–143
Marsh MT, Schilling DA (1994) Equity measurement in facility location analysis: a review and framework.
Eur J Oper Res 74(1):1–17
Mayhew LD, Leonardi G (1982) Equity, efficiency, and accessibility in urban and regional health-care
systems. Environ Plan A 14(11):1479–1507
McAllister DM (1976) Equity and efficiency in public facility location. Geogr Anal 8(1):47–63
Melo MT, Nickel S, Saldanha-da-Gama F (2009) Facility location and supply chain management—a review.
Eur J Oper Res 196(2):401–412
Mestre A, Oliveira M, Barbosa-Póvoa A (2009) Aplicação do modelo hierárquico Multi-Serviço ao planeamento hospitalar na Região de Saúde do Norte (Application of the hierarchical multi-service model
for hospital Planning in the Portuguese North Health Region). Internal report, Lisbon
123
348
A. M. Mestre et al.
Morgan G, Henrion M (1990) Uncertainty: a guide to dealing with uncertainty in quantitative risk and
policy analysis. Cambridge University Press, New York
Narula SC (1986) Minisum hierarchical location-allocation problems on a network: a survey. Ann Oper
Res 6:257–272
Oliveira MD, Bevan G (2006) Modelling the redistribution of hospital supply to achieve equity taking
account of patient’s behaviour. Health Care Manag Sci 9(1):19–30
Oliveira MD, Pinto CG (2005) Health care reform in Portugal: an evaluation of the NHS experience. Health
Econ 14(S1):S203–S220
Owen SH, Daskin MS (1998) Strategic facility location: A review. Eur J Oper Res 111(3):423–447
Portuguese National Institute of Statistics (2006) XIV census. Accessed info service: http://www.ine.pt
Rahman S, Smith DK (2000) Use of location-allocation models in health service development planning in
developing nations. Eur J Oper Res 123(3):437–452
ReVelle CS, Eiselt HA (2005) Location analysis: a synthesis and survey. Eur J Oper Res 165(1):1–19
Rice N, Smith P (2001) Capitation and risk adjustment in health care financing: an international progress
report. Milbank Q 79(1):81–113
Rosko MD, Broyles R (1988) The economics of health care: a reference handbook Greenwood Press,
New York
Ruth RJ (1981) A mixed integer programming model for regional planning of a hospital inpatient service.
Manag Sci 27(5):521–533
Sahin G, Süral H (2007) A review of hierarchical facility location models. Comput Oper Res 34(8):2310–
2331
Salema M, Póvoa A, Novais A (2009) A strategic and tactical model for closed-loop supply chains. OR
Spectr 31(3):573–599
Savas ES (1978) Equity in providing public-services. Manag Sci 24(8):800–808
Smith HK, Harper PR, Potts CN, Thyle A (2009a) Planning sustainable community health schemes in rural
areas of developing countries. Eur J Oper Res 193(3):768–777
Smith HK, Laporte G, Harper PR (2009b) Locational analysis: highlights of growth to maturity. J Oper Res
Soc 60:140–148
Smith H, Harper P, Potts C (2010) Bicriteria efficiency/equity hierarchical location models. Working paper,
152075. University of Southampton
Teixeira J, Antunes A, Peeters D (2007) An optimization-based study on the redeployment of a secondary
school network. Environ Plan B 34(2):296–315
Truelove M (1993) Measurement of spatial equity. Environ Plan C 11(1):19–34
ViaMichelin Maps and route planner. http://www.viamichelin.co.uk. Accessed 2006
123
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