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\fancyhead[C]{Improved Cuckoo Search for TTP} % Custom header text
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\title{\vspace{-15mm}\fontsize{17pt}{10pt}\selectfont\textbf{An Improved Cuckoo Search algorithm for the Traveling Tournament Problem}} % Article title
\author{
\large
\textsc{Sandile Saul, Adewumi Aderemi Oluyinka}\thanks{The financial assistance of the National Research Foundation (NRF) towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at, are those of the author and are not necessarily to be attributed to the NRF.}\\[2mm] % Your name
\normalsize School of Mathematics, Statistics and Computer Science\\ \normalsize University of Kwazulu-Natal,
Westville Campus,\\ \normalsize Durban, 4000,
South Africa\\ % Your institution
\normalsize \href{mailto:[email protected]}{[email protected]} % Your email address
\vspace{-5mm}
}
\date{}
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\begin{document}
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% ABSTRACT
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\begin{abstract}
\noindent Scheduling of professional sports is one of many researched practical problems in combinatorial optimization.
\textbf{Keywords:} Sport Scheduling, Traveling Tournament Problem, Double Round-Robin Tournament, Improved Cuckoo Search.
\end{abstract}
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% ARTICLE CONTENTS
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\begin{multicols}{2} % Two-column layout throughout the main article text
\section{Introduction}
\lettrine[nindent=0em,lines=3]{S}cheduling is one of many researched practical problems in combinatorial optimization. It involves developing a timetable of events within a specified time frame and giv
\begin{equation}
\textit{Minimize} \ \sum _{i=1}^{n}\sum_{k=1}^{2(n-1)} d_{ij}x_{ijk}
\end{equation}
Subject to the following constraints:
\begin{equation}
\sum_{j=1}^{n} (x_{i,j,k} + x_{j,i,k})= 1\:\:\:\:
\end{equation}
\mbox{ $\forall{1\leq{i} \leq{n}}$} \ \ \mbox{$ \forall{1\leq{k}\leq{2(n-1)}}$}
\begin{equation}
\sum_{k=1}^{2(n-1)} x_{i,j,k} = 1\:\:\:\: \forall{{1\leq{i}\leq{n}}}, \:\:\forall{{1\leq{j}\leq{n}}}, \:\:i\neq{j}
\end{equation}
\begin{equation}
x_{i,j,k} + x_{j,i,k+1} \leq{1}\:\:\:\: \forall{{1\leq{i\neq{j}}\leq{n}}},
\end{equation}
\mbox{$\forall{{1\leq{k}\leq{2n-3}}}$}
\begin{equation}
x_{j,i,k} + x_{i,j,k+1} \leq{1}\:\:\:\: \forall{{1\leq{i\neq{j}}\leq{n}}},
\end{equation}
\mbox{$\forall{{1\leq{k}\leq{2n-3}}}$}
\begin{equation}
\sum_{j=1}^{n}\sum_{s=k}^{k+3} x_{i,j,s} \leq{3}\:\:\:\: \forall{{1\leq{i}\leq{n}}}, \:\:\forall{{1\leq{k}\leq{2n-5}}}
\end{equation}
\begin{equation}
\sum_{j=1}^{n}\sum_{s=k}^{k+3} x_{j,i,s} \leq{3}\:\:\:\: \forall{{1\leq{i}\leq{n}}}, \:\:\forall{{1\leq{k}\leq{2n-5}}}
\end{equation}
Where:\\
$n$ : represents the number of teams\\
$i$ : a team index\\
$j$ : a team index\\
$k$ : represents a time slot or a round\\
$d_{ij}$ : is the distance between team \textit{i's} home venue and team \textit{j's} home.\\
\begin{equation*}
x_{ijk} = \left\{
\begin{array}{ll}
0, & \text{if team $i$ is playing at home }\\
& \text{ in slot or round $k$,}\\
1, & \text{Otherwise.}\\
\end{array} \right\}
\end{equation*}
\section{Methods 1}
\begin{center}
\begin{figure}[h!]
\centering
\begin{tikzpicture}[%
back line/.style={densely dotted},
cross line/.style={preaction={draw=white, -,line width=8pt}}]
\node (A) {$8$};
\node (B)[below of=A] {$1$};
\node (C) [node distance=1.5cm, right of=B, below of=B]{$2$};
\node (D) [node distance=1.5cm, left of=C, below of=B]{$7$};
\node (E) [node distance=1.5cm, below of=C]{$3$};
\node (F) [node distance=3.0cm, left of=E]{$6$};
\node (G) [node distance=0.8cm, left of=E, below of=E]{$4$};
\node (H) [node distance=0.8cm, right of=G, below of=F]{$5$};
\draw[cross line] (B) -- (C);
\draw[cross line] (C) -- (E);
\draw[cross line] (E) -- (G);
%\draw[cross line] (G) -- (H);
\draw[cross line] (H) -- (F);
\draw[cross line] (F) -- (D);
\draw[cross line] (D) -- (B);
\draw[<->, dashed] (C) to node {} (D);
\draw[<->, dashed] (F) to node {} (E);
\draw[<->, dashed] (G) to node {} (H);
\draw[<->, dashed] (A) to node {} (B);
\end{tikzpicture}
\caption{Polygon method for $n$ = 8.}
%Figure 1: Polygon method for $n$ = 8.
\end{figure}
\end{center}
\section{Methods 2}
\begin{table}{H}
\caption{Results produced by the execution of the polygon method.}
\centering
\begin{tabular}{rrrrrrrrr}
\toprule(r){1-8}
$T_i \textbackslash k$ & 1 & 2 & 3 & 4 & 5 & 6 & 7\\
\midrule
1 & 8 & 6 & 4 & 2 & 7 & 5 & 3\\
2 & 7 & 5 & 3 & 1 & 6 & 4 & 8\\
3 & 6 & 4 & 2 & 7 & 5 & 8 & 1\\
4 & 5 & 3 & 1 & 6 & 8 & 2 & 7\\
5 & 4 & 2 & 7 & 8 & 3 & 1 & 6\\
6 & 3 & 1 & 8 & 4 & 2 & 7 & 5\\
7 & 2 & 8 & 5 & 3 & 1 & 6 & 4\\
8 & 1 & 7 & 6 & 5 & 4 & 3 & 2\\
\bottomrule
\end{tabular}
\end{table}
%------------------------------------------------
\section{Results}
\begin{table}[H]
\caption{Example table}
\centering
\begin{tabular}{llr}
\toprule
\multicolumn{2}{c}{Name} \\
\cmidrule(r){1-2}
First name & Last Name & Grade \\
\midrule
John & Doe & $7.5$ \\
Richard & Miles & $2$ \\
\bottomrule
\end{tabular}
\end{table}
\begin{thebibliography}{99} % Bibliography - this is intentionally simple in this template
\bibitem[Figueredo and Wolf, 2009]{Figueredo:2009dg}
Figueredo, A.~J. and Wolf, P. S.~A. (2009).
\newblock Assortative pairing and life history strategy - a cross-cultural
study.
\newblock {\em Human Nature}, 20:317--330.
\end{thebibliography}
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\end{multicols}
\end{document}
答案1
纠正两处打字错误后问题解决。
而不是\begin{figure}[h!]输入\begin{figure}[H]并替换\begin{table}{H}为\begin{table}[H]。
如果要介绍具有不同隶属关系的不同作者，请使用\and内部\author命令：
\author{
\large
\textsc{Sandile Saul}\thanks{The financial assistance of the National Research Foundation (NRF) towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at, are those of the author and are not necessarily to be attributed to the NRF.}\\[2mm] % Your name
\normalsize School of Mathematics, Statistics and Computer Science\\ \normalsize University of Kwazulu-Natal,
Westville Campus,\\ \normalsize Durban, 4000,
South Africa\\ % Your institution
\normalsize \href{mailto:[email protected]}{[email protected]} % Your email address
\and %<-------------------------------
\large
\textsc{Adewumi Aderemi Oluyinka}\\[2mm] % Your name
\normalsize School of Mathematics, Statistics and Computer Science\\ \normalsize University of Kwazulu-Natal,
Westville Campus,\\ \normalsize Durban, 4000,
South Africa\\ % Your institution
\normalsize \href{mailto:[email protected]}{[email protected]} % Your email address
}
