From: Karine Deschinkel Date: Fri, 3 Jul 2015 12:46:53 +0000 (+0200) Subject: ok ingrid corrections X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/LiCO.git/commitdiff_plain/f728a52ec5b5d3a0b90f03b90777f454b88b21be ok ingrid corrections --- diff --git a/PeCO-EO/articleeo.tex b/PeCO-EO/articleeo.tex index d4ae9d9..9676c99 100644 --- a/PeCO-EO/articleeo.tex +++ b/PeCO-EO/articleeo.tex @@ -36,7 +36,7 @@ distributed among sensor nodes in each subregion. The novelty of our approach lies essentially in the formulation of a new mathematical optimization model based on the perimeter coverage level to schedule sensors' activities. Extensive simulation experiments demonstrate that PeCO can offer longer lifetime -coverage for WSNs in comparison with some other protocols. +coverage for WSNs compared to other protocols. \begin{keywords} Wireless Sensor Networks, Area Coverage, Energy efficiency, Optimization, Scheduling. @@ -48,34 +48,34 @@ coverage for WSNs in comparison with some other protocols. \label{sec:introduction} The continuous progress in Micro Electro-Mechanical Systems (MEMS) and wireless -communication hardware has given rise to the opportunity to use large networks +communication hardware has given rise to the opportunity of using large networks of tiny sensors, called Wireless Sensor Networks (WSN)~\citep{akyildiz2002wireless,puccinelli2005wireless}, to fulfill monitoring tasks. A WSN consists of small low-powered sensors working together by communicating with one another through multi-hop radio communications. Each node can send the data it collects in its environment, thanks to its sensor, to the -user by means of sink nodes. The features of a WSN made it suitable for a wide -range of application in areas such as business, environment, health, industry, +user by means of sink nodes. The features of a WSN makes it suitable for a wide +range of applications in areas such as business, environment, health, industry, military, and so on~\citep{yick2008wireless}. Typically, a sensor node contains three main components~\citep{anastasi2009energy}: a sensing unit able to measure physical, chemical, or biological phenomena observed in the environment; a processing unit which will process and store the collected measurements; a radio -communication unit for data transmission and receiving. +communication unit for data transmission and reception. The energy needed by an active sensor node to perform sensing, processing, and -communication is supplied by a power supply which is a battery. This battery has +communication is provided by a power supply which is a battery. This battery has a limited energy provision and it may be unsuitable or impossible to replace or -recharge it in most applications. Therefore it is necessary to deploy WSN with +recharge in most applications. Therefore it is necessary to deploy WSN with high density in order to increase reliability and to exploit node redundancy thanks to energy-efficient activity scheduling approaches. Indeed, the overlap of sensing areas can be exploited to schedule alternatively some sensors in a low power sleep mode and thus save energy. Overall, the main question that must -be answered is: how to extend the lifetime coverage of a WSN as long as possible +be answered is: how is it possible to extend the lifetime coverage of a WSN as long as possible while ensuring a high level of coverage? These past few years many energy-efficient mechanisms have been suggested to retain energy and extend the lifetime of the WSNs~\citep{rault2014energy}. -This paper makes the following contributions. +This paper makes the following contributions : \begin{enumerate} \item A framework is devised to schedule nodes to be activated alternatively such that the network lifetime is prolonged while ensuring that a certain @@ -88,7 +88,7 @@ This paper makes the following contributions. architecture. \item A new mathematical optimization model is proposed. Instead of trying to cover a set of specified points/targets as in most of the methods proposed in - the literature, we formulate an integer program based on perimeter coverage of + the literature, we formulate a mixed-integer program based on the perimeter coverage of each sensor. The model involves integer variables to capture the deviations between the actual level of coverage and the required level. Hence, an optimal schedule will be obtained by minimizing a weighted sum of these @@ -125,9 +125,9 @@ to the objective of coverage for a finite number of discrete points called targets, and barrier coverage~\citep{HeShibo,kim2013maximum} focuses on preventing intruders from entering into the region of interest. In \citep{Deng2012} authors transform the area coverage problem into the target -coverage one taking into account the intersection points among disks of sensors -nodes or between disk of sensor nodes and boundaries. In -\citep{Huang:2003:CPW:941350.941367} authors prove that if the perimeters of +coverage one, taking into account the intersection points among disks of sensors +nodes or between disks of sensor nodes and boundaries. In +\citep{Huang:2003:CPW:941350.941367} authors prove that if the perimeters of the sensors are sufficiently covered it will be the case for the whole area. They provide an algorithm in $O(nd~log~d)$ time to compute the perimeter-coverage of each sensor. $d$ denotes the maximum number of sensors that are neighbors to a @@ -139,7 +139,7 @@ sensor, and $n$ is the total number of sensors in the network. {\it In PeCO The major approach to extend network lifetime while preserving coverage is to divide/organize the sensors into a suitable number of set covers (disjoint or non-disjoint) \citep{wang2011coverage}, where each set completely covers a -region of interest, and to activate these set covers successively. The network +region of interest, and to successively activate these set covers. The network activity can be planned in advance and scheduled for the entire network lifetime or organized in periods, and the set of active sensor nodes decided at the beginning of each period \citep{ling2009energy}. In fact, many authors propose @@ -162,20 +162,20 @@ algorithms~\citep{ChinhVu,qu2013distributed,yangnovel} each sensor decides of its own activity scheduling after an information exchange with its neighbors. The main interest of such an approach is to avoid long range communications and thus to reduce the energy dedicated to the communications. Unfortunately, since -each node has only information on its immediate neighbors (usually the one-hop -ones) it may make a bad decision leading to a global suboptimal solution. +each node has information on its immediate neighbors only (usually the one-hop +ones), it may make a bad decision leading to a global suboptimal solution. Conversely, centralized algorithms~\citep{cardei2005improving,zorbas2010solving,pujari2011high} always -provide nearly or close to optimal solution since the algorithm has a global +provide nearly optimal solutions since the algorithm has a global view of the whole network. The disadvantage of a centralized method is obviously its high cost in communications needed to transmit to a single node, the base station which will globally schedule nodes' activities, data from all the other sensor nodes in the area. The price in communications can be huge since long range communications will be needed. In fact the larger the WSN, the higher the communication energy cost. {\it In order to be suitable for large-scale - networks, in PeCO protocol the area of interest is divided into several + networks, in the PeCO protocol the area of interest is divided into several smaller subregions, and in each one, a node called the leader is in charge of - selecting the active sensors for the current period. Thus PeCO protocol is + selecting the active sensors for the current period. Thus the PeCO protocol is scalable and a globally distributed method, whereas it is centralized in each subregion.} @@ -195,14 +195,14 @@ practiced techniques for solving linear programs with too many variables, have also been used~\citep{castano2013column,doi:10.1080/0305215X.2012.687732,deschinkel2012column}. {\it In the PeCO protocol, each leader, in charge of a subregion, solves an - integer program which has a twofold objective: minimize the overcoverage and + integer program which has a twofold objective: minimizing the overcoverage and the undercoverage of the perimeter of each sensor.} The authors in \citep{Idrees2} propose a Distributed Lifetime Coverage Optimization (DiLCO) protocol, which maintains the coverage and improves the lifetime in WSNs. It is an improved version of a research work presented in~\citep{idrees2014coverage}. First, the area of interest is partitioned into -subregions using a divide-and-conquer method. DiLCO protocol is then distributed +subregions using a divide-and-conquer method. The DiLCO protocol is then distributed on the sensor nodes in each subregion in a second step. Hence this protocol combines two techniques: a leader election in each subregion, followed by an optimization-based node activity scheduling performed by each elected @@ -211,8 +211,8 @@ decomposed into 4 phases: information exchange, leader election, decision, and sensing. The simulations show that DiLCO is able to increase the WSN lifetime and provides improved coverage performance. {\it In the PeCO protocol, a new mathematical optimization model is proposed. Instead of trying to cover a set - of specified points/targets as in DiLCO protocol, we formulate an integer - program based on perimeter coverage of each sensor. The model involves integer + of specified points/targets as in the DiLCO protocol, we formulate an integer + program based on the perimeter coverage of each sensor. The model involves integer variables to capture the deviations between the actual level of coverage and the required level. The idea is that an optimal scheduling will be obtained by minimizing a weighted sum of these deviations.} @@ -280,7 +280,7 @@ west side of sensor~$u$, with the following respective coordinates in the sensing area~: $(v_x,v_y)$ and $(u_x,u_y)$. From the previous coordinates the euclidean distance between nodes~$u$ and $v$ is computed as follows: $$ - Dist(u,v)=\sqrt{\vert u_x - v_x \vert^2 + \vert u_y-v_y \vert^2}, + Dist(u,v)=\sqrt{(u_x - v_x)^2 + (u_y-v_y)^2}, $$ while the angle~$\alpha$ is obtained through the formula: \[ @@ -342,7 +342,7 @@ above is thus given by the sixth line of the table. \end{table} In the PeCO protocol, the scheduling of the sensor nodes' activities is -formulated with an mixed-integer program based on coverage +formulated with a mixed-integer program based on coverage intervals~\citep{doi:10.1155/2010/926075}. The formulation of the coverage optimization problem is detailed in~Section~\ref{cp}. Note that when a sensor node has a part of its sensing range outside the WSN sensing field, as in @@ -396,13 +396,13 @@ of the application. \label{figure4} \end{figure} -We define two types of packets to be used by PeCO protocol: +We define two types of packets to be used by the PeCO protocol: \begin{itemize} \item INFO packet: sent by each sensor node to all the nodes inside a same subregion for information exchange. \item ActiveSleep packet: sent by the leader to all the nodes in its subregion to transmit to them their respective status (stay Active or go Sleep) during - sensing phase. + the sensing phase. \end{itemize} Five statuses are possible for a sensor node in the network: @@ -511,10 +511,10 @@ criteria for the leader are (in order of priority): \begin{enumerate} \item larger number of neighbors; \item larger remaining energy; -\item and then in case of equality, larger index. +\item and then, in case of equality, larger indexes. \end{enumerate} Once chosen, the leader collects information to formulate and solve the integer -program which allows to construct the set of active sensors in the sensing +program which allows to build the set of active sensors in the sensing stage. \section{Perimeter-based Coverage Problem Formulation} @@ -536,7 +536,7 @@ First, the following sets: \item $I_j$ designates the set of coverage intervals (CI) obtained for sensor~$j$. \end{itemize} -$I_j$ refers to the set of coverage intervals which have been defined according +$I_j$ refers to the set of coverage intervals which has been defined according to the method introduced in Subsection~\ref{CI}. For a coverage interval $i$, let $a^j_{ik}$ denote the indicator function of whether sensor~$k$ is involved in coverage interval~$i$ of sensor~$j$, that is: @@ -610,10 +610,10 @@ $V_{i}^{j}=l^{i}-l$. $\alpha^j_i$ and $\beta^j_i$ are nonnegative weights selected according to the relative importance of satisfying the associated level of coverage. For example, -weights associated with coverage intervals of a specified part of a region may +weights associated with coverage intervals of the specified part of a region may be given by a relatively larger magnitude than weights associated with another region. This kind of mixed-integer program is inspired from the model developed -for brachytherapy treatment planning for optimizing dose distribution +for brachytherapy treatment planning to optimize dose distribution \citep{0031-9155-44-1-012}. The choice of the values for variables $\alpha$ and $\beta$ should be made according to the needs of the application. $\alpha$ should be large enough to prevent undercoverage and so to reach the highest @@ -667,10 +667,10 @@ coverage task. This value corresponds to the energy needed by the sensing phase, obtained by multiplying the energy consumed in the active state (9.72 mW) with the time in seconds for one period (3600 seconds), and adding the energy for the pre-sensing phases. According to the interval of initial energy, a sensor may -be active during at most 20 periods. Information exchange to update the coverage +be active during at most 20 periods. the information exchange to update the coverage is executed every hour, but the length of the sensing period could be reduced -and adapted dynamically. On the one hand a small sensing period would allow to -be more reliable but would have result in higher communication costs. On the +and adapted dynamically. On the one hand a small sensing period would allow the network to +be more reliable but would have resulted in higher communication costs. On the other hand the choice of a long duration may cause problems in case of nodes failure during the sensing period. @@ -681,7 +681,7 @@ so as to prevent the non-coverage for the interval~$i$ of the sensor~$j$. On the other hand, $\beta^j_i$ is assigned to a value which is slightly lower so as to minimize the number of active sensor nodes which contribute in covering the interval. Subsection~\ref{sec:Impact} investigates more deeply how the values of -both parameters affect the performance of PeCO protocol. +both parameters affect the performance of the PeCO protocol. The following performance metrics are used to evaluate the efficiency of the approach. @@ -741,7 +741,7 @@ approach. \subsection{Simulation Results} In order to assess and analyze the performance of our protocol we have -implemented PeCO protocol in OMNeT++~\citep{varga} simulator. The simulations +implemented the PeCO protocol in OMNeT++~\citep{varga} simulator. The simulations were run on a DELL laptop with an Intel Core~i3~2370~M (1.8~GHz) processor (2 cores) whose MIPS (Million Instructions Per Second) rate is equal to 35330. To be consistent with the use of a sensor node based on Atmels AVR ATmega103L @@ -789,15 +789,15 @@ GAF~\citep{xu2001geography}, consists in dividing the monitoring area into fixed squares. Then, during the decision phase, in each square, one sensor is chosen to remain active during the sensing phase. The last one, the DiLCO protocol~\citep{Idrees2}, is an improved version of a research work we presented -in~\citep{idrees2014coverage}. Let us notice that PeCO and DiLCO protocols are +in~\citep{idrees2014coverage}. Let us notice that the PeCO and DiLCO protocols are based on the same framework. In particular, the choice for the simulations of a partitioning in 16~subregions was made because it corresponds to the configuration producing the best results for DiLCO. Of course, this number of subregions should be adapted according to the size of the area of interest and the number of sensors. The protocols are distinguished from one another by the formulation of the integer program providing the set of sensors which have to be -activated in each sensing phase. DiLCO protocol tries to satisfy the coverage of -a set of primary points, whereas PeCO protocol objective is to reach a desired +activated in each sensing phase. The DiLCO protocol tries to satisfy the coverage of +a set of primary points, whereas the objective of the PeCO protocol is to reach a desired level of coverage for each sensor perimeter. In our experimentations, we chose a level of coverage equal to one ($l=1$). @@ -807,8 +807,8 @@ Figure~\ref{figure5} shows the average coverage ratio for 200 deployed nodes obtained with the four protocols. DESK, GAF, and DiLCO provide a slightly better coverage ratio with respectively 99.99\%, 99.91\%, and 99.02\%, compared to the 98.76\% produced by PeCO for the first periods. This is due to the fact that at -the beginning DiLCO and PeCO protocols put to sleep status more redundant -sensors (which slightly decreases the coverage ratio), while the two other +the beginning the DiLCO and PeCO protocols put more redundant +sensors to sleep status (which slightly decreases the coverage ratio), while the two other protocols activate more sensor nodes. Later, when the number of periods is beyond~70, it clearly appears that PeCO provides a better coverage ratio and keeps a coverage ratio greater than 50\% for longer periods (15 more compared to @@ -825,13 +825,13 @@ allows later a substantial increase of the coverage performance. \subsubsection{Active Sensors Ratio} -Having the less active sensor nodes in each period is essential to minimize the +Minimizing the number of active sensor nodes in each period is essential to minimize the energy consumption and thus to maximize the network lifetime. Figure~\ref{figure6} shows the average active nodes ratio for 200 deployed nodes. We observe that DESK and GAF have 30.36~\% and 34.96~\% active nodes for -the first fourteen rounds, and DiLCO and PeCO protocols compete perfectly with +the first fourteen rounds, and the DiLCO and PeCO protocols compete perfectly with only 17.92~\% and 20.16~\% active nodes during the same time interval. As the -number of periods increases, PeCO protocol has a lower number of active nodes in +number of periods increases, the PeCO protocol has a lower number of active nodes in comparison with the three other approaches and exhibits a slow decrease, while keeping a greater coverage ratio as shown in Figure \ref{figure5}. @@ -848,13 +848,13 @@ The effect of the energy consumed by the WSN during the communication, computation, listening, active, and sleep status is studied for different network densities and the four approaches compared. Figures~\ref{figure7}(a) and (b) illustrate the energy consumption for different network sizes and for -$Lifetime95$ and $Lifetime50$. The results show that PeCO protocol is the most +$Lifetime95$ and $Lifetime50$. The results show that the PeCO protocol is the most competitive from the energy consumption point of view. As shown by both figures, PeCO consumes much less energy than the other methods. One might think that the resolution of the integer program is too costly in energy, but the results show that it is very beneficial to lose a bit of time in the selection of sensors to activate. Indeed the optimization program allows to reduce significantly the -number of active sensors and so the energy consumption while keeping a good +number of active sensors and also the energy consumption while keeping a good coverage level. Let us notice that the energy overhead when increasing network size is the lowest with PeCO. @@ -870,14 +870,14 @@ size is the lowest with PeCO. \subsubsection{Network Lifetime} -We observe the superiority of both PeCO and DiLCO protocols in comparison with +We observe the superiority of both the PeCO and DiLCO protocols in comparison with the two other approaches in prolonging the network lifetime. In Figures~\ref{figure8}(a) and (b), $Lifetime95$ and $Lifetime50$ are shown for different network sizes. As can be seen in these figures, the lifetime -increases with the size of the network, and it is clearly largest for DiLCO and +increases with the size of the network, and it is clearly larger for the DiLCO and PeCO protocols. For instance, for a network of 300~sensors and coverage ratio greater than 50\%, we can see on Figure~\ref{figure8}(b) that the lifetime is -about twice longer with PeCO compared to DESK protocol. The performance +about twice longer with PeCO compared to the DESK protocol. The performance difference is more obvious in Figure~\ref{figure8}(b) than in Figure~\ref{figure8}(a) because the gain induced by our protocols increases with time, and the lifetime with a coverage over 50\% is far longer than with 95\%. @@ -892,7 +892,7 @@ time, and the lifetime with a coverage over 50\% is far longer than with 95\%. \label{figure8} \end{figure} -Figure~\ref{figure9} compares the lifetime coverage of DiLCO and PeCO protocols +Figure~\ref{figure9} compares the lifetime coverage of the DiLCO and PeCO protocols for different coverage ratios. We denote by Protocol/50, Protocol/80, Protocol/85, Protocol/90, and Protocol/95 the amount of time during which the network can satisfy an area coverage greater than $50\%$, $80\%$, $85\%$, @@ -916,13 +916,13 @@ sizes. Table~\ref{my-labelx} shows network lifetime results for different values of $\alpha$ and $\beta$, and a network size equal to 200 sensor nodes. On the one -hand, the choice of $\beta \gg \alpha$ prevents the overcoverage, and so limit +hand, the choice of $\beta \gg \alpha$ prevents the overcoverage, and also limits the activation of a large number of sensors, but as $\alpha$ is low, some areas may be poorly covered. This explains the results obtained for {\it Lifetime50} with $\beta \gg \alpha$: a large number of periods with low coverage ratio. On the other hand, when we choose $\alpha \gg \beta$, we favor the coverage even if -some areas may be overcovered, so high coverage ratio is reached, but a large -number of sensors are activated to achieve this goal. Therefore network +some areas may be overcovered, so ahigh coverage ratio is reached, but a large +number of sensors are activated to achieve this goal. Therefore the network lifetime is reduced. The choice $\alpha=0.6$ and $\beta=0.4$ seems to achieve the best compromise between lifetime and coverage ratio. That explains why we have chosen this setting for the experiments presented in the previous @@ -973,7 +973,7 @@ lifetime, coverage ratio, active sensors ratio, and energy consumption. We plan to extend our framework so that the schedules are planned for multiple sensing periods. We also want to improve the integer program to take into account heterogeneous sensors from both energy and node characteristics point of -views. Finally, it would be interesting to implement PeCO protocol using a +views. Finally, it would be interesting to implement the PeCO protocol using a sensor-testbed to evaluate it in real world applications. diff --git a/PeCO-EO/reponse.tex b/PeCO-EO/reponse.tex index 9d11094..1fb3f07 100644 --- a/PeCO-EO/reponse.tex +++ b/PeCO-EO/reponse.tex @@ -73,7 +73,7 @@ methodology uses existing methods and the original contribution lies only in the application of these methods for the coverage scheduling problem.\\ \textcolor{blue}{\textbf{\textsc{Answer:} To the best of our knowledge, no - integer linear programming based on perimeter coverage has been already + integer linear programming based on perimeter coverage has ever been proposed in the literature. As specified in the paper, in Section 4, it is inspired from a model developed for brachytherapy treatment planning for optimizing dose distribution. In this model the deviation between an actual @@ -86,9 +86,9 @@ application of these methods for the coverage scheduling problem.\\ assumption made on the selection criteria for the leader seems too vague. \\ \textcolor{blue}{\textbf{\textsc{Answer:} The selection criteria for the leader - inside each subregion is explained in page~9, at the end of Section~3.3 - After information exchange among the sensor nodes in the subregion, each - node will have all the information needed to decide if it will the leader or + inside each subregion is explained page~9, at the end of Section~3.3 + After the information exchange among the sensor nodes in the subregion, each + node will have all the information needed to decide if it will be the leader or not. The decision is based on selecting the sensor node that has the larger number of one-hop neighbors. If this value is the same for many sensors, the node that has the largest remaining energy will be selected as a leader. If @@ -139,9 +139,9 @@ results showing how the algorithm performs with different alphas and betas.\\ for alpha and beta. Table 4 presents the results obtained for a WSN of 200~sensor nodes. It explains the value chosen for the simulation settings in Table~2. \\ \indent The choice of alpha and beta should be made according - to the needs of the application. Alpha should be enough large to prevent - undercoverage and so to reach the highest possible coverage ratio. Beta - should be enough large to prevent overcoverage and so to activate a minimum + to the needs of the application. Alpha should be large enough to prevent + undercoverage and thus to reach the highest possible coverage ratio. Beta + should be enough large to prevent overcoverage and thus to activate a minimum number of sensors. The values of $\alpha_{i}^{j}$ can be identical for all coverage intervals $i$ of one sensor $j$ in order to express that the perimeter of each sensor should be uniformly covered, but $\alpha_{i}^{j}$ @@ -151,9 +151,9 @@ results showing how the algorithm performs with different alphas and betas.\\ as $\alpha$ is low, some areas may be poorly covered. This explains the results obtained for $Lifetime_{50}$ with $\beta \gg \alpha$: a large number of periods with low coverage ratio. With $\alpha \gg \beta$, we favor the - coverage even if some areas may be overcovered, so high coverage ratio is + coverage even if some areas may be overcovered, so a high coverage ratio is reached, but a large number of sensors are activated to achieve this goal. - Therefore network lifetime is reduced. The choice $\alpha=0.6$ and + Therefore the network lifetime is reduced. The choice $\alpha=0.6$ and $\beta=0.4$ seems to achieve the best compromise between lifetime and coverage ratio.}}\\ @@ -192,10 +192,10 @@ may be an issue if this approach is used in an application that requires high coverage ratio. \\ \textcolor{blue}{\textbf{\textsc{Answer:} Your remark is very interesting. Indeed, - Figures 8(a) and (b) highlight this result. PeCO protocol allows to achieve + Figures 8(a) and (b) highlight this result. The PeCO protocol allows to achieve a coverage ratio greater than $50\%$ for far more periods than the others three methods, but for applications requiring a high level of coverage - (greater than $95\%$), DiLCO method is more efficient. It is explained at + (greater than $95\%$), the DiLCO method is more efficient. It is explained at the end of Section 5.2.4.}}\\ %%%%%%%%%%%%%%%%%%%%%% ENGLISH and GRAMMAR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -283,7 +283,7 @@ how should this common duration should be chosen?\\ do not have the same Quality of Service requirements. In our case, information exchange is executed every hour, but the length of the sensing period could be reduced and adapted dynamically. On the one hand, a small - sensing period would allow to be more reliable but would have higher + sensing period would allow the network to be more reliable but would have higher communication costs. On the other hand, the choice of a long duration may cause problems in case of nodes failure during the sensing period. Several explanations on these points are given throughout the paper. In @@ -320,8 +320,7 @@ and explain how the protocol is built to optimize these objectives. \\ \textcolor{blue}{\textbf{\textsc{Answer:} Right. The mixed Integer Linear Program adresses a multiobjective problem, where the goal is to minimize - overcoverage and undercoverage for each coverage interval of a sensor. As - far as we know, representing the objective function as a weighted sum of + overcoverage and undercoverage for each coverage interval of a sensor. To the best of our knowledge, representing the objective function as a weighted sum of criteria to be minimized in case of multicriteria optimization is a classical method. In Section 5, the comparison of protocols with a large variety of performance metrics allows to select the most appropriate method @@ -387,7 +386,7 @@ of nodes &&&&relaxation &B\&B tree &\\ \medskip \\ It is noteworthy that the difference of memory used with GLPK between the resolution of the IP and its LP-relaxation is very weak (not more than 0.1 -MB). The size of the branch and bound tree dos not exceed 3 nodes. This result +MB). The size of the branch and bound tree does not exceed 3 nodes. This result leads one to believe that the memory use with CPLEX\textregistered for solving the IP would be very close to that for the LP-relaxation, that is to say around 100 Kb for a subregion containing $S=10$ sensors. Moreover the IP seems to have @@ -399,7 +398,7 @@ Optimization, issn 1572-5286). \item the subdivision of the region of interest. To make the resolution of integer programming tractable by a leader sensor, we need to limit the number of nodes in each subregion (the number of variables and constraints of the - integer programming is directly depending on the number of nodes and + integer programming directly depends on the number of nodes and neigbors). It is therefore necessary to adapt the subdvision according to the number of sensors deployed in the area and their sensing range (impact on the number of coverage intervals). @@ -487,8 +486,8 @@ A discussion about memory consumption has been added in Section 5.2}} \textcolor{blue}{\textbf{\textsc{Answer:} For minimizing the objective function, $M_{i}^{j}$ and $V_{i}^{j}$ should be set to the smallest possible value - such that the inequalities are satisfied. It is explained in the answer 4 - for the reviewer 1. But, at optimality, constraints are not necessary + such that the inequalities are satisfied. It is explained in answer 4 + for reviewer 1. But, at optimality, constraints are not necessary satisfied with equality. For instance, if a sensor $j$ is overcovered, there exists at least one of its coverage interval (say $i$) for which the number of active sensors (denoted by $l^{i}$) covering this part of the perimeter