ANT COLONY OPTIMIZATION FOR THE UNEQUAL-AREA
FACILITY LAYOUT PROBLEM
Sadan Kulturel-Konak and Abdullah Konak
Penn State Berks, Reading, PA, U.S.A.
Keywords: Unequal Area Facility Layout, Relax Flexible Bay Structure, Ant Colony Optimization.
Abstract: In this paper, an ant colony optimization (ACO) approach is proposed to solve the Facility Layout Problem
(FLP) with unequal area departments. The flexible bay structure (FBS) is relaxed by allowing empty spaces
in bays, which results in more flexibility while assigning departments in bays. The comparative results show
that the ACO approach is very promising.
1 INTRODUCTION
The Facility Layout Problem (FLP) is generally
defined as locating N departments in an area of size
W×H. The inputs of the problem include department
area a
i
and minimum side length l
i
min
requirements
for each department i as well as material flow f
ij
and
material handling cost c
ij
(per unit flow per unit
distance travelled) between each department pair i
and j. The goal is to minimize the total material
handling cost, which is generally expressed as
follows:
11
() ()
NN
ij ij ij
iji
F
scfds
==+
=
∑∑
(1)
where d
ij
(s) is the distance between the centre points
of departments i and j for a given layout s. The
decision variables of the FLP include determining
department centres (x
i
, y
i
) and department shapes for
each department i. Satisfying the area requirements
of the departments, the boundaries of the layout and
restrictions on the departments’ shapes are the
problem constraints. The output of the FLP is called
block layout, which specifies relative location and
shape of each department in the area.
In this paper, an ant colony inspired algorithm is
proposed to solve the FLP with unequal area
departments in the flexible bay structure (FBS). In
the FBS, departments are located only in parallel-
bays with varying width, bays are bounded by
straight aisles on both sides, and departments are
restricted to be located only in one bay. Recently,
Komarudin and Wong (2010), Wong and
Komarudin (2010), and Kulturel-Konak and Konak
(2011a) have proposed ACO approaches to solve the
unequal area FLP. In this paper, an ACO approach
for the relaxed FBS, called ACO-RFBS, is
developed. The relaxed FBS (RFBS) concept was
originally proposed by Kulturel-Konak and Konak
(2011b) to remedy the drawbacks of the FBS. The
RFBS allows empty spaces in bays, which results in
more flexibility while assigning departments in bays.
Being different from the Particle Swarm
Optimization (PSO) by Kulturel-Konak and Konak
(2011b), the ACO-RFBS uses a different encoding
scheme, a dynamic penalty handling method, and a
two-phase diversification scheme. Moreover, in this
paper, facility areas are expanded, and the proposed
approach is used to solve the problems with
expanded areas to demonstrate the advantages of the
RFBS.
2 THE ACO-RFBS
2.1 Solution Construction Definition
In the ACO-RFBS, first a layout sketch is
constructed by filling bays one department at a time,
from bottom to top. A layout sketch defines the
relative locations of the departments within bays.
After creating a layout sketch, the actual locations
and shapes of the departments are calculated
according to the RFBS as described by Kulturel-
Konak and Konak (2011b). Figure 1 demonstrates a
273
Kulturel-Konak S. and Konak A..
ANT COLONY OPTIMIZATION FOR THE UNEQUAL-AREA FACILITY LAYOUT PROBLEM.
DOI: 10.5220/0003627302730277
In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 273-277
ISBN: 978-989-8425-83-6
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

step-by-step example of constructing a layout sketch
with five departments.
Figure 1: An example of layout construction where (x)
represents admissible cells to assign departments.
As demonstrated in the example in Figure 1, three
types of department assignments are admissible
while adding an unassigned department i to a partial
layout sketch as follows:
• (i, j, 1): department j is the first department in
the leftmost bay of a partial layout sketch. As a
result of this assignment, department i is located
immediately to the left of department j.
• (i, j, 2): department j is the last department in a
bay. As a result of this assignment, department i
is located immediately above department j.
• (i, j, 3): department j is the first department in
the rightmost bay of a partial layout sketch. As a
result of this assignment, department i is located
immediately to the right of department j.
Pheromone
τ (i, j, k) is defined as the favourability of
assignment (i, j, k). Let A be the set of all admissible
assignments. While constructing a layout sketch, an
admissible assignment (i, j, k) is randomly selected
from A, and department i is added to the sketch
according to the assignment rules defined above.
The probability of selecting an admissible
assignment (i, j, k) from A is given as follows:
1
1
(,,)
(, , ) (, )
(, , )
(, ,) (, )
xyz A
ijk ij
pi jk
x
yz xy
ββ
β
β
τη
τη
−
−
∈
=
∑
(2)
where
η (i, j) is the problem specific heuristic
information, which is defined as a function of the
normalized flows between departments i and j as
follows:
,
(, ) 1
max{ }
ij
pq
pq
Nf
ij
f
η
×
=+
(3)
Unlike the standard ACO (Dorigo et al., 1996;
Dorigo and Gambardella, 1997), only a single
parameter, 0<
β <1, is used in the ACO-RFBS to
scale the relative importance of the pheromone and
the problem specific heuristic information. To do so,
the pheromone and heuristic information values are
normalized in the same range. Layout construction
initially starts with an empty sketch. While assigning
the first department, however, equation (2) cannot be
used because A is an empty set. The first department
is randomly selected with the following probability,
(
)
()
1,..., ,
1,..., 1,..., ,
(, ,1) (, ,3)
()
(,,1) (,,3)
iNji
iNlNli
ij ij
pj
il il
ττ
ττ
=≠
==≠
+
=
+
∑
∑∑
(4)
where p(j) denotes the probability of selecting
department j as the first department. In equation (4),
only pheromones
τ (*, j, 1) and τ (*, j, 3) are
considered while calculating probability of selecting
department j. Therefore, the layout sketch is likely to
start with a department that might yield good
solutions if it is located as the first department in a
bay.
Procedure Solution_Construction()
Step 1. Set A={}, U={1,…,N} and calculate p(i) for
i=1,…,N.
Step 2. Randomly select a department i with
probability p(i) to assign to the layout sketch. Let
i
+
denote the selected department.
Step 3. Set U=U\{i
+
} and A={(i, i
+
, k): i∈U, k∈{1,
2, 3}}.
Step 4. Calculate p(i, j, k) for all (i, j, k)∈A, and
randomly select an assignment. Let (i
+
, j
+
, k
+
)
denote the selected assignment.
Step 5. Set U=U\{i
+
}, A=A\{(i
+
, j, k):(i
+
, j, k) ∈A},
A=A\{(i, j
+
, k
+
):(i, j
+
, k
+
) ∈A}, A=A∪{(i, i
+
, k):
i∈U, k∈{k
+
,2}}.
Step 6. If U≠{}, then go to Step 4.
Step 7. Create the actual layout from the sketch.
2.2 Solution Evaluation
Although the FBS representation is relaxed in this
paper, some solutions may still have departments
with impractical shapes, such as a very narrow/long
rectangular department. In addition, the width of the
layout may exceed the maximum allowed width of
the area because adjusted bay widths are wider than
regular bay widths. The ACO-RFBS uses the
maximum aspect ratio, which is defined as the ratio
of a department’s longer side to its shorter side, to
quantify the infeasibility of solutions with respect to
department shapes. Therefore, a small-sized and a
large-sized department can be penalized in the same
scale. Let α
i
(s) represent the aspect ratio of
department i for solution s and let α
i
be the given
ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications
274

maximum aspect ratio of department i. Let W(s) be
the width of the layout for solution s. Solution s is
said to be feasible if and only if α
i
(s) ≤ α
i
for each
department i and W(s)≤W. Infeasible solutions are
dynamically penalized using the near feasibility
threshold (NFT) concept (Kulturel-Konak et al.,
2004).
2.3 Local Search
After evaluating the fitness of the solutions in an
iteration, a local search attempts to improve the best
solution of the iteration, s
*
, where new solutions are
created from s
*
by swap and insert operators.
Operator swap(i, j) swaps the positions of
departments i and j. Operator insert(i, b, k) inserts
department i into the k
th
position of bay b. The insert
operator changes the relative locations of the bay
breaks in a layout. The swap and insert operators are
randomly selected in each loop of the local search
and performed for all possible combinations. If a
better solution is found, s
*
is updated, and the local
search continues until no improvements possible.
2.4 Pheromone Update, Diversification,
and Overall Algorithm Evaluation
In each iteration, μ solutions are generated as
described in the previous section, pheromone values
are updated based on the best feasible solution s
**
or
the best solution of the iteration s
*
as follows:
**
**
*
(, , )
is available,
(, , | )
(, , )
( , , ) otherwise.
(, , | )
ijk
s
ijks
ijk
ijk
ijks
ρτ
ω
τ
ρτ
ω
×+
⎧
⎪
⎪
⎪
=
⎨
⎪
×+
⎪
⎪
⎩
(5)
where
ρ <1 is the evaporation parameter and
ω (i,j,k|s) is a binary function such that ω (i, j, k | s)=1
if assignment (i, j, k) is used to construct solution s
and
ω (i, j, k | s)=0, otherwise.
During the search, if s
**
has not been updated for
a certain number of iterations, new solutions cannot
be generated. The ACO-RFBS uses a two-phase
diversification schema when the search stagnates in
such cases as follows:
*
**
*
**
if
(1, )
and ,
(, , )
if
max{0, ( , , )}
and .
s
u
s
u
gg
UN
g
g
ijk
gg
Nijk
g
g
τ
τ
>
≤
=
>
−
>
⎧
⎪
⎪
⎨
⎪
⎪
⎩
(6)
where g
u
is the number of consecutive iterations
such that s
**
has not been updated, g
s
is the number
of the consecutive iterations in which the same s
*
is
obtained, and g
*
and g
**
are diversification
parameters. Observing the same s
*
in the last g
*
iterations indicates stagnation of the search.
Therefore, all pheromone values are randomly reset
between one and N to restart the search at a different
location in the search space. If the search is
stagnated without improving s
**
in the last
consecutive g
**
iterations, the pheromone values are
reversed in the second case of equation (6). The
search is terminated after performing g
d
diversifications.
Procedure ACO-RFBS (µ,
ρ , β , g
d
, g
*
, g
**
)
Step 1. g=0, g
u
=0, g
s
=0, s
**
=∅, τ (i, j, k)=N for all i
and j, and k=1, 2, 3.
Step 2. Generate µ solutions using Procedure
Create_Solution() and calculate the fitness of
each solution.
Step 3. Identify s
*
and if s
*
is different than the best
solution of the previous iteration, set g
s
=0. Apply
the local search on s
*
if it is not equal to s
**
or s
*
of the previous iteration. If a better solution is
found, replace s
*
. Update s
**
and F
min
if
necessary. Set g
u
=0 if s
**
is updated.
Step 4. Update pheromone values using (5).
Step 5. If one of the conditions in equation (6) is
satisfied, apply diversification and set g
s
=0.
Step 6. Set g=g+1, g
s
= g
s
+1, g
u
= g
u
+1. If g
d
number
of second phase diversifications have been
performed, then stop and return s
**
; else, go to
Step 2.
3 COMPUTATIONAL
EXPERIMENTS
To compare the performance of the ACO-RFBS,
seven test problems ranging from twelve to 35
departments are used as given in Table 1. All these
problems have been previously solved in the
literature using the FBS. Additional information
about these problems as well as their best RFBS
solutions can be found in (Kulturel-Konak and
Konak, 2011b). Herein this paper, these problems
were first solved with their original dimensions
given in the literature, and then, they were solved
again with their relaxed dimensions in which the
layout widths were increased about 10% allowing
empty spaces in bays. In addition, the problems were
solved using horizontal and vertical running bays. In
problems Tam20, Tam30, SC30, and SC35,
ANT COLONY OPTIMIZATION FOR THE UNEQUAL-AREA FACILITY LAYOUT PROBLEM
275

Table 1: Properties of the test problems.
Problem Relaxed Area Best Known Reference
Best
Known
RFBS
Nug12 5×3 262.00
(Kulturel-Konak and Konak,
2011a)
257.50
Nug15 4×5 524.75 524.75
AB20(4) 35×20 5073.82
(Liu and Meller, 2007)
5336.36
Tam20 44×35 9003.82
(Kulturel-Konak and Konak,
2011a)
8753.57
Tam30 50×40 19667.45 19462.41
SC30 18×12 3679.85
(Wong and Komarudin,
2010)
3443.34
SC35 20×15 3604.00
(Liu and Meller, 2007)
3700.75
Table 2: Solutions found by the ACO-RFBS for the test problems.
Original Area
Relaxed Area
Problem Best
Imp (%) over Best-
known
Average
CPU
Sec
Best
Imp (%) over Best-known
RFBS
CPU Sec
Nug12 257.50 1.75 257.50 57 253.00 1.75 53
Nug15 524.75 0.00 524.75 120 511.50 2.53 121
AB20(4) 5336.36 -5.17 5336.36 1940 5023.23 5.87 288
Tam20 8753.57 2.86 8778.15 311 8727.45 0.30 327
Tam30 19462.41 1.05 19528.96 1881 19462.41 0.00 1669
SC30 3443.34 6.87 3499.20 1655 3259.61 5.34 1797
SC35 3700.75 -2.68 3971.76 2393 3607.60 2.52 3102
Imp (%) =Percent improvement from the previously reported best-solution.
the corresponding areas of the facilities are larger
than the total areas of the departments. The ACO-
RFBS allows empty spaces in bays by allocating the
empty spaces at the top and bottom of the bays. The
ACO-RFBS was coded in C and all runs performed
on a PC with 3.0 GHz Intel Quad-Core CPU and
32GB memory. The average CPU times in ten
replications are given in Table 2. After initial
experiments to determine the parameters, the
following parameter values were used:
μ = 50, ρ =
0.97,
β = 0.7, g
d
=3, g
*
=30, and g
**
=500.
In Table 2, ACO-RFBS results are compared to
their best-known RFBS solutions as well as their
best-known solutions. It should be noted that the
best-known solutions of several FLP test problems
(i.e., Nug12, Tam20, Tam30, and SC30) were
improved by the ACO-RFBS in this paper despite
the limitations of the FBS as stated in the
introduction section. These improvements indicate
that the proposed ACO-RFBS is effective.
Moreover, when the department widths were
relaxed, the proposed ACO-RFBS was able to
improve the best solutions for all problems
excluding Tam30. Note that such improvements may
not be achieved using the original FBS
representation. Therefore, these results demonstrate
an advantage of the RFBS over the original FBS.
4 CONCLUSIONS
In this paper, an ACO algorithm is proposed to solve
FLP with relaxed FBS and compared with the
existing methods in the literature with promising
results. With the ability of incorporating problem
specific heuristic information into the search
process, the ACO approach is well suited to
effectively solve various facility layout problems. In
this paper, it is demonstrated that the relaxed FBS
may result in a block layout with a lower material
handling cost by expanding the width of a facility.
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