Geoscience Reference
In-Depth Information
Sect.
11.4
):
Minimize
X
t2T
X
X
c
ijt
x
ijt
(11.33)
i2I
j2J
subject to
X
i2J
x
ijt
D
1; t
2
T;j
2
J
(11.34)
X
x
ijt
nx
iit
; t
2
T; i
2
J
(11.35)
j2J
X
x
iit
D
p
t
; t
2
T
(11.36)
i2J
x
iit
x
ii
;t1
; t
D
2;:::;
j
T
j
;i
2
J
(11.37)
x
ijt
0; t
2
T; i
2
J; j
2
J;
(11.38)
where 1
p
1
p
2
:::
p
jT j
D
p.
Constraints of type (
11.37
) were first proposed for a multi-period facility location
problem by Roodman and Schwarz (
1975
,
1977
). The latter paper was pioneering
in the assumption that a set of facilities may be operating before the beginning
of the planning horizon. These are the facilities that can be removed. Therefore,
the possibility of adapting an existing system to predictable changes in some
parameters, becomes explicitly considered in the models. The set of locations I can
now be partitioned into two subsets: I
c
and I
o
. The former represents the facilities
that are operating before the beginning of the planning horizon; the latter represents
the set of locations for new facilities. A more comprehensive model for the multi-
period facility location problem emerges:
Minimize
(
11.23
)
subject to
(
11.24
)-(
11.27
)
y
it
y
i;t1
; t
D
2;:::;
j
T
j
;i
2
I
c
(11.39)
y
it
y
i;t1
; t
D
2;:::;
j
T
j
;i
2
I
o
:
(11.40)
Roodman and Schwarz (
1977
) were also pioneering by considering a maximum
number of customers that can be served by each facility in each period and assumed
that not all facilities can serve all customers. These aspects are easily accommodated
in the above model if we replace (
11.25
)by(
11.30
). As mentioned before, the
latter constraints would be later considered by Canel and Khumawala (
1997
). The
research done by Roodman and Schwarz (
1977
) extends the work by the same
authors published 2 years before (Roodman and Schwarz
1975
) in which a pure
phase-out problem had been considered.
The above models allow the removal of an existing facility before the beginning
of period 1 with no costs imputed to the planning horizon. Imposing that the existing
facilities must operate in at least one period, can be easily done by setting y
i1
D
1,
i
2
I
c
.
