deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index 08c5ae7..9694e15 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
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\begin{eqnarray}  \label{e.hmcr}  \hat{x}_{m+1,j} & = & \left\{ \begin{array}{ll} x_{A,j} & \mbox{{ with probability HMCR}} $HMCR$}}  \\ \theta \in \bm{\Phi}_j & \mbox{{ with probability (1-HMCR),}} (1-$HMCR$),}}  \end{array}\right.  \end{eqnarray}  where $A\sim {\cal U}(1,2,\ldots,m)$, and $\bm{\Phi}=\{\bm{\Phi}_1,\bm{\Phi}_2,\ldots,\bm{\Phi}_m\}$ stands for the set of feasible values for each decision variable. 
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\begin{eqnarray}  \label{e.par}  x_{m+1,j} & = & \left\{ \begin{array}{ll} x_{m+1,j}\pm \varphi_j \varrho & \mbox{{ with probability PAR}} $PAR$}}  \\ x_{m+1,j} & \mbox{{ with probability (1-PAR).}} (1-$PAR$).}}  \end{array}\right.  \end{eqnarray}  The pitch adjustment is often used to improve solutions and to escape from local optima. This mechanism concerns shifting the neighboring values of some decision variable in the harmony, where $\varrho$ is an arbitrary distance bandwidth, and $\varphi_j\sim {\cal U}(0,1)$. 
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\begin{eqnarray}  \label{e.hmcr_sghs}  x_{m+1,j} & = & \left\{ \begin{array}{ll} x_{A,j}\pm \varphi_j \varrho & \mbox{{ with probability HMCR}} $HMCR$}}  \\ \theta \in \bm{\Phi}_j & \mbox{{ with probability (1-HMCR)}.} (1-$HMCR$)}.}  \end{array}\right.  \end{eqnarray} 
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\begin{equation}  \label{e.position_tmp_bat}  \tilde{\textbf{x}}^{t+1}=\textbf{x}^t_i+\textbf{v}^{t+1}_i, \tilde{\textbf{x}}^{t+1}=\textbf{x}^t_i+\textbf{v}^{t+1}_i.  \end{equation}  Further, we apply a random walk with probability $p_r$ (also known as ``pulse rate"):  \begin{eqnarray}  \label{e.random_walk_bat}  \tilde{\textbf{x}}^{t+1} & = & \left\{ \begin{array}{ll} \tilde{\textbf{x}}^{t+1}+\epsilon\bar{A} & \mbox{{ with probability $p_r$}} \\  \tilde{\textbf{x}}^{t+1} & \mbox{{ with probability (1-$p_r$)}.}  \end{array}\right.  \end{eqnarray}  %\begin{equation}