%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}}
-}
-
+\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)
%continuously and effectively when monitoring a certain area (or
%region) of interest.
Coverage and lifetime are two paramount problems in Wireless Sensor Networks
-(WSNs). In this paper, a method called Multiround Distributed Lifetime Coverage
+(WSNs). In this paper, a method called Multiround Distributed Lifetime Coverage
Optimization protocol (MuDiLCO) is proposed to maintain the coverage and to
improve the lifetime in wireless sensor networks. The area of interest is first
-divided into subregions and then the MuDiLCO protocol is distributed on the
-sensor nodes in each subregion. The proposed MuDiLCO protocol works in periods
-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
-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.
-
+divided into subregions and then the MuDiLCO protocol is distributed on the
+sensor nodes in each subregion. The proposed MuDiLCO protocol works in periods
+during which sets of sensor nodes are scheduled, with one set for each round of
+a period, to remain active during the sensing phase and thus ensure coverage so
+as to maximize the WSN lifetime. \textcolor{blue}{The decision process is
+ carried out by a leader node, which solves an optimization problem to produce
+ the best representative sets to be used during the rounds of the sensing
+ phase. The optimization problem formulated as an integer program is solved to
+ optimality through a Branch-and-Bound method for small instances. For larger
+ instances, the best feasible solution found by the solver after a given time
+ limit threshold is considered.}
+%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.
+%\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.
\end{abstract}
\begin{keyword}
Wireless Sensor Networks, Area Coverage, Network Lifetime,
Optimization, Scheduling, Distributed Computation.
-
\end{keyword}
\end{frontmatter}
The remainder of the paper is organized as follows. The next section
% Section~\ref{rw}
-reviews the related works in the field. Section~\ref{pd} is devoted to the
+reviews the related works in the field. Section~\ref{pd} is devoted to the
description of MuDiLCO protocol. Section~\ref{exp} shows the simulation results
obtained using the discrete event simulator OMNeT++ \cite{varga}. They fully
-demonstrate the usefulness of the proposed approach. Finally, we give
+demonstrate the usefulness of the proposed approach. Finally, we give
concluding remarks and some suggestions for future works in
Section~\ref{sec:conclusion}.
\item Sensors scheduling algorithm implementation, i.e. centralized or
distributed/localized algorithms.
\item The objective of sensor coverage, i.e. to maximize the network lifetime or
- to minimize the number of sensors during a sensing round.
+ to minimize the number of active sensors during a sensing round.
\item The homogeneous or heterogeneous nature of the nodes, in terms of sensing
or communication capabilities.
\item The node deployment method, which may be random or deterministic.
The major approach is to divide/organize the sensors into a suitable number of
cover sets where each set completely covers an interest region and to activate
these cover sets successively. The centralized algorithms always provide nearly
-or close to optimal solution since the algorithm has global view of the whole
+or close to optimal solution since the algorithm has global view of the whole
network. Note that centralized algorithms have the advantage of requiring very
low processing power from the sensor nodes, which usually have limited
-processing capabilities. The main drawback of this kind of approach is its
+processing capabilities. The main drawback of this kind of approach is its
higher cost in communications, since the node that will make the decision needs
-information from all the sensor nodes. Moreover, centralized approaches usually
-suffer from the scalability problem, making them less competitive as the network
-size increases.
+information from all the sensor nodes. \textcolor{blue} {Exact or heuristic
+ approaches are designed to provide cover sets.
+%(Moreover, centralized approaches usually
+%suffer from the scalability problem, making them less competitive as the network
+%size increases.)
+Contrary to exact methods, heuristic ones can handle very large and centralized
+problems. They are proposed to reduce computational overhead such as energy
+consumption, delay, and generally allow to increase the network lifetime.}
The first algorithms proposed in the literature consider that the cover sets are
disjoint: a sensor node appears in exactly one of the generated cover
-sets~\cite{abrams2004set,cardei2005improving,Slijepcevic01powerefficient}. In
-the case of non-disjoint algorithms \cite{pujari2011high}, sensors may
-participate in more than one cover set. In some cases, this may prolong the
+sets~\cite{abrams2004set,cardei2005improving,Slijepcevic01powerefficient}. In
+the case of non-disjoint algorithms \cite{pujari2011high}, sensors may
+participate in more than one cover set. In some cases, this may prolong the
lifetime of the network in comparison to the disjoint cover set algorithms, but
-designing algorithms for non-disjoint cover sets generally induces a higher
+designing algorithms for non-disjoint cover sets generally induces a higher
order of complexity. Moreover, in case of a sensor's failure, non-disjoint
-scheduling policies are less resilient and reliable because a sensor may be
+scheduling policies are less resilient and reliable because a sensor may be
involved in more than one cover sets.
%For instance, the proposed work in ~\cite{cardei2005energy, berman04}
-In~\cite{yang2014maximum}, the authors have considered a linear programming
+In~\cite{yang2014maximum}, the authors have considered a linear programming
approach to select the minimum number of working sensor nodes, in order to
-preserve a maximum coverage and to extend lifetime of the network. Cheng et
+preserve a maximum coverage and to extend lifetime of the network. Cheng et
al.~\cite{cheng2014energy} have defined a heuristic algorithm called Cover Sets
Balance (CSB), which chooses a set of active nodes using the tuple (data
coverage range, residual energy). Then, they have introduced a new Correlated
-Node Set Computing (CNSC) algorithm to find the correlated node set for a given
-node. After that, they proposed a High Residual Energy First (HREF) node
-selection algorithm to minimize the number of active nodes so as to prolong the
-network lifetime. Various centralized methods based on column generation
-approaches have also been
-proposed~\cite{castano2013column,rossi2012exact,deschinkel2012column}.
+Node Set Computing (CNSC) algorithm to find the correlated node set for a given
+node. After that, they proposed a High Residual Energy First (HREF) node
+selection algorithm to minimize the number of active nodes so as to prolong the
+network lifetime. Various centralized methods based on column generation
+approaches have also been
+proposed~\cite{gentili2013,castano2013column,rossi2012exact,deschinkel2012column}.
+\textcolor{blue}{In~\cite{gentili2013}, authors highlight the trade-off between
+ the network lifetime and the coverage percentage. They show that network
+ lifetime can be hugely improved by decreasing the coverage ratio.}
\subsection{Distributed approaches}
%{\bf Distributed approaches}
\cite{Ye03} or regulated \cite{cardei2005maximum} over time.
The MuDiLCO protocol (for Multiround Distributed Lifetime Coverage Optimization
-protocol) presented in this paper is an extension of the approach introduced
+protocol) presented in this paper is an extension of the approach introduced
in~\cite{idrees2014coverage}. In~\cite{idrees2014coverage}, the protocol is
-deployed over only two subregions. Simulation results have shown that it was
+deployed over only two subregions. Simulation results have shown that it was
more interesting to divide the area into several subregions, given the
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
+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{blue}{The idea is
+ to take advantage of the pre-sensing phase to plan the sensor's activity for
+ several rounds instead of one, thus saving energy. In addition, when the
+ optimization problem becomes more complex, its resolution is stopped after a
+ given time threshold}.
\iffalse
hypothesis, a complete coverage of a convex area implies connectivity among the
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.
+%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.
+
+\indent Instead of working with the coverage area, we consider for each sensor a
+set of points called primary points~\cite{idrees2014coverage}. We assume that
+the sensing disk defined by a sensor is covered if all the primary points of
+this sensor are covered. By knowing the position of wireless sensor node
+(centered at the the position $\left(p_x,p_y\right)$) and it's sensing range
+$R_s$, we define up to 25 primary points $X_1$ to $X_{25}$ as decribed on
+Figure~\ref{fig1}. The optimal number of primary points is investigated in
+section~\ref{ch4:sec:04:06}.
+
+The coordinates of the primary points are defined as follows:\\
+%$(p_x,p_y)$ = point center of wireless sensor node\\
+$X_1=(p_x,p_y)$ \\
+$X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\
+$X_3=( p_x + R_s * (-1), p_y + R_s * (0)) $\\
+$X_4=( p_x + R_s * (0), p_y + R_s * (1) )$\\
+$X_5=( p_x + R_s * (0), p_y + R_s * (-1 )) $\\
+$X_6=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
+$X_7=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
+$X_8=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
+$X_9=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
+$X_{10}= ( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (0)) $\\
+$X_{11}=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (0))$\\
+$X_{12}=( p_x + R_s * (0), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
+$X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
+$X_{14}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (\frac{1}{2})) $\\
+$X_{15}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (\frac{1}{2})) $\\
+$X_{16}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (\frac{- 1}{2})) $\\
+$X_{17}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (\frac{- 1}{2})) $\\
+$X_{18}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (0)) $\\
+$X_{19}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (0)) $\\
+$X_{20}=( p_x + R_s * (0), p_y + R_s * (\frac{1}{2})) $\\
+$X_{21}=( p_x + R_s * (0), p_y + R_s * (-\frac{1}{2})) $\\
+$X_{22}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{\sqrt{3}}{2})) $\\
+$X_{23}=( p_x + R_s * (\frac{- 1}{2}), p_y + R_s * (\frac{\sqrt{3}}{2})) $\\
+$X_{24}=( p_x + R_s * (\frac{- 1}{2}), p_y + R_s * (\frac{-\sqrt{3}}{2})) $\\
+$X_{25}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{-\sqrt{3}}{2})) $.
+
+
+%\begin{figure} %[h!]
+%\centering
+% \begin{multicols}{2}
+%\centering
+%\includegraphics[scale=0.28]{fig21.pdf}\\~ (a)
+%\includegraphics[scale=0.28]{principles13.pdf}\\~(c)
+%\hfill \hfill
+%\includegraphics[scale=0.28]{fig25.pdf}\\~(e)
+%\includegraphics[scale=0.28]{fig22.pdf}\\~(b)
+%\hfill \hfill
+%\includegraphics[scale=0.28]{fig24.pdf}\\~(d)
+%\includegraphics[scale=0.28]{fig26.pdf}\\~(f)
+%\end{multicols}
+%\caption{Wireless Sensor Node represented by (a) 5, (b) 9, (c) 13, (d) 17, (e) 21 and (f) 25 primary points respectively}
+%\label{fig1}
+%\end{figure}
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[scale=0.375]{fig26.pdf}
+ \label{fig1}
+ \caption{Wireless sensor node represented by up to 25~primary points}
+\end{figure}
%By knowing the position (point center: ($p_x,p_y$)) of a wireless
%sensor node and its $R_s$, we calculate the primary points directly
\subsection{Background idea}
%%RC : we need to clarify the difference between round and period. Currently it seems to be the same (for me at least).
-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
-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,
-Decision, and Sensing. 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
-task. 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!]
-\centering \includegraphics[width=100mm]{Modelgeneral.pdf} % 70mm
+%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
+%implemented in each subregion in a distributed way.
+
+\textcolor{blue}{The WSN area of interest is, at first, divided into
+ regular homogeneous subregions using a divide-and-conquer algorithm. Then, our protocol will be executed in a distributed way in each
+ subregion simultaneously to schedule nodes' activities for one sensing
+ period. Sensor nodes are assumed to be deployed almost uniformly and with high
+ density over the region. The regular subdivision is made so that the number
+ of hops between any pairs of sensors inside a subregion is less than or equal
+ to 3.}
+
+As can be seen in Figure~\ref{fig2}, our protocol works in periods fashion,
+where each period is divided into 4~phases: Information~Exchange,
+Leader~Election, Decision, and Sensing. Each sensing phase may be itself
+divided into $T$ rounds \textcolor{blue} {of equal duration} 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 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}[t!]
+\centering \includegraphics[width=125mm]{Modelgeneral.pdf} % 70mm
\caption{The MuDiLCO protocol scheme executed on each node}
\label{fig2}
\end{figure}
% set cover responsible for the sensing task.
%For each round a set of sensors (said a cover set) is responsible for the sensing task.
-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.
%This protocol is reliable against an unexpected node failure, because it works in periods.
%%RC : why? I am not convinced
- 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. \textcolor{blue}{The duration of the rounds is a predefined
+ parameter. Round duration should be long enough to hide the system control
+ overhead and short enough to minimize the negative effects in case of node
+ failures.}
+
%%RC so if there are at least one failure per period, the coverage is bad...
%%MS if we want to be reliable against many node failures we need to have an
%% overcoverage...
\subsection{Leader Election phase}
-This step consists in choosing the Wireless Sensor Node Leader (WSNL), which
+This step consists in choosing the Wireless Sensor Node Leader (WSNL), which
will be responsible for executing the coverage algorithm. Each subregion in the
area of interest will select its own WSNL independently for each period. All
-the sensor nodes cooperate to elect a WSNL. The nodes in the same subregion
-will select the leader based on the received information from all other nodes
-in the same subregion. The selection criteria are, in order of importance:
-larger number of neighbors, larger remaining energy, and then in case of
-equality, larger index. Observations on previous simulations suggest to use the
-number of one-hop neighbors as the primary criterion to reduce energy
-consumption due to the communications.
+the sensor nodes cooperate to elect a WSNL. The nodes in the same subregion
+will select the leader based on the received information from all other nodes in
+the same subregion. The selection criteria are, in order of importance: larger
+number of neighbors, larger remaining energy, and then in case of equality,
+larger index. Observations on previous simulations suggest to use the number of
+one-hop neighbors as the primary criterion to reduce energy consumption due to
+the communications.
%the more priority selection factor is the number of $1-hop$ neighbors, $NBR j$, which can minimize the energy consumption during the communication Significantly.
%The pseudo-code for leader election phase is provided in Algorithm~1.
\subsection{Decision phase}
-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
-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.
+Each WSNL will \textcolor{blue}{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. $T$ cover sets will be produced, 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 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
+%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}
+%solve an integer program
-\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
+%\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
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
-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
sensors by $J$. Only sensors able to be alive during at least one round are
involved in the integer program.
Our coverage optimization problem can then be formulated as follows:
\begin{equation}
- \min \sum_{t=1}^{T} \sum_{p=1}^{P} \left(W_{\theta}* \Theta_{t,p} + W_{U} * U_{t,p} \right) \label{eq15}
+ \min \sum_{t=1}^{T} \sum_{p=1}^{|P|} \left(W_{\theta}* \Theta_{t,p} + W_{U} * U_{t,p} \right) \label{eq15}
\end{equation}
Subject to
\end{equation}
\begin{equation}
- \sum_{t=1}^{T} X_{t,j} \leq \floor*{RE_{j}/E_{R}} \hspace{6 mm} \forall j \in J, t = 1,\dots,T
+ \sum_{t=1}^{T} X_{t,j} \leq \floor*{RE_{j}/E_{R}} \hspace{10 mm}\forall j \in J\hspace{6 mm}
\label{eq144}
\end{equation}
The first group of constraints indicates that some primary point $p$ should be
covered by at least one sensor and, if it is not always the case, overcoverage
-and undercoverage variables help balancing the restriction equations by taking
+and undercoverage variables help balancing the restriction equations by taking
positive values. The constraint given by equation~(\ref{eq144}) guarantees that
the sensor has enough energy ($RE_j$ corresponds to its remaining energy) to be
alive during the selected rounds knowing that $E_{R}$ is the amount of energy
required to be alive during one round.
-There are two main objectives. First, we limit the overcoverage of primary
-points in order to activate a minimum number of sensors. Second we prevent the
-absence of monitoring on some parts of the subregion by minimizing the
-undercoverage. The weights $W_\theta$ and $W_U$ must be properly chosen so as
-to guarantee that the maximum number of points are covered during each round.
+There are two main objectives. First, we limit the overcoverage of primary
+points in order to activate a minimum number of sensors. Second we prevent the
+absence of monitoring on some parts of the subregion by minimizing the
+undercoverage. The weights $W_\theta$ and $W_U$ must be properly chosen so as
+to guarantee that the maximum number of points are covered during each round.
%% MS W_theta is smaller than W_u => problem with the following sentence
-In our simulations priority is given to the coverage by choosing $W_{U}$ very
+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.
-This integer program can be solved using two approaches:
+\textcolor{blue}{The size of the problem depends on the number of variables and
+ constraints. The number of variables is linked to the number of alive sensors
+ $A \subseteq J$, the number of rounds $T$, and the number of primary points
+ $P$. Thus the integer program contains $A*T$ variables of type $X_{t,j}$,
+ $P*T$ overcoverage variables and $P*T$ undercoverage variables. The number of
+ constraints is equal to $P*T$ (for constraints (\ref{eq16})) $+$ $A$ (for
+ constraints (\ref{eq144})).}
+%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{Optimization solver for Multiround Lifetime Coverage Optimization}
-\label{glpk}
-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.
+\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 sensor node~$s_j$ 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 Integer Program Algorithm($T,J$)}\;
+ \emph{$s_j.status$ = COMMUNICATION}\;
+ \emph{Send $ActiveSleep()$ packet 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}
-%\textcolor{red}{\textbf{\textsc{Answer:} ali }}
+\iffalse
+\textcolor{red}{This integer program can be solved using two approaches:}
+\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.}
+\fi
+
+\iffalse
-\subsection{Genetic Algorithm (GA) for Multiround Lifetime Coverage Optimization}
+\subsection{\textcolor{red}{Genetic Algorithm 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 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. Compared to GLPK optimization solver, GA provides a near optimal solution with acceptible execution time, while GLPK provides optimal solution but it requires high execution time for large problem.
+\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.}
-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:
+\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!]
+\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}$}
+ \SetKwInput{Input}{\textcolor{red}{Input}}
+ \SetKwInput{Output}{\textcolor{red}{Output}}
+ \Input{ \textcolor{red}{$ 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} \;
- \ForEach {Individual $ind$ $\in$ $S_{pop}$} {
- \emph{Generate Randomly Chromosome $\left\{\left(X_{1,1},\dots, X_{t,j}, \dots, X_{T,J}\right)\right\}_{t \in T, j \in J}$}\;
+ \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{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}{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 $(P, J, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind})$}\;
+ \emph{\textcolor{red}{Evaluate Individual $(P, J, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind})$}}\;
}
- \While{ Stopping criteria is not satisfied }{
+ \While{\textcolor{red}{ 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{\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{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{\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{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{\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{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}{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{\textcolor{red}{Replacement $(P, J, T, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind} )$ }}\;
}
- \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($T, J$)}
+ \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 \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.
+
+\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
-\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 \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 \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.
+\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!]
- \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 $\Theta^{ind} = \left\lbrace \Theta_{1,1}^{ind}, \dots, \Theta_{t,p}^{ind}, \dots, \Theta_{T,P}^{ind} \right\rbrace$}
+ \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{$t\leftarrow 1$ \KwTo $T$}{
- \For{$p\leftarrow 1$ \KwTo $P$}{
+ \For{\textcolor{red}{$t\leftarrow 1$ \KwTo $T$}}{
+ \For{\textcolor{red}{$p\leftarrow 1$ \KwTo $P$}}{
% \For{$i\leftarrow 0$ \KwTo $I_j$}{
- \emph{$SUM\leftarrow 0$}\;
- \For{$j\leftarrow 1$ \KwTo $J$}{
- \emph{$SUM \leftarrow SUM + (\alpha_{j,p}^{ind} \times X_{t,j}^{ind})$ }\;
+ \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 { SUM = 0} {
- \emph{$U_{t,p}^{ind} \leftarrow 0$}\;
- \emph{$\Theta_{t,p}^{ind} \leftarrow 1$}\;
+ \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{$U_{t,p}^{ind} \leftarrow SUM -1$}\;
- \emph{$\Theta_{t,p}^{ind} \leftarrow 0$}\;
+ \emph{\textcolor{red}{$U_{t,p}^{ind} \leftarrow SUM -1$}}\;
+ \emph{\textcolor{red}{$\Theta_{t,p}^{ind} \leftarrow 0$}}\;
}
}
}
-\emph{return $U^{ind}, \Theta^{ind}$ } \;
+\emph{\textcolor{red}{return $U^{ind}, \Theta^{ind}$ }} \;
\caption{O-U-Coverage}
\label{OU}
-\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.
+\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 \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 \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 \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.
+\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!]
\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 \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 \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 \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 \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{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 \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 \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.
+\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}
+\fi
+%% EXPERIMENTAL STUDY
\section{Experimental study}
\label{exp}
\subsection{Simulation setup}
-We conducted a series of simulations to evaluate the efficiency and the
-relevance of our approach, using the discrete event simulator OMNeT++
-\cite{varga}. 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.
+We conducted a series of simulations to evaluate the efficiency and the
+relevance of our approach, using the discrete event simulator OMNeT++
+\cite{varga}. 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
-250~nodes deployed over a $50 \times 25~m^2 $ sensing field. More
-precisely, the deployment is controlled at a coarse scale in order to ensure
-that the deployed nodes can cover the sensing field with the given sensing
-range.
+250~nodes deployed over a $50 \times 25~m^2 $ sensing field. More precisely,
+the deployment is controlled at a coarse scale in order to ensure that the
+deployed nodes can cover the sensing field with the given sensing range.
%%RC these parameters are realistic?
%% maybe we can increase the field and sensing range. 5mfor Rs it seems very small... what do the other good papers consider ?
% [1ex] adds vertical space
%\hline
$W_{U}$ & $|P|^2$ \\
-$P_c$ & 0.95 \\
-$P_m$ & 0.6 \\
-$S_{pop}$ & 50
+%$P_c$ & 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}
-
-\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
-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}. 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.
-During the decision phase, in each square, one sensor is then chosen to remain
-active during the sensing phase time.
+
+\textcolor{blue}{Our protocol is declined into four versions: MuDiLCO-1,
+ MuDiLCO-3, MuDiLCO-5, and MuDiLCO-7, corresponding respectively to $T=1,3,5,7$
+ ($T$ the number of rounds in one sensing period). Since the time resolution
+ may be prohibitive when the size of the problem increases, a time limit
+ threshold has been fixed when solving large instances. In these cases, the
+ solver returns the best solution found, which is not necessary the optimal
+ one. In practice, we only set time limit values for the three largest network
+ sizes when $T=7$, using the following respective values (in second): 0.03 for
+ 150~nodes, 0.06 for 200~nodes, and 0.08 for 250~nodes.
+% Table \ref{tl} shows time limit values.
+ These time limit thresholds have been set empirically. The basic idea consists
+ in considering the average execution time to solve the integer programs to
+ optimality, then in dividing this average time by three to set the threshold
+ value. After that, this threshold value is increased if necessary so that
+ the solver is able to deliver a feasible solution within the time limit. In
+ fact, selecting the optimal values for the time limits will be investigated in
+ the future.}
+%In Table \ref{tl}, "NO" indicates that the problem has been solved to optimality without time limit.}
+
+%\begin{table}[ht]
+%\caption{Time limit values for MuDiLCO protocol versions }
+%\centering
+%\begin{tabular}{|c|c|c|c|c|}
+% \hline
+% WSN size & MuDiLCO-1 & MuDiLCO-3 & MuDiLCO-5 & MuDiLCO-7 \\ [0.5ex]
+%\hline
+% 50 & NO & NO & NO & NO \\
+% \hline
+%100 & NO & NO & NO & NO \\
+%\hline
+%150 & NO & NO & NO & 0.03 \\
+%\hline
+%200 & NO & NO & NO & 0.06 \\
+% \hline
+% 250 & NO & NO & NO & 0.08 \\
+% \hline
+%\end{tabular}
+
+%\label{tl}
+
+%\end{table}
+
+ 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. During the decision phase,
+ in each square, one sensor is then chosen to remain active during the sensing
+ phase time.
Some preliminary experiments were performed to study the choice of the number of
subregions which subdivides the sensing field, considering different network
% is used to refer this table in the text
\end{table}
-For the sake of simplicity we ignore the energy needed to turn on the radio, to
+For the sake of simplicity we ignore the energy needed to turn on the radio, to
start up the sensor node, to move from one status to another, etc.
%We also do not consider the need of collecting sensing data. PAS COMPRIS
-Thus, when a sensor becomes active (i.e., it has already chosen its status), it can
-turn its radio off to save battery. MuDiLCO uses two types of packets for
-communication. The size of the INFO packet and Active-Sleep packet are 112~bits
-and 24~bits respectively. The value of energy spent to send a 1-bit-content
+Thus, when a sensor becomes active (i.e., it has already chosen its status), it
+can turn its radio off to save battery. MuDiLCO uses two types of packets for
+communication. The size of the INFO packet and Active-Sleep packet are 112~bits
+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
+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.
-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
less than $E_{R}=36~\mbox{Joules}$, the minimum energy needed for the node to
-stay alive during one round. This value has been computed by multiplying the
+stay alive during one round. This value has been computed by multiplying the
energy consumed in active state (9.72 mW) by the time in second for one round
-(3600 seconds). According to the interval of initial energy, a sensor may be
+(3600 seconds). According to the interval of initial energy, a sensor may be
alive during at most 20 rounds.
\subsection{Metrics}
% 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 $M$ 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
+consumed by the sensors (EC) comes through taking into consideration four main
energy factors. The first one , denoted $E^{\scriptsize \mbox{com}}_m$,
-represents the energy consumption spent by all the nodes for wireless
-communications during period $m$. $E^{\scriptsize \mbox{list}}_m$, the next
-factor, corresponds to the energy consumed by the sensors in LISTENING status
-before receiving the decision to go active or sleep in period $m$.
+represents the energy consumption spent by all the nodes for wireless
+communications during period $m$. $E^{\scriptsize \mbox{list}}_m$, the next
+factor, corresponds to the energy consumed by the sensors in LISTENING status
+before receiving the decision to go active or sleep in period $m$.
$E^{\scriptsize \mbox{comp}}_m$ refers to the energy needed by all the leader
nodes to solve the integer program during a period. Finally, $E^a_t$ and $E^s_t$
indicate the energy consumed by the whole network in round $t$.
\end{enumerate}
-\subsection{Results and analysis}
+\subsection{Performance analysis for different number of primary points}
+\label{ch4:sec:04:06}
+
+In this section, we study the performance of MuDiLCO-1 approach for different
+numbers of primary points. The objective of this comparison is to select the
+suitable number of primary points to be used by a MuDiLCO protocol. In this
+comparison, MuDiLCO-1 protocol is used with five primary point models, each
+model corresponding to a number of primary points, which are called Model-5 (it
+uses 5 primary points), Model-9, Model-13, Model-17, and Model-21.
+
+%\begin{enumerate}[i)]
+
+%\item {{\bf Coverage Ratio}}
+\subsubsection{Coverage ratio}
+
+Figure~\ref{Figures/ch4/R2/CR} shows the average coverage ratio for 150 deployed
+nodes. As can be seen, at the beginning the models which use a larger number of
+primary points provide slightly better coverage ratios, but latter they are the
+worst.
+%Moreover, when the number of periods increases, coverage ratio produced by Model-9, Model-13, Model-17, and Model-21 decreases in comparison with Model-5 due to a larger time computation for the decision process for larger number of primary points.
+Moreover, when the number of periods increases, the coverage ratio produced by
+all models decrease due to dead nodes. However, Model-5 is the one with the
+slowest decrease due to lower numbers of active sensors in the earlier periods.
+% smaller time computation of decision process for a smaller number of primary points.
+Overall this model is slightly more efficient than the other ones, because it
+offers a good coverage ratio for a larger number of periods.
+%\parskip 0pt
+\begin{figure}[t!]
+\centering
+ \includegraphics[scale=0.5] {R2/CR.pdf}
+\caption{Coverage ratio for 150 deployed nodes}
+\label{Figures/ch4/R2/CR}
+\end{figure}
+
+
+%\item {{\bf Network Lifetime}}
+\subsubsection{Network lifetime}
+
+Finally, we study the effect of increasing the number of primary points on the lifetime of the network.
+%In Figure~\ref{Figures/ch4/R2/LT95} and in Figure~\ref{Figures/ch4/R2/LT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different network sizes.
+As highlighted by Figures~\ref{Figures/ch4/R2/LT}(a) and
+\ref{Figures/ch4/R2/LT}(b), the network lifetime obviously increases when the
+size of the network increases, with Model-5 which leads to the largest lifetime
+improvement.
+
+\begin{figure}[h!]
+\centering
+\centering
+\includegraphics[scale=0.5]{R2/LT95.pdf}\\~ ~ ~ ~ ~(a) \\
+
+\includegraphics[scale=0.5]{R2/LT50.pdf}\\~ ~ ~ ~ ~(b)
+
+\caption{Network lifetime for (a) $Lifetime_{95}$ and (b) $Lifetime_{50}$}
+ \label{Figures/ch4/R2/LT}
+\end{figure}
+
+Comparison shows that Model-5, which uses less number of primary points, is the
+best one because it is less energy consuming during the network lifetime. It is
+also the better one from the point of view of coverage ratio, as stated
+before. Therefore, we have chosen the model with five primary points for all the
+experiments presented thereafter.
+
+%\end{enumerate}
+
+% MICHEL => TO BE CONTINUED
+
+\subsection{Experimental results and analysis}
\subsubsection{Coverage ratio}
\begin{figure}[ht!]
\centering
- \includegraphics[scale=0.5] {R/CR.pdf}
+ \includegraphics[scale=0.5] {F/CR.pdf}
\caption{Average coverage ratio for 150 deployed nodes}
\label{fig3}
\end{figure}
+\iffalse
+\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.}
+\fi
+
+
\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
-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
-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
-\includegraphics[scale=0.5]{R/ASR.pdf}
+\includegraphics[scale=0.5]{F/ASR.pdf}
\caption{Active sensors ratio for 150 deployed nodes}
\label{fig4}
\end{figure}
+%\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
-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
-\includegraphics[scale=0.5]{R/SR.pdf}
+\includegraphics[scale=0.5]{F/SR.pdf}
\caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
\label{fig6}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tabular}{cl}
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R/EC95.pdf}} & (a) \\
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{F/EC95.pdf}} & (a) \\
\verb+ + \\
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R/EC50.pdf}} & (b)
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{F/EC50.pdf}} & (b)
\end{tabular}
\caption{Energy consumption for (a) $Lifetime_{95}$ and
(b) $Lifetime_{50}$}
The results show that MuDiLCO is the most competitive from the energy
consumption point of view. The other approaches have a high energy consumption
-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.
\subsubsection{Execution time}
-
+\label{et}
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 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
\begin{figure}[ht!]
\centering
-\includegraphics[scale=0.5]{R/T.pdf}
+\includegraphics[scale=0.5]{F/T.pdf}
\caption{Execution Time (in seconds)}
\label{fig77}
\end{figure}
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}
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R/LT95.pdf}} & (a) \\
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{F/LT95.pdf}} & (a) \\
\verb+ + \\
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R/LT50.pdf}} & (b)
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{F/LT50.pdf}} & (b)
\end{tabular}
\caption{Network lifetime for (a) $Lifetime_{95}$ and
(b) $Lifetime_{50}$}
