Geoscience Reference
In-Depth Information
Then, the last part of this section is devoted to test the efficiency of the last approach
by providing some preliminary numerical experiments.
10.5.1
A Three-Index Formulation
In order to introduce this formulation let A denote the given set of n sites and
identify these with the integers 1;:::;n, i.e., A
Df
1;:::;n
g
. We assume without
loss of generality that the set of candidate sites for new facilities is identical to the
set of clients. Let C
D
.c
ij
/
i;jD1;:::;n
be the given non-negative n
n cost matrix,
where c
ij
denotes the cost of satisfying the demand of client i from a facility located
at site j.Letp
n be the number of facilities to be located. Each client i has a
demand a
i
that must be served and each server j has an upper bound b
j
on the
capacity that it can fulfill. We assume further that assignment is binary, that is, the
demand of each client must be served by a unique server.
A solution to the location problem is given by a set of p sites; we use X
A,
with
j
X
jD
p, to denote a solution. Then, the problem consists of finding the set
of sites X with
j
X
jD
p, which can supply the overall demand at a minimum cost
with respect to the ordered median objective function.
A natural way to attack the formulation of the discrete ordered median problem
is to use variables that keep track of the order of the transportation costs from each
client and its server. This approach gives rise to a formulation with three-index
variables, one for the order and the remaining two indices, for the client-server
allocation. In order to formulate this model we consider a set of -weights, where
i
can be seen as a correction factor to the
ith
-position with i
D
1;:::;n. In addition,
we define the following set of variables:
8
<
1; if client i is supplied by server j and is the k-th
cheapest cost allocation
0; otherwise,
x
ij
D
8
i;j;k
D
1;:::;n;
:
y
j
D
1; if the server at j is open
0; otherwise,
8
j
D
1;:::;n:
Hence, the formulation of the model is:
X
n
X
n
X
n
k
c
ij
x
ij
minimize
(10.33)
iD1
jD1
kD1
n
n
X
X
x
ij
D
1;
8
i
D
1;:::;n
subject to
(10.34)
jD1
kD1
