Geoscience Reference
In-Depth Information
function of the CDOMP (i.e. the capacitated ordered median problem), using the
variables
u
jk
,as
P
jD1
P
kD1
j
.c
.k/
c
.k1/
/
u
jk
:
First of all, we need to impose the following group of sorting constraints on the
u
jk
-variables:
u
jC1;k
u
jk
j
D
1;:::;n
1
I
k
D
1;:::;G:To guarantee that
exactly p servers will be opened among the n possibilities, we consider constraint
(
10.37
) defined in the previous formulation.
Then, we need to ensure that demand is covered and capacity is satisfied. For
these reasons we introduce the variables x
ij
:
x
ij
D
1; if the client i is allocated to server j
0; otherwise
(10.42)
(binary allocation) and constraints
P
jD1
x
ij
D
1; i
D
1;:::;n(each client is just
assigned to one server) and
P
iD1
a
i
x
ij
b
j
y
j
; j
D
1;:::;n(all the demand and
capacity requirements must be satisfied and clients can only be assigned to servers
which are open).
In addition, the relationship that links the variables
u
and x is:
P
jD1
u
jk
D
P
iD1
P
jWc
ij
c
.k/
x
ij
: The meaning being clear. The number of allocations with a
cost at least c
.k/
must be equal to the number of servers that support demand from
facilities at a cost greater than or equal to c
.k/
.
Summing up all these constraints and the objective function, the CDOMP can be
formulated as
X
n
X
G
minimize
j
.c
.k/
c
.k1/
/
u
jk
(10.43)
jD1
kD1
n
X
subject to
x
ij
D
1;
8
i
D
1;:::;n
(10.44)
jD1
X
n
a
i
x
ij
b
j
y
j
;
8
j
D
1;:::;n
(10.45)
iD1
x
ij
y
j
8
i;j
D
1;:::;n
(10.46)
X
n
y
j
D
p
(10.47)
jD1
X
n
X
n
X
u
jk
D
x
ij
;
8
k
D
1;:::;G
(10.48)
jD1
iD1
j
1:::;n
c
ij
D
c
.k/
u
jC1k
u
jk
;
8
j
D
1;:::;n
1
I
k
D
1;:::;G (10.49)
u
jk
2f
0;1
g
;
8
j
D
1;:::;n
I
k
D
1;:::;G
(10.50)
x
ij
;y
j
2f
0;1
g
;
8
i;j
D
1;:::;n:
(10.51)
