154 Agent-Based Modeling and Simulation with Swarm
5
2.5
0
2.5
5
5
2.5
0
2.5
5
0
20
40
60
80
5
2.5
0
2.5
FIGURE 6.16:
Rastrigin’s function (F 8).
10
5
0
5
10
10
5
0
5
10
5
6
7
0
5
0
5
FIGURE 6.17:
Griewangk’s function (F 9).
sp e c ific. One of the reasons PSO has gathered so much attention is the ten-
dency of its individuals to proceed directly toward the targe t. This feature
resembles the behavior of BUGS, which is described in Section 6.6.
In the chapter “T he Optimal Allocation of Trials” in his book [52], Holland
ascribes the success of EC to the balance of “exploitation,” through search of
known regions, with “explora tion,” thro ugh search, at finite r isks, of unknown
regions. PSO is adept at managing such subtle balances. These stochastic fa c -
tors enable PSO to ma ke thorough searches of the relatively promising regions
and, due to the momentum of sp e e d, also allows effective searches o f unknown
regions. Theoretical research is currently underway to derive optimized values
for PSO parameters by mathematical a nalysis, for stability and convergence
(see [18, 72]).
6.4.3 Examples of PSO applications
PSO has been applied to an ana ly sis of trembling of the human body [34].
Trembling has two types, ordinary shivering and the type of shaking that is
Particle S warm Simulation 155
1e-005
0.0001
0.001
0.01
0.1
1
10
0 5 10 15 20 25 30 35 40 45 50
log(Fitness)
Generation
original Rastrigin function
GA
PSO
PSO with Gaussian
FIGURE 6.18:
Standard PSO versus PSO with a Gaussian mutation for F 8.
caused by Parkinson’s disease or o ther illnesses. The authors used a combi-
nation of PSO and a neural network to distinguish between the types. The
sigmoid function given below was optimized with PSO in a layered network
with 60 input units, 12 hidden nodes, and 2 output units, thus:
output =
1
1 + e
−k
P
w
i
x
i
,
where x
i
and w
i
were the inputs and weights to each of the hidden layers
and output layers, respectively. O ptimizatio n of the weight indirectly causes
changes in the network structure. Ten healthy controls and twelve patients
took part in this experiment. The system succeeded in distinguishing correctly
between the types of shaking in the subjects with 100% accuracy.
PSO has been applied to pro blems of electric power network s [86]. In their
research, the experiments were conducted employing s e lec tion procedures that
were effective for standard PSO and an extended version (EPSO) with a self-
adaptive feature. The problem of “losses” in electric power networks refers to
searching out the series of control actions needed to minimize power losses.
The objective function for this included the level of excitation of generators
and adjustments to the connections to transformers and condensers, i.e., the
control variables included both continuous and discrete types. The maximum
power flow and the pe rmitted voltage level were imposed as b oundary con-
ditions, and the algorithm searched for the solution with the minimum loss.
Miranda and Fonseca [81 ] conducted a comparative experiment with EPSO
and simulated annealing (SA), conducting 270 runs in each system and com-
paring the mean of the r e sults. EPSO rapidly identified a solution that was
close to the optimal one. SA converged mo re slowly. Comparison of the mean
square errors indicated that SA did not have as high a probability of arriv-
ing at the optimal solution as EPSO. PSO has also been successfully applied
156 Agent-Based Modeling and Simulation with Swarm
1e-006
1e-005
0.0001
0.001
0.01
0.1
0 5 10 15 20 25 30 35 40 45 50
log(Fitness)
Generation
generalized Rastrigin function
GA
PSO
PSO with Gaussian
FIGURE 6.19:
Standard PSO versus PSO with a Gaussian mutation for F 9.
to the ec onomic load dispatch (ELD) problem for lea st cost power genera-
tion [42, 95]. These findings indicate that PSO can be trusted as a sufficiently
robust method for solving real problems.
Practical research has als o been conducted applying PSO to optimize
the mixing of materials for the production of valuable excretions by micro-
organisms [7 2]. The authors compared PSO with traditiona l methods of exper -
imental design, finding that the mixture indicated by PSO resulted in more
than a doubling of performa nce . When materials of low quality were used,
the search efficiency was quite poor in the initial stages, but ultimately, PSO
provided super ior results. These findings confirmed that PSO offers good ro-
bustness against changes in the environment.
6.5 ABC algorithm
Bees, along with ants, are well-known examples of social insects (Fig. 6.20).
Bees are classified into three types: employed bees, onlooker b e e s, and scout
bees. Employed bees fly in the vicinity of feeding sites they h ave identified,
sending informatio n about food to onlooker b e e s. Onlooker bees use the in-
formation from employed bees to perform selective searches for the best food
sources from the feeding site. When information about a feeding site is not
updated for a given period of time, its employed bees abandon it and become
scout bees that search for a new feeding site. The objective of a bee colony
is to find the highest-rated feeding sites. The population is approximately
half employed bees and scout bees (about 10–15% of the total) ; the rest are
onlooker bees.
Particle S warm Simulation 157
FIGURE 6.20: A bee colony.
The waggle dance (a series of movements) performed by employed be e s to
transmit information to onlooker bees is well known (Fig. 6.21). T he dance
involves s haking the hindquarters and indicating the angle with which the sun
will be positioned when flying straight to the food source, with the sun rep-
resented as straight up. For example, a waggle da nce performed horizontally
and to the right with respect to the nest co mbs means “fly with the sun at 90
degrees to the left.” The speed of shaking the rea r indicates the distance to
the food; when the rear is shaken quickly, the food source is very near, and
when shaken slowly it is far away. Communicatio n via similar dances is also
performed with regard to pollen and water collection, as well as the selection
of locations for new hives.
The Artificial Bee Colony (ABC) algorithm [65, 66], initially pro posed by
Karaboga et al., is a swarm optimization algorithm that mimics the fora ging
behavior of honey bees. Since ABC was designed, it has been proved that
ABC, with fewer control para meters, is very effective and competitive with
other search techniques such as Genetic Alg orithm (GA), Particle Swarm Op-
timization (PSO), and Differential Evolution (DE).
In ABC algorithms, an artificial swarm is divided into employed bees,
onlooker bees, and scouts. N d-dimensional solutions to the problem are ran-
domly initialized in the domain and referred to as food sources. Each employed
bee is assigned to a spe c ific food source x
i
and searches for a new food source
158 Agent-Based Modeling and Simulation with Swarm
60 degrees
60 degrees
food source
nest
sun
FIGURE 6.21: Waggle dance.
v
i
by using the following operator:
v
ij
= x
ij
+ rand(−1, 1) × (x
ij
− x
kj
), (6.8)
where k ∈ {1, 2, ··· , N}, k 6= i, and j ∈ {1, 2, ··· , d} are randomly chosen
indices. v
ij
is the jth element of the vector v
i
. If the trail to a food source is
outside of the domain, it is reset to an accepta ble value. The v
i
obtained is
then evaluated and put into comp e tition with x
i
for survival. The bee prefers
the better food source. Unlike employed b e e s, each onlooker bee chooses a
preferable s ource according to the food source’s fitness to do further searches
in the food spa c e using eq. (6.8). This preference scheme is based on the fitness
feedback information fr om employed be e s. In classic ABC [65], the probability
of the food source x
i
that ca n be exploited is expressed as
p
i
=
fit
i
P
N
j=1
fit
j
, (6.9)
where fit
i
is the fitness of the ith food s ource, x
i
. For the sake of simplicity,
we assume that the fitness value is non-negative and tha t the lar ger, the
better. If the trail v
i
is superior to x
i
in terms of profitability, this onlooker
bee informs the relevant employed bee associated with the ith food source,
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