Geoscience Reference
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from district j allocated to a facility of type s located at i; x
12
ij
is the number of
patients that are transferred from a health center in location i to a hospital in location
j. Finally, q
s
is the exogenous capacity of a facility with service type s, c
ij
is the
minimum distance between locations i and j,andd
j
is the number of individuals
of population j. Then a mixed-integer programming formulation to minimize total
distance traveled is as follows:
minimize
X
i2I
X
c
ij
.x
01
ij
C
x
02
ij
C
x
12
ij
/
(21.20)
j2J
subject to
X
i2I
.x
01
ij
C
x
02
ij
/
D
d
j
8
j
2
J
(21.21)
X
ij
D
X
i2I
x
12
x
01
ij
8
j
2
J
(21.22)
i2I
X
x
01
ij
q
1
y
i1
8
i
2
I
(21.23)
j2J
X
.x
02
ij
C
x
12
ij
/
q
2
y
i2
8
i
2
I
(21.24)
j2J
X
y
is
D
p
s
s
2
S
(21.25)
i2I
y
i1
C
y
i2
1
8
i
2
I
(21.26)
0
x
01
ij
d
j
8
i
2
I;j
2
J
(21.27)
0
x
02
ij
d
j
8
i
2
I;j
2
J
(21.28)
0
x
12
ij
q
1
8
i
2
I;j
2
J
(21.29)
y
is
2f
0;1
g
8
i
2
I;s
2
S:
(21.30)
NarulaandOgbu(
1979
) proposed heuristic procedures for tackling this model.
Some examples of hierarchical facility location models include Hodgson (
1988
)
for primary care facilities, Smith et al. (
2009
,
2013
) for community health facilities,
and Mestre et al. (
2012
) for regional and central hospitals. Typically, these models
can be solved by commercial solvers. Galvão et al. (
2002
) applied a three-level
hierarchical model for the delivery of perinatal care in the municipality of Rio
de Janeiro with service referrals, and Galvão et al. (
2006
) extended this model to
include capacitated facilities. The increased complexity of the models motivated the
use of Lagrangian relaxation based procedures.
