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Use of Location-Allocation Models in Health Service Development Planning in Developing Nations

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There is considerable evidence that because of poor geographical accessibility, basic health care does not reach the majority of the population in developing nations. Despite the view that mathematical methods of locational analysis are too sophisticated for use in many of these nations, several studies have demonstrated the usefulness of such methods in the locational decision-making process. This paper reviews the use of location-allocation models in health service development planning in the developing nations. The purpose of this review is to examine the suitability of these methods for designing health care systems and their relevance to overall development problems in such countries.
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Invited Review
Use of location-allocation models in health service development
planning in developing nations
Shams-ur Rahman
a
, David K. Smith
b,*
a
The Graduate School of Management, The University of Western Australia, Nedlands, WA 6907, Australia
b
School of Mathematical Sciences, University of Exeter, North Park Road, Exeter, EX4 4QE, UK
Received 1 November 1998; accepted 1 June 1999
Abstract
There is considerable evidence that because of poor geographical accessibility, basic health care does not reach the
majority of the population in developing nations. Despite the view that mathematical methods of locational analysis are
too sophisticated for use in many of these nations, several studies have demonstrated the usefulness of such methods in
the locational decision-making process. This paper reviews the use of location-allocation models in health service
development planning in the developing nations. The purpose of this review is to examine the suitability of these
methods for designing health care systems and their relevance to overall development problems in such coun-
tries. Ó2000 Elsevier Science B.V. All rights reserved.
Keywords: Location; Health; Developing countries; Government
1. Introduction
The role of location analysis in planning ser-
vices for regional development is well known. One
of the tools for such analysis is quantitative loca-
tion-allocation modeling. It provides a framework
for investigating service accessibility problems,
comparing the quality (in terms of eciency) of
previous locational decisions, and generating al-
ternatives either to suggest more ecient service
systems or to improve existing systems. In the
context of developing nations, locational decisions
are generally taken locally by government ocers
or by local elected leaders or by both. In the ab-
sence of any formal analysis and generation of
alternatives, the ®nal decision may be made on
political or pragmatic considerations. As a result
the decisions can very often be far from optimal
[24,57]. Despite the view that mathematical meth-
ods of locational analysis are too sophisticated for
use in most developing countries, many studies
have demonstrated the usefulness of such methods
in the locational decision-making process. In this
paper we review a number of location-allocation
studies that have been undertaken for health ser-
vice development planning in developing nations.
European Journal of Operational Research 123 (2000) 437±452
www.elsevier.com/locate/dsw
*
Corresponding author. Tel.: +44-1392-264478; fax: +44-
1392-264460.
E-mail addresses: srahman@ecel.uwa.edu.au (S.-u. Rah-
man), d.k.smith@exeter.ac.uk (D.K. Smith).
0377-2217/00/$ - see front matter Ó2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 289-1
There are two objectives of this review paper.
First, we examine the role of location-allocation
models in planning health care systems and sec-
ond, consider their relevance to development
problems in developing nations. This paper is
divided into four sections. In Section 2 the
relationship between health and economic devel-
opment is discussed. In Section 3, public facilities,
the structure of the health care system and
location-allocation models are discussed and we
develop a framework for reviewing the studies.
The actual review is given in Section 4 and the
paper is concluded in Section 5 with a discussion
on future research directions.
2. Health and economic development
Despite remarkable reductions in the incidence
of disease and mortality rates over the last four
decades, the degree of ill health in developing na-
tions remains enormous [81]. Available data on
health indicate that extremely high levels of pre-
ventable illness prevail in these countries. For in-
stance, at least ®ve million children are blinded or
crippled each year, most of them as a result of
polio, a disease for which eective vaccines are
available. Infant mortality in many developing
countries is extremely high. Therefore, further
improvement in health will largely depend on the
capacity of health systems to deliver primary
health care (PHC) services. This view has been
widely accepted and well documented. The inter-
national conference on primary health care at
Alma-Ata ([1], p. 7) declared that ``Primary health
care is the key to achieving an acceptable level of
health ... in the foreseeable future and in the spirit
of social justice. It is equally true for all countries
... For developing countries in particular, it is a
burning necessary''. Gradually extending the PHC
services to the rural people of developing nations,
the following objectives could be achieved [37].
·To increase equity in the distribution of health
bene®ts and reduce morbidity and mortality.
·To reduce population growth rates in the longer
run.
·To stimulate economic growth through a health-
ier population.
There is evidence to believe that the develop-
ment of health is essential for economic develop-
ment. For instance, a study in Malaysia
demonstrated that reduction of malaria in a rub-
ber estate caused output per worker to rise 17-fold
[5]. It is widely acknowledged that it is no longer
tenable to draw a distinction between economic
and health development. The development of
health along with other social improvements is
necessary to achieve economic goals. Corre-
spondingly, economic development is necessary to
achieve most of the social goals. The importance
of health and its relation to development in de-
veloping nations can be observed in a statement of
a former President of the Institute of the US Na-
tional Academy of Sciences. He said that ``Dis-
eases in developing countries take such a terrible
toll in human suering and economic loss that they
are at the heart of the whole problem of develop-
ment ...The developing countries may become the
never-to-be-developed countries unless the burden
of illness can be greatly eased'' [39]. Evidently,
development in health is a precondition for eco-
nomic development of a nation. In a simple ex-
ample of the economic bene®ts of health provision,
Dinwiddy and Teal ([17], p. 216) write: ``immuni-
sation programmes for contagious diseases ...
bene®t far more people than the number actually
receiving the treatment''.
3. Public facilities, health delivery systems, and
location-allocation models
There are two principal categories of central
location problems, whose objectives are often dif-
ferent. Trying to ®nd the best locations for public
facilities, such as schools, parks, utilities, sports
and health centres is a dierent problem from that
of locating private facilities, such as banks, shops
and private leisure facilities. (This paper is only
concerned with the location of central facilities.
Hodgart [31] has classi®ed location problems into
three groups, those locating central facilities, semi-
desirable facilities and obnoxious facilities.) Cen-
tral facilities are facilities to which people must
travel to receive the service, or from which a ser-
vice is provided to the whole community of inter-
438 S.-u. Rahman, D.K. Smith / European Journal of Operational Research 123 (2000) 437±452
est. It is generally felt that the closer (usually
measured in terms of distance or time) the facilities
are to the users, the better the service provided.
Research for locating health facilities in the
context of the problems of developing nations has
developed two categories of models. Some re-
search has been directed towards the location of
components of a health care system in which fa-
cilities are considered to be of one type (with re-
spect to the level of service provided). We refer to
these models as single-level location-allocation
models (SLAMs). Patel [54] and Berghmans et al.
[7] give some examples. However, it is widely rec-
ognised that most health care systems in develop-
ing countries are organised as hierarchical systems,
just as they are in industrialised nations. A patient
in a typical developed country goes to see a doctor
or nurse, relatively close to his or her home. If
necessary, the patient attends a hospital which
provides facilities which are not available from the
doctor. Some patients will then progress to a third
level of hierarchy, a specialist hospital. The same is
true in developing countries. For example, the
primary health care delivery system in Bangladesh
consists of Community Clinics (CCs), Health and
Family Welfare Centres (HFWC); and Thana
Health Complexes (THCs). In this system, Com-
munity Clinics are considered to be the ®rst point
of contact between rural people and the health
system. A patient whose medical problems cannot
be dealt with at such a clinic may be referred to a
HFWC or THC, and a patient who goes to a THC
may be referred to a HFWC. The system consist-
ing of these three distinct types of facilities may be
termed a ``3-hierarchical system''. Generalising
this notion, it may be stated that a health care
system that consists of f(P2) distinct types of
facilities which collectively deliver services can be
called an f-hierarchical system. Narula [49] and
Tien et al. [75] give detailed descriptions of these
hierarchies, and the interested reader should refer
to these papers. Addressing the location problem
on a regional level, some researchers have con-
sidered the problem as a hierarchical system. We
call these hierarchical location-allocation models
(HLAM). The studies by Banerji and Fisher [3]
and Harvey et al. [29] are some examples of
HLAMs.
A location-allocation model is a method for
®nding optimal sites for facility locations. The
method involves simultaneously selecting a set of
locations for facilities and assigning spatially dis-
tributed sets of demands to these facilities to op-
timise some speci®ed measurable criterion. The
most important issue raised in the process of
solving location problems is the selection of a
suitable criterion or objective function. The for-
mulation of an objective function depends greatly
on the ownership of the organisation, both
whether public or private and the nature of the
facilities, as has been mentioned earlier. For in-
stance, private sector facilities are often located to
ful®l precisely stated objectives, such as to mini-
mise cost or to maximise pro®t. In contrast, the
goals and objectives of public facility location are
more dicult to capture [63,67]. For example, if
the problem is to locate emergency ambulance
services, a possible criterion would be to minimise
the average distance (or time) an ambulance must
travel in order to reach a random incident. An-
other appropriate criterion could be to minimise
the maximum distance that the ambulance must
travel to reach an accident. (An advertisement for
IBM in 1978 drew attention to a planning decision
made in the 19th century; ``Horses pulling a ®re
wagon, it was once decided, can run no further
than 10 city blocks. For that reason, ®re stations in
many American cities were located 20 blocks
apart.'' [42].) There have been comprehensive re-
views of the general problem of locating emer-
gency facilities (not simply in developing
countries) in [64,65].
Dierent interpretations of the goal of maxi-
mum public welfare lead to a number of possible
location-allocation problems.
One of the most popular models for public fa-
cility location problems is the p-median problem
[28,62]. (It has also been useful in planning control
of disease [73].) The problem can be de®ned in the
following manner: given discrete demands locate a
number (por less) of facilities so that the total
weighted travel distance or time between facilities
and demand centres is minimised.
The p-median problem is attractive, since the
smaller the total (or average) weighted travel
distance (time), the more convenient for the
S.-u. Rahman, D.K. Smith / European Journal of Operational Research 123 (2000) 437±452 439
users to get to the nearest facility. It is assumed
that all users of the facility choose to travel to
the closest one. However, this model does not
take the Ôworst caseÕsituations into account and
so it may result in solutions which are not ac-
ceptable from a service point of view. Solutions
which minimise the weighted travel distance may
be inequitable, forcing a few users to travel far.
This may mean that the remoter users do not
actually travel to their nearest facility, or to any
facility. It has frequently been observed that the
usage of service facilities declines rapidly when
the travel time (distance) exceeds some critical
value. This is a typical situation with the use of
rural health facilities in developing nations
[58,72]. Therefore it is quite reasonable to con-
sider maximum distance or time constraints in
formulating a location problem. This leads to
the formulation of the location set covering
problem (LSCP) [78]. Here the problem can be
de®ned as: ®nd the minimum number of facilities
and their locations such that each and every
demand centre is covered by at least one facility
within a given maximal service distance (time).
This formulation has been used in a developed
country for locating kidney dialysis machines, a
form of treatment for which the patient must
make frequent, repeated journeys [22]. A related
problem, known as the pq-median problem, is
concerned with ®nding an ecient set of facility
locations which can be associated with districting
the catchment areas for two or more levels of
facility. Some heuristic approaches to it are de-
scribed in [70] and an exact algorithm is found
in [71].
In reality there may not be enough resources to
provide the number of facilities which would be
required by the optimum solution to this problem.
This is particularly the case in developing nations.
Then, the decision maker may abandon the goal of
total coverage and attempt instead to locate the
facilities in such a way that as few people as pos-
sible lie outside the desired service distance. This
means the problem is to maximise coverage within
a desired service distance by locating a ®xed
number of facilities. This problem is referred to as
the maximal covering location problem (MCLP)
[12].
The mathematical formulations of these basic
problems are well documented in the literature,
and will not be repeated here. Over the last two
decades many variants of these models have been
developed ([9,14,38,42,79]). Some of these variants
allow for multiple or sequential objectives (for
instance [21]). This paper reviews studies which
have addressed the health facility location prob-
lem in developing nations. However, the discus-
sion is limited to the network forms of the
p-median and covering problems and their vari-
ants. For models addressing the same problems on
a plane, see [18,19] whose models are p-median
HLAMs dealing with health facility location in
Turkey. Some researchers have used other meth-
ods to design health systems in developing coun-
tries. For instance, Bhatnagar [8] showed how
interactive graphics could be successfully applied
in locating service centres in rural India. Hodgson
[34] applied a model in the health care system in
rural India which is based on a negative expo-
nential adoption of Reilly's retail gravitational
law. This re¯ects the unwillingness of patients to
make a journey greater than a certain distance or
time. The model, however, is both data intensive
and data sensitive. This casts serious doubts about
its applicability in a developing nation. More re-
cently, Massam and Malezenski [43], and Datta
and Bandyopadhyay [15] demonstrated the role of
decision support systems (DSSs) on health plan-
ning and location in Zambia and India, respec-
tively. This review of the use location-allocation
models in developing nations follows the frame-
work shown in Fig. 1.
4. Review of location-allocation studies
In this section we review several health facility
location-allocation studies conducted in the con-
text of developing nations. These studies were de-
signed to:
·Find a set of optimal sites.
·Locate optimal sites in a new area.
·Measure the eectiveness of past location deci-
sions.
·Improve existing location patterns.
440 S.-u. Rahman, D.K. Smith / European Journal of Operational Research 123 (2000) 437±452
4.1. Finding optimal site(s)
Perhaps the earliest location-allocation study
which explicitly addressed the problem of locating
health facilities in a developing country is one by
Gould and Lienbach [27]. In this study, the
problem of locating hospitals and determining
their capacities (in terms of number of beds) in the
western part of Guatemala was considered as a p-
median problem on the existing road network. The
transportation algorithm was used to solve the
problem [16]. The population centres were treated
as surplus nodes while the hospital capacities were
de®ned as quantities to be ®lled. The transporta-
tion costs from a population centre to the hospital
were calculated considering the most ecient ser-
vice route available. However, no mathematical
formulation of the problem was given. First, the
model was solved to locate three regional hospitals
of equal capacity for 18 population centres. The
optimal set of hospitals produced the shortest total
travel distance. However, there were some awk-
ward ¯ow patterns between communities and
hospitals. It appeared that some communities were
not assigned to the closest hospital, because these
were all of the same size. Changing hospital ca-
pacities eliminated this unacceptable ¯ow. Apart
from ®nding the size and sites of hospitals based
on the existing road networks, the study made
useful suggestions relating to the impact of a
change in the transportation network on the lo-
cation of centres and the assignment of commu-
nities to them. Such comments may be seen as
simple sensitivity analysis, and as pointers to the
answers to ``What if ...?'' questions.
A similar type of study was conducted to lo-
cate rural health clinics in the Eastern Region of
Upper Volta by Mehretu et al. in 1983 [45]. The
objective was to locate clinics such that the total
weighted travel distance between clinics and vil-
lages was minimised subject to the constraint that
no one would travel more than a maximum dis-
tance of 5 km. Their problem was a modi®ed p-
median problem, de®ned as the p-median problem
with maximum distance constraints [38,78]. First,
635 villages in the study area were arbitrarily
grouped into 94 village clusters referred to as
programming units. Then the facilities were lo-
cated in each programming unit separately (a
geographically constrained problem) using the
Teitz and Bart [74] algorithm. In all, 222 health
clinics were necessary to cover all the villages.
However, one is left with the impression that the
same objective could have been achieved using
fewer facilities if only one problem were to be
solved for the entire region. It is a common fea-
ture of models whose objective is minimum cost,
not simply location models, that extra constraints
increase the cost; so restricting the solutions,
programming unit by programming unit, will
Fig. 1. Framework for the review.
S.-u. Rahman, D.K. Smith / European Journal of Operational Research 123 (2000) 437±452 441
almost certainly give a higher cost and more fa-
cilities than a less constrained problem.
The study by Patel [54] concentrated on the
choice of locations for service centres in rural In-
dia. The list of centres included public health
centres, primary schools, agricultural extension
oces, post oces and fair-price shops. These
were considered to be essential to provide the
minimal infrastructure, education and health fa-
cilities for rural people. 44 out of 237 villages were
chosen as the potential sites for service centres.
The criteria used included population, growth
potential, proximity to highways, the nature of
adjoining areas and the existence of a village
market. Then the problem was formulated as a
constrained minimisation problem. No service
centre had to be greater than a predetermined
Ômaximum distanceÕfrom the community that is
served. There was a budget constraint, which made
the minimisation of Ômaximum distanceÕa dicult
integer nonlinear programming problem. (The
nonlinearity was essentially combinatorial, and
could have been removed at the cost of many extra
variables and constraints.) For ®xed distance, the
cost could be minimised as (eectively) a dual
problem, using an algorithm similar to one used by
Toregas et al. [78] for locating emergency service
facilities. Part of Patel's paper deals with the
problems of including new transport infrastruc-
ture, as well as the resources for health provision.
The method of Lemke et al. [40] was used to solve
this for eight values of maximum distance, yielding
an empirical plot of cost versus maximum dis-
tance. It was shown that despite a cut in the
original budget of 75%, the maximum distance had
to be increased by only 20%, from ®ve miles to six
miles. Patel's analysis showed that the original
budget for the project was far too large for the
basic location problem. Therefore, with optimal
location, the service level of ®ve miles could have
been obtained with a cost of only 42% of the
original budget. There was no feasible solution for
smaller service levels.
Eaton et al. [21] in their study described the
development, use and implementation of loca-
tion-allocation techniques for ambulance deploy-
ment in Santo Domingo, capital of the
Dominican Republic. Here the problem was
considered as an LSCP with back-up cover which
sought to maximise the multiple coverage of de-
mand within a speci®ed response time, with the
minimum number of ambulances. The problem
was solved (by hand, because computers were not
available!) for 214 demand areas in Santo Do-
mingo using several maximal response times. The
study suggested that between 8 and 23 vehicles
would be necessary to cover all demands within
5±10 minutes. The study achieved two main
bene®ts. First, it provided the Ministry of Health
with a basis for establishing an emergency medi-
cal service system in Santo Domingo. Second, the
con®dence in Operational Research techniques
was enhanced by the leading role of Dominican
participants in this study. Reid et al. [61] used a
similar model to ®nd locations of medical supply
centres in Ecuador.
Bennett et al. [6] demonstrated how the use of
location analysis could be successfully integrated
into health centre planning in rural Colombia. In
the late 1970s the local health planners of the state
of Valle del Cauca, Colombia initiated a study to
determine rural health centres from which to re-
cruit rural health workers (promontoras) and at
which to base ambulances. They considered sev-
eral factors such as the availability of water and
electricity in a village, and the maximum travel
distance from health centre to any village while
determining health centres. Based upon a detailed
data survey and using their intuitive judgement
local health planners determined 24 health centres
which were accessible to 78% of the population.
The authors worked in consultation with the local
health planners. Their paper formulated the
problem as an MCLP which was solved by a
Ôgreedy adding with substitution' (GAS) heuristic
and tested for optimality by linear programming.
The results from the GAS heuristic showed that it
required only 15 centres to cover the same per-
centage of the population. With 24 optimally lo-
cated health centres, the same number as proposed
by the local planners, 90% of the population could
be covered. The study illustrated that location
analysis could be combined with the intuitive
judgement of the planner in developing countries.
Oppong [52] studied the eect of changes to
communication links due to the rainy season in a
442 S.-u. Rahman, D.K. Smith / European Journal of Operational Research 123 (2000) 437±452
tropical country. This paper was concerned with
the location of health facilities in Suhum District,
Ghana. The problem was formulated both as a p-
median problem and as an MCLP. The author
developed a number of scenarios for the decision-
makers to consider. The use of such scenarios
opens the way to multi-criteria decision analysis,
with trade-os between several performance mea-
sures. However, little mention is made of the
possibility of formulating the problem in this way,
which would seem to be a more appropriate
approach.
Rahman and Smith [56] conducted a study to
®nd suitable sites for new health facilities with
respect to the existing facilities to improve the
accessibility of the overall health system to people
in Tangail Thana (Sub-district), Bangladesh. The
study region has an area of 210 square kilometres
and consists of 8 unions and a municipality. (A
union is the lowest administrative zone in Ban-
gladesh.) According to the Government of Ban-
gladesh's plan, 30 Community Clinics were to be
located to serve about 240 000 people living in 212
villages. Those villages, without a health facility,
which have a population of 500 or more, and have
potable water and electricity were considered to be
the potential sites for Community Clinics. The
location-allocation problem was then considered
in two ways. First, locating the facilities taking one
union at a time (similar to Mehretu et al. [45]).
These problems are referred to as geographically
constrained problems. Second, locating all facili-
ties at a time by solving one global problem (a
geographically unconstrained problem). Commu-
nity Clinics were located solving an MCLP and
four sets of location sites were suggested. Gener-
ally, the solutions of the geographically uncon-
strained problem were found to be more ecient
than the solutions of the constrained problems.
The study suggested that implementation of one of
the solutions of the unconstrained problem would
make the health delivery system 60% less costly,
while serving the entire population with a maxi-
mum travel distance of 2 km. The objective of the
study however, was not to suggest a single optimal
decision, rather to develop and test feasible deci-
sion processes in the light of the Government's
health plan.
The studies above focused only on the accessi-
bility component of the health care coverage
problem. Tien and El-Tell [76] proposed a p-me-
dian model which simultaneously addressed two
components, accessibility and availability. Avail-
ability as a measure has been used in health service
problems in at least two senses. In developed
countries, it frequently refers to the ratio of health
service professionals to a given population. In the
paper availability was measured in terms of the
time for which a physician was available in health
centres and village clinics. These professionals
were only available for part of the working week,
and divided their time between the clinics. The
model was applied to the Mafraq district of Jor-
dan. The problem was to locate village clinics and
health centres and to identify a relationship be-
tween them. This relationship was based on the
organisational attachment of one or more (village)
clinics to every health centre and on the presence
of a health centre-based physician at each clinic. It
was necessary to locate the health centres at the
most populated villages having the essential sup-
port services (electricity, water, telephone and
good road transport). The problem was formu-
lated as a zero±one integer programming problem,
based on Revelle and Swain's formulation [62]
with extra constraints which used the availability
measure. It was solved on the MPSX package as a
relaxed linear programme. The results demon-
strated that by both reallocating the villages to the
clinics and the clinics to the existing health centres
considerable improvement (in terms of coverage)
could be made.
The above studies focused on locating compo-
nents of health care system in which facilities are
considered to be one type. Thus, they belong to the
class of models which has been identi®ed as single-
level location-allocation models (SLAMs). The
model by Tien and El-Tell [76] which claimed to be
a quasihierarchical location-allocation model, is
essentially a location routing problem of the gen-
eral nature described by Narula [50]. Banerji and
Fisher [3] described the application of hierarchical
location analysis for integrated area planning in
Andhra Pradesh, India. The problem of locating
health facilities in rural villages is a part of the
larger study. Here, a successively inclusive facility
S.-u. Rahman, D.K. Smith / European Journal of Operational Research 123 (2000) 437±452 443
hierarchy was assumed. The proposed formulation
was based on a combination of the LSCP and the
p-median problem. However, no mathematical
formulation was given. Later, Weaver [80] gave a
mathematical formulation of their problem. First
the LSCP was solved to determine the number of
facilities required at each level, for a given maxi-
mum allowable distance at each level of hierarchy.
Then, given the number of facilities at each level,
the p-median problem was solved to determine the
optimal locations of the facilities. The procedure
involved here was ®rst to locate the highest-level
facility and then to move down the hierarchy. This
strategy of successively locating some higher to
lower level facilities is termed the top-down ap-
proach [35]. The LSCP and p-median problem
was solved using the Banerji [4] heuristic and the
Teitz and Bart [74] heuristic respectively. The
health care delivery system described here is an
example of a 2/I/Ulocation-allocation model. The
notation for the models is explained in detail in
Appendix A.
Like Banerji and Fisher [3], both the studies by
Harvey et al. [29] and Moore and ReVelle [47]
considered the problem of locating facilities as
hierarchical systems. Their subject of the ®rst of
these was the identi®cation of nodal hierarchies of
growth centres in Sierra Leone. Each centre pro-
vides 35 dierent functions and delivering health
care was one such. A successively inclusive facility
hierarchy was assumed. Repeated use of a p-me-
dian algorithm proposed by Hung and Brown [36]
gave a solution to the problem. Unlike Banerji and
Fisher [3] a bottom-up approach was used for lo-
cating facilities. This means, ®rst the problem was
solve to locate the lowest-level facility and then
moved up the hierarchy to locate the higher-level
facilities. In this study the problem of choosing ®ve
dierent types of growth centres is an example of
5/I/Ulocation-allocation model.
Moore and ReVelle [47] extended the MCLP
and applied it to a two-tier hierarchical health care
delivery system in Honduras. The problem was to
simultaneously locate a ®xed number of clinics and
hospitals . To maximise the population with clinic
services available within a distance standard set for
clinics, and with hospital services available within
a hospital distance standard. Out of 144 demand
nodes, 62 nodes (those with over 50 people) were
chosen as the feasible facility sites. Of these only 28
(those with over 100 people) were considered fea-
sible for the higher level facilities (hospitals). The
problem was solved using a commercially available
linear programming package (MPSX). The results
were represented as a curve of population coverage
versus the investment in facilities, instead of the
number of facilities at each level. This was done in
order to present the results in a two dimensional
manner (coverage and investment). The hierarchy
of the health system was considered to be succes-
sively inclusive. The problem can be categorized as
a2/I/Ulocation-allocation model.
4.2. Locating facilities in a new area
Berghmans et al. [7] reported on a study which
dealt with a problem of locating health centres in a
completely new city (Yanbu al Sinaya) in Saudi
Arabia. Here the health centres were considered to
be the ®rst point of contact between the popula-
tion and the health system. Taking into consider-
ation four quantitative criteria the problem was
formulated as a LSCP. These were the ratio of
doctors to population, the maximum service dis-
tance S, the equity in service (de®ned by insisting
``that all health centres should be similar in sta
and equipment'') and the number of doctors per
centre. First, the projected population of the city
was distributed between 36 regions, approximately
equally. The city was then described by a graph
with 36 vertices of equal weight, each vertex rep-
resenting the `central point' of a region. Two ver-
tices were joined by a link if they corresponded to
two adjacent regions. The problem consisted of
®nding the number of centres and their location
anywhere on the links of the graph so that all the
vertices were contained within the maximum ser-
vice distance S. The problem was solved using the
Gar®nkel and Nemhauser [25] heuristic for dier-
ent values of S. Because the centres did not need to
be located at vertices, the solution space was ef-
fectively in®nite, and heuristic approaches essen-
tial. The results were compared with solutions of
the problem which assumed that all centres would
be restricted to the existing vertices. In general, the
444 S.-u. Rahman, D.K. Smith / European Journal of Operational Research 123 (2000) 437±452
study provided the local health consultants with a
cost/bene®t analysis using accessibility to the
health care delivery system (S) as the main
parameter.
4.3. Measuring the eectiveness of past locational
decisions
There are studies which have explicitly ad-
dressed the eciency of earlier locational deci-
sions. Among these are those by Rushton and
Krishnamurthi [66] and Rahman and Smith [57].
In both these studies, the problem of locating op-
timal sites was considered as a p-median problem.
In their study, Rushton and Krishnamurthi [66]
computed the locational eciency of health facil-
ities that had actually been chosen in the state of
Karnataka, India during the period between 1971
and 1981. They found that the selected health fa-
cilities from 1971 to 1976 were 70% ecient com-
pared to the optimal sites, 77% ecient in the
period 1976±1979 and 62% ecient from 1979 to
1981.
The study by Rahman and Smith [57] was de-
signed to ®nd sites for Health and Family Welfare
Centres (HFWC) in Tangail Thana in Bangladesh.
HFWCs are meant to organise immunisation ac-
tivities, treat diarrhoeal diseases and fever cases,
and work for the family planning programmes in
the rural areas. According to government policy,
ten HFWCs are to be opened to serve the area.
However, seven are already operating in the
area. In addition, there are three rural dispensaries
(RD) in the area which according to the govern-
mentÕs plan will be upgraded and converted to
HFWCs. This means that sites for 10 HFWCs
have already been decided and there remained no
scope for siting any new facilities. Therefore, the
study concentrated on the following analysis:
·Compare the locational eciency of seven exist-
ing HFWCs with seven optimal HFWCs.
·Compare the locational eciency of seven exist-
ing plus three proposed HFWCs with seven ex-
isting plus three optimal HFWCs.
The optimal locations of seven facilities found by
the p-median method meant that the mean dis-
tance traveled by users (the objective function in
this case) was 1.9 km while the existing facilities
needed an average journey of 3.1 km ± 65% more.
This study also demonstrated that in a system with
four optimally located HFWCs, the number of
person-kilometers travelled is approximately equal
to the number of person-kilometers travelled in the
existing system with seven facilities. The system
with seven existing plus three proposed facilities
produced a mean distance travelled of 2.3 km. In
the system with seven existing plus seven optimal
facilities found by the p-median method the mean
fell to 1.7 km, an improvement of 26%. Other
relevant studies include those by Logan [41] in
Sierra Leone, Ayemi et al. [2] in Nigeria and Op-
pong and Hodgson [53] in Ghana.
The studies by Rushton and Krishnamurthi
[66], Logan [41], Ayemi et al. [2] and Rahman and
Smith [57] are examples of SLAMs. Some studies
which belong to the HLAM category and which
also addressed the eciency issue are those by
Fisher and Rushton [24] and Hodgson and
Valadares [32]. Both these studies considered the
location-allocation problem as a p-median prob-
lem and solved it using the Teitz and Bart [74]
heuristic. While conducting location analysis for
health facilities along with other services such as
police stations and district headquarters in Juna-
gadh, India, Fisher and Rushton [24] found that
compared with the optimal system, the existing
system was 1.22 times inecient. This means that
the average distance travelled in the existing sys-
tem is 1.22 times that in the optimal system. In
Hodgson and ValadaresÕ[32] study the spatial in-
eciency of the existing health system in Goa,
India was found to be even greater (1.56 times). In
both these studies the problems were considered as
a series of single-level, p-median problems, solved
in a step manner (top-down or bottom-up ap-
proach). Hodgson [33] demonstrated that the si-
multaneous approach produces a better solution
than either stepwise method.
4.4. Improving existing systems
Okafor [51] conducted a study to ®nd a site
(from four possible sites) for a hospital which is to
be added to an existing health delivery system with
S.-u. Rahman, D.K. Smith / European Journal of Operational Research 123 (2000) 437±452 445
three hospitals in Bendel state, Nigeria. The
problem was developed and solved as a transpor-
tation problem similar to one developed by Gould
and Leinbach [27]. The capacities of the existing
hospitals were included in the transportation for-
mulation. However, it appears from the study that
ap-median type formulation would have been
more appropriate. With the limited choice, this
was not an especially dicult problem, although
there was scope for sensitivity analysis. Eaton et al.
[20], on the other hand, used an MCLP method to
locate facilities in Colombia. The purpose of the
study was to ®nd sites for new health centres
which, if added to the 28 existing sites, would most
improve population coverage. Mehretu [44] in a
study in Bourkina Faso used the p-median method
to locate new health centres in an existing system
that provided primary health care. However, the
government planners imposed two constraints.
These are: (1) every village with a population of
730 and over will be assigned one primary Com-
munity Clinic; (2) every rural household should
have access to a health facility within a maximum
of 5 km.
Mehrez et al. [46] conducted a study to locate a
new hospital in Israel using location-allocation
models in conjunction with the Analytic Hierarchy
Process (AHP) approach. First the problem was
analysed using the p-median method and LSCP
both on the plane and on networks. Then the AHP
was applied to evaluate the optimal sites using a
set of criteria which include: minisum (p-median)
objective function; service availability to remote
settlements; improving employment; contribution
for population diversity; using the existing infra-
structure.
5. Conclusion and future research directions
The location-allocation models and methods
reviewed in this paper have been formulated either
as p-median problems or covering problems. The
objective of the p-median problem is to locate a
given number of facilities so that the total travel
distance (or time) between facilities and demand
points is minimised. Location analysis using p-
median formulations seems to be one of the most
popular approaches to rural health facility loca-
tion planning in developing countries (the majority
of the studies described have used this formula-
tion). The objective of LSCP is to locate the min-
imum number of facilities such that each demand
point has a facility within a given maximal service
distance (or time) and the objective of MCLP is to
locate a ®xed number of facilities to maximise the
total demand within a maximum service distance
(or time). When LSCP was ®rst introduced, its
simplicity of problem statement and its straight-
forward solution technique quickly established it
as a widely recognised model for the location of
public facilities. Chaiken [11] reported that Tore-
gas and ReVelle's [77] model had been acquired by
many national and international organisations.
An underlying assumption of most of the lo-
cation models described in this paper is that the
health facilities being sited are uncapacitated (i.e.
each has an in®nite capacity to serve consumer
demand). This paper has identi®ed three excep-
tions, one the paper by Okafor [51], that by Gould
and Lienbach [27] where the location of hospitals
was investigated but the capacity constraint made
it insoluble, and the work of Heller et al. [30] on
the capacitated p-median problem. Solving unca-
pacitated problems means that the work load at
the facilities may vary substantially. It is typical of
developing countries that rural health facilities are
organised with similar medical equipment and
employ the same number of health personnel. (For
instance, each Community Clinic intended to de-
liver primary health care at the village/ward level
in Bangladesh is to be manned by a Health As-
sistant and a Family Welfare Assistant [58].) Al-
lowing variation in the demand for services at the
facilities could be handled either by solving the
problem directly as a capacitated location problem
or by allocating health personnel according to the
demand at the facilities. Recently, Current and
Storbeck [13], and Pirkul and Schilling [55] have
suggested capacitated versions of the classical
covering models which would be useful for health
facility location planning with capacity constraints
in developing nations. For solving the capacitated
p-median problem one can make use of location-
allocation formulations suggested by Goodchild
[26].
446 S.-u. Rahman, D.K. Smith / European Journal of Operational Research 123 (2000) 437±452
The studies reviewed in this paper have been
applied to ®nd optimal sites of facilities, assess the
eectiveness of previous location decisions, and
generate viable alternatives for action by decision
makers. An assessment of the eectiveness of past
locational decisions can provide information re-
garding what could be achieved using the same
resources. This type of study has limited use, since
to relocate an existing system even partly to im-
prove eciency could be infeasible both politically
and economically in the context of developing
nations. Referring to a study in Honduras, Moore
([48], p. 135), reported that relocation of auxiliary
nurse posts would be politically infeasible since
``the resentment created by closing facilities would
probably be greater than any goodwill created by
opening others''. In the absence of formal analysis
for locational decisions, quite often ®nal decisions
are made by the local politicians. Evidence from
the studies show that these decisions are generally
far from optimal. In order to prevent such inter-
ventions locational models can play an important
role in generating alternative location arrange-
ments and predicting the cost-eectiveness of en-
tirely new service systems or existing systems
which are to be expanded using new facilities. The
same view has been shared by Rushton ([68], p.
113): ``So long as the ledger is blank, politicians
have the opportunity to vie with one another to
bring spoils back to their constituents, but with the
accounts open and enough detail to count the costs
and returns for alternative locations, politicians
have to pause before they intervene to counter the
(bureaucrats') recommendations''.
Most of studies reviewed have used some kind
of single criterion objective functions. In reality
however, most location decisions are complex
problems and require multi-criteria objective
models. In a few of the cases described, secondary
objectives have been included as constraints, as a
simple way of handling multi-dimensional mea-
sures of performance. This may not always be
adequate. It seems, therefore, that there is a ne-
cessity to develop multiple criteria location models
to develop health systems in developing nations.
This is in line with the ®ndings of other research-
ers. For instance, Erkut and Neuman [23] men-
tioned that Ôcurrent models can be used to generate
a small number of candidate sites, but the ®nal
selection of a site is a complex problem and should
be approached using multi-objective decision
making toolsÕ.
Understanding a problem and taking actions to
improve the problem situation may follow the
following steps:
1. Problem conceptualisation and de®nition.
2. Model (conceptual and quantitative) develop-
ment.
3. Analysis of the model.
4. Evaluation of results.
5. Implementation of results.
Most of the studies reviewed in this paper have
addressed Steps 1±4. Only a handful of studies
claimed implementation. In spite of the great po-
tential of the location-allocation models for de-
veloping and improving health systems, it is not
clear why only a few studies were implemented.
Rahman and Smith [59] suggested the following
necessary conditions for proper implementation of
OR studies in developing nations:
1. The involvement of local analysts from the be-
ginning by attachment to the project team, irre-
spective of whether or not the project was
initiated locally. The successful completion
and implementation will depend on the knowl-
edge and experience of such local people.
2. Eective communication with local decision-
makers to demonstrate the bene®ts of using
the OR approach.
Point 1 has been frequently advocated in the
OR literature [60,69]. As regards to point 2 in the
context of location-allocation studies in develop-
ing countries, some successful eorts were reported
in [46,56,21,6]. However, both are to be considered
as an integral part of any OR project in developing
nations. Point 1 is necessary to allow for successful
transfer of every phase of the planning and model
for further use; point 2 is required for acceptance
of the implementation and diusion of OR ideas
amongst planners and decision makers.
There will be health needs in the less-developed
countries of the world for many years to come. The
modelling skills of the operational research scien-
tist in studying the location and use of health cen-
tres will continue to be of value to the planners and
decision-makers. The same scientist's communica-
S.-u. Rahman, D.K. Smith / European Journal of Operational Research 123 (2000) 437±452 447
tion skills will be essential to involve the consumers
and their communities in the provision of suitably
located facilities. This paper has shown that this
has happened in the past, is happening now, and,
hopefully, will continue to happen in the future.
Acknowledgements
We are very grateful for the suggestions of two
anonymous referees, one of whom has identi®ed
several references for us, and to the comments of
several of our colleagues.
Appendix A
Based on the relationship between various lev-
els of hierarchies, Tien et al. [75] recommended
three types of service hierarchies. These are: suc-
cessively inclusive, locally inclusive and succes-
sively exclusive hierarchies. A hierarchical system
in which the facilities at any level oer all the
services oered by the facilities of a lower level is
said to have a successively inclusive hierarchy. This
has been illustrated in Fig. 2. Here, the facility at
location 4 (level 3 facility) oers type 1, 2 and 3
services to all locations and the facility at location
3 (level 2 facility) oers type 1 and 2 services to all
locations. In health care systems it is generally
assumed that the facility hierarchy is successively
inclusive. However, there are examples which
could be locally inclusive as well. In a locally in-
clusive hierarchy a facility at any level oers all
services only to the location where it is located and
only the highest order service to all other locations
(Fig. 2 (ii)). Fig. 2 (ii) shows that the facility at
location 4 (level 3 facility) oers type 1, 2 and 3
services to location 4 and oers only type 3 service
to all other locations. Likewise, the facility at 3
(level 2 facility) oers type 1 and 2 services to lo-
cation 3 and oers only type 2 services to other
locations. The health care system discussed by
Calvo and Marks [10] is an example of such an
hierarchy. In a successively exclusive hierarchy a
facility at level moers only type mservices to all
locations. This has been illustrated in Fig. 2 (iii).
Fig. 2. (i) Successively inclusive hierarchy. (ii) Locally inclusive hierarchy. (iii) Successively exclusive hierarchy.
448 S.-u. Rahman, D.K. Smith / European Journal of Operational Research 123 (2000) 437±452
Production±distribution systems and solid waste
disposal systems are generally assumed to have
successively exclusive hierarchies. In a production±
distribution system a product is manufactured at a
factory and stored at warehouses from which it is
transferred to the retail shops. Here each facility
oers only one type of service.
Narula [49] proposed a classi®cation scheme
based upon the number of types of facilities in a
hierarchy and the ¯ow of service between locations
and types of facilities. The ¯ow and the network
can be divided into four categories. The ¯ow may
be integrated (I) or discriminating (D). A ¯ow is
said to be integrated if it occurs from any lower
level 0;1;2;...;fÿ1facility to any higher level
1;2;...ffacility. A ¯ow is discriminating when
it occurs from any lower level facility mto the next
higher level facility m1 only.
The network may be unipath (U) or multipath
(M). In a unipath network the positive degree of
every node is less than or equal to 1. In a multipath
network the positive degree of at least one of the
nodes is greater than or equal to 2. The degree of a
node is the number of lines incident at that node.
These categories may be combined to give four
options which are preceded by the number of fa-
cilities to yield
f/I/U location-allocation models.
f/I/M location-allocation models.
f/D/U location-allocation models.
f/D/M location-allocation models.
Appendix B
1. The p-median problem [62]
minimise X
n
i1
X
n
j1
aidijxij
subject to X
n
j1
xij 18i;
X
n
j1
xjj p;
xij 6xjj 8i;j;
xij 0;18i;j;
where xij 1 if demand iis assigned to a facility j,
xij 0 otherwise, nthe number of demand points,
aithe population of demand i,dij the shortest
distance between iand j,pthe number of facilities
to be located.
As an integer-programming problem, there are
n2zero±one variables, although reducing the
number of possible sites for facilities can make this
smaller.
2. The LCSP [78]
minimise X
n
j1
xj
subject to X
j2Ni
xjP1;8i;
xj0;18j;
where xj1 if a facility is located at j,xj0
otherwise, Ni fjjdij 6Sgis the set of facilities
which are eligible to provide cover to demand i,S
the maximal service distance, n;dij are as before.
This problem only has nzero±one variables,
which makes it suitable for large problems, al-
though the ninequality constraints may make it
computationally dicult.
3. The MCLP [12]
maximise X
n
i1
aiyi
subject to X
j2Ni
xjPyi;16i6n;
X
n
j1
xjp;
xj0;18j;
yi0;18i;
where xj1 if a facility is located at j,xj0
otherwise, yi1 if demand from i is covered by a
facility, yi0 otherwise, Ni fjjdij 6Sgis the set
of facilities which are eligible to provide cover to
demand i,Sthe maximal service distance, n,ai,dij ,
pare as before.
This problem only has 2nzero±one variables,
which makes it suitable for large problems
S.-u. Rahman, D.K. Smith / European Journal of Operational Research 123 (2000) 437±452 449
than the p-median problem. The constraints are
``awkward''.
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... The government of India has recently passed the National Medical Commission Bill (NMC Bill) to create a world class medical education system that ensures adequate supply of high quality of medical professionals in the country while being flexible enough to adapt to the changing needs of a transforming nation [1]. This move is not surprising considering that India has always adhered to the tenet which asserts that the health of the country's population influences its overall development and its economic prosperity [2]. The country has taken positive steps in strengthening its commitment towards Universal Health Coverage (UHC) proposed by World Health Organization (WHO) [3] by launching the ambitious Ayushman Bharat Health Scheme (Pradhan Mantri Jan Arogya Yojana) in 2018 [4]. ...
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