+\begin{enumerate}[i)]
+\item {{\bf Coverage Ratio}}
+%\subsubsection{Coverage Ratio}
+%\label{ch4:sec:04:02:01}
+
+Figure~\ref{Figures/ch4/R1/CR} shows the average coverage ratio for 150 deployed nodes.
+\parskip 0pt
+\begin{figure}[h!]
+\centering
+ \includegraphics[scale=0.8] {Figures/ch4/R1/CR.pdf}
+\caption{Coverage ratio for 150 deployed nodes}
+\label{Figures/ch4/R1/CR}
+\end{figure}
+It can be seen that DiLCO protocol (with 4, 8, 16 and 32 subregions) gives nearly similar coverage ratios during the first thirty periods.
+DiLCO-2 protocol gives a coverage ratio very close to the other protocols for the first 10 periods, and then the coverage decreases until the death of the network in the period $18^{th}$. In case of only 2 subregions, the energy consumption is high and the network is rapidly disconnected.
+As can be seen in Figure~\ref{Figures/ch4/R1/CR}, as the number of subregions increases, the coverage preservation for the area of interest increases for a larger number of periods. Coverage ratio decreases when the number of periods increases due to dead nodes. Although some nodes are dead, thanks to DiLCO-8, DiLCO-16, and DiLCO-32 protocols, other nodes are preserved to ensure the coverage. Moreover, when we have a dense sensor network, it leads to maintain the coverage for a larger number of periods. DiLCO-8, DiLCO-16, and DiLCO-32 protocols are slightly more efficient than other protocols, because they subdivide the area of interest into 8, 16 and 32~subregions; if one of the subregions becomes disconnected, the coverage may be still ensured in the remaining subregions.
+
+\item {{\bf Active Sensors Ratio}}
+%\subsubsection{Active Sensors Ratio}
+
+Figure~\ref{Figures/ch4/R1/ASR} shows the average active nodes ratio for 150 deployed nodes.
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=0.8]{Figures/ch4/R1/ASR.pdf}
+\caption{Active sensors ratio for 150 deployed nodes }
+\label{Figures/ch4/R1/ASR}
+\end{figure}
+
+The results presented in the figure show that increasing the number of subregions lead to the increase of the number of active nodes. The DiLCO-16 and DiLCO-32 protocols have a larger number of active nodes, but they both preserve the coverage for a larger number of periods. The advantage of the DiLCO-16 and DiLCO-32 protocols are that even if a network is disconnected in one subregion, the other ones usually continue the optimization process, and this extends the lifetime of the network.
+
+\item {{\bf Stopped simulation runs}}
+%\subsubsection{The percentage of stopped simulation runs}
+
+Figure~\ref{Figures/ch4/R1/SR} illustrates the percentage of stopped simulation runs per period for 150 deployed nodes. DiLCO-2 is the approach which stops first because it applies the optimization on only two subregions and the high energy consumption accelerate the network disconnection. Thus, as explained previously, in case of DiLCO-16 and DiLCO-32 which have many subregions, the optimization effectively continues as long as a subnetwork in a subregion is still connected. This longer partial coverage optimization participates in extending the network lifetime.
+\begin{figure}[t]
+\centering
+\includegraphics[scale=0.8]{Figures/ch4/R1/SR.pdf}
+\caption{Percentage of stopped simulation runs for 150 deployed nodes }
+\label{Figures/ch4/R1/SR}
+\end{figure}
+
+
+
+\item {{\bf Energy Consumption}}
+%\subsubsection{The Energy Consumption}
+
+We measure the energy consumed by the sensors during the communication, listening, computation, active, and sleep modes for different network densities and compare it for different subregions. Figures~\ref{Figures/ch4/R1/EC}(a) and~\ref{Figures/ch4/R1/EC}(b) illustrate the energy consumption for different network sizes for $Lifetime_{95}$ and $Lifetime_{50}$. The results show that DiLCO-16 and DiLCO-32 are the most competitive from the energy consumption point of view. The other approaches have a high energy consumption due to the energy consumed during the different modes of the sensor node.
+
+\begin{figure}[h!]
+\centering
+ %\begin{multicols}{1}
+\centering
+\includegraphics[scale=0.8]{Figures/ch4/R1/EC95.pdf}\\~ ~ ~ ~ ~(a) \\
+%\vfill
+\includegraphics[scale=0.8]{Figures/ch4/R1/EC50.pdf}\\~ ~ ~ ~ ~(b)
+
+%\end{multicols}
+\caption{Energy consumption for (a) $Lifetime_{95}$ and (b) $Lifetime_{50}$}
+\label{Figures/ch4/R1/EC}
+\end{figure}
+
+As shown in Figures~\ref{Figures/ch4/R1/EC}(a) and~\ref{Figures/ch4/R1/EC}(b), DiLCO-2 consumes more energy than the other versions of DiLCO, especially for large sizes of network. This is easy to understand since the bigger the number of sensors involved in the integer program, the larger the computation time to solve the optimization problem, as well as the higher energy consumed during the communication. In fact, the distribution of the computation over many subregions greatly reduces the number of communications, the time of listening and computation.
+
+\item {{\bf Execution Time}}
+%\subsubsection{Execution Time}
+
+In this experiment, the execution time of the distributed optimization approach has been studied. Figure~\ref{Figures/ch4/R1/T} gives the average execution times in seconds for the decision phase (solving of the optimization problem) during one period. They are given for the different approaches and various numbers of sensors. The original execution time is computed as described in section \ref{ch4:sec:04:04}. \\ \\ \\ \\
+
+
+
+\begin{figure}[t]
+\centering
+\includegraphics[scale=0.8]{Figures/ch4/R1/T.pdf}
+\caption{Execution Time (in seconds)}
+\label{Figures/ch4/R1/T}
+\end{figure}
+
+We can see from Figure~\ref{Figures/ch4/R1/T} that DiLCO-32 has very low execution times in comparison with other DiLCO versions because it is distributed on larger number of small subregions. Conversely, DiLCO-2 requires to solve an optimization problem considering half the nodes in each subregion and thus presents high execution times. Overall, to be able to deal with very large networks, a distributed method is clearly required.
+
+\item {{\bf Network Lifetime}}
+%\subsubsection{The Network Lifetime}
+
+In Figures~\ref{Figures/ch4/R1/LT}(a) and \ref{Figures/ch4/R1/LT}(b), network lifetime, $Lifetime_{95}$ and $Lifetime_{50}$ respectively, are illustrated for different network sizes.
+
+
+\begin{figure}[h!]
+\centering
+\centering
+\includegraphics[scale=0.8]{Figures/ch4/R1/LT95.pdf}\\~ ~ ~ ~ ~(a) \\
+
+\includegraphics[scale=0.8]{Figures/ch4/R1/LT50.pdf}\\~ ~ ~ ~ ~(b)
+
+\caption{Network lifetime for (a) $Lifetime_{95}$ and (b) $Lifetime_{50}$}
+ \label{Figures/ch4/R1/LT}
+\end{figure}
+
+For DiLCO-2 protocol, execution times quickly become unsuitable for a sensor network, and the energy consumed during the communication, seems to be huge because it is distributed over only two subregions. As highlighted by Figures~\ref{Figures/ch4/R1/LT}(a) and \ref{Figures/ch4/R1/LT}(b), the network lifetime obviously increases when the size of the network increases. The network lifetime also increases with the number of subregions, but only up to a given number. Thus we can see that DiLCO-16 leads to the larger lifetime improvement and not DiLCO-32. In fact, DilCO-32 protocol puts in active mode a larger number of sensor nodes especially near the borders of the subdivisions. It means that distributing the protocol in each node and subdividing the sensing field into many subregions, which are managed independently and simultaneously, is a relevant way to maximize the lifetime of a network.
