International Journal of Engineering Materials and Manufacture (2019) 4(1) 27-32
https://doi.org/10.26776/ijemm.04.01.2019.04
S. Kamaruddin and M. A. H. A. Latif
Department of Manufacturing and Materials Engineering
International Islamic University Malaysia
PO Box 10, 50728 Kuala Lumpur, Malaysia
E-mail: shafie@iium.edu.my
Reference: Kamararuddin and Latif (2019). Application of the Bees Algorithm for Constrained Mechanical Design Optimisation
Problem. International Journal of Engineering Materials and Manufacture, 4(1), 27-32.
Application of the Bees Algorithm for Constrained Mechanical Design
Optimisation Problem
Shafie Kamaruddin and Mohd Arif Hafizi Abd Latif
Received: 12 November 2018
Accepted: 09 January 2019
Published: 01 March 2019
Publisher: Deer Hill Publications
© 2019 The Author(s)
Creative Commons: CC BY 4.0
ABSTRACT
Optimisation is a technique or procedure to find the optimal or feasible solution whether it is to minimise or maximise
by comparing other possible solutions until the best solution is found. Nowadays, many optimisation algorithms have
been introduced due to the advancement of technology such as Teaching Learning Based Optimisation (TLBO), Ant
Colony Optimisation (ACO), Particle Swarm Optimisation (PSO) and the Bees Algorithm. The Bees Algorithm is
considered as one of the best optimisation algorithms because it has been successfully solved different type
optimisation problem from in various field. It is inspired by the foraging behaviour of honeybees in nature. This study
applies the Bees Algorithm to minimise the mass of disc clutch brake in its design. To find the optimal solution for
the multiple disc clutch design, the Bees Algorithm is applied and found better result compared to other optimisation
algorithms that already have been used.
Keywords: Bees Algorithm, Optimisation Algorithm, Multiple Disc Clutch Problem
1 INTRODUCTION
In this era, many problems need optimisation regardless to minimise or maximise. For examples, minimise (cost,
weight, time) and maximise (profit, performance). Optimisation technique is a procedure to find the optimal solution
whether it is minimisation or maximisation by comparing other possible solutions until the satisfactory solution is
found. This technique finds the combination of parameters or variables to maximise or minimise objective functions
subject to some constraints. One of the methods to solve the optimisation problem is using optimisation algorithms.
There are many types of algorithms available today to find the optimal solution such as Ant-Colony algorithm, the
Bees Algorithm, Teaching-Learning Based Optimisation and Particle Swarm Optimisation.
The Bees Algorithm is considered as among the best optimisation algorithms because it has been successfully
solved different types of optimisation problems (Hussein et al., 2016). It is inspired by the foraging behaviour of
honeybees in nature. Multiple Disc Clutch Brake problem is one of the common problems faced in real world problem
and it has been applied by other algorithms. This problem requires finding the best combination of design variables
(optimal design) at a minimum weight. Despite it has been applied by other algorithms, applying the Bees Algorithm
on this problem would be interesting in term of exploring the capability of the Bees Algorithm in different problem.
The significant of this research output would contribute in finding best design variables at minimum weight compared
with other algorithms, which will improve the efficiency in product design management. For this reason, this problem
was selected to test the performance of the Bees Algorithm. This study also provides other alternative algorithm in
solving this problem. The main objective of this problem is to find the best combination of variables of Multiple
Clutch Brake design with minimum mass and satisfy all the constraints.
2 THE BEES ALGORITHM
2.1 Bees in Nature
A colony of bees always has some of its population to be used as scout bees to scrutinize the surrounding of their
hive for potential patches of flower. The process of foraging starts when the scout bees are sent to the space and
move randomly from one patch to another patch. Then, the scout bees return to the hive and inform other bees in
Application of the Bees Algorithm for Constrained Mechanical Design Optimisation Problem
28
the hive about the quality and location of a food source by performing a dance called “waggle dance”. It will recruit
other bees to exploit high potential location while other scout bees will continue to find new patches (Pham et al.,
2009).
In this foraging behaviour, the bees need to divide their workforce to each task with proper number of bees.
They also must flexible enough to adapt to any environment changes. There are many mechanisms on how a colony
of bees regulates the division of labour between the scouts and the recruits (Beekman et al., 2007). Based on these
behaviours, several bees inspired algorithms have been established such as Artificial Bee Colony (ABC) algorithm, Bee
Colony Optimisation algorithm (BCO) and the Bees Algorithm (Karaboga & Basturk, 2007; Nikolić & Teodorović,
2013; Pham & Castellani, 2015). Although, all these algorithms are similar in term of inspiration from bees foraging
behaviour, but each algorithm has its own work mechanism. This study focuses on one of the algorithms, which is
the Bees Algorithm.
2.2 Description of the Bees Algorithm
The Bees Algorithm uses the same concept as the food foraging behaviour of honey bees in the nature. It was
introduced by a group of researchers from Cardiff University (Pham et al., 2006). It works by sending the scouts bees
to explore for potential solutions. Then, the potential sites found are exploited and better solution will attract more
recruits. This process continues iteratively until the best solution has been found. The flow chart of the Bees Algorithm
is shown in Figure 1 (Pham & Castellani, 2009).
The first step of the Bees Algorithm is setting the parameters which are number of scout bees (n), number of best
sites selected out of n sites(m), number of elite sites selected out of best sites(e), number of bees recruited for elite
sites (nep), number of bees recruited for the other selected sites (nsp), patches size (ngh) and number of unselected
scout bees (n-m). The next step of the Bees Algorithm is sending n scout bees randomly (random initialisation) across
the search space. After that, each position visited by the scout bee is evaluated via fitness function and ranked
according to it fitness value. Once the positions visited by the scout bees have been ranked, local search is performed
by sending recruit bees (nep and nsp) to selected elite sites and best sites. Elite sites attract more recruit bees compared
to best sites. These recruit bees are placed randomly across patch size of the selected sites. If the recruit bees found
better fitness value compared to the followed scout bee of the selected site, it will replace the scout bee as the new
scout bee. If the recruit bees failed to find better fitness value, the scout bee remains as scout bee for that patch. As
for the remaining unselected scout bees, they are sent randomly across the space to find new patches (global search).
At the end of each iteration, a new population is formed consists of best recruit bees from each patch and unselected
scout bees. The stopping criterion can be set either based on predefined threshold or predefined number of iteration
(Pham & Castellani, 2009).
Figure 1: Flowchart of the Bees Algorithm
Kamaruddin and latif (2019): International Journal of Engineering Materials and Manufacture, 4(1), 27-32
29
2.3 Multiple Disc Clutch Problem
The objective is to minimise the mass of the multiple disc clutch brake design, which consists of five discrete variables;
inner radius, ri, outer radius, ro, thickness of the discs, t, actuating forces, F, and number of friction surfaces, Z (Rao
et al., 2011). Figure 2 shows the schematic of Multiple Disc Clutch used for this application. The objectives function
and constraints for this problem are shown in Appendix.
Figure 2: Multiple Disc Clutch Schematic
3 METHODOLOGY
This study started with background study or reviewing the articles, journals and books related to the problem. The
articles are about the optimisation algorithm, specifically the Bees Algorithm and the mechanical design problem,
which is multiple disc clutch.
Then, writing the code in open source software known as R-Software and run the code. Most of optimisation
algorithms require fine-tuning of parameters to find the best solution. Thus, several set of parameters were selected
for this experiment. For each set of parameters, the algorithm was run 100 times. The result of best fitness value,
worst fitness value, mean, standard deviation and successful rate found over 100 runs of each set of parameters were
recorded. The criteria in determining the best result is based on the mean of fitness values found over 100 runs. The
set of parameters that generated the best result are shown in Table 1.
The stopping criterion for this experiment was set based on number of function evaluations, which are 1000
function evaluations. This means the algorithm was stopped once it reached maximum number of evaluations for
each run. Next, results of this experiment were compared with other optimisation algorithms available in the
literature. The best result found was compared with other optimisation algorithms available in the literature. Most
of results in the literature do not provide the standard deviation value of their results. Thus, the comparison is in
term of mean value only. The value of variables and constraints are also shown and compared between those
different optimisation algorithms.
Table 1: Set of Parameters of the Bees Algorithm
Parameters Value
n 14
e 2
nep 11
ngh 3
m 4
nsp 4
Application of the Bees Algorithm for Constrained Mechanical Design Optimisation Problem
30
4 RESULTS AND DISCUSSIONS
4.1 Results
The multiple disc clutch brake design problem was also solved by NSGA-II, TLBO, ABC, and APSO. Table 2 shows
the statistical comparisons of other optimisation algorithm results with the Bees Algorithm. The best value of the mass
of the multiple disc clutch brake found by the Bees Algorithm is 0.313657 kg. This value is similar with TLBO and
ABC, but better compared to the results of NSGA-II and APSO.
Table 3 shows the combination of variables found by the Bees Algorithm corresponding to the best fitness value
over 100 runs. In most cases, the combination of variables is similar with other algorithms as the best fitness value is
similar with TLBO and ABC. Despite statistical result, comparison shows that TLBO achieved better successful rate
(SR) compared to the Bees Algorithm, but the Bees Algorithm showed less variability of the result with standard
deviation only 0.008631. In terms of other statistical results such as worst and mean fitness values found, the Bees
Algorithm also shows better results compared with other optimisation algorithms. Figure 3 and Figure 4 show the
comparison of convergence rate for the Bees Algorithm and ABC algorithm respectively. From Figure 4, it is observed
that the convergence rate for the TLBO method is faster than ABC in the early stage of the search. Then, both found
almost similar fitness value until at the end of the search. This could have contributed to better successful rate of
TLBO compared to ABC algorithm and the Bees Algorithm.
Table 2: Result Comparison with other Optimisation Algorithms
Best Worst Mean SR SD
TLBO (Rao et al., 2011) 0.313657 0.392071 0.327166 0.67 N/A
ABC (Rao et al., 2011) 0.313657 0.352864 0.324751 0.54 N/A
NSGA-II (Debb & Srivinasan, 2006) 0.4704 N/A N/A N/A N/A
APSO (Ben Guedria, 2016) 0.337181 0.716313 0.506829 N/A 0.09767
Bees Algorithm 0.313657 0.345022 0.320761 0.41 0.008631
Table 3: Values of Objectives Functions, Design Variables and Constraints
Variables TLBO
(Rao et al., 2011)
NSGA-II
(Debb &
Srivinasan,
2006)
ABC
(Rao et al., 2011)
APSO
(Ben Guedria,
2016)
Bees
Algorithm
x1 70 70 N/A 76 70
x2 90 90 N/A 96 90
x3 1 1.5 N/A 1 1
x4 810 1000 N/A 840 990
x5 3 3 N/A 3 3
f(x) 0.313657 0.4704 0.313657 0.337181 0.313657
g1 0 0 N/A 0 0
g2 24 22 N/A 24 20
g3 0.919428 0.9005 N/A 0.922273 999.902
g4 9830.371 9.7906 N/A 9.824211 9999793
g5 7894.697 7.894 N/A 7.738378 7894.697
g6 0.702013 3.3527 N/A 1.396611 3.23794
g7 37706.25 60.625 N/A 48.84837 59418.75
g8 14.29799 11.6473 N/A 13.60339 11.76206
Kamaruddin and latif (2019): International Journal of Engineering Materials and Manufacture, 4(1), 27-32
31
Figure 3: Convergence plots for the Multiple Disc Clutch Brake using Bees Algorithm
Figure 4: Convergence plots for the Multiple Disc Clutch Brake using the TLBO and ABC algorithms (Rao et al., 2011).
4.2 Discussions
The results of the Bees Algorithm were compared with the results of other algorithms to compare it performance.
Some of these optimisation algorithms also share almost similar work mechanism with Bees Algorithm. The ABC,
APSO and the Bees Algorithm are in the same category of Swarm-based optimisation. The Bees Algorithms and ABC
use the concept of exploration and exploitation, while APSO focuses more on exploration. Meanwhile, NSGA-II and
TLBO have their own search mechanisms. TLBO uses the concept of learning process in a class with certain number
of student (population) and subject (variable) and NGSA is a type of evolutionary algorithm.
As shown it Table II, it is observed that three out of five (ABC, TLBO and The Bees Algorithm) optimisation
algorithms able to find similar best solution, which is 0.313657 kg. The remaining two algorithms (NGSA-II and
APSO) found worse solutions where, APSO obtained 0.337181 kg and NSGA-II obtained 0.4704 kg. In term of mean’s
value and standard deviation’s value, the Bees Algorithm achieved better value, which is 0.320761kg and 0.008631
respectively. The successfulness of the Bees Algorithm and ABC algorithm in finding the best solution could due their
similarity of work.
5 CONCLUSIONS
In this work, a swarm based optimisation algorithm named the Bees Algorithm is applied to constrained mechanical
design problem (Multiple Clutch Brake problem) and its performance is compared with other algorithms. The results
obtained show that the Bees Algorithm found similar best fitness value compared with other algorithms at similar
computational effort (number of function evaluation). Despite the best fitness, value found is similar with other
optimisation algorithms such as TLBO algorithm and ABC algorithm, the Bees Algorithm has fewer variations with
better mean and standard deviation. In conclusion, this research showed:
1. The successfulness of applying the Bees Algorithm to find the best solutions for mechanical design
optimisation problem.
2. Despite the best fitness value is similar with other algorithms, the Bees Algorithm found better results in term
of mean and standard deviation which are 0.320761kg and 0.008631 respectively.
3. The capability of applying the Bees Algorithm to more complex optimisation problem.
Application of the Bees Algorithm for Constrained Mechanical Design Optimisation Problem
32
ACKNOWLEDGEMENT
The authors are grateful to the Department of Manufacturing and Materials Engineering, International Islamic
University Malaysia for giving the opportunity to complete this study.
APPENDIX: OBJECTIVE FUNCTIONS AND CONSTRAINTS
The objective is to minimize the mass of the multiple disc clutch brake using five discrete variables: inner radius
(ri=60, 61, 62, . . . , 80), outer radius (ro= 90, 91, 92, . . . , 110), thickness of discs (t = 1, 1.5, 2, 2.5, 3), actuating
force (F = 600, 610, 620, . . . , 1000) and number of friction surfaces (Z = 2, 3, 4, 5, 6, 7, 8, 9).
Minimise: f (x) = π(ro
2
− r i
2
)t(Z +1)ρ
Subject to: g1(x) = ro −ri − ∆r ≥ 0,
g2(x) = lmax−(Z +1)(t +δ) ≥ 0,
g3(x) = pmax −prz ≥ 0,
g4(x) = pmaxvsrmax−przvsr ≥ 0,
g5(x) = vsrmax−vsr ≥ 0,
g6(x) = Tmax −T ≥ 0,
g7(x) = Mh−sMs ≥ 0,
g8(x) = T ≥ 0,
where Mh =
2
3
𝜇𝐹𝑍
𝑟𝑜
3−𝑟𝑖
3
𝑟𝑜
2−𝑟𝑖
2 , prz =
𝐹
𝜋(𝑟𝑜
2−𝑟𝑖
2)
, vsr =
2𝜋𝑛(𝑟𝑜
3−𝑟𝑖
3)
90(𝑟𝑜
2−𝑟𝑖
2)
, T =
𝐼𝑧𝜋𝑛
30(𝑀ℎ+𝑀𝑓 )
,
∆r = 20mm, tmax = 3mm, tmin = 1.5mm, lmax = 30mm, Zmax = 10, vsrmax = 10 m/s, 𝜇= 0.5, s = 1.5, Ms = 40
N m, Mf = 3 N m, n = 250 rpm, pmax =1MPa, Iz = 55 kg mm
2
, Tmax = 15 s, Fmax = 1000N, rimin = 55mm,
romax = 110mm.
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