Update by Ali

[JournalMultiPeriods.git] / article.tex

diff --git a/article.tex b/article.tex
index 6707012e2633a7c36d5b1826ef5c7e23690f2f00..00a0b3581dba4f8dab07e06056ad813cbc4d8a89 100644 (file)
--- a/article.tex
+++ b/article.tex
@@ -73,15 +73,25 @@
 %% \author[label1,label2]{}
 %% \address[label1]{}
 %% \address[label2]{}
 %% \author[label1,label2]{}
 %% \address[label1]{}
 %% \address[label2]{}

-\author{Ali Kadhum Idrees, Karine Deschinkel, \\
-Michel Salomon, and Rapha\"el Couturier}
+%\author{Ali Kadhum Idrees, Karine Deschinkel, \\
+%Michel Salomon, and Rapha\"el Couturier}
+

 %\thanks{are members in the AND team - DISC department - FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France.
 % e-mail: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}% <-this % stops a space
 %\thanks{}% <-this % stops a space
  
 %\thanks{are members in the AND team - DISC department - FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France.
 % e-mail: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}% <-this % stops a space
 %\thanks{}% <-this % stops a space
  

-\address{FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France. \\ 
-e-mail: ali.idness@edu.univ-fcomte.fr, \\
-$\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}
+%\address{FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France. \\ 
+%e-mail: ali.idness@edu.univ-fcomte.fr, \\
+%$\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}
+
+
+\author{Ali Kadhum Idrees$^{a,b}$, Karine Deschinkel$^{a}$, \\
+Michel Salomon$^{a}$ and Rapha\"el Couturier $^{a}$ \\
+  $^{a}${\em{FEMTO-ST Institute, UMR 6174 CNRS, \\
+  University Bourgogne Franche-Comt\'e, Belfort, France}} \\ 
+  $^{b}${\em{Department of Computer Science, University of Babylon, Babylon, Iraq}}
+}  
+

 
 \begin{abstract}
 %One of  the fundamental challenges in Wireless Sensor Networks (WSNs)
 
 \begin{abstract}
 %One of  the fundamental challenges in Wireless Sensor Networks (WSNs)


@@ -98,7 +108,7 @@ during which sets of sensor nodes are scheduled to remain active for a number of
 rounds  during the  sensing phase,  to  ensure coverage  so as  to maximize  the
 lifetime of  WSN.  The decision process is  carried out by a  leader node, which
 solves an  integer program to  produce the best  representative sets to  be used
 rounds  during the  sensing phase,  to  ensure coverage  so as  to maximize  the
 lifetime of  WSN.  The decision process is  carried out by a  leader node, which
 solves an  integer program to  produce the best  representative sets to  be used

-during the rounds  of the sensing phase. Compared  with some existing protocols,
+during the rounds  of the sensing phase. \textcolor{red}{The integer program is solved by either GLPK solver or Genetic Algorithm (GA)}. 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.
 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.


@@ -106,7 +116,7 @@ network lifetime and improve the coverage performance.
 \end{abstract}
 
 \begin{keyword}
 \end{abstract}
 
 \begin{keyword}

-Wireless   Sensor   Networks,   Area   Coverage,   Network   lifetime,
+Wireless   Sensor   Networks,   Area   Coverage,   Network   Lifetime,

 Optimization, Scheduling, Distributed Computation.
 
 \end{keyword}
 Optimization, Scheduling, Distributed Computation.
 
 \end{keyword}


@@ -286,7 +296,7 @@ computation complexity. Compared to our previous paper, in this one we study the
 possibility of dividing  the sensing phase into multiple rounds  and we also add
 an  improved  model  of energy  consumption  to  assess  the efficiency  of  our
 approach. In fact, in this paper we make a multiround optimization, while it was
 possibility of dividing  the sensing phase into multiple rounds  and we also add
 an  improved  model  of energy  consumption  to  assess  the efficiency  of  our
 approach. In fact, in this paper we make a multiround optimization, while it was

-a single round optimization in our previous work.
+a single round optimization in our previous work. \textcolor{red}{In addition, a metaheuristic based GA is proposed to solve our multiround optimization}.

 
 \iffalse
    
 
 \iffalse
    


@@ -521,8 +531,7 @@ active nodes.
 Instead  of working  with a  continuous coverage  area, we  make it  discrete by
 considering for each sensor a set of points called primary points. Consequently,
 we assume  that the sensing disk  defined by a sensor  is covered if  all of its
 Instead  of working  with a  continuous coverage  area, we  make it  discrete by
 considering for each sensor a set of points called primary points. Consequently,
 we assume  that the sensing disk  defined by a sensor  is covered if  all of its

-primary points are covered. The choice of number and locations of primary points
-is the subject of another study not presented here.
+primary points are covered. The choice of number and locations of primary points is the subject of another study not presented here.

 
 %By  knowing the  position (point  center: ($p_x,p_y$))  of  a wireless
 %sensor node  and its $R_s$,  we calculate the primary  points directly
 
 %By  knowing the  position (point  center: ($p_x,p_y$))  of  a wireless
 %sensor node  and its $R_s$,  we calculate the primary  points directly


@@ -638,12 +647,71 @@ consumption due to the communications.
 
 \subsection{Decision phase}
 
 
 \subsection{Decision phase}
 

-Each  WSNL will solve  an integer  program to  select which  cover sets  will be
+Each  WSNL will \textcolor{red}{ execute an optimization algorithm (see section \ref{oa})} to  select which  cover sets  will be

 activated in  the following  sensing phase  to cover the  subregion to  which it
 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
+belongs.  The \textcolor{red}{optimization algorithm} 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.  
+
+%solve  an integer  program
+
+\subsection{Sensing phase}
+
+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.
+
+% In each round during the sensing phase, there is a cover set of sensor nodes,  in which  the active  sensors will  execute  their sensing  task  to preserve maximal  coverage and lifetime in the subregion and this will continue until finishing the round $T$ and starting new period. 
+
+\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 \textcolor{red}{Optimization 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{\textcolor{red}{ Optimization Algorithm for Multiround Lifetime Coverage Optimization}}
+\label{oa}
+As shown in Algorithm~\ref{alg:MuDiLCO}, the leader will execute an optimization algorithm based on an integer program. The  integer program  is based on  the model

 proposed by  \cite{pedraza2006} with some modifications, where  the objective is
 to find  a maximum  number of disjoint  cover sets.   To fulfill this  goal, the
 authors proposed an integer  program which forces undercoverage and overcoverage
 proposed by  \cite{pedraza2006} with some modifications, where  the objective is
 to find  a maximum  number of disjoint  cover sets.   To fulfill this  goal, the
 authors proposed an integer  program which forces undercoverage and overcoverage


@@ -762,56 +830,173 @@ In our simulations priority is given  to the coverage by choosing $W_{U}$ very
 large compared to $W_{\theta}$.
 %The Active-Sleep packet includes the schedule vector with the number of rounds that should be applied by the receiving sensor node during the sensing phase.
 
 large compared to $W_{\theta}$.
 %The Active-Sleep packet includes the schedule vector with the number of rounds that should be applied by the receiving sensor node during the sensing phase.
 

-\subsection{Sensing phase}
+\textcolor{red}{This integer program can be solved using two approaches:}

 
 

-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.
+\subsection{\textcolor{red}{Optimization solver for Multiround Lifetime Coverage Optimization}}
+\label{glpk}
+\textcolor{red}{The modeling language for Mathematical Programming (AMPL)~\cite{AMPL} is  employed to generate the integer program instance  in a  standard format, which  is then read  and solved  by the optimization solver  GLPK (GNU  linear Programming Kit  available in  the public domain) \cite{glpk} through a Branch-and-Bound method. We named the protocol which is based on GLPK solver in the decision phase as MuDiLCO.}

 
 

-% In each round during the sensing phase, there is a cover set of sensor nodes,  in which  the active  sensors will  execute  their sensing  task  to preserve maximal  coverage and lifetime in the subregion and this will continue until finishing the round $T$ and starting new period. 

 
 

-\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)}
+
+
+\subsection{\textcolor{red}{Genetic Algorithm for Multiround Lifetime Coverage Optimization}}
+\label{GA}
+\textcolor{red}{Metaheuristics  are a generic search strategies for exploring search spaces for solving the complex problems. These strategies have to dynamically balance between the exploitation of the accumulated search experience and the exploration of the search space. On one hand, this balance can find regions in the search space with high-quality solutions. On the other hand, it prevents waste too much time in regions of the search space which are either already explored or don’t provide high-quality solutions. Therefore,  metaheuristic provides an enough good solution to an optimization problem, especially with incomplete  information or limited computation capacity \cite{bianchi2009survey}. Genetic Algorithm (GA) is one of the population-based metaheuristic methods that simulates the process of natural selection \cite{hassanien2015applications}.  GA starts with a population of random candidate solutions (called individuals or phenotypes) . GA uses genetic operators inspired by natural evolution, such as selection, mutation, evaluation, crossover, and replacement so as to improve the initial population of candidate solutions. This process repeated until a stopping criterion is satisfied. In comparison with GLPK optimization solver, GA provides a near optimal solution with acceptable execution time, as well as it requires a less amount of memory especially for large size problems. GLPK provides optimal solution, but it requires higher execution time and amount of memory for large problem.}
+
+\textcolor{red}{In this section, we present a metaheuristic based GA to solve our multiround lifetime coverage optimization problem. The proposed GA provides a near optimal sechedule for multiround sensing per period. The proposed GA is based on the mathematical model which is presented in Section \ref{oa}. Algorithm \ref{alg:GA} shows the proposed GA to solve the coverage lifetime optimization problem. We named the new protocol which is based on GA in the decision phase as GA-MuDiLCO. The proposed GA can be explained in more details as follow:}
+
+\begin{algorithm}[h!]    
+       
+ \small
+ \SetKwInput{Input}{\textcolor{red}{Input}}
+ \SetKwInput{Output}{\textcolor{red}{Output}}
+ \Input{ \textcolor{red}{$ P, J, T, S_{pop}, \alpha_{j,p}^{ind}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind}, Child_{t,j}^{ind}, Ch.\Theta_{t,p}^{ind}, Ch.U_{t,p}^{ind_1}$}}
+ \Output{\textcolor{red}{$\left\{\left(X_{1,1},\dots, X_{t,j}, \dots, X_{T,J}\right)\right\}_{t \in T, j \in J}$}}
+

   \BlankLine
   %\emph{Initialize the sensor node and determine it's position and subregion} \; 
   \BlankLine
   %\emph{Initialize the sensor node and determine it's position and subregion} \; 

+  \ForEach {\textcolor{red}{Individual $ind$ $\in$ $S_{pop}$}} {
+     \emph{\textcolor{red}{Generate Randomly Chromosome $\left\{\left(X_{1,1},\dots, X_{t,j}, \dots, X_{T,J}\right)\right\}_{t \in T, j \in J}$}}\;
+     
+     \emph{\textcolor{red}{Update O-U-Coverage $\left\{(P, J, \alpha_{j,p}^{ind}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind})\right\}_{p \in P}$}}\;
+     
+  
+     \emph{\textcolor{red}{Evaluate Individual $(P, J, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind})$}}\;  
+  }
+  
+  \While{\textcolor{red}{ Stopping criteria is not satisfied} }{
+  
+  \emph{\textcolor{red}{Selection $(ind_1, ind_2)$}}\;
+    \emph{\textcolor{red}{Crossover $(P_c, X_{t,j}^{ind_1}, X_{t,j}^{ind_2}, Child_{t,j}^{ind_1}, Child_{t,j}^{ind_2})$}}\;
+    \emph{\textcolor{red}{Mutation $(P_m, Child_{t,j}^{ind_1}, Child_{t,j}^{ind_2})$}}\;
+   
+   
+   \emph{\textcolor{red}{Update O-U-Coverage $(P, J, \alpha_{j,p}^{ind}, Child_{t,j}^{ind_1}, Ch.\Theta_{t,p}^{ind_1}, Ch.U_{t,p}^{ind_1})$}}\;
+  \emph{\textcolor{red}{Update O-U-Coverage $(P, J, \alpha_{j,p}^{ind}, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2})$}}\;  
+ 
+\emph{\textcolor{red}{Evaluate New Individual$(P, J, Child_{t,j}^{ind_1}, Ch.\Theta_{t,p}^{ind_1}, Ch.U_{t,p}^{ind_1})$}}\;  
+ \emph{\textcolor{red}{Replacement $(P, J, T, Child_{t,j}^{ind_1}, Ch.\Theta_{t,p}^{ind_1}, Ch.U_{t,p}^{ind_1}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind}  )$ }}\;
+ 
+ \emph{\textcolor{red}{Evaluate New Individual$(P, J, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2})$}}\;  
+  
+ \emph{\textcolor{red}{Replacement $(P, J, T, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind}  )$ }}\;

   
   

-  \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{\textcolor{red}{$\left\{\left(X_{1,1},\dots,X_{t,j},\dots,X_{T,J}\right)\right\}$ =
+            Select Best Solution ($S_{pop}$)}}\;
+ \emph{\textcolor{red}{return X}} \;
+\caption{\textcolor{red}{GA($T, J$)}}
+\label{alg:GA}
+
+\end{algorithm}
+
+
+\begin{enumerate} [I)]
+
+\item \textcolor{red}{\textbf{Representation:} Since the proposed GA's goal is to find the optimal schedule of the sensor nodes which take the responsibility of monitoring the subregion for $T$ rounds in the sensing phase, the chromosome is defined as a schedule for alive  sensors and each chromosome contains $T$ rounds. The proposed GA uses binary representation, where each round in the schedule includes J genes, the total alive sensors in the subregion. Therefore, the gene of such a chromosome is a schedule of a sensor. In other words, The genes corresponding to active nodes have the value of one, the others are zero. Figure \ref{chromo} shows solution representation in the proposed GA.}
+%[scale=0.3]
+\begin{figure}[h!]
+\centering
+ \includegraphics [scale=0.35] {rep.pdf} 
+\caption{Candidate Solution representation by the proposed GA. }
+\label{chromo}
+\end{figure} 
+
+
+
+\item \textcolor{red}{\textbf{Initialize Population:} The initial population is randomly generated and each chromosome  in the GA population represents a possible sensors schedule solution to cover the entire subregion for $T$ rounds during current period. Each sensor in the chromosome is given a random value (0 or  1) for all rounds. If the random value is 1, the remaining  energy of this sensor should be adequate to activate this sensor during the current round. Otherwise, the value is set to 0. The energy constraint is applied for each sensor during all rounds. }
+
+
+\item \textcolor{red}{\textbf{Update O-U-Coverage:} 
+After creating the initial population, The overcoverage $\Theta_{t,p}$ and undercoverage $U_{t,p}$ for each candidate solution are computed (see Algorithm \ref{OU}) so as to use them in the next step.}
+
+\begin{algorithm}[h!]                

   
   

- %   \emph{return X} \;
-\caption{MuDiLCO($s_j$)}
-\label{alg:MuDiLCO}
+ \SetKwInput{Input}{\textcolor{red}{Input}}
+ \SetKwInput{Output}{\textcolor{red}{Output}}
+ \Input{ \textcolor{red}{parameters $P, J, ind, \alpha_{j,p}^{ind}, X_{t,j}^{ind}$}}
+ \Output{\textcolor{red}{$U^{ind} = \left\lbrace U_{1,1}^{ind}, \dots, U_{t,p}^{ind}, \dots, U_{T,P}^{ind} \right\rbrace$ and $\Theta^{ind} = \left\lbrace \Theta_{1,1}^{ind}, \dots, \Theta_{t,p}^{ind}, \dots, \Theta_{T,P}^{ind} \right\rbrace$}}
+
+  \BlankLine
+
+  \For{\textcolor{red}{$t\leftarrow 1$ \KwTo $T$}}{
+  \For{\textcolor{red}{$p\leftarrow 1$ \KwTo $P$}}{
+     
+ %    \For{$i\leftarrow 0$ \KwTo $I_j$}{
+       \emph{\textcolor{red}{$SUM\leftarrow 0$}}\;
+         \For{\textcolor{red}{$j\leftarrow 1$ \KwTo $J$}}{
+              \emph{\textcolor{red}{$SUM \leftarrow SUM + (\alpha_{j,p}^{ind} \times X_{t,j}^{ind})$ }}\;
+         }
+         
+         \If { \textcolor{red}{SUM = 0}} {
+         \emph{\textcolor{red}{$U_{t,p}^{ind} \leftarrow 0$}}\;
+         \emph{\textcolor{red}{$\Theta_{t,p}^{ind} \leftarrow 1$}}\;
+         }
+         \Else{
+         \emph{\textcolor{red}{$U_{t,p}^{ind} \leftarrow SUM -1$}}\;
+         \emph{\textcolor{red}{$\Theta_{t,p}^{ind} \leftarrow 0$}}\;
+         }
+     
+     }
+     
+  }
+\emph{\textcolor{red}{return $U^{ind}, \Theta^{ind}$ }} \;
+\caption{O-U-Coverage}
+\label{OU}

 
 \end{algorithm}
 
 
 \end{algorithm}
 

+
+
+\item \textcolor{red}{\textbf{Evaluate Population:}
+After creating the initial population, each individual is evaluated and assigned a fitness value according to the fitness function is illustrated in Eq. \eqref{eqf}. In the proposed GA, the optimal (or near optimal) candidate solution, is the one with the minimum value for the fitness function. The lower the fitness values been assigned to an individual, the better opportunity it gets survived.  In our works, the function rewards  the decrease in the sensor nodes which cover the same primary point and penalizes the decrease to zero in the sensor nodes which cover the primary point. }
+
+\begin{equation}
+ F^{ind} \leftarrow  \sum_{t=1}^{T} \sum_{p=1}^{P} \left(W_{\theta}* \Theta_{t,p} + W_{U} * U_{t,p}  \right)    \label{eqf} 
+\end{equation}
+
+
+\item \textcolor{red}{\textbf{Selection:} In order to generate a new generation, a portion of the existing population is elected based on a fitness function that ranks the fitness of each candidate solution and preferentially select the best solutions. Two parents should be selected to the mating pool.  In the proposed GA-MuDiLCO algorithm, the first parent is selected by using binary tournament selection to select one of the parents \cite{goldberg1991comparative}. In this method,  two individuals are chosen at random from the population and the better of the two
+individuals is selected. If they have similar fitness values, one of them will be selected randomly. The best individual in the population is selected as a second parent.}
+
+
+
+\item \textcolor{red}{\textbf{Crossover:} Crossover is a genetic operator used to take more than one parent solutions and produce a child solution from them. If crossover probability $P_c$ is 100$\%$, then the crossover operation takes place between two individuals. If it is 0$\%$, the  two selected individuals in the mating pool will be the new chromosomes without crossover. In the proposed GA, a two-point crossover is used. Figure \ref{cross} gives an example for a two-point crossover for 8 sensors in the subregion and the schedule for 3 rounds.}
+
+
+\begin{figure}[h!]
+\centering
+ \includegraphics [scale = 0.3] {crossover.pdf} 
+\caption{Two-point crossover. }
+\label{cross}
+\end{figure} 
+
+
+\item \textcolor{red}{\textbf{Mutation:}
+Mutation is a divergence operation which introduces random modifications.  The purpose of the mutation is to maintain diversity within the population and prevent premature convergence. Mutation is used to add new genetic information (divergence) in order to achieve a global search over the solution search space and avoid to fall in local optima. The mutation operator in the proposed GA-MuDiLCO works as follow: If mutation probability $P_m$ is 100$\%$, then the mutation operation takes place on the new individual. The round number is selected randomly within (1..T) in the schedule solution. After that one sensor within this round is selected randomly within (1..J). If the sensor is scheduled as active "1", it should be rescheduled to sleep "0". If the sensor is scheduled as sleep, it rescheduled to active only if it has adequate remaining energy.}
+
+
+\item \textcolor{red}{\textbf{Update O-U-Coverage for children:}
+Before evaluating each new individual, Algorithm \ref{OU} is called for each new individual to compute the new undercoverage $Ch.U$ and overcoverage $Ch.\Theta$ parameters. }
+ 
+\item \textcolor{red}{\textbf{Evaluate New Individuals:}
+Each new individual is evaluated using Eq. \ref{eqf} but with using the new undercoverage $Ch.U$ and overcoverage $Ch.\Theta$ parameters of the new children.}
+
+\item \textcolor{red}{\textbf{Replacement:}
+After evaluation of new children, Triple Tournament Replacement (TTR) will be applied for each new individual. In TTR strategy, three individuals are selected
+randomly from the population. Find the worst from them and then check its fitness with the new individual fitness. If the fitness of the new individual is better than the fitness of  the worst individual, replace the new individual with the worst individual. Otherwise, the replacement is not done. }
+
+ 
+\item \textcolor{red}{\textbf{Stopping criteria:}
+The proposed GA-MuDiLCO stops when the stopping criteria is met. It stops after running for an amount of time in seconds equal to \textbf{Time limit}. The \textbf{Time limit} is the execution time obtained by the optimization solver GLPK for solving the same size of problem. The best solution will be selected as a schedule of sensors for $T$ rounds during the sensing phase in the current period.}
+
+
+
+\end{enumerate} 
+
+
+

 \section{Experimental study}
 \label{exp}
 \subsection{Simulation setup}
 \section{Experimental study}
 \label{exp}
 \subsection{Simulation setup}


@@ -859,19 +1044,23 @@ Sensing time for one round & 60 Minutes \\
 $E_{R}$ & 36 Joules\\
 $R_s$ & 5~m   \\     
 %\hline
 $E_{R}$ & 36 Joules\\
 $R_s$ & 5~m   \\     
 %\hline

-$W_{\Theta}$ & 1   \\
+$W_{\theta}$ & 1   \\

 % [1ex] adds vertical space
 %\hline
 % [1ex] adds vertical space
 %\hline

-$W_{U}$ & $|P|^2$
+$W_{U}$ & $|P|^2$ \\
+$P_c$ & 0.95   \\ 
+$P_m$ & 0.6 \\
+$S_{pop}$ & 50

 %inserts single line
 \end{tabular}
 \label{table3}
 % is used to refer this table in the text
 \end{table}
   
 %inserts single line
 \end{tabular}
 \label{table3}
 % is used to refer this table in the text
 \end{table}
   

-Our protocol  is declined into  four versions: MuDiLCO-1,  MuDiLCO-3, MuDiLCO-5,
+\textcolor{red}{Our first protocol based GLPK optimization solver 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
 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
+rounds in one sensing period). The second protocol based GA is declined into  four versions: GA-MuDiLCO-1,  GA-MuDiLCO-3, GA-MuDiLCO-5,
+and  GA-MuDiLCO-7 for the same reason of the first protocol. After extensive experiments, we chose the dedicated values for the parameters $P_c$, $P_m$, and $S_{pop}$ because they gave the best results}.  In  the following, we will make comparisons with

 two other methods. The first method, called DESK and proposed by \cite{ChinhVu},
 is  a   full  distributed  coverage   algorithm.   The  second   method,  called
 GAF~\cite{xu2001geography}, consists in dividing  the region into fixed squares.
 two other methods. The first method, called DESK and proposed by \cite{ChinhVu},
 is  a   full  distributed  coverage   algorithm.   The  second   method,  called
 GAF~\cite{xu2001geography}, consists in dividing  the region into fixed squares.


@@ -948,7 +1137,7 @@ and 24~bits  respectively.  The  value of energy  spent to send  a 1-bit-content
 message is  obtained by using  the equation in  ~\cite{raghunathan2002energy} to
 calculate  the energy cost  for transmitting  messages and  we propose  the same
 value for receiving the packets. The energy  needed to send or receive a 1-bit
 message is  obtained by using  the equation in  ~\cite{raghunathan2002energy} to
 calculate  the energy cost  for transmitting  messages and  we propose  the same
 value for receiving the packets. The energy  needed to send or receive a 1-bit

-packet is equal to $0.2575~mW$.
+packet is equal to 0.2575~mW.

 
 The initial energy of each node  is randomly set in the interval $[500;700]$.  A
 sensor node  will not participate in the  next round if its  remaining energy is
 
 The initial energy of each node  is randomly set in the interval $[500;700]$.  A
 sensor node  will not participate in the  next round if its  remaining energy is


@@ -1011,7 +1200,7 @@ network, and $R$ is the total number of subregions in the network.
   % New version with global loops on period
   \begin{equation*}
     \scriptsize
   % New version with global loops on period
   \begin{equation*}
     \scriptsize

-    \mbox{EC} = \frac{\sum\limits_{m=1}^{M_L} \left[ \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m \right) +\sum\limits_{t=1}^{T_m} \left( E^{a}_t+E^{s}_t \right) \right]}{\sum\limits_{m=1}^{M_L} T_m},
+    \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_m} \left( E^{a}_t+E^{s}_t \right) \right]}{\sum\limits_{m=1}^{M} T_m},

   \end{equation*}
 
 
   \end{equation*}
 
 


@@ -1030,7 +1219,7 @@ network, and $R$ is the total number of subregions in the network.
 % Old version -> where $M_L$ and  $T_L$ are respectively the number of  periods and rounds during
 %$Lifetime_{95}$ or  $Lifetime_{50}$. 
 % New version
 % Old version -> where $M_L$ and  $T_L$ are respectively the number of  periods and rounds during
 %$Lifetime_{95}$ or  $Lifetime_{50}$. 
 % New version

-where  $M_L$ is  the number  of periods  and  $T_m$ the  number of  rounds in  a
+where  $M$ is  the number  of periods  and  $T_m$ 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  $E^{\scriptsize  \mbox{com}}_m$,
 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  $E^{\scriptsize  \mbox{com}}_m$,


@@ -1040,7 +1229,7 @@ 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 $E^s_t$
 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 consummed by the whole network in round $t$.
+indicate the energy consumed by the whole network in round $t$.

 
 %\item {Network Lifetime:} we  have defined the network  lifetime as the  time until all
 %nodes  have  been drained  of  their  energy  or each  sensor  network monitoring  an area has become  disconnected.
 
 %\item {Network Lifetime:} we  have defined the network  lifetime as the  time until all
 %nodes  have  been drained  of  their  energy  or each  sensor  network monitoring  an area has become  disconnected.


@@ -1057,9 +1246,9 @@ indicate the energy consummed by the whole network in round $t$.
 
 \end{enumerate}
 
 
 \end{enumerate}
 

-\section{Results and analysis}
+\subsection{Results and analysis}

 
 

-\subsection{Coverage ratio} 
+\subsubsection{Coverage ratio} 

 
 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
 
 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


@@ -1080,55 +1269,58 @@ rounds, and thus should extend the network lifetime.
 
 \begin{figure}[ht!]
 \centering
 
 \begin{figure}[ht!]
 \centering

- \includegraphics[scale=0.5] {R1/CR.pdf} 
+ \includegraphics[scale=0.5] {R/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} 
+\textcolor{red}{ We
+can see that for the first thirty nine rounds GA-MuDiLCO provides a little bit better coverage ratio  than MuDiLCO. Both DESK and GAF provide a coverage
+which is a little bit better than the one of MuDiLCO and GA-MuDiLCO for the first thirty rounds because they activate a larger number of nodes during sensing phase. After that GA-MuDiLCO provides a coverage ratio near to the  MuDiLCO and better than DESK and GAF. GA-MuDiLCO gives approximate solution with activation a larger number of nodes than MuDiLCO during sensing phase while it activates a less number of nodes in comparison with both DESK and GAF. MuDiLCO and GA-MuDiLCO clearly outperform DESK and GAF for
+a number of periods between 31 and 103. This is because they optimize the coverage and the lifetime in a wireless sensor network by selecting the best representative sensor nodes to take the responsibility of coverage during the sensing phase.}
+
+
+
+\subsubsection{Active sensors ratio} 

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

 nodes all along the network lifetime. It appears that up to round thirteen, DESK
 and GAF have  respectively 37.6\% and 44.8\% of nodes  in ACTIVE status, whereas
 nodes all along the network lifetime. It appears that up to round thirteen, DESK
 and GAF have  respectively 37.6\% and 44.8\% of nodes  in ACTIVE status, whereas

-MuDiLCO clearly  outperforms them  with only 24.8\%  of active nodes.  After the
-thirty-fifth round, MuDiLCO exhibits larger numbers of active nodes, which agrees
-with  the  dual  observation  of  higher  level  of  coverage  made  previously.
-Obviously, in  that case DESK  and GAF have  less active nodes, since  they have
-activated many nodes  at the beginning. Anyway, MuDiLCO  activates the available
-nodes in a more efficient manner.
+MuDiLCO clearly outperforms them  with only 24.8\%  of active nodes. \textcolor{red}{GA-MuDiLCO activates a number of sensor nodes larger than MuDiLCO but lower than both DESK and GAF. GA-MuDiLCO-1, GA-MuDiLCO-3, and GA-MuDiLCO-5 continue in providing a larger number of active sensors until the forty-sixth round after that it provides less number of active nodes due to the died nodes. GA-MuDiLCO-7 provides a larger number of sensor nodes and maintains a better coverage ratio compared to MuDiLCO-7 until the fifty-seventh round.  After the thirty-fifth round, MuDiLCO exhibits larger numbers of active nodes compared with DESK  and GAF, which agrees with  the  dual  observation  of  higher  level  of  coverage  made  previously}.
+Obviously, in that case DESK  and GAF have less active nodes, since  they have activated many nodes  at the beginning. Anyway, MuDiLCO  activates the available nodes in a more efficient manner. \textcolor{red}{GA-MuDiLCO activates near optimal number of sensor nodes also in efficient manner compared with both DESK  and GAF}.

 
 \begin{figure}[ht!]
 \centering
 
 \begin{figure}[ht!]
 \centering

-\includegraphics[scale=0.5]{R1/ASR.pdf}  
+\includegraphics[scale=0.5]{R/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}
+%\textcolor{red}{GA-MuDiLCO activates a sensor nodes larger than MuDiLCO but lower than both DESK and GAF }
+
+
+\subsubsection{Stopped simulation runs}

 %The results presented in this experiment, is to show the comparison of our MuDiLCO protocol with other two approaches from the point of view the stopped simulation runs per round. Figure~\ref{fig6} illustrates the percentage of stopped simulation
 %runs per round for 150 deployed nodes. 
 
 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
 %The results presented in this experiment, is to show the comparison of our MuDiLCO protocol with other two approaches from the point of view the stopped simulation runs per round. Figure~\ref{fig6} illustrates the percentage of stopped simulation
 %runs per round for 150 deployed nodes. 
 
 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

-phase. GAF  stops secondly for the  same reason than  DESK.  MuDiLCO overcomes
-DESK and GAF because the  optimization process distributed on several subregions
-leads  to coverage  preservation and  so extends  the network  lifetime.  Let us
-emphasize that the  simulation continues as long as a network  in a subregion is
-still connected.
+phase. GAF  stops secondly for the  same reason than  DESK. \textcolor{red}{GA-MuDiLCO  stops thirdly for the  same reason than  DESK and GAF.} \textcolor{red}{MuDiLCO and GA-MuDiLCO overcome}
+DESK and GAF because \textcolor{red}{they activate less number of sensor nodes, as well as }the optimization process distributed on several subregions leads to coverage  preservation and  so extends  the network  lifetime.  
+Let us emphasize that the  simulation continues as long as a network  in a subregion is still connected. 

 
 %%% The optimization effectively continues as long as a network in a subregion is still connected. A VOIR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \begin{figure}[ht!]
 \centering
 
 %%% The optimization effectively continues as long as a network in a subregion is still connected. A VOIR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \begin{figure}[ht!]
 \centering

-\includegraphics[scale=0.5]{R1/SR.pdf} 
+\includegraphics[scale=0.5]{R/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}
+\subsubsection{Energy consumption} \label{subsec:EC}

 
 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


@@ -1139,9 +1331,9 @@ network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.
 \begin{figure}[h!]
   \centering
   \begin{tabular}{cl}
 \begin{figure}[h!]
   \centering
   \begin{tabular}{cl}

-    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/EC95.pdf}} & (a) \\
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{R/EC95.pdf}} & (a) \\

     \verb+ + \\
     \verb+ + \\

-    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/EC50.pdf}} & (b)
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{R/EC50.pdf}} & (b)

   \end{tabular}
   \caption{Energy consumption for (a) $Lifetime_{95}$ and 
     (b) $Lifetime_{50}$}
   \end{tabular}
   \caption{Energy consumption for (a) $Lifetime_{95}$ and 
     (b) $Lifetime_{50}$}


@@ -1150,23 +1342,19 @@ network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.
 
 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

-due  to activating a  larger number  of redundant  nodes as  well as  the energy
-consumed during  the different  status of the  sensor node. Among  the different
-versions of our protocol, the MuDiLCO-7  one consumes more energy than the other
-versions. This is  easy to understand since the bigger the  number of rounds and
-the number of  sensors involved in the integer program are,  the larger the time
-computation to solve the optimization problem is. To improve the performances of
-MuDiLCO-7, we  should increase the  number of subregions  in order to  have less
-sensors to consider in the integer program.
-
+due  to activating a  larger number  of redundant  nodes as  well as  the energy consumed during  the different  status of the  sensor node. Among  the different versions of our protocol, the MuDiLCO-7  one consumes more energy than the other
+versions. This is  easy to understand since the bigger the  number of rounds and the number of  sensors involved in the integer program are,  the larger the time computation to solve the optimization problem is. To improve the performances of MuDiLCO-7, we  should increase the  number of subregions  in order to  have less sensors to consider in the integer program.
+\textcolor{red}{As shown in Figure~\ref{fig7}, GA-MuDiLCO consumes less energy than both DESK and GAF, but a little bit higher than MuDiLCO  because it provides a near optimal solution by activating a larger number of nodes during the sensing phase.  GA-MuDiLCO consumes less energy in comparison with MuDiLCO-7 version, especially for the dense networks. However, MuDiLCO protocol and GA-MuDiLCO protocol are the most competitive from the energy
+consumption point of view. The other approaches have a high energy consumption
+due to activating a larger number of redundant nodes.}

 %In fact,  a distributed optimization decision, which produces T rounds, on the subregions is  greatly reduced the cost of communications and the time of listening as well as the energy needed for sensing phase and computation so thanks to the partitioning of the initial network into several independent subnetworks and producing T rounds for each subregion periodically. 
 
 
 %In fact,  a distributed optimization decision, which produces T rounds, on the subregions is  greatly reduced the cost of communications and the time of listening as well as the energy needed for sensing phase and computation so thanks to the partitioning of the initial network into several independent subnetworks and producing T rounds for each subregion periodically. 
 
 

-\subsection{Execution time}
+\subsubsection{Execution time}

 
 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
 
 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
+seconds (needed to solve optimization problem) for different values of $T$. The modeling language for Mathematical Programming (AMPL)~\cite{AMPL} is  employed to generate the Mixed Integer Linear Program instance  in a  standard format, which  is then read  and solved  by the optimization solver  GLPK (GNU  linear Programming Kit  available in  the public domain) \cite{glpk} through a Branch-and-Bound method. The

 original execution time  is computed on a laptop  DELL with Intel Core~i3~2370~M
 (2.4 GHz)  processor (2  cores) and the  MIPS (Million Instructions  Per Second)
 rate equal to 35330. To be consistent  with the use of a sensor node with Atmels
 original execution time  is computed on a laptop  DELL with Intel Core~i3~2370~M
 (2.4 GHz)  processor (2  cores) and the  MIPS (Million Instructions  Per Second)
 rate equal to 35330. To be consistent  with the use of a sensor node with Atmels


@@ -1177,7 +1365,7 @@ for different network sizes.
 
 \begin{figure}[ht!]
 \centering
 
 \begin{figure}[ht!]
 \centering

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

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


@@ -1195,7 +1383,7 @@ optimization problem.
 
 %While MuDiLCO-1, 3, and 5 solves the optimization process with suitable execution times to be used on wireless sensor network because it distributed on larger number of small subregions as well as it is used acceptable number of round(s) T.  We think that in distributed fashion the solving of the optimization problem to produce T rounds in a subregion can be tackled by sensor nodes. Overall, to be able to deal with very large networks, a distributed method is clearly required.
 
 
 %While MuDiLCO-1, 3, and 5 solves the optimization process with suitable execution times to be used on wireless sensor network because it distributed on larger number of small subregions as well as it is used acceptable number of round(s) T.  We think that in distributed fashion the solving of the optimization problem to produce T rounds in a subregion can be tackled by sensor nodes. Overall, to be able to deal with very large networks, a distributed method is clearly required.
 

-\subsection{Network lifetime}
+\subsubsection{Network lifetime}

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


@@ -1210,14 +1398,14 @@ 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
 energy consumption,  since network lifetime and energy  consumption are directly
 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
 energy consumption,  since network lifetime and energy  consumption are directly

-linked.
-
+linked. \textcolor{red}{As can be seen in these figures, the lifetime increases with the size of the network, and it is clearly largest for the MuDiLCO
+and the GA-MuDiLCO protocols. GA-MuDiLCO prolongs the network lifetime obviously in comparison with both DESK and GAF, as well as the MuDiLCO-7 version for $lifetime_{95}$.  However, comparison shows that MuDiLCO protocol and GA-MuDiLCO protocol, which use distributed optimization over the subregions are the best ones because they are robust to network disconnection during the network lifetime as well as they consume less energy in comparison with other approaches.}

 \begin{figure}[t!]
   \centering
   \begin{tabular}{cl}
 \begin{figure}[t!]
   \centering
   \begin{tabular}{cl}

-    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/LT95.pdf}} & (a) \\
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{R/LT95.pdf}} & (a) \\

     \verb+ + \\
     \verb+ + \\

-    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/LT50.pdf}} & (b)
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{R/LT50.pdf}} & (b)

   \end{tabular}
   \caption{Network lifetime for (a) $Lifetime_{95}$ and 
     (b) $Lifetime_{50}$}
   \end{tabular}
   \caption{Network lifetime for (a) $Lifetime_{95}$ and 
     (b) $Lifetime_{50}$}


@@ -1252,7 +1440,7 @@ scheduling.
 The activity  scheduling in each subregion  works in periods,  where each period
 consists of four  phases: (i) Information Exchange, (ii)  Leader Election, (iii)
 Decision Phase to plan the activity  of the sensors over $T$ rounds, (iv) Sensing
 The activity  scheduling in each subregion  works in periods,  where each period
 consists of four  phases: (i) Information Exchange, (ii)  Leader Election, (iii)
 Decision Phase to plan the activity  of the sensors over $T$ rounds, (iv) Sensing

-Phase itself divided into T rounds.
+Phase itself divided into $T$ rounds.

 
 Simulations  results show the  relevance of  the proposed  protocol in  terms of
 lifetime, coverage  ratio, active  sensors ratio, energy  consumption, execution
 
 Simulations  results show the  relevance of  the proposed  protocol in  terms of
 lifetime, coverage  ratio, active  sensors ratio, energy  consumption, execution