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%title{Efficient heuristic building disjoint cover sets \\for target coverage problem in wireless sensor networks}
\title{ in wireless sensor networks}
\author{\authorname{Ali Khadum\sup{1},Karine Deschinkel\sup{1},Michel Salomon\sup{1}, Raphaël Couturier \sup{1}}
\affiliation{\sup{1}FEMTO-ST Institute, UMR CNRS, University of Franche-Comte, Belfort, France}
\email{\{ali.khadum, karine.deschinkel, michel.salomon, raphael.couturier\}@univ-fcomte.fr}
}
\author{\authorname{~~}
\affiliation{~~~~~}
\email{~~~~~~~}
}

\keywords{area coverage, wireless sensor networks, lifetime optimization.}
%\vspace{3cm}
\abstract{}

\onecolumn \maketitle \normalsize 
%\vspace{3cm}
\section{\uppercase{Introduction}}
\label{sec:introduction}
%\vspace{1cm}
\noindent Recent years have witnessed significant advances in wireless sensor
networks which emerge as one of the most promising technologies for
the 21st century~\cite{asc02}. In fact, they present huge potential in
several domains ranging from health care applications to military
applications. 
A sensor network is composed of a large number of tiny sensing devices deployed in a region of interest. Each device has processing and wireless communication capabilities, which enable to sense its environment, to compute, to store information and to deliver report messages to a base station. 
%These sensor nodes run on batteries with limited capacities. To achieve a long life of the network, it is important to conserve battery power. Therefore, lifetime optimisation is one of the most critical issues in wireless sensor networks.
One of the main design challenges in Wireless Sensor Networks (WSN) is to prolong the system lifetime, while achieving acceptable quality of service for applications. Indeed, sensor nodes
have limited resources in terms of memory, energy and computational powers. 

%\medskip
Since sensor nodes have limited battery life and without being able to replace 
batteries, especially in remote and hostile environments,
it is desirable that a WSN should be deployed with 
high density and thus redundancy can be exploited to increase 
the lifetime of the network. In such a high density network, if all sensor nodes 
were to be activated at the same time, the lifetime would be reduced. Consequently, 
future software may need to adapt appropriately to achieve acceptable quality of service for applications.  
In this paper we concentrate on area coverage problem, with the objective of maximizing the network lifetime by using an adaptive scheduling. Area of interest is divided into subregions and an activity scheduling for sensor nodes is planned for each subregion.
Our scheduling scheme works in period which includes a discovery phase to exchange information between sensors of the subregion, then a sensor is chosen in suitable manner to carry out a coverage strategy.  This coverage strategy involves the resolution of an integer program which provides the activation of the sensors for the $T$ next rounds, where $T$ is a parameter to adjust in efficient way.


The remainder  of the paper is  organized as follows.
Section~\ref{rw} reviews the related work in the field. 
Section \ref{pd} is devoted to the scheduling strategy for energy-efficient coverage. 
Section \ref{cp} gives the coverage model formulation which is used to schedule the activation of sensors.
Section \ref{exp} shows the simulation results conducted on OMNET++, that fully demonstrate the usefulness of the proposed approach. Finally, we give concluding remarks in Section~\ref{sec:conclusion}.

\section{\uppercase{Related work}}
\label{rw}
\noindent 
This section is dedicated to the various approaches proposed in the literature for the coverage lifetime maximization problem where the objective is to optimally schedule sensors'activities in order to extend network lifetime  in a randomly deployed network. As this problem is subject to a wide range of interpretations, we suggest to recall main definitions and assumptions related to our work. 

{\bf Coverage}
%\begin{itemize}
%\item Area Coverage: The main objective is to cover an area. The area coverage requires
%that the sensing range of working Active nodes cover the whole targeting area, which means any
%point in target area can be covered~\cite{Mihaela02,Raymond03}.

%\item Target Coverage: The objective is to cover a set of targets. Target coverage means that the discrete target points can be covered in any time. The sensing range of working Active nodes only monitors a finite number of discrete points in targeting area~\cite{Mihaela02,Raymond03}. 

%\item Barrier Coverage An objective to determine the maximal support/breach paths that traverse a sensor field. Barrier coverage is expressed as finding one or more routes with starting position and ending position when the targets pass through the area deployed with sensor nodes~\cite{Santosh04,Ai05}.
%\end{itemize}

The most discussed coverage problems in literature can be classified into two types \cite{} : area coverage and targets coverage. An area coverage problem is to find a minimum number of sensors to work such that each physical point in the area is monitored by at least a working sensor. Target coverage problem is to cover only a finite number of discrete points called targets. 
 Our work will concentrate on the area coverage by design and implement a strategy which efficiently select the active nodes that must maintain both sensing coverage and network connectivity and in the same time improve the lifetime of the wireless sensor network. But requiring that all physical points are covered may be too strict, specially where the sensor network is not dense. 
Our approach represents an area covered by a sensor as a set of principle points and tries to maximize the total number of principles points that are covered in each round, while minimizing overcoverage (points covered by multiple active sensors simultaneously).\\
{\bf Lifetime}\\
Various definitions exist for the lifetime of a sensor network. Main definitions proposed in the literature are related to the remaining energy of the nodes \cite{} or to the percentage of coverage \cite{}. The lifetime of the network is mainly defined as the amount of time that the network can satisfy its coverage objective (the amount of time that the network can cover a given percentage of its area or targets of interest) . In our simulation we assume that the network is alive until all sensor nodes are died and we measure the coverage ratio during the process.

{\bf Activity scheduling}\\
Activity scheduling is to schedule the activation and deactivation of nodes 'sensor units. The basic objective is to decide which sensors are in which states (active or sleeping mode) and for how long a time such that the application coverage requirement can be guaranteed and network lifetime can be prolonged. Various approaches, including centralized, distributed and localized algorithms, have been proposed for activity scheduling.  In the distributed algorithms, each node in the network autonomously makes decisions on whether to turn on or turn off itself only using local neighbor information. In centralized algorithms, a central controller (node or base station) informs every sensor of the time intervals to be activated.

{\bf Distributed approaches}

Some distributed algorithms have been developed in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02}. Distributed algorithms typically operate in roundsf predetermined duration. At the beginning of each round, a sensor exchange information with its neighbors and makes a decision to either turn on or go to sleep for the round. This decision is basically based on simple greedy criteria like the largest uncovered area \cite{Berman05efficientenergy}, maximum uncovered targets \cite{1240799}. 
In \cite{Tian02}, the sheduling scheme is divided into rounds, where each round has a self-scheduling phase followed by a sensing phase. Each sensor broadcasts  a message to its neighbors containing node ID and node location at the beginning of each round. Sensor determines its status by a rule named off-duty eligible rule which tells him to turn off if its sensing area is covered by its neighbors. A back-off scheme is introduced to let each sensor delay the decision process with a random period of time, in order to avoid that nodes make conflicting decisions simultaneously and that a part of the area is no longer covered.
\cite{Prasad:2007:DAL:1782174.1782218} propose a model for capturing the dependencies between different cover sets and propose localized heuristic based on this dependency. The algorithm consists of two phases, an initial setup phase during which each sensor calculates and prioritize the covers and a sensing phase  during which each sensor first decides its on/off status and then remains on or off  for the rest of the duration. 
Authors in \cite{chin2007} propose a novel distributed heuristic named distributed Energy-efficient Scheduling for k-coverage (DESK) so that the energy consumption among all the sensors is balanced, and network lifetime is maximized while the coverage requirements being maintained. This algorithm works in round, requires only 1-sensing-hop-neigbor information, and a sensor decides its status (active/sleep) based on its perimeter coverage computed through the k-Non-Unit-disk coverage algorithm  proposed in \cite{Huang:2003:CPW:941350.941367}.\\

Some others approaches  do not consider synchronized and predetermined period of time where the sensors are active or not. Each sensor maintains its own timer and its time wake-up is randomized \cite{Ye03} or regulated \cite{cardei05} over time. 
%A ecrire \cite{Abrams:2004:SKA:984622.984684}p33


%The scheduling information is disseminated throughout the network and only sensors in the active state are responsible
%for monitoring all targets, while all other nodes are in a low-energy sleep mode. The nodes decide cooperatively which of them will remain in sleep mode for a certain
%period of time.

 %one way of increasing lifeteime is by turning off redundant nodes to sleep mode to conserve energy while active nodes provide essential coverage, which improves fault tolerance. 

%In this paper we focus on centralized algorithms because distributed algorithms are outside the scope of our work. Note that centralized coverage algorithms have the advantage of requiring very low processing power from the sensor nodes which have usually limited processing capabilities. Moreover, a recent study conducted in \cite{pc10} concludes that there is a threshold in terms of network size to switch from a localized to a centralized algorithm. Indeed the exchange of messages in large networks may consume  a considerable amount of energy in a localized approach compared to a centralized one. 
{\bf Centralized approaches}\\
Power efficient centralized schemes differ according to several criteria \cite{Cardei:2006:ECP:1646656.1646898}, such as the coverage objective (target coverage or area coverage), the node deployment method (random or deterministic) and the heterogeneity of sensor nodes (common sensing range, common battery lifetime). The major approach is to divide/organize the sensors into a suitable number of set covers where each set completely covers an interest region and to activate these set covers successively. 

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. For instance Slijepcevic and Potkonjak \cite{Slijepcevic01powerefficient} propose an algorithm which allocates sensor nodes in mutually independent sets to monitor an area divided into several fields. Their algorithm constructs a cover set by including in priority the sensor nodes which cover critical fields, that is to say fields that are covered by the smallest number of sensors. The time complexity of their heuristic is $O(n^2)$ where $n$ is the number of sensors.  ~\cite{cardei02} present
a graph coloring technique to achieve energy savings 
by organizing the sensor nodes into a maximum number of disjoint 
dominating sets which are activated successively. The dominating 
sets do not guarantee the coverage of the whole region of interest. 
Abrams et al.\cite{Abrams:2004:SKA:984622.984684} design three approximation algorithms for a variation of the set k-cover problem, where the objective is
to partition the sensors into covers such that the number of
covers that include an area, summed over all areas, is maximized. Their work builds upon previous work in~\cite{Slijepcevic01powerefficient} and the generated cover sets do not provide complete coverage of the monitoring zone. 


%examine the target coverage problem by disjoint cover sets but relax the requirement that every  cover set monitor all the targets and try to maximize the number of times the targets are covered by the partition. They propose various algorithms and establish approximation ratio.

In~\cite{Cardei:2005:IWS:1160086.1160098}, the authors propose a heuristic to
compute the disjoint set covers (DSC). In order to compute the maximum number of covers, they
first transform DSC into a maximum-flow problem ,
which is then formulated as a mixed integer programming problem
(MIP). Based on the solution of the MIP, they design a heuristic
to compute the final number of covers. The results show a slight performance
improvement in terms of the number of produced DSC in comparison to~\cite{Slijepcevic01powerefficient} but it incurs
higher execution time due to the complexity of the mixed integer programming resolution.
 %Cardei and Du \cite{Cardei:2005:IWS:1160086.1160098} propose a method to efficiently compute the maximum number of disjoint set covers such that each set can monitor all targets. They first transform the problem into a maximum flow problem which is formulated as a mixed integer programming (MIP). Then their heuristic uses the output of the MIP to compute disjoint set covers. Results show that these heuristic provides a number of set covers slightly larger compared to \cite{Slijepcevic01powerefficient} but with a larger execution time due to the complexity of the mixed integer programming resolution.
Zorbas et al. \cite{Zorbas2007} present B\{GOP\}, a centralized coverage algorithm introducing sensor candidate categorisation depending on their coverage status and the notion of critical target to call targets that are associated with a small number of sensors. The total running time of their heuristic is $0(m n^2)$ where $n$ is the number of sensors, and $m$ the number of targets. Compared to algorithm's results of Slijepcevic and Potkonjak \cite{Slijepcevic01powerefficient}, their heuristic produces more cover sets with a slight growth rate in execution time.
%More recently Manju and Pujari\cite{Manju2011}

In the case of non-disjoint algorithms \cite{Manju2011}, 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 incurs a higher order of complexity. Moreover in case of a sensor's failure, non-disjoint scheduling policies are less resilient and less reliable because a sensor may be involved in more than one cover sets. For instance, Cardei et al.~\cite{cardei05bis} present a linear programming (LP) solution
and a greedy approach to extend
the sensor network lifetime by organizing the sensors into a
maximal number of non-disjoint cover sets. Simulation results show that by allowing sensors to 
participate in multiple sets, the network lifetime
increases compared with related work~\cite{Cardei:2005:IWS:1160086.1160098}. In~\cite{berman04}, the authors have formulated the lifetime problem and suggested another (LP) technique to solve this problem. A centralized provably near
optimal solution based on the Garg-K\"{o}nemann algorithm~\cite{garg98} is also proposed.

{\bf Our contribution}
%{decoupage de la region en sous region, selection de noeud leader, formulation %et resolution du probleme de couverture, planification périodique
There are three main questions which should be answered  to build a scheduling strategy. We give a brief answer to these three questions to describe our approach before going into details in the subsequent sections.
\begin{itemize}
\item {\bf How must be planned the 
phases for information exchange, decision and sensing over time?}
Our algorithm partitions the time line into a number of periods. Each period contains 4 phases : information Exchange, Leader Election, Decision, and Sensing. Our work further divides sensing phase into a number of rounds of predetermined length.
\item {\bf What are the rules to decide which node has to turn on or off?}
Our algorithm tends to limit the overcoverage of points of interest to avoid turning on too much sensors covering the same areas at the same time, and tries to prevent undercoverage. The decision is a good compromise between these two conflicting objectives and is made for the next $T$ rounds of sensing. In our experimentations we will check which value of $T$ is the most appropriate.
\item {\bf Which node should make such decision ?}
As mentioned in \cite{pc10}, both centralized and distributed algorithms have their own advantages and disadvantages. Centralized coverage algorithms have the advantage of requiring very low processing power from the sensor nodes which have usually limited processing capabilities. Distributed algorithms are very adaptable to the dynamic and scalable nature of sensors network. Authors in \cite{pc10} concludes that there is a threshold in terms of network size to switch from a localized to a centralized algorithm. Indeed the exchange of messages in large networks may consume  a considerable amount of energy in a localized approach compared to a centralized one. Our work does not consider only one leader to compute and to broadcast the schedule decision to all the sensors. When the size of network increases, the network is divided in many subregions and the decision is made by a leader in each subregion.
\end{itemize}

%compromis entre centralize et distribue (voir article \cite{pc10})

\section{\uppercase{Distributed coverage model}}
\label{pd}
We consider a randomly and uniformly deployed network consisting of static wireless sensors. The wireless sensors are deployed in high density to ensure initially a full coverage of the interested area. We assume that all nodes are homogeneous in terms of energy, communication, and processing capabilities.  The location information is available to the sensor node either through hardware such as embedded GPS or through location discovery algorithms.
The area of interest can be divided using the divide-and-conquer strategy into smaller area called subregions and then our coverage algorithm will be implemented in each subregion simultaneously. Our algorithm works in period fashion as in figure \ref{fig:4}. 
%Given the interested Area $A$, the wireless sensor nodes set $S=\lbrace  s_1,\ldots,s_N \rbrace $ that are deployed randomly and uniformly in this area such that they are ensure a full coverage for A. The Area A is divided into regions $A=\lbrace A^1,\ldots,A^k,\ldots, A^{N_R} \rbrace$. We suppose that each sensor node $s_i$ know its location and its region. We will have a subset $SSET^k =\lbrace s_1,...,s_j,...,s_{N^k} \rbrace $ , where $s_N = s_{N^1} + s_{N^2} +,\ldots,+ s_{N^k} +,\ldots,+s_{N^R}$. Each sensor node $s_i$ has the same initial energy $IE_i$ in the first time and the current residual energy $RE_i$ equal to $IE_i$  in the first time for each $s_i$ in A. \\ 

\begin{figure}[ht!]
\centering
\includegraphics [width=70mm]{Modelgeneral.eps}
\caption{Multi-Round Coverage Protocol}
\label{fig:4}
\end{figure} 

Each period is divided into 4 phases : INFO Exchange, Leader Election, Decision, and Sensing. The sensing phase works also in rounds. For each round there is exactly one set cover responsible for sensing task. If a working node fails unexpectedly in the interval of time of the round, the sensing task of the network will be affected temporarily, since a new set cover will take charge of the sensing task in the next round. The energy consumption and some other constraints can easily be taken into account since the sensors can update and then exchange the information (including their residual energy) at the beginning of each period. However, the preprocessing phase (discovery, leader selection, decision) are energy consuming for some nodes even when they not join the network to monitor the area. We describe each phase in more detail.



\subsection{\textbf Discovery Phase}

Each sensor node $j$  sends its position, remaining energy $RE_j$, number of local neighbours $NBR_j$  to all wireless sensor nodes in its subregion by using INFO packet and listen to the packets sent from other nodes. After that, each node will have information about all the sensor nodes in the subregion. In our model, the remaining energy corresponds to the time that a sensor can live in the active mode.


%\subsection{\textbf Working Phase:}

%The working phase works in rounding fashion. Each round include 3 steps described as follow :

\subsection{\textbf Leader Selection Phase}
This step includes choosing the Wireless Sensor Node Leader (WSNL) which will be responsible of executing coverage algorithm  to choose the list of active sensor nodes that contribute in covering the subregion.
% The sensors in the same region are capable to communicate with each others using a routing protocol provided by the simulator OMNET++ in order to provide multi-hop communication protocol.
The WSNL will be chosen based on the number of local neighbours $NBR_j$  of sensor node $s_j$ and it's remaining energy $RE_j$.
If we have more than one node has the same $NBR_j$ and $RE_j$, this leads to choose WSNL based on the largest index among them. Each subregion in the area of interest will select its WSNL independently for each period. 


\subsection{\textbf Decision Phase}
The WSNL will execute the algorithm  PEC to select which sensors will be activated in the next rounds to cover the subregion. WSNL will send Active-Sleep packet to each sensor in the subregion based on algorithm's results. 
%The main goal in this step after choosing the WSNL is to produce the best representative active nodes set that will take the responsibility of covering the whole region $A^k$ with minimum number of sensor nodes to prolong the lifetime in the wireless sensor network. For our problem, in each round we need to select the minimum set of sensor nodes to improve the lifetime of the network and in the same time taking into account covering the region $A^k$ . We need an optimal solution with tradeoff between our two conflicting objectives.
%The above region coverage problem can be formulated as a Multi-objective optimization problem and we can use the Binary Particle Swarm Optimization technique to solve it. 
\\

\subsection{\textbf Sensing Phase}
This phase can be divided in many rounds. Let be $T$ the number of rounds. $T$ is an adjustable parameter. Active sensors in each round will execute their sensing task.
%The algorithm will produce the best representative set of the active nodes that will take the mission of covering the sub region $A^k$  in Sensing step .    
Since that we use a homogeneous wireless sensor network, we will assume that the cost of keeping a node awake for sensing task is the same for all wireless sensor nodes in the network.  



%\subsection{Sensing coverage model}
%\label{pd}

%\noindent We try to produce an adaptive scheduling which allows sensors to operate alternatively so as to prolong the network lifetime. For convenience, the notations and assumptions are described first.
%The wireless sensor node use the  binary disk sensing model by which each sensor node will has a certain sensing range is reserved within a circular disk called radius $R_s$.
\noindent We consider a boolean disk coverage model which is the most widely used sensor coverage model in the literature. Each sensor has a constant sensing range $R_s$. All space points within a disk centered at the sensor with the radius of the sensing range is said to be covered by this sensor. We also assume that the communication range is at least twice of the sening range. In fact, Zhang and Zhou ~\cite{Zhang05} prove that if the tranmission range is at least twice of the sensing range, a complete coverage of a convex area implies connectivity amnong the working nodes in the active mode.
%To calculate the coverage ratio for the area of interest, we can propose the following coverage model which is called Wireless Sensor Node Area Coverage Model to ensure that all the area within each node sensing range are covered. We can calculate the positions of the points in the circle disc of the sensing range of wireless sensor node based on the Unit Circle in figure~\ref{fig:cluster1}:

%\begin{figure}[h!]
%\centering
%\begin{tabular}{cc}
%%\includegraphics[scale=0.25]{fig1.pdf}\\ %& \includegraphics[scale=0.10]{1.pdf} \\
%%(A) Figure 1 & (B) Figure 2
%\end{tabular}
%\caption{Unit Circle in radians. }
%\label{fig:cluster1}
%\end{figure}

%By using the Unit Circle in figure~\ref{fig:cluster1}, 
%We choose to representEach wireless sensor node will be represented into a selected number of principle points by which we can know if the sensor node is covered or not.
% Figure ~\ref{fig:cluster2} shows the selected principle points that represents the area of the sensor node and according to the sensing range of the wireless sensor node.

\noindent Instead of working with area coverage, we consider for each sensor a set of points called principal points. And we assume the sensing disk defined by a sensor is covered  if all principal points of this sensor are covered.

%\begin{figure}[h!]
%\centering
%\begin{tabular}{cc}
%%\includegraphics[scale=0.25]{fig2.pdf}\\ %& \includegraphics[scale=0.10]{1.pdf} \\
%%(A) Figure 1 & (B) Figure 2
%\end{tabular}
%\caption{Wireless Sensor Node Area Coverage Model.}
%\label{fig:cluster2}
%\end{figure}



\noindent By knowing the position (point center :($p_x,p_y$) of the Wireless sensor node and its $R_s$ , we calculate the principle points directly based on proposed model. We use these principle points (that can be increased or decreased as if it is necessary) as references to ensure that the monitoring area of the region is covered by the selected set of sensors instead of using the all points in the area.

 \begin{figure}[h!]
%\centering
% \begin{multicols}{6}
\centering
%\includegraphics[scale=0.10]{fig21.pdf}\\~ ~ ~(a)
%\includegraphics[scale=0.10]{fig22.pdf}\\~ ~ ~(b)
\includegraphics[scale=0.2]{principles13.eps}
%\includegraphics[scale=0.10]{fig25.pdf}\\~ ~ ~(d)
%\includegraphics[scale=0.10]{fig26.pdf}\\~ ~ ~(e)
%\includegraphics[scale=0.10]{fig27.pdf}\\~ ~ ~(f)
%\end{multicols} 
\caption{Wireless Sensor node represented by 13 principle points }
\label{fig3}
\end{figure}

\noindent We can calculate the positions of the selected principle points in the circle disk of the sensing range of wireless sensor node in figure ~\ref{fig3} as follow:\\
$p_x,p_y$ = 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 * (0)) $\\
$X_7=( p_x + R_s *  (\frac{\sqrt{2}}{2}), p_y + R_s * (0))$\\
$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 * (\frac{\sqrt{2}}{2})) $\\
$X_{11}=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
$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})) $\\




\section{\uppercase{Coverage problem formulation}}
\label{cp}
%We can formulate our optimization problem as energy cost minimization by minimize the number of active sensor nodes and maximizing the coverage rate at the same time in each $A^k$ . This optimization problem can be formulated as follow: Since that we use a homogeneous wireless sensor network, we will assume that the cost of keeping a node awake is the same for all wireless sensor nodes in the network. We can define the decision parameter  $X_j$ as in \eqref{eq11}:\\


%To satisfy the coverage requirement, the set of the principal points that will represent all the sensor nodes in the monitored region as $PSET= \lbrace P_1,\ldots ,P_p, \ldots , P_{N_P^k} \rbrace $, where $N_P^k = N_{sp} * N^k $ and according to the proposed model in figure ~\ref{fig:cluster2}. These points can be used by the wireless sensor node leader which will be chosen in each region in A to build a new parameter $\alpha_{jp}$  that represents the coverage possibility for each principal point $P_p$ of each wireless sensor node $s_j$ in $A^k$ as in \eqref{eq12}:\\


\noindent Our model is based on the model proposed by \cite{pedraza2006} where the objective is to find a maximum number of disjoint cover sets. To accomplish this goal, authors propose a integer program  which forces undercoverage and overcoverage of targets to become minimal at the same time. They use variables $x_{s,l}$ to indicate if the sensor $s$ belongs to cover set $l$. In our model, we consider binary variables $X_{j,t}$ which determine the activation of sensor $j$ in round $t$. We replace the constraint guarantying that each sensor is a member of only one cover of the entire set of disjoint covers by a constraint specifying that the sum of energy consumed by the activation of sensor during several rounds is less than or equal to the remaining energy of the sensor. We also consider principle points as targets.  \\
\noindent For a principle point $p$, let $\alpha_{jp}$ denote the indicator function of whether the point $p$ is covered, that is, \\
\begin{equation}
\alpha_{jp} = \left \{ 
\begin{array}{l l}
  1 & \mbox{if the principal point $p$ is covered} \\
 & \mbox{by active sensor node $j$} \\
  0 & \mbox{Otherwise}\\
\end{array} \right.
%\label{eq12} 
\end{equation}
The number of sensors that are covering point $p$ during a round $t$ is equal to $\sum_{j \in J} \alpha_{jp} * X_{j,t}$ where :
\begin{equation}
X_{j,t} = \left \{ 
\begin{array}{l l}
  1& \mbox{if sensor $s_j$  is active during round } t\\
  0 &  \mbox{Otherwise}\\
\end{array} \right.
%\label{eq11} 
\end{equation}
We define the Overcoverage variable $\Theta_{p,t}$ .\\

\begin{equation}
 \Theta_{p,t} = \left \{ 
\begin{array}{l l}
  0            & \mbox{if point p is not }\\
&\mbox{covered during round } t\\
  \left( \sum_{j \in J} \alpha_{jp} * X_{j,t} \right)- 1 & \mbox{Otherwise}\\
\end{array} \right.
\label{eq13} 
\end{equation}


\noindent$\Theta_{p}$ represents the number of active sensor nodes minus one that cover the principle point $p$.\\
The Undercoverage variable $U_{p,t}$ of the principle point $p$ is defined as follow :\\

\begin{equation}
U_{p,t} = \left \{ 
\begin{array}{l l}
  1 &\mbox{if point } $p$ \mbox{ is not covered during round } $t$\\
  0 & \mbox{Otherwise}\\
\end{array} \right.
\label{eq14} 
\end{equation}

\noindent Our coverage optimization problem can be formulated as follow.\\
\begin{equation} \label{eq:ip2r}
\left \{
\begin{array}{ll}
\min \sum_{p \in P} (w_{\theta,t} \Theta_{p,t} + w_{u,t} U_{p,t})&\\
\textrm{subject to :}&\\
\sum_{j \in J}  \alpha_{jp} X_{j,t} - \Theta_{p,t}+ U_{p,t} =1, &\forall p \in P, \forall t \in T\\
%\label{c1} 
\sum_{t \in T} X_{j,t} \leq \frac{RE_j}{e_t} &\forall j \in J \\
%\label{c2}
\Theta_{p,t}\in \mathbb{N} , &\forall p \in P, \forall t \in T\\
U_{p,t} \in \{0,1\}, &\forall p \in P, \forall t \in T \\
X_{j,t} \in \{0,1\}, &\forall j \in J, \forall t \in T
\end{array}
\right.
\end{equation}
\begin{itemize}
\item $X_{j,t}$ : indicating whether or not sensor $j$ is active in round $t$(1 if yes and 0 if not)
\item $\Theta_{p,t}$ : {\it overcoverage}, the number of sensors minus one that are covering point $p$ in round $t$
\item $U_{p,t}$ : {\it undercoverage}, indicating whether or not point $p$ is being covered (1 if not covered and 0 if covered) in round $t$
\end{itemize}
The first group of constraints indicates that some point $p$ should be covered by at least one sensor in every round $t$ and, if it is not always the case, overcoverage and undercoverage variables help balance the restriction equation by taking positive values. Second group of contraints ensures for each sensor that the amount of energy consumed during its activation periods will be less than or equal to its remaining energy.
There are two main objectives. We limit overcoverage of principle points in order to activate a minimum number of sensors and we prevent that parts of the subregion are not monitored by minimizing undercoverage. The weights $w_{\theta,t}$ and $w_{u,t}$ must be properly chosen so as to guarantee that the maximum number of points are covered during each round. 
 
%In equation \eqref{eq15}, there are two main objectives: the first one using  the Overcoverage parameter to minimize the number of active sensor nodes in the produced final solution vector $X$ which leads to improve the life time of wireless sensor network. The second goal by using the  Undercoverage parameter  to maximize the coverage in the region by means of covering each principle point in $SSET^k$.The two objectives are achieved at the same time. The constraint which represented in equation \eqref{eq16} refer to the coverage function for each principle point $P_p$ in $SSET^k$ , where each $P_p$ should be covered by
%at least one sensor node in $A^k$. The objective function in \eqref{eq15} involving two main objectives to be optimized simultaneously, where optimal decisions need to be taken in the presence of trade-offs between the two conflicting main objectives in \eqref{eq15} and this refer to that our coverage optimization problem is a multi-objective optimization problem and we can use the BPSO to solve it. The concept of Overcoverage and Undercoverage inspired from ~\cite{Fernan12} but we use it with our model as stated in subsection \ref{Sensing Coverage Model} with some modification to be applied later by BPSO.
%\subsection{Notations and assumptions}

%\begin{itemize}
%\item $m$ : the number of targets
%\item $n$ : the number of sensors
%\item $K$ : maximal number of cover sets
%\item $i$ : index of target ($i=1..m$)
%\item $j$ : index of sensor ($j=1..n$)
%\item $k$ : index of cover set ($k=1..K$)
%\item $T_0$ : initial set of targets
%\item $S_0$ : initial set of sensors
%\item $T $ : set of targets which are not covered by at least one cover set
%\item $S$ : set of available sensors
%\item $S_0(i)$ : set of sensors which cover the target $i$
%\item $T_0(j)$ : set of targets covered by sensor $j$
%\item $C_k$ : cover set of index $k$
%\item $T(C_k)$ : set of targets covered by the cover set $k$
%\item $NS(i)$ : set of  available sensors which cover the target $i$
%\item $NC(i)$ : set of cover sets which do not cover the target $i$
%\item $|.|$ : cardinality of the set

%\end{itemize}
\section{\uppercase{Simulation Results}}
\label{exp}
In this section, we evaluate the efficiency of PEC through conducting some simulations measuring the network lifetime with different number of sensors and different number of rounds for the sensing phase.
Coverage ratio measures how much area of a sensor field is covered. In our case, the coverage ratio is regarded as the number of principle points covered among the set of all prinicple points. 
\label{Simulation Results}
\section{\uppercase{Conclusions}}
\label{sec:conclusion}
In this paper, we have addressed the problem of lifetime optimization in wireless sensor networks. This is a very 
natural and important problem, as sensor nodes
have limited resources in terms of memory, energy and computational power.
%energy-efficiency is crucial in power-limited wireless sensor network. 
To cope with this problem, 
%an efficient centralized energy-aware algorithm is presented and analyzed. Our algorithm seeks to 
%Energy-efficiency is crucial in power-limited wireless sensor network, since sensors have significant power constraints (battery life). In this paper we have investigated the problem of 



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