The international journal of advanced manufacturing technology, tập 60, số 1 4, 2012

Int J Adv Manuf Technol (2012) 60:1–10
DOI 10.1007/s00170-011-3582-1

ORIGINAL ARTICLE

Levitation characteristics of a squeeze-film air journal
bearing at its normal modes
Chao Wang & Y. H. Joe Au

Received: 13 January 2011 / Accepted: 10 August 2011 / Published online: 8 September 2011
# Springer-Verlag London Limited 2011

Abstract A tubular squeeze-film journal bearing was
designed such that it flexed its shell at its normal modes
producing a triangular modal shape. The shell motion was
created by a single-layer piezoelectric actuator powered at
75 V AC with a 75 V DC offset and the driving frequency
coincided with the modal frequency of the bearing. The
paper provided a theory that shows the existence of a
positive pressure in a squeeze film responsible for the
levitation phenomenon. The various modes of vibration of
the tubular bearing, made from AL2024-T3, were obtained
from a finite element model implemented in ANSYS. Two
normal modes, the 13th and 23rd, at the respective
theoretical frequencies of 16.37 and 25.64 kHz, were
identified for further investigation by experiments with
respect to the squeeze-film thickness and its load-carrying
capacity. While the bearing at both modes could cause
levitation, the 13th mode has a greater load-carrying
capacity because its modal shape produced a much lower

end leakage.
Keywords Single-layer piezoelectric actuator .
Squeeze-film air bearing . Mode shape . Natural frequency .
Elastic hinge

1 Introduction
Bearings today need to be able to run at very high speed,
providing high positional accuracy for the structure that it
C. Wang : Y. H. J. Au (*)
Advanced Manufacturing and Enterprise Engineering,
School of Engineering and Design, Brunel University,
Uxbridge, UK
e-mail:

supports, and requiring very little or no maintenance. For
this to happen, bearings must have tight tolerances and very
low or zero friction during operation [1]. This pushes many
traditional contact-type bearings to their limits as they often
fail due to friction, generating heat and causing wear.
By comparison, existing non-contact bearings fare better
because of their very low or zero friction. But some have
their own problems, too. For example, the fact that
aerostatic bearings require an air supply means having to
use a separate air compressor and connecting hoses. This
makes the installation bulky. Aerodynamic and hydrodynamic bearings cannot support loads at zero speed. Both
hydrodynamic and hydrostatic bearings may cause contamination to the workpieces and the work environment
because of the use of lubricating fluid.
A potential solution to the abovementioned problems is
the new squeeze-film air bearing. It works on the rapid
squeeze action of an air film to produce separation between

two metal surfaces. This has the benefit of being compact
with a very simple configuration because it does not require
an external pressurized air supply, can support loads at zero
speed and is free of contamination.
The general theory of the operation of squeeze-film type
bearings has been documented by, for example, Salbu [2]
who provided a basic account of the principle of squeezefilm bearing. Since its publication, a variety of bearing
designs based on this principle had been proposed and
used. The squeeze-film action was created using a
piezoelectric actuator; in the 1960s, some of the ideas had
even been patented in the USA [3–6].
Scranton [11], in his 1987 patent, summarized the
limitations of the design in the 1960s as: (1) the conforming
surfaces are rigid and heavy; (2) the transducer which
drives the surface of the bearing must be correspondingly
massive; (3) the power consumption is high; (4) the noise is


2

in the audible range; (5) the oscillatory force causes
excessive vibration of the object supported by the bearing.
Scranton suggested bending the piezoelectric actuators in
order to excite a flexural vibration mode on bearing.
However, he only showed a sketch of the fundamental
concept without any implementation.
The design of the squeeze-film air bearing created by
Yoshimoto [7] in 1993 involved a counterweight and two
stack piezoelectric actuators. The disadvantages of such a
design are that the counterweight adds to the load and the

stack actuators are more expensive than their single-layer
counterparts, not to mention the higher power consumption.
Yoshimoto [7] and Storlaski [8, 9] produced designs that
used what are called ‘elastic hinges’—in order to create
localised reduction in stiffness—resulting in a greater
deformation and hence a greater variation of the squeezefilm thickness. But designing elastic hinges is a complex
matter and they cost more to manufacture. All the designs
reported in [7–9] excited the bearings at frequencies below
that of their fundamental. In 2006, Yoshimoto [10] reported
the research on a newer design in which the bearing was
driven by two piezoelectric actuators at the fundamental
frequency, at 23.7 kHz, of the bearing. When driven at this
frequency, the oscillating amplitude of the bearing plate was
significantly increased and, being ultrasonic, the bearing was
quiet in operation. Yoshimoto's work has led to the question
of whether better performance can be achieved by driving a
bearing at a modal frequency above the fundamental. This
paper attempts to explore such a possibility.
The purposes of work reported in this paper are:
1. To develop a model that affirms the existence of
positive pressure developed in a squeeze-film air
bearing and to produce an approximate working
formula for estimating the pressure;
2. To develop a finite element model for a single-layer
piezoelectric actuator that incorporates realistic boundary conditions;
3. To develop a finite element model for the squeeze-film
air bearing to study its modal shapes at various normal
modes and to identify desirable modes for acceptable
bearing performance;
4. To determine by experiments the levitation performance

of the squeeze-film air bearing at the desirable modes in
respect of the air-film thickness and load-carrying
capacity.

2 Principle of squeeze-film air bearings
Consider two parallel plates of infinite lateral dimensions
separated by a gap of h0; one of the plates oscillates
sinusoidal normal to the other at a frequency ω with an

Int J Adv Manuf Technol (2012) 60:1–10

amplitude a. If the oscillating frequency is very high and
the gap is very small, then, edge leakage of air is
insignificant, and in addition, the process can be regarded
as adiabatic. Thus,
pV g ¼ K;
where p is the pressure, V the volume, γ the adiabatic
constant equal to 1.4 for air and K the constant.
Suppose the moving plate is at the initial distance of h0
from the stationary plate, at which the air pressure in the
squeeze film is ambient, denoted as po, then
po Vog ¼ K:

ð1Þ

At time t, the plate moves to h ¼ ho þ a sinðwt Þat which
the pressure in the air film has changed (p+po), where p is
the gauge pressure; thus, the equation of state becomes
ðp þ po ÞV g ¼ K:

ð2Þ

Dividing Eq. 1 by Eq. 2 and rearranging to obtain the
pressure ratio as
 Àg
p
V
¼
À 1:
ð3Þ
p0
Vo
Since the volume is proportional to the gap height, Eq. 3
can be rewritten in terms of the ratio of gap heights, as
 Àg
p
h
¼
À 1:
ð4Þ
p0
ho
The plate moves sinusoidally such that the gap height at
time t is governed by h ¼ ho þ a sinðwtÞ, which on
substitution into Eq. 4 gives


p
h0 þ a sinðwt Þ Àg
¼

À 1:
ð5Þ
p0
ho
Introducing the non-dimensional parameters to Eq. 5,
namely the amplitude ratio " ¼ hao , and the time ratio t ¼ Tt ,
where T is the period of oscillation related to the angular
frequency ω by T ¼ 2p
w , Eq. 5 can be simplified to
p
¼ ð1 þ " sinð2pt ÞÞÀg À 1:
p0

ð6Þ

It is possible to show that the mean pressure ratio over a
cycle of oscillation is positive, which means that the
squeeze film exerts a lifting force on the plate thus causing
it to float. The proof is given below.
Using binomial expansion, the pressure ratio, Eq. 6, can
be represented by the infinite series
pr ¼

1
X
p
Àg ðÀg À 1Þ . . . ðÀg À n þ 1Þ n n
" sin ð2pt Þ:
¼
p0

n!
n¼1

ð7Þ


Int J Adv Manuf Technol (2012) 60:1–10

3

The coefficients of the terms in this series are successively -γ, -γ(−γ-1), -γ(−γ-1)(− γ-2), etc. Since γ is
positive, the sign of the coefficient alternates: negative
when n is odd and positive when n is even.
The mean pressure ratio pr is obtained by integrating,
with respect to τ, the series (7) term by term over the nondimensional time interval τ=[0, 1]. In mathematical terms,
Z
1
X
Àg ðÀg À 1Þ . . . ðÀg À n þ 1Þ n 1 n
À
sin ð2pt Þdt:
"
pr ¼
n!
0
n¼1
ð8Þ
n

For odd powers of sin (2πτ), that is, when n=2m+1,

(m=0, 1, 2, …)
Z 1
sin2mþ1 ð2pt Þdt ¼ 0:
ð9Þ
0

For even powers, that is, when n=2m,
1

sin2m ð2ptÞdt ¼

0

1:3:5 Á Á Á ð2m À 3Þð2m À 1Þ
:
2:4:6 Á Á Á ð2m À 2Þ2m

À

m¼1

Á

2m!

1:3:5 Á Á Á ð2m À 3Þð2m À 1Þ 2m
Á" :
2:4:6 Á Á Á ð2m À 2Þ2m

ð11Þ

The series (11) contains only even power terms, and so is
positive, thus confirming the existing of a levitation force in
the squeeze film whose gap oscillates at high frequency in a
sinusoidal manner.
Furthermore, since each term in Eq. (11) is positive,
Àg ðÀgÀ1ÞÁÁÁðÀgÀ2mþ1Þ
2m
g ðgþ1ÞÁÁÁðgþ2mÀ1Þ
,
and
Eq.
(11) can
2m

the factor

À
pr

¼

can be replaced by

Á

Cmþ1 ð2m þ gÞð2m þ g þ 1Þ
¼
:
Cm

ð2m þ 2Þ2
For γ=1.4, CCmþ1
< 1 for all positive integer values of m. In
m
addition, since ε2 ≤1, the series (11a) converges. The largest
coefficient is C1, whose value is 0.840, as calculated earlier;
other coefficients have values that are smaller than C1.
This suggests another infinite series which defines the
upper bound mean pressure ratio; this series is
1
P
À
C1 Á "2m and is a geometric series whose sum is
pru ¼
m¼1

pru ¼

0:840 Á "2
1 À "2

ð12Þ

Alternatively, applying numerical integration to Eq. (6) over
the non-dimensional time period τ=[0.1] with γ=1.4 and ε=
0.1 to 0.7 in 0.1 increments, the corresponding values of the
mean pressure ratio pr were obtained. The relationship
between pr and ε is as shown in Fig. 1. Also shown on the
graph is the upper bound mean pressure ratio pru calculated
from Eq. (12). It is noted that up to an amplitude ratio ε of 0.4,

the error in the mean pressure ratio pr is less than about 1.3%.

3 Modelling of the proposed squeeze-film air bearing
3.1 Configuration of the bearing

be re-written as
Figure 2 shows a squeeze-film air bearing. It consists of a
guideway and a squeeze-film air journal bearing with three

1
X
g ðg þ 1Þ . . . ðg þ 2m À 1Þ
m¼1

1:3:5 Á Á Á ð2m À 3Þð2m À 1Þð2m þ 1Þ
:
2:4:6 Á Á Á ð2m À 2Þð2mÞð2m þ 2Þ

The ratio of the two coefficients, after simplifying, is

À

1
X
Àg ðÀg À 1Þ . . . ðÀg À 2m þ 1Þ

g ðg þ 1Þ Á Á Á ðg þ 2m À 1Þðg þ 2mÞðg þ 2m þ 1Þ
ð2m þ 2Þ!
Á

ð10Þ

Substituting Eqs. 9 and 10 into (8) gives the mean
pressure ratio as
pr ¼

Cmþ1 ¼

2m!

1:3:5 Á Á Á ð2m À 3Þð2m À 1Þ 2m
Á" :
2:4:6 Á Á Á ð2m À 2Þ2m

0.9

ð11aÞ

The theoretical mean pressure ratio can be calculated
using Eq. (11a) or by performing numerical integration on
Eq. (6). However, it would be helpful to be able to use a
simpler formula for estimating the mean pressure ratio. The
following derivation shows the formula.
In Eq. (11a), the coefficient of the first term of the series
Þ 1
is C1 ¼ g ðgþ1
2! Á 2; and for γ = =1.4C1 =0.840.
Similarly the coefficient for the mth term is given by
ðgþ2mÀ1Þ 1:3:5ÁÁÁð2mÀ3Þð2mÀ1Þ
Á 2:4:6ÁÁÁð2mÀ2Þ2m ,and of the (m+1)th

Cm ¼ g ðgþ1ÞÁÁÁ2m

0.8
Mean pressure ratio

Z

term by

0.7
0.6
0.5
0.4

Exact

0.3

Approximate

0.2
0.1
0
0.1

0.3

0.5

0.7

Amplitude ratio

Fig. 1 Mean pressure ratio versus amplitude ratio—exact solution Eq.
(7) versus approximate solution Eq. (12)


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Int J Adv Manuf Technol (2012) 60:1–10

Fig. 2 Squeeze air film bearing
and its components: a guideway,
a bearing and six piezoelectric
actuators bonded to the three flat
surfaces

longitudinal flats on the circumference 120° apart. On each
flat surface were bonded two single-layer piezoelectric
actuators, Fig. 2.
The guideway, made from structural carbon steel, is a
round rod fixed at one end and free at the other with an
overhang of 130 mm; the short overhang is desired to avoid
sagging due to its own weight. The diameter of the round
rod is 19.99 mm and the surface was ground finished. The
natural frequency of the round guideway set up as a
cantilever was about 800 Hz.
The bearing, made from the material AL2024-T3, has a
diameter of 20.02 mm, a length of 60 mm and a thickness
of 2 mm. Three fins, each 20 mm long, are positioned 120°

apart on the outer circumference of the bearing. These fins
are designed to provide a desirable modal shape when
excited by actuators. The desirable modal shape is that of a
‘triangle’ (see Fig. 4 where the scale for radial displacement
is grossly magnified) in the cross-sectional plane of
maximum vibration. This enables the air gap underneath
the actuators to behave effectively as a squeeze air film.
The number of flats on the circumference of a tubular
bearing has to be odd. This way, loading in any direction
can be effectively countered by the presence of a squeeze
air film. Three flats, and hence three fins, are preferred to
five or more flats, because of the larger static and dynamic
deformation that can be produced under the same driving
conditions.
The design does not rely on complex elastic hinges to
provide local flexibility, as used by other researchers [7–9],
in order to achieve greater deflection of elements. Being a
simpler design, its manufacturing cost is much lower and
the bearing can be adequately driven by a single-layer
piezoelectric actuator (0.5 mm thick) with little power to
provide the sinusoidal squeeze-film motion [10]. Furthermore, the simple geometry of the design makes for
subsequent simpler finite-element analysis (FEA).

3.2 Experimental setup
Figure 3 shows a schematic diagram of the experimental
setup. The items of equipment used were:
1. A signal generator—0 to 15 V peak-to-peak and 0 to
100 kHz (S J Electronics)
2. An actuator driver—ENP-1-1U (Echo Electronics)
3. An actuator driver monitor—ENP-50U (Echo Electronics)

4. A capacitive displacement sensor and a gauging
module—MicroSense 6810; measurement bandwidth
up to 100 kHz; measurement ranges from 20 μm to
2 mm; resolution 0.25 nm rms at 5 kHz over 50-μm
measurement range (Ixthus)
5. A data acquisition card—PXI 6110 (National Instruments)
The signal generator created a sinusoidal wave which
was amplified by the actuator driver and shaped by the
actuator monitor to provide an excitation signal, with a
75 V DC offset and a 75 V peak-to-zero AC sinusoid. This
excitation signal was used to drive the single-layer
piezoelectric actuators. The vibration response of the
structure caused by the actuators was measured by the
capacitive displacement sensor, whose output was sampled
into a PC via the data acquisition card controlled by a
LabVIEW programme.
3.3 Modal analysis
Modal analysis can determine the theoretical vibration
characteristics, in terms of natural frequencies and mode
shapes, of a structure or a machine component. The natural
frequencies and the mode shapes are important parameters
in the design of a structure for dynamic loading conditions.
It is believed that certain mode shapes enhance the
effectiveness of the squeeze air film in journal bearings.


Int J Adv Manuf Technol (2012) 60:1–10

5

Fig. 3 Schematics of the experimental setup

These mode shapes have geometry that maximizes the
amplitude ratio ε and minimizes the end leakage.
The tubular bearing was FEA modelled in ANSYS to
establish its various modal frequencies and modal
shapes. The finite element Solid5 (3D Coupled-Field
Solid) was chosen in order to study accurately the
coupling between the piezoelectric actuator and the
bearing flat.
From the FEA modal modelling, two candidate mode
shapes were identified to have the desired geometry,
namely the 13th and the 23rd modes at the respective
theoretical natural frequencies of 16.37 and 25.64 kHz.
The mode shapes are shown in Fig. 4 where the red end
of the colour spectrum denotes greater deformation. It is
observed that:
1. Both modes produce flexing of the shell on the sleeve
between pairs of fins which remain in the same angular
orientation during the vibration; both mode shapes are
triangular.
2. At mode 13, the outer edges of the round sleeve do not
appear to deform much while the middle section
deforms noticeably (Fig. 4c).
3. At mode 23, the outer edges of the round sleeve deform
noticeably while the middle section deforms not as
much (Fig. 4d).

kind of motion cannot produce effective squeeze-film
action.

3.4 Static and dynamic analysis
3.4.1 Static analysis—computer modelling and simulation
The purpose of the static analysis was to determine the
static deformation of the sleeve bearing when a 75 V DC
voltage (0 V on the bottom and 75 V on the top surfaces of
piezoelectric layer) was applied to the six single-layer
piezoelectric actuators. Figure 5 shows the result of the
analysis, from which a maximum radial deformation of
0.124 μm is seen to occur in the middle section of the
sleeve.
The abovementioned analysis was repeated for other
driving voltages and Fig. 6 shows the relationship between
the maximum static deformation and the voltage input,
which is observed to be linear [9].
In the FEA modelling process, the force of the
piezoelectric actuators as it varies with the driving voltage
was accurately represented. This is unlike the approximations that most other researchers, for example [9] made by
assuming that a maximum blocking force exists for all
boundary conditions.
3.4.2 Dynamic analysis

To create these mode shapes, all six piezoelectric
actuators need to be driven in synchronisation at the natural
frequency of the mode shape.
From the FEA modelling, it was observed that the mode
shapes produced at other modes besides modes 13 and 23
tend to create bending or twisting of the tubular wall. This

Dynamic analysis is used to determine the dynamic
response of a structure under the dynamic excitation force.

The dynamic excitation forces are from the expansion and
the compression of the piezoelectric actuators when they
are loaded with an AC voltage (75 V) on top of a DC offset


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Int J Adv Manuf Technol (2012) 60:1–10

Fig. 4 End view of mode
shapes: a upper left—mode 13
(16.37 kHz); b upper right—
mode 23 (25.64 kHz); Side view
of mode shapes: c lower
left—mode 13; d lower
right—mode 23

(75 V). The excitation frequency should be coincident with
one of the natural frequencies for either mode 13 or 23, as
identified in Section 3.3, in order to achieve maximum
dynamic response.
A dynamic experiment was performed to verify the
bearing's natural frequencies and mode shapes at modes 13
and 23. The bearing was placed on a horizontal flat surface,
as shown in Fig. 7, and was supported at two positions near
the bottom edge. These points of contact were chosen to
coincide with the nodal points (of no displacement) of the
bearing.
According to the FEA simulation for mode 13, the fins
on the tubular bearing have translation motion. To verify

this prediction, with the bearing excited at mode 13, the
displacement amplitude at three points—top, middle and
bottom—along the length of the fin was measured with the
capacitive displacement sensor. The results confirm the
prediction from FEA simulation. Further measurements

made on the other two fins showed the same results but
stationary (zero displacement amplitude) at the both ends.
For this reason, subsequent dynamic experiments only
attempted to measure the displacement amplitude at one
single position, as shown in Fig. 7.
Figure 8 shows the dynamic deflection of a point on the
fin of the bearing as measured by the capacitive displacement sensor. Measurements were made ten times and it is
the average that is shown on the graph; the corresponding
error bar represents ±2 standard errors. The narrow extent
of the error bars suggests good measurement repeatability
and high precision of the displacement amplitude obtained.
To correctly locate the natural frequency, the actuators
were driven to excite the bearing over a range of
frequencies from 16.28 to 16.55 kHz at three different
levels of AC voltage, namely 75, 65 and 55 V. The
experimental natural frequency for mode 13, from
Fig. 8, is 16.32 kHz at which the displacement amplitude
on the fin is the greatest, for example at 75 V AC, the
displacement is 2.88 μm. Since the 75 V AC gives the
greatest displacement amplitude, which in turn produces the

Deflection Unit: µm

0.14

0.12
0.1
0.08
0.06
0.04
0.02
0
45

55

65

75

DC Unit: V

Fig. 5 Static deformation of the bearing when a 75 V DC voltage was
applied to the six actuators

Fig. 6 Static deformation varies linearly with the applied DC voltage


Int J Adv Manuf Technol (2012) 60:1–10

7

Deflection Unit: µm

2.5

2
1.5
75V AC
1

65V AC
55V AC

0.5
0
24.8

25

25.2

25.4

25.6

Frequency Unit:kHz

Fig. 9 Deflection on the fin of the bearing at mode 23 versus the
excitation frequency for the three levels of AC input

Fig. 7 Setup for the dynamic response measurement

greatest mean pressure ratio, Fig. 1, this condition was going
to be used for driving the bearing subsequently.
The experiment was repeated for mode 23. The

measurement point, in this case, was near the end of the
sleeve where the deformation is observed to be significant,
Fig. 4b. The results are shown in Fig. 9. It is noted that the
experimental natural frequency for mode 23 is 25.32 kHz
From the FEA model, the theoretical displacement
amplitude at the measurement point was also obtained for
the different driving conditions. Figures 10 and 11 show the
comparison between the theoretical and experimental
displacement amplitude at modes 13 and 23, respectively.

4 Load-carrying capacity experiments
In these experiments, the bearing was inserted into the
round guideway, set up as a cantilever with an overhanging
length of 130 mm giving a natural frequency of about
800 Hz, as mentioned in Section 3.1. As shown in Fig. 12,

a mass was hung onto a wire attached to a fin of the bearing
and the bearing was excited at a number of frequencies
around a particular mode.
The capacitive displacement sensor was positioned at the
top of the tubular bearing diametrically opposite to the
hanging weight with the capacitive displacement sensor
right over the point of maximum amplitude of oscillation as
predicted by the FEA modelling. Thus, for mode 13, the
point was at the mid-span of the bearing's length and for
mode 23, at the edge of the bearing (Fig. 4c, d). The sensor
was zeroed when the bearing was at rest on the round
guideway. As the bearing was excited by the surfacemounted piezoelectric actuators, the bearing began to float.
This resulted in a non-zero displacement output comprising
an alternating component superimposed on a mean component. This mean component is the mean film thickness; and

the minimum film thickness is the mean minus the
amplitude of oscillation. In the experiments, the minimum
film thickness was computed over a thousand cycles of
oscillations.
For mode 13, Fig. 13 shows the relationship between the
minimum film thickness and load at four different excitation frequencies at and around the natural frequency of
mode 13. At the natural frequency 16.32 kHz, the minimum
film thickness is greater than those at other frequencies

3
2.5

75 V AC

3.5

65 V AC

3

55 V AC

2
1.5
1
0.5
0
16.25

Deflection Unit: µm

Deflection

Unit: µm

3.5

2.5
2
1.5

Theory

1

Experiment

0.5

16.3

16.35 16.4
16.45 16.5
16.55
Excitation frequency Unit: kHz

16.6

0
55

60

65

70

75

AC Unit:V

Fig. 8 Deflection on the fin of the bearing at mode 13 versus the
excitation frequency for the three levels of AC input; the error bars
represent ±2 standard errors

Fig. 10 Comparison between theoretical and experimental displacement amplitude at mode 13 (DC=75 V and variable AC)


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Int J Adv Manuf Technol (2012) 60:1–10
Unit: um

2

8
Excitation frequency
=16.220KHz"

7

Experiment
Theory

1
0.5
0
55

60

65
AC Unit:V

70

75

Fig. 11 Comparison between theoretical and experimental displacement amplitude at mode 23 (DC=75 V and variable AC)

below a load of about 2 N. However, when the load is
increased beyond 2.5 N, its film thickness becomes about
the same as those at other frequencies. An explanation
could be that with increasing load through adding mass, the
natural frequency of the bearing/mass system shifts away
from its original value and so the bearing is no longer being
excited at its true natural frequency.
Figure 14 shows the relationship between the minimum
film thickness and load at four different excitation frequencies at and around the natural frequency of mode 23. At the
natural frequency of 25.32 kHz, the minimum film

thickness is also greater in the range of loads experimented.
When the minimum film thickness at modes 13 and 23
are placed side by side, the difference in values is striking,
showing that mode 13 is a far superior mode in terms of
load-carrying capacity. The comparison is made in Fig. 15
where the bearing was excited at the natural frequency of
modes 13 and 23.
According to Fig. 1, the pressure ratio increases with the
amplitude ratio. An increase in amplitude ratio corresponds
to a decrease in minimum film thickness; an increase in
pressure ratio means an increase in the load-carrying
capacity of the bearing. Therefore, it can be reasoned that
as the minimum film thickness decreases, the bearing
stiffness increases.
Fig. 12 Direction of loading by
hanging masses

Minimum air film thickness

6

1.5

16.430KHz

5

Excitation frequency
=16.320KHz

4
3

16.520KHz

2

1
0

1

1.5

2

2.5
3
Load Unit: N

3.5

4

Fig. 13 Minimum film thickness of bearing versus load at four
excitation frequencies around mode 13

When the load of the bearing was increased from 0 to
2 N, the frequency of the modal peak at mode 13 was
observed to increase from 16.32 to 16.98 kHz. Figure 16

compares the two sets of results obtained, one at the fixed
frequency of 16.32 kHz and the other at the resonant
frequency which varied between 16.32 and 16.98 kHz as
the loading changed. It is noted that the latter always
produces a greater minimum film thickness.

5 Discussion
The model of an air film between two flat plates using the
ideal gas law assuming adiabatic process proves theoretically the existence of a mean positive pressure and that this
pressure increases as the amplitude ratio, Fig. 1. It does not,
however, attempt to model the pressure leakage on the
edges the squeeze-film air bearing. Often, it is argued that
when the plates oscillate at a very high frequency, the
leakage effect can be ignored and the adiabatic process
holds true.
On the issue of the end leakage, driving the bearing
at its natural frequency particularly at higher modes is
beneficial because the natural frequency tends to be
high. The design described in the paper was operated at
modes 13 and 23 at the natural frequencies of 16.37
Minimum air film thickness Unit: µm

Deflection Unit: µm

2.5

1.4
1.2
1

Excitation frequency
25.697KHz

0.8

Excitation frequency
25.477KHz

0.6
0.4

Excitation frequency
25.383KHz

0.2

Excitation frequency
25.322KHz

0

1

1.2

1.4

1.6
1.8
Load Unit: N

2

2.2

Fig. 14 Minimum film thickness of bearing versus load at four
excitation frequencies around mode 23


Minimum film thickness Unit: µm

Int J Adv Manuf Technol (2012) 60:1–10

9

8
7
6
5
4
Mode shape 13th

3

Mode shape 23th
2
1
0

1

1.5

2

2.5
3
Load unit: N

3.5

4

Fig. 15 Comparison between modes 13 and 23 in the load-carrying
capacity

Minimum air film thickness

Unit: µm

and 25.64 kHz, respectively. This compares favourably
with the designs by Stolarski [9] and by Yoshimoto [7],
both driving their designs at a frequency lower than or at
the fundamental frequency.
On the issue of the mean pressure, according to
Fig. 1, higher mean pressure is achieved by higher
amplitude ratio, which means that the structure must be
such designed that the bearing surface can have large
displacement. The proposed design allows this to happen
with relative ease.

Between modes 13 and 23, mode 13 has superior
performance. This is because of its lower end leakage
due to the deflection geometry: both ends of the sleeve
hardly deform while the shell in the middle section of
the sleeve under the actuators are made to flex thus
creating a squeeze film. The ring of stagnant air film at
both ends of the squeeze film minimises the leakage
effect.
In the FEA modelling of the bearing sleeve, the
piezoelectric actuators are accurately represented as a
unit that expands and contracts with the driving voltage.
In addition, the interaction with sleeve as the actuator
moves is also accounted for by including the material
properties of the two parts. Consequently, the analysis is
more accurate.

8

7
6
5
Frequency without
adjustment
Frequency with
adjustment

4
3
2
1

0

1

1.5

2

2.5
3
Load Unit: N

3.5

4

Fig. 16 Relationship between the minimum film thickness and the
load applied

The static analysis shows the linear relationship
between the input DC voltage and the deformation on
the bearing. The same result was also obtained by
Stolarski [9].
Given the same driving condition, the dynamic response
is much bigger than the static response. In particular, when
driven at the mode 13 natural frequency, the maximum
displacement at the fin is roughly 3 μm. However, when
excitation frequency drifts from the natural frequency, the
amplitude falls, Fig. 8. The same conclusion can be drawn
for mode 23. The similar result was also obtained by

Yoshimoto [10], who observed vibration amplitudes of
about 1.5 μm at the excitation frequency of 23.7 kHz and at
70 V AC.

6 Conclusion
The advantage of a squeeze-film air bearing system is its
compactness because it does not require an externally
pressurized air supply system. The advantage of the tubular
squeeze-film air bearing, as reported in this paper, is its
simple design, not involving any elastic hinges, which can
be difficult to manufacture.
The findings from this research are summarized as
follows:
1. The theory developed from using the ideal gas law
shows the existence of a positive pressure in the tubular
squeeze-film air bearing that causes levitation.
2. The positive pressure at any amplitude ratio can be
estimated using the approximate formula, Eq. (12),
with an error of less than about 1.3% up to the
amplitude ratio of 0.4.
3. Two normal modes, at the 13th and 23rd, of the bearing
were identified to have the desired geometry of the
modal shape, namely that of a triangle. The
corresponding theoretical natural frequencies were
found to be 16.37 and 25.64 kHz, a result confirmed
also by experiments.
4. From the FEA analysis, the maximum radial deformation of the bearing when driven at 75 V DC was
observed to be 0.124 μm.
5. When the bearing was driven at 75 V AC with 75 V DC
offset, the displacement response was 2.88 μm (Fig. 8)

and 1.98 μm (Fig. 9) for modes 13 and 23, respectively.
The measurements were highly repeatable as is evident
from the small extent of the error bars in Fig. 8.
6. The load-carrying experiments show that when driven
at the natural frequency in either mode 13 or 23, the
squeeze-air film was the thickest. However, comparing
between the two modes, mode 13 has superior
levitation performance than mode 23 (Fig. 15) because


10

Int J Adv Manuf Technol (2012) 60:1–10

the former has a modal shape (Fig. 4a) that reduces end
leakage of the squeeze-air film.

References
1. Stolarski TA, Chai W (2006) ‘Load-carrying capacity generation
in squeeze film action’. Int J Mech Sci 48(3):736–741
2. Salbu E (1964) ‘Compressible squeeze films and squeeze
bearings’. J Basic Eng 86:355–366
3. Emmerich CL (1967) “Piezoelectric oscillating bearing,” US
patent No.3351393, no. 3351393, November
4. Warnock LF Jr (1967) “Dynamic gas film supported inertial
instrument,”US patent No. 3339421, no. 3339421, September

5. Scranton N, Robert A (1987) (South Salem, “Planar and
cylindrical oscillating pneumatodynamic bearings,” US patent
No. 4666315, no. 4666315, May

6. Farron TBR, John R (1969) “Squeeze film bearings,” US patent
No.3471205, no. 3471205, October
7. Yoshimoto S, Anno Y, Sato Y, Hamanaka K (1995) Floating
characteristics of squeeze-film gas bearing with elastic hinges for
linear motion guide. Int J JSME 60(11):2109–2115
8. Stolarski TA, Chai W (2006) ‘Self-levitation sliding air contact’.
Int J Mech Sci 48(1):601–620
9. Stolarski TA, Woolliscroft SP (2007) ‘Performance of a self-lifting
linear air contact’. J Mech Eng Sci 221:1103–1115
10. Yoshimoto S, Kobayashi H, Miyatake M (2006) ‘Floating
characteristics of a squeeze-film bearing for a linear motion guide
using ultrasonic vibration’. J Tribol Int 40(5):503–511
11. Scranton RA (1987) “Planar and cylindrical oscillating pneumatodynamic bearings”, US patent No. 4666315, May


Int J Adv Manuf Technol (2012) 60:11–27
DOI 10.1007/s00170-011-3614-x

CRITICAL REVIEW

A review of intelligent approaches for designing dynamic
and robust layouts in flexible manufacturing systems
Ghorbanali Moslemipour & Tian Soon Lee & Dirk Rilling

Received: 28 January 2011 / Accepted: 25 August 2011 / Published online: 14 September 2011
# Springer-Verlag London Limited 2011

Abstract Facility layout problem is associated with the
arrangement of facilities in a plant. It is a critical issue in the
early stages of designing a manufacturing system because it

affects the total manufacturing cost significantly. Dynamic and
robust layouts are flexible enough to cope with fluctuations
and uncertainties in product demands in volatile environment
of flexible manufacturing systems. Since the facility layout is a
hard combinatorial optimization problem, intelligent
approaches are the most appropriate methods for solving the
large size of this problem in reasonable computational time. In
this paper, first of all, dynamic and robust layouts are surveyed.
After a quick look of different mathematical models, including
quadratic assignment, quadratic set covering, mixed integer
programming, and graph theoretic models, the various solution
methods especially intelligent approaches along with their
advantages and disadvantages are surveyed. Finally, after
review of hybrid algorithms, the conclusion of this paper is
reported.
Keywords Facility layout problem . Dynamic layout .
Robust layout . Mathematical models . Intelligent
approaches

1 Introduction
A flexible manufacturing system (FMS) is formed by using at
least four automated and multifunctional machines, which are
linked together mechanically by an automated material
handling system and electronically by a distributed computer
G. Moslemipour : T. S. Lee (*) : D. Rilling
Faculty of Engineering and Technology, Multimedia University,
75450, Melaka, Malaysia
e-mail:

control system. The facility layout problem (FLP) is one of the

most critical issues in designing the FMS. The research on the
FLP has been formally started since the early 1950s. A facility
is a piece of equipment such as a workstation in a
manufacturing system or a department in an organization,
which makes it possible to produce goods or provide
particular kinds of services easily. The FLP can be defined
as an optimization problem so that facilities are assigned to
locations in such a way that the total material handling cost
(MHC) is minimized. In fact, the MHC is one of the most
appropriate measures to evaluate the efficiency of a facility
layout so that it must be minimized for an optimal layout.
According to Tompkins et al. [152], the MHC forms 20–50%
of the total manufacturing costs and it can be decreased by at
least 10–30% by an efficient layout design. In a manufacturing system, the FLP has a significant effect on manufacturing costs, work in process, lead times and productivity
[31]. In designing the FLP, two approaches can be
considered. The first one classifies the environment of the
FLP into certain and uncertain types where the problem data
such as product demands is deterministic and stochastic
respectively. The second one categorizes the FLP into static
and dynamic, which are corresponding to single period and
multi-period time planning horizons, respectively. The FLP is
a combinatorial optimization problem (COP). The COP can
be defined as an optimization problem with a finite number
of feasible solutions [160]. The COPs are nondeterministic
polynomial (NP)-complete problems so that they can be
solved in a computational time, which is exponentially
proportional to the size of the problem. There is no single
best approach to solve the FLP. It needs to select a method
according to the characteristics of the problem such as size,
linearity, and non-linearity. In general, resolution approaches

can be categorized into exact (optimal) methods, heuristic
methods, and intelligent approaches.


12

In the following sections, this paper elaborates more on:
dynamic and robust layouts, mathematical models, and
resolution approaches. This paper ends with the conclusion.

2 Dynamic and robust layouts
In the static facility layout problem (SFLP), the flow of
materials is deterministic and constant over the entire time
planning horizon. Since it is very difficult to forecast the
product demands for a long period of time, the planning
horizon is divided into several time periods (weeks,
months, or years) so that each period has different and
fixed product demand requirements. By doing so, the SFLP
becomes a multi-period layout problem named dynamic
facility layout problem (DFLP). Actually, in the DFLP, the
flow of materials is deterministic and constant for each
period, but it changes from period to period. The product
demands are usually obtained by using inaccurate techniques such as forecasting methods or historical trends.
Therefore, it would be more realistic if we consider the
product demands as stochastic variables. Considering
uncertainty in the product demands in both of the
aforementioned static (single period) and dynamic (multiperiod) FLPs leads to two stochastic FLPs called stochastic
static facility layout problem and stochastic dynamic
facility layout problem, respectively. Designing dynamic
and robust layouts are the two approaches to cope with

fluctuations and uncertainties in product demands in the
FLP. In dynamic layout design method, an optimal layout is
designed for each period of the multi-period planning horizon
so that the total material handling and rearrangement cost is
minimized. Using this approach, the layout of facilities can be
changed from period to period in accordance with changes in
product demands. Therefore, although this method has the
advantage of having optimal layout for each period, but it
suffers from the disadvantage of having the facility rearrangement cost. Actually, in the dynamic approach, considering
each period as a stage, the multi-period problem can be
considered as a multi-stage dynamic system with optimal
behaviour from stage to stage. On the other hand, in the robust
approach, a robust layout, which is not necessarily an optimal
layout for a particular time period, is designed as the best
layout over the entire time planning horizon. Unlike the
dynamic approach, the robust approach has the advantage of
lack of rearrangement cost and the disadvantage of not having
an optimal layout for each period. Rosenblatt and Lee [137]
and Kouvelis et al. [80] defined the robustness of a layout as
the number of times that the layout falls within a prespecified percentage of the optimal solution for different sets
of product demand scenarios. A review of some previous
works on dynamic and robust layouts is summarized in
Tables 1 and 2, respectively.

Int J Adv Manuf Technol (2012) 60:11–27

3 Mathematical models
There are different kinds of mathematical models to
formulate the FLP such as quadratic assignment problem
(QAP) [78], quadratic set covering problem (QSP) [14],

mixed integer programming (MIP) [73], and graph theoretic
(GT) [45] models.
The FLP with discrete representation can be modelled by
QAP when the equal-sized facilities are assigned to the
same number of known locations. In discrete representation, the shop floor is divided into a number of equal-sized
squares as facility locations so that unequal area facilities
are assigned to sets of the locations. The QAP is also an
appropriate model to design a block layout where the
relative location of each facility is determined. The paper of
Loiola and Abreu [101] surveys this model.
The FLP with unequal-sized facilities and discrete
representation can be formulated as QSP. In this problem
the entire area, which is taken up by all facilities is divided
into several smaller blocks so that each facility is assigned
to just one location and each block is considered to have at
most one facility. For more information about this model,
please refer to Ref. [89].
The MIP model consists of a linear objective function of
the mixture of integer and non-integer decision variables
subject to a number of linear equality and inequality
constraints. It is an appropriate model to formulate the
FLP with continuous representation and unequal-sized
facilities. In continuous representation, the dimensions of
facilities don’t take integer values and the facilities can be
located anywhere on the plant floor so that the real
optimum solution can be found. The MIP is also a suitable
model to design a detailed layout where the location of aisle
structure and pick-up/drop-off (P/D) points can also be
determined. A review of this model can be found in Refs.
[31, 100, 118].

In the GT model, the FLP is modelled as a graph. This
graph is formed by a number of vertices (nodes) and edges,
which represent the facilities and the adjacency of each pair
of facilities respectively. The adjacency between facilities,
which is represented by using the edge weight, is known in
advance [45]. The weights are given as an ‘adjacency
matrix’ usually named a ‘relativity chart’ or a ‘relationship
chart’. In a graph network without any loop, the elements
(weights) of the main diagonal of the adjacency matrix are
zero. The edge weights can also be either the benefit or the
cost of two adjacent facilities. If they represent benefit, then
the objective is to obtain an arrangement of facilities so that
the total benefit is maximized. It is impossible to find an
optimum solution for a FLP with unequal-sized facilities
even in small size by using the graph model similar to the
QAP models [115]. For more information and review of
this model, please refer to Refs. [18, 44, 57, 59].


Int J Adv Manuf Technol (2012) 60:11–27
Table 1 Review of dynamic
layout

DP dynamic programming,
CRAFT computerized relative
allocation of facilities technique,
QAP quadratic assignment
problem, MIP mixed integer
programming, EFD expected
flow density, DHOPE dynamic

heuristically operated placement
evolution, DFBC dynamic from
between chart, W-W Wagner–
Whitin, GA genetic algorithm,
TS tabu search, ACO ant colony
optimization, CSA clonal
selection algorithm, DEA data
envelopment analysis

13

Authors (year)

Approach

Rosenblatt [136]

DP and CRAFT

Kouvelis and Kiran [79]
Balakrishnan et al. [10]

DP
QAP

Palekar et al. [125]

QAP

Montreuil and Laforge [119]

McLean et al. [114]
Drolet [32]

Scenario tree
Virtual layout

Irani et al. [68]
Venkatadri et al. [158] Montreuil et al. [121]

Hybrid layout
Fractal layout

Urban [155]
Lacksonen [90]

CRAFT and steepest-decent pair-wise interchange
QAP and MIP

Urban [156]

QAP

Yang and Peters [162]

EFD

Balakrishnan and Cheng [6]
Kochhar and Heragu [76]

A comprehensive survey on DFLP

DHOPE

Balakrishnan et al. [8]

Dynamic pair-wise exchange heuristic and DP

Chang et al. [20]
Krishnan et al. [81]

Symbiotic evolutionary algorithm
DFBC-improved W-W and GA

Kulturel-Konak et al. [87]
Kulturel-Konak [85]

TS
Survey

Krishnan et al. [82]
Chen and Rogers [21]
Berna and Attila [17]
Bashiri and Dehghan [11]

Mathematical models

In general, the above-mentioned mathematical models
can be formulated as P=(S, Ω, f) where S is a solution space
defined over a certain number of discrete decision variables
Xi (i=1,…,n), Ω is a set of constraints in terms of the
decision variables, and an objective function f: S→R+,

which can be either maximized or minimized [105]. A
feasible solution s ∈ S can also be considered as the value
of the decision variable that meets all constraints in Ω. In
minimization problem, the solution s* ∈ S is an optimum
solution if and only if f (s*)≤f (s), ∀s ∈ S. The solution with

ACO
CSA
DEA

the best objective function value in comparison with its
nearby points or over the entire feasible solution space is
named a local or global optimum solution, respectively. In
FLP, the decision variables are the solution of the problem
so that they determine the location of the facilities. The
objective functions can be minimization of space costs,
handling costs, rearrangement costs, total flow distance,
backtracking and bypassing, traffic congestion and shape
irregularities. Maximizing the number of in-sequence
movements and closeness ratio can also be considered as

Table 2 Review of robust layout

B&B branch and bound, CLT
central limit theorem

Authors (year)

Approach

Kouvelis et al. [80]
Montreuil et al. [120]

QAP–B&B
Holonic layout, different machines are spread over the shop floor

Benjaafar and Sheikhzadeh [16]
Azadivar and Wang [5]
Smith and Norman [144]
Aiello and Enea [3]
Kulturel-Konak et al. [86]
Enea et al. [35]
Braglia et al. [19]
Norman and Smith [123]
Irappa and Madhusudanan [69]

Duplicating the same facilities
GA
GA-CLT
Fuzzy model
TS—minimizing the region under the total MHC curve
Fuzzy model
Average of the flow of materials between facilities
CLT—minimizing the region under the total MHC curve
QAP—average of the deterministic product demands for each period


14

the other objectives. A multi-objective problem includes

one or more of these objectives. The constraints can include
area constraints (space allocated and facilities location),
position constraints (the clearance between facilities,
orientation, P/D points and non-overlapping) and budget
constraints.

4 Solution methods
The FLPs can be solved by using the following methods:
4.1 Exact methods
Exact (optimal) algorithms are useful approaches to find an
optimal solution for the small-sized FLPs. They include
branch and bound, cutting plane, and dynamic programming
(DP) algorithms described as follows:
4.1.1 Branch and bound
Branch and bound (B&B) solves a problem in such a way
that at each iteration, the current problem is branched into
smaller sub-problems. The branches with non-improving
solution or infeasible solution are pruned. Finally, when all
branches have been pruned, the optimal solution (if any) is
found. The first two B&B solution methods were proposed
by [51, 92]. This method can be used to solve the smallsized FLPs (up to 16 facilities) formulated by the QAP in a
reasonable computational time [163]. Tavakkoli-Moghaddam
et al. [149] concluded that this algorithm cannot solve the
inter and intra-cell layout problem including more than ten
parts, nine machines and three cells in a reasonable
computational time.
4.1.2 Cutting plane
Cutting plane algorithm was proposed by Bazaraa and
Sherali [15]. This algorithm adds a constraint to the model
to cut off the continuous regions. The FLP formulated by

QAP with maximum 25 facilities can be solved optimally
by cutting plane algorithms [104]. Lacksonen and Enscore
[91] found that the cutting plane algorithm has the best
performance in comparison with computerized relative
allocation of facilities technique (CRAFT), B&B and DP
algorithms for solving the DFLP modelled by the QAP.
Lacksonen [90] also used the cutting plane algorithm to
solve the DFLP formulated by QAP.

Int J Adv Manuf Technol (2012) 60:11–27

large number (M!)T of possible solutions must be evaluated
to find the optimum solution. For example, for a DFLP with
six facilities and five periods, very large possible solutions
(1.93×1014) must be evaluated. Therefore, it can be used
only to solve the small-sized problems. Rosenblatt [136]
solved the DFLP with six equal-sized facilities and five
periods by using DP and CRAFT. Kouvelis and Kiran [79]
solved the stochastic DFLP by using DP. Urban [154]
assessed the performance of different lower-bound
approaches for the DP to solve the DFLP. According to
Balakrishnan and Cheng [6], DP uses a recursive relationship to solve the problem. Balakrishnan et al. [8] combined
Urban’s method with DP for solving the DFLP. In DP, each
period in the planning horizon forms a stage and each static
layout forms a state.
4.2 Heuristic algorithms
Heuristic algorithms are sometimes called sub-optimal or
approximated approaches. They can also be named as
computerized layout algorithms [46]. Heuristic methods are
suitable for solving the FLPs with high and low flow

dominance respectively. These algorithms can produce a
good-quality solution for the FLPs with equal and unequalsized facilities in a very low computational time [89]. A
review of heuristic methods can be found in [89, 115]. In
general, heuristic algorithms are categorized into construction and improvement (local search) algorithms as follows:
4.2.1 Construction algorithms
Construction algorithms such as plant layout analysis and
evaluation techniques, computerized relationship layout
planning and automated layout design programme produce
a single solution from scratch in such a way that they select
and locate one facility at a time alternatively. This
procedure continues until all facilities are located and the
layout is completed. The objective of the algorithms can be
considered as minimizing the total cost of materials flow
between the facilities. These algorithms are the fastest
heuristic algorithms [11]. The construction algorithms are
regarded to be the simplest and the oldest heuristic methods
to solve the QAP models from a conceptual and implementation standpoint, but they do not generate solutions
with reasonable quality [142]. A computational analysis
provided by Liggett [98] shows that it is possible to find
better solutions at low cost by using these algorithms.
4.2.2 Improvement algorithms

4.1.3 Dynamic programming
DP algorithm is usually used to solve the DFLP with M
facilities and T time periods. It can be proved that a very

Improvement algorithms such as CRAFT, computerized
facilities design and REVISED HILLER need a randomly
produced initial solution, which is improved by using some

Int J Adv Manuf Technol (2012) 60:11–27

approaches such as pair-wise or multi-pair-wise exchanges.
Urban [155] suggested a CRAFT-equivalent heuristic
method according to the steepest-decent pair-wise interchange approach with less computational time than the DP
and QAP models. An extension of CRAFT named CRAFTM was proposed by Heragu [61] to solve re-layout
problems such as DFLP in regard to the fixed and variable
rearrangement costs.
4.3 Intelligent approaches
The intelligent or meta-heuristic algorithms have incremental ability to solve a variety of hard COPs such as the FLP
with discontinuous, non-differentiable, stochastic, or highly
non-linear objective functions by finding very good-quality
solutions in reasonable computational time. ‘A metaheuristic is a set of algorithmic concepts that can be used
to define heuristic methods applicable to a wide set of
different problems’ [29]. The following papers are good
surveys on the intelligent approaches for solving the FLP:
[31, 60, 85, 108, 142]. The intelligent approaches including
genetic algorithm (GA), tabu search (TS), simulated
annealing (SA), ant colony optimization (ACO), artificial
immune system (AIS), greedy randomised adaptive search
procedure (GRASP), particle swarm optimization (PSO),
expert systems (ES), fuzzy systems (FS) and artificial
neural networks (ANN) are described in the following
subsections:
4.3.1 Genetic algorithm
The natural process of evolution in human beings is
simulated in evolutionary algorithms [48]. One of the best
kinds of these algorithms is GA, which is developed by
Holland [63]. GA starts with a population of randomly

generated initial solutions named chromosomes. Each
chromosome consists of genes, which are usually represented by binary digits. The initial population evolves
through successive iterations into an optimal solution. Each
iteration (generation) of this algorithm is formed by four
stages, including selection, evaluation, crossover and
mutation. Using the selection procedure, a group of
individuals (chromosomes) from the current population
are selected at random as parents to generate the children
(offspring) for the next generation. Using the evaluation
procedure, the chromosomes are evaluated by using their
objective function ( fitness) values. The chromosomes with
higher fitness value have higher likelihood to be selected.
New chromosomes (children or offspring) are made by
either combining two present chromosomes using the
crossover procedure or improving a single chromosome
using the mutation procedure. The individuals in the current
population with the best fitness values automatically

15

survive to the next generation as elite children. The new
generations are made by choosing some chromosomes and
rejecting some others in order to prevent changes in the
population size. Finally, the best chromosome (solution) is
obtained after several iterations.
Conway and Venkataramanan [24] used the GA, which
was better than the DP, to solve the DFLP by considering a
budget constraint on shifting costs. Balakrishnan and
Cheng [7] developed a nested-loop GA to solve the
large-sized problems by using a point-to-point cross over

to expand the search space. In this approach, the
population diversity is increased by using ‘mutation’ and
‘generational replacement’. Azadivar and Wang [5] proposed an approach using GA to design the best layout for
the stochastic FLP by considering the product demands
and routings as random variables. Smith and Norman
[144] minimized a statistical percentage of total MHCs to
design a block layout in a manufacturing system by
considering uncertain product demands and unequal area
facilities. Finally, they used GA to solve the problem.
Ficko et al. [42] solved a multi-row unequal-sized FLP in
FMS by using GA so that the optimal number of rows was
also found. Hu et al. [65] used GA to solve a cell layout
problem by considering material handling system and cells
with fixed shape and known P/D points simultaneously.
Wu and Ji [161] proposed an improved GA to solve the
NP-hard QAP by balancing the convergence of the
searching process and the population diversity. The
average distance between individuals in a population is
named diversity that enables the algorithm to search in a
larger solution space. Krishnan et al. [82] proposed three
mathematical models for designing a facility layout in an
uncertain environment by considering multiple product
demand scenarios. Finally, they solved their models by
GA. Krishnan et al. [83] developed a three-stage riskbased approach to design a FLP by considering the
forecasted product demand as a random variable. They
also used GA to solve the FLP in both of the first and the
third stages.
4.3.2 Tabu search
TS was proposed by [52] and starts with an initial solution
»

s0 as the best current solution s (i.e. s0 ¼ s ¼ s ). At each
iteration, a new solution s′ is produced during a local search
process in the neighbourhood of the current solution s. If
the solution s′ is better than the current solution s, then, it is
»
considered as the best current solution (i.e.s0 ¼ s ¼ s ). In
*
order to find the optimal solution s , the just found
solutions, which are ‘taboo’ and forbidden to be visited,
are stored in a ‘tabu list’, including long-term and shortterm flexible memories. The number of these taboo
solutions is named the memory (tabu list) size. For keeping


16

the size of the tabu list constant, the oldest member must be
removed from the list. The above-mentioned instructions
are repeated until termination criterion is fulfilled.
Kaku and Mazzola [72] presented a TS approach for
solving the DFLP. Kulturel-Konak et al. [86] considered
uncertainty in production volume and flexibility in
product routing due to changes in product design or
product demand in the stochastic FLP simultaneously.
They also found flexible bay structured layouts by using a
TS solution method. Marvin et al. [107] compared the
performance of TS, SA, and GA on different types of the
FLP under three conditions viz time limited, solution
limited and unrestricted. They found TS as the best in
most cases. McKendall [112] proposed three TS algorithms, including the simple basic TS, adding diversification and intensification strategies to the simple basic TS,
and the probabilistic TS for solving the dynamic space

allocation problem. Liang and Chao [96] developed a
multi-searching TS algorithm for solving the FLP by using
the efficient strategies based on intensification and
diversification approaches. Surya [148] proposed an
improved tabu search algorithm, including three levels
viz intensification, reconstruction, and solution acceptance for solving the FLP modelled as the QAP formulation. Scholz et al. [141] proposed a TS approach
represented by the slicing tree, including four kinds of
neighbourhood moves to solve an unequal-sized FLP.
McKendall and Hakobyan [113] solved a continuous
large-sized DFLP with unequal-sized and free orientation
facilities using a boundary search technique to generate
the initial solution improved by a TS approach. Samarghandi
and Eshghi [139] solved the single row layout problem with
unequal-sized facilities by using a TS with an adaptive
memory for intensification and diversification approaches,
which are used to find solutions in the neighbourhood of a
good solution and the solutions that have not found yet,
respectively. These approaches use the long-term memory in
the tabu list.
4.3.3 Simulated annealing
SA is a simulation of physical annealing process of
solids in statistical mechanics that has been used to
solve hard COPs since the early 1980s. Metropolis et al.
[116] suggested a useful technique to simulate the thermal
motion of atoms during a cooling process. Kirkpatrick, et
al. [75] proposed the first SA algorithm by generalizing
the metropolis’s approach and replacing the atom’s energy
with the cost function. In physical annealing process in
thermodynamics, the perfect structure of crystals can be
obtained by melting a solid and then reducing the

temperature very slowly so that the crystal can reach this

Int J Adv Manuf Technol (2012) 60:11–27

minimum energy level named ground state. Comparing the
COP with the physical annealing process shows that the
solution space, the optimal solution and the objective
function in this problem are equivalent of the state space,
ground state and atom’s energy in physical annealing,
respectively.
SA starts with a known or randomly generated initial
solution s0. This algorithm consists of two loops, including
inner loop and outer loop. At each iteration of the inner loop,
a solution (e.g. s′) in the neighbouring area of the best current
solution (e.g. s), which is obtained in the previous iteration, is
generated by using a local search technique such as random
descent pair-wise exchange method. This solution (e.g. s′) is
evaluated by the objective (cost) function f. It is accepted as
the current best solution if f (s′)≤f (s). In the case of f (s′)>f
(s) it is also accepted if x ∈ (0,1)≤P, where x is a randomly
generated number and P ¼ expðð f ðsÞ À f ðs0 ÞÞ=Tel Þ is the
probability of accepting this non-improving neighbouring
solution (e.g. s′). The outer loop starts with a high value of a
control parameter named temperature. At each iteration of
this loop or in other words, at each temperature, the inner
loop must be repeated until the system reaches the steady
state or thermal equilibrium. The temperature is reduced
gradually while the outer loop of the algorithm is repeated.
Therefore, P is high at the initial stages of the algorithm (Pin ≈
1) and it approaches to a very small value at the final stages

of the algorithm (Pf ≈ 0). The current value of the temperature
in the iteration el of the outer loop is calculated using Tel ¼
T0 ael (el=0,1…,elmax) where, α ∈ (0.80, 0.99) is the cooling
ratio, elmax denotes the maximum number of iterations of the
outer loop, and T0 denotes the initial value of the temperature,
which can be calculated as T0 ¼ À0:1f ðs0 Þ= lnð0:25Þ
Mckendal et al. [111]. There is no fixed rule to determine
the initial value of temperature. A very high initial temperature leads to wasting computational time while a very low
value decreases the quality of the final solution. If the
temperature goes down very quickly, the search might be
fixed in a worse local optimal solution. On the other hand, if
the cooling process is too slow, the algorithm might not have
converged when it gets to the stop measure. In addition to
elmax, some other termination criteria for the algorithm are
minimal temperature, no improvement for a number of
iterations, etc.
A comprehensive survey on SA-based algorithm for
solving single and multi-objective optimization problems
can also be found in [147]. Abdelghani [1] proposed the
general SA for solving the intra-cell layout problem in the
cellular manufacturing system (CMS). Leonardo et al. [94]
proposed a SA-based algorithm for solving the FLP by
regarding facilities shape and area, input/output points,
and orientations in a continual plane. Baykasoglu and
Gindy [13] proposed a SA algorithm to solve a DFLP with


Int J Adv Manuf Technol (2012) 60:11–27

17

the same shaped and equal-sized facilities. They proved
that this algorithm is more efficient than GA algorithms
from computational time standpoint. McKendall et al.
[111] solved the DFLP by using two SA approaches. The
first approach is a direct version and the second one is the
improvement of the first approach by adding look-ahead/
look-back procedure. Ashtiani et al. [4] proposed a multistart SA to solve the DFLP with regard to product mix and
demand. Dong et al. [28] suggested a SA for solving the
DFLP with capability of removing/adding facilities in
different periods, which is converted to a shortest path
problem by using some rules. Sahin et al. [138] developed
a SA for solving the DFLP by considering budget constraint.

constant and Lk represents the total distance (tour length)
for the ant k.
t ij ðt þ nÞ ¼ r:t ij ðtÞ þ Δt ij

Δt ij ¼

m
X

Δt ij k

ð1Þ

ð2Þ

k¼1

Δt kij

¼

8
Q
>
>
>
< Lk
>
>
>
:

if ant k uses edge ði; jÞ in its
tour in the time interval ðt; t þ nÞ

0

ð3Þ

otherwise

4.3.4 Ant colony optimization
ACO algorithm takes inspiration from the social behaviour
of real ants to find the shortest path from the nest to the
food source. As the ant moves along a randomly selected
path, it lays a volatile value of a chemical substance named

pheromone on the path. Using the smell of the pheromone
as an indirect communication named stigmergy, which is
proposed by the French zoologist [53], the other ants follow
the path and thereby, the amount of pheromone on the path
is increased. Finally, the ants find the shortest path from the
nest to the food source. Different types of the ACO
algorithms have been proposed such as ant system (AS),
elitist AS, ant-q, ant colony system, max–min ant system
(MMAS), rank-based AS, ants, best–worst ant system, and
hyper-cube AS [105].
AS algorithm proposed by Corne et al. [25] is the first
type of ACO algorithms that has been used to find the
shortest tour in the travelling salesman problem (TSP),
which contains n known cities, so that each city to be
visited just one time. According to Marco et al. [105, 106],
n
P
at each iteration of the algorithm, each of the m ¼
bi ðtÞ
i¼1

ants moves from city i at time t to city j at time t +1, where
bi(t) (i=1,…,n) denotes the number of artificial ants in city i
at time t and m is the total number of ants in the problem.
The tour of each ant is completed during n iterations named
a cycle, which its time interval is equal to (t, t+n). The
cities, which have already been visited, are stored in the
tabu list tabuk to prevent revisiting. The trail intensity is
updated according to the Eq. (1) where, the coefficient ρ
stands for the amount of trail after evaporation in the time

interval (t, t+n), τij(t) is the trail intensity on edge (i, j) at
time t, ρ<1, and the initial trail intensity τij(0) is a small
positive number. Δτij can be defined as Eq. 2 where, Δτijk
represents the trail value, which is deposited on for each
unit of length of the edge (i, j) by the ant k in the time
interval (t, t+n). Δτijk is given by Eq. 3 where, Q is a

The ant k moves from the current city i to the nearest city
j by using the edge (i, j) with the highest trail intensity τij (t)
and visibility hij ¼ d1ij (dij is the length of the edge (i, j)) with
a probability as given in Eq. 4 where, N is the set of cities
and the parameters α and β indicate the significance of trail
against visibility.
8 Â
à a  Ãb
>
t ij ðtÞ hij
>
>
< PÂ
Ãa  Ãb if j; k 2 fN À tabuk g
t ij ðtÞ hij
Pijk ðtÞ ¼
>
k
>
>
:
0
otherwise

ð4Þ

The algorithm is repeated until either the number of
tours (tour counter) reaches to a user defined value
(stagnation performance) or the same tour is made by all
ants.
Corry and Kozan [26] solved the DFLP with fixed shape
an unequal-sized facilities by using an ACO algorithm,
which is better than the reduced integer programming
approach. Solimanpur et al. [146] modelled the inter-cell
layout problem by using QAP and solved it by their
proposed ACO approach. McKendall and Shang [110]
introduced three different types of ACO approaches to
solve the DFLP. The first one (HAS I) is originated from the
HAS–QAP, which is proposed by Gambardella et al. [47],
with suitability for the DFLP. The second one (HAS II) is a
combination of the HAS I and SA meta-heuristic
approaches. Finally, the third approach (HAS III) adds the
forward/backward approach to the pair-wise exchange
heuristic method. Baykasoglu et al. [12] used an ant colony
solution method to solve a DFLP by considering the
unconstrained and budget constrained cases. Chen and
Rogers [21] developed two models for a dynamic multiobjective layout problem, including the distance-based
(quantitative) and the adjacency-based (qualitative) objectives. They used the HAS–QAP, which was proposed by
Gambardella et al. [47] for solving this problem. Solimanpur


18

et al. [145] proposed an approach based on ACO to solve the
cell formation problem in the CMS. Komarudin, and Kuan
[77] developed a kind of ACO algorithm named AS, which
is represented by a slicing tree, for solving unequal area
FLPs. Ning et al. [122] proposed a new solution approach by
using a continuous dynamic search method for the MMAS
algorithm for solving a dynamic multi-objective problem
named construction site layout planning. The two papers,
which survey the ACO on the COPs and the FLPs, are
Thangavel et al.’s [151] and Phen and Kuan’s [126],
respectively.
4.3.5 GRASP algorithms
GRASP algorithm consists of two separate stages, including
randomised construction stage to produce an initial feasible
solution from scratch and local improvement stage to search
better solutions in the neighbourhood of the initial feasible
solution. The best solution obtained at the end of all
iterations is kept as a final solution. At each iteration of the
construction stage, one component of the incomplete initial
feasible solution is randomly selected from a restricted
candidate list and it is added to this solution. The restricted
candidate list includes a number of the best candidate
components, which are selected amongst the initial components set, according to their suitability by using a greedy
function. This function shows an increase in the objective
(cost) function for each component, which is added to the
incomplete initial feasible solution. Therefore, the best
candidate components are the ones, which lead to minimum
increase in the cost function. This list is updated during the
iteration of the construction stage. The iteration of this stage
continues until the initial solution is constructed.

Feo and Resende [41] were the first, which solved the
QSP by using the GRASP. For the first time, it was also used
to solve the QAP in 1994 by Li et al. [95]. Urban [156]
proposed a solution approach to solve the large size of the
DFLP by using GRASP and an initialized multi-greedy
procedure. Urban et al. [157] solved the machine layout
problem modelled by the QAP without needing for locating
machines in functional or cellular layout. Abdinnour and
Hadley [2] generated an initial solution for solving a multifloor FLP by using the GRASP and a non-linear optimization approach. Then, the initial solution is got better by using
the interchange and shift techniques in the TS algorithm. The
papers from Feo and Resende [40], Pitsoulis and Resende
[127] and Resende and Ribeiro [133] can be referred as good
surveys on GRASP.
4.3.6 Particle swarm optimization
PSO was proposed by Kennedy and Eberhart [74]. It is a
swarm intelligence technique like ACO and a randomly

Int J Adv Manuf Technol (2012) 60:11–27

generated population based like GA. Considering the
behaviour of a group of birds that are randomly searching
for the only food in an area can help us to understand how
the algorithm works. Each bird named ‘particle’ can be
regarded as a single possible solution of the optimization
problem and the food corresponds to the global optimal
value of the objective function called the fitness value. This
algorithm is started with a randomly generated population of
particles as the initial potential solutions, which are improved
during several iterations (time steps) to obtain the optimal
solution. Each particle in a multidimensional search space

has its own velocity, which directs the particle around the
search space by modifying its position in the area. In the
birds flocking scenario, each bird (particle) finds the food
(the global optimal solution) in such a way that it follows the
nearest bird to the food (current optimal solution). Using this
scenario and considering the objective function (fitness)
value as a measure to evaluate the solutions, PSO uses pbest
and gbest fitness values to update the particles’ velocity in
the next iteration. The personal best ( pbest) is the best
fitness value, which a particle has achieved personaly so far.
The global best ( gbest) is the best fitness value, which has
found so far by any particle in the population. Instead of
using the gbest value, the local best (lbest) value, which has
been found so far by any particle in its neighbouring area,
can also be considered. According to Rezazadeh et al. [134]
and Wang et al. [159], in a PSO including m particles, the
position and velocity of the particle i at iteration t are
updated by using Eqs. 5 and 6, respectively. The ddimensional vectors Xit ¼ ðxti1 ; xti2 ; . . . ; xtid Þ and Vit ¼
ðvti1 ; vti2 ; . . . ; vtid Þ (i=1, 2…,m) represent the positional and
‘flying’ velocity coordinates of the ith particle at iteration t.
The positional coordinates of particle i associated with
its pbest and gbest fitness values at iteration t are
also represented byPit ¼ ðpti1 ; pti2 ; . . . ; ptid Þ, and Pgt ¼
ð ptg1 ; ptg2 ; . . . ; ptgd Þ, respectively.

vti

¼

þ c1 r1 ðptÀ1

À xtÀ1
wvtÀ1
i
i
i Þþ
tÀ1
c2 r2 ðptÀ1
g À xi Þ

xti ¼ xtÀ1
þ vtÀ1
i
i

!
ð5Þ

ð6Þ

In Eqs. 5 and 6, t represents the iteration (time step)
number, ω is the inertia weight, r1, r2 are random numbers
uniformly distributed in [0, 1], and c1, c2, which are usually
equal to 2, are the cognition learning factor and social
learning factors respectively. In order to prevent the
particles’ velocities grow to infinity, the suitable values of
ω, c1 and c2 must be chosen [23]. For more information
about the algorithm and its application, the good surveys on
PSO approach presented by Engelbrecht [36] and Poli et al.

Int J Adv Manuf Technol (2012) 60:11–27

[128] can be referred. Rezazadeh et al. [134] proposed an
extended discrete particle swarm optimization algorithm for
solving the DFLP. They used a SA algorithm having only
the outer loop named semi-annealing approach as an
effective local search technique.
4.3.7 Artificial immune system
The biological immune system protects the human body
against foreign invaders such as viruses and bacteria called
antigens. The molecules named antibodies, which recognize
the presence of an antigen, are rapidly increased by cloning
during the clonal selection process. The affinity of the new
cloned antibodies is improved by mutations, which in turn,
leads to neutralization and elimination of the antigen. The
probability of selecting antibodies for mutation is proportional to their affinity to the antigen. After mutation, the
receptor editing process is started by eliminating some
percentage of the ineffective antibodies and introducing the
same percentage of the new ones. The simulation of the
biological immune system leads to developoment of a new
intelligent algorithm named AIS. The AIS starts with a
randomly generated population of individuals (antibodies)
as the possible solutions. At each iteration of this algorithm,
first, the affinity of each antibody is calculated by using the
objective function of the problem. Next, a number of
antibodies with the best affinity value are selected and
cloned. Then, each clone is mutated and the improved
antibodies are preserved for next generation. Finally, using
the receptor editing process a pre-specified number of
antibodies with low affinity value are replaced with the new

ones, which are generated at random. Berna and Attila [17]
proposed a kind of AIS named clonal selection algorithm
for solving the DFLP. Satheesh Kumar et al. [140] proposed
an AIS algorithm to solve the unidirectional loop layout
problem by minimizing the total congestion of all parts and
minimizing the maximum congestion amongst a part
family.
4.3.8 Expert system
Artificial intelligence (AI) tries to teach machines the
attributes of human being intelligence. Intelligence means
the ability of learning, understanding, and thinking about
things. One of the most important branches of the AI is the
ES which can be used to solve problems by using the
knowledge obtained from human experts. ES can be
regarded as a computer programme, which makes a
decision like a human expert. User interface, working
memory, knowledge acquisition, knowledge based and
inference engine are the basic modules of the ES.
To solve the FLP, ‘intelligent facility layout planning and
analysis system’ (IFLAPS) has been developed by [88]. This

19

system consists of two major units. The first unit is an ES,
including three types of assignment rules, which are used to
determine the adjacent facilities. The second one is a pattern
recognition system includes some production rules to identify
the first facility, which must be assigned in the shop floor.
Some other ESs which have been developed for designing
the FLPs are facilities design expert system [43], multiple

criteria decision making (MCDM) [103], knowledge-based
machine layout [62], knowledge-based decision support
system [58] and a neuro-based expert system [22]. The
paper written by Liao [97] can be a good review on the ES.
4.3.9 Fuzzy system
The classical binary logic based on the two values 0 (false) and
1 (true) is not sufficient to describe human reasoning. Instead,
the fuzzy logic, which uses the whole interval between 0 (false)
and 1 (true), can be used for describing human reasoning. The
fuzzy set theory was originally proposed by [165]. A fuzzy
decision-making system is formed by four major components,
including fuzzification, knowledge base, decision rules, and
defuzzification. In the fuzzification process, the crisp values of
input and output variables are transformed into linguistic
variables. The knowledge base includes the membership
functions according to the experts’ decision making. The
decision rules contains a number of ‘IF–THEN’ fuzzy rules
with the connective ‘AND’. Finally, in the defuzzification
process, the fuzzy outputs are transformed into crisp values by
using the centre-of-area method [166].
The fuzzy logic is an easy way to cope with uncertainties.
For example, in the FLP, closeness ratio between facilities is
often described by linguistic words such as ‘absolutely
necessary’, ‘very important’, and ‘undesirable’. For more
details about fuzzy systems the comprehensive book written by
George and Yuan [50] is introduced. Grobelny [54, 55] solved
the FLP with the same number of facilities and locations so
that the total MHC is minimized. They used the linguistic
variables ‘high’ and ‘near’ to represent the values of the flow
of materials between facilities and distance between locations

respectively. In other words, when the flow of materials
between facilities is ‘high’, it is better to place the facilities
‘near’ to each other in order to minimize the MHC. Evans et
al. [39] designed a block layout with unequal-sized facilities.
They used the fuzzy linguistic variables to express the
qualitative relationships of ‘closeness’ and ‘importance’
between facilities. Raoot and Rakshit [131] proposed a fuzzy
approach to design an optimal facility layout. They used the
linguistic variables to express the qualitative and quantitative
parameters. Raoot and Rakshit [132] proposed a MCDM by
using linguistic variables for defining the relationship
between facilities in the multi-objective FLP. Gen et al. [49]
suggested a multi-objective fuzzy model for a cluster layout
problem with unequal-sized facilities. Dweiri and Meier [34]


20

solved the discrete FLP by considering the flow of materials
and the information flow as fuzzy variables. They used the
analytical hierarchy process to develop an activity relationship chart for designing better layouts. Grobenlyn [56]
proposed a fuzzy method to deal with uncertainties in product
demands in the FLP. Aiello and Enea [3] minimized the total
MHC in a single row layout problem by considering limited
capacity of production for each facility and stochastic
demands, which are described by fuzzy numbers. Deb and
Bhattacharyya [27] proposed a fuzzy decision support system
to design the FLP by considering P/D points. They defined
the flow of material and information as linguistic variables.
4.3.10 Artificial neural networks

Simulation of biological neural networks has been led to a
computational approach in the AI named ANN. In fact, the
ANN is a simplified model of human brain, including a group
of artificial neurons, which is represented by nodes. The nodes
are connected together by using weighted links. In this
network, each neuron receives input signals from the environment or other neurons. The influence of each input is
represented by its weight value. A weighted sum of the input
signals is computed and compared with a threshold by the
neuron. If it is bigger than the threshold, the output of the
neurone is fired and transmited to the other neurons in the
network, otherwise it is not fired. The neural network can be
trained by adjusting the weights by making a comparison
between its output and a target. Mcculloch and Pitts [109]
developed the first model of a simple ANN by using an
electric circuit. Solving COPs by using ANNs has been
started with the first neural network, which is proposed by
Hopfield and Tank [64] to solve the TSP. Tsuchiya et al. [153]
proposed a near-optimal parallel approach based on a twodimensional neural network for solving the FLPs. For more
information about different methodologies and applications of
ANNs, the paper written by Liao [97] can be referred.
The advantages and disadvantages of the abovementioned intelligent approaches are given in Table 3 [37,
48, 66, 84, 129].

5 Hybrid algorithms
The combination of various solution approaches named hybrid
algorithms has been proposed to solve FLPs. Huntley and
Brown [67]combined a high-level GA with SA to obtain a
hybrid algorithm. Mahdi et al. [102] proposed a hybrid
algorithm by combining SA, GA and Hitchcock’s exact
method to solve the FLP. Mir and Imam [117] solved an

unequal-sized FLP by using a hybrid approach, including SA
to generate an initial solution and an analytical search method
to determine the optimal location of facilities. Lee and Lee

Int J Adv Manuf Technol (2012) 60:11–27

[93] proposed a hybrid algorithm by combining SA, TS and
GA to find the global solution for a layout problem with fixed
shape and unequal-sized facilities. Erel et al. [38] developed a
new three stages hybrid algorithm to solve a DFLP. In the
first stage, they selected some feasible layouts as potential
solutions. In the second stage, the feasible solutions from the
first stage were used to find the shortest path by DP. Finally,
in the third stage, a local improvement process was used to
improve the solutions obtained from the preceding stage.
They concluded that their new algorithm can solve the largesized COPs. Balakrishnan et al. [9] proposed a hybrid GA by
using the pair-wise exchange heuristics and DP approach to
solve a DFLP. They compared their algorithm with some
other approaches and concluded that it has better performance
than GA and the preceding SA procedures. Rodriguez et al.
[135] suggested a hybrid meta-heuristic algorithm to solve a
DFLP. In this method, a TS approach is used to improve the
offspring in the GA. They concluded that their approach
performs better than other meta-heuristic solution methods.
Dunker et al. [33] presented a hybrid algorithm by combining
GA and DP approaches to solve a FLP with unequal area
facilities. Ji et al. [71] proposed a hybrid GA by combining a
new selection scheme and a descent local search approach to
solve the QAPs. Ramkumar and Ponnambalam [130]
proposed a population-based hybrid ant colony system to

solve the FLP formulated by QAP. Sirirat and Peerayuth [143]
combined DP, Bender’s decomposition, SA, GA and TS to
generate a hybrid algorithm for solving a DFLP with 20
facilities and five periods. Teo and Ponnambalam [150]
proposed a hybrid meta-heuristic approach for solving a
single row FLP by using ACO and PSO. Drezner [30] solved
the QAP by using a memetic algorithm proposed by Norman
and Moscato [124], which is the combination of GA and TS.
Liu and Abraham [99] proposed an effective hybrid algorithm
to solve QAP by combining PSO and fuzzy system
techniques in such a way that the position and velocity
vectors in PSO were represented by fuzzy matrices. Jeong et
al. [70] used the fuzzy system approach to control the
temperature and the local search repetition of SA, which led
to improve the performance of conventional SA. Yuying et al.
[164] developed a hybrid chaotic ant swarm optimization
approach by proposing pre-selection and discrete recombination operators in the CASO to improve the computational
time, solution accuracy and stability. The review of the
previous works on the intelligent and hybrid approaches are
summarized in the Appendix.

6 Conclusions
In this paper, we have presented a review of dynamic and
robust layout along with their mathematical models and
solution approaches with the following conclusions:


Int J Adv Manuf Technol (2012) 60:11–27

21

Table 3 Advantages and disadvantages of intelligent approaches
Approach

Advantages

Disadvantage

GA

The ability of:
1. Solving different kinds of COPs

1. Very slow
2. Finding sub-optimal solution

2. Finding a global best solution

3. Converging even to local optima is not guaranteed

3. Combining with other algorithms

4. The crossover and mutation rates affects the stability and convergence
5. Dependency of the evaluation performance on the gene coding method

Using the flexible memory to retain the history
of the search process
1. Low computational time
2. Free of local optima

The obtained solution is not necessarily an optimal solution

TS
SA

Dependency of the solution quality on the maximum iteration number of the inner
loop (cooling schedule) and the initial temperature

3. Easy for implementation
4. Convergent property
ACO

Scalability, robustness and flexibility in
dynamic environments

1. It is not easy for coding
2. Parameters initializations by trial and errors or at random

GRASP

1.
2.
1.
2.
3.
4.

Simple and easy to implement
Good initial solution generator
Easy to implement, fast and cheap

Having few parameters to adjust
Efficient global search approach
Having simple structure

Dependency of the solution quality on both neighbourhood and initial solutions

1.
2.
1.
2.

It explores new search areas
Free of local optima
Ability to solve unstructured problems
Controlling a symbolic data

Finding sub-optimal solution rather than the exact optimal solution

PSO

AIS
ES

FS

ANN

Having weak local search ability

1. Lacking of common sense requiring for decision making

2. Inability to response creatively

3. Adapt ability to new knowledge
4. Providing expert degree advisory services
for users to improve productivity

3. Need for change of the knowledge base to adapt to environmental changes

5.Low dependency on human experts
6. Cost effectual when human knowledge is
not cheap or available
1. Using unclear linguistic terms in the fuzzy
rules
2. Capability of solving uncertain problems
3. Availability of many commercial packages

4. Making incorrect decisions because of probable errors in the knowledge base

1.
2.
3.
4.

Difficulty in
1. Figuring their internal operations out
2. Prediction of their future performance
3.Training

Having learning ability
Adaptable to changes

High speed
Imitation of human thinking process

1. Prior knowledge requirements
2. Difficulty in estimating the membership function
3. Interpreting the fuzzy rules and defuzzification of the output in different ways

GA genetic algorithm, TS tabu search, ACO ant colony optimization, SA simulated annealing, AIS artificial immune system, GRASP greedy
randomised adaptive search procedure, PSO particle swarm optimization, ES expert systems, FS fuzzy systems, ANN artificial neural networks

Design of dynamic and robust layouts is considered to
deal with volatile environments. The dynamic approach has the advantage of having optimal layout
for each period, but it suffers from the disadvantage of
having the facility rearrangement cost. On the other
hand, the robust approach has the advantage of lack of
rearrangement cost and the disadvantage of not having
an optimal layout for each period.
The QAP and MIP are the two well-known models,
which are suitable for modelling the block layout with
discrete representation and the detailed layout with

continuous representation, respectively. QAP and GT
are suitable models for equal-sized FLPs whereas QSP
and MIP are appropriate formulations for unequalsized layout problems.
Exact (optimal) algorithms are useful approaches to
find an optimal solution for the small-sized FLPs.
Heuristic methods are suitable for solving the FLPs
with high and low flow dominance, respectively.
Regarding intelligent approaches, GA, TS, SA and ACO
algorithms have been widely used to solve different kinds

of FLPs, including block and detailed layouts with discrete


22

Int J Adv Manuf Technol (2012) 60:11–27

and continues representation, large-sized facilities and
realistic constraints in a reasonable computational time.
From computational time point of view, SA has better
performance than GA and TS. ACO is appropriate to solve
more complex COPs including dynamic, stochastic and
multi-objective problems. It can generate a higher quality
solution by raising the number of ants and iterations,
which in turn the computational time is also increased.
ES and FS are suitable for solving the multi-objectives
FLPs while evolutionary algorithms like GA are
appropriate techniques to solve the single-objective
problems. Fuzzy systems can also be used to deal with
uncertainties particularly for situations in which some

parameters such as closeness ratio between facilities
should be described by linguistic words rather than an
exact definition.
In general, hybrid algorithms overcome the weakness
of some solution method by using the strength of other
techniques to improve the solution quality. SA is a
good local search approach in hybrid algorithms. ACO
and GRASP are good approaches to construct initial
solutions for other algorithms such as GA, SA and TS

in which the quality of solution is significantly affected
by the initial solution. Fuzzy systems are suitable
techniques to combine with other algorithms such as
SA and PSO to control their parameters.

Appendix

Table 4 Review of intelligent and hybrid approaches on FLP
Authors (year)

GA

TS

SA

ACO

GRASP

PSO

AIS

ES

FS

ANN

Hybrid

√

Fisher and Nof [43]

Description
Facilities design expert system

Grobelny [54, 55]

√

MHC

Evans et al. [39]

√

Linguistics

Kumara and Kashyab [88]

√

IFLAPS

Malakooti, and Tsurushima [103]

√

Multiple criteria decision making (MCDM)

√

Feo and Resende [41]

QSP
√

Heragu and Kusiak [62]

Knowledge-based machine layout
√

Huntley and Brown [67]

GA–SA

Raoot and Rakshit [131]

√

Linguistics

Raoot and Rakshit [132]

√

MCDM

Conway and Venkataramanan [24]

√

DFLP budget constraint
√

Li et al. [95]

QAP
√

Gen et al. [49]

Intra-cell layout
√

Feo and Resende [40]

Survey
√

Dweiri and Meier [34]

Analytical hierarchy process
√

Tsuchiya [153]

Parallel ANN

√

Harraz [58]
Mavridou and Pardalos [108]

Fuzzy model

√

Abdelghani [1]

√

Knowledge-based decision support system

√

Survey

√

Kaku and Mazzola [72]

DFLP
√

Grobenlyn [56]

Fuzzy demands

√

Leonardo et al. [94]

Detailed layout
√

Urban [156]

DFLP
√

Mahdi et al. [102]
√

Chung [22]

Neuro-based expert system

√

Gambardella et al. [47]
Urban et al. [157]
Abdinnour and Hadley [2]

SA–GA
HAS–QAP

√

QAP

√

Multi-floor FLP

Balakrishnan and Cheng [7]

√

Nested-loop GA

Azadivar and Wang [5]

√

Stochastic FLP

Smith and Norman [144]

√

Stochastic FLP
√

Aiello and Enea [3]
Baykasoglu and Gindy [13]
Mir and Imam [117]

Fuzzy demands

√

DFLP
√

SA–analytical search


Int J Adv Manuf Technol (2012) 60:11–27

23

Table 4 (continued)
Authors (year)

GA

TS

SA

ACO

GRASP

PSO

AIS

ES

FS

ANN

Hybrid

√

Pitsoulis and Resende [127]

Survey
√

Lee and Lee [93]
√

Resende and Ribeiro [133]

Description

GA–TS–SA
Survey

Erel et al. [38]

√

DP–local search

Balakrishnan et al. [9]

√

GA–DP

Ficko et al. [42]

√

Multi-row FLP
√

Kulturel-Konak et al. [86]

Stochastic FLP

Corry and Kozan [26]

√

Solimanpur et al. [146]

√

DFLP
QAP

√

Rodriguez et al. [135]
√

Deb and Bhattacharyya [27]

Decision support system

√

Engelbrecht [36]

TS–GA
Survey

√

Liao [97]

√

Survey

Dunker et al. [33]

√

GA–DP

Ji et al. [71]

√

GA-local search

√

Marvin et al. [107]

Comparing TS–SA–GA

McKendall et al. [111]

√

Suman and Kumar [147]

√

DFLP
Survey

McKendall and Shang [110]

√

HAS I–II–III

Baykasoglu et al. [12]

√

DFLP

Thangavel et al. [151]

√

Survey
√

Ramkumar and Ponnambalam [130]
Hu et al. [65]

√

Wu and Ji [161]

√

Population-based hybrid ant colony system
Cell layout
QAP

√

Ashtiani et al. [4]

DFLP

√

Poli et al. [128]

survey

Sirirat and Peerayuth [143]

√

DP–SA–GA–TS

Liu and Abraham [99]

√

PSO–FS

Krishnan et al. [82]

√

Stochastic FLP

McKendall [112]

√

Liang and Chao [96]

√

Three TSs
Multi-searching
√

Phen and Kuan [126]

Survey

Teo and Ponnambalam [150]

√

Drezner [30]

√

Krishnan et al. [83]

√

ACO–PSO
Memetic algorithm–GA–TS
Risk-based approach

Surya [148]

√

Scholz et al. [141]

√

Improved tabu search
Slicing tree
√

Dong et al. [28]

DFLP
√

Chen and Rogers [21]

HAS–QAP
√

Rezazadeh et al. [134]

DFLP

Berna and Attila [17]

√

Satheesh Kumar et al. [140]

√

DFLP
Loop layout

Yuying et al. [164]

√

Hybrid chaotic ant swarm optimization

Jeong et al. [70]

√

FS–SA

McKendall and Hakobyan [113]

√

Samarghandi and Eshghi [139]

√

Sahin et al. [138]

Boundary search technique
Single-row layout
√

DFLP

Solimanpur et al. [145]

√

Cell formation

Komarudin, and Kuan [77]

√

Ant system

Ning et al. [122]

√

MMAS


24

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