X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/JournalMultiPeriods.git/blobdiff_plain/4e0783212d2ef38d47c78e94e3053b88999aef79..bc1490b3187eb4303ac1eece3ea4f89f0bb5905a:/article.tex?ds=inline

diff --git a/article.tex b/article.tex
index 6707012..a0472dd 100644
--- a/article.tex
+++ b/article.tex
@@ -73,15 +73,25 @@
 %% \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
  
-\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)
@@ -106,7 +116,7 @@ network lifetime and improve the coverage performance.
 \end{abstract}
 
 \begin{keyword}
-Wireless   Sensor   Networks,   Area   Coverage,   Network   lifetime,
+Wireless   Sensor   Networks,   Area   Coverage,   Network   Lifetime,
 Optimization, Scheduling, Distributed Computation.
 
 \end{keyword}
@@ -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
-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
@@ -812,6 +821,165 @@ Active-Sleep packet is obtained.
 
 \end{algorithm}
 
+%\textcolor{red}{\textbf{\textsc{Answer:}   ali   }}
+
+
+\section{Genetic Algorithm (GA) for Multiround Lifetime Coverage Optimization}
+\label{GA}
+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 donât 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 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{pd}. 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}{Input}
+ \SetKwInput{Output}{Output}
+ \Input{ $ 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}$}
+ \Output{$\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} \; 
+  \ForEach {Individual $ind$ $\in$ $S_{pop}$} {
+     \emph{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{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{Evaluate Individual $(P, J, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind})$}\;  
+  }
+  
+  \While{ Stopping criteria is not satisfied }{
+  
+  \emph{Selection $(ind_1, ind_2)$}\;
+    \emph{Crossover $(P_c, X_{t,j}^{ind_1}, X_{t,j}^{ind_2}, Child_{t,j}^{ind_1}, Child_{t,j}^{ind_2})$}\;
+    \emph{Mutation $(P_m, Child_{t,j}^{ind_1}, Child_{t,j}^{ind_2})$}\;
+   
+   
+   \emph{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{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{Evaluate New Individual$(P, J, Child_{t,j}^{ind_1}, Ch.\Theta_{t,p}^{ind_1}, Ch.U_{t,p}^{ind_1})$}\;  
+ \emph{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{Evaluate New Individual$(P, J, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2})$}\;  
+  
+ \emph{Replacement $(P, J, T, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind}  )$ }\;
+  
+      
+  }
+  \emph{$\left\{\left(X_{1,1},\dots,X_{t,j},\dots,X_{T,J}\right)\right\}$ =
+            Select Best Solution ($S_{pop}$)}\;
+ \emph{return X} \;
+\caption{GA-MuDiLCO($s_j$)}
+\label{alg:GA}
+
+\end{algorithm}
+
+
+\begin{enumerate} [I)]
+\item \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 next phase, the chromosome is defined as a schedule for alive  sensors and each chromosome contains $T$ rounds. 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.eps} 
+\caption{Candidate Solution representation by the proposed GA. }
+\label{chromo}
+\end{figure} 
+
+
+
+\item \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 current round. Otherwise, the value is set to 0. The energy constraint is applied for each sensor during all rounds. 
+
+
+\item \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!]                
+  
+ \SetKwInput{Input}{Input}
+ \SetKwInput{Output}{Output}
+ \Input{ parameters $P, J, ind, \alpha_{j,p}^{ind}, X_{t,j}^{ind}$}
+ \Output{$U^{ind} = \left\lbrace U_{1,1}^{ind}, \dots, U_{t,p}^{ind}, \dots, U_{T,P}^{ind} \right\rbrace$ and $\Theta^{ind} = \left\lbrace \Theta_{1,1}^{ind}, \dots, \Theta_{t,p}^{ind}, \dots, \Theta_{T,P}^{ind} \right\rbrace$}
+
+  \BlankLine
+
+  \For{$t\leftarrow 1$ \KwTo $T$}{
+  \For{$p\leftarrow 1$ \KwTo $P$}{
+     
+ %    \For{$i\leftarrow 0$ \KwTo $I_j$}{
+     	\emph{$SUM\leftarrow 0$}\;
+         \For{$j\leftarrow 1$ \KwTo $J$}{
+              \emph{$SUM \leftarrow SUM + (\alpha_{j,p}^{ind} \times X_{t,j}^{ind})$ }\;
+         }
+         
+         \If { SUM = 0} {
+         \emph{$U_{t,p}^{ind} \leftarrow 0$}\;
+         \emph{$\Theta_{t,p}^{ind} \leftarrow 1$}\;
+         }
+         \Else{
+         \emph{$U_{t,p}^{ind} \leftarrow SUM -1$}\;
+         \emph{$\Theta_{t,p}^{ind} \leftarrow 0$}\;
+         }
+     
+     }
+     
+  }
+\emph{return $U^{ind}, \Theta^{ind}$ } \;
+\caption{O-U-Coverage}
+\label{OU}
+
+\end{algorithm}
+
+
+
+\item \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 get 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 \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 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 \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.eps} 
+\caption{Two-point crossover. }
+\label{cross}
+\end{figure} 
+
+
+\item \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 oprator in the proposed GA-MuDiLCO works as follow: If mutation probability $P_m$ is 100$\%$, then the mutation operation takes place on the 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 \textbf{Update O-U-Coverage for children:}
+Before evalute 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 \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 \textbf{Replacement:}
+After evaluatation 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 \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 divided by two. 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}
@@ -859,7 +1027,7 @@ Sensing time for one round & 60 Minutes \\
 $E_{R}$ & 36 Joules\\
 $R_s$ & 5~m   \\     
 %\hline
-$W_{\Theta}$ & 1   \\
+$W_{\theta}$ & 1   \\
 % [1ex] adds vertical space
 %\hline
 $W_{U}$ & $|P|^2$
@@ -948,7 +1116,7 @@ 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
-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
@@ -1011,7 +1179,7 @@ network, and $R$ is the total number of subregions in the network.
   % 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*}
 
 
@@ -1030,7 +1198,7 @@ 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
-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  $E^{\scriptsize  \mbox{com}}_m$,
@@ -1040,7 +1208,7 @@ 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 $E^s_t$
-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.
@@ -1057,9 +1225,9 @@ indicate the energy consummed by the whole network in round $t$.
 
 \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
@@ -1080,12 +1248,12 @@ rounds, and thus should extend the network lifetime.
 
 \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} 
 
-\subsection{Active sensors ratio} 
+\subsubsection{Active sensors ratio} 
 
 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
@@ -1101,12 +1269,12 @@ nodes in a more efficient manner.
 
 \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} 
 
-\subsection{Stopped simulation runs}
+\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. 
 
@@ -1123,12 +1291,12 @@ still connected.
 
 \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} 
 
-\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
@@ -1139,9 +1307,9 @@ network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.
 \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+ + \\
-    \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) $Lifetime_{50}$}
@@ -1162,11 +1330,11 @@ sensors to consider in the integer program.
 %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
-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
@@ -1177,7 +1345,7 @@ for different network sizes.
 
 \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} 
@@ -1195,7 +1363,7 @@ 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.
 
-\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 $Lifetime_{95}$
@@ -1215,9 +1383,9 @@ linked.
 \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+ + \\
-    \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) $Lifetime_{50}$}
@@ -1252,7 +1420,7 @@ 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
-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
