![Real-World Applications of Genetic Algorithms
4
points in the factorial portion of the design which is given in Equation (1) (Montgomery,
2001; Clarke & Kempson, 1997).
( )1/4
c
α n
= (1)
where nc is number of points in the cube portion of the design (nc = 2k, k is number of
factors). Since there are four parameters/factors in this experiment, the nc number is equal to
24 (= 16) points, and α=2 according to Equation (1).
An experimental design matrix revealed in Table 1 consists of sets of coded conditions
expressed in natural values (Istadi & Amin, 2006a) with a two-level full factorial design (nc),
star points (ns) and centre points (n0). Based on this table, the experiments for obtaining the
responses of CH4 conversion (X(CH4)), C2 hydrocarbons selectivity (S(C2)) and C2
hydrocarbons yield (Y(C2)) were carried out at the corresponding independent variables.
Number experimental data were used for validating the hybrid ANN-GA model of the
catalytic-plasma CO2 OCM process. Sequence of the experimental work was randomized in
order to minimize the effects of uncontrolled factors. The experimental data from catalytic-
plasma reactor operation with respect to combination of four factors including their respected
responses (plasma-catalytic reactor performances: CH4 conversion, C2 hydrocarbons
selectivity, C2 hydrocarbons yield, and H2 selectivity) are presented in Table 2.
Factors Range and levels
-α -1 0 +1 +α
CH4/CO2 Ratio (X1), [-] 0.8 1.5 2.5 3.5 4.2
Discharge voltage (X2), kV 12.5 13.5 15.0 16.5 17.5
Total feed flow rate (X3), cm3/min 18 25 35 45 52
Reactor temperature (X4), oC 81 150 250 350 418
Note: -1 (low level value); +1 (high level value); 0 (centre point); +α and -α (star points)
Table 1. Central Composite Design with fractional factorial design for the catalytic DBD
plasma reactor (Istadi, 2006)
2.2 Simultaneous modelling and multi-objective optimization
The integrated ANN-GA strategy meets the objective based on two steps: (a) development
of an ANN-based process model which has inputs of process operating parameters of
plasma – catalytic reactor, and output(s) of process output/response variable(s), i.e. yield of
C2hydrocarbons or hydrogen, or methane conversion; and (b) development of GA technique
for multi-objective optimization of the ANN model. Input space of the ANN model is
optimized using the GA technique such that the optimal response(s) or objective(s) are
obtained corresponding to the optimal process parameters. The developed simultaneous
algorithm is presented in a hybrid Algorithm of ANN-GA schematically for simultaneous
modeling and optimization.
In the GA, a population of strings (called chromosomes), which encode individual solutions
towards an optimization problem, adjusts toward better solutions. The solutions are
represented in binary strings. The evolution begins from a population of randomly](https://image.slidesharecdn.com/2057398-250604150423-81238427/85/Realworld-Applications-Of-Genetic-Algorithms-Olympia-Roeva-22-320.jpg)
![Different Tools on Multi-Objective Optimization of a Hybrid Artificial
Neural Network – Genetic Algorithm for Plasma Chemical Reactor Modelling 7
Therefore, an input vector from the training set is applied to the network input nodes, and
subsequently outputs of the hidden and output nodes are computed. The outputs are
computed as follows: (a) the weighted sum of all the node-specific input is evaluated, which
is then transformed using a nonlinear activation function (f), such as tangent-sigmoid
(tansig) and linear (purelin) transfer functions for hidden and output layers, respectively; (b)
the outputs from the output nodes {yi,k} are then compared with their target values {ti,k}, and
the difference is used to compute the MSE (Equation 2); (c) upon the MSE computation, the
weight matrices WH and WO are updated using the corresponding method (Levenberg-
Marquardt) (Hagan & Menhaj, 1994; Yao et al., 2005).
In the back-propagation training method, the input x and target t values were normalized
linearly to be within the range [-1 1]. The normalization of inputs and outputs leads to
avoidance of numerical overflows due to very large or very small weights (Razavi et al.,
2003; Bowen et al., 1998; Yao et al., 2005). This normalization was performed to prevent
mismatch between the influence of some input values to the network weights and biases.
Network training was performed using Levenberg-Marquardt algorithm due to its fast
convergence and reliability in locating the global minimum of the mean-squared error
(MSE) (Levenberg-Marquardt) (Hagan & Menhaj, 1994; Yao et al., 2005). The transfer
function at the hidden layer nodes is tangent sigmoid, which is nonlinear but differentiable.
The output node utilizes the linear transfer function so that the input values n equal to the
output values y. The normalized output values yn are retransformed to its original range
(Razavi et al., 2003; Bowen et al., 1998; Yao et al., 2005).
Fig. 1. A schematic diagram of the multi-layered perceptron (MLP) in feed-forward neural
network with back-propagation training (X1: CH4/CO2 ratio; X2: discharge voltage; X3: total
feed flow rate; X4: reactor temperature; yo
1: CH4 conversion; yo
2: C2 hydrocarbons selectivity;
yo
3: Hydrogen selectivity; and yo
4: C2 hydrocarbons yield)](https://image.slidesharecdn.com/2057398-250604150423-81238427/85/Realworld-Applications-Of-Genetic-Algorithms-Olympia-Roeva-25-320.jpg)
![Real-World Applications of Genetic Algorithms
8
In terms of multi-objective optimization, GA was used for solving the scalar optimization
problem based on the principle of survival of the fittest during the evolution. The GA
implements the “survival of the fittest” and “genetic propagation of characteristics”
principles of biological evolution for searching the solution space of an optimization
problem. In nature, individuals must adapt to the frequent changing environment in order
to survive. The GA is one of the strategic randomized search techniques, which are well
known for its robustness in finding the optimal or near-optimal solution since it does not
depend on gradient information in its walk of life to find the best solution. Various kinds of
algorithm were reported by previous researchers (Tarca et al., 2002; Nandi et al., 2002, 2004;
Kundu et al., 2009; Bhatti et al., 2011).
The GA uses and manipulates a population of potential solutions to find optimal solutions.
The generation is complete after each individual in the population has performed the
genetic operators. The individuals in the population will be better adapted to the
objective/fitness function, as they have to survive in the subsequent generations. At each
step, the GA selects individuals at random from the current population to be parents and
uses them to produce the children for the next generation. Over successive generation, the
population evolves toward an optimal solution. The GA uses three main types of rules at
each step to create the next generation from the current population: (a) Selection rules select
the individuals, called parents, that contribute to the population at the next generation; (b)
Crossover rules combine two parents to form children for the next generation; (c) Mutation
rules apply random changes to individual parents to form children.
The detail stepwise procedures for the hybrid ANN-GA algorithm for simultaneous
modelling and optimization are described below and are depicted schematically in Figure 2:
Step 1. (Development of an ANN-based model): Specify input and output experimental
data of the system used for training and testing the ANN-based model. Create the
network architecture involving input, hidden and output layers. Investigate the
optimal network architecture (optimal number of hidden layer) and make sure that
the network is not overfitted.
Step 2. (Training of the ANN-based model): Normalize the experimental input and output
data to be within the range [-1 1]. The normalization is performed to prevent
mismatch between the influence of some input values to the network weights and
biases. Train the network using the normalized data by utilizing a robust training
algorithm (Levenberg-Marquardt).
Step 3. (Initialization of solution population): Set the initial generation index (Gen) to zero
and the number of population (Npop). Set the number of independent variables
(nvars). Generate a random initial population of Npop individuals. Each individual
possesses vector entries with certain length or called as genes which are divided into
many segments based on the number of decision variables (nvars).
Step 4. (Fitness computation): In this step the performance (fitness) of the solution vector
in the current population is computed by using a fitness function. Normalize the
solution vector xj to be within the range [-1 1]. Next, the vector xj is entered as
inputs vector to the trained ANN-based model to obtain the corresponding outputs
yj, yj=f(xj,W, b). Re-transform the output vector yj to the original values that are
subsequently utilized to compute the fitness value/scores of the solution.](https://image.slidesharecdn.com/2057398-250604150423-81238427/85/Realworld-Applications-Of-Genetic-Algorithms-Olympia-Roeva-26-320.jpg)
![odour of blood, they stood with haggard eyes and heaving breasts,
without having the power to turn away their looks.
There soon remained but one pirate. This man contemplated for a
moment the heap of bodies which lay before him; then, drawing the
dagger from the breast of him who had preceded him, he said with a
smile,—
"A fellow is lucky to die in such good company; but where the devil
do we go to after death? Bah! what a fool I am! I shall soon know!"
And with a gesture quick as thought he stabbed himself.
He fell instantly quite dead.
This frightful slaughter did not last more than a quarter of an hour.
[1]
Not one of the pirates had struck twice; all were killed by the first
blow.
"The dagger is mine!" said Eagle Head, drawing it smoking from the
still palpitating body of the last bandit. "It is a good weapon for a
warrior;" and he placed it coolly in his belt, after having wiped it
upon the grass.
The bodies of the pirates were scalped, and borne out of the camp.
They were abandoned to the vultures and the urubus, for whom
they would furnish an ample feast, and who, attracted by the odour
of blood, were already hovering over them, uttering lugubrious cries
of joy.
The formidable troop of Captain Waktehno was thus annihilated.
Unfortunately there were other pirates in the prairies.
After the execution, the Indians re-entered their huts carelessly; for
them it had only been one of those spectacles to which they had
been for a long time accustomed, and which have no effect upon
their nerves.](https://image.slidesharecdn.com/2057398-250604150423-81238427/85/Realworld-Applications-Of-Genetic-Algorithms-Olympia-Roeva-79-320.jpg)
![[1] All this scene is historical, and strictly true; the author was
present in Apacheria, at a similar execution.
CHAPTER XV.
THE PARDON.
The interview between the general and his niece was most touching.
The old soldier, so roughly treated for some time past, was delighted
to press to his bosom the innocent child who constituted his whole
family, and who, by a miracle, had escaped the misfortunes that had
assailed her.
For a long time they forgot themselves in a delightful interchange of
ideas; the general anxiously inquiring how she had lived while he
was a prisoner—the young girl questioning him upon the perils he
had run, and the ill-treatment he had suffered.
"Now, uncle," she said at length, "what is your intention?"
"Alas! my child," he replied, in a tone of sadness, and stifling a sigh;
"we must without delay leave these terrible countries, and return to
Mexico."
The heart of the young girl throbbed painfully, although she inwardly
confessed the necessity for a prompt return. To leave the prairies
would be to leave him she loved—to separate herself, without hope
of a reunion, from a man whose admirable character every minute
passed in sweet intercourse had made her more duly appreciate,
and who had now become indispensable to her life and her
happiness.
"What ails thee, my child? You are sad, and your eyes are full of
tears," her uncle asked, pressing her hand affectionately.](https://image.slidesharecdn.com/2057398-250604150423-81238427/85/Realworld-Applications-Of-Genetic-Algorithms-Olympia-Roeva-81-320.jpg)
Realworld Applications Of Genetic Algorithms Olympia Roeva
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- 5. REAL-WORLD APPLICATIONS OF GENETIC ALGORITHMS Edited by Olympia Roeva
- 6. Real-World Applications of Genetic Algorithms Edited by Olympia Roeva Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2012 InTech All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license, which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. As for readers, this license allows users to download, copy and build upon published chapters even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. Notice Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Marina Jozipovic Technical Editor Teodora Smiljanic Cover Designer InTech Design Team First published February, 2012 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Real-World Applications of Genetic Algorithms, Edited by Olympia Roeva p. cm. ISBN 978-953-51-0146-8
- 9. Contents Preface IX Chapter 1 Different Tools on Multi-Objective Optimization of a Hybrid Artificial Neural Network – Genetic Algorithm for Plasma Chemical Reactor Modelling 1 Nor Aishah Saidina Amin and I. Istadi Chapter 2 Application of Bio-Inspired Algorithms and Neural Networks for Optimal Design of Fractal Frequency Selective Surfaces 27 Paulo Henrique da Fonseca Silva, Marcelo Ribeiro da Silva, Clarissa de Lucena Nóbrega and Adaildo Gomes D’Assunção Chapter 3 Evolutionary Multi-Objective Algorithms 53 Aurora Torres, Dolores Torres, Sergio Enriquez, Eunice Ponce de León and Elva Díaz Chapter 4 Evolutionary Algorithms Based on the Automata Theory for the Multi-Objective Optimization of Combinatorial Problems 81 Elias D. Niño Chapter 5 Evolutionary Techniques in Multi-Objective Optimization Problems in Non-Standardized Production Processes 109 Mariano Frutos, Ana C. Olivera and Fernando Tohmé Chapter 6 A Hybrid Parallel Genetic Algorithm for Reliability Optimization 127 Ki Tae Kim and Geonwook Jeon Chapter 7 Hybrid Genetic Algorithm-Support Vector Machine Technique for Power Tracing in Deregulated Power Systems 147 Mohd Wazir Mustafa, Mohd Herwan Sulaiman, Saifulnizam Abd. Khalid and Hussain Shareef
- 10. VI Contents Chapter 8 Hybrid Genetic Algorithm for Fast Electromagnetic Synthesis 165 Artem V. Boriskin and Ronan Sauleau Chapter 9 A Hybrid Methodology Approach for Container Loading Problem Using Genetic Algorithm to Maximize the Weight Distribution of Cargo 183 Luiz Jonatã Pires de Araújo and Plácido Rogério Pinheiro Chapter 10 Hybrid Genetic Algorithms for the Single Machine Scheduling Problem with Sequence-Dependent Setup Times 199 Aymen Sioud, MarcGravel and Caroline Gagné Chapter 11 Genetic Algorithms and Group Method of Data Handling- Type Neural Networks Applications in Poultry Science 219 Majid Mottaghitalb Chapter 12 New Approaches to Designing Genes by Evolution in the Computer 235 Alexander V. Spirov and David M. Holloway Chapter 13 Application of Genetic Algorithms and Ant Colony Optimization for Modelling of E. coli Cultivation Process 261 Olympia Roeva and Stefka Fidanova Chapter 14 Multi-Objective Genetic Algorithm to Automatically Estimating the Input Parameters of Formant-Based Speech Synthesizers 283 Fabíola Araújo, Jonathas Trindade, José Borges, Aldebaro Klautau and Igor Couto Chapter 15 Solving Timetable Problem by Genetic Algorithm and Heuristic Search Case Study: Universitas Pelita Harapan Timetable 303 Samuel Lukas, Arnold Aribowo and Milyandreana Muchri Chapter 16 Genetic Algorithms for Semi-Static Wavelength-Routed Optical Networks 317 R.J. Durán, I. de Miguel, N. Merayo, P. Fernández, J.C. Aguado, A. Bahillo, R. de la Rosa and A. Alonso Chapter 17 Surrogate-Based Optimization 343 Zhong-Hua Han and Ke-Shi Zhang
- 13. Preface Genetic Algorithms are a part of Evolutionary Computing, which is a rapidly growing area of Artificial Intelligence. The popularity of Genetic Algorithms is reflected in the increasing amount of literature devoted to theoretical works and real-world applications in both scientific and engineering areas. The useful application and the proper combination of the different Genetic Algorithms with the various optimization algorithms is still an open research topic. This book addresses some of the most recent issues, with the theoretical and methodological aspects, of evolutionary multi-objective optimization problems and the various design challenges using different hybrid intelligent approaches. Multi- objective optimization has been available for about two decades, and its application in real-world problems is continuously increasing. Furthermore, many applications function more effectively using a hybrid systems approach. Hybridization of Genetic Algorithms is getting popular due to their capabilities in handling different problems involving complexity, noisy environment, uncertainty, etc. The book presents hybrid techniques based on Artificial Neural Network, Fuzzy Sets, Automata Theory, other metaheuristic or classical algorithms, etc. The volume examines various examples of algorithms in different real-world application domains as graph growing problem, speech synthesis, traveling salesman problem, scheduling problems, antenna design, genes design, modeling of chemical and biochemical processes etc. The book, organized in 17 chapters, begins with several applications of Hybrid Genetic Algorithms in wide range of problems. Further, some applications of Genetic Algorithms and other heuristic search methods are presented. The objective of Chapter 1 is to model and to optimize the process performances simultaneously in the plasma-catalytic conversion of methane such that the optimal process performances are obtained at the given process parameters. A Hybrid Artificial Neural Network-Genetic Algorithm (ANN-GA) is successfully developed to model, to simulate and to optimize simultaneously a catalytic-dielectric-barrier discharge plasma reactor. The integrated ANN-GA method facilitates powerful modeling and multi-objectives optimization for co-generation of synthesis gas, C2 and higher hydrocarbons from methane and carbon dioxide in a dielectric barrier discharge plasma reactor.
- 14. X Preface Chapter 2 presents a new fast and accurate electromagnetic optimization technique combining full-wave method of moments, bio-inspired algorithms, continuous Genetic Algorithm and Particle Swarm Optimization, and multilayer perceptrons Artificial Neural Networks. The proposed optimization technique is applied for optimal design of frequency selective surfaces with fractal patch elements. A fixed frequency selective surface screen geometry is chosen a priori and then a smaller subset of frequency selective surface design variables is optimized to achieve a desired bandstop filter specification. The main contribution of the Chapter 3 is the test of the Hybrid MOEA-HCEDA Algorithm and the quality index based on the Pareto front used in the graph drawing problem. The Pareto front quality index printed on each generation of the algorithm showed a convergent curve. The results of the experiments show that the algorithm converges. A graphical user interface is constructed providing users with a tool for a friendly and easy to use graphs display. The automatic drawing of optimized graphs makes it easier for the user to compare results appearing in separate windows, giving the user the opportunity to choose the graph design which best suits their needs. Chapter 4 studies metaheuristics based on the Automata Theory for the multi-objective optimization of combinatorial problems. The SAMODS (Simulated Annealing inspired Algorithm), SAGAMODS (Evolutionary inspired Algorithm) and EMODS (using Tabu Search) algorithms are presented. Presented experimental results of each proposed algorithm using multi-objective metrics from the specialized literature show that the EMODS has the best performance. In some cases the behavior of SAMODS and SAGAMODS tend to be the same – similar error rate. Chapter 5 presents a Hybrid Genetic Algorithm (Genetic Algorithm linked to a Simulated Annealing) intended to solve the Flexible Job-Shop Scheduling Problem procedure able to schedule the production in a Job-Shop manufacturing system. The authors show that this Hybrid Genetic Algorithm yields more solutions in the Approximate Pareto Frontier than other algorithms. A platform and programming language independent interface for search algorithms has been used as a guide for the implementation of the proposed hybrid algorithm. Chapter 6 suggests mathematical programming models and a Hybrid Parallel Genetic Algorithm (HPGA) for reliability optimization with resource constraints. The considered algorithm includes different heuristics such as swap, 2-opt, and interchange for an improvement solution. The experimental results of HPGA are compared with the results of existing meta-heuristics. The suggested algorithm presents superior solutions to all problems and found that the performance is superior to existing meta-heuristics. Chapter 7 discusses the effectiveness of Genetic Algorithms in determining the optimal values of hyper-parameters of Least Squares-Support Vector Machines to solve power tracing problem. The developed hybrid Genetic Algorithm-Support Vector Machines (GA-
- 15. Preface XI SVM) adopts real and reactive power tracing output determined by Superposition method as an estimator to train the model. The results show that GA-SVM gives good accuracy in predicting the generators’ output and compared well with Superposition method and load flow study. Chapter 8 provides an insight into the general reasoning behind selection of the Genetic Algorithms control parameters, discuss the ways of boosting the algorithm efficiency, and finally introduce a simple Global-local Hybrid Genetic Algorithms capable of fast and reliable optimization of multi-parameter and multi-extremum functions. The effectiveness of the proposed algorithm is demonstrated by numerical examples, namely: synthesis of linear antenna arrays with pencil-beam and flat-top patterns. Chapter 9 introduces a hybrid methodology, the Heuristics Backtracking, an approach that combines a search algorithm, the backtracking, integer linear programming and Genetic Algorithms to solve the three dimensional knapsack loading problem considering weight distribution. The authors show that the Heuristics Backtracking achieved good results without the commonly great trade-off between the utilization of container and a good weight distribution. Some benchmark tests taken from literature are used to validate the performance and efficiency of the Heuristics Backtracking methodology as well as its applicability to cutting-stock problems. Chapter 10 introduces two Hybrid Genetic Algorithms to solve the sequence-dependent setup times single machine problem. The proposed approaches are essentially based on adapting highly specialized genetic operators to the specificities of the studied problem. The numerical experiments demonstrate the efficiency of the hybrid algorithms for this problem. A natural conclusion from these experimental results is that Genetic Algorithms may be robust and efficient alternative to solve this problem. Chapter 11 presents the Group Method of Data Handling-type Neural Network with Genetic Algorithm used to develop the early egg production in broiler breeder. By means of the Group Method of Data Handling Algorithm, a model can be represented as a set of quadratic polynomials. Genetic Algorithms are deployed to assign the number of neurons (polynomial equations) in the network and to find the optimal set of appropriate coefficients of the quadratic expressions. Chapter 12 discusses some of the computational issues for evolutionary searches to find gene-regulatory sequences. Here the retroGenetic Algorithm technique is introduced. Proposed Genetic Algorithm crossover operator is inspired by retroviral recombination and in vitro DNA shuffling mechanisms to copy blocks of genetic information. The authors present particular results on the efficiency of retroGenetic Algorithm in comparison with the standard Genetic Algorithm. Chapter 13 examines the use of Genetic Algorithms and Ant Colony Optimization for parameter identification of a system of nonlinear differential equations modeling the fed-batch cultivation process of the bacteria E. coli. The results from both
- 16. XII Preface metaheuristics Genetic Algorithms and Ant Colony Optimization are compared using the modified Hausdorff distance metric, in place of most common used – least squares regression. Analyzing of average results authors conclude that the Ant Colony Optimization algorithm performs better for the considered problem. Chapter 14 presents a brief description about the estimation problem of a formant synthesizer, such as the Klatt. The combination of its input parameters to the imitation of human voice is not a simple task, because a reasonable number of parameters have to be combined and each of them has an interval of acceptable values that must be carefully adjusted to produce a specific voice. The authors conclude that it is necessary to develop a more efficient mechanism for evaluating the quality of the generated voice as a whole, and include it in the Genetic Algorithm speech framework. Chapter 15 discusses about how Genetic Algorithm and heuristic search can solve the scheduling problem. As a case study the “Universitas Pelita Harapan” timetable is considered. The authors propose the architecture design of the system and show some experiments implementing the system. The objective of Chapter 16 is to show a set of single-objective and multi-objective Genetic Algorithms, designed by the Optical Communications Group at the University of Valladolid, to optimize the performance of semi-static Wavelength-Routed Optical Networks (WRONs). The fundamentals of those algorithms, i.e., the chromosome structures, their translation, the optimization goals and the genetic operators employed are described. Moreover, a number of simulation results are also included to show the efficiency of Genetic Algorithms when designing WRONs. Finally, Chapter 17 gives an overview of existing surrogate modeling techniques and issues about how to use them for optimization. Surrogate modeling techniques are of particular interest for engineering design when high-fidelity, thus expensive analysis codes (e.g. computation fluid dynamics and computational structural dynamics) are used. The book is designed to be of interest to a wide spectrum of readers. The authors hope that the readers will find this book useful and inspiring. Olympia Roeva Institute of Biophysics and Biomedical Engineering Bulgarian Academy of Sciences Sofia, Bulgaria
- 19. 1 Different Tools on Multi-Objective Optimization of a Hybrid Artificial Neural Network – Genetic Algorithm for Plasma Chemical Reactor Modelling Nor Aishah Saidina Amin1,* and I. Istadi2 1Chemical Reaction Engineering Group, Faculty of Chemical Engineering, Universiti Teknologi Malaysia, Johor Bahru, 2Laboratory of Energy and Process Engineering, Department of Chemical Engineering, Diponegoro University, Jl. Prof. H. Soedarto, SH., Semarang, 1Malaysia 2Indonesia 1. Introduction Simultaneous modeling and optimization allows a cost-effective alternative to cover large number of experiments. The model should be able to improve overall process performance particularly for the complex process. A hybrid Artificial Neural Network - Genetic Algorithm (ANN-GA) was developed to model, to simulate, and to optimize simultaneously a catalytic–plasma reactor. The present contribution is intended to develop an ANN-GA method to facilitate simultaneous modeling and multi-objective optimization for co- generation of synthesis gas, C2 and higher hydrocarbons from methane and carbon dioxide in a dielectric-barrier discharge (DBD) plasma reactor. The hybrid approach simplifies the complexity in process modeling the DBD plasma reactor. A hybrid of ANN-GA method has been used for integrated process modelling and multi- objectives optimization. The detail hybrid algorithm for simultaneous modelling and multi- objective optimization has been developed in previous publication which focused on plasma reactor application (Istadi & Amin, 2005, 2006, 2007). They reported that the hybrid ANN- GA technique is a powerful method for process modelling and multi-objectives optimization (Nandi et al., 2002, 2004; Ahmad et al., 2004; Stephanopoulos & Han, 1996; Huang et al., 2003; Radhakrishnan & Suppiah, 2004; Fissore et al., 2004; Nandi et al., 2002, 2004; Ahmad et al., 2004; Kundu et al., 20009; Marzbanrad & Ebrahimi, 2011; Bhatti et al., 2011). The method is better than other technique such as response surface methodology (RSM) (Istadi & Amin, 2006, 2007), particularly for complex process model. The RSM proposes a quadratic model as empirical model for representing the effect of independent variables toward the targeting response. Therefore, all models which may not follow the quadratic trend are forced to the * Corresponding Author
- 20. Real-World Applications of Genetic Algorithms 2 quadratic model. Disadvantage of the RSM method is then improved by the hybrid ANN- GA. In the later method, an empirical mathematical modelling of catalytic cracking was conducted by ANN strategy, while the multi-objectives optimization of operating conditions to reach optimal responses was performed using GA method. In terms of single-response optimization applications, the selection of optimization method is very important to design an optimal catalyst as well as the relations between process parameters and catalytic performances (Wu et al., 2002). Pertaining to the catalyst design, some previous researchers introduced ANN to design the catalysts (Hattori & Kito, 1991, 1995; Hou et al., 1997). The ANN is feasible for modeling and optimization, and consequently, large number experiments can be avoidable (Wu et al., 2002). According to the complex interaction among the catalyst compositions, the process parameters and the metal- support interaction with no clear reaction mechanism as in CO2 OCM process, the empirical models are more useful in the catalyst design especially in the optimization studies. The reason is that the phenomenological modeling of interactions in the catalyst design is very complex. Unfortunately, a single-response optimization is usually insufficient for the real CO2 OCM process due to the fact that most responses, i.e. methane conversion, product selectivity and product yield, are dependent during the process. Therefore, simultaneous modeling and multi-objective optimization techniques in complex plasma reactor is worthy. A simultaneous multi-objective optimization is more realistic than a single-response from reliability point of view. Empirical and pseudo-phenomenological modeling approaches were employed by previous researchers (Wu et al., 2002; Larentis et al., 2001; Huang et al., 2003) for optimizing the catalytic process. The empirical modeling is efficient for the complex process optimization, but the drawback is that the model has no fundamental theory or actual phenomena meaning. Pertaining to multi-objective optimization, a graphical multi-responses optimization technique was implemented by previous researchers for xylitol crystallization from synthetic solution (de Faveri et al., 2004), but it was not useful for more than two independent variables or highly nonlinear models. In another study, a generalized distance approach technique was developed to optimize process variables in the production of protoplast from mycelium (Muralidhar et al., 2003). The optimization procedure was carried out by searching independent variables that minimize the distance function over the experimental region in the simultaneous optimal critical parameters. Recently, robust and efficient technique of elitist Non-dominated Sorting Genetic Algorithm (NSGA) was used to obtain solution of the complex multi-objective optimization problem (Huang et al., 2003; Nandasana et al., 2003; Zhao et al., 2000; Nandi et al., 2004). A hybrid GA with ANN was also developed (Huang et al., 2003) to design optimal catalyst and operating conditions for O2 OCM process. In addition, a comprehensive optimization study of simulated moving bed process was also reported using a robust GA optimization technique (Zhang et al., 2002b). Several methods are available for solving multi-objective optimization problem, for example, weighted sum strategy (The MathWorks, 2005; Youness, 2004; Istadi, 2006), ε- constraint method (Yu et al., 2003; The MathWorks, 2005; Youness, 2004), goal attainment method (Yu et al., 2003; The MathWorks, 2005), NSGA (Nandasana et al., 2003; Zhang et al., 2002b; Yu et al., 2003), and weighted sum of squared objective function (WSSOF) (Istadi & Amin, 2006b, 2007; Istadi, 2006) to obtain the Pareto set. The NSGA method has several advantages (Zhang et al., 2002b): (a) its efficiency is relatively insensitive to the shape of the
- 21. Different Tools on Multi-Objective Optimization of a Hybrid Artificial Neural Network – Genetic Algorithm for Plasma Chemical Reactor Modelling 3 Pareto-optimal front; (b) problems with uncertainties, stochasticities, and discrete search space can be handled efficiently; (c) spread of the Pareto set obtained is excellent, and (d) involves a single application to obtain the entire Pareto set. Among the methods, the NSGA is the most powerful method for solving a complex multi-responses optimization problem. In the multi-objective optimization of the CO2 OCM process, the goal attainment combined with hybrid ANN-GA method was used to solve the optimization of catalytic-plasma process parameters. The multi-objective optimization strategy was combined simultaneously with ANN modelling and GA optimization algorithm. The multi-objective optimization deals with generation and selection of non-inferior solution points or Pareto- optimal solutions of the responses / objectives corresponding to the optimal operating parameters. The DBD plasma-catalytic coupling of methane and carbon dioxide is an intricate process within the plasma-catalytic reactor application. A hybrid ANN-GA modelling and multi-objective optimization was developed to produce a process model that simulated the complex DBD plasma – catalytic process. There were no previous researchers focused on the simultaneous modelling and multi-objective optimization of DBD plasma – catalytic reactor using the hybrid ANN-GA. The objective of this chapter is to model and to optimize the process performances simultaneously in the DBD plasma-catalytic conversion of methane to higher hydrocarbons such that the optimal process performances (CH4 conversion and C2 hydrocarbons yield) are obtained at the given process parameters. In this Chapter, multi-objective optimization of two cases, i.e. C2 hydrocarbon yield and C2 hydrocarbons selectivity, and C2 hydrocarbons yield and CH4 conversion, to produce a Pareto Optimal solution is considered. In the process modeling, a number of experimental data was needed to validate the model. The ANN-based model required more example data which were noise-free and statistically well- distributed. Therefore, design of experiment was performed using central composite design with full factorial design for designing the training and test data sets. The method was chosen in order to provide a wider covering region of parameter space and good consideration of variable interactions in the model. This chapter is organized according to sections 1, 2, 3 and 4. After Introduction in section 1, section 2 covers design of experiment and strategy for simultaneous modeling and optimization including hybrid ANN-GA algorithm. In section 3, multi-objective optimization of methane conversion to higher hydrocarbons process over plasma – catalytic reactor is applied. In this section, ANN simulation of the DBD plasma – catalytic reactor performance is also presented with respect to the two cases. The final section, section 4 offers conclusions about the chapter. 2. Design of experiment, modeling, and optimization strategies 2.1 Central composite design for design of experiment Central Composite Design for four factors was employed for designing the experimental works in which variance of the predicted response Y at some point X is only a function of distance from the point to the design centre (Montgomery, 2001). Hence, the variance of Y remained unchanged when the design is rotated about the centre. In the design, standard error, which depends on the coordinates of the point on the response surface at which Y is evaluated and on the coefficients β, is the same for all points that are same distance from the central point. The value of α for star point with respect to design depends on the number of
- 22. Real-World Applications of Genetic Algorithms 4 points in the factorial portion of the design which is given in Equation (1) (Montgomery, 2001; Clarke & Kempson, 1997). ( )1/4 c α n = (1) where nc is number of points in the cube portion of the design (nc = 2k, k is number of factors). Since there are four parameters/factors in this experiment, the nc number is equal to 24 (= 16) points, and α=2 according to Equation (1). An experimental design matrix revealed in Table 1 consists of sets of coded conditions expressed in natural values (Istadi & Amin, 2006a) with a two-level full factorial design (nc), star points (ns) and centre points (n0). Based on this table, the experiments for obtaining the responses of CH4 conversion (X(CH4)), C2 hydrocarbons selectivity (S(C2)) and C2 hydrocarbons yield (Y(C2)) were carried out at the corresponding independent variables. Number experimental data were used for validating the hybrid ANN-GA model of the catalytic-plasma CO2 OCM process. Sequence of the experimental work was randomized in order to minimize the effects of uncontrolled factors. The experimental data from catalytic- plasma reactor operation with respect to combination of four factors including their respected responses (plasma-catalytic reactor performances: CH4 conversion, C2 hydrocarbons selectivity, C2 hydrocarbons yield, and H2 selectivity) are presented in Table 2. Factors Range and levels -α -1 0 +1 +α CH4/CO2 Ratio (X1), [-] 0.8 1.5 2.5 3.5 4.2 Discharge voltage (X2), kV 12.5 13.5 15.0 16.5 17.5 Total feed flow rate (X3), cm3/min 18 25 35 45 52 Reactor temperature (X4), oC 81 150 250 350 418 Note: -1 (low level value); +1 (high level value); 0 (centre point); +α and -α (star points) Table 1. Central Composite Design with fractional factorial design for the catalytic DBD plasma reactor (Istadi, 2006) 2.2 Simultaneous modelling and multi-objective optimization The integrated ANN-GA strategy meets the objective based on two steps: (a) development of an ANN-based process model which has inputs of process operating parameters of plasma – catalytic reactor, and output(s) of process output/response variable(s), i.e. yield of C2hydrocarbons or hydrogen, or methane conversion; and (b) development of GA technique for multi-objective optimization of the ANN model. Input space of the ANN model is optimized using the GA technique such that the optimal response(s) or objective(s) are obtained corresponding to the optimal process parameters. The developed simultaneous algorithm is presented in a hybrid Algorithm of ANN-GA schematically for simultaneous modeling and optimization. In the GA, a population of strings (called chromosomes), which encode individual solutions towards an optimization problem, adjusts toward better solutions. The solutions are represented in binary strings. The evolution begins from a population of randomly
- 23. Different Tools on Multi-Objective Optimization of a Hybrid Artificial Neural Network – Genetic Algorithm for Plasma Chemical Reactor Modelling 5 generated individuals and grows to produce next generations. In each generation, the fitness of each individual in the new population is evaluated and scored (recombination and mutation) to form a new population. During the fitness evaluation, the resulted ANN model is used. The new population is then used in the next iteration. The algorithm terminates when either a maximum generations number has been reached, or a best fitness level has been approached for the population. The multi-objective optimization can be formulated by converting the problem into a scalar single-objective optimization problem which is solvable by unconstrained single-response optimization technique. Many methods can be used for converting the problems into scalar optimization problem, such as weighted sum of squared objective functions (WSSOF), goal attainment, weighted sum strategy, and ε-constraint method. Schematic diagram of the feed-forward ANN used in this model development is depicted in Figure 1. Detail stepwise procedure used for the hybrid ANN-GA modelling and multi- objectives optimization is modified from the previous publications (Istadi, 2006; Istadi & Amin, 2007). The modified algorithm is described in this section and is depicted schematically in Figure 2. The fit quality of the ANN model was checked by a correlation coefficient (R) or a determination coefficient (R2) and Mean Square Error (MSE). The ANN model generated was repeated until the R2 reached higher than 0.90. The commonly employed error function to check the fit quality of the model is the MSE as defined in Equation (2). ( ) 2 , , 1 1 1 p i N k K i k i k p i k MSE t y N K = = = = = − (2) where Np and K denote the number of patterns and output nodes used in the training, i denotes the index of the input pattern (vector), and k denotes the index of the output node. Meanwhile, ti,k and yi,k express the desired (targeted or experimental) and predicted values of the kth output node at ith input pattern, respectively. With respect to the ANN modelling, a feed-forward ANN model was used in this model development which was trained using back-propagation training function. In general, four steps are developed in the training process: assemble the training data, create the network object, train the network, and simulate the network response to new inputs. The schematic of the feed-forward neural network used in the model development is depicted in Figure 1. As shown, the network consists of three layers nodes, i.e. input, hidden, and output layers comprising four numbers of each processing nodes. Each node in the input layer is linked to all nodes in the hidden layer and simultaneously the node in the hidden layer is linked to all nodes in the output layer using weighting connections (W). The weights are adjusted in the learning process in which all the patterns of input-output are presented in the learning phase repeatedly. In addition, the feed-forward neural network architecture also addresses the bias nodes which are connected to all nodes in subsequent layer, and they provide additional adjustable parameters (weights) for the fitting. From Figure 1, WH and WO denote the weights between input and hidden nodes and between hidden and output nodes, respectively. Meanwhile, yH and yO denote the outputs vector from hidden and output layers, respectively. In this system, bH and bO signify the
- 24. Real-World Applications of Genetic Algorithms 6 scalar bias corresponding to hidden and output layers, respectively. The weighted input (W) is the argument of the activation/transfer function f, which produces the scalar output y. The activation function net input is a summing function (nH or nO) which is the sum of the weighted input (WH or WO) and the bias b. In order that the ANN network accurately approximates the nonlinear relationship existing between the process inputs and outputs, it needs to be trained in a manner such that a pre-specified error function is minimized. There are many learning algorithms available and the most popular and successful learning algorithm used to train multilayer network is back-propagation scheme. Any output point can be obtained after this learning phase, and good results can be achieved. Process variables Responses/ Dependent variables (%) CH4/CO2 ratio (X1) Discharge voltage (X2) Total feed flow rate (X3) Reactor Temperature (X4) X(CH4) (Y1) S(C2+) (Y2) S(H2) (Y3) Y(C2+) (Y4) 3.5 16.5 45 150 21.45 26.13 13.24 5.61 3.5 16.5 25 150 23.48 33.41 12.13 7.85 * 3.5 13.5 45 350 18.76 28.43 13.16 5.33 1.5 16.5 25 350 27.55 27.47 8.11 7.57 3.5 13.5 25 350 20.22 35.21 12.87 7.12 1.5 13.5 45 150 23.11 26.98 8.01 6.24 1.5 16.5 45 350 28.03 24.45 7.48 6.85 * 1.5 13.5 25 150 30.02 24.15 8.54 7.25 0.8 15.0 35 250 32.14 12.54 5.17 4.03 4.2 15.0 35 250 21.12 34.77 13.99 7.34 2.5 12.5 35 250 18.55 29.76 10.22 5.52 2.5 17.5 35 250 41.32 28.01 10.12 11.57 2.5 15.0 18 250 38.65 31.77 11.32 12.28 * 2.5 15.0 52 250 20.88 30.00 11.56 6.26 2.5 15.0 35 81 25.49 28.04 9.87 7.15 2.5 15.0 35 418 26.74 32.55 10.41 8.70 2.5 15.0 35 250 25.77 31.33 11.55 8.07 2.5 15.0 35 250 23.41 30.74 9.87 7.20 2.5 15.0 35 250 25.14 29.65 10.44 7.45 * 2.5 15.0 35 250 26.11 28.14 9.54 7.35 Note: X, S, and Y denote conversion, selectivity and yield, respectively, and C2+ comprises C2H4, C2H6, C2H2, C3H8. * These data were used as test set. X1 (CH4/CO2 feed ratio); X2 (Discharge voltage, kV); X3 (Total feed flow rate, cm3/min); X4 (Reactor wall temperature, oC); Pressure: 1 atm; Catalyst loading: 5 gram; Frequency: 2 kHz (pulse) Table 2. Experimental data of hybrid catalytic DBD plasma reactor at low temperature (Istadi, 2006)
- 25. Different Tools on Multi-Objective Optimization of a Hybrid Artificial Neural Network – Genetic Algorithm for Plasma Chemical Reactor Modelling 7 Therefore, an input vector from the training set is applied to the network input nodes, and subsequently outputs of the hidden and output nodes are computed. The outputs are computed as follows: (a) the weighted sum of all the node-specific input is evaluated, which is then transformed using a nonlinear activation function (f), such as tangent-sigmoid (tansig) and linear (purelin) transfer functions for hidden and output layers, respectively; (b) the outputs from the output nodes {yi,k} are then compared with their target values {ti,k}, and the difference is used to compute the MSE (Equation 2); (c) upon the MSE computation, the weight matrices WH and WO are updated using the corresponding method (Levenberg- Marquardt) (Hagan & Menhaj, 1994; Yao et al., 2005). In the back-propagation training method, the input x and target t values were normalized linearly to be within the range [-1 1]. The normalization of inputs and outputs leads to avoidance of numerical overflows due to very large or very small weights (Razavi et al., 2003; Bowen et al., 1998; Yao et al., 2005). This normalization was performed to prevent mismatch between the influence of some input values to the network weights and biases. Network training was performed using Levenberg-Marquardt algorithm due to its fast convergence and reliability in locating the global minimum of the mean-squared error (MSE) (Levenberg-Marquardt) (Hagan & Menhaj, 1994; Yao et al., 2005). The transfer function at the hidden layer nodes is tangent sigmoid, which is nonlinear but differentiable. The output node utilizes the linear transfer function so that the input values n equal to the output values y. The normalized output values yn are retransformed to its original range (Razavi et al., 2003; Bowen et al., 1998; Yao et al., 2005). Fig. 1. A schematic diagram of the multi-layered perceptron (MLP) in feed-forward neural network with back-propagation training (X1: CH4/CO2 ratio; X2: discharge voltage; X3: total feed flow rate; X4: reactor temperature; yo 1: CH4 conversion; yo 2: C2 hydrocarbons selectivity; yo 3: Hydrogen selectivity; and yo 4: C2 hydrocarbons yield)
- 26. Real-World Applications of Genetic Algorithms 8 In terms of multi-objective optimization, GA was used for solving the scalar optimization problem based on the principle of survival of the fittest during the evolution. The GA implements the “survival of the fittest” and “genetic propagation of characteristics” principles of biological evolution for searching the solution space of an optimization problem. In nature, individuals must adapt to the frequent changing environment in order to survive. The GA is one of the strategic randomized search techniques, which are well known for its robustness in finding the optimal or near-optimal solution since it does not depend on gradient information in its walk of life to find the best solution. Various kinds of algorithm were reported by previous researchers (Tarca et al., 2002; Nandi et al., 2002, 2004; Kundu et al., 2009; Bhatti et al., 2011). The GA uses and manipulates a population of potential solutions to find optimal solutions. The generation is complete after each individual in the population has performed the genetic operators. The individuals in the population will be better adapted to the objective/fitness function, as they have to survive in the subsequent generations. At each step, the GA selects individuals at random from the current population to be parents and uses them to produce the children for the next generation. Over successive generation, the population evolves toward an optimal solution. The GA uses three main types of rules at each step to create the next generation from the current population: (a) Selection rules select the individuals, called parents, that contribute to the population at the next generation; (b) Crossover rules combine two parents to form children for the next generation; (c) Mutation rules apply random changes to individual parents to form children. The detail stepwise procedures for the hybrid ANN-GA algorithm for simultaneous modelling and optimization are described below and are depicted schematically in Figure 2: Step 1. (Development of an ANN-based model): Specify input and output experimental data of the system used for training and testing the ANN-based model. Create the network architecture involving input, hidden and output layers. Investigate the optimal network architecture (optimal number of hidden layer) and make sure that the network is not overfitted. Step 2. (Training of the ANN-based model): Normalize the experimental input and output data to be within the range [-1 1]. The normalization is performed to prevent mismatch between the influence of some input values to the network weights and biases. Train the network using the normalized data by utilizing a robust training algorithm (Levenberg-Marquardt). Step 3. (Initialization of solution population): Set the initial generation index (Gen) to zero and the number of population (Npop). Set the number of independent variables (nvars). Generate a random initial population of Npop individuals. Each individual possesses vector entries with certain length or called as genes which are divided into many segments based on the number of decision variables (nvars). Step 4. (Fitness computation): In this step the performance (fitness) of the solution vector in the current population is computed by using a fitness function. Normalize the solution vector xj to be within the range [-1 1]. Next, the vector xj is entered as inputs vector to the trained ANN-based model to obtain the corresponding outputs yj, yj=f(xj,W, b). Re-transform the output vector yj to the original values that are subsequently utilized to compute the fitness value/scores of the solution.
- 27. Different Tools on Multi-Objective Optimization of a Hybrid Artificial Neural Network – Genetic Algorithm for Plasma Chemical Reactor Modelling 9 Fig. 2. Flowchart of the hybrid ANN-GA algorithms for modelling and optimization Step 5. (Scaling the fitness scores): Scale/rank the raw fitness scores to values in a range that is suitable for the selection function. In the GA, the selection function uses the scaled fitness values to choose the parents for the next generation. The range of the scaled values influences performance of the GA. If the scaled values vary too widely, the individuals with the highest scaled values reproduce too rapidly, taking over the
- 28. Real-World Applications of Genetic Algorithms 10 population gene pool too quickly, and preventing the GA from searching other areas of the solution space. On the other hand, if the scaled values vary only a little, all individuals have approximately the same chance of reproduction and the search will progress slowly. The scaling function used in this algorithm scales the raw scores based on the rank of each individual instead of its score. Because the algorithm minimizes the fitness function, lower raw scores have higher scaled values. Step 6. (Parents selection): Choose the parents based on their scaled values by utilizing the selection function. The selection function assigns a higher probability of selection to individuals with higher scaled values. An individual can be selected more than once as a parent. Step 7. (Reproduction of children): Reproduction options determine how the GA creates children for the next generation from the parents. Elite count (Echild) specifies the number of individuals with the best fitness values that are guaranteed to survive to the next generation. Set elite count to be a positive integer within the range: 1 ≤ Echild ≤ Npop. These individuals are called elite children. Crossover fraction (Pcross) specifies the fraction of each population, other than elite children, that are produced by crossover. The remaining individuals in the next generation are produced by mutation. Set crossover fraction to be a fraction between 0 and 1. - Crossover: Crossover enables the algorithm to extract the best genes from different individuals by selecting genes from a pair of individuals in the current generation and recombines them into potentially superior children for the next generation with the probability equal to crossover fraction (Pcross) from Step 7. - Mutation: Mutation function makes small random changes in the individuals, which provide genetic diversity and thereby increases the likelihood that the algorithm will generate individuals with better fitness values. Step 8. (Replaces the current population with the children): After the reproduction is performed and the new children are obtained, the current populations are replaced with the children to form the next generation. Step 9. Update/increment the generation index): Increment the generation index by 1: Gen=Gen+1. Step 10. (Repeat Steps 4-9 until convergence is achieved): Repeat the steps 4-9 on the new generation until the convergences are met. The GA uses the following five criteria to determine when the algorithm stops: • Generations: the algorithm stops when the number of generation reaches the maximum value (Genmax). • Fitness limit: the algorithm stops when the value of the fitness function for the best point in the current population is less than or equal to Fitness limit. • Time limit: the algorithm stops after running for an amount of time in seconds equal to Time limit. • Stall generations: the algorithm stops if there is no improvement in the objective function for a sequence of consecutive generations of length Stall generations. • Stall time limit: the algorithm stops if there is no improvement in the objective function during an interval of time in seconds equal to Stall time limit.The algorithm stops if any one of these five conditions is met. Step 11. (Assign the top ranking of children to the optimal solution vector): After the GA convergence criteria is achieved, the children possessing top ranking of fitness value is assigned to the optimized population or decision variable vector, x*.
- 29. Different Tools on Multi-Objective Optimization of a Hybrid Artificial Neural Network – Genetic Algorithm for Plasma Chemical Reactor Modelling 11 There is a vector of objectives, F(X) = {F1(X), F2(X),…, FM(X)} where M denotes the number of objectives, that must be considered in chemical engineering process. The optimization techniques are developed to find a set of decision parameters, X={X1, X2, …, XN} where N is the number of independent variables. As the number of responses increases, the optimal solutions are likely to become complex and less easily quantified. Therefore, the development of multi-objectives optimization strategy enables a numerically solvable and realistic design problem (Wu et al., 2002; Yu et al., 2003). In this method, a set of design goals, F* = {F1*, F2*, ..., FM*} is associated with a set of objectives, F(X) = {F1(X), F2(X),…, FM(X)}. The multi-objectives optimization formulation allows the objectives to be under- or over- achieved which is controlled by a vector of weighting coefficient, w={w1, w2, ..., wM}. The optimization problem is formulated as follow: 1 1 1 , x 2 2 2 inimize subject to m F (x) - w γ F * F (x) - w γ F * γ γ ∈ Ω ≤ ≤ (3) Specification of the goals, (F1*, F2*), defines the goal point. The weighting vector defines the direction of search from the goal point to the feasible function space. During the optimization, γ is varied which changes the size of the feasible region. The constraint boundaries converge to the unique solution point (F1s, F2s). 3. Results and discussion 3.1 Development and testing of artificial neural network – Genetic algorithm model In developing a phenomenological model, it is mandatory to consider detailed kinetics of stated multiple reactions in the conservation equations. However, due to the tedious procedures involved in obtaining the requisite kinetic information within phenomenological model, the empirical data-based ANN-GA modelwas chosen for maximizing the process performances. In this study, simultaneous modeling and multi-objectives optimization of catalytic-plasma reactor for methane and carbon dioxide conversions to higher hydrocarbons (C2) and hydrogen was done. The purpose of multi-objectives optimization is to maximize the process performances simultaneously, i.e. CH4 conversion (Y1) and C2 hydrocarbons yield (Y4). Accordingly, four parameters namely CH4/CO2 ratio (X1), discharge voltage (X2), total feed flow rate (X3), and reactor temperature (X4), generate input space of the ANN model. In the ANN model, the four parameters and four targeted responses (CH4 conversion (yo 1), C2 hydrocarbons selectivity (yo 2), Hydrogen selectivity (yo 3), and C2 hydrocarbons yield (yo 4) were developed and simulated. Regarding the simultaneous modeling and optimization using the ANN-GA method (Figure 2), accuracy of the hybrid method was validated by a set of simple discrete data extracted from a simple quadratic equation (i.e. y= -2x2 + 15x + 5). From the testing, the determination coefficient (R2) of the method closes to 1 means the empirical method (ANN-GA) has a good fitting, while the relative error of the optimized results (comparison between GA results and analytical solution) are below 10%. In this chapter, Multi Input and Multi Output (MIMO) system with 4 inputs and 4 outputs of the ANN model was developed. Prior to the network training, numbers of experimental data (Table 2) were supplied into the training. The data were obtained based on the
- 30. Real-World Applications of Genetic Algorithms 12 experimental design (central composite design) as revealed in Tables 1 and 2. In each network training, the training data set was utilized for adjusting the weight matrix set, W. The performance of the ANN model is considered as fitness tests of the model, i.e. MSE, R, and epoch number (epochs). Comparison of the ANN model performance for various topologies was performed. The MSE decreases and R increases with increasing number of nodes in the hidden layer. However, increasing number of hidden layer takes more time in computation due to more complexity of the model. Therefore, optimization of layer number structure is important step in ANN modeling. The ANN model fitness in terms of comparison between targeted (t) and predicted (y) performances is shown in Figures 3 and 4. In the figures, the ANN models are fit well to the experimental data which is demonstrated by high determination coefficients (R2) of 0.9975 and 0.9968 with respect to CH4 conversion (y1) and C2 hydrocarbons yield (y2) models, respectively. The high R2 and low MSE value implies a good fitting between the targeted (experimental) and the predicted (calculated) values. Therefore, the ANN-based models are suitable for representing the plasma-catalytic conversion of methane and carbon dioxide to higher hydrocarbons. From the simulation, the hybrid ANN-GA algorithm is supposed to be powerful for simultaneous modeling and optimizing process conditions of the complex process as inline with the previous literatures (Istadi & Amin, 2006, 2007) with similar algorithm. The R2 by this method is high enough (higher than 0.95). The ANN-GA model has advantageous on the fitted model which is a complex non linear model. This is to improve the weaknesses of the response surface methodology that is forced to quadratic model. 3.2. Multi-objective oOptimization of DBD plasma - Catalytic reactor performances In this study, simultaneous modeling and multi-objective optimization of catalytic-plasma reactor for methane and carbon dioxide conversions to higher hydrocarbons (C2) and hydrogen was performed. The multi-objective optimization is aimed to maximize the CH4 conversion (Y1) and C2 hydrocarbons yield (Y4) simultaneously. Accordingly, four respected parameters, namely CH4/CO2 ratio (X1), discharge voltage (X2), total feed flow rate (X3), and reactor temperature (X4) are optimized stated as input space of the ANN model. In the ANN model, the four parameters and four targeted responses (CH4 conversion (yo 1), C2 hydrocarbons selectivity (yo 2), hydrogen selectivity (yo 3), and C2 hydrocarbons yield (yo 4)) were developed and simulated. In this case, two responses or objectives can be optimized simultaneously to obtain optimum four respected process parameters, i.e. CH4 conversion and C2 hydrocarbons yield (yo 1 and yo 4), CH4 conversion and C2hydrocarbon selectivity (yo 1 and yo 2), or CH4 conversion and hydrogen selectivity (yo 1and yo 3). For maximizing F1 and F4 (CH4 conversion and C2hydrocarbons yield, respectively), the actual objective functions are presented in Equation 4 which is one of the popular approaches for inversion (Deb, 2001; Tarafder et al., 2005). The equation was used due to the default of the optimization function is minimization. , 1 1 i i o F F = + (4) where Fi,o denotes the real objective functions, while Fi is the inverted objective functions for minimization problem.
- 31. Different Tools on Multi-Objective Optimization of a Hybrid Artificial Neural Network – Genetic Algorithm for Plasma Chemical Reactor Modelling 13 For the multi-objectives optimization, the decision variables/operating parameters bound were chosen from the corresponding bounds in the training data as listed in Table 3. Meanwhile, Table 4 lists the numerical parameter values used in the GA for all optimization runs. In this optimization, rank method was used for fitness scaling, while stochastic tournament was used for selection method to specify how the GA chooses parents for the next generation. Meanwhile, scattered method was chosen for crossover function and uniform strategy was selected for mutation function. From the 40 numbers of population size, two of them are elite used in the next generation, while 80% of the rest population was used for crossover reproduction and 20% of them was used for mutation reproduction with 5% rate. Operating Parameters Bounds CH4/CO2 feed ratio 1.5 ≤ X1 ≤ 4.0 Discharge voltage (kV) 12 ≤ X2 ≤ 17 Total feed flow rate (cm3/min) 20 ≤ X3 ≤ 40 Reactor temperature (oC) 100 ≤ X4 ≤ 350 Table 3. Operating parameters bound used in multi-objectives optimization of DBD plasma reactor without catalyst Computational Parameters Values Population size 40 Elite count 2 Crossover fraction 0.80 Number of generation 20 Fitness scaling function fitscalingrank Selection function selectiontournament Crossover function crossoverscattered Mutation function mutationuniform Mutation probability 0.05 Table 4. Computational parameters of GA used in the multi-objectives optimization The Pareto optimal solutions owing to the simultaneous CH4 conversion and C2 hydrocarbons yield and the corresponding four process parameters are presented in Figure 5. The Pareto optimal solutions points are obtained by varying the weighting coefficient (wk) in Equation (3) (goal attainment method) and performing the GA optimization corresponding to each wk so that the γ reaches its minimum value (Fk(x)-wk.γ ≤ Fk) (goal). From Figure 5, it was found in the Pareto optimal solution that if CH4 conversion improves, C2hydrocarbons yield deteriorates or vice versa. Theoretically, all sets of non- inferior/Pareto optimal solutions are acceptable. The maximum CH4 conversion and C2 hydrocarbons yield of 48 % and 15 %, respectively are recommended at corresponding optimum process parameters of CH4/CO2 feed ratio 3.6, discharge voltage 15 kV, total feed flow rate 20 cm3/min, and reactor temperature of 147 oC. Larger CH4 amount in the feed and higher feed flow rate enhance the C2+ hydrocarbons yield which is corroborated with the results of Eliasson et al. (2000). From the Pareto optimal solutions and the corresponding optimal operating parameters, the suitable operating conditions ranges for DBD plasma
- 32. Real-World Applications of Genetic Algorithms 14 reactor owing to simultaneous maximization of CH4 conversion and C2hydrocarbons yield can be recommended easily. Fig. 3. Comparison of targeted (experimental) and predicted (calculated) CH4 conversion of the ANN model (R2=0.9975) (* : test set data) Fig. 4. Comparison of targeted (experimental) and predicted (calculated) C2 hydrocarbons yield of the ANN model (R2=0.9968) (* : test set data)
- 33. Different Tools on Multi-Objective Optimization of a Hybrid Artificial Neural Network – Genetic Algorithm for Plasma Chemical Reactor Modelling 15 Fig. 5. Pareto optimal solutions with respect to multi-objectives optimization of CH4 conversion (Y1) and C2hydrocarbons yield (Y2). 3.3 Effect of hybrid catalytic-plasma DBD reactor for CH4 and CO2 conversions When a gas phase consisting electrically neutral species, electrons, ions and other excited species flow through the catalyst bed, the catalyst particles become electrically charged. The charge on the catalyst surface, together with other effects of excited species in the gas discharge leads to the variations of electrostatic potential of the catalyst surface. The chemisorption and desorption performances of the catalyst therefore may be modified in the catalyst surface (Jung et al., 2004; Kraus et al., 2001). Effects of these modifications on methane conversion are dependent on the amount and concentration of surface charge and the species present at the catalyst surface (Kim et al., 2004). The combining DBD plasma and a heterogeneous catalyst are possible to activate the reactants in the discharge prior to the catalytic reaction, which should have positive influences on the reaction conditions. Comparison of the application of DBD plasma technology in CH4 and CO2 conversion with catalyst is studied in this research. Since most of the energetic electrons are required to activate the CH4 and CO2 gases in a discharge gap, special consideration must be taken in the designing a reactor that maximizes the contact time between the energetic electrons and the neutral feed gas species. The catalyst located in the discharge gap is an alternative way to increase the time and area of contact between gas molecules and energetic electrons in addition to other modification of electronic properties. The energetic electrons determine the chemistry of the conversions of both gases (Eliasson et al., 2000; Yao et al., 2000; Zhou et al., 1998). The nature of dielectric and electrode surfaces is also an important factor for products distribution of CH4 and CO2 conversions using the DBD.
- 34. Real-World Applications of Genetic Algorithms 16 In the catalytic DBD plasma reactor system, the catalyst acts as a dielectric material. Most of the discharge energy is used to produce and to accelerate the electrons generating highly active species (metastable, radicals and ions). The combined action of catalysts and a non- equilibrium gas discharge leads to an alternative method for production of syngas and hydrocarbons from CH4 and CO2. When an electric field is applied across the packed dielectric layer, the catalyst is polarized and the charge is accumulated on the dielectric surface. An intense electric field is generated around each catalyst pellet contact point resulting in microdischarges between the pellets. The microdischarges in the packed-bed of catalyst produced energetic electrons rather than ions. The microdischarges induced a significant enrichment of electrons that were essential for the sustainability of plasmas. Methane and carbon dioxide were chemically activated by electron collisions. Liu et al. (1997) concluded that the electronic properties of catalysts have an important role in oxidative coupling of methane using DBD plasma reactor. The electronic properties and catalytic properties can be expected to be changed if the catalyst is electrically charged. From the non-catalytic DBD plasma reactor, it is shown that the plasma process seems to be less selective than conventional catalytic processes, but it has high conversion. The conventional catalytic reactions on the other hand can give high selectivity, but they require a certain gas composition, an active catalyst, and high temperature condition (endothermic reaction). In the hybrid catalysis-plasma, the catalyst has important roles such as increasing the reaction surface area, maintaining and probably increasing the non-equilibrium properties of gas discharge, acting as a dielectric-barrier material, and improving the selectivity and efficiency of plasma processes by surface reactions. The catalyst placed in the plasma zone can influence the plasma properties due to the presence of conductive surfaces in the case of metallic catalysts (Heintze & Pietruszka, 2004; Kizling & Järås, 1996). The catalyst can also change the reaction products due to surface reactions. The heating and electronic properties of the catalyst by the plasma induce chemisorption of surface species. A synergy between the catalyst and the plasma is important so that the interactions lead to improved reactant conversions and higher selectivity to the desired products. However until now, the exact role of the catalyst in the DBD plasma reactor is still not clear from the chemistry point of view. Even the kind of plasma reactor determines the product selectivity (Gordon et al., 2001). The most significant influence of the plasma was observed at low temperatures (Liu et al., 2001b) at which the catalysts were not active. At higher temperatures the catalysts became active; nonetheless, the plasma catalytic effect was still observed (Huang et al., 2000). 3.4. Simulation of DBD plasma - Catalytic reactor performances This section demonstrates ANN simulation for the effect of operating parameters (X1, X2, X3, X4) in catalytic DBD plasma reactor on CH4 conversion (y1) and C2 hydrocarbons yield (y4). The simulation results were presented in three dimensional surface graphics (Figures 6 to 13). From the results, the CH4 conversion and C2 hydrocarbons yield are affected by CH4/CO2 feed ratio, discharge voltage, total feed flow rate, and reactor wall temperature from the ANN-based model simulation. Figures 6, 7, 8, and 9 simulates the effect of discharge voltage, CH4/CO2 feed ratio, total feed flow rate, and reactor temperature on the methane conversion. Increasing the discharge voltage improves methane conversion significantly. That is true because energy of energetic
- 35. Different Tools on Multi-Objective Optimization of a Hybrid Artificial Neural Network – Genetic Algorithm for Plasma Chemical Reactor Modelling 17 Fig. 6. Effect of discharge voltage (X2) and CH4/CO2 ratio (X1) toward methane conversion (y1) Fig. 7. Effect of total flow rate (X3) and CH4/CO2 ratio (X1) toward methane conversion (y1) electrons is dependent on the discharge voltage. Higher the discharge voltage, higher the energy of electrons flows from high voltage electrode to ground electrode. Increasing the CH4 concentration in the feed favors the selectivity of C2 hydrocarbons and hydrogen significantly, but the C2 hydrocarbons yield is slightly affected due to the decrease of CH4 conversion. It is suggested that the CH4 concentration in the feed is an important factor for the total amount of hydrocarbons produced. However, increasing CH4/CO2 ratio to 4 reduces the methane conversion considerably and leads to enhanced C2 hydrocarbons
- 36. Real-World Applications of Genetic Algorithms 18 selectivity and H2/CO ratio. It is confirmed that CO2 as co-feed has an important role in improving CH4 conversion by contributing some oxygen active species from the CO2. This phenomenon is corroborated with the results of Zhang et al. (2001). Effect of total feed flow rate on methane conversion is displayed in Figures 7 and 8. From the figures, total feed flow rate has significant effect on methane conversion. Higher the total feed flow rate, lower methane conversion. This is due to primarily from short collision of energetic electrons with feed gas during flow through the plasma reactor. Therefore, only a few reactant molecules that has been cracked by the energetic electrons. Fig. 8. Effect of total flow rate (X3) and discharge voltage (X2) toward methane conversion (y1) Fig. 9. Effect of reactor temperature (X4) and discharge voltage (X2) toward methane conversion (y1)
- 37. Different Tools on Multi-Objective Optimization of a Hybrid Artificial Neural Network – Genetic Algorithm for Plasma Chemical Reactor Modelling 19 Figures 10, 11, 12, and 13presents the effect of discharge voltage, CH4/CO2 feed ratio, total feed flow rate, and reactor temperature on the C2 hydrocarbons yield. The yield of gaseous hydrocarbons (C2) increases with the CH4/CO2 feed ratio as exhibited in Figure. It is possible to control the composition of C2 hydrocarbons and hydrogen products by adjusting the CH4/CO2 feed ratio. Increasing CH4/CO2 ratio above 2.5 exhibits dramatic enhancement of C2hydrocarbons yield and lowers CH4 conversion slightly. In this work, the composition of the feed gas is an essential factor to influence the product distribution. Obviously, more methane in the feed will produce more light hydrocarbons. In comparison with non-catalytic DBD plasma reactor, the enhancement of reactor performance is obtained when using the hybrid catalytic-DBD plasma reactor (Istadi, 2006). The CH4 conversion, C2 hydrocarbons selectivity, C2 hydrocarbons yield and H2 selectivity of catalytic DBD plasma reactor is higher than that without catalyst (Istadi, 2006). The catalyst located in the discharge gap can increase the time and area of contact in addition to other modification of electronic properties. Therefore, collision among the energetic electrons and the gas molecules is intensive. Through the hybrid system, the chemisorption and desorption performances of the catalyst may be modified in the catalyst surface (Jung et al., 2004; Kraus et al., 2001) which is dependent on the amount and concentration of surface charge and the species on the catalyst surface (Kim et al., 2004). The results enhancement was also reported by Eliasson et al. (2000) over DBD plasma reactor with high input power 500 W (20 kV and 30 kHz) that the zeolite catalyst introduction significantly increased the selectivity of light hydrocarbons compared to that in the absence of zeolite. Varying the discharge power/voltage affects predominantly on methane conversion and higher hydrocarbons (C2) yield and selectivity. At high discharge voltage the CH4 conversion becomes higher than that of CO2 as presented in Table 2, since the dissociation energy of CO2 (5.5 eV) is higher than that of CH4 (4.5 eV) as reported by Liu et al. (1999a). More plasma species may be generated at higher discharge voltage. Previous researchers suggested that the conversions of CH4 and CO2 were enhanced with discharge power in a catalytic DBD plasma reactor (Caldwell et al., 2001; Eliasson et al., 2000; Zhang et al., 2001) and non-catalytic DBD plasma reactor (Liu et al., 2001b). From Figures10 and 12, the yield of C2 hydrocarbons decreases slightly with the discharge voltage which is corroborated with the results of Liu et al. (2001b). This means that increasing discharge power may destroy the light hydrocarbons (C2-C3). In this research, the lower range of discharge power (discharge voltage 12 - 17 kV and frequency 2 kHz) does not improve the H2 selectivity over DBD plasma reactor although the catalyst and the heating was introduced in the discharge space as exhibited in Figures 9 and 13. Eliasson et al. (2000) reported that higher discharge power is necessary for generating higher selectivity to higher hydrocarbons (C5+) over DBD plasma reactor with the presence of zeolite catalysts. Higher discharge power is suggested to be efficient for methane conversion. As the discharge power increases, the bulk gas temperature in the reaction zone may also increase. The total feed flow rate also affects predominantly on residence time of gases within the discharge zone in the catalytic DBD plasma reactor. Therefore, the residence time influences collisions among the gas molecules and the energetic electrons. Increasing the total feed flow rate reduces the residence time of gases and therefore decreases the C2 hydrocarbons yield dramatically as demonstrated in Figures 11 and 12. A lower feed flow rate is beneficial for producing high yields light hydrocarbons (C2+) and synthesis gases with higher H2/CO
- 38. Real-World Applications of Genetic Algorithms 20 ratio as reported by Li et al. (2004c). The hydrogen selectivity is also affected slightly by the total feed flow rate within the range of operating conditions. Indeed, the total feed flow rate affects significantly on the methane conversion rather than yield of C2 hydrocarbons. Actually, the low total feed flow rate (high residence time) leads to high intimate collision among the gas molecules, the catalyst and high energetic electrons. The high intensive collisions favor the methane and carbon dioxide conversions to C2+ hydrocarbons. From Figures 9 and 13, it is evident that the current range of reactor temperature only affects the catalytic - DBD plasma reactor slightly. The methane conversion and C2 hydrocarbons yield is only affected slightly by reactor wall temperature over the CaO-MnO/CeO2 catalyst. This may be due to the altering of the catalyst surface phenomena and the temperature of energetic electrons is quite higher than that of reactor temperature. The adsorption- desorption, heterogeneous catalytic and electronic properties of the catalysts may change the surface reaction activity when electrically charged. However, the chemistry and physical phenomena at the catalyst surface cannot be determined in the sense of traditional catalyst. Some previous researchers implied that the synergistic effect of catalysis-plasma only occurred at high temperature where the catalyst was active. Huang et al. (2000) and Heintze & Pietruszka (2004) pointed out that the product selectivity significantly improved only if the temperature was high enough for the catalytic material to become itself active. Zhang et al. (2001) also claimed that the reactor wall temperature did not significantly affect the reaction activity (selectivity) over zeolite NaY catalyst under DBD plasma conditions at the temperature range tested (323-423 K). Particularly, increasing the wall temperature at the low temperature range tested did not affect the reaction activity under plasma conditions. In contrast, some other researchers suggested that the synergistic effect of catalysis – plasma may occur at low temperature. Based on the ANN-based model simulation, it can be suggested that low total feed flow rate, high CH4/CO2 feed ratio, high discharge voltage and proper reactor temperature are suitable for producing C2+ hydrocarbons and synthesis gas over catalytic DBD plasma reactor. Fig. 10. Effect of discharge voltage (X2) and CH4/CO2 ratio (X1) toward C2 hydrocarbons yield (y4)
- 39. Different Tools on Multi-Objective Optimization of a Hybrid Artificial Neural Network – Genetic Algorithm for Plasma Chemical Reactor Modelling 21 Fig. 11. Effect of total feed flowrate (X3) and CH4/CO2 ratio (X1) toward C2 hydrocarbons yield (y4) Fig. 12. Effect of total feed flowrate (X3) and discharge voltage (X2) toward C2 hydrocarbons yield (y4)
- 40. Real-World Applications of Genetic Algorithms 22 Fig. 13. Effect of reactor temperature (X4) and discharge voltage (X2) toward C2 hydrocarbons yield (y4) 4. Conclusions A hybrid ANN-GA was successfully developed to model, to simulate and to optimize simultaneously a catalytic–DBD plasma reactor. The integrated ANN-GA method facilitates powerful modeling and multi-objective optimization for co-generation of synthesis gas, C2 and higher hydrocarbons from methane and carbon dioxide in a DBD plasma reactor. The hybrid approach simplified the complexity in process modeling of the DBD plasma reactor. In the ANN model, the four parameters and four targeted responses (CH4 conversion (yo 1), C2 hydrocarbons selectivity (yo 2), hydrogen selectivity (yo 3), and C2 hydrocarbons yield (yo 4) were developed and simulated. In the multi-objectives optimization, two responses or objectives were optimized simultaneously for optimum process parameters, i.e. CH4 conversion (yo 1) and C2 hydrocarbons yield (yo 4). Pareto optimal solutions pertaining to simultaneous CH4 conversion and C2 hydrocarbons yield and the corresponding process parameters were attained. It was found that if CH4 conversion improved, C2 hydrocarbons yield deteriorated, or vice versa. Theoretically, all sets of non-inferior/Pareto optimal solutions were acceptable. From the Pareto optimal solutions and the corresponding optimal operating parameters, the suitable operating condition range for DBD plasma reactor for simultaneous maximization of CH4 conversion and C2 hydrocarbons yield could be recommended easily. The maximum CH4 conversion and C2 hydrocarbons yield of 48 % and 15 %, respectively were recommended at corresponding optimum process parameters of CH4/CO2 feed ratio 3.6, discharge voltage 15 kV, total feed flow rate 20 cm3/min, and reactor temperature of 147 oC. 5. Abbreviations ANN : artificial neural network GA : genetic algorithm
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- 45. 2 Application of Bio-Inspired Algorithms and Neural Networks for Optimal Design of Fractal Frequency Selective Surfaces Paulo Henrique da Fonseca Silva1, Marcelo Ribeiro da Silva2, Clarissa de Lucena Nóbrega2 and Adaildo Gomes D’Assunção2 1Federal Institute of Education, Science and Technology of Paraiba, IFPB, 2Federal University of Rio Grande do Norte, UFRN, Brazil 1. Introduction Technological advances in the field of microwave and communication systems and the increase of their commercial applications in recent years have resulted in more stringent requirements for innovative design of microwave passive devices, such as: antennas, filters, power splitters and couplers, frequency selective surfaces, etc. To be competitive in the commercial marketplace, microwave engineers may be using computer-aided design (CAD) tools to minimize cost and design cycle times. Modern CAD tools have become an integral part of the microwave product cycle and demand powerful optimization techniques combined with fast and accurate models so that the optimal solutions can be achieved, eventually guaranteeing first-pass design success. The target of microwave device design is to determine a set of physical parameters to satisfy certain design specifications (Mohamed, 2005). Early methods of designing and optimizing microwave devices by hand are time and labor intensive, limit complexity, and require significant expertise and experience. Many of the important developments in microwave engineering were made possible when complex electromagnetic characteristics of microwave devices were represented in terms of circuit equivalents, lumped elements and transmission lines. Circuit simulators using empirical/analytical models are simple and efficient, reduce optimization time, but have limited accuracy or validity region. Although circuit simulator is still used today it suffers from some severe limitations (the most serious of them is that it considers only fundamental mode interactions) and requires corrections in the form of post manufacturing tuning (Fahmi, 2007). While developments in circuit simulators were taking place, numerical electromagnetic (EM) techniques were also emerging. With the computational power provided by modern computers, the use of accurate full-wave electromagnetic models by EM simulators for design and optimization of microwave devices became possible. By using full-wave electromagnetic methods higher order modes are taken into consideration and microwave devices can be rigorously characterized in the designs so that simulation and experimental results are in close agreement. This is particularly of interest for the rapid large scale
- 46. Real-World Applications of Genetic Algorithms 28 production of low-cost high performance microwave devices reducing or eliminating the need of post manufacturing tuning (Bandler et al., 1994; Fahmi, 2007). The EM simulators can simulate microwave device structures of arbitrary geometrical shapes and ensure a satisfactory degree of accuracy up to millimeter wave frequencies (Mohamed, 2005). These simulators are based on EM field solvers whose function is to solve the EM problem of the structure under analysis, which is described by the Maxwell´s equations. Thus, the design of electromagnetic structures is usually a very challenging task due to the complexity of the models involved. In the majority of cases, there are no simple analytical formulas to describe the performance of new microwave devices. However, the use of EM field solver for device optimization is still a time consuming procedure and need heavy computations. For complex problems, resulting in very long design cycles, this computational cost may be prohibitive (Haupt & Werner, 2007). Actually, many approaches are available to implement optimization using full-wave methods. For instance, the exploitation of commercial EM software packages inside the optimization loop of a general purpose optimization program. New techniques, such as geometry capture (Bandler et al., 1996) (suitable for automated EM design of arbitrary three- dimensional structures), space mapping (Bandler et al., 1994) (alternative design schemes combining the speed of circuit simulators with the accuracy of EM solvers), adjoint network concept (Nikolova et al., 2004), global optimization techniques based on bio-inspired algorithms, knowledge based methods, and artificial neural networks (ANNs), establish a solid foundation for efficient optimization of microwave device structures (Haupt & Werner, 2007; Zhang & Gupta, 2000; Silva et al., 2010a). This chapter presents a new fast and accurate EM optimization technique combining full- wave method of moments (MoM), bio-inspired algorithms, continuous genetic algorithm (GA) and particle swarm optimization (PSO), and multilayer perceptrons (MLP) artificial neural networks. The proposed optimization technique is applied for optimal design of frequency selective surfaces with fractal patch elements. A fixed FSS screen geometry is choose a priori and then optimizing a smaller subset of FSS design variables to achieve a desired bandstop filter specification. A frequency selective surface (FSS) is a two-dimensional array of periodic metallic elements on a dielectric layer or two-dimensional arrays of apertures within a metallic screen. This surface exhibits total reflection or transmission for patch and aperture elements, respectively. The most important parameters that will determine the overall frequency response of a FSS are: element shape, cell size, orientation, and dielectric layer properties. FSSs have been widely used as spatial filters for plane waves in a variety of applications, such as: microwave, optical, and infrared filters, bandpass radomes, microwave absorbers, polarizers, dichroic subreflectors, antenna systems, etc. (Munk, 2000). Several authors proposed the design of FSS using fractals. In this chapter, different fractal geometries are considered, such as: Koch, Dürer’s pentagon, and Sierpinski. While the use of space-filling fractal properties (e.g., Koch, Minkowski, Hilbert) reduce the overall size of the FSS elements (Oliveira et al., 2009; Campos et al., 2010), the attractive features of certain self- similar fractals (e.g., Sierpinski, Gosper, fractal tree, etc.) have received attention of microwave engineers to design multiband FSS. Many others self-similar geometries have been explored in the design of dual-band and dual polarized FSS (Gianvittorio et al., 2001).
- 47. Application of Bio-Inspired Algorithms and Neural Networks for Optimal Design of Fractal Frequency Selective Surfaces 29 The self-similarity property of these fractals enables the design of multiband fractal elements or fractal screens (Gianvittorio et al., 2003). Furthermore, as the number of fractal iterations increases, the resonant frequencies of these periodic structures decrease, allowing the construction of compact FSSs (Cruz et al., 2009). In addition, an FSS with fractal elements present resonant frequency that is almost independent of the plane-wave incidence angle. There is no closed form solution directly from a given desired frequency response to the corresponding FSS with fractal elements. The analysis of scattering characteristics from FSS devices requires the application of rigorous full-wave techniques. Besides that, due to the computational complexity of using a full-wave simulator to evaluate the FSS scattering variables, many electromagnetic engineers still use trial-and-error process until to achieve a given design criteria. Obviously this procedure is very laborious and human dependent. On the other hand, calculating the gradient of the scattering coefficients in terms of the FSS design variables is quite difficult. Therefore, optimization techniques are required to design practical FSSs with desired filter specifications. Some authors have been employed neural networks, PSO, and GA for FSS design and optimization (Manara et al., 1999; Hussein & El- Ghazaly, 2004; Silva et al., 2010b). The main computational drawback for EM optimization of FSSs based on bio-inspired algorithms relies on the repetitive evaluation of numerically expensive fitness functions. Due the expensive computation to calculate the scattering variables for every population member at multiple frequencies over many generations, several schemes are available to improve the GA performance for optimal design of FSSs, such as: the use of fast full-wave methods, micro-genetic algorithm, which aims to reduce the population size, and parallel GA using parallel computation. However, despite of these improvements done on the EM optimization using genetic algorithms, all the same several hours are required for expensive computational simulations of GA optimization (Haupt & Werner, 2007; Silva et al., 2010b). The application of ANNs as approximate fitness evaluation tools for genetic algorithms, though suggest often, had seldom been put to practice. The combination of ANNs and GAs has been applied mainly for the construction of optimized neural networks through GA- based optimization techniques. Few applications of ANNs to GA processing have been reported for EM optimization of microwave devices. The advantages of the MoM-ANN-GA/PSO optimization technique are discussed in terms of convergence and computational cost. This technique is applied for optimal design of bandstop FSS spatial filters with fractal elements considering the resonant frequency (fr) and bandwidth (BW) bandstop specifications. Some FSS prototypes with fractal elements are built and measured. The accuracy of the proposed optimization technique is verified by means of comparisons between theoretical and experimental results. 2. An overview of bio-inspired optimization technique The idea of blending full-wave methods, artificial neural networks, and bio-inspired optimization algorithms for electromagnetic optimization of FSS spatial filters was first proposed in 2007 (Silva et al., 2007). This optimization technique named MoM-ANN-GA replaces the computational intensive full-wave method of moments simulations by a fast and accurate MLP neural network model of FSS spatial filter, which is used to compute the cost (or fitness) function in the genetic algorithm iterations.
- 48. Real-World Applications of Genetic Algorithms 30 The proposed bio-inspired EM optimization technique starts with the definition of a FSS screen geometry that is choose a priori. A full-wave parametric analysis is carried out for accurate EM characterization of FSS spatial filter scattering properties. From obtained EM dataset, a MLP network is trained to establish the complicated relationships between FSS design variables and frequency response. Then, in order to overcome the computational requirements associated with full-wave numerical simulations, the developed MLP model is used for fast and accurate evaluation of fitness function into bio-inspired algorithm simulations. From the optimal design of FSS parameters, FSS prototypes are fabricated and measured for verification of optimization methodology. Fig. 1 gives a “big picture” overview of proposed bio-inspired EM optimization technique. Fig. 1. An overview of proposed bio-inspired optimization technique This section is a brief introduction that provides an overview of the proposed optimization technique to be presented. The overview includes fundamentals of multilayer perceptrons, continuous genetic algorithm, and particle swarm optimization. 2.1 Artificial neural networks Since the beginning of the 1990s, the artificial neural networks have been used as a flexible numerical tool, which are efficient for modeling of microwave devices. In the CAD applications related to microwave engineering, the use of ANNs as nonlinear models becomes very common. Neural network models trained by accurate EM data (obtained through measurements or by EM simulations) are used for fast and accurate design/optimization of microwave devices. In addition, the use of previously established knowledge in the microwave area (as empirical models) combined with the neural networks, results in a major reliability of the resulting hybrid model – with a major ability to learn nonlinear input-output mappings, as well as to generalize responses, when new values of the input design variables are presented. Another important advantage is the data amount reduction necessary for the neural networks training. Some hybrid modeling techniques have been proposed for the use with empirical models and neural networks, such as: Source Difference Method, PKI (Prior Knowledge Input), KBNN (Knowledge Based
- 49. Application of Bio-Inspired Algorithms and Neural Networks for Optimal Design of Fractal Frequency Selective Surfaces 31 Neural Network), and SMANN (Space Mapping Artificial Neural Network) (Zhang & Gupta, 2000). Versatility, efficient computation, reduced memory occupation, stability of learning algorithms, and generalization from representative data, are some characteristics that have motivated the use of neural networks in many areas of microwave engineering as models for complex ill-defined input-output mappings in new, not well-known microwave devices (Santos et al., 1997; Patnaik & Mishra, 2000; Zhang & Gupta, 2000). As mentioned previously, the electromagnetic behavior of a microwave device is extremely complex and simple empirical model cannot accurately describe its behavior under all conditions. Only with a detailed full-wave device model, more accurate results can be found. In general, the quality of simulation is decided by the accuracy of device models. On the other hand, a very detailed model would naturally slow down the program. A compromise between accuracy and speed of computation has to be struck. Using neural networks enables to overcome this problem (Silva et al., 2010a). The multilayer perceptrons is the most used artificial neural network for neuromodeling applications. Multilayer perceptrons artificial neurons are based on the nonlinear model proposed by (McCulloch & Pitts, 1943; Rosenblatt, 1958, as cited in Haykin, 1999). In this model, neurons are signal processing units composed by a linear combiner and an activation function, that can be linear or nonlinear, as shown in Fig. 2. Fig. 2. Nonlinear model of an artificial neuron The input signals are defined as i x , i i 0,1, ,N = , where Ni is the number of input units. The output of linear combiner corresponds to the neuron level of internal activity j net , as defined in (1). The information processed by neuron is storage in weights ji w , 1, ,Nj j = , where Nj is the number of neurons in a given neural network layer; 0 1 x = ± is the polarization potential (or threshold) applied to the neurons. The neuron output signal j y is the value of the activation function ( ) ϕ ⋅ in response to the neuron activation potential j net , as defined in (2). Ni j ji i i 0 w net x = = ⋅ (1) j j y ( ) net ϕ = (2)
- 50. Real-World Applications of Genetic Algorithms 32 Multilayer perceptrons presents a feed forward neural network (FNN) configuration with neurons set into layers. Each neuron of a layer is connected to those of the previous layer, as illustrated in Fig. 3. Signal propagation occurs from input to output layers, passing through the hidden layers of the FNN. Hidden neurons represent the input characteristics, while output neurons generate the neural network responses (Haykin, 1999). Fig. 3. Feed forward neural network configuration with two hidden layers The design of a MLP model consists by three main steps: i) configuration – how layers are organized and connected; ii) supervised learning – how information is stored in neural network; iii) generalization test – how neural network produces reasonable outputs for inputs not found in the training set (Haykin, 1999). In this work, we use feed forward neural networks and supervised learning to develop MLP neural network models. In the computational simulation of supervised error-correcting learning, a training algorithm is used for the adaptation of neural network synaptic weights. The instantaneous error (n ) e , as defined in (3), represents the difference between the desired response (n) d , and the neural network output ( ) n y , at the n-th iteration, corresponding to the presentation of the n-th training example, ( ) (n); (n) x d . Training examples variables are normalized to present unitary maximum absolute value. So, when using a given MLP model, prior scaling and de-scaling operations may be performed into input and output signals of MLP neural network, according to (4) and (5), respectively. ( ) ( ) ( ) n n n = − e y d (3) max / = x x x (4) max = ⋅ y y y (5)
- 51. Application of Bio-Inspired Algorithms and Neural Networks for Optimal Design of Fractal Frequency Selective Surfaces 33 Supervised learning has as objective the minimization of the sum squared error SSE(t), given in (6), where the index t, represents the number of training epochs (one complete presentation of all training examples, 1,2, , n N = , where N is the total number of examples, is called an epoch). 2 1 1 1 1 ( ) ( ) 2 Nj N j j n j SSE t e n N N = = = ⋅ (6) Currently, there are several algorithms for the training of MLP neural networks. The most popular training algorithms are those derived from back-propagation algorithm (Rumelhart, Hinton, & Williams, 1986, as cited in Haykin, 1999). Among the family of back-propagation algorithms, the RPROP algorithm shows to be very efficient in solving complex modeling learning tasks. After neural network training, we hope that MLP weights will storage the representative information contained on training dataset. The trained neural network is tested in order to verify its capability of generalizing to new values that do not belong to the training dataset. Therefore, the MLP neural network operates like a “black box” model inside a given region of interest, which was previously defined when the training dataset was generated. 2.2 Bio-inspired optimization algorithms Bio-inspired algorithms, which are stochastic population-based global search methods inspired by nature, such as simulated annealing (SA), genetic algorithm and particle swarm optimization are effective for optimization problems with a large number of design variables and inexpensive fitness function evaluation (Haupt, 1995; Haupt & Werner, 2007; Kennedy & Eberhart, 1995). However, the main computational drawback for optimization of microwave devices relies on the repetitive evaluation of numerically expensive fitness functions. Finding a way to shorten the optimization cycle is highly desirable (Silva et al., 2010b). For instance, several GA schemes are available in order to improve its performance, such as: the use of fast full-wave methods, micro-genetic algorithm, which aims to reduce the population size, and parallel GA using parallel computation (R. L. Haupt & Sue, 2004). Bio-inspired algorithms start with an initial population of candidate individuals for the optimal solution. Assuming an optimization problem with Nvar input variables and Npop individuals, the population at the i-th iteration is represented as a matrix P(i)Npop×Nvar of floating-point elements, denoted by , i m n p , with each row corresponding to an individual. Under GA and PSO jargons, the individuals are named chromosomes and particles (or agents), respectively. 2.2.1 Continuous genetic algorithm Continuous genetic algorithm is very similar to the binary-GA but works with floating-point variables. Continuous-GA chromosomes are defined in (7) as a vector with Nvar floating- point optimization variables. Each chromosome is evaluated by means of its associated cost, which is computed through the cost function E given in (8).
- 52. Real-World Applications of Genetic Algorithms 34 ,1 ,2 , var (i, ) , , , , 1,2, , i i i m m m N chromosome m p p p m Npop = = (7) ( ) cos (i, ) (i, ) t m chromosome m = E (8) Based on the cost associated to each chromosome, the population evolves through generations with the application of genetic operators, such as: selection, crossover and mutation. Flow chart shown in Fig. 4(a) gives an overview of continuous-GA. Mating step includes roulette wheel selection presented in (Haupt & Werner, 2007; R. L. Haupt & Sue, 2004). Population selection is performed after the Npop chromosomes are ranked from lowest to highest costs. Then, the Nkeep most-fit chromosomes are selected to form the mating pool and the rest are discarded to make room for the new offspring. Mothers and fathers pair in a random fashion through the blending crossover method (R. L. Haupt & Sue, 2004). Each pair produces two offspring that contain traits from each parent. In addition, the parents survive to be part of the next generation. After mating, a fraction of chromosomes in the population will suffer mutation. Then, the chromosome variable selected for real-value mutation is added to a normally distributed random number. Most users of continuous-GA add a normally distributed random number to the variable selected for mutation with a constant standard deviation (R. L. Haupt & Sue, 2004). In particular, we propose a new real-value mutation operator for continuous-GA as given in (9), where pmax and pmin are constant values defined according to the limits of the region of interest composed by input parameters. Function randn() returns a normal distribution with mean equal to zero and standard deviation equal to one. This mutation operator was inspired by simulating annealing cooling schedules (R. L. Haupt & Sue, 2004). It is used to improve continuous-GA convergence at the neighbourhood of global minimum. The quotient function Q given in (10) is crescent when the number of iterations increases and the global cost decreases. Thus, similar to the decrease of temperature in a simulating annealing algorithm, the standard deviation is decreased when the number of continuous-GA iterations is increased. The parameter A is a constant value and B is a value of cost function neighbour to the global minimum. The continuous-GA using the real-value mutation definition given in (9) and (10) is denominated improved genetic algorithm. ( ) ( ) max min 1 , , () , cos ( ) i i m n m n p p p p randn Q i global t i + − = + ⋅ (9) ( ) ( ) 2 , cos ( ) , cos ( ) log cos ( ) , cos ( ) A global t i B Q i global t i A i global t i global t i B ≥ = + ⋅ < (10) 2.2.2 Particle swarm optimization Particle swarm optimization was first formulated in 1995 (Kennedy & Eberhart, 1995). The thought process behind the algorithm was inspired by social behavior of animals, such as bird flocking or fish schooling. PSO is similar to continuous-GA since it begins with a
- 53. Application of Bio-Inspired Algorithms and Neural Networks for Optimal Design of Fractal Frequency Selective Surfaces 35 random initial population. Unlike GA, PSO has no evolution operators such as crossover and mutation. Each particle moves around the cost surface with an individual velocity. The implemented PSO algorithm updates the velocities and positions of the particles based on the local and global best solutions, according to (11) and (12), respectively. ( ) ( ) ( ) ( ) 1 , 0 , 1 1 , , 2 2 , , localbes i globalbest i i i i i m n m n m n m n m n m n v C r v r p p r p p + = + Γ ⋅ ⋅ − + Γ ⋅ ⋅ − (11) 1 1 , , , i i i m n m n m n p p v + + = + (12) Here, , m n v is the particle velocity; , m n p is the particle variables; 0 r , 1 r and 2 r are independent uniform random numbers; 1 Γ is the cognitive parameter and 2 Γ is the social parameter; ( ) , localbest i m n p is the best local solution and ( ) , globalbest i m n p is the best global solution; C is the constriction parameter (Kennedy & Eberhart, 1995). If the best local solution has a cost less than the cost of the current global solution, then the best local solution replaces the best global solution. PSO is a very simple bio-inspired algorithm, easy to implement and with few parameters to adjust. Flow chart shown in Fig. 4(b) gives an overview of PSO algorithm. Fig. 4. Flow charts of (a) continuous-GA and (b) PSO algorithm. 3. FSS design considerations Frequency selective surfaces (FSSs) are used in many commercial and military applications. Usually, conducting patches and isotropic dielectric layers are used to build these FSS structures. FSS frequency response is entirely determined by the geometry of the structure in one period called a unit cell. In this section is presented some considerations about the design of FSS with fractal elements for operation at the X-band (8–12 GHz) and Ku-band (12–18 GHz). FSS fabrication and measurement procedures are summarized.
- 54. Real-World Applications of Genetic Algorithms 36 3.1 Design of FSS using fractal geometries FSS with fractal elements has attracted the attention of microwave engineering researchers because of its particular/special features. The design of a FSS with pre-fractal elements is a very competitive solution that enables the fabrication of compact spatial filters, with better performances when compared to conventional structures (Oliveira, et al., 2009; Campos et al., 2010). Several fractal iterations can be used to design a FSS with multiband frequency response associated to the self-similarity contained in the structure. Various self-similar fractals elements (e.g., Koch, Sierpinski, Minkowski, Dürer’s pentagon) were previously used to design multiband FSSs (Gianvittorio et al., 2003; Cruz et al., 2009; Trindade et al., 2011). Fig. 5 illustrates the considered periodic array in this chapter. The periodicity of the elements is given by tx=Wc, in the x axis, and ty=Lc, in the y axis, where Wc is the width and Lc is the length of the unit cell element; in addition, W is the width and L is the length of the patch. The design of fractal patch elements depend of desired FSS filter specifications, such as: bandstop attenuation, resonant frequency, quality factor, fabrication restrictions, etc. Fig. 5. Periodic array of fractal patch elements 3.1.1 Koch island fractal The Koch island fractal patch elements were obtained assuming a rectangular construction, fractal iteration-number (or level), k=0,1,2, and a variable fractal iteration-factor 1 / r a = , where a belongs to interval 3.05 10.0 a ≤ ≤ . The geometry of the Koch island fractal patch elements is shown in Fig. 6, considering for k=0,1,2, and a=4. The rectangular patch element (fractal initiator) dimensions are (mm): W=4.93, L= 8.22, tx=8.22, and ty=12.32.
- 55. Another Random Scribd Document with Unrelated Content
- 56. After surveying him for an instant with disdain, the pirate was turning his back upon him, when an idea occurred to him, which made him, on the contrary,—advance towards the savant, upon whose shoulder he somewhat roughly laid his hand. At this rude salutation, the poor doctor drew himself up in a fright, letting fall both plants and spade. "Holla! my good fellow," said the captain, in a jeering tone, "what madness possesses you to be herbalizing thus at all hours of the day and night?" "How!" the doctor replied, "what do you mean by that?" "Zounds! it's plain enough! Don't you know it is not far from midnight?" "That is true," the savant remarked ingenuously; "but there is such a fine moon." "Which you, I suppose, have taken for the sun," said the pirate, with a loud laugh; "but," he added, becoming all at once serious, "that is of no consequence now; although half a madman, I have been told that you are a pretty good doctor." "I have passed my examinations," the doctor replied, offended by the epithet applied to him. "Very well! you are just the man I want, then." The savant bowed with a very ill grace; it was evident he was not much flattered by the attention. "What do you require of me?" he asked; "are you ill?" "Not I, thank God! but one of your friends, who is at this moment my prisoner, is; so please to follow me." "But——" the doctor would fain have objected. "I admit of no excuses; follow me, or I will blow your brains out. Besides, don't be afraid, you run no risk; my men will pay you all the respect science is entitled to."
- 57. As resistance was impossible, the worthy man did as he was bidden with a good grace—with so good a grace, even, that for a second he allowed a smile to stray across his lips, which would have aroused the suspicions of the pirate if he had perceived it. The captain commanded the savant to walk on before him, and both thus reached the river. At the instant they quitted the place where this conversation, had taken place, the branches of a bush parted slowly, and a head, shaved with the exception of a long tuft of hair at the top, on which was stuck an eagle's feather, appeared, then a body, and then an entire man, who bounded like a jaguar in pursuit of them. This man was Eagle Head. He was a silent spectator of the embarkation of the two whites, saw them enter the grotto, and then, in his turn, disappeared in the shade of the woods, after muttering to himself in a low voice the word— "Och!" (good) the highest expression of joy in the language of the Comanches. The doctor had plainly only served as a bait to attract the pirate, and cause him to fall into the snare laid by the Indian chief. Now, had the worthy savant any secret intelligence with Eagle Head? That is what we shall soon know. On the morrow, at daybreak, the pirate ordered a close battue to be made in the environs of the grotto; but no track existed. The captain rubbed his hands with joy; his expedition had doubly succeeded, since he had managed to return to his cavern without being followed. Certain of having nothing to dread, he was unwilling to keep about him so many men in a state of inactivity; placing, therefore, his troop provisionally under the command of Frank, a veteran bandit, in
- 58. whom he had perfect confidence, he only retained ten chosen men with him, and sent away the rest. Although the affair he was now engaged in was interesting, and his success appeared certain, he was not, on that account, willing to neglect his other occupations, and maintain a score of bandits in idleness, who might, at any moment, from merely having nothing else to do, play him an ugly turn. It is evident that the captain was not only a prudent man, but was thoroughly acquainted with his honourable associates. When the pirates had left the grotto, the captain made a sign to the doctor to follow him, and conducted him to the general. After having introduced them to each other with that ironical politeness in which he was such a master, the bandit retired, leaving them together. Only before he departed, the captain drew a pistol from his belt, and clapping it to the breast of the savant— "Although you may be half a madman," he said, "as you may, nevertheless, have some desire to betray me, observe this well, my dear sir; at the least equivocal proceeding that I see you attempt, I will blow your brains out; you are warned, so now act as you think proper." And replacing his pistol in his belt, he retired with one of his eloquent sneers on his lips. The doctor listened to this admonition with a very demure countenance, but with a sly smile, which, in spite of himself, glided over his lips, but which, fortunately, was not perceived by the captain. The general and his Negro, Jupiter, were confined in a compartment of the grotto at some distance from the outlet. They were alone, for the captain had deemed it useless to keep guards constantly with them.
- 59. Both seated upon a heap of leaves, with heads cast down and crossed arms, they were reflecting seriously, if not profoundly. At sight of the savant, the dismal countenance of the general was lighted up by a fugitive smile of hope. "Ah, doctor, is that you?" he said, holding out to him a hand which the other pressed warmly hut silently, "have I reason to rejoice or to be still sad at your presence?" "Are we alone?" the doctor asked, without answering the general's question. "I believe so," he replied, in a tone of surprise; "at all events, it is easy to satisfy yourself." The doctor groped all round the place, carefully examined every corner; he then went back to the prisoners. "We can talk," he said. The savant was habitually so absorbed by his scientific calculations, and was naturally so absent, that the prisoners had but little confidence in him. "And my niece?" the general asked, anxiously. "Be at ease on her account; she is in safety with a hunter named Loyal Heart, who has a great respect for her." The general breathed a sigh of relief; this good news had restored him all his courage. "Oh!" he said, "of what consequence is my being a prisoner? Now I know my niece is safe, I can suffer anything." "No, no," said the doctor, warmly, "on the contrary, you must escape from this place tomorrow, by some means." "Why?" "Answer me in the first place." "I ask no better than to do so."
- 60. "Your wounds appear slight; are they progressing towards cure?" "I think so." "Do you feel yourself able to walk? "Oh, yes!" "But let us understand each other. I mean, are you able to walk a distance?" "I believe so, if it be absolutely necessary." "Eh! eh!" said the Negro, who, up to this moment had remained silent, "am I not able to carry my master when he can walk no longer?" The general pressed his hand. "That's true, so far," said the doctor; "all is well, only you must escape." "I should be most glad to do so, but how?" "Ah! that," said the savant, scratching his head, "is what I do not know, for my part! But be at ease, I will find some means; at present, I don't know what." Steps were heard approaching, and the captain appeared. "Well!" he asked, "how are your patients going on?" "Not too well!" the doctor replied. "Bah! bah!" the pirate resumed; "all that will come round; besides, the general will soon be free, then he can get well at his ease. Now, doctor, come along with me; I hope I have left you and your friend long enough together to have said all you wish." The doctor followed him without reply, after having made the general a parting sign to recommend prudence. The day passed away without further incident.
- 61. The prisoners looked for the night with impatience; in spite of themselves, a confidence in the doctor had gained upon them—they hoped. Towards evening the worthy savant reappeared. He walked with a deliberate step, his countenance was cheerful, he held a torch in his hand. "What is there fresh, doctor?" the general asked; "you appear to be quite gay." "In fact, general, I am so," he replied with a smile, "because I have found the means of securing your escape—not forgetting my own." "And those means?" "Are already half executed," he said, with a little dry smile, which was peculiar to him when he was satisfied. "What do you mean by that?" "By Galen! something very simple, but which you never would guess: all our bandits are asleep, we are masters of the grotto." "That may be possible; but if they should wake?" "Don't trouble yourself about that; they will wake, of that there is no doubt, but not within six hours at least." "How the devil can you tell that?" "Because I took upon myself to send them to sleep; that is to say, at their supper I served them with a decoction of opium, which brought them down like lumps of lead, and they have all been snoring ever since like so many forge bellows." "Oh, that is capital!" said the general. "Is it not?" the doctor observed, modestly. "By Galen, I was determined to repair the mischief I had done you by my negligence! I am not a soldier, I am but a poor physician; I have made use of my proper weapons; you see that in certain cases they are as good as others."
- 62. "They are a hundred times better! Doctor, you are a noble fellow!" "Well, come, let us lose no time." "That is true, let us be gone; but the captain, what have you done with him?" "Oh, as to him, the devil only knows where he is. He left us after dinner without saying anything to anybody; but I have a shrewd suspicion I know where he is gone, and am much mistaken if we do not see him presently." "All, then, is for the best; lead on." The three men set off at once. In spite of the means employed by the doctor, the general and the Negro were not quite at ease. They arrived at the compartment which now served as a dormitory for the bandits; they were lying about asleep in all directions. The fugitives passed safely through them. When they arrived at the entrance of the grotto, at the moment they were about to unfasten the raft to cross the river, they saw, by the pale rays of the moon, another raft, manned by fifteen men, who steadily directed their course towards them. Their retreat was cut off. How could they possibly resist such a number of adversaries? "What a fatality!" the general murmured, despondingly. "Oh!" said the doctor, piteously, "a plan of escape that cost me so much trouble to elaborate!" The fugitives threw themselves into a cavity of the rocks, to avoid being seen, and there waited the landing of the newcomers, whose manoeuvres appeared more and more suspicious.
- 63. CHAPTER XIII. THE LAW OF THE PRAIRIES. A considerable space of ground, situated in front of the grotto inhabited by Loyal Heart, had been cleared, the trees cut down, and from a hundred and fifty to two hundred huts erected. The whole tribe of the Comanches was encamped on this spot. Among trappers, hunters, and redskin warriors there existed the best possible understanding. In the centre of this temporary village, where the huts of buffalo hides painted of different colours were arranged with a degree of symmetry, one much larger than the others, surmounted by scalps fixed to long poles, and in which a large fire was continually kept up, served as the council lodge. The greatest bustle prevailed in the village. The Indian warriors were armed and in their war paint, as if preparing to march to battle. The hunters had dressed themselves in their best costumes, and cleaned their arms with the greatest care, as if expecting soon to make use of them. The horses completely caparisoned, stood hobbled, and held by half a score warriors, ready to be mounted. Hunters and redskins were coming and going in a busy, preoccupied manner. A rare and almost unknown thing among Indians, sentinels were placed at regular distances to signal the approach of a stranger, whoever he might be. In short, everything denoted that one of the ceremonies peculiar to the prairies was about to take place. But, strange to say, Loyal
- 64. Heart, Eagle Head, and Black Elk were absent. Belhumeur alone watched over the preparations that were being made, talking, the while, to the old Comanche chief Eshis, or the Sun. But their countenances were stern, their brows thoughtful, they appeared a prey to an overpowering preoccupation. It was the day fixed upon by the captain of the pirates for Doña Luz to be delivered up to him. Would the captain venture to come? or was his proposition anything more than a rodomontade? Those who knew the pirate, and their number was great—almost all having suffered by his depredations—inclined to the affirmative. This man was endowed, and it was the only quality they acknowledged in him, with a ferocious courage and an iron will. If once he had affirmed he would do a thing, he did it, without regard to anybody or any danger. And then, what had he to dread in coming a second time amongst his enemies? Did he not hold the general in his power? the general, whose life answered for his own; all knew that he would not hesitate to sacrifice him to his safety. It was about eight o'clock in the morning, a brilliant sun shed its dazzling rays in profusion upon the picture we have endeavoured to describe. Doña Luz left the grotto, leaning upon the arm of the mother of Loyal Heart, and followed by Nô Eusebio. The two women were sad and pale, their faces looked worn, and their red eyes showed they had been weeping. As soon as Belhumeur perceived them, he advanced towards them, bowing respectfully. "Has not my son returned yet?" the old lady asked, anxiously.
- 65. "Not yet," the hunter replied, "but keep up your spirits, señora, it will not be long before he is here." "Good God! I do not know why, but it seems as if he must be detained at a distance from us by some untoward event." "No, señora, I should know if he were so. When I left him last night, for the purpose of tranquillizing you, and executing the orders he gave me, he was in an excellent situation; therefore, believe me, be reassured, and, above all, have confidence." "Alas!" the poor woman murmured, "I have lived for twenty years in continual agony, every night dreading not to see my son on the morrow; my God! will you not then have pity on me!" "Have comfort, dear señora," said Doña Luz, affectionately, and with a gentle kiss: "Oh! I know that if Loyal Heart at this moment be in danger, it is to save my poor uncle; my God!" she added, fervently, "grant that he may succeed!" "All will soon be cleared up, ladies, be assured by me, and you know I would not deceive you." "Yes," said the old lady, "you are good, you love my son, and you would not be here if he had anything to dread." "You judge me rightly, señora, and I thank you for it. I cannot, at the present moment, tell you anything, but I implore you to have a little patience; let it suffice for you to know that he is labouring to render the señorita happy." "Oh! yes," said the mother, "always good, always devoted!" "And therefore was he named Loyal Heart," the maiden murmured, with a blush. "And never was name better merited," the hunter exclaimed proudly. "A man must have lived a long time with him, and know him as well as I know him, in order to appreciate him properly." "Thanks, in my turn, for all you say of my son, Belhumeur," the old lady replied, pressing the callous hand of the hunter.
- 66. "I speak nothing but the truth, señora; I am only just, that is all. Oh! things would go on well in the prairies if all hunters were like him." "Good heavens! time passes, will he never come?" she murmured, looking around with feverish impatience. "Very soon, señora." "I wish to be the first to see him and salute him on his arrival!" "Unfortunately that is impossible." "Why so?" "Your son charged me to beg you, as well as Señora Luz, to retire into the grotto; he is anxious that you should not be present at the scene that is about to take place here." "But," said Doña Luz, anxiously, "how shall I know if my uncle be saved or not?" "Be assured, señorita, that you shall not remain in uncertainty long. But I beg you not to remain here. Go in, go in." "Perhaps it will be best to do so," the old lady observed. "Let us be obedient, darling," she added, smiling on the girl; "let us go in, since my son requires it." Doña Luz followed her without resistance, but casting furtive looks behind her, to try if she could catch a glimpse of him she loved. "How happy are those who have mothers!" murmured Belhumeur, stifling a sigh, and looking after the two women, who disappeared in the shade of the grotto. All at once the Indian sentinels uttered a cry, which was immediately repeated by a man placed in front of the council lodge. At this signal the Comanche chiefs arose and left the hut in which they were assembled. The hunters and Indian warriors seized their arms, ranged themselves on either side of the grotto, and waited.
- 67. A cloud of dust rolled towards the camp with great rapidity, but was soon dispersed, and revealed a troop of horsemen riding at full speed. These horsemen, for the most part, wore the costume of Mexican gambusinos. At their head, upon a magnificent horse, black as night, came a man whom all immediately recognized. This was Captain Waktehno, who came audaciously at the head of his troop, to claim the fulfilment of the odious bargain he had imposed three days before. Generally, in the prairies, when two troops meet, or when warriors or hunters visit a village, it is the custom to execute a sort of fantasia, by rushing full speed towards each other, yelling and firing off guns. On this occasion, however, nothing of the kind took place. The Comanches and the hunters remained motionless and silent, awaiting the arrival of the pirates. This cold, stern reception did not astonish the captain; though his eyebrows were a little contracted, he feigned not to perceive it, and entered the village intrepidly at the head of his band. When he arrived in front of the chiefs drawn up before the council lodge, the twenty horsemen stopped suddenly, as if they had been changed into statues of bronze. This bold manoeuvre was executed with such dexterity that the hunters, good judges of horsemanship, with difficulty repressed a cry of admiration. Scarcely had the pirates halted, ere the ranks of the warriors placed on the right and left of the lodge deployed like a fan, and closed behind them. The twenty pirates found themselves by this movement, which was executed with incredible quickness, enclosed within a circle formed of more than five hundred men, well armed and equally well mounted.
- 68. The captain felt a slight tremor of uneasiness at the sight of this manoeuvre, and he almost repented having come. But surmounting this involuntary emotion, he smiled disdainfully; he believed he was certain he had nothing to fear. He bowed slightly to the chiefs ranged before him, and addressed Belhumeur in a firm voice,— "Where is the girl?" he demanded. "I do not know what you mean," the hunter replied, in a bantering tone; "I do not believe that there is any young lady here upon whom you have any claim whatever." "What does this mean? and what is going on here?" the captain muttered, casting around a look of defiance. "Has Loyal Heart forgotten the visit I paid him three days ago?" "Loyal Heart never forgets anything," said Belhumeur, in a firm tone; "but the question is not of him now. How can you have the audacity to present yourself among us at the head of a set of brigands?" "Well," said the captain jeeringly, "I see you want to answer me by an evasion. As to the menace contained in the latter part of your sentence, it is worth very little notice." "You are wrong; for since you have committed the imprudence of throwing yourself into our hands, we shall not be simple enough, I warn you, to allow you to escape." "Oh, oh!" said the pirate; "what game are we playing now?" "You will soon learn." "I can wait," the pirate replied, casting around a provoking glance. "In these deserts, where all human laws are silent," the hunter replied, in a loud clear voice, "the law of God ought to reign in full vigour. This law says, 'An eye for an eye, a tooth for a tooth.'" "What follows?" said the pirate, in a dry tone.
- 69. "During ten years," Belhumeur continued impassively, "at the head of a troop of bandits, without faith and without law, you have been the terror of the prairies, pillaging and assassinating white men and red men; for you are of no country, plunder and rapine being your only rule; trappers, hunters, gambusinos, or Indians, you have respected no one, if murder could procure you a piece of gold. Not many days ago you took by assault the camp of peaceful Mexican travellers, and massacred them without pity. This career of crime must have an end, and that end has now come. We have Indians and hunters assembled here to try you, and apply to you the implacable law of the prairies." "An eye for an eye, a tooth for a tooth," the assembled Indians and hunters cried, brandishing their arms. "You deceive yourselves greatly, my masters," the pirate answered, with assurance, "If you believe I shall hold my throat out peaceably to the knife, like a calf that is being led to the shambles. I suspected what would happen, and that is why I am so well accompanied. I have with me twenty resolute men, who well know how to defend themselves. You have not got us yet." "Look around you, and see what is left for you to do." The pirate cast a look behind him, and saw five hundred guns levelled at his band. A shudder passed through his limbs, a mortal pallor covered his face, the pirate understood that he was confronted by a terrible danger; but after a second of reflection, he recovered all his coolness, and addressing the hunter, he replied in a jeering voice:— "What is the use of all these menaces, which do not frighten me? You know very well that I am screened from all your violence. You have told me that I attacked some Mexican travellers a few days ago, but you are not ignorant that the most important of those travellers has fallen into my power. Dare but to touch a single hair of my head, and the general, the uncle of the girl you would in vain ravish from my power, will immediately pay with his life for the insult
- 70. you offer me. Believe me, then, my masters, you had better cease endeavouring to terrify me; give up to me with a good grace her whom I come to demand, or I swear to you, by God, that within an hour the general will be a dead man." All at once a man broke through the crowd, and placing himself in front of the pirate, said— "You are mistaken, the general is free!" That man was Loyal Heart. A hum of joy resounded from the ranks of the hunters and Indians, whilst a shudder of terror agitated the pirates. CHAPTER XIV. THE CHASTISEMENT. The general and his two companions had not remained long in a state of uncertainty. The raft, after several attempts, came to shore at last, and fifteen men, armed with guns advanced, and rushed into the grotto, uttering loud cries. The fugitives ran towards them with joy; for they recognized at the head of them Loyal Heart, Eagle Head, and Black Elk. This is what had happened. As soon as the doctor had entered the grotto with the captain, Eagle Head, certain of having discovered the retreat of the pirates, had rejoined his friends, to whom he imparted the success of his stratagem, Belhumeur had been despatched to Loyal Heart, who had hastened to come. All, in concert, had resolved to attack the bandits
- 71. in their cavern, whilst other detachments of hunters and redskin warriors, spread about the prairies, and concealed among the rocks should watch the approaches to the grottos and prevent the escape of the pirates. We have seen the result of this expedition. After having devoted the first moment entirely to joy, and the pleasure of having succeeded without a blow being struck, the general informed his liberators that half a score bandits were sleeping in the grotto, under the influence of the worthy doctor's opium. The pirates were strongly bound and carried away; then, after calling in the various detachments, the whole band again bent their way to the camp. Great had been the surprise of the captain at the exclamation of Loyal Heart; but that surprise was changed into terror, when he saw the general, whom he thought so safely guarded by his men, standing before him. He saw at once that all his measures were defeated, and his tricks circumvented, and that this time he was lost without resource. The blood mounted to his throat, his eyes darted lightning, and turning towards Loyal Heart, he said, in a hoarse loud voice— "Well played! but all is not yet ended between us. By God's help I shall have my revenge!" He made a gesture as if to put his horse in motion; but Loyal Heart held it resolutely by the bridle. "We have not done yet," he remarked. The pirate looked at him for an instant with eyes injected with blood, and then said in a voice broken by passion, whilst urging on his horse to oblige the hunter to quit his hold. "What more do you want with me?"
- 72. Loyal Heart, thanks to a wrist of iron, still held the horse, which plunged furiously. "You have been brought to trial," he replied, "and the law of the prairies is about to be applied to you." The pirate uttered a terrible, sneering, maniac laugh, and tore his pistols from his belt:— "Woe be to him who touches me!" he cried, with rage, "give me way!" "No," the impassive hunter replied, "you are fairly taken, my master; this time you shall not escape me." "Die then!" cried the pirate, aiming one of his pistols at Loyal Heart. But, quick as thought, Belhumeur, who had watched his movements closely, threw himself before his friend with a swiftness increased tenfold by the seriousness of the situation. The shot was fired. The ball struck the Canadian, who fell bathed in his blood. "One!" cried the pirate, with a ferocious laugh. "Two!" screamed Eagle Head, and with the bound of a panther, he leaped upon the pirate's horse behind him. Before the captain could make a movement to defend himself, the Indian seized him with his left hand, by the long hair, of which he formed a tuft, and pulled him backwards violently, with his head downwards. "Curses on you!" cried the pirate, in vain endeavouring to free himself from his enemy. And then took place a scene which chilled the spectators with horror. The horse, which Loyal Heart had left his hold of, when at liberty, furious with being urged on by its master and checked by Loyal Heart, and with the double weight imposed upon it, sprang forward, mad with rage, breaking and overturning in its course every object
- 73. that opposed its passage. But it still carried, clinging to its sides, the two men struggling to kill each other, and who on the back of the terrified animal writhed about like serpents. Eagle Head had, as we have said, pulled back the head of the pirate; he placed his knee against his loins, uttered his hideous war cry, and flourished with a terrible gesture his knife around the brow of his enemy. "Kill me, then, vile wretch!" the pirate cried, and with a rapid effort he raised his left hand, still armed with a pistol, but the bullet was lost in space. The Comanche chief fixed his eyes upon the captain's face. "Thou art a coward!" he said, with disgust, "and an old woman, who is afraid of death!" At the same time he pushed the bandit forcibly with his knee, and plunged the knife into his skull. The captain uttered a piercing cry, which arose into the air, mingled with the howl of triumph of the chief. The horse stumbled over a root; the two enemies rolled upon the ground. Only one rose up. It was the Comanche chief, who brandished the bleeding scalp of the pirate. But the latter was not dead. Almost mad with pain and fury, and blinded with the blood which trickled into his eyes, he arose and rushed upon his adversary, who had no expectation of such an attack. Then, with limbs entwined, each endeavoured, by strength and artifice, to throw his antagonist, and plunge into his body the knife with which he was armed.
- 74. Several hunters sprang forward to separate them, but when they reached them all was over. The captain lay upon the ground with the knife of Eagle Head buried to the hilt in his heart. The pirates, held in awe by the white hunters and the Indian warriors who surrounded them, did not attempt a resistance, which they knew would be useless. When he saw his captain fall, Frank, in the name of his companions, proclaimed that they surrendered. At a signal from Loyal Heart they laid down their arms and were bound. Belhumeur, the brave Canadian, whose devotedness had saved the life of his friend, had received a serious wound, but, happily, it was not mortal. He had been instantly lifted up and carried into the grotto, where the mother of the hunter paid him every attention. Eagle Head approached Loyal Heart, who stood pensive and silent, leaning against a tree. "The chiefs are assembled round the fire of council," he said, "and await my brother." "I follow, my brother," the hunter replied, laconically. When the two men entered the hut, all the chiefs were assembled; among them were the general, Black Elk, and several other trappers. The calumet was brought into the middle of the circle by the pipe bearer; he bowed respectfully towards the four cardinal points, and then presented the long tube to every chief in his turn. When the calumet had made the round of the circle, the pipe bearer emptied the ashes into the fire, murmuring some mystic words, and then retired. Then the old chief named the Sun, arose, and after saluting the members of the council, said—
- 75. "Chiefs and warriors, listen to the words which my lungs breathe and which the Master of Life has placed in my heart. What do you purpose doing with the twenty prisoners who are now in your hands? Will you release them that they may continue their life of murder and rapine? that they may carry off your wives, steal your horses, and kill your brothers? Will you conduct them to the stone villages of the great white hearts of the East? The route is long, abounding in dangers, traversed by mountains and rapid rivers; the prisoners may escape in the journey, or may surprise you in your sleep and massacre you. And then, you know, warriors, when you have arrived at the stone villages, the long knives will release them, for there exists no justice for red men. No, warriors, the Master of Life, who has, at length, delivered up these men into our power, wills that they should die. He has marked the term of their crimes. When we find a jaguar or a grizzly bear upon our path, we kill them; these men are more cruel than jaguars or grizzlies, they owe a reckoning for the blood they have shed, an eye for an eye, a tooth for a tooth. Let them, then, be fastened to the stake of torture. I cast a necklace of red wampums into the council. Have I spoken well, men of power?" After these words, the old chief sat down again. There was a moment of solemn silence. It was evident that all present approved of his advice. Loyal Heart waited for a few minutes; he saw that nobody was preparing to reply to the speech of the Sun; then he arose:— "Comanche chiefs and warriors, and you white trappers, my brothers," he said in a mild, sad tone, "the words pronounced by the venerable sachem are just; unfortunately, the safety of the prairies requires death of our prisoners. This extremity is terrible, but we are forced to submit to it, if we desire to enjoy the fruit of our rude labours in peace. But if we find ourselves constrained to apply the implacable law of the desert, let us not show ourselves barbarians by choice; let us punish, since it must be so, but let us punish like men of heart, and not like cruel men. Let us prove to these bandits
- 76. that we are executing justice, that in killing them it is not for the purpose of avenging ourselves, but the whole of society. Besides, their chief, by far the most guilty of them, has fallen before the courage and weapons of Eagle Head. Let us be clement without ceasing to be just. Let us leave them the choice of their death. No useless torture. The Master of Life will smile upon us, he will be content with his red children, to whom he will grant abundance of game in their hunting grounds. I have spoken: have I spoken well, men of power?" The members of the council had listened attentively to the words of the young man. The chiefs had smiled kindly at the noble sentiments he had expressed; for all, both Indians and trappers, loved and respected him. Eagle Head arose. "My brother, Loyal Heart has spoken well," said he; "his years are few in number, but his wisdom is great. We are happy to find an opportunity of proving our friendship for him; we seize it with eagerness. We will do what he desires." "Thank you!" Loyal Heart replied warmly; "thank you, my brothers! The Comanche nation is a great and noble nation, which I love; I am proud of having been adopted by it." The council broke up, and the chiefs left the lodge. The prisoners, collected in a group, were strictly guarded by a detachment of warriors. The public crier called together all the members of the tribe, and the hunters dispersed about the village. When all were assembled, Eagle Head arose to speak, and, addressing the pirates, said— "Dogs of palefaces, the council of the great chiefs of the powerful nation of the Comanches, whose vast hunting grounds cover a great part of the earth, has pronounced your fate. Try, after having lived like wild beasts, not to die like timid old women; be brave, and then,
- 77. perhaps, the Master of Life will have pity on you, and will receive you after death into the eskennane,—that place of delights where the brave who have looked death in the face hunt during eternity." "We are ready," replied Frank, unmoved; "fasten us to the stakes, invent the most atrocious tortures; you will not see us blench." "Our brother, Loyal Heart," the chief continued, "has interceded for you. You will not be fastened to the stake; the chiefs leave to yourselves the choice of your death." Then was awakened that characteristic trait in the manners of the whites, who, inhabiting the prairies for any length of time, end by forsaking the customs of their ancestors, and adopt those of the Indians. The proposition made by Eagle Head was revolting to the pride of the pirates. "By what right," Frank cried, "does Loyal Heart intercede for us? Does he fancy that we are not men? that tortures will be able to draw from us cries and complaints unworthy of us? No! no! lead us to punishment; whatever you can inflict upon us will not be so cruel as what we make the warriors of your nation undergo when they fall into our hands." At these insulting words a sensation of anger pervaded the ranks of the Indians, whilst the pirates, on the contrary, uttered cries of joy and triumph. "Dogs! rabbits!" they shouted; "Comanche warriors are old women, who ought to wear petticoats!" Loyal Heart advanced, and silence was re-established. "You have wrongly understood the words of the chief," he said; "in leaving you the choice of your death, it was not an insult, but a mark of respect that he paid you. Here is my dagger; you shall be unbound, let it pass from hand to hand, and be buried in all your hearts in turn. The man who is free, and without hesitation kills himself at a single blow, is braver than he who, fastened to the stake
- 78. of torture, and unable to endure the pain, insults his executioner in order to receive a prompt death." A loud acclamation welcomed these words of the hunter. The pirates consulted among themselves for an instant with a look, then, with one spontaneous movement, they made the sign of the cross, and cried with one voice— "We accept your offer!" The crowd, an instant before, so tumultuous and violent, became silent and attentive, awed by the expectation of the terrible tragedy which was about to be played before them. "Unbind the prisoners," Loyal Heart commanded. This order was immediately executed. "Your dagger!" said Frank. The hunter gave it to him. "Thank you, and farewell!" said the pirate, in a firm voice; and, opening his vestments, he deliberately, and with a smile, as if he enjoyed death, buried the dagger up to the hilt in his heart. A livid pallor gradually invaded his countenance, his eyes rolled in their orbits, and casting round wild and aimless glances, he staggered like a drunken man, and rolled upon the ground. He was dead. "My turn!" cried the pirate next him, and plucking the still reeking dagger from the wound, he plunged it into his heart. He fell upon the body of the first victim. After him came the turn of another, then another, and so on; not one hesitated, not one displayed weakness,—all fell smiling, and thanking Loyal Heart for the death they owed to him. The spectators were awestruck by this terrible execution; but, fascinated by the frightful spectacle,—drunk, so to say, with the
- 79. odour of blood, they stood with haggard eyes and heaving breasts, without having the power to turn away their looks. There soon remained but one pirate. This man contemplated for a moment the heap of bodies which lay before him; then, drawing the dagger from the breast of him who had preceded him, he said with a smile,— "A fellow is lucky to die in such good company; but where the devil do we go to after death? Bah! what a fool I am! I shall soon know!" And with a gesture quick as thought he stabbed himself. He fell instantly quite dead. This frightful slaughter did not last more than a quarter of an hour. [1] Not one of the pirates had struck twice; all were killed by the first blow. "The dagger is mine!" said Eagle Head, drawing it smoking from the still palpitating body of the last bandit. "It is a good weapon for a warrior;" and he placed it coolly in his belt, after having wiped it upon the grass. The bodies of the pirates were scalped, and borne out of the camp. They were abandoned to the vultures and the urubus, for whom they would furnish an ample feast, and who, attracted by the odour of blood, were already hovering over them, uttering lugubrious cries of joy. The formidable troop of Captain Waktehno was thus annihilated. Unfortunately there were other pirates in the prairies. After the execution, the Indians re-entered their huts carelessly; for them it had only been one of those spectacles to which they had been for a long time accustomed, and which have no effect upon their nerves.
- 80. The trappers, on the contrary, notwithstanding the rough life they lead, and the frequency with which they see blood shed—either their own or that of other people, dispersed silently and noiselessly, with hearts oppressed by the spectacle of this frightful butchery. Loyal Heart and the general directed their steps towards the grotto. The ladies, shut up in the interior of the cavern, were ignorant of the terrible drama that had been played, and of the sanguinary expiation which had terminated it.
- 81. [1] All this scene is historical, and strictly true; the author was present in Apacheria, at a similar execution. CHAPTER XV. THE PARDON. The interview between the general and his niece was most touching. The old soldier, so roughly treated for some time past, was delighted to press to his bosom the innocent child who constituted his whole family, and who, by a miracle, had escaped the misfortunes that had assailed her. For a long time they forgot themselves in a delightful interchange of ideas; the general anxiously inquiring how she had lived while he was a prisoner—the young girl questioning him upon the perils he had run, and the ill-treatment he had suffered. "Now, uncle," she said at length, "what is your intention?" "Alas! my child," he replied, in a tone of sadness, and stifling a sigh; "we must without delay leave these terrible countries, and return to Mexico." The heart of the young girl throbbed painfully, although she inwardly confessed the necessity for a prompt return. To leave the prairies would be to leave him she loved—to separate herself, without hope of a reunion, from a man whose admirable character every minute passed in sweet intercourse had made her more duly appreciate, and who had now become indispensable to her life and her happiness. "What ails thee, my child? You are sad, and your eyes are full of tears," her uncle asked, pressing her hand affectionately.
- 82. "Alas! dear uncle," she replied, in a plaintive tone; "how can I be otherwise than sad after all that has happened within the last few days? My heart is oppressed." "That is true. The frightful events of which we have been the witnesses and the victims are more than enough to make you sad; but you are still very young, my child. In a short time these events will only remain in your thoughts as the remembrance of facts which, thanks to Heaven! you will not have to dread in future." "Then shall we depart soon?" "Tomorrow, if possible. What should I do here now? Heaven itself declares against me, since it obliges me to renounce this expedition, the success of which would have made the happiness of my old age; but God is not willing that I should be consoled. His will be done!" he added, in a tone of resignation. "What do you mean, dear uncle?" the maiden asked, eagerly. "Nothing that can interest you at present, my child. You had better, therefore, be ignorant of it, and that I should suffer alone. I am old. I am accustomed to sorrow," he said, with a melancholy smile. "My poor uncle!" "Thank you for the kindness you evince, my child; but let us quit this subject that saddens you; let us speak a little, if you please, of the worthy people to whom we owe so many obligations." "Of Loyal Heart?" Doña Luz murmured, with a blush. "Yes," the general replied. "Loyal Heart and his mother; the excellent woman whom I have not yet been able to thank, on account of the wound of poor Belhumeur, and to whom it is due, you say, that you have not suffered any privations." "She has had all the cares of a tender mother for me!" "How can I ever acquit myself towards her and her noble son? She is blessed in having such a child! Alas! that comfort is not given to me —I am alone!" the general said, letting his head sink into his hands.
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