70
4 Layout of a Single Floor
4.4.3 Second Stage
The second-stage model is a SOCO formulation of the UA-FLP in which the values
of the variables (α ij , β ij ) have already been fixed based on the solution of the first
stage. The second-stage model arises from the formulation in Sect. 4.2 and is as
follows:
minimize
1≤i
x
ij + d
y
ij )
s.t. d
x
ij ≥ x i − x j , d
x
ij ≥ x j − x i , 1 ≤ i < j ≤ n,
d
y
ij ≥ y i − y j , d
y
ij ≥ y j − y i , 1 ≤ i < j ≤ n,
h
min
i
≤ h i ≤ h
max
i , 1 ≤ i ≤ n,
w
min
i
≤ w i ≤ w
max
i
, 1 ≤ i ≤ n,
w i + h i ≥
w i − h i
2
√
A i
2
, 1 ≤ i ≤ n
w i ≤ ρ i h i and h i ≤ ρ i w i , 1 ≤ i ≤ n,
1
2
(w i − w F ) ≤ x i ≤
1
2
(w F − w i ), 1 ≤ i < j ≤ n,
1
2
(h i − h F ) ≤ y i ≤
1
2
(h F − h i ), 1 ≤ i < j ≤ n,
linear nonoverlap constraints,
where the linear nonoverlap constraints are obtained from (4.8) according to the
solution of the first stage.
4.5 Flexible Bay Structure
A flexible bay structure is a version of the UA-FLP in which the departments are
located in parallel bays with varying widths; this special case arises in the design
of manufacturing facilities. We illustrate the concept of a flexible bay structure
in Fig. 4.7. We observe that the column-wise bay structure is similar to the row
structure in row FLPs. However, a fundamental difference is that the width of each
flexible bay depends on the total area of the departments assigned to that bay,
whereas in row FLPs, the heights of the rows (and of the departments) are equal
and fixed.
The bays have straight aisles on both sides, and departments are not allowed to
span multiple bays. This structure restricts the set of feasible solutions, but it has
4 Layout of a Single Floor
4.4.3 Second Stage
The second-stage model is a SOCO formulation of the UA-FLP in which the values
of the variables (α ij , β ij ) have already been fixed based on the solution of the first
stage. The second-stage model arises from the formulation in Sect. 4.2 and is as
follows:
minimize
1≤i
x
ij + d
y
ij )
s.t. d
x
ij ≥ x i − x j , d
x
ij ≥ x j − x i , 1 ≤ i < j ≤ n,
d
y
ij ≥ y i − y j , d
y
ij ≥ y j − y i , 1 ≤ i < j ≤ n,
h
min
i
≤ h i ≤ h
max
i , 1 ≤ i ≤ n,
w
min
i
≤ w i ≤ w
max
i
, 1 ≤ i ≤ n,
w i + h i ≥
w i − h i
2
√
A i
2
, 1 ≤ i ≤ n
w i ≤ ρ i h i and h i ≤ ρ i w i , 1 ≤ i ≤ n,
1
2
(w i − w F ) ≤ x i ≤
1
2
(w F − w i ), 1 ≤ i < j ≤ n,
1
2
(h i − h F ) ≤ y i ≤
1
2
(h F − h i ), 1 ≤ i < j ≤ n,
linear nonoverlap constraints,
where the linear nonoverlap constraints are obtained from (4.8) according to the
solution of the first stage.
4.5 Flexible Bay Structure
A flexible bay structure is a version of the UA-FLP in which the departments are
located in parallel bays with varying widths; this special case arises in the design
of manufacturing facilities. We illustrate the concept of a flexible bay structure
in Fig. 4.7. We observe that the column-wise bay structure is similar to the row
structure in row FLPs. However, a fundamental difference is that the width of each
flexible bay depends on the total area of the departments assigned to that bay,
whereas in row FLPs, the heights of the rows (and of the departments) are equal
and fixed.
The bays have straight aisles on both sides, and departments are not allowed to
span multiple bays. This structure restricts the set of feasible solutions, but it has