Geoscience Reference
In-Depth Information
The objective function (
21.36
) maximizes the amount of demand that is covered
twice within t
1
. Each node must be covered at least once within time t
2
as assured by
constraints (
21.37
). Constraint (
21.38
) states that a proportion Ǜ of the demand must
be covered within t
1
. A location can only be covered twice within t
1
if it is covered
once, as expressed by constraints (
21.39
)and(
21.40
). Exactly p ambulances
must be located in total (
21.41
) and only p
i
can be located at node i (
21.42
).
Constraints (
21.43
)and(
21.44
) define the domains of the decision variables. The
model (
21.36
)-(
21.44
) has been tackled in Gendreau et al. (
1997
) by a tabu search
heuristic.
21.3.1.2
Considering Ambulance Utilization
In practice, ambulances are not always available when they are needed. Therefore,
the strategic and tactical level planning has to take some aggregated data from the
operational level into account (if possible): the utilization of ambulances. For this
situation, the expected coverage of a region can be determined. When the number
of ambulances to be placed is fixed and the expected coverage is to be maximized,
the problem can be formulated as the maximum expected location covering problem
(MEXCLP) proposed by Daskin (
1983
).
The set of demand nodes is denoted by J, and each node has a demand d
j
. I is
the set of possible ambulance locations, and the maximum number of ambulances
that can be located is bounded by p. In the original model, we have I
D
J
D
f
1;:::;n
g
. The probability that an ambulance is occupied is defined by P and P
k
is the probability that k ambulances are busy at the same time. If node j
2
J is
covered by k ambulances, E
j
k
D
d
j
.1
P
k
/ gives the corresponding expected
covered demand and E
j
k
E
j
k1
D
d
j
.1
P/P
k1
is the marginal contribution
of the kth ambulance to this expected value. A decision variable y
i
is considered
representing the number of ambulances to locate at node i. Moreover, we use set
K
Df
1;:::;n
g
in order to refer to the number of times that a node is covered by an
ambulance. A decision variable x
jk
is equal to 1 if node j is covered k times and 0
otherwise. In addition,
ij
is a binary parameter with:
ij
D
(
1 if t
ij
T
.an ambulance at i covers demands at j/
(21.45)
0 otherwise
Here, t
ij
states the driving time from node i to node j and T expresses the maximal
allowed driving time. The MEXCLP can be written as follows:
maximize
X
k2K
X
d
j
.1
P/P
k1
x
jk
(21.46)
j2J
subject to
X
k2K
x
jk
X
i2I
ij
y
i
8
j
2
J
(21.47)
