Geoscience Reference
In-Depth Information
in the services offered by these facilities. For instance, a hospital can usually
provide all the services offered in a clinic. Moreover, some health systems require a
referral from a general practitioner before a patient can ask for service at a hospital.
Hierarchical location models can incorporate such characteristics. Sahin and Süral
(
2007
) provided a comprehensive review on hierarchical systems with a discussion
of modeling approaches and applications.
Hierarchical systems are commonly classified as successively inclusive or
exclusive: in a
successively inclusive hierarchy
, a facility at some level provides
all the services offered by lower level facilities (Calvo and Marks
1973
;Narula
1984
). This is a typical structure for healthcare facilities. Conversely, a
successively
exclusive hierarchy
implies that facilities at each level offer a service that is unique
to that level (Tien et al.
1983
). This is the case for specialized service facilities. We
now assume that I
D
J
Df
1;:::;n
g
, i.e., in each district j
2
J there is exactly one
potential location i
2
I for a facility. The formulation provided by Calvo and Marks
(
1973
) for a successively inclusive hierarchy is an assignment based p-median type
model with an objective function that quantifies the total distance traveled:
minimize
X
i2I
X
c
ij
X
s2S
d
js
x
ijs
(21.15)
j2J
subject to
X
i2I
x
ijs
D
1
8
j
2
J;s
2
S
(21.16)
x
iis
x
ijs
8
i
2
I;j
2
J;s
2
S
(21.17)
X
x
iik
D
X
s2SWsk
p
s
8
k
2
S
(21.18)
i2I
x
ijs
2f
0;1
g
8
i
2
I;j
2
J;s
2
S:
(21.19)
where c
ij
is the distance between location i and district j, x
ijs
is a binary variable
equal to 1 if individuals residing in district j that require service type s are assigned
to location i and 0 otherwise, d
js
is the number of individuals residing in district
j and requiring service type s,andp
s
is the number of facilities offering type s
services to be located. Constraints (
21.16
) ensure that all districts are assigned to
a facility for all services. Constraints (
21.17
) ensure that assignments are done to
open facilities only, and constraints (
21.18
) specify the possible number of self-
assignments (i.e., the assignment of the groups of individuals residing at a location
to the facility at that location). Finally, constraints (
21.19
) are the variable domain
constraints.
NarulaandOgbu(
1979
) developed a two-level hierarchical model with an
approach based on network flows where p
1
health centers (level s
D
1)andp
2
hospitals (level s
D
2) are to be located among the population centers, and a
proportion of patients, , at health centers are transferred to hospitals. In each
location, at most one facility type can be located. y
is
is a binary variable equal to 1
if a facility of service type s is located in location i. x
0s
ij
is the number of patients
