+\begin{itemize}
+\item $X_{t,j}$: indicates whether or not the sensor $j$ is actively sensing
+ during round $t$ (1 if yes and 0 if not);
+\item $\Theta_{t,p}$ - {\it overcoverage}: the number of sensors minus one that
+ are covering the primary point $p$ during round $t$;
+\item $U_{t,p}$ - {\it undercoverage}: indicates whether or not the primary
+ point $p$ is being covered during round $t$ (1 if not covered and 0 if
+ covered).
+\end{itemize}
+
+%The first group of constraints indicates that some primary point $p$ should be covered by at least one sensor and, if it is not always the case, overcoverage and undercoverage variables help balancing the restriction equations by taking positive values.
+The constraint given by equation~(\ref{eq144}) guarantees that the sensor has enough energy ($RE_j$ corresponds to its remaining energy) to be alive during the selected rounds knowing that $E_{th}$ is the amount of energy required to be alive during one round.
+
+%There are two main objectives. First, we limit the overcoverage of primary points in order to activate a minimum number of sensors. Second we prevent the absence of monitoring on some parts of the subregion by minimizing the undercoverage. The weights $W_\theta$ and $W_U$ must be properly chosen so as to guarantee that the maximum number of points are covered during each round. In our simulations, priority is given to the coverage by choosing $W_{U}$ very large compared to $W_{\theta}$.
+
+
+
+
+
+\section{Experimental Study and Analysis}
+\label{ch5:sec:04}
+
+\subsection{Simulation Setup}
+\label{ch5:sec:04:01}
+We conducted a series of simulations to evaluate the efficiency and the
+relevance of our approach, using the discrete event simulator OMNeT++
+\cite{ref158}. We performed the optimization in the same manner than in chapter 4, considering the same energy model. The simulation parameters are summarized in Table~\ref{tablech4}. Each experiment for a network is run over 25~different random topologies and the results presented hereafter are the average of these 25 runs.
+We performed simulations for five different densities varying from 50 to
+250~nodes deployed over a $50 \times 25~m^2 $ sensing field. More
+precisely, the deployment is controlled at a coarse scale in order to ensure
+that the deployed nodes can cover the sensing field with the given sensing
+range.
+
+Our protocol is declined into four versions: MuDiLCO-1, MuDiLCO-3, MuDiLCO-5, and MuDiLCO-7, corresponding respectively to $T=1,3,5,7$ ($T$ the number of rounds in one sensing period). In the following, we will make comparisons with three other methods. DESK \cite{DESK}, GAF~\cite{GAF}, and DiLCO~\cite{Idrees2}, where MuDiLCO-1 is the same of DiLCO.
+%Some preliminary experiments were performed in chapter 4 to study the choice of the number of subregions which subdivides the sensing field, considering different network sizes. They show that as the number of subregions increases, so does the network lifetime. Moreover, it makes the MuDiLCO protocol more robust against random network disconnection due to node failures. However, too many subdivisions reduce the advantage of the optimization. In fact, there is a balance between the benefit from the optimization and the execution time needed to solve it. Therefore,
+We set the number of subregions to 16 rather than 32 as explained in section \ref{ch4:sec:04:05}.
+We use the modeling language and the optimization solver which are mentioned in section \ref{ch4:sec:04:02}.
+%In addition, the energy consumption model is presented in chapter 4, section \ref{ch4:sec:04:03}.
+
+\subsection{Metrics}
+\label{ch5:sec:04:02}
+To evaluate our approach we consider the following performance metrics
+\begin{frame}{}
+\begin{enumerate}[i)]
+
+\item {{\bf Coverage Ratio (CR)}:} The coverage ratio can be calculated by:
+\begin{equation*}
+\scriptsize
+\mbox{CR}(\%) = \frac{\mbox{$n^t$}}{\mbox{$N$}} \times 100,
+\end{equation*}
+where $n^t$ is the number of covered grid points by the active sensors of all
+subregions during round $t$ in the current sensing phase.
+% and $N$ is the total number of grid points in the sensing field of the network. In our simulations $N = 51 \times 26 = 1326$ grid points.
+
+\item{{\bf Number of Active Sensors Ratio (ASR)}:} The Active Sensors
+ Ratio for round t is defined as follows:
+\begin{equation*}
+\scriptsize \mbox{$ASR^t$}(\%) = \frac{\sum\limits_{r=1}^R
+ \mbox{$A_r^t$}}{\mbox{$|J|$}} \times 100,
+\end{equation*}
+where $A_r^t$ is the number of active sensors in the subregion $r$ during round
+$t$ in the current sensing phase.
+%, $|J|$ is the total number of sensors in the network, and $R$ is the total number of subregions in the network.
+
+\item {{\bf Energy Consumption (EC)}:} EC can be computed as follows:
+
+ \begin{equation*}
+ \scriptsize
+ \mbox{EC} = \frac{\sum\limits_{m=1}^{M} \left[ \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m \right) +\sum\limits_{t=1}^{T} \left( E^{a}_t+E^{s}_t \right) \right]}{\sum\limits_{m=1}^{M} T},
+ \end{equation*}
+
+The energy factors of above equation are described in section \ref{ch4:sec:04:04}. $E^a_t$ and $E^s_t$
+indicate the energy consumed by the whole network in round $t$ of the sensing phase.
+
+%where $M$ is the number of periods and $T$ the number of rounds in a period~$m$, both during $Lifetime_{95}$ or $Lifetime_{50}$. The total energy consumed by the sensors (EC) comes through taking into consideration four main energy factors.
+%The first one , denoted $E^{\scriptsize \mbox{com}}_m$, represents the energy consumption spent by all the nodes for wireless communications during period $m$. $E^{\scriptsize \mbox{list}}_m$, the next factor, corresponds to the energy consumed by the sensors in LISTENING status before receiving the decision to go active or sleep in period $m$. $E^{\scriptsize \mbox{comp}}_m$ refers to the energy needed by all the leader nodes to solve the integer program during a period. Finally, $E^a_t$ and $E^s_t$ indicate the energy consumed by the whole network in round $t$.
+
+ %$\right\}$
+
+\item {{\bf Execution Time}:}
+\item {{\bf Stopped simulation runs}:} \makebox(0,0){\put(0,2.2\normalbaselineskip){%
+ $\left.\rule{0pt}{2.3\normalbaselineskip}\right\}$ Described in section \ref{ch4:sec:04:04}.}}
+\item {{\bf Network Lifetime}:}
+
+\end{enumerate}
+
+\end{frame}
+
+\subsection{Results Analysis and Comparison }
+\label{ch5:sec:04:03}
+
+
+\begin{enumerate}[i)]
+
+\item {{\bf Coverage Ratio}}
+%\subsection{Coverage ratio}
+%\label{ch5:sec:03:02:01}
+
+Figure~\ref{fig3} shows the average coverage ratio for 150 deployed nodes. We
+can notice that for the first thirty rounds both DESK and GAF provide a coverage
+which is a little bit better than the one of MuDiLCO.
+
+This is due to the fact that, in comparison with MuDiLCO which uses optimization
+to put in sleep status redundant sensors, more sensor nodes remain active with
+DESK and GAF. As a consequence, when the number of rounds increases, a larger
+number of node failures can be observed in DESK and GAF, resulting in a faster
+decrease of the coverage ratio. Furthermore, our protocol allows to maintain a
+coverage ratio greater than 50\% for far more rounds. Overall, the proposed
+sensor activity scheduling based on optimization in MuDiLCO maintains higher
+coverage ratios of the area of interest for a larger number of rounds. It also
+means that MuDiLCO saves more energy, with fewer dead nodes, at most for several
+rounds, and thus should extend the network lifetime.
+
