Update bt Ali

[ThesisAli.git] / CHAPITRE_04.tex

diff --git a/CHAPITRE_04.tex b/CHAPITRE_04.tex
index 9e478cc8ba8d30c39e06a8df123b0c1fdc079a2a..d09e8b34eb46ae66635ff6f3244e868c8d993141 100644 (file)
--- a/CHAPITRE_04.tex
+++ b/CHAPITRE_04.tex
@@ -17,13 +17,13 @@ lifetime of  WSN. The decision process is  carried out by a  leader node, which
 during the rounds  of the sensing phase. Compared  with some existing protocols, simulation  results based  on  multiple criteria  (energy consumption,  coverage
 ratio, and  so on) show that  the proposed protocol can  prolong efficiently the network lifetime and improve the coverage performance.
 
 during the rounds  of the sensing phase. Compared  with some existing protocols, simulation  results based  on  multiple criteria  (energy consumption,  coverage
 ratio, and  so on) show that  the proposed protocol can  prolong efficiently the network lifetime and improve the coverage performance.
 

-\section{MuDiLCO protocol description}
+\section{MuDiLCO Protocol Description}

 \label{ch4:sec:02}
 \noindent In this section, we introduce the MuDiLCO protocol which is distributed on each subregion in the area of interest. It is based on two energy-efficient
 mechanisms: subdividing the area of interest into several subregions (like cluster architecture) using divide and conquer method, where the sensor nodes cooperate within each subregion as independent group in order to achieve a network leader election; and sensor activity scheduling for maintaining the coverage and prolonging the network lifetime, which are applied periodically. MuDiLCO protocol uses the same assumptions and network model that presented in chapter 3, section \ref{ch3:sec:02:01} and it has been used the primary point coverage model which is described in the same chapter, section \ref{ch3:sec:02:02}.
 
  
 \label{ch4:sec:02}
 \noindent In this section, we introduce the MuDiLCO protocol which is distributed on each subregion in the area of interest. It is based on two energy-efficient
 mechanisms: subdividing the area of interest into several subregions (like cluster architecture) using divide and conquer method, where the sensor nodes cooperate within each subregion as independent group in order to achieve a network leader election; and sensor activity scheduling for maintaining the coverage and prolonging the network lifetime, which are applied periodically. MuDiLCO protocol uses the same assumptions and network model that presented in chapter 3, section \ref{ch3:sec:02:01} and it has been used the primary point coverage model which is described in the same chapter, section \ref{ch3:sec:02:02}.
 
  

-\subsection{Background Idea}
+\subsection{Background Idea and Algorithm}

 \label{ch4:sec:02:02}
 The area of  interest can be divided using  the divide-and-conquer strategy into
 smaller  areas,  called  subregions,  and  then our MuDiLCO  protocol will be
 \label{ch4:sec:02:02}
 The area of  interest can be divided using  the divide-and-conquer strategy into
 smaller  areas,  called  subregions,  and  then our MuDiLCO  protocol will be


@@ -31,9 +31,15 @@ implemented in each subregion in a distributed way.
 
 As can be seen  in Figure~\ref{fig2},  our protocol  works in  periods fashion,
 where  each is  divided  into 4  phases: Information~Exchange,  Leader~Election,
 
 As can be seen  in Figure~\ref{fig2},  our protocol  works in  periods fashion,
 where  each is  divided  into 4  phases: Information~Exchange,  Leader~Election,

-Decision, and Sensing.  Each sensing phase may be itself divided into $T$ rounds
+Decision, and Sensing. The information exchange among wireless sensor nodes is described in chapter 3, section \ref{ch3:sec:02:03:01}. The leader election in each subregion is explained in chapter 3, section \ref{ch3:sec:02:03:02}, but the difference in that the elected leader in each subregion is for each period. In decision phase, each WSNL will solve an integer  program to select which  cover sets  will be
+activated in  the following  sensing phase  to cover the  subregion to  which it belongs.  The integer  program will produce $T$ cover sets,  one for each round. The WSNL will send an Active-Sleep  packet to each sensor in the subregion based on the algorithm's results, indicating if the sensor should be active or not in
+each round  of the  sensing phase. Each sensing phase may be itself divided into $T$ rounds

 and for each round a set of sensors (a cover set) is responsible for the sensing
 and for each round a set of sensors (a cover set) is responsible for the sensing

-task. In this way a multiround optimization  process is performed  during each
+task. Each sensor node in the subregion will
+receive an Active-Sleep packet from WSNL, informing it to stay awake or to go to
+sleep for  each round of the sensing  phase.  Algorithm~\ref{alg:MuDiLCO}, which
+will be  executed by each node  at the beginning  of a period, explains  how the
+Active-Sleep packet is obtained. In this way a multiround optimization  process is performed  during each

 period  after  Information~Exchange  and  Leader~Election phases,  in  order  to
 produce $T$ cover sets that will take the mission of sensing for $T$ rounds.
 \begin{figure}[ht!]
 period  after  Information~Exchange  and  Leader~Election phases,  in  order  to
 produce $T$ cover sets that will take the mission of sensing for $T$ rounds.
 \begin{figure}[ht!]


@@ -43,36 +49,66 @@ produce $T$ cover sets that will take the mission of sensing for $T$ rounds.
 \end{figure} 
 
 
 \end{figure} 
 
 

-This protocol minimizes the impact of unexpected node failure (not due to batteries
-running out of energy), because it works in periods. 
+This protocol minimizes the impact of unexpected node failure (not due to batteries running out of energy), because it works in periods. 

 
 

- On the one hand, if a node failure is detected before making the decision, the node will not participate to this phase, and, on the other hand, if the node failure occurs after the decision, the sensing  task of the network will be temporarily affected:  only during  the period of sensing until a new period starts.
+On the one hand, if a node failure is detected before making the decision, the node will not participate to this phase, and, on the other hand, if the node failure occurs after the decision, the sensing  task of the network will be temporarily affected:  only during  the period of sensing until a new period starts.

 
 The  energy consumption  and some  other constraints  can easily  be  taken into account,  since the  sensors  can  update and  then  exchange their  information (including their residual energy) at the beginning of each period.  However, the pre-sensing  phases (Information  Exchange, Leader  Election, and  Decision) are energy  consuming for some  nodes, even  when they  do not  join the  network to monitor the area.
 
 
 The  energy consumption  and some  other constraints  can easily  be  taken into account,  since the  sensors  can  update and  then  exchange their  information (including their residual energy) at the beginning of each period.  However, the pre-sensing  phases (Information  Exchange, Leader  Election, and  Decision) are energy  consuming for some  nodes, even  when they  do not  join the  network to monitor the area.
 

+ 
+
+\begin{algorithm}[h!]                
+ % \KwIn{all the parameters related to information exchange}
+%  \KwOut{$winer-node$ (: the id of the winner sensor node, which is the leader of current round)}
+  \BlankLine
+  %\emph{Initialize the sensor node and determine it's position and subregion} \; 
+  
+  \If{ $RE_j \geq E_{R}$ }{
+      \emph{$s_j.status$ = COMMUNICATION}\;
+      \emph{Send $INFO()$ packet to other nodes in the subregion}\;
+      \emph{Wait $INFO()$ packet from other nodes in the subregion}\; 
+      %\emph{UPDATE $RE_j$ for every sent or received INFO Packet}\;
+      %\emph{ Collect information and construct the list L for all nodes in the subregion}\;
+      
+      %\If{ the received INFO Packet = No. of nodes in it's subregion -1  }{
+      \emph{LeaderID = Leader election}\;
+      \If{$ s_j.ID = LeaderID $}{
+        \emph{$s_j.status$ = COMPUTATION}\;
+        \emph{$\left\{\left(X_{1,k},\dots,X_{T,k}\right)\right\}_{k \in J}$ =
+          Execute Integer Program Algorithm($T,J$)}\;
+        \emph{$s_j.status$ = COMMUNICATION}\;
+        \emph{Send $ActiveSleep()$ to each node $k$ in subregion a packet \\
+          with vector of activity scheduling $(X_{1,k},\dots,X_{T,k})$}\;
+        \emph{Update $RE_j $}\;
+      }          
+      \Else{
+        \emph{$s_j.status$ = LISTENING}\;
+        \emph{Wait $ActiveSleep()$ packet from the Leader}\;
+        % \emph{After receiving Packet, Retrieve the schedule and the $T$ rounds}\;
+        \emph{Update $RE_j $}\;
+      }  
+      %  }
+  }
+  \Else { Exclude $s_j$ from entering in the current sensing phase}
+  
+ %   \emph{return X} \;
+\caption{MuDiLCO($s_j$)}
+\label{alg:MuDiLCO}
+
+\end{algorithm}
+
+

 
 

-These phases can be described in more details as follow:
-\subsection{Information Exchange Phase}
-\label{ch4:sec:02:02:01}
-The information exchange among the wireless sensor nodes is similar to that one which is described in chapter 3, sections \ref{ch3:sec:02:03:01}.

 
 

-\subsection{Leader Election phase}
-\label{ch4:sec:02:02:02}
-The leader election in each subregion is similar to that one which is described in chapter 3, sections\ref{ch3:sec:02:03:02}, but the difference in that the elected leader in each subregion is for each period. 
+\subsection{Primary Points based Multiround Coverage Problem Formulation}
+%\label{ch4:sec:02:02}

 
 

-\subsection{Decision phase}
-\label{ch4:sec:02:02:03}
-Each WSNL will solve  an integer  program to  select which  cover sets  will be
-activated in  the following  sensing phase  to cover the  subregion to  which it
-belongs.  The integer  program will produce $T$ cover sets,  one for each round.
-The WSNL will send an Active-Sleep  packet to each sensor in the subregion based
-on the algorithm's results, indicating if  the sensor should be active or not in
-each round  of the  sensing phase.  The  integer program  is based on  the model
+According to our algorithm~\ref{alg:MuDiLCO}, the integer program is based on the model

 proposed by  \cite{ref156} with some modifications, where  the objective is
 proposed by  \cite{ref156} with some modifications, where  the objective is

-to find  a maximum  number of disjoint  cover sets.   To fulfill this  goal, the
+to find  a maximum number of disjoint cover sets. To fulfill this  goal, the

 authors proposed an integer  program which forces undercoverage and overcoverage
 of  targets to  become minimal  at  the same  time.  They  use binary  variables
 authors proposed an integer  program which forces undercoverage and overcoverage
 of  targets to  become minimal  at  the same  time.  They  use binary  variables

-$x_{jl}$ to indicate if  sensor $j$ belongs to cover set $l$.   In our model, we
+$x_{jl}$ to indicate if  sensor $j$ belongs to cover set $l$. In our model, we

 consider binary  variables $X_{t,j}$ to determine the  possibility of activating
 sensor $j$ during round $t$ of  a given sensing phase.  We also consider primary
 points as targets.  The  set of primary points is denoted by  $P$ and the set of
 consider binary  variables $X_{t,j}$ to determine the  possibility of activating
 sensor $j$ during round $t$ of  a given sensing phase.  We also consider primary
 points as targets.  The  set of primary points is denoted by  $P$ and the set of


@@ -180,56 +216,9 @@ 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 $W_{\theta}$.
 
 In our simulations priority is given  to the coverage by choosing $W_{U}$ very
 large compared to $W_{\theta}$.
 

-\subsection{Sensing phase}
-\label{ch4:sec:02:02:04}
-The sensing phase consists of $T$ rounds. Each sensor node in the subregion will
-receive an Active-Sleep packet from WSNL, informing it to stay awake or to go to
-sleep for  each round of the sensing  phase.  Algorithm~\ref{alg:MuDiLCO}, which
-will be  executed by each node  at the beginning  of a period, explains  how the
-Active-Sleep packet is obtained.
-
-\begin{algorithm}[h!]                
- % \KwIn{all the parameters related to information exchange}
-%  \KwOut{$winer-node$ (: the id of the winner sensor node, which is the leader of current round)}
-  \BlankLine
-  %\emph{Initialize the sensor node and determine it's position and subregion} \; 
-  
-  \If{ $RE_j \geq E_{R}$ }{
-      \emph{$s_j.status$ = COMMUNICATION}\;
-      \emph{Send $INFO()$ packet to other nodes in the subregion}\;
-      \emph{Wait $INFO()$ packet from other nodes in the subregion}\; 
-      %\emph{UPDATE $RE_j$ for every sent or received INFO Packet}\;
-      %\emph{ Collect information and construct the list L for all nodes in the subregion}\;
-      
-      %\If{ the received INFO Packet = No. of nodes in it's subregion -1  }{
-      \emph{LeaderID = Leader election}\;
-      \If{$ s_j.ID = LeaderID $}{
-        \emph{$s_j.status$ = COMPUTATION}\;
-        \emph{$\left\{\left(X_{1,k},\dots,X_{T,k}\right)\right\}_{k \in J}$ =
-          Execute Integer Program Algorithm($T,J$)}\;
-        \emph{$s_j.status$ = COMMUNICATION}\;
-        \emph{Send $ActiveSleep()$ to each node $k$ in subregion a packet \\
-          with vector of activity scheduling $(X_{1,k},\dots,X_{T,k})$}\;
-        \emph{Update $RE_j $}\;
-      }          
-      \Else{
-        \emph{$s_j.status$ = LISTENING}\;
-        \emph{Wait $ActiveSleep()$ packet from the Leader}\;
-        % \emph{After receiving Packet, Retrieve the schedule and the $T$ rounds}\;
-        \emph{Update $RE_j $}\;
-      }  
-      %  }
-  }
-  \Else { Exclude $s_j$ from entering in the current sensing phase}
-  
- %   \emph{return X} \;
-\caption{MuDiLCO($s_j$)}
-\label{alg:MuDiLCO}
-
-\end{algorithm}
-
-

 
 

+ 
+ 

 
 \section{Experimental Study and Analysis}
 \label{ch4:sec:03}
 
 \section{Experimental Study and Analysis}
 \label{ch4:sec:03}


@@ -237,10 +226,8 @@ Active-Sleep packet is obtained.
 \subsection{Simulation Setup}
 \label{ch4:sec:03:01}
 We  conducted  a  series of  simulations  to  evaluate  the efficiency  and  the
 \subsection{Simulation Setup}
 \label{ch4:sec:03:01}
 We  conducted  a  series of  simulations  to  evaluate  the efficiency  and  the

-relevance  of   our  approach,  using  the  discrete   event  simulator  OMNeT++
-\cite{ref158}.     The simulation     parameters    are     summarized     in
-Table~\ref{table3}.  Each experiment  for  a network  is  run over  25~different
-random topologies and  the results presented hereafter are  the average of these
+relevance  of our  approach,  using  the  discrete   event  simulator  OMNeT++
+\cite{ref158}. The simulation  parameters are summarized in Table~\ref{table3}.  Each experiment  for  a network  is  run over  25~different random topologies and  the results presented hereafter are  the average of these

 25 runs.
 %Based on the results of our proposed work in~\cite{idrees2014coverage}, we found as the region of interest are divided into larger subregions as the network lifetime increased. In this simulation, the network are divided into 16 subregions. 
 We  performed  simulations for  five  different  densities  varying from  50  to
 25 runs.
 %Based on the results of our proposed work in~\cite{idrees2014coverage}, we found as the region of interest are divided into larger subregions as the network lifetime increased. In this simulation, the network are divided into 16 subregions. 
 We  performed  simulations for  five  different  densities  varying from  50  to


@@ -377,11 +364,16 @@ indicate the energy consumed by the whole network in round $t$.
 
 
 
 
 
 

-\subsection{Results analysis and Comparison }
+\subsection{Results Analysis and Comparison }

 \label{ch4:sec:03:02}
 
 \label{ch4:sec:03:02}
 

-\subsection{Coverage ratio} 
-\label{ch4:sec:03:02:01}
+
+\begin{enumerate}[(i)]
+
+\item {{\bf Coverage Ratio}}
+%\subsection{Coverage ratio} 
+%\label{ch4: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.  
 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.  


@@ -399,13 +391,16 @@ rounds, and thus should extend the network lifetime.
 
 \begin{figure}[ht!]
 \centering
 
 \begin{figure}[ht!]
 \centering

- \includegraphics[scale=0.5] {Figures/ch4/R1/CR.pdf}   
+ \includegraphics[scale=0.6] {Figures/ch4/R1/CR.pdf}   

 \caption{Average coverage ratio for 150 deployed nodes}
 \label{fig3}
 \end{figure} 
 
 \caption{Average coverage ratio for 150 deployed nodes}
 \label{fig3}
 \end{figure} 
 

-\subsection{Active sensors ratio} 
-\label{ch4:sec:03:02:02}
+
+\item {{\bf Active sensors ratio}}
+%\subsection{Active sensors ratio} 
+%\label{ch4:sec:03:02:02}
+

 It is crucial to have as few active nodes as possible in each round, in order to 
 minimize    the    communication    overhead    and   maximize    the    network
 lifetime. Figure~\ref{fig4}  presents the active  sensor ratio for  150 deployed
 It is crucial to have as few active nodes as possible in each round, in order to 
 minimize    the    communication    overhead    and   maximize    the    network
 lifetime. Figure~\ref{fig4}  presents the active  sensor ratio for  150 deployed


@@ -420,13 +415,15 @@ nodes in a more efficient manner.
 
 \begin{figure}[ht!]
 \centering
 
 \begin{figure}[ht!]
 \centering

-\includegraphics[scale=0.5]{Figures/ch4/R1/ASR.pdf}  
+\includegraphics[scale=0.6]{Figures/ch4/R1/ASR.pdf}  

 \caption{Active sensors ratio for 150 deployed nodes}
 \label{fig4}
 \end{figure} 
 
 \caption{Active sensors ratio for 150 deployed nodes}
 \label{fig4}
 \end{figure} 
 

-\subsection{Stopped simulation runs}
-\label{ch4:sec:03:02:03}
+\item {{\bf Stopped simulation runs}}
+%\subsection{Stopped simulation runs}
+%\label{ch4:sec:03:02:03}
+

 Figure~\ref{fig6} reports the cumulative  percentage of stopped simulations runs
 per round for 150 deployed nodes. This figure gives the  breakpoint for each method.  DESK stops first,  after approximately 45~rounds, because it consumes the
 more energy by  turning on a large number of redundant  nodes during the sensing
 Figure~\ref{fig6} reports the cumulative  percentage of stopped simulations runs
 per round for 150 deployed nodes. This figure gives the  breakpoint for each method.  DESK stops first,  after approximately 45~rounds, because it consumes the
 more energy by  turning on a large number of redundant  nodes during the sensing


@@ -439,30 +436,37 @@ still connected.
 
 \begin{figure}[ht!]
 \centering
 
 \begin{figure}[ht!]
 \centering

-\includegraphics[scale=0.5]{Figures/ch4/R1/SR.pdf} 
+\includegraphics[scale=0.6]{Figures/ch4/R1/SR.pdf} 

 \caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
 \label{fig6}
 \end{figure} 
 
 \caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
 \label{fig6}
 \end{figure} 
 

-\subsection{Energy consumption} \label{subsec:EC}
-\label{ch4:sec:03:02:04}
+
+ 
+\item {{\bf Energy consumption}} \label{subsec:EC} 
+%\subsection{Energy consumption} 
+%\label{ch4:sec:03:02:04}
+

 We  measure  the  energy  consumed  by the  sensors  during  the  communication,
 listening, computation, active, and sleep status for different network densities
 We  measure  the  energy  consumed  by the  sensors  during  the  communication,
 listening, computation, active, and sleep status for different network densities

-and   compare   it   with   the  two   other   methods.    Figures~\ref{fig7}(a)
+and   compare   it   with   the  two   other   methods.  Figures~\ref{fig7}(a)

 and~\ref{fig7}(b)  illustrate  the  energy  consumption,  considering  different
 network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.
 
 and~\ref{fig7}(b)  illustrate  the  energy  consumption,  considering  different
 network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.
 

+ 

 \begin{figure}[h!]
 \begin{figure}[h!]

-  \centering
-  \begin{tabular}{cl}
-    \parbox{9.5cm}{\includegraphics[scale=0.5]{Figures/ch4/R1/EC95.pdf}} & (a) \\
-    \verb+ + \\
-    \parbox{9.5cm}{\includegraphics[scale=0.5]{Figures/ch4/R1/EC50.pdf}} & (b)
-  \end{tabular}
-  \caption{Energy consumption for (a) $Lifetime_{95}$ and 
-    (b) $Lifetime_{50}$}
-  \label{fig7}
-\end{figure} 
+\centering
+ %\begin{multicols}{1}
+\centering
+\includegraphics[scale=0.6]{Figures/ch4/R1/EC95.pdf}\\~ ~ ~ ~ ~(a) \\
+%\vfill
+\includegraphics[scale=0.6]{Figures/ch4/R1/EC50.pdf}\\~ ~ ~ ~ ~(b)
+
+%\end{multicols} 
+\caption{Energy consumption for (a) $Lifetime_{95}$ and (b) $Lifetime_{50}$}
+\label{fig7}
+\end{figure}
+

 
 The  results  show  that  MuDiLCO  is  the  most  competitive  from  the  energy
 consumption point of view.  The  other approaches have a high energy consumption
 
 The  results  show  that  MuDiLCO  is  the  most  competitive  from  the  energy
 consumption point of view.  The  other approaches have a high energy consumption


@@ -476,8 +480,11 @@ MuDiLCO-7, we  should increase the  number of subregions  in order to  have less
 sensors to consider in the integer program.
 
 
 sensors to consider in the integer program.
 
 

-\subsection{Execution time}
-\label{ch4:sec:03:02:05}
+
+ \item {{\bf Execution time}}
+%\subsection{Execution time}
+%\label{ch4:sec:03:02:05}
+

 We observe  the impact of the  network size and of  the number of  rounds on the
 computation  time.   Figure~\ref{fig77} gives  the  average  execution times  in
 seconds (needed to solve optimization problem) for different values of $T$. The original execution time is computed as described in chapter 3, section \ref{ch3:sec:04:02}.
 We observe  the impact of the  network size and of  the number of  rounds on the
 computation  time.   Figure~\ref{fig77} gives  the  average  execution times  in
 seconds (needed to solve optimization problem) for different values of $T$. The original execution time is computed as described in chapter 3, section \ref{ch3:sec:04:02}.


@@ -486,7 +493,7 @@ seconds (needed to solve optimization problem) for different values of $T$. The
 
 \begin{figure}[ht!]
 \centering
 
 \begin{figure}[ht!]
 \centering

-\includegraphics[scale=0.5]{Figures/ch4/R1/T.pdf}  
+\includegraphics[scale=0.6]{Figures/ch4/R1/T.pdf}  

 \caption{Execution Time (in seconds)}
 \label{fig77}
 \end{figure} 
 \caption{Execution Time (in seconds)}
 \label{fig77}
 \end{figure} 


@@ -502,8 +509,12 @@ reduce the  energy-overhead due  to the three  pre-sensing phases, on  the other
 hand  a leader  node may  waste a  considerable amount  of energy  to  solve the
 optimization problem.
 
 hand  a leader  node may  waste a  considerable amount  of energy  to  solve the
 optimization problem.
 

-\subsection{Network lifetime}
-\label{ch4:sec:03:02:06}
+
+
+\item {{\bf Network lifetime}}
+%\subsection{Network lifetime}
+%\label{ch4:sec:03:02:06}
+

 The next  two figures,  Figures~\ref{fig8}(a) and \ref{fig8}(b),  illustrate the
 network lifetime  for different network sizes,  respectively for $Lifetime_{95}$
 and  $Lifetime_{50}$.  Both  figures show  that the  network  lifetime increases
 The next  two figures,  Figures~\ref{fig8}(a) and \ref{fig8}(b),  illustrate the
 network lifetime  for different network sizes,  respectively for $Lifetime_{95}$
 and  $Lifetime_{50}$.  Both  figures show  that the  network  lifetime increases


@@ -515,21 +526,27 @@ lifetime for a  coverage over 95\% is greater than 38\%  when switching from GAF
 to MuDiLCO-3.  The  slight decrease that can be observed  for MuDiLCO-7 in case
 of  $Lifetime_{95}$  with  large  wireless  sensor  networks  results  from  the
 difficulty  of the optimization  problem to  be solved  by the  integer program.
 to MuDiLCO-3.  The  slight decrease that can be observed  for MuDiLCO-7 in case
 of  $Lifetime_{95}$  with  large  wireless  sensor  networks  results  from  the
 difficulty  of the optimization  problem to  be solved  by the  integer program.

-This  point was  already noticed  in subsection  \ref{subsec:EC} devoted  to the
+This  point was  already noticed  in \ref{subsec:EC} devoted  to the

 energy consumption,  since network lifetime and energy  consumption are directly
 linked.
 
 energy consumption,  since network lifetime and energy  consumption are directly
 linked.
 

-\begin{figure}[t!]
-  \centering
-  \begin{tabular}{cl}
-    \parbox{9.5cm}{\includegraphics[scale=0.5]{Figures/ch4/R1/LT95.pdf}} & (a) \\
-    \verb+ + \\
-    \parbox{9.5cm}{\includegraphics[scale=0.5]{Figures/ch4/R1/LT50.pdf}} & (b)
-  \end{tabular}
-  \caption{Network lifetime for (a) $Lifetime_{95}$ and 
-    (b) $Lifetime_{50}$}
+
+\begin{figure}[h!]
+\centering
+% \begin{multicols}{0}
+\centering
+\includegraphics[scale=0.6]{Figures/ch4/R1/LT95.pdf}\\~ ~ ~ ~ ~(a) \\
+%\hfill 
+\includegraphics[scale=0.6]{Figures/ch4/R1/LT50.pdf}\\~ ~ ~ ~ ~(b)
+
+%\end{multicols} 
+\caption{Network lifetime for (a) $Lifetime_{95}$ and (b) $Lifetime_{50}$}

   \label{fig8}
   \label{fig8}

-\end{figure} 
+\end{figure}
+
+
+
+\end{enumerate}

 
 
 \section{Conclusion}
 
 
 \section{Conclusion}


@@ -559,3 +576,8 @@ subregion, that can be solved  more easily. Nevertheless, results also show that
 it is not possible to plan the activity of sensors over too many rounds, because
 the resulting optimization problem leads to too high resolution times and thus to
 an excessive energy consumption.
 it is not possible to plan the activity of sensors over too many rounds, because
 the resulting optimization problem leads to too high resolution times and thus to
 an excessive energy consumption.

+
+
+
+
+ 
\ No newline at end of file