Facility Layout Problems
I. History
1.1 Introduction
Montreuil's Mixed-Integer Programming Formulation
A mixed-integer programming formulation for the facility layout problem was presented by
Montreuil in 199098 at a material handling research conference (a slight modification of the model
is presented in Tompkins et al., 131 pp. 344-350). The model uses a distance-based objective but is
not based on the traditional discrete (QAP) framework. Instead, it utilizes a continuous
representation of a layout. The notation and general formulation follow. Note that a specialized case
of this model was developed by Heragu and Kusiak
orientation are specified a priori.
55
where the departmentlength, width, and
1.2 Type of facility layout problems
1.2.1 Single facility
Generally speaking, the multi-floor layout problem is more complicated than its
single-floor counterpart since the former involves vertical flow and lifts.
1.2.2 Multi-floor facility layout problem
As in most computer-based layout algorithms, in MULTIPLE the layout is
represented as a matrix. Each element of the matrix corresponds to a grid square (or
grid) of specified area, and the space required by each department is expressed as an
integer number of grids. To construct the layout, we propose to use a spacefilling
curve which simply visits all the grids on a floor. To ensure that a department is not
split, all the grids as-signed to a department must be contiguous, i.e., each grid must
be adjacent to another grid that has been assigned to the same department. A
spacefilling curve can guarantee that no departments will be split because a separate
curve is used for each floor and, within each floor, the curve visits the "neighbors" of
a grid before visiting other grids.
1.2.3 Multi-floor and single-floor difference
1. The impact of vertical flow must be considered.
Note that vertical travel occurs via access to a life, which we define as any
vertical material handling device. Thus, if two departments arc located on
different floors, vertical travel will be necessary, and due to the location of a life,
the horizontal travel between the two departments may be also increase.
2. The determination of area feasibility.
Problem Definition:
2.1 Problem Model:
Minimize
f
( i,j)F
ij
 (xij  xij  yij  yij )  M(h   h   v   v  )
subject to
X i  xi  xi  X i
 i C
(1)
Y i  yi  y i  Y i
 i C
(2)
P i  2( x i  x i  y i  y i )  P i
 i C
(3)
x i  xi  x t
 i C
(4)
y i  yi  y t
 i C
(5)
(i,j)  F
(6)
(i,j)  F
(7)
i  C
(8)
i  C
(9)
Ri|j  R j|i  Ri/j  R j/  3
i  (i  j ; i, j  C)
(10)
Ri|j ,Ri/j  0 or 1
(i  j  C)
(11)
x i  x j  MRi|j
(i  j  C)
(12)
y i  y j  MR j|i
(i  j  C)
(13)
xi  x j  xij  xij

ij
yi  y j  y  y

ij
0  x i  BH  h  -h 
0  y i  BH  v  v
-
Where
fij represents interstation flow between input/output station of cell i and input/output station
of cell j.
M is a large positive integer.
C is the set of cells, F is the set of flows underscore and bar represent lower and upper
bounds respectively.
Xi and Yi represent position independent length and width dimensions of cell i.
Pi represents perimeter of cell i.
x i and y i represent lower length and width co-ordinate values for cell i.
x i and y i represent upper length and width co-ordinates.
xi and yi represent input/output station location co-ordinates of cell i.
BH and BV represent facilities envelope upper limit co-ordinate values
Ri|j =0 if cell i is imposed to be to the west of (left of) cell j.
=1 if the cell above imposition is not enforced.
xij , yij , h  and v  are positive numbers representing positive components of xi-xj, yi-yj,
penalty variable for crossing BH and BV respectively
xij , yij , h  and v  are positive numbers representing negative components of xi-xj, yi-yj,
penalty variable for crossing BH and BV respectively.
II. Applications
Application 1
Source:
Banerjee, P., Montreuil, B., Moodie, C.,L., Kashyap, R.,L., 1992. A modeling of interactive
facilities layout designer reasoning using qualitative patterns. International Journal of Production
Research, 30(3), 433-453.
Description:
The first case is entiled ‘Hypo Manufacturing Company’. The data for the case is shown in
Table1. A starting solution for this experiment is the layout show in Fig3. The initial rectilinear
inter-station flow travel score for this solution state is 24665. The context setting agent sets context
agent sets context 1 and informs the scheduling and governing agents. An aggregate scheduling of
the QLA set is performed by the scheduling agent.
Application 2
Source:
Bozer, Y.,A., Meller, R.,D., Erlebacher, S.,J., 1994. An improvement type layout algorithm for
single and multiple oor facilities. Management Sci, 40 (7), 918-932.
Description:
The problem is based on a 15-department, three-floor facility with six existing (or potential)
lifts. The departmental area data are shown in Table 1. (We assume there are no upper bounds
imposed on depart-ment areas.) Department 15 (the receiving/ shipping department) is fixed in its
current location in the initial layout, which is shown in Figure 5a.
Table 1 Area requirement for the multi-floor
example problem
Department
Current Area
Minimum Area
1
15
12
2
3
4
5
6
7
8
10
9
7
9
25
25
15
7
6
5
7
22
22
13
9
10
11
12
13
14
15
10
25
10
15
6
9
25
7
22
9
13
4
17
25
Figure 5a
Application 3
Source:
Y.A., Bozer, R.D., Meller, 1997. A reexamination of the distance-based facility layout problem.
International journal of production research, 29, 549-560.
Description:
The first data set consists of nine departments, with four small, three medium and two large
departments of area 9, 16, and 36 units, respectively. The building is 13units long and 12units wide.
For the second example, we use a 12-department problem with two large, two medium, and
eight small departments of area 16, 4, and 1 units, respectively. The building is 8 units long and 6
units wide.
Thus for both examples the total area required is equal to the total area available in the facility.
Also, the department sizes were chosen so that square departments are possible and ideal for the
QAP-approach (note that departments with area 36 would ideally be diamond shaped; however, the
facility area prohibits this choice).
Application 4
Source:
Tam, K. Y., CHAN, S. K., 1998. Solving facility layout problems with geometric constraints using
parallel genetic algorithms: experimentation and findings. International Journal of Production
Research, 36(12), 3253-3272.
Description:
The factors considered in our comparison are (1) parallel GA algorithm; (2)problem size; and (3) number of
processors. The first factor includes the four parallel GAs, namely the centralized implementation (CENT), the
semi-distrubted parallel GA (SEMI), the distributed parallel model (DIST), and the totally distributed parallel GA
(TOTAL). For problem size (factor 2), layouts for 5, 7, 10 and 20 facilities are compared. Since algorithm (factor
1) and number of processors (factor 3) are not independent, the latter is nested in the former in our experiment.
III.
Reference
[1] Banerjee, P., Montreuil, B., Moodie, C.,L., Kashyap, R.,L., 1992. A modeling of interactive
[2]
[3]
[4]
[5]
facilities layout designer reasoning using qualitative patterns. International Journal of
Production Research, 30(3), 433-453.
Sherali, H., D., Fraticelli, Barbara M., P., Meller, D., 2003. Enhanced Model Formulations for
Optimal Facility Layout. Operations research, 51(4), 629-644.
Meller, R.D., Bozer, Y.A., 1996. A new simulated annealing algorithm for facility layout
problem. International journal of production research, 34(6), 1675-1692.
Meller, R. D., Chen, Sherali, W., H.,D., 2007. Applying the sequence-pair representation to
optimal facility layout designs. Operations Research Letters, 35(5), 651-659.
Meller, R. D., K. Y. Gau. 1996. The facility layout problem: Recenta nde mergingtr endsa ndp
erspectives.J. Manufactur-ing Systems ,15(5), 351-366.
[6] Meller, R., D., Narayanan, V., Vance, P.,H., 1999. Optimal facility layout design. Operations
esearch letters, 23(3-5), 117-127.
[7] Y.A., Bozer, R.D., Meller, 1997. A reexamination of the distance-based facility layout problem.
International journal of production research, 29, 549-560.