The document describes a two-stage optimization strategy called the Eagle Strategy (ES) that combines global and local search algorithms to improve search efficiency. It evaluates applying ES to differential evolution (DE), a popular evolutionary algorithm. ES first uses randomization like Levy flights for global exploration, then switches to DE for intensive local search around promising solutions. The authors validate ES-DE on test functions, finding it requires only 9.7-24.9% of the function evaluations of pure DE. They also apply it to real-world pressure vessel and gearbox design problems, achieving solutions with 14.9-17.7% fewer function evaluations than pure DE.
Constructing a classification model is important in machine learning for a particular task. A
classification process involves assigning objects into predefined groups or classes based on a
number of observed attributes related to those objects. Artificial neural network is one of the
classification algorithms which, can be used in many application areas. This paper investigates
the potential of applying the feed forward neural network architecture for the classification of
medical datasets. Migration based differential evolution algorithm (MBDE) is chosen and
applied to feed forward neural network to enhance the learning process and the network
learning is validated in terms of convergence rate and classification accuracy. In this paper,
MBDE algorithm with various migration policies is proposed for classification problems using
medical diagnosis.
Fuzzy clustering has been widely studied and applied in a variety of key areas of science and
engineering. In this paper the Improved Teaching Learning Based Optimization (ITLBO)
algorithm is used for data clustering, in which the objects in the same cluster are similar. This
algorithm has been tested on several datasets and compared with some other popular algorithm
in clustering. Results have been shown that the proposed method improves the output of
clustering and can be efficiently used for fuzzy clustering.
MULTIPROCESSOR SCHEDULING AND PERFORMANCE EVALUATION USING ELITIST NON DOMINA...ijcsa
Task scheduling plays an important part in the improvement of parallel and distributed systems. The problem of task scheduling has been shown to be NP hard. The time consuming is more to solve the problem in deterministic techniques. There are algorithms developed to schedule tasks for distributed environment, which focus on single objective. The problem becomes more complex, while considering biobjective.This paper presents bi-objective independent task scheduling algorithm using elitist Nondominated
sorting genetic algorithm (NSGA-II) to minimize the makespan and flowtime. This algorithm generates pareto global optimal solutions for this bi-objective task scheduling problem. NSGA-II is implemented by using the set of benchmark instances. The experimental result shows NSGA-II generates efficient optimal schedules.
Optimal rule set generation using pso algorithmcsandit
Classification and Prediction is an important resea
rch area of data mining. Construction of
classifier model for any decision system is an impo
rtant job for many data mining applications.
The objective of developing such a classifier is to
classify unlabeled dataset into classes. Here
we have applied a discrete Particle Swarm Optimizat
ion (PSO) algorithm for selecting optimal
classification rule sets from huge number of rules
possibly exist in a dataset. In the proposed
DPSO algorithm, decision matrix approach was used f
or generation of initial possible
classification rules from a dataset. Then the propo
sed algorithm discovers important or
significant rules from all possible classification
rules without sacrificing predictive accuracy.
The proposed algorithm deals with discrete valued d
ata, and its initial population of candidate
solutions contains particles of different sizes. Th
e experiment has been done on the task of
optimal rule selection in the data sets collected f
rom UCI repository. Experimental results show
that the proposed algorithm can automatically evolv
e on average the small number of
conditions per rule and a few rules per rule set, a
nd achieved better classification performance
of predictive accuracy for few classes.
An Automatic Medical Image Segmentation using Teaching Learning Based Optimiz...idescitation
Nature inspired population based evolutionary algorithms are very popular with
their competitive solutions for a wide variety of applications. Teaching Learning based
Optimization (TLBO) is a very recent population based evolutionary algorithm evolved
on the basis of Teaching Learning process of a class room. TLBO does not require any
algorithmic specific parameters. This paper proposes an automatic grouping of pixels into
different homogeneous regions using the TLBO. The experimental results have
demonstrated the effectiveness of TLBO in image segmentation.
Constructing a classification model is important in machine learning for a particular task. A
classification process involves assigning objects into predefined groups or classes based on a
number of observed attributes related to those objects. Artificial neural network is one of the
classification algorithms which, can be used in many application areas. This paper investigates
the potential of applying the feed forward neural network architecture for the classification of
medical datasets. Migration based differential evolution algorithm (MBDE) is chosen and
applied to feed forward neural network to enhance the learning process and the network
learning is validated in terms of convergence rate and classification accuracy. In this paper,
MBDE algorithm with various migration policies is proposed for classification problems using
medical diagnosis.
Fuzzy clustering has been widely studied and applied in a variety of key areas of science and
engineering. In this paper the Improved Teaching Learning Based Optimization (ITLBO)
algorithm is used for data clustering, in which the objects in the same cluster are similar. This
algorithm has been tested on several datasets and compared with some other popular algorithm
in clustering. Results have been shown that the proposed method improves the output of
clustering and can be efficiently used for fuzzy clustering.
MULTIPROCESSOR SCHEDULING AND PERFORMANCE EVALUATION USING ELITIST NON DOMINA...ijcsa
Task scheduling plays an important part in the improvement of parallel and distributed systems. The problem of task scheduling has been shown to be NP hard. The time consuming is more to solve the problem in deterministic techniques. There are algorithms developed to schedule tasks for distributed environment, which focus on single objective. The problem becomes more complex, while considering biobjective.This paper presents bi-objective independent task scheduling algorithm using elitist Nondominated
sorting genetic algorithm (NSGA-II) to minimize the makespan and flowtime. This algorithm generates pareto global optimal solutions for this bi-objective task scheduling problem. NSGA-II is implemented by using the set of benchmark instances. The experimental result shows NSGA-II generates efficient optimal schedules.
Optimal rule set generation using pso algorithmcsandit
Classification and Prediction is an important resea
rch area of data mining. Construction of
classifier model for any decision system is an impo
rtant job for many data mining applications.
The objective of developing such a classifier is to
classify unlabeled dataset into classes. Here
we have applied a discrete Particle Swarm Optimizat
ion (PSO) algorithm for selecting optimal
classification rule sets from huge number of rules
possibly exist in a dataset. In the proposed
DPSO algorithm, decision matrix approach was used f
or generation of initial possible
classification rules from a dataset. Then the propo
sed algorithm discovers important or
significant rules from all possible classification
rules without sacrificing predictive accuracy.
The proposed algorithm deals with discrete valued d
ata, and its initial population of candidate
solutions contains particles of different sizes. Th
e experiment has been done on the task of
optimal rule selection in the data sets collected f
rom UCI repository. Experimental results show
that the proposed algorithm can automatically evolv
e on average the small number of
conditions per rule and a few rules per rule set, a
nd achieved better classification performance
of predictive accuracy for few classes.
An Automatic Medical Image Segmentation using Teaching Learning Based Optimiz...idescitation
Nature inspired population based evolutionary algorithms are very popular with
their competitive solutions for a wide variety of applications. Teaching Learning based
Optimization (TLBO) is a very recent population based evolutionary algorithm evolved
on the basis of Teaching Learning process of a class room. TLBO does not require any
algorithmic specific parameters. This paper proposes an automatic grouping of pixels into
different homogeneous regions using the TLBO. The experimental results have
demonstrated the effectiveness of TLBO in image segmentation.
A Mathematical Programming Approach for Selection of Variables in Cluster Ana...IJRES Journal
Data clustering is a common technique for statistical data analysis; it is defined as a class of
statistical techniques for classifying a set of observations into completely different groups. Cluster analysis
seeks to minimize group variance and maximize between group variance. In this study we formulate a
mathematical programming model that chooses the most important variables in cluster analysis. A nonlinear
binary model is suggested to select the most important variables in clustering a set of data. The idea of the
suggested model depends on clustering data by minimizing the distance between observations within groups.
Indicator variables are used to select the most important variables in the cluster analysis.
AUTOMATIC TRANSFER RATE ADJUSTMENT FOR TRANSFER REINFORCEMENT LEARNINGgerogepatton
This paper proposes a novel parameter for transfer reinforcement learning to avoid over-fitting when an
agent uses a transferred policy from a source task. Learning robot systems have recently been studied for
many applications, such as home robots, communication robots, and warehouse robots. However, if the
agent reuses the knowledge that has been sufficiently learned in the source task, deadlock may occur and
appropriate transfer learning may not be realized. In the previous work, a parameter called transfer rate
was proposed to adjust the ratio of transfer, and its contribution include avoiding dead lock in the target
task. However, adjusting the parameter depends on human intuition and experiences. Furthermore, the
method for deciding transfer rate has not discussed. Therefore, an automatic method for adjusting the
transfer rate is proposed in this paper using a sigmoid function. Further, computer simulations are used to
evaluate the effectiveness of the proposed method to improve the environmental adaptation performance in
a target task, which refers to the situation of reusing knowledge.
A Combined Approach for Feature Subset Selection and Size Reduction for High ...IJERA Editor
selection of relevant feature from a given set of feature is one of the important issues in the field of
data mining as well as classification. In general the dataset may contain a number of features however it is not
necessary that the whole set features are important for particular analysis of decision making because the
features may share the common information‟s and can also be completely irrelevant to the undergoing
processing. This generally happen because of improper selection of features during the dataset formation or
because of improper information availability about the observed system. However in both cases the data will
contain the features that will just increase the processing burden which may ultimately cause the improper
outcome when used for analysis. Because of these reasons some kind of methods are required to detect and
remove these features hence in this paper we are presenting an efficient approach for not just removing the
unimportant features but also the size of complete dataset size. The proposed algorithm utilizes the information
theory to detect the information gain from each feature and minimum span tree to group the similar features
with that the fuzzy c-means clustering is used to remove the similar entries from the dataset. Finally the
algorithm is tested with SVM classifier using 35 publicly available real-world high-dimensional dataset and the
results shows that the presented algorithm not only reduces the feature set and data lengths but also improves the
performances of the classifier.
MOCANAR: A Multi-Objective Cuckoo Search Algorithm for Numeric Association Ru...csandit
Extracting association rules from numeric features
involves searching a very large search space. To
deal with this problem, in this paper a meta-heuris
tic algorithm is used that we have called
MOCANAR. The MOCANAR is a Pareto based multi-object
ive cuckoo search algorithm which
extracts high quality association rules from numeri
c datasets. The support, confidence,
interestingness and comprehensibility are the objec
tives that have been considered in the
MOCANAR. The MOCANAR extracts rules incrementally,
in which, in each run of the algorithm, a
small number of high quality rules are made. In thi
s paper, a comprehensive taxonomy of meta-
heuristic algorithm have been presented. Using this
taxonomy, we have decided to use a Cuckoo
Search algorithm because this algorithm is one of t
he most matured algorithms and also, it is simple
to use and easy to comprehend. In addition, until n
ow, to our knowledge this method has not been
used as a multi-objective algorithm and has not bee
n used in the association rule mining area. To
demonstrate the merit and associated benefits of th
e proposed methodology, the methodology has
been applied to a number of datasets and high quali
ty results in terms of the objectives were
extracted
MARKOV CHAIN AND ADAPTIVE PARAMETER SELECTION ON PARTICLE SWARM OPTIMIZERijsc
Particle Swarm Optimizer (PSO) is such a complex stochastic process so that analysis on the stochastic
behavior of the PSO is not easy. The choosing of parameters plays an important role since it is critical in
the performance of PSO. As far as our investigation is concerned, most of the relevant researches are
based on computer simulations and few of them are based on theoretical approach. In this paper,
theoretical approach is used to investigate the behavior of PSO. Firstly, a state of PSO is defined in this
paper, which contains all the information needed for the future evolution. Then the memory-less property of
the state defined in this paper is investigated and proved. Secondly, by using the concept of the state and
suitably dividing the whole process of PSO into countable number of stages (levels), a stationary Markov
chain is established. Finally, according to the property of a stationary Markov chain, an adaptive method
for parameter selection is proposed.
Population based optimization algorithms improvement using the predictive par...IJECEIAES
A new efficient improvement, called Predictive Particle Modification (PPM), is proposed in this paper. This modification makes the particle look to the near area before moving toward the best solution of the group. This modification can be applied to any population algorithm. The basic philosophy of PPM is explained in detail. To evaluate the performance of PPM, it is applied to Particle Swarm Optimization (PSO) algorithm and Teaching Learning Based Optimization (TLBO) algorithm then tested using 23 standard benchmark functions. The effectiveness of these modifications are compared with the other unmodified population optimization algorithms based on the best solution, average solution, and convergence rate.
How to optimize with the help of the Particle Swarm Optimization Technique and xlOptimizer ? This brief tutorial will enable you to solve any optimization problem with the application of Particle Swarm Optimization Method. After a brief introduction about the method the tutorial will show you the steps that you will need to follow for application of PSO in optimization even if you do not know any programming.(Some basic knowledge of MS Excel 2010 and later is required).
For three decades, many mathematical programming methods have been developed to solve optimization problems. However, until now, there has not been a single totally efficient and robust method to coverall optimization problems that arise in the different engineering fields.Most engineering application design problems involve the choice of design variable values that better describe the behaviour of a system.At the same time, those results should cover the requirements and specifications imposed by the norms for that system. This last condition leads to predicting what the entrance parameter values should be whose design results comply with the norms and also present good performance, which describes the inverse problem.Generally, in design problems the variables are discreet from the mathematical point of view. However, most mathematical optimization applications are focused and developed for continuous variables. Presently, there are many research articles about optimization methods; the typical ones are based on calculus,numerical methods, and random methods.
The calculus-based methods have been intensely studied and are subdivided in two main classes: 1) the direct search methods find a local maximum moving a function over the relative local gradient directions and 2) the indirect methods usually find the local ends solving a set of non-linear equations, resultant of equating the gradient from the object function to zero, i.e., by means of multidimensional generalization of the notion of the function’s extreme points from elementary calculus given smooth function without restrictions to find a possible maximum which is to be restricted to those points whose slope is zero in all directions. The real world has many discontinuities and noisy spaces, which is why it is not surprising that the methods depending upon the restrictive requirements of continuity and existence of a derivative, are unsuitable for all, but a very limited problem domain. A number of schemes have been applied in many forms and sizes. The idea is quite direct inside a finite search space or a discrete infinite search space, where the algorithms can locate the object function values in each space point one at a time. The simplicity of this kind of algorithm is very attractive when the numbers of possibilities are very small. Nevertheless, these outlines are often inefficient, since they do not complete the requirements of robustness in big or highly-dimensional spaces, making it quite a hard task to find the optimal values. Given the shortcomings of the calculus-based techniques and the numerical ones the random methods have increased their popularity.
Feature selection using modified particle swarm optimisation for face recogni...eSAT Journals
Abstract
One of the major influential factors which affects the accuracy of classification rate is the selection of right features. Not all features have vital role in classification. Many of the features in the dataset may be redundant and irrelevant, which increase the computational cost and may reduce classification rate. In this paper, we used DCT(Discrete cosine transform) coefficients as features for face recognition application. The coefficients are optimally selected based on a modified PSO algorithm. In this, the choice of coefficients is done by incorporating the average of the mean normalized standard deviations of various classes and giving more weightage to the lower indexed DCT coefficients. The algorithm is tested on ORL database. A recognition rate of 97% is obtained. Average number of features selected is about 40 percent for a 10 × 10 input. The modified PSO took about 50 iterations for convergence. These performance figures are found to be better than some of the work reported in literature.
Keywords: Particle swarm optimization, Discrete cosine transform, feature extraction, feature selection, face recognition, classification rate.
AN IMPROVED ITERATIVE METHOD FOR SOLVING GENERAL SYSTEM OF EQUATIONS VIA GENE...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
MEDICAL DIAGNOSIS CLASSIFICATION USING MIGRATION BASED DIFFERENTIAL EVOLUTION...cscpconf
Constructing a classification model is important in machine learning for a particular task. A
classification process involves assigning objects into predefined groups or classes based on a
number of observed attributes related to those objects. Artificial neural network is one of the
classification algorithms which, can be used in many application areas. This paper investigates
the potential of applying the feed forward neural network architecture for the classification of
medical datasets. Migration based differential evolution algorithm (MBDE) is chosen and
applied to feed forward neural network to enhance the learning process and the network
learning is validated in terms of convergence rate and classification accuracy. In this paper,
MBDE algorithm with various migration policies is proposed for classification problems using
medical diagnosis.
COMPARISON BETWEEN THE GENETIC ALGORITHMS OPTIMIZATION AND PARTICLE SWARM OPT...IAEME Publication
Close range photogrammetry network design is referred to the process of placing a set of
cameras in order to achieve photogrammetric tasks. The main objective of this paper is tried to find
the best location of two/three camera stations. The genetic algorithm optimization and Particle
Swarm Optimization are developed to determine the optimal camera stations for computing the three
dimensional coordinates. In this research, a mathematical model representing the genetic algorithm
optimization and Particle Swarm Optimization for the close range photogrammetry network is
developed. This paper gives also the sequence of the field operations and computational steps for this
task. A test field is included to reinforce the theoretical aspects.
A Mathematical Programming Approach for Selection of Variables in Cluster Ana...IJRES Journal
Data clustering is a common technique for statistical data analysis; it is defined as a class of
statistical techniques for classifying a set of observations into completely different groups. Cluster analysis
seeks to minimize group variance and maximize between group variance. In this study we formulate a
mathematical programming model that chooses the most important variables in cluster analysis. A nonlinear
binary model is suggested to select the most important variables in clustering a set of data. The idea of the
suggested model depends on clustering data by minimizing the distance between observations within groups.
Indicator variables are used to select the most important variables in the cluster analysis.
AUTOMATIC TRANSFER RATE ADJUSTMENT FOR TRANSFER REINFORCEMENT LEARNINGgerogepatton
This paper proposes a novel parameter for transfer reinforcement learning to avoid over-fitting when an
agent uses a transferred policy from a source task. Learning robot systems have recently been studied for
many applications, such as home robots, communication robots, and warehouse robots. However, if the
agent reuses the knowledge that has been sufficiently learned in the source task, deadlock may occur and
appropriate transfer learning may not be realized. In the previous work, a parameter called transfer rate
was proposed to adjust the ratio of transfer, and its contribution include avoiding dead lock in the target
task. However, adjusting the parameter depends on human intuition and experiences. Furthermore, the
method for deciding transfer rate has not discussed. Therefore, an automatic method for adjusting the
transfer rate is proposed in this paper using a sigmoid function. Further, computer simulations are used to
evaluate the effectiveness of the proposed method to improve the environmental adaptation performance in
a target task, which refers to the situation of reusing knowledge.
A Combined Approach for Feature Subset Selection and Size Reduction for High ...IJERA Editor
selection of relevant feature from a given set of feature is one of the important issues in the field of
data mining as well as classification. In general the dataset may contain a number of features however it is not
necessary that the whole set features are important for particular analysis of decision making because the
features may share the common information‟s and can also be completely irrelevant to the undergoing
processing. This generally happen because of improper selection of features during the dataset formation or
because of improper information availability about the observed system. However in both cases the data will
contain the features that will just increase the processing burden which may ultimately cause the improper
outcome when used for analysis. Because of these reasons some kind of methods are required to detect and
remove these features hence in this paper we are presenting an efficient approach for not just removing the
unimportant features but also the size of complete dataset size. The proposed algorithm utilizes the information
theory to detect the information gain from each feature and minimum span tree to group the similar features
with that the fuzzy c-means clustering is used to remove the similar entries from the dataset. Finally the
algorithm is tested with SVM classifier using 35 publicly available real-world high-dimensional dataset and the
results shows that the presented algorithm not only reduces the feature set and data lengths but also improves the
performances of the classifier.
MOCANAR: A Multi-Objective Cuckoo Search Algorithm for Numeric Association Ru...csandit
Extracting association rules from numeric features
involves searching a very large search space. To
deal with this problem, in this paper a meta-heuris
tic algorithm is used that we have called
MOCANAR. The MOCANAR is a Pareto based multi-object
ive cuckoo search algorithm which
extracts high quality association rules from numeri
c datasets. The support, confidence,
interestingness and comprehensibility are the objec
tives that have been considered in the
MOCANAR. The MOCANAR extracts rules incrementally,
in which, in each run of the algorithm, a
small number of high quality rules are made. In thi
s paper, a comprehensive taxonomy of meta-
heuristic algorithm have been presented. Using this
taxonomy, we have decided to use a Cuckoo
Search algorithm because this algorithm is one of t
he most matured algorithms and also, it is simple
to use and easy to comprehend. In addition, until n
ow, to our knowledge this method has not been
used as a multi-objective algorithm and has not bee
n used in the association rule mining area. To
demonstrate the merit and associated benefits of th
e proposed methodology, the methodology has
been applied to a number of datasets and high quali
ty results in terms of the objectives were
extracted
MARKOV CHAIN AND ADAPTIVE PARAMETER SELECTION ON PARTICLE SWARM OPTIMIZERijsc
Particle Swarm Optimizer (PSO) is such a complex stochastic process so that analysis on the stochastic
behavior of the PSO is not easy. The choosing of parameters plays an important role since it is critical in
the performance of PSO. As far as our investigation is concerned, most of the relevant researches are
based on computer simulations and few of them are based on theoretical approach. In this paper,
theoretical approach is used to investigate the behavior of PSO. Firstly, a state of PSO is defined in this
paper, which contains all the information needed for the future evolution. Then the memory-less property of
the state defined in this paper is investigated and proved. Secondly, by using the concept of the state and
suitably dividing the whole process of PSO into countable number of stages (levels), a stationary Markov
chain is established. Finally, according to the property of a stationary Markov chain, an adaptive method
for parameter selection is proposed.
Population based optimization algorithms improvement using the predictive par...IJECEIAES
A new efficient improvement, called Predictive Particle Modification (PPM), is proposed in this paper. This modification makes the particle look to the near area before moving toward the best solution of the group. This modification can be applied to any population algorithm. The basic philosophy of PPM is explained in detail. To evaluate the performance of PPM, it is applied to Particle Swarm Optimization (PSO) algorithm and Teaching Learning Based Optimization (TLBO) algorithm then tested using 23 standard benchmark functions. The effectiveness of these modifications are compared with the other unmodified population optimization algorithms based on the best solution, average solution, and convergence rate.
How to optimize with the help of the Particle Swarm Optimization Technique and xlOptimizer ? This brief tutorial will enable you to solve any optimization problem with the application of Particle Swarm Optimization Method. After a brief introduction about the method the tutorial will show you the steps that you will need to follow for application of PSO in optimization even if you do not know any programming.(Some basic knowledge of MS Excel 2010 and later is required).
For three decades, many mathematical programming methods have been developed to solve optimization problems. However, until now, there has not been a single totally efficient and robust method to coverall optimization problems that arise in the different engineering fields.Most engineering application design problems involve the choice of design variable values that better describe the behaviour of a system.At the same time, those results should cover the requirements and specifications imposed by the norms for that system. This last condition leads to predicting what the entrance parameter values should be whose design results comply with the norms and also present good performance, which describes the inverse problem.Generally, in design problems the variables are discreet from the mathematical point of view. However, most mathematical optimization applications are focused and developed for continuous variables. Presently, there are many research articles about optimization methods; the typical ones are based on calculus,numerical methods, and random methods.
The calculus-based methods have been intensely studied and are subdivided in two main classes: 1) the direct search methods find a local maximum moving a function over the relative local gradient directions and 2) the indirect methods usually find the local ends solving a set of non-linear equations, resultant of equating the gradient from the object function to zero, i.e., by means of multidimensional generalization of the notion of the function’s extreme points from elementary calculus given smooth function without restrictions to find a possible maximum which is to be restricted to those points whose slope is zero in all directions. The real world has many discontinuities and noisy spaces, which is why it is not surprising that the methods depending upon the restrictive requirements of continuity and existence of a derivative, are unsuitable for all, but a very limited problem domain. A number of schemes have been applied in many forms and sizes. The idea is quite direct inside a finite search space or a discrete infinite search space, where the algorithms can locate the object function values in each space point one at a time. The simplicity of this kind of algorithm is very attractive when the numbers of possibilities are very small. Nevertheless, these outlines are often inefficient, since they do not complete the requirements of robustness in big or highly-dimensional spaces, making it quite a hard task to find the optimal values. Given the shortcomings of the calculus-based techniques and the numerical ones the random methods have increased their popularity.
Feature selection using modified particle swarm optimisation for face recogni...eSAT Journals
Abstract
One of the major influential factors which affects the accuracy of classification rate is the selection of right features. Not all features have vital role in classification. Many of the features in the dataset may be redundant and irrelevant, which increase the computational cost and may reduce classification rate. In this paper, we used DCT(Discrete cosine transform) coefficients as features for face recognition application. The coefficients are optimally selected based on a modified PSO algorithm. In this, the choice of coefficients is done by incorporating the average of the mean normalized standard deviations of various classes and giving more weightage to the lower indexed DCT coefficients. The algorithm is tested on ORL database. A recognition rate of 97% is obtained. Average number of features selected is about 40 percent for a 10 × 10 input. The modified PSO took about 50 iterations for convergence. These performance figures are found to be better than some of the work reported in literature.
Keywords: Particle swarm optimization, Discrete cosine transform, feature extraction, feature selection, face recognition, classification rate.
AN IMPROVED ITERATIVE METHOD FOR SOLVING GENERAL SYSTEM OF EQUATIONS VIA GENE...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
MEDICAL DIAGNOSIS CLASSIFICATION USING MIGRATION BASED DIFFERENTIAL EVOLUTION...cscpconf
Constructing a classification model is important in machine learning for a particular task. A
classification process involves assigning objects into predefined groups or classes based on a
number of observed attributes related to those objects. Artificial neural network is one of the
classification algorithms which, can be used in many application areas. This paper investigates
the potential of applying the feed forward neural network architecture for the classification of
medical datasets. Migration based differential evolution algorithm (MBDE) is chosen and
applied to feed forward neural network to enhance the learning process and the network
learning is validated in terms of convergence rate and classification accuracy. In this paper,
MBDE algorithm with various migration policies is proposed for classification problems using
medical diagnosis.
COMPARISON BETWEEN THE GENETIC ALGORITHMS OPTIMIZATION AND PARTICLE SWARM OPT...IAEME Publication
Close range photogrammetry network design is referred to the process of placing a set of
cameras in order to achieve photogrammetric tasks. The main objective of this paper is tried to find
the best location of two/three camera stations. The genetic algorithm optimization and Particle
Swarm Optimization are developed to determine the optimal camera stations for computing the three
dimensional coordinates. In this research, a mathematical model representing the genetic algorithm
optimization and Particle Swarm Optimization for the close range photogrammetry network is
developed. This paper gives also the sequence of the field operations and computational steps for this
task. A test field is included to reinforce the theoretical aspects.
The well-known Vehicle Routing Problem (VRP) consist of assigning routeswith a set
ofcustomersto different vehicles, in order tominimize the cost of transport, usually starting from a central
warehouse and using a fleet of fixed vehicles. There are numerousapproaches for the resolution of this kind of
problems, being the metaheuristic techniques the most used, including the Genetic Algorithms (AG). The
number of approachesto the different parameters of an AG (selection, crossing, mutation...) in the literature is
such that it is not easy to take a resolution of a VRP problem directly. This paper aims to simplify this task by
analyzing the best known approaches with standard VRP data sets, and showing the parameter configurations
that offer the best results.
Optimization of Mechanical Design Problems Using Improved Differential Evolut...IDES Editor
Differential Evolution (DE) is a novel evolutionary
approach capable of handling non-differentiable, non-linear
and multi-modal objective functions. DE has been consistently
ranked as one of the best search algorithm for solving global
optimization problems in several case studies. This paper
presents an Improved Constraint Differential Evolution
(ICDE) algorithm for solving constrained optimization
problems. The proposed ICDE algorithm differs from
unconstrained DE algorithm only in the place of initialization,
selection of particles to the next generation and sorting the
final results. Also we implemented the new idea to five versions
of DE algorithm. The performance of ICDE algorithm is
validated on four mechanical engineering problems. The
experimental results show that the performance of ICDE
algorithm in terms of final objective function value, number
of function evaluations and convergence time.
Optimization of Mechanical Design Problems Using Improved Differential Evolut...IDES Editor
Differential Evolution (DE) is a novel evolutionary
approach capable of handling non-differentiable, non-linear
and multi-modal objective functions. DE has been consistently
ranked as one of the best search algorithm for solving global
optimization problems in several case studies. This paper
presents an Improved Constraint Differential Evolution
(ICDE) algorithm for solving constrained optimization
problems. The proposed ICDE algorithm differs from
unconstrained DE algorithm only in the place of initialization,
selection of particles to the next generation and sorting the
final results. Also we implemented the new idea to five versions
of DE algorithm. The performance of ICDE algorithm is
validated on four mechanical engineering problems. The
experimental results show that the performance of ICDE
algorithm in terms of final objective function value, number
of function evaluations and convergence time.
Performance Analysis of Genetic Algorithm as a Stochastic Optimization Tool i...paperpublications3
Abstract: Engineering design problems are complex by nature because of their critical objective functions involving many variables and Constraints. Engineers have to ensure the compatibility with the imposed specifications keeping the manufacturing costs low. Moreover, the methodology may vary according to the design problem.
The main issue is to choose the proper tool for optimization. In the earlier days, a design problem was optimized by some of the conventional optimization techniques like gradient Search, evolutionary optimization, random search etc. These are known as classical methods.
The method is to be properly Chosen depending on the nature of the problem- an incorrect choice may sometimes fail to give the optimal solution. So the methods are less robust.
Now-a-days soft-computing techniques are being widely used for optimizing a function. These are more robust. Genetic algorithm is one such method. It is an effective tool in the realm of stochastic optimization (non-classical). The algorithm produces many strings and generation to reach the optimal point.
The main objective of the paper is to optimize engineering design problems using Genetic Algorithm and to analyze how the algorithm reaches the optima effectively and closely. We choose a mathematical expression for the objective function in terms of the design variables and optimize the same under given constraints using GA.
MOCANAR: A MULTI-OBJECTIVE CUCKOO SEARCH ALGORITHM FOR NUMERIC ASSOCIATION RU...cscpconf
Extracting association rules from numeric features involves searching a very large search space. To
deal with this problem, in this paper a meta-heuristic algorithm is used that we have called
MOCANAR. The MOCANAR is a Pareto based multi-objective cuckoo search algorithm which
extracts high quality association rules from numeric datasets. The support, confidence,
interestingness and comprehensibility are the objectives that have been considered in the
MOCANAR. The MOCANAR extracts rules incrementally, in which, in each run of the algorithm, a
small number of high quality rules are made. In this paper, a comprehensive taxonomy of metaheuristic
algorithm have been presented. Using this taxonomy, we have decided to use a Cuckoo
Search algorithm because this algorithm is one of the most matured algorithms and also, it is simple
to use and easy to comprehend. In addition, until now, to our knowledge this method has not been
used as a multi-objective algorithm and has not been used in the association rule mining area. To
demonstrate the merit and associated benefits of the proposed methodology, the methodology has
been applied to a number of datasets and high quality results in terms of the objectives were
extracted
Evolutionary Computing is a research area within computer science. As the name suggest, it is a special flavour of computing, which draws inspiration from the process of natural evolution. The fundamental metaphor of evolutionary computing relates this powerful natural evolution to a particular style of problem solving – that of trial and error.
An Improved Iterative Method for Solving General System of Equations via Gene...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
An Improved Iterative Method for Solving General System of Equations via Gene...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
Traveling Salesman Problem (TSP) is a kind of NPHard problem which cant be solved in polynomial time for
asymptotically large values of n. In this paper a balanced combination of Genetic algorithm and Simulated Annealing is used. To
improve the performance of finding optimal solution from huge
search space, we have incorporated the use of tournament and
rank as selection operator. And Inver-over operator Mechanism
for crossover and mutation . To illustrate it more clearly an
implementation in C++ (4.9.9.2) has been done.
Index Terms—Genetic Algorithm (GA) , Simulated Annealing
(SA) , Inver-over operator , Lin-Kernighan algorithm , selection
operator , crossover operator , mutation operator.
Cuckoo Search Algorithm: An IntroductionXin-She Yang
This presentation explains the fundamental ideas of the standard Cuckoo Search (CS) algorithm, which also contains the links to the free Matlab codes at Mathswork file exchanges and the animations of numerical simulations (video at Youtube). An example of multi-objective cuckoo search (MOCS) is also given with link to the Matlab code.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Two-Stage Eagle Strategy with Differential Evolution
1. arXiv:1203.6586v1 [math.OC] 29 Mar 2012
Two-Stage Eagle Strategy with Differential Evolution
Xin-She Yang1 and Suash Deb2
1) Mathematics and Scientific Computing,
National Physical Laboratory, Teddington TW11 0LW, UK.
2) Department of Computer Science & Engineering,
C. V. Raman College of Engineering,
Bidyanagar, Mahura, Janla, Bhubaneswar 752054, INDIA.
Abstract
Efficiency of an optimization process is largely determined by the search algorithm and its
fundamental characteristics. In a given optimization, a single type of algorithm is used in most
applications. In this paper, we will investigate the Eagle Strategy recently developed for global
optimization, which uses a two-stage strategy by combing two different algorithms to improve
the overall search efficiency. We will discuss this strategy with differential evolution and then
evaluate their performance by solving real-world optimization problems such as pressure vessel
and speed reducer design. Results suggest that we can reduce the computing effort by a factor
of up to 10 in many applications.
Keywords: eagle strategy; bio-inspired algorithm; differential evolution; optimization.
Reference to this paper should be made as follows:
Yang, X. S. and Deb, S., (2012). ‘Two-Stage Eagle Strategy with Differential Evolution’,
Int. J. Bio-Inspired Computation, Vol. 4, No. 1, pp.1–5.
1 Introduction
Metaheuristic optimization and computational modelling have become popular in engineering design
and industrial applications. The essence of such paradigm is the efficient numerical methods and
search algorithms. It is no exaggeration to say that how numerical algorithms perform will largely
determine the performance and usefulness of modelling and optimization tools (Baeck et al.,1997;
Yang, 2010).
Among all optimization algorithms, metaheuristic algorithms are becoming powerful for solving
tough nonlinear optimization problems (Kennedy and Eberhart, 1995; Price et al., 2005; Yang,
2008; Cui and Cai, 2009). The aim of developing modern metaheuristic algorithms is to enable the
capability of carrying out global search, and good examples of nature-inspired metaheuristics are
particle swarm optimisation (PSO) (Kennedy and Eberhart, 1995) and Cuckoo Search (Yang and
Deb, 2010a). Most metaheuristic algorithms have relatively high efficiency in terms of finding global
optimality.
The efficiency of metaheuristic algorithms can be attributed to the fact that they are designed
to imitate the best features in nature, especially the selection of the fittest in biological systems
which have evolved by natural selection over millions of years. In real-world applications, most data
have noise or associated randomness to a certain degree, some modifications to these algorithms are
1
2. Objective functions f1(x), ..., fN(x)
Initialization and random initial guess xt=0
while (stop criterion)
Global exploration by randomization
Evaluate the objectives and find a promising solution
Intensive local search around a promising solution
via an efficient local optimizer
if (a better solution is found)
Update the current best
end
Update t = t + 1
end
Post-process the results and visualization.
Figure 1: Pseudo code of the eagle strategy.
often required, in combination with some form of averaging or reformulation of the problem. There
exist some algorithms for stochastic optimization, and the Eagle Strategy (ES), develop by Yang
and Deb, is one of such algorithms for dealing with stochastic optimization (Yang and Deb, 2010b).
In this paper, we will investigate the Eagle Strategy further by hybridizing it with differential
evolution (Storn, 1996; Storn and Price, 1997; Price et al., 2005). We first validate the ES by
some multimodal test functions and then apply it to real-world optimization problems. Case studies
include pressure vessel design and gearbox speed reducer design. We will discuss the results and
point out directions for further research.
2 Eagle Strategy
Eagle strategy developed by Xin-She Yang and Suash Deb (Yang and Deb, 2010b) is a two-stage
method for optimization. It uses a combination of crude global search and intensive local search
employing different algorithms to suit different purposes. In essence, the strategy first explores the
search space globally using a L´evy flight random walk, if it finds a promising solution, then an
intensive local search is employed using a more efficient local optimizer such as hill-climbing and
downhill simplex method. Then, the two-stage process starts again with new global exploration
followed by a local search in a new region.
The advantage of such a combination is to use a balanced tradeoff between global search which
is often slow and a fast local search. Some tradeoff and balance are important. Another advantage
of this method is that we can use any algorithms we like at different stages of the search or even at
different stages of iterations. This makes it easy to combine the advantages of various algorithms so
as to produce better results.
It is worth pointing that this is a methodology or strategy, not an algorithm. In fact, we can
use different algorithms at different stages and at different time of the iterations. The algorithm
used for the global exploration should have enough randomness so as to explore the search space
diversely and effectively. This process is typically slow initially, and should speed up as the system
converges (or no better solutions can be found after a certain number of iterations). On the other
hand, the algorithm used for the intensive local exploitation should be an efficient local optimizer.
The idea is to reach the local optimality as quickly as possible, with the minimal number of function
evaluations. This stage should be fast and efficient.
2
3. 3 Differential Evolution
Differential evolution (DE) was developed by R. Storn and K. Price by their nominal papers in
1996 and 1997 (Storn, 1996; Storn and Price, 1997). It is a vector-based evolutionary algorithm,
and can be considered as a further development to genetic algorithms. It is a stochastic search
algorithm with self-organizing tendency and does not use the information of derivatives. Thus, it is
a population-based, derivative-free method. Another advantage of differential evolution over genetic
algorithms is that DE treats solutions as real-number strings, thus no encoding and decoding is
needed.
As in genetic algorithms, design parameters in a d-dimensional search space are represented as
vectors, and various genetic operators are operated over their bits of strings. However, unlikely
genetic algorithms, differential evolution carries out operations over each component (or each di-mension
of the solution). Almost everything is done in terms of vectors. For example, in genetic
algorithms, mutation is carried out at one site or multiple sites of a chromosome, while in differential
evolution, a difference vector of two randomly-chosen population vectors is used to perturb an ex-isting
vector. Such vectorized mutation can be viewed as a self-organizing search, directed towards
an optimality. This kind of perturbation is carried out over each population vector, and thus can be
expected to be more efficient. Similarly, crossover is also a vector-based component-wise exchange
of chromosomes or vector segments.
For a d-dimensional optimization problem with d parameters, a population of n solution vectors
are initially generated, we have xi where i = 1, 2, ..., n. For each solution xi at any generation t, we
use the conventional notation as
xt
i = (xt
1,i, xt
2,i, ..., xt
d,i), (1)
which consists of d-components in the d-dimensional space. This vector can be considered as the
chromosomes or genomes.
Differential evolution consists of three main steps: mutation, crossover and selection.
Mutation is carried out by the mutation scheme. For each vector xi at any time or generation
t, we first randomly choose three distinct vectors xp, xq and xr at t, and then generate a so-called
donor vector by the mutation scheme
vt+1
i = xt
p + F(xt
q − xt
r), (2)
where F 2 [0, 2] is a parameter, often referred to as the differential weight. This requires that the
minimum number of population size is n 4. In principle, F 2 [0, 2], but in practice, a scheme with
F 2 [0, 1] is more efficient and stable. The perturbation = F(xq − xr) to the vector xp is used to
generate a donor vector vi, and such perturbation is directed and self-organized.
The crossover is controlled by a crossover probability Cr 2 [0, 1] and actual crossover can be
carried out in two ways: binomial and exponential. The binomial scheme performs crossover on
each of the d components or variables/parameters. By generating a uniformly distributed random
number ri 2 [0, 1], the jth component of vi is manipulated as
ut+1
j,i = vj,i if ri Cr (3)
otherwise it remains unchanged. This way, each component can be decided randomly whether to
exchange with donor vector or not.
Selection is essentially the same as that used in genetic algorithms. It is to select the most fittest,
and for minimization problem, the minimum objective value.
Most studies have focused on the choice of F, Cr and n as well as the modification of (2). In
fact, when generating mutation vectors, we can use many different ways of formulating (2), and this
leads to various schemes with the naming convention: DE/x/y/z where x is the mutation scheme
(rand or best), y is the number of difference vectors, and z is the crossover scheme (binomial or
exponential). The basic DE/Rand/1/Bin scheme is given in (2). For a detailed review on different
schemes, please refer to Price et al. (2005).
3
4. 4 ES with DE
As ES is a two-stage strategy, we can use different algorithms at different stage. The large-scale
coarse search stage can use randomization via L´evy flights In the context of metaheuristics, the so-called
L´evy distribution is a distribution of the sum of N identically and independently distribution
random variables (Gutowski, 2001; Mantegna, 1994; Pavlyukevich, 2007).
This distribution is defined by a Fourier transform in the following form
FN(k) = exp[−N|k|
5. ]. (4)
The inverse to get the actual distribution L(s) is not straightforward, as the integral
L(s) =
1
Z 1
0
cos(s)e−
11. = 1, the above integral becomes the Cauchy distribution. When
12. = 2, it
becomes the normal distribution. In this case, L´evy flights become the standard Brownian motion.
For the second stage, we can use differential evolution as the intensive local search. We know
DE is a global search algorithm, it can easily be tuned to do efficient local search by limiting new
solutions locally around the most promising region. Such a combination may produce better results
than those by using pure DE only, as we will demonstrate this later. Obviously, the balance of local
search (intensification) and global search (diversification) is very important, and so is the balance
of the first stage and second stage in the ES.
5 Validation
Using our improved ES with DE, we can first validate it against some test functions which are highly
nonlinear and multimodal.
There are many test functions, here we have chosen the following 5 functions as a subset for our
validation.
Ackley’s function
f(x) = −20 exp h −
1
5
vuut
1
d
d
Xi=1
x2
i i
−exp h1
d
d
Xi=1
cos(2xi)i + 20 + e, (6)
where d = 1, 2, ..., and −32.768 xi 32.768 for i = 1, 2, ..., d. This function has the global mini-mum
f = 0 at x = (0, 0, ..., 0).
The simplest of De Jong’s functions is the so-called sphere function
f(x) =
d
Xi=1
x2
i , −5.12 xi 5.12, (7)
whose global minimum is obviously f = 0 at (0, 0, ..., 0). This function is unimodal and convex.
Rosenbrock’s function
f(x) =
d−1
Xi=1 h(xi − 1)2 + 100(xi+1 − x2
i )2i, (8)
4
13. whose global minimum f = 0 occurs at x = (1, 1, ..., 1) in the domain −5 xi 5 where
i = 1, 2, ..., d. In the 2D case, it is often written as
f(x, y) = (x − 1)2 + 100(y − x2)2, (9)
which is often referred to as the banana function.
Schwefel’s function
f(x) = −
d
Xi=1
xi sin p|xi|, −500 xi 500, (10)
whose global minimum f −418.9829n occurs at xi = 420.9687 where i = 1, 2, ..., d.
Shubert’s function
f(x) = h
K
Xi=1
i cos i + (i + 1)xi
·h
K
Xi=1
i cosi + (i + 1)yi, (11)
which has multiple global minima f −186.7309 for K = 5 in the search domain −10 x, y 10.
Table I summarizes the results of our simulations, where 9.7% corresponds to the ratio of the
number of function evaluations in ES to the number of function evaluations in DE. That is the
computational effort in ES is only about 9.7% of that using pure DE. As we can see that ES with
DE is significantly better than pure DE.
Table 1: Ratios of computational time
Functions ES/DE
Ackley (d = 8) 24.9%
De Jong (d = 16) 9.7%
Rosenbrock (d = 8) 20.2%
Schwefel (d = 8) 15.5%
Shubert 19.7%
6 Design Benchmarks
Now we then use the ES with DE to solve some real-world case studies including pressure vessel and
speed reducer problems.
6.1 Pressure Vessel Design
Pressure vessels are literally everywhere such as champagne bottles and gas tanks. For a given
volume and working pressure, the basic aim of designing a cylindrical vessel is to minimize the total
cost. Typically, the design variables are the thickness d1 of the head, the thickness d2 of the body,
the inner radius r, and the length L of the cylindrical section (Coello, 2000; Cagnina et al., 2008).
This is a well-known test problem for optimization and it can be written as
minimize f(x) = 0.6224d1rL + 1.7781d2r2
+3.1661d2
1L + 19.84d2
1r, (12)
subject to the following constraints
g1(x) = −d1 + 0.0193r 0
g2(x) = −d2 + 0.00954r 0
g3(x) = −r2L − 4
3 r3 + 1296000 0
g4(x) = L − 240 0.
(13)
5
14. The simple bounds are
0.0625 d1, d2 99 × 0.0625, (14)
and
10.0 r, L 200.0. (15)
Table 2: Comparison of number of function evaluations
Case study Pure DE ES ES/DE
Pressure vessel 15000 2625 17.7%
Speed reducer 22500 3352 14.9%
Recently, Cagnina et al. (2008) used an efficient particle swarm optimiser to solve this problem
and they found the best solution f 6059.714 at
x (0.8125, 0.4375, 42.0984, 176.6366). (16)
This means the lowest price is about $6059.71.
Using ES, we obtained the same results, but we used significantly fewer function evaluations,
comparing using pure DE and other methods. This again suggests ES is very efficient.
6.2 Speed Reducer Design
Another important benchmark is the design of a speed reducer which is commonly used in many
mechanisms such as a gearbox (Golinski, 1973). This problem involves the optimization of 7 vari-ables,
including the face width, the number of teeth, the diameter of the shaft and others. All
variables are continuous within some limits, except x3 which only takes integer values.
f(x) = 0.7854x1x2
2(3.3333x2
3 + 14.9334x3 − 43.0934)
−1.508x1(x2
6 + x2
7) + 7.4777(x3
6 + x3
7)
+0.7854(x4x2
6 + x5x2
7) (17)
g1(x) =
27
x1x2
2x3
− 1 0, (18)
g2(x) =
397.5
x1x2
2x2
3
− 1 0 (19)
g3(x) =
1.93x3
4
x2x3x4
6
− 1 0, (20)
g4(x) =
1.93x3
5
x2x3x4
7
− 1 0 (21)
g5(x) =
6r745.0x4
1.0
110x3
x2x3 2
+ 16.9 × 106 − 1 0 (22)
g6(x) =
7r745.0x5
1.0
85x3
x2x3 2
+ 157.5 × 106 − 1 0 (23)
g7(x) =
x2x3
40
− 1 0 (24)
g8(x) =
5x2
x1
− 1 0 (25)
6
15. g9(x) =
x1
12x2
− 1 0 (26)
g10(x) =
1.5x6 + 1.9
x4
− 1 0 (27)
g11(x) =
1.1x7 + 1.9
x5
− 1 0 (28)
where the simple bounds are 2.6 x1 3.6, 0.7 x2 0.8, 17 x3 28, 7.3 x4 8.3,
7.8 x5 8.4, 2.9 x6 3.9, and 5.0 x7 5.5.
In one of latest studies, Cagnina et al. (2008) obtained the following solution
x = (3.5, 0.7, 17, 7.3, 7.8, 3.350214, 5.286683) (29)
with fmin = 2996.348165.
Using our ES, we have obtained the new best
x = (3.5, 0.7, 17, 7.3, 7.8, 3.34336449, 5.285351) (30)
with the best objective fmin = 2993.7495888. We can see that ES not only provides better solutions
but also finds solutions more efficiently using fewer function evaluations.
7 Discussions
Metaheuristic algorithms such as differential evolution and eagle strategy are very efficient. We
have shown that a proper combination of these two can produce even better performance for solving
nonlinear global optimization problems. First, we have validated the ES with DE and compared
their performance. We then used them to solve real-world optimization problems including pressure
vessel and speed reducer design. Same or better results have been obtained, but with significantly
less computational effort.
Further studies can focus on the sensitivity studies of the parameters used in ES and DE so as to
identify optimal parameter ranges for most applications. Combinations of ES with other algorithms
may also prove fruitful. Furthermore, convergence analysis can provide even more profound insight
into the working of these algorithms.
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