Geoscience Reference
In-Depth Information
ambulances can be placed. Additionally, instead of a single maximum response
time, two values, t
1
and t
2
, are considered with t
2
t
1
. Note that t
2
is equivalent to
T since all demand must be covered by an ambulance located within time t
2
. Finally,
a proportion Ǜ is defined for which the demand must also be fulfilled within t
1
time
units by some of the ambulances (which can be the same ambulances or different
ones). Consider now a complete graph whose nodes correspond to the elements in
I
[
J, and whose edges
f
i;j
g
with i
2
I and j
2
J are weighted with the travel
time t
ij
between these two nodes. Further, let d
j
denote the demand at node j
2
J,
and define the following two coefficients for i
2
I and j
2
J:
ij
D
(
1 if t
ij
t
1
.j is covered by location i within time t
1
/
(21.34)
0 otherwise
and
ij
D
(
1 if t
ij
t
2
.j is covered by location i within time t
2
/
(21.35)
0 otherwise
Two sets of decision variables can be considered: y
i
denotes the (integer) number
of ambulances to locate at i
2
I (bounded by p
i
), and x
jk
is a binary variable equal
to 1 if j is covered at least k times within t
1
for k
2f
1;2
g
and 0 otherwise. The
double standard model (DSM) proposed by Gendreau et al. (
1997
) is the following:
maximize
X
j2J
d
j
x
j2
(21.36)
subject to
X
i2I
ij
y
i
1
8
j
2
J
(21.37)
X
d
j
x
j1
Ǜ
X
j2J
d
j
(21.38)
j2J
X
ij
y
i
x
j1
C
x
j2
8
j
2
J
(21.39)
i2I
x
j2
x
j1
8
j
2
J
(21.40)
X
y
i
D
p
(21.41)
i2I
y
i
p
i
8
i
2
I
(21.42)
x
j1
;x
j2
2f
0;1
g
8
j
2
J
(21.43)
Z
0
y
i
2
8
i
2
I:
(21.44)
