Geoscience Reference
In-Depth Information
predefined locations as in the QAP model. Again, the objective is to minimize the
total travel distance. The model presented here goes back to Montreuil (
1991
)and
has been linearized and explained in detail by Tompkins et al. (
2010
).
The following parameters are given: B
a
and B
b
represent the length and width
of the building, respectively. The lower and upper limits on the length and width
of OU j are given by L
l
j
;L
j
;W
j
and W
j
, respectively. P
j
and P
j
are lower and
upper limits on the perimeter of OU j, respectively.M represents a sufficiently large
number (Big M). Again, f
jk
is the flow between two OUs j and k.Furthermore,
the following decision variables are defined: Ǜ
j
and LJ
j
are the x- and y-coordinates
of the centroid of OU j. The x-coordinates of the left and right sides of OU j are
defined by a
0
j
and a
0
j
, respectively. The y-coordinates of the bottom and top of OU
j are represented by b
0
j
and b
0
j
, respectively. Furthermore, the binary variables
z
jk
(
z
jk
) are considered which are equal to 1 if OU j is strictly to the right (top) of OU
k and 0 otherwise. The layout problem can be formulated as follows:
minimize
X
j2J
f
jk
Ǜ
jk
C
Ǜ
jk
C
LJ
jk
C
LJ
jk
X
(21.64)
k2J
subject to Ǜ
j
Ǜ
k
D
Ǜ
jk
Ǜ
jk
8
j;k
2
J;j
¤
k
(21.65)
LJ
j
LJ
k
D
LJ
jk
LJ
jk
8
j;k
2
J;j
¤
k
(21.66)
L
l
j
a
0
j
a
0
j
L
j
8
j
2
J
(21.67)
W
j
b
0
j
b
0
j
W
j
8
j
2
J
(21.68)
P
j
2
a
0
j
a
0
j
C
b
0
j
b
0
j
P
j
8
j
2
J
(21.69)
0
a
0
j
a
0
j
B
a
8
j
2
J
(21.70)
0
b
0
j
b
0
j
B
b
8
j
2
J
(21.71)
Ǜ
j
D
0:5
a
0
j
C
a
0
j
8
j
2
J
(21.72)
LJ
j
D
0:5
b
0
j
C
b
0
j
8
j
2
J
(21.73)
a
0
k
a
0
j
C
M
1
z
jk
8
j;k
2
J;j
¤
k
(21.74)
b
0
k
b
0
j
C
M
1
z
jk
8
j;k
2
J;j
¤
k
(21.75)
z
jk
C
z
kj
C
z
jk
C
z
kj
1
8
j;k
2
J;j <k
(21.76)
Ǜ
j
;LJ
j
;a
0
j
;a
0
j
;b
0
j
;b
0
j
0
8
j
2
J
(21.77)
Ǜ
jk
;Ǜ
jk
;LJ
jk
;LJ
jk
0
8
j;k
2
J;j
¤
k
(21.78)
z
jk
;
z
jk
2f
0;1
g
8
j;k
2
J;j
¤
k:
(21.79)
