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\title{\LARGE \bf
Analysis and Enhancement of the Simulated Binary Crossover*
}

%\author{ \parbox{3 in}{\centering Huibert Kwakernaak*
%         \thanks{*Use the $\backslash$thanks command to put information here}\\
%         Faculty of Electrical Engineering, Mathematics and Computer Science\\
%         University of Twente\\
%         7500 AE Enschede, The Netherlands\\
%         {\tt\small h.kwakernaak@autsubmit.com}}
%         \hspace*{ 0.5 in}
%         \parbox{3 in}{ \centering Pradeep Misra**
%         \thanks{**The footnote marks may be inserted manually}\\
%        Department of Electrical Engineering \\
%         Wright State University\\
%         Dayton, OH 45435, USA\\
%         {\tt\small pmisra@cs.wright.edu}}
%}

\author{\IEEEauthorblockN{Joel Chac\'on,
Carlos Segura}\\
\IEEEauthorblockA{Center for Research in Mathematics, Jalisco S/N, Col. Valenciana CP: 36023 Guanajuato, Gto, M\'exico}\\
\IEEEauthorblockA{joel.chacon@cimat.mx, carlos.segura@cimat.mx}% <-this % stops an unwanted space
}

% \author{Joel Chac\'on Castillo$^{1}$ and  Carlos Segura$^{2}$,
% \\{Center for Research in Mathematics, Jalisco S/N, Col. Valenciana CP: 36023 Guanajuato, Gto, M\'exico}\\
% {joel.chacon@cimat.mx, carlos.segura@cimat.mx}% <-this % stops an unwanted space
% % and Arturo Hen\'andez$^3$% <-this % stops a space
% %\thanks{*This work was not supported by any organization}% <-this % stops a space
% %\thanks{$^{1}$H. Kwakernaak is with Faculty of Electrical Engineering, Mathematics and Computer Science,
% %        University of Twente, 7500 AE Enschede, The Netherlands
% %        {\tt\small h.kwakernaak at papercept.net}}%
% %\thanks{$^{2}$P. Misra is with the Department of Electrical Engineering, Wright State University,
% %        Dayton, OH 45435, USA
% %        {\tt\small p.misra at ieee.org}}%
% }


\begin{document}



\maketitle
\thispagestyle{empty}
\pagestyle{empty}


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\begin{abstract}
\input{Abstract.tex}
\end{abstract}

\section{Introduction}
\label{Introduction}
\input{Introduction.tex}

\section{Literature Review}
\label{LiteratureReview}
\input{LiteratureReview.tex}

% Please add the following required packages to your document preamble:
% \usepackage{multirow}
% \usepackage{graphicx}
\begin{table*}[t]
\centering

\caption{Statistical Information of Metrics with two objectives}
\label{tab:Metrics_2}
\resizebox{\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{} & \multicolumn{6}{c|}{NSGA-II} & \multicolumn{6}{c|}{MOEA/D} & \multicolumn{6}{c|}{SMS-EMOA} \\ \cline{2-19} 
 & 1 & 2 & 3 & 4 & 5 & DE & 1 & 2 & 3 & 4 & 5 & DE & 1 & 2 & 3 & 4 & 5 & DE \\ \hline
Average HV & 0.88 & 0.90 & 0.90 & 0.91 & 0.93 & \textbf{0.94} & 0.87 & 0.87 & 0.87 & 0.90 & \textbf{0.91} & \textbf{0.91} & 0.88 & 0.89 & 0.87 & 0.91 & 0.92 & \textbf{0.93} \\ \hline
Best Counts HV & 2 & 1 & 0 & 1 & 8 & \textbf{11} & 2 & 0 & 2 & 2 & 8 & \textbf{9} & 0 & 1 & 1 & 5 & 6 & \textbf{10} \\ \hline
\multicolumn{1}{|l|}{Average Best Difference HV} & \multicolumn{1}{l|}{0.068} & \multicolumn{1}{l|}{0.057} & \multicolumn{1}{l|}{0.053} & \multicolumn{1}{l|}{0.039} & \multicolumn{1}{l|}{0.019} & \multicolumn{1}{l|}{\textbf{0.017}} & \multicolumn{1}{l|}{0.053} & \multicolumn{1}{l|}{0.048} & \multicolumn{1}{l|}{0.049} & \multicolumn{1}{l|}{0.024} & \multicolumn{1}{l|}{\textbf{0.013}} & \multicolumn{1}{l|}{0.014} & \multicolumn{1}{l|}{0.074} & \multicolumn{1}{l|}{0.064} & \multicolumn{1}{l|}{0.081} & \multicolumn{1}{l|}{0.045} & \multicolumn{1}{l|}{0.028} & \multicolumn{1}{l|}{\textbf{0.019}} \\ \hline
Average IGD+ & 0.12 & 0.09 & 0.11 & 0.07 & 0.06 & \textbf{0.05} & 0.14 & 0.12 & 0.14 & 0.09 & 0.08 & \textbf{0.07} & 0.13 & 0.11 & 0.14 & 0.08 & 0.07 & \textbf{0.05} \\ \hline
Best Counts IGD+ & 2 & 1 & 1 & 1 & 8 & \textbf{10} & 3 & 0 & 2 & 3 & 6 & \textbf{9} & 0 & 2 & 0 & 3 & \textbf{9} & \textbf{9} \\ \hline
\multicolumn{1}{|l|}{Average Best Difference IGD+} & \multicolumn{1}{l|}{0.086} & \multicolumn{1}{l|}{0.052} & \multicolumn{1}{l|}{0.077} & \multicolumn{1}{l|}{0.035} & \multicolumn{1}{l|}{0.021} & \multicolumn{1}{l|}{\textbf{0.016}} & \multicolumn{1}{l|}{0.075} & \multicolumn{1}{l|}{0.059} & \multicolumn{1}{l|}{0.072} & \multicolumn{1}{l|}{0.025} & \multicolumn{1}{l|}{0.019} & \multicolumn{1}{l|}{\textbf{0.008}} & \multicolumn{1}{l|}{0.093} & \multicolumn{1}{l|}{0.071} & \multicolumn{1}{l|}{0.101} & \multicolumn{1}{l|}{0.038} & \multicolumn{1}{l|}{0.030} & \multicolumn{1}{l|}{\textbf{0.017}} \\ \hline
\end{tabular}%
}
\end{table*}

\section{Proposal}
\label{Proposal}
\input{Proposal.tex}
% Please add the following required packages to your document preamble:
% \usepackage{multirow}
% \usepackage{graphicx}
\begin{table*}[t]
\centering
\caption{Statistical Information of Metrics with three objectives}
\label{tab:Metrics_3}
\resizebox{\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{} & \multicolumn{6}{c|}{NSGA-II} & \multicolumn{6}{c|}{MOEA/D} & \multicolumn{6}{c|}{SMS-EMOA} \\ \cline{2-19} 
 & 1 & 2 & 3 & 4 & 5 & DE & 1 & 2 & 3 & 4 & 5 & DE & 1 & 2 & 3 & 4 & 5 & DE \\ \hline
Average HV & \textbf{0.87} & 0.84 & \textbf{0.87} & \textbf{0.87} & \textbf{0.87} & 0.85 & 0.84 & 0.84 & 0.84 & \textbf{0.86} & \textbf{0.86} & 0.85 & 0.90 & 0.89 & 0.88 & \textbf{0.91} & \textbf{0.91} & \textbf{0.91} \\ \hline
Best Counts HV & 1 & 2 & 1 & 4 & 4 & \textbf{7} & 1 & 2 & 1 & 2 & 5 & \textbf{8} & 3 & 2 & 0 & 2 & 5 & \textbf{7} \\ \hline
\multicolumn{1}{|l|}{Average Best Difference HV} & \multicolumn{1}{l|}{0.019} & \multicolumn{1}{l|}{0.047} & \multicolumn{1}{l|}{0.020} & \multicolumn{1}{l|}{\textbf{0.014}} & \multicolumn{1}{l|}{\textbf{0.014}} & \multicolumn{1}{l|}{0.032} & \multicolumn{1}{l|}{0.036} & \multicolumn{1}{l|}{0.041} & \multicolumn{1}{l|}{0.038} & \multicolumn{1}{l|}{0.016} & \multicolumn{1}{l|}{\textbf{0.013}} & \multicolumn{1}{l|}{0.027} & \multicolumn{1}{l|}{0.038} & \multicolumn{1}{l|}{0.038} & \multicolumn{1}{l|}{0.049} & \multicolumn{1}{l|}{\textbf{0.019}} & \multicolumn{1}{l|}{0.027} & \multicolumn{1}{l|}{\textbf{0.019}} \\ \hline
Average IGD+ & 0.13 & 0.16 & 0.13 & \textbf{0.12} & \textbf{0.12} & 0.13 & 0.15 & 0.14 & 0.15 & \textbf{0.11} & \textbf{0.11} & 0.13 & 0.11 & 0.11 & 0.13 & \textbf{0.09} & \textbf{0.09} & 0.13 \\ \hline
Best Counts IGD+ & 0 & 2 & 2 & 4 & 3 & \textbf{8} & 2 & 2 & 0 & 2 & 4 & \textbf{9} & 1 & 3 & 0 & 3 & 5 & \textbf{7} \\ \hline
\multicolumn{1}{|l|}{Average Best Difference IGD+} & \multicolumn{1}{l|}{0.029} & \multicolumn{1}{l|}{0.061} & \multicolumn{1}{l|}{0.027} & \multicolumn{1}{l|}{0.023} & \multicolumn{1}{l|}{\textbf{0.020}} & \multicolumn{1}{l|}{0.032} & \multicolumn{1}{l|}{0.053} & \multicolumn{1}{l|}{0.048} & \multicolumn{1}{l|}{0.053} & \multicolumn{1}{l|}{\textbf{0.015}} & \multicolumn{1}{l|}{\textbf{0.015}} & \multicolumn{1}{l|}{0.030} & \multicolumn{1}{l|}{0.047} & \multicolumn{1}{l|}{0.040} & \multicolumn{1}{l|}{0.062} & \multicolumn{1}{l|}{\textbf{0.020}} & \multicolumn{1}{l|}{0.024} & \multicolumn{1}{l|}{0.069} \\ \hline
\end{tabular}%
}
\end{table*}
\begin{table*}[t]
\centering
\caption{Summary of Statistical Tests}
\label{tab:statistical_Tests}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{16}{|c|}{NSGA-II} \\ \hline
 & \multicolumn{3}{c|}{1} & \multicolumn{3}{c|}{2} & \multicolumn{3}{c|}{3} & \multicolumn{3}{c|}{4} & \multicolumn{3}{c|}{5} \\ \hline
 & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ \\ \hline
\textbf{HV-2obj} & 16 & 29 & 47 & 6 & 61 & 25 & 28 & 19 & 45 & 31 & 23 & 38 & \textbf{54} & 3 & 35 \\ \hline
\textbf{HV-3obj} & 15 & 19 & 42 & 12 & 50 & 14 & 17 & 15 & 44 & \textbf{33} & 10 & 33 & 26 & 9 & 41 \\ \hline
\textbf{IGD-2obj} & 14 & 30 & 48 & 4 & 60 & 28 & 25 & 17 & 50 & 33 & 19 & 40 & \textbf{52} & 2 & 38 \\ \hline
\textbf{IGD-3obj} & 14 & 18 & 44 & 13 & 44 & 19 & 18 & 15 & 43 & \textbf{33} & 15 & 28 & 23 & 9 & 44 \\ \hline
% \end{tabular}
% \end{table*}

% \begin{table*}[]
% \centering
% \caption{My caption}
% \label{my-label}
% \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\hline
\multicolumn{16}{|c|}{MOEA/D} \\ \hline
 & \multicolumn{3}{c|}{1} & \multicolumn{3}{c|}{2} & \multicolumn{3}{c|}{3} & \multicolumn{3}{c|}{4} & \multicolumn{3}{c|}{5} \\ \hline
 & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ \\ \hline
\textbf{HV-2obj} & 15 & 33 & 44 & 10 & 60 & 22 & 25 & 26 & 41 & 39 & 18 & 35 & \textbf{57} & 9 & 26 \\ \hline
\textbf{HV-3obj} & 10 & 22 & 44 & 12 & 39 & 25 & 11 & 19 & 46 & 24 & 10 & 42 & \textbf{38} & 5 & 33 \\ \hline
\textbf{IGD-2obj} & 16 & 31 & 45 & 9 & 60 & 23 & 23 & 27 & 42 & 37 & 17 & 38 & \textbf{57} & 7 & 28 \\ \hline
\textbf{IGD-3obj} & 12 & 22 & 42 & 13 & 43 & 20 & 13 & 24 & 39 & 30 & 9 & 37 & \textbf{40} & 10 & 26 \\ \hline
% \end{tabular}
% \end{table*}

% \begin{table*}[]
% \centering
% \caption{My caption}
% \label{my-label}
% \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\hline
\multicolumn{16}{|c|}{SMS-EMOA} \\ \hline
 & \multicolumn{3}{c|}{1} & \multicolumn{3}{c|}{2} & \multicolumn{3}{c|}{3} & \multicolumn{3}{c|}{4} & \multicolumn{3}{c|}{5} \\ \hline
 & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ & $\uparrow$ & $\downarrow$ & $\longleftrightarrow$ \\ \hline
\textbf{HV-2obj} & 9 & 35 & 48 & 7 & 43 & 42 & 16 & 31 & 45 & 41 & 9 & 42 & \textbf{53} & 8 & 31 \\ \hline
\textbf{HV-3obj} & 7 & 21 & 48 & 9 & 35 & 32 & 13 & 21 & 42 & 27 & 6 & 43 & \textbf{31} & 4 & 41 \\ \hline
\textbf{IGD-2obj} & 10 & 34 & 48 & 15 & 48 & 29 & 12 & 33 & 47 & 41 & 12 & 39 & \textbf{55} & 6 & 31 \\ \hline
\textbf{IGD-3obj} & 8 & 20 & 48 & 13 & 30 & 33 & 9 & 19 & 48 & 22 & 5 & 49 & \textbf{27} & 5 & 44 \\ \hline
\end{tabular}
\end{table*}

\section{Experimental validation}
\label{Experimental_Validation}
\input{ExperimentalValidation.tex}



\section{Conclusions}
\label{Conclusions}

In the usual scheme of EAs, the crossover is one of the most important operators.
%
In most cases, crossover operators provide a large degree of exploration when the content of the population is diverse.
%
However, when a low diversity degree is reached, they tend to  promote intensification.
%
Since the behavior of operators does not usually depend on the stopping criterion, losing diversity in a gradual way might be a complex task.
%
Thus, some parameterizations might be suitable for some stopping criterion but not for others.
%
This paper proposes extending the well-known SBX to incorporate the stopping criterion and elapsed generations as one of its inputs.
%
First, the standard version of SBX and some actions that can be done to alter its exploration capabilities are identified.
%
Specifically, the SBX is composed by three key components that are related with diversity issues.
%
The first controls the amount of variables that are inherited intact.
%
The second is the probability of interchanging a given variable between offspring.
%
Finally, the last component is related with the aperture of the distribution index.
%
These tree features are adapted by taking into account both the stopping criterion and elapsed generations, with the aim of inducing an additional degree of exploration in the first stages of the optimization.
%
The experimental validation is carried out with long-term executions and the popular WFG, DTLZ and UF problems.
%
This validation shows that a dynamic distribution index provides significantly better and more robust results than the standard SBX with all the tested MOEAs.
%
Additionally, adapting the probability of interchanging variables between offspring provides benefits and by adapting this probability simultaneously with the distribution index performance can be further improved.
%
In the case of three objectives, state-of-the-art DE operators could also be outperformed by our proposal.
%
No additional parameterizations were required to devise the proposed operators, meaning that robust results could be obtained with the additional user efforts.
%An additional advantage is that does not required a distribution index a difference of the normal SBX.

Several lines of the future work might be explored.
%
First, we would like to devise some strategies to adaptively manage the distribution index.
%
Meassuring the diversity of the population to alter the behavior of the operators seems a plausible approach.
%
Additionally, using these kinds of operators together with some of the specific methods that has been devised to avoid premature convergence might bring additional benefits.
%
Finally, we plan to use the principles that governed the design of our proposals to devise novel vector-wise operators. 
%


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%\section*{APPENDIX}
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%\section*{ACKNOWLEDGMENT}
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