Part I – Swarm Intelligence technicalities
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General introduction
Swarm Intelligence (SI)
Particle Swarm Optimization (PSO) & Differential Evolution (DE)
Artificial Bee Colony (ABC)
Artificial super-Bee enhanced Colony (AsBeC)
ABC vs. AsBeC on benchmark test functions
Part II – Engineering optimization
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Contextualization of real-like problems on turbomachinery
Implementing of bee colony for optimization purposes
Introduction of other techniques:
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Gradient Descent (GD)
Interpolated Random Walk (IRW)
Genetic Algorithm (GeDEA)
Artificial Neural Network (ANN)
Overall comparisons
Conclusive remarks
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The applicative field of numerical optimization in engineering is
normally characterized by simulation based problems heavily time
and resource consuming (CFD, FEM, non-linear models, etc.)
There is a strong need for fast techniques allowing to optimize
many parameters under very few function evaluations.
Since the simulated objective function shape and properties are
generally not well-known, the most widespread techniques lay in
the class of metaheuristic methods and Artificial Intelligence (AI).
Mainly diffused and advanced are evolutionary algorithms (GA, ES
and EP) and Artificial Neural Networks (ANN) as surrogate metamodels, but also simpler approaches like random path-based
methods (Hill Climber, Simulated Annealing) have been and are
still used.
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Another new and nature-inspired strategy to be considered is the
promising Swarm Intelligence (SI). At first, swarm methods could
not appear suited, since a colony needs multiple function
evaluation at every optimization step, without any guaranteed
improvement of the solution. Although some researchers have
proven brilliant performance .
SI class gathers a lot of different algorithms, among which Particle
Swarm Optimization (PSO), Differential Evolution (DE) and
especially other very recent developments like Artificial Bee
Colony (ABC) seems to offer excellent qualities.
Starting from ABC and limiting the total number of function
evaluations, my research was focused on modifying the original
algorithm in order to increase its speed and solution accuracy. The
final up-to-date version of ABC is the subject of an in-dept
scientific paper and is called AsBeC.
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By definition is “the collective behavior of decentralized, selforganized systems, natural or artificial”. The expression was
introduced by Gerardo Beni and Jing Wang in 1989, in the context
of cellular robotic systems. In principle, it should be a multi-agent
self-organized system that shows some intelligent behavior.
SI systems are population-based and consist typically of a
collection of simple agents or bird-like-objects (boids), interacting
locally with one another and with their environment. The agents
follow very simple rules to move in their neighborhood and
although there is no centralized control structure the interactions
between such agents let emerge an "intelligent" global behavior.
The inspiration often comes from nature, especially biological
systems. Natural examples of SI include ant colonies, bird flocking,
animal herding, bacterial growth, and fish schooling.
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Ant colony optimization (ACO)
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Gravitational search algorithm (GSA)
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Glowworm Swarm Optimization (GSO) & Firefly Algorithm (FA)
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Intelligent water drops (IWD) & River Formation Dynamics (RFD)
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Stochastic diffusion search (SDS)
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Firstly introduced by James Kennedy and Russel Ebhart in 1995, it is
an algorithm capable of optimizing non-linear and
multidimensional problems. It usually reaches good solutions
efficiently while requiring minimal parameterization.
The basic concept is to create a swarm of particles which move in
the problem space searching for the place which best suits their
fitness function. There are two main ideas behind its optimization
properties:
 A single particle can determine how good its current position is.
It benefits not only from its space exploration knowledge but
also from the knowledge shared by the other particles
 A stochastic factor in each particle's velocity makes them move
through unknown problem space regions. This property
combined with a good initial distribution of the swarm enable an
extensive exploration of the problem space
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Good explorative skills but poor local search for refinement, so slow
convergence rate, and possibility to get trapped in local minima if
the swarm clusters to early (premature convergence or collapse)
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Local Best
This variation reduces the sharing of information between
particles to a smaller neighborhood, overlapping the
congregations in order to enable convergence to the global
best. This version is slower to converge but it is less susceptible
to local minima[
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Inertia Weight
This variation aims to balance the exploitation of good solutions
and the exploration of new areas, by multiplying the
momentum component in velocity formulation by a specific
inertia weight 0.9<w<1.2
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Antennas
Biomedical
Control
Design
Distribution Networks & Artificial Neural Networks
Electronics and Electromagnetics
Engines and Motors
Fuzzy and Neuro-fuzzy
Image, Graphics , Video and Visualization
Metallurgy
Power Systems and Plants
Prediction and Forecasting
Robotics
Scheduling
Signal Processing
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Differential Evolution optimizes a problem by iteratively trying to
improve a candidate solution with regard to a given measure of
quality. Typical example of metaheuristic:
 It make no assumptions about the problem being optimized
 It can search very large spaces
 It does not guarantee an optimal solution is ever found
DE is used for multidimensional real-valued functions but does not
use the gradient. DE can therefore be used on optimization
problems that are discontinuous, noisy, change over time, etc.
DE maintains a population of candidate solutions and creates new
candidates by combining existing ones according to a simple
formulae. If the new position of an agent is an improvement it is
accepted and forms part of the population, otherwise it is simply
discarded.
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The algorithm was developed by Karaboga in 2005. It is one of the
newest and most promising nature-inspired metaheuristic, which
combines PSO and DE. It reproduces the behavior of a honey bee
colony searching the best nectar source into a target area.
Some bees (employees) are each assigned to a food source and
search the space near it (exploration). Then they come back to the
hive and communicate by dancing the position of the best food
sources found to other bees (onlookers), that help the first ones in
the most promising regions (exploitation). Nectar sources that reveal
themselves non-productive are abandoned in place of eventual
new fruitful positions, investigated by a travelling bee (scout).
In optimization context food sources represents input configurations
and the comparisons among them is based on the objective
function to optimize; non-productive food sources represent
configuration not improved for some time.
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ABC algorithm tries to balance exploration and exploitation,
offering worthy global and local search skills at once. If compared
with other competitive methods (genetic, PSO and its variants and
also FA) ABC demonstrates high-quality, speed, robustness and
flexibility for a great variety of optimization problems.
The main qualities of the algorithm are the following:
 Simple and easy to implement
 It can be parallelized
 It can be hybridized
 It needs few control parameters
 It is flexible and robust to wide range of problems
While the deficiencies can be outlined in:
 No exploitation of the history of points analyzed
 Local search and refinement skills are less efficient with respect
to global search attitude
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The bee movement in for the food source j is based on the
modification on a single parameter i, chosen randomly between all
the possible ones. Another food source k≠j is chosen randomly and
the new position xjnew(i) for the bee associated to the food source j is:
For as regards onlookers, they are assigned to food sources by a
stochastic rule, assuming a certain probability pj related with a fitness
value of the configuration xj of the food source j:
Where SN is the food sources number and fit(xj) is inversely
proportional to the objective function f(xj). Usually fit is set as:
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Since the original paper by Karaboga many researches on the
topic were developed, but no one underlines a performance gain
even with few function evaluations. This framework motivates the
willingness of introducing and analyzing modifications effective with
small bee colonies and few iterations.
Some of the improvements here applied exploits the basic
principles brought in standard ABC by other authors, but some
others introduce novel ideas. These technologies allow to address
ABC deficiencies.
The technologies presented try to speed up the best solutions in
their neighborhood, without clustering the swarm and leading to
premature overall convergence. In fact, the aim of this work is to
improve the local search skills of original ABC (exploitation) without
worsening its global attitude (exploration), especially during the first
search phases.
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The technologies have been classified into two main groups, that
explain the name of the new algorithm: Artificial super-Bee
enhanced Colony (AsBeC).
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Enhancements
These are modifications that do not alter the architecture of the
original ABC, but make it work in a slight different way to match
specific goals, such as improve the velocity on the short
optimization period:
 Each squad of bees can have more time to evolve their nectar
sources (Postponed hive dance)
 Exploration can be privileged setting more than one parameter
to change (Multiple parameter selection)
 For small swarms, the exploitation of the best food sources can
be privileged, penalizing always the worst ones (Strictly biased
onlooker assignment )
 The scout can be relocated in a range that depends on the
position of the food sources (Smart scout repositioning)
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2.
Hybridizations: super-bee concept
These technologies alter the original pseudo-random movement of
the bees, trying to accelerate the optimization process and its
accuracy. Therefore with these modification a bee assumes new
abilities and it will be called super-bee:
 The local behavior of the objective function can be estimated
by linearity (Opposite principle)
 A further evolution is to approximate local concavity of the
objective function (Second order interpolation)
 Data history can be used to make a prediction of the next best
search direction (Prophet)
All the possible combination of technologies were tested in order to
capture all the interactions between them. A statistical analysis on
results obtained for an extensive benchmark test bed allows to
select the best combination among the dominating solutions.
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A set of 10 analytical mathematical test functions have been
selected as a benchmark. Even if this set is far from represent a
good sample of real-world numerical optimizations, it tries to gather
many characteristics that appear in engineering problems. It
contains unimodal, multimodal, separable and not-separable
functions with domain dimensions between 5 and 50. It contains
functions with few far local minima, thousands of closed local
minima, stochastic noise and very narrow holes.
Function name
Characteristics
Dimension
Sphere
Dixon – Price
Schwefel
Stochastic Styblinski Tang (15% noise)
Levy
Rastrigin
Perm
Rosenbrock
Ackley
Griewank
US
UN
MS
MS
MS
MS
MN
MN
MN
MN
50
20
5
5
10
10
5
10
10
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Range for each
dimension
-100<xi<100
-10<xi<10
-500<xi<500
-5<xi<5
-100<xi<100
-10<xi<10
-5<xi<5
-5<xi<5
-20<xi<70
-600<xi<600
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Sphere
Dixon-Price
Schwefel
Styblinski Tang
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Levy
Perm
Rastrigin
Rosenbrock
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Ackley
Griewank
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For each test function and for each configuration of technologies
were performed 300 runs with a colony of 16 bees, limit parameter
equal to 10 and 100 overall iterations, corresponding to a maximum
of 1600 function evaluation. MATLAB® coding.
We analyzed the gain G with respect to the standard ABC, intended
to be a delta performance estimator and defined as:
Starting from the previous it is possible to derive the Mean
Logarithmic Gain (MLG) over all the benchmark functions:
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Postponed hive dance
Check=3
Opposition principle
+
Second order
interpolation
Strictly biased
onlooker assignment
Prophet
Step=0.5
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Since a modern workstation offers great calculation power thanks
to numerous processing units, it is straightforward to take
advantage of this technology even without make use of distributed
computing or clusters. As a consequence, the serial AsBeC code
have been modified into parallel versions.
Number of onlookers, employees and food sources is taken equal
to 8. The same optimization procedure can be carried out in up to 8
times less with a swarm of 16 elements. In case where function
evaluation is the bottleneck with respect to the threads creation
and communications, then parallelization factor is close to 8.
Three possible parallelization of bee-colony based algorithm,
already presented in literature, are considered. They will be
implemented together with AsBeC technologies.
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Multi-start parallel approach
It is the simplest way to take advantages from parallelization,
consisting in running many independent instances of the
optimization process in parallel, with different random seeds.
Multi-swarm parallel approach
It is thought to be a better way to exploit the Multi-Start parallel
approach considering the same number of total function
evaluations. Multi-Swarm comprises communication among the
different colony that are running in parallel.
Bee-by-Bee parallel approach
In the BbB half the colony moves all together in parallel. Losses
in performance are expected since there is no improvement
communication during the 8 parallel runs and no sequential
adjourning of upgraded food sources. The colony convergence
slows down but its explorative skills are intensified. This
approach is affordable when time bottleneck are not in threads
communication but in function evaluation.
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Tests with 10^5 function evaluations
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Rastrigin function 2D, -2<xi<3
AsBeC
ABC
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Modern aeronautic Low Pressure gas Turbines (LPTs) for aeronautics
are already characterized by high quality standards, thus they offer
very narrow margins of improvement. Typical design process starts
with a Concept Design (CD) phase, defined using mean-line 1D and
other low-order tools, and evolves through a Preliminary Design (PD)
phase, which allows the geometric definition in details.
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In this framework, the intensive application and tuning of
multidisciplinary high-performance and multi-objective
optimization strategies is the only way to properly handle the
complicated peculiarities of the design.
During the years, different strategies and algorithms that have been
implemented, from the simplest to the forefront ones:
 A basic gradient method;
 A path-based semi-random second order method, Interpolated
Random Walk (IRW);
 Multi-objective Genetic Diversity Evolutionary Algorithm
(GeDEA, University of Padua, Prof. E. Benini and Dr. L. Dal Mas);
 A multi-objective response surface approach based on Artificial
Neural Network (ANN) and Latin Optimal Hypercube (LOH);
 The brand new AsBeC algorithm, SI of bee colony.
Parallelization, speedup arrangements and hybrid strategies.
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PD phase was selected as a real-like design benchmark to illustrate
results. In this phase, 3D blade’s local geometries (typically 5%, 50%
and 95%) are refined by means of Q3D CFD simulations (from ~15 [s]
to ~30 [m] per run).
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In the PD framework, two different type of optimization problems
have been addressed in single row environment:
 Fitting operations 3D/Q3D
It ensures a reliable geometry optimization, consisting in
overlapping the isentropic 3D/Q3D Mach profiles. Challenging 5
dimensional quasi mono-objective optimization problem,
characterized by jagged and very large boundaries not wellknown. The solution may not be unique and the domain space
usually presents many minima with close objective function.
 Geometrical optimization
Core of the PD phase. Inherently multi-objective with strongly
contrasting targets, but easy-knowable boundaries to set for
feasibility. Typically multidisciplinary, at least aero-mechanical,
it is a 6 dimensional problem and 3 reference objective are set:
efficiency, target area and Mach+Convergence.
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Fitting 3D/Q3D
Geometrical
optimization
Radius at Leading Edge
Axial Chord
Tangential Chord
Unguided Turning
Inlet Blade angle
Inlet Wedge Angle
Leading Edge Radius
Exit Blade Angle
Radius at Trailing Edge
Number of Blades(*)
Throat(*)
Leading Edge Eccentricity
Trailing Edge Eccentricity
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HUB section is selected as real-like example benchmark for ABC vs.
AsBeC comparisons. 24 identical runs was performed for serial and
MS versions and then averaged; at least 8 runs for BbB.
Fitting results are presented for ~100 function evaluations (serial)
and ~550 (parallel), corresponding to 30 [m] of machine time.
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Fitting problem represent one of the severest test case to be fine
solved quickly for population-based algorithms, due to boundary
settings. Path-based algorithm (IRW) are advantaged.
Bee colony range independence is impressive, higher for AsBeC
than ABC, especially if compared to GeDEA. To prove this
statement, three set of Boundaries have been considered and
optimization procedures re-performed for averaged results.
Range
Da
K33
K66
Large
5°
20% 20% 20% 10%
4.00E-03
Narrow
3°
5%
5%
1%
3.75E-06
Custom
5°
15% 10% 15%
3%
3.38E-04
5%
KTE
DPout
V [-]
Algorithm
ABC
AsBeC
ABC BbB
AsBeC BbB
GeDEA
Standard deviation of
final ErrorRatio [-]
against Range setting
1.27
0.23
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