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Researchers have been inspired by nature in many different ways. A newpopulation based search method called the Bee Algorithm (BA) is proposed. Theswarm based algorithm described in this paper goes beyond the traditional design andrevolves around mimicking the food foraging behaviour of honey bees which has beenapplied to solve unconstrained functions in the domain of continuous space. Theauthors believe this algorithm is very efficient and robust, as indicated by theextensive simulation results on numerous benchmarking problems.
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Technical Note
Bee Algorithm
A Novel Approach to Function Optimisation
D.T. Pham , A. Ghanbarzadeh , E. Koc , S. Otri , S. Rahim , M. Zaidi
The Manufacturing Engineering Centre
Cardiff University
Queen’s University
The Parade
Newport Road
Cardiff CF24 3AA
Technical Note: MEC 0501 Dec 2005
Bee Algorithm
A Novel Approach to Function Optimisation
D.T. Pham , A. Ghanbarzadeh , E. Koc , S. Otri , S. Rahim , M. Zaidi
The Manufacturing Engineering Centre
Cardiff University
Queen’s University
The Parade
Newport Road
Cardiff CF24 3AA
Technical Note: MEC 0501 Dec 2005
2
Table of Content
Abstract ............................................................................................................................ 4
1.Introduction ................................................................................................................... 5
2.Literature Review and Related Works .......................................................................... 6
3.The Bee Algorithm ...................................................................................................... 9
3.1.Natural Bees ........................................................................................................... 9
3.2.The Proposed Bee Algorithm .............................................................................. 11
3.3.A Study of Some Characteristics of Bee Algorithm ............................................ 15
3.3.1.Population Size of the Bee Algorithm .......................................................... 15
3.3.2.Elitism .......................................................................................................... 17
3.3.3.Neighbourhood Search ................................................................................. 18
4.Simulation and Comparison ...................................................................................... 25
5.Conclusion and Discussion ......................................................................................... 30
Acknowledgement ......................................................................................................... 30
References ...................................................................................................................... 39
3
Technical Note MEC 0501
Bee Algorithm
A Novel Approach to Function Optimisation
D.T. Pham , A. Ghanbarzadeh , E. Koc , S. Otri , S. Rahim , M. Zaidi
The Manufacturing Engineering Centre
Cardiff University
Queen’s University
The Parade
Newport Road
Cardiff CF24 3AA
Abstract
Researchers have been inspired by nature in many different ways. A new
population based search method called the Bee Algorithm (BA) is proposed. The
swarm based algorithm described in this paper goes beyond the traditional design and
revolves around mimicking the food foraging behaviour of honey bees which has been
applied to solve unconstrained functions in the domain of continuous space. The
authors believe this algorithm is very efficient and robust, as indicated by the
extensive simulation results on numerous benchmarking problems.
Keywords: Bee Algorithm, Continuous Function Optimization, Swarm Intelligence,
Search Strategies
4
1. Introduction
Researchers have been inspired by nature in many different ways. Classical
optimization methods encounter great difficulty when faced with the challenge of
solving hard problems with acceptable levels of time and precision. It is generally
believed that NP-hard problems cannot be solved to optimality within polynomially
bounded computation times, thus generating much interest in approximation
algorithms that find near-optimal solutions within reasonable running times. The
swarm based algorithm described in this paper goes beyond the traditional design and
revolves around mimicking the food foraging behaviour of scout bees.
Section 2 details the literature review and previous works related to bees. Section 3
describes the foraging behaviour of natural bees as well as the core idea of our Bee
Algorithm. In section 4, the simulation work is discussed. Section 5 details the results
and benchmarking problems which were used to test the robustness and agility of the
bee algorithm. Section 6 gives the concluding remarks and further discussions.
5
2. Literature Review and Related Works
A recent trend is the introduction of non-classical stochastic search optimization
algorithms. The swarm based evolutionary algorithms (EA) mimic nature’s
evolutionary principles to drive their search towards an optimal solution. One of the
most striking differences from direct search algorithms such as hill climbing and
random walk is that EAs use a population of solutions for every iteration, instead of a
single solution. Since a population of solutions are processed in every iteration, the
outcome is also a population of solutions. If an optimization problem has a single
optimum, all EA population members can be expected to converge to that optimum
solution. However, if an optimization problem has multiple optimal solutions, an EA
can be used to capture multiple optimal solutions in its final population. These
approaches include Ant Colony Optimization (ACO) [4], Genetic Algorithm (GA) [3]
and Particle Swarm Optimization (PSO) [5].
Common to all population based search methods is a strategy that generates variations
of the tuning parameters. Most search methods use a greedy criterion to make this
decision, which accepts the new parameter if and only if it reduces the value of the
objective or cost function. One of the most successful metaheuristic is the ACO, which
is built on real ant colony behaviour. Ants are capable of finding the shortest path from
the food source to their nest using a chemical substance called pheromone. The
pheromone is deposited on the ground as they walk and the probability that a passing
stray ant will follow this trail depends on the quantity of pheromone laid. ACO was
6
first used for continuous optimization by Bilchev [1] and further attempts were made
in [1,2].
The Genetic Algorithm is based on artificial selection and genetic recombination. This
works by selecting two parents from the current population and then by applying
genetic operators mutation and crossover to create a new gene pool set. They
efficiently exploit historical information to speculate on new search areas with
improved performance [3]. When applied to optimization problems, GA has the
advantage of performing global search and may hybridize with domain-dependant
heuristics for better results. [2] describes such a hybrid algorithm of ACO with GA for
continuous function optimization.
The idea of honey bees has been used inadequately in different topics mostly in
discrete space. BeeHive, BeeAdHoc and Bee Algorithm [8, 9, 6] are all algorithms
inspired from the bee swarming behaviour. A model generated by studying the
allocation of bees to different flower patches to maximize the nectar intake is
described in [6]. This was subsequently applied to distribute web applications at
hosting centres. Another model borrowing from the principles of bee communication is
presented in [8]. The artificial bee agents are used in packet switching networks to find
suitable paths between nodes by updating the routing table. Two types of agents are
used short distance bee agents which disseminate routing information by travelling
within a restricted number of hops and long distance bee agents which travel to all
nodes of the network. Though the paper talks in terms of bees, it only loosely follows
their natural behaviour.
7
In [7], the author describes a virtual bee algorithm where the objective function is
transformed into virtual food. The paper is like a black box in that no information
about how the transformation from objective function to food source or agent
communication is highlighted. Nor are there any comparative results to check the
validity of the algorithm.
8
3. The Bee Algorithm
3.1. Natural Bees
A colony of honey bees can be explained as a large and diffuse creature which can
extend itself over great distances (more than 10 km) and in multiple directions
simultaneously to exploit a vast number of food sources.[12,13] A colony can only be
successful by deploying its foragers to good fields. In principle, flower patches that
provide plentiful amounts of nectar or pollen that are easy to collect with less energy
usage, should receive more bees, whereas, patches with less nectar or pollen should
receive fewer bees. [10, 11]
Similar to other social insect colonies, honey bees also have the same constituents for
self-organization as classified in [10]. Self organization relies on four basic
ingredients: (1) Positive feedback, (2) negative feedback, (3) the amplification of
fluctuations random walks, errors- and (4) multiple interactions. In bee colony, the
objective is to increase the amount of honey by bringing more nectar, pollen and water
with good enough quality. Like other insects, achievement is not easy task in a short
term harvesting season for that sized animals. Thus, self-organization has a vital role in
the colony. For positive feedback bees have a special dancing behaviour which was
discovered by Karl von Frisch. [12] This waggle dance is essential for exploitation of
good sources, so it has a vital role in colony. Negative feedback is another issue which
is balancing the positive feedback effect in colony. During harvesting when the bees
detect that there is no more nectar appears in a flower patch bees will be abandoned
and they follow other dances on dance floor. Randomness is also one of the essential
9
part of the bee colony, especially in searching process. Although, bees have some
senses to decrease the chance factor they still follow the basic random search idea.
And individuals use others information that presented by a dance, as a multiple
interaction part.[10,11,12,13]
The foraging process begins in a colony by sending scout bees to search for adequate
flower patches. Scout bees search randomly through their journey from one patch to
another. Moreover, during the whole harvesting season, a colony continues its
exploration, keeping a percentage of the whole population as scout bees. [13]
When they return to the hive, those who found a source which is above a certain
threshold (a combination of some constituents, such as sugar percentage regarding the
source) deposit their nectar or pollen and go to the dance floor to perform their waggle
dance. [12]
This mysterious dance is essential for colony communication, and contains three vital
pieces of information regarding flower patches: direction, distance, and quality of
source. [11, 12] This information guides the colony to send its bees to flower patches
precisely, without any supervisory leader or blueprint. Each individual’s knowledge of
the outside environment is gleaned solely from the waggle dance. This dance gives the
colony a chance to evaluate different patches simultaneously in addition to minimising
the energy usage rate.[11] After waggle dancing on the dance floor, the dancer bee
(i.e. the scout bee) goes back to the flower patch with follower bees that were waiting
inside the hive. This allows the colony to gather food quickly and efficiently.
While harvesting the source, the bees monitor the food level, which is necessary to
decide upon the next waggle dance when they return to the hive. [11] If the food
10
source is still good enough and calls for more recruitment, then that patch will be
advertised by making a waggle dance and recruiting more bees to that source.
3.2. The Proposed Bee Algorithm
The Bee Algorithm is inspired by the natural foraging behaviour of honey bees to find
the optimal solution. Fig 1 shows the pseudo code for a simple Bee Algorithm.
The algorithm starts by initialising a set of parameters namely the: number of scout
bees (n), number of elite bees (e), number of selected regions out of n points (m),
number of recruited bees around elite regions, number of recruited bees around other
selected (m-e) regions, and stopping criteria Then, N bees are placed on the search
space randomly, similar to scout bees. Every bee on the problem space evaluates the
fitness of its field in step 2. Similar to the natural searching (or scouting) process
carried out by scout bees.
Subsequently, in step 4, elite bees that have better fitness are selected and saved for the
next population. Then, in neighbourhood search (step 5 to 7), the algorithm starts by
assigning elite bees positions to search directly around them and other positions will be
chosen, either the best ones or stochastically using a roulette wheel proportional to
fitness. In step 6, the bees search around these points within the boundaries and their
individual fitness is evaluated. More bees will be recruited around elite points and
fewer bees will be recruited around the remaining selected points. Recruitment is one
of the essential hypotheses of the bee algorithm along with the scout bee concept, both
of which are used in nature.
11
Fig.1. Pseudo code of the bee algorithm
However, in step 7, for each site only one representative bee with the highest fitness
will be selected. In nature there is no such restriction. Bees behave more flexibly, but
here in term of optimisation and efficiency it is enough to select only one
representative from each site. In step 8, the remaining bees will be sent randomly
around the search space. Similar to honey bees, the algorithm keeps its random
searching part by sending scout bees to find new potential solutions. These steps will
be repeated until certain criterion is met.
At the end of each iteration, the colony will organise itself by having three parts to its
new population - elite bees, representatives from each selected site, and remaining bees
assigned to random search.
12
1. Initialise population with random solutions.
2. Evaluate fitness of the population.
3. While (stopping criteria not met) //Forming new population.
4. Select elite bees.
5. Select sites for neighbourhood search.
6. Recruit bees around selected sites and evaluate fitness.
7. Select the fittest bee from each site.
8. Assign remaining bees to search randomly and evaluate their fitness.
9. End While
Fig.2. Flowchart of basic bee algorithm
13
Initialize a scout bee population (n)
Evaluate fitness of the population
Select sites for neighbourhood search (m-e)
Assign remaining bees to random search
( n – m – e )
New Population of Scout Bees
( e + m + ( n – m –e ) )
Select elite bees (e)
Determine the Neighbourhood Range
Recruit the bees around selected sites
Neighbourhood
Search
Select fittest bees from each site
Fig.3. Simple example of bee algorithm.
14
*
*
*
*
*
*
*
*
*
*
Graph 1. Initialise population with random solutions and
evaluate the fitness.
*
*
*
*
*
*
*
Graph 2. Select elite bees “▪”.
*
*
*
*
*
Graph 4. Define neighbourhood range.
*
*
*
*
*
* *
* *
*
*
* *
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
Graph 5. Recruit bees around selected sites.
*
*
*
*
*
*
*
*
*
*
Graph 6. Select the fittest from each site “*
*
*
*
*
*
*
*
*
*
*
Graph 7. Assign remaining bees to search randomly and
evaluate their fitness
*
*
*
*
*
*
*
*
*
*
Graph 8. New population with “previous elite bee”,
representative bees and randomly distributed bees
*
*
*
*
*
Graph. 3. Select sites for neighbourhood search“▪”
and“▫”.
3.3. A Study of Some Characteristics of Bee Algorithm
In this section different parameters of algorithm and their effect will be discussed.
3.3.1. Population Size of the Bee Algorithm
Population (n) is one of the key parameters in the Bee Algorithm. Effect of changing
population size on the mean number of iteration to get the answer, number of
evaluated points and reliability of successfulness of the algorithm are shown in the
figures 4, 5 and 6.
Population (n)
460
430
400
370
340
310
280
250
220
190
160
130
100
70
40
Mean Iterations
500
400
300
200
100
0
Fig.4. Population size versus Mean iteration
15
In fig.4, with increasing the population size the iteration required to obtain the answer
will reduce.
Population (n)
460
430
400
370
340
310
280
250
220
190
160
130
100
70
40
Mean Generated Points
40000
30000
20000
10000
Fig.5. Population size versus Number of function evaluation
According to fig.5, with the increase of population size, number of function evaluation
will increase which is predictable.
16
Successfulness
0
20
40
60
80
100
0 50 100 150 200
n ( Population )
Fails in 1000 runs
Fig.6 Successfulness for different population size
To have a good reliability, a minimum size of population is required (Fig. 6). In order
to have less number of iteration to get the answer, population should be large, to get a
reasonable number of function evaluations, population has to be as less as possible and
for have a good reliability population should be more than a size. These three
constitutions will give a good range to choose the size of population.
3.3.2. Elitism
Number of elites as long as it is more than one, does not have major effect on the
performance of the Bee Algorithm, and thus the number of elites can be a small
number more than zero.
17
3.3.3. Neighbourhood Search
Neighbourhood search is an essential concept for all evolutionary algorithms as well as
the bee algorithm. In the bee algorithm, searching process in neighbourhood range is
similar to foraging field exploitation of natural bees. As explained above, when a scout
bee find any good enough foraging field, she advertise it back into the hive for
recruiting more bees to that field. This behaviour is helpful to bring more nectar into
the hive. Hence, this fruitful method might be also helpful for engineering optimization
problems.
Harvesting process also includes a monitoring process which is necessary for decision
making for a waggle dance back into the hive to recruit more bees to that site. In the
bee algorithm, this monitoring process can be used as neighbourhood search.
Principally, when a scout bee finds a good field (good solution), she advertise her field
to more bees. Subsequently, those bees fly to that source take piece of nectar and turn
back to hive again. Depending on the quality, source can be advertised by some of the
bees that already know the source. In proposed bee algorithm, this behaviour has been
used as a neighbourhood search. As explained above, from each foraging site (or
neighbourhood site) only one bee is chosen. This bee must have the best solution
information about that field. Thus, algorithm can create some solutions which related
to previous ones.
*
*
*
*
*
Best Bee
*
*
*
*
*
18
AA
B
Fig.7. Graphical explanation of neighbourhood search
However, neighbourhood search based on random distribution of bees in a predefined
neighbourhood range. For every selected site, bees are randomly distributed to find a
better solution. As shown in fig. 7, only the best bee is chosen to advertise her source
and the centre of the neighbourhood field shifted to best bees` position (from A to B).
Moreover, during harvesting process other constituents must also be taken into account
for increasing the efficiency; number of recruited bees in neighbourhood range and
range width.
Number of recruited bees around selected sites, should be defined properly. When the
number is increased then the number of function evaluation will also be increased.
Vice verse, when the number decreases, chance to find a good solution will also be
decreased.
This problem also depends on the neighbourhood range. If range can be arranged
adequately, then number of recruited bees will depend on the complexity of a solution
space. This will be discussed later with more details.
3.3.3.1. Site Selection
Two deferent techniques have been implemented: Probabilistic Selection and Best Site
Selection.
19
In the probabilistic selection, roulette wheel method has been used and sites with better
fitness have more chance to be selected, but in the best selection, the best sites
according to fitness will be selected. Of course in this report different combinations of
selection of two methods from pure probabilistic selection ( q=0 ) to pure best
selection ( q=1 ) have been investigated and mean iterations required to rich the answer
and successfulness are shown in Fig.8. and 9.
Mean Iterations to get the answer in 1000 runs
0
200
400
600
800
1000
1200
0 50 100 150 200 250 300
n ( Population )
Iterations
q=0
q=0.1
q=0.5
q=0.9
q=1
Fig.8. Mean iteration required for Different combination of selection
20
Successfulness
0
20
40
60
80
100
0 50 100 150 200
n ( Population )
Fails in 1000 runs
q=0
q=0.1
q=0.5
q=0.9
q=1
Fig.9. Successfulness of different combinations of selection methods
Regarding to results, best selection will present better results and also is more simple,
so Best Selection technique is recommended to use.
3.3.3.2. Neighbourhood Range
Neighbourhood range is another variable which needs to tune for different types of
problem spaces. And also, it is necessary to discuss different strategies for increasing
the robustness and quality of the algorithm. In this section, different strategies have
been discussed.
Four different strategies have been applied to improve the efficiency and robustness of
the bee algorithm. These are (1) Fixed neighbourhood region width strategy, (2) region
changing according to iteration and (3) hybrid strategies between previous two
21
strategies, for instance, first strategy has been used up to 50th iteration then second
strategy have been used up to end (or up to optimum point).
In first strategy, for all selected sites neighbourhood width has been fixed to a certain
range that is enough to deal with the complexity of a problem space. In this strategy,
beginning from the first iteration all the bees harvest on the same size of the fields
which is defined as “
” in equation (1).
α
=
)2(_ ndStrategyRangeoodNeighbourh
; (1)
However, in second strategy, region changes proportional to iteration. All sites have
the same range value and this range will be narrowed down depends on iteration as
shown in equation (2). This strategy has been established on an idea to increase the
accuracy of the solution.
iteration
)3(_
α
=
rdStrategyRangeoodNeighbourh
; (2)
Finally, a third strategy has been implemented as a combination of first two strategies.
This model has been implemented to improve the efficiency of prior strategies.
Neighbourhood range strategies have been tested using two functions that have also
been used in benchmarking experiments presented in table 1. These tests run
independently 10 times and mean of this 10 run was presented on Fig. 9. Also number
of iteration set up to both tests as 1000 iterations.
22
3902
3902.5
3903
3903.5
3904
3904.5
3905
3905.5
3906
3906.5
0 10 20 30 40 50
Itera tion
Fitness
1st Strategy
2nd Strategy
3rd Strategy
Fig.10. Comparison of different neighbourhood strategies for De Jong Function.
In Fig. 10, three different strategies have been experimented on De Jong test function.
The following parameters were set for this test.- population is 15, selected sites is 5,
elite site is 2, bees around elite points are 4, bees around selected points are 2.
However, neighbourhood spread 0.01 is defined as a initial range for all strategies.
Also for first strategy, first 20 iteration algorithm will follow the first strategy (fixed
ranges) and after that follows second strategy (narrowing down ranges).
Clearly, first strategy reaches the minimum before then other strategies. But, second
and third strategies are also gives similar results for this relatively simple problem
space as well as first strategy. Thus, it does not make sense to use a complex method
like second and third methods. So, the first strategy is a simple and an efficient method
for neighbourhood search
23
3.3.3.3. Recruitment
Recruitment can be done by different methods, which in this research recruitment
proportion to the fitness of the selected sites and equal distribution of bees on the
selected sites have investigated, which for simplicity the second method, equal
distribution, has been chosen. Although numbers of recruited bees are fixing for each
site, but the elite sites will recruits more bees (nep) and other selected sites recruit less
amount of bees (nsp).
24
4. Simulation and Comparison
Two standard functions problems were selected to test the bee algorithm and another
ten for benchmarking. Since the Bee Algorithm searches for the maximum, the
function is inverted for minimization problems.
Shekel’s Foxholes (Fig. 11), a 2D equation from De Jong’s test suite (function 5) is
chosen as the first function for testing the algorithm (Eq. 3.).
==
+
=
25
1
2
1
6
5
)(
1
998.119
)(
1
jiiji
ax
j
xf
(3)
=
323232...3232323232
32160...321601632
a
ij
536.65536.65
i
X
119)32,32)(()max(
55
=
ff
The following parameters were set for this test.- population n = 45, selected sites m =
3, elite site e = 1, neighbourhood spread ngh = 0.6, bees round elite points nep = 7,
bees around selected points nsp = 2.
Fig.11. Inverted Shekel’s Foxholes
25
Fig.12 shows the number of function evaluations to average fitness for 100
independent runs. It can be seen that after approximately 1200 function evaluations,
the Bee Algorithm reaches close proximity of the optimum.
Inverted Shekel's Foxholes
25
35
45
55
65
75
85
95
105
115
125
0 500 1000 1500 2000
Generated Points ( Mean number of Function Evaluations )
Fitness
Fig.12. Result for fitness average and Generated points
To test the reliability of the algorithm, Inverted Schwefel’s function with six
dimensions was chosen (Eq.4). In Fig. 13, a two dimensional model is shown to
highlight the complexity of this function.
xx
i
i
i
xf sin()(
6
1
=
=
(4)
500500
i
X
6:1,9687.420;9829.418*6)(max
===
iXxf
i
The following parameters were set for this test.- population n = 500, selected sites
m=15, elite site e = 5, neighbourhood spread ngh = 0.01, bees around elite points nep=
50, bees around selected points nsp = 30.
26
Fig.13. 2D Schwefel’s function
Inverted Schwefel's Function ( 6 dim )
1550
1650
1750
1850
1950
2050
2150
2250
2350
2450
2550
0 500,000 1,000,000 1,500,000 2,000,000 2,500,000 3,000,000
Generated Points ( mean number of function evaluations )
Fitness
Fig.14. Generated points to Avg. fitness
Fig. 14 illustrates the function evaluations versus average fitness for 100 independent
runs. It can be seen that after approximately 3,000,000 function evaluations, the bee
algorithm reaches close to the optimum.
To compare the Bee Algorithm with other methods, eight functions [2] of various
dimensions were chosen to test the robustness and speed. Test functions, their
intervals, and the global optimum for each have been described in Table 1.
27
Table 1. Test Functions.
Table 2. Bee Algorithms` comparison results between different algorithms
In Table 2. a comparison of the eight functions examined by bee algorithm with
deterministic simplex method and the stochastic simulated annealing optimisation
No Reference Interval Test Function Global Optimum
1 De Jong
X[-
2.048,2.048
]
2
1
2
2
2
1
)1()(100)93.3905(max
xxx
F
=
X[1,1]
F=3905.93
2Goldstein &
Price X[-2,2]
)]273648123218()32(30[
)]361431419()1(1[min
2
2212
2
11
2
21
2
2212
2
11
2
21
xxxxxxxx
xxxxxxxx
X
F
++++
++++++=
X[0,-1]
F=3
3 Branin X[-5,10]
22
7
8
1
,10,6,7
22
5
,
22
7
4
1.5
,1
)cos()1()(min
2
1
2
1
2
12
XfedXcba
efedcbaF
xxxx
====
==
+++=
X[-22/7,12.275]
X[22/7,2.275]
X[66/7,2.475]
F=0.3977272
4Martin &
Gaddy X[0,10]
2
21
2
21
)3/)10(()(min
++=
xxxx
F
X[5,5]
F=0
5 Rosenbrock -1 X[-1.2,1.2]
X[-10,10]
2
1
2
2
2
1
)1()(100min
xxx
F
+=
X[1,1]
F=0
6 Rosenbrock - 2 X[-1.2,1.2]
})1()(100{min
2
1
2
3
1
1
2
xxx
i
ii
F
+=
=+
X[1,1,1,1]
F=0
7Hyper sphere
model
X[-
5.12,5.12]
=
=
6
1
2
min
i
i
x
F
X[0,0,0,0,0,0]
F=0
8 Griewangk X[-512,512]
+
+
=
==
10
1
10
1
2
1cos
4000
1.0
1
max
i
i
i
i
i
F
xx
X[0,0,0,0,0,0,0,0,
0,0]
F=10
func
no
SIMPSA NE SIMPSA GA ANT Bee Algorithm
success %
mean no
of func
evals
success %
mean no
of func
evals
success %
mean
no of
func
evals
success %
mean
no of
func
evals
success %
mean
no of
func
evals
1 *** ***** *** ***** 100 10160 100 6000 100 49
2 *** ***** *** ***** 100 5662 100 5330 100 998.9
3 *** ***** *** ***** 100 7325 100 1936 100 1657.4
4 *** ***** *** ***** 100 2844 100 1688 100 525.76
5a 100 10780 100 4508 100 10212 100 6842 100 898
5b 100 12500 100 5007 *** ***** 100 7505 100 2306
6 99 21177 94 3053 *** ***** 100 8471 100 29185
7 *** ***** *** ***** 100 15468 100 22050 100 7112.9
8 *** ***** *** ***** 100 200000 100 50000 100 1846.8
28
procedure (SIMPSA and NE SIMPSA), genetic algorithm (GA) and ant colony
approach (ANT)is shown. The mean number of function evaluations was obtained
from 100 number independent runs and compared with other available results.
The first test function was De Jong, for which the Bee Algorithm could find the
optimum 120 times faster than ANTS and 207 times faster than GA, with convergence
for all them of 100%. The second function was Goldstein and Price, for which the Bee
Algorithm reaches the optimum almost 5 times faster than ANT and GA with 100%
reliability for all three methods. In the Branin function, there are 15% improvement
compares with ANT and 77% over GA, again with 100% convergence.
Function 5 and 6, were Rosenbrock functions with two and four dimensions. In
two dimensions Rosenbrock function in different intervals, the Bee Algorithm delivers
a very good performance with 100% reliability and good improvement over other
methods (at least two times less than other methods). In case of four dimensions, the
Bee Algorithm needed more function evaluations to get the optimum with 100%
success. NE SIMPSA could find the optimum in 10 times fewer function evaluations
but the success was 94% and ANT method found the optimum with 100% success and
3.5 times faster than Bee Algorithm. Test function 7 was Hyper Sphere model with six
dimensions which bee algorithm needed half of the function evaluations compares
with GA and one third of those required for ANT method as reliability was 100%. Last
test function was a ten dimensional problem known as Griewangk and Bee Algorithm
shows huge differences with GA and ANT. Bee Algorithm could reach the optimum
10 times faster than GA and 25 times faster than ANT with 100% success.
29
5. Conclusion and Discussion
This paper presented a new algorithm for continuous function optimization.
Experimental results on uni-modal and multi-modal functions in n-dimensions showed
that the proposed algorithm has remarkable robustness and agility producing a 100%
success rate. It converges to the maximum or minimum without becoming stuck at
local optima. The Bee Algorithm outperformed other techniques that were compared in
terms of speed and accuracy of the results obtained. One of the drawbacks of the
algorithm would be the number of tuneable parameters used. However, it was possible
to set the parameter values with a few trials.
Other evolutionary algorithms converge quickly making use of gradient information.
However, our algorithm makes little use of this gradient field information in the
neighbourhood, thus helping to escape from being stuck at local optima. Further work
should address the reduction of tuneable parameters and the incorporation of better
learning mechanisms.
Acknowledgement
The authors would like to thank MEC to provide the facility to do the research.
30
Appendix A Program List
Main :
// Bee2D.cpp : Defines the entry point for the console application.
#include "stdafx.h"
#include <stdlib.h>
#include <time.h>
#include <math.h>
#include <stdio.h>
#include <conio.h>
#include <iostream.h>
#include <limits.h>
#include <fstream.h>
#include <iomanip.h>
#include "params.h"
#include "func.h"
void main()
{
cout<<"Started..................\n";
int i,d,j,k,temp2,aa[50],ranSearchBees,counter1,runs,fail,iter,loop1; //50 max
number of selected sites out of n
double temp1 ;
double nb,sum_fit;
double randselection;
int a=20;
int gen[1000],ite[1000]; // Generated Points
int R=100; // number of runs
double nghx[dim]; // Neighborhood X[]-Direction ( m )
double q; // random number which indicates Exploration or Exploitation
int Flag_a = 1; // 1: Equal distribution of bees in selected sites and more in ellit
sites
// 2: Proportion to fitness distribution of bees in selected sites
double bPos[dim][pop],bNghPos[dim][pop],fit[pop],bNghFit[pop],sortedFit[pop];
double percentsort[pop],candidx[dim][pop];
double totalsort = 0.0,shiftSortedFit[pop],sumsort[pop],bPosSort[dim][pop];
double gen_fit[700][100],mgen_fit[1000];
ofstream Result;
Result.open("Result.txt");
31
// Different random numer each time
srand( (unsigned)time( NULL ) );
for(runs=0; runs<R; runs++) { //number of runs
//Random distribution
for(i=0;i<n;i++)
{
for(d=0;d<dim;d++)
bPos[d][i]=randfunc(start_x[d],end_x[d]) ;
fit[i]=func(bPos,i);
}
ranSearchBees=n-m-e; // Number of bees search randomly
//imax number of iteration
for(iter=0; iter<imax ;iter++)
{
for(d=0;d<dim;d++)
nghx[d] = ngh;
//sort fit & pos
funcSort(fit, sortedFit, bPos, bPosSort, n);
counter1=0;
for(k=0;k<e;k++)
{
for(d=0;d<dim;d++)
bPos[d][counter1]=bPosSort[d][k];
counter1++;
}
//Roulette wheel selection to choose sites or choosing best sites
q=myrandom(); // choose a number b/w 0 and 1
for (i=0;i<=n;i++) sumsort[i] = 0.0;
if (sortedFit[n-1]<0.0)
for(k=0;k<n;k++)
shiftSortedFit[k]= sortedFit[k]-sortedFit[n-1]+1;
else
for(k=0;k<n;k++)
shiftSortedFit[k] = sortedFit[k];
if(q < q0) { // Choosing best m
for(i=0;i<m;i++)
for(d=0;d<dim;d++)
candidx[d][i] = bPosSort[d][i];
32
}
else { // Choosing sites probabalistically
for (i=0;i<m;i++)
{
if( i < e ) // choosing the best (e) sites
{
for(d=0;d<dim;d++)
candidx[d][i] =
bPosSort[d][i];
shiftSortedFit[i]=0.0;
continue;
}
// choosing the (m-e) sites using ( R/W)
randselection = myrandom();
totalsort = 0.0;
for (k=0;k<n;k++) totalsort +=
shiftSortedFit[k];
for (k=0;k<n;k++) percentsort[k] =
shiftSortedFit[k]/totalsort;
for (k=1;k<=n;k++)
sumsort[k] = sumsort[k-1] +
percentsort[k-1];
for (temp2=0;temp2<n;temp2++)
{
if (sumsort[temp2]<randselection &&
sumsort[temp2+1]>=randselection)
{
for(d=0;d<dim;d++)
candidx[d][i] = bPosSort[d]
[temp2];
shiftSortedFit[temp2]=0.0;
break;
}
}
}
}
///////////////////////////////
// changing number of bees around each selected point
//condition for max values of n1 & n2
if (n1>(n-m-e)) n1 = n-m-e;
33
if (n2>(n-m-e)) n2 = n-m-e;
if(Flag_a == 1) // Equal distribution of bees in selected sites
{
for(i=0;i<m;i++)
{
if(i<e)
aa[i]=n2; // Number of bees around each ellite point/s
else
aa[i]=n1; // Number of bees around other selected
point
}
}
else // Proportion to fitness distribution of bees in selected sites
{
sum_fit=0;
if (sortedFit[n-1]<0.0)
{
for(i=0;i<m;i++)
sum_fit=sum_fit+func(candidx,i)-sortedFit[n-1];
for(i=0;i<m;i++)
{
nb=((func(candidx,i)-sortedFit[n-
1])/sum_fit)*a+.5;
aa[i]=nb; // Number of bees around each
selected point
if(aa[i] < 1)
aa[i]=1;
}
}
else
{
for(i=0;i<m;i++)
sum_fit=sum_fit+func(candidx,i);
for(i=0;i<m;i++)
{
nb=(func(candidx,i)/sum_fit)*a+.5;
aa[i]=nb; // Number of bees around each
selected point
if(aa[i]<1) aa[i]=1;
}
}
}
// Search in the neighbourhood
34
temp1=-INT_MAX;
for(k=0;k<m;k++)//k site
{
for(j=0;j<aa[k];j++) //j recruited bee
{
for(d=0;d<dim;d++)//d dimension
{
if ((candidx[d][k]-nghx[d])<start_x[d]) //
boundry check (left)
bNghPos[d]
[j]=randfunc(start_x[d],candidx[d][k]+nghx[d]);
else if ((candidx[d][k]+nghx[d])>end_x[d])//
boundry check (right)
bNghPos[d][j]=randfunc(candidx[d][k]-
nghx[d],end_x[d]);
else
bNghPos[d][j]=randfunc(candidx[d][k]-
nghx[d] , candidx[d][k]+nghx[d]);
}
}// end of recruitment
for(j=0;j<aa[k];j++) // evaluate fitness for recruited bees
bNghFit[j]=func(bNghPos,j);
for(j=0;j<aa[k];j++) // choosing the rep bee
if(bNghFit[j]>= temp1 )
{
temp1=bNghFit[j];
temp2=j;
} // end of choosing the rep bee
for(d=0;d<dim;d++)
bPos[d][counter1]=bNghPos[d][temp2];
counter1++; // next member of the new list
temp1=-INT_MAX; //
} // end of Neighbourhood Search
for(k=0;k<ranSearchBees;k++) //start of rand search for rest of
bees
{
for(d=0;d<dim;d++)
bPos[d]
[counter1]=randfunc(start_x[d],end_x[d]);
counter1++;
}
35
for(j=0;j<n;j++) // evalute the fitness of the new list
fit[j]=func(bPos,j);
gen_fit[iter][runs]=-fit[0];
} //end iter = imax
fprintf(Array,"%d, %f, %f, %f \n",runs+1,fit[0],bPos[0][0],bPos[1][0]);
} // end runs
Result.close();
cout<<"\n finished..................\n";
}
Subroutines:
// Parameters used in main.cpp
#define dim 2 // Dimension of function
// Variables used
#define pop 500 // max num of population
int imax=100;
int n= 45; // Number of Scout Bees
int m= 3; // Number of selected Locations
int n1 =2; // Number of Bees around each selected locations ( except the elite
location )
int n2 =7; // Number of Bees around each ellit locations
int e =1; // Elite point/s
double ngh=.06; // Neighbourhood search domain
double q0=0.90; // Balancing b/w Exploration ( Probabalistically choosing m
sites)
// and Exploitation ( choosing best m sites )
// Function used in main.cpp
#include <iostream.h>
double start_x[]={-65.536,-65.536};
double end_x[]={65.536,65.536};
double ans = -0.9982;
double func(double x[][pop], int i ) // Definition of Fitness Function
{
double y;
double a[25][2]={{-32,-32},{-16,-32},{0,-32},{16,-32},{32,-32},{-32,-16},{-16,-16},
{0,-16},{16,-16},{32,-16},{-32,0},{-16,0},{0,0},{16,0},{32,0},{-32,16},{-16,16},
{0,16},{16,16},{32,16},{-32,32},{-16,32},{0,32},{16,32},{32,32}};
36
y =0.002;
for(int j=0;j<25;j++)
{
y=y+1/((j+1)+pow(x[0][i]-a[j][0],6)+pow(x[1][i]-a[j][1],6));
}
y=-1/y;
return y;
}
double randfunc( double xs, double xe ) // Definition of Randon number Generator
Function
{
double randnum;
randnum=rand() ;
randnum=xs+randnum*(xe-xs) / RAND_MAX;
return randnum;
}
double myrandom()
{
double r;
int M,x;
M = 10000;
x= M-2;
r = (1.0+(rand()%x))/M;
return r;
}
void funcSort(double inP1[],double oP1[],double inP2[][pop],double oP2[][pop],int
size)
{
int temp2;
double temp1=-INT_MAX;
for( int j=0;j<size;j++) //sort
{
for(int k=0;k<size;k++)
if(inP1[k]> temp1 )
{
temp1=inP1[k];
temp2=k;
}
oP1[j]=temp1;
for(int d=0;d<dim;d++)
oP2[d][j]=inP2[d][temp2];
temp1=-INT_MAX;
37
inP1[temp2]=-INT_MAX;
} //end sort
}
38
References
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40
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This paper describes a form of dynamical computational system—the ant colony—and presents an ant colony model for continuous space optimisation problems. The ant colony metaphor is applied to a real world heavily constrained engineering design problem. It is capable of accelerating the search process and finding acceptable solutions which otherwise could not be discovered by a GA. By integrating the Pareto optimality concept within the selection mechanism in GAs and Ant Colony it is possible to treat both hard and soft constraints. Hard constraints participate in a penalty term while soft constraints become part of a multi-criteria formulation of the problem.
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Many engineering applications often involve the minimization of some objective functions. In the case of multilevel optimizations or functions with many local minimums, the optimization becomes very difficult. Biology-inspired algorithms such as genetic algorithms are more effective than conventional algorithms under appropriate conditions. In this paper, we intend to develop a new virtual bee algorithm (VBA) to solve the function optimizations with the application in engineering problems. For the functions with two-parameters, a swarm of virtual bees are generated and start to move randomly in the phase space. These bees interact when they find some target nectar corresponding to the encoded values of the function. The solution for the optimization problem can be obtained from the intensity of bee interactions. The simulations of the optimization of De Jong’s test function and Keane’s multi-peaked bumpy function show that the one agent VBA is usually as effective as genetic algorithms and multiagent implementation optimizes more efficiently than conventional algorithms due to the parallelism of the multiple agents. Comparison with the other algorithms such as genetic algorithms will also be discussed in detail.
Conference Paper
Bees organize their foraging activities as a social and com- municative effort, indicating both the direction, distance and quality of food sources to their fellow foragers through a "dance" inside the bee hive (on the "dance floor"). In this paper we present a novel routing algorithm, BeeHive, which has been inspired by the communicative and evaluative methods and procedures of honey bees. In this algorithm, bee agents travel through network regions called foraging zones .O n their way their information on the network state is delivered for updating the local routing tables. BeeHive is fault tolerant, scalable, and relies com- pletely on local, or regional, information, respectively. We demonstrate through extensive simulations that BeeHive achieves a similar or better performance compared to state-of-the-art algorithms.