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%\title{Response to the reviewers of \bf "Perimeter-based Coverage Optimization to Improve Lifetime in Wireless Sensor Networks"}
%\author{Ali Kadhum Idrees, Karine Deschinkela, Michel Salomon and Raphael Couturier}

\begin{document}

\begin{flushright}
\today
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\vspace{-0.5cm}\hspace{-2cm}FEMTO-ST Institute, UMR 6714 CNRS

\hspace{-2cm}University Bourgogne Franche-Comt\'e

\hspace{-2cm}IUT Belfort-Montb\'eliard, BP 527, 90016 Belfort Cedex, France.

\bigskip

\begin{center}
Detailed changes and addressed issues in the revision of the article

``Perimeter-based Coverage Optimization \\
to Improve Lifetime in Wireless Sensor Networks''\\

by Ali Kadhum Idrees, Karine Deschinkel, Michel Salomon, and Raph\"ael Couturier

\medskip

\end{center}
Dear Editor and Reviewers,

First of all, we would like to thank you very much for your kind help to improve
our article  named: ``Perimeter-based Coverage Optimization  to Improve Lifetime
in  Wireless  Sensor Networks''.  We  highly  appreciate the  detailed  valuable
comments of the reviewers on our  article. The suggestions are quite helpful for
us and we incorporate them in the revised article. We are happy to submit to you
a revised version that considers most of your remarks and suggestions to improve
the quality of our article.

As below, we  would like to clarify  some of the points raised  by the reviewers
and we hope the reviewers and the  editors will be satisfied by our responses to
the comments and the revision for the original manuscript.

%Journal: Engineering Optimization
%Reviewer's Comment to the Author Manuscript id GENO-2015-0094
%Title: \bf "Perimeter-based Coverage Optimization to Improve Lifetime in Wireless Sensor Networks"
%Authors: Ali Kadhum Idrees, Karine Deschinkela, Michel Salomon and Raphael Couturier

\section*{Response to Reviewer No. 1 Comments}

This paper  proposes a  scheduling technique  for WSN  to maximize  coverage and
network lifetime.  The novelty of this  paper is the integration  of an existing
perimeter  coverage   measure  with  an  existing   integer  linear  programming
model. Here are few comments:\\

\noindent {\bf  1.} The  paper makes  use of  the existing  integer optimization
model  to govern  the state  of  each sensor  node  within the  WSN to  maximize
coverage  and network  lifetime. This  formulation  of the  coverage problem  is
different from the literature in the  sense that they use the perimeter coverage
measures to  optimize coverage  as opposed to  the targets/points  coverage. The
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
    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
    dose  distribution  and  a  required  dose distribution  in  each  organ  is
    minimized. In  WSN the deviations between  the actual level of  coverage and
    the required  level are minimized.  Outside this parallel between  these two
    applications the mathematical formulation is completely different.}}\\

\noindent  {\bf  2.}  The  theory   seems  mathematically  sound.  However,  the
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
    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
    there exists  sensors with the same  number of neighbors and  the same value
    for the remaining energy, the sensor node that has the largest index will be
    finally selected as a leader. }}\\

%{\bf In fact, we gave a high priority to the number of neighbors to reduce the communication energy consumption - PAS CLAIR  }}.\\

\noindent  {\bf  3.}  The  communication and  information  sharing  required  to
cooperate and make these decisions was not discussed.\\

\textcolor{blue}{\textbf{\textsc{Answer:}  The   communication  and  information
    sharing required to  cooperate and make these decisions is  discussed at the
    end of page  8.  Position coordinates, remaining energy, sensor  node ID and
    number of one-hop neighbors are exchanged.}}\\

\noindent  {\bf  4.}  The  definitions  of  the undercoverage  and  overcoverage
variables are not  clear. I suggest adding some information  about these values,
since  without it,  you cannot  understand  how M  and  V are  computed for  the
optimization problem.\\

\textcolor{blue}{\textbf{\textsc{Answer:} The  perimeter of  each sensor  may be
    cut in parts called coverage intervals (CI). The level of coverage of one CI
    is defined as  the number of active sensors neighbors  covering this part of
    the perimeter. If a given level of  coverage $l$ is required for one sensor,
    the  sensor is  said to  be undercovered  (respectively overcovered)  if the
    level  of coverage  of one  of its  CI is  less (respectively  greater) than
    $l$. In  other terms, we  define undercoverage and overcoverage  through the
    use of  variables $M_{i}^{j}$  and $V_{i}^{j}$  for one  sensor $j$  and its
    coverage interval  $i$. If the sensor  $j$ is undercovered, there  exists at
    least  one of  its CI  (say  $i$) for  which  the number  of active  sensors
    (denoted by  $l^{i}$) covering this part  of the perimeter is  less than $l$
    and in this  case : $M_{i}^{j}=l-l^{i}$, $V_{i}^{j}=0$. On  the contrary, if
    the sensor $j$ is overcovered, there exists at least one of its CI (say $i$)
    for which  the number of active  sensors (denoted by $l^{i}$)  covering this
    part of the perimeter is greater than  $l$ and in this case : $M_{i}^{j}=0$,
    $V_{i}^{j}=l^{i}-l$.   This  explanation  has  been  added in the penultimate
    paragraph of Section~4.}}\\

\noindent {\bf  5.} Can you mathematically  justify how you chose  the values of
alpha and  beta? This is  not very clear. I  would suggest possibly  adding more
results showing how the algorithm performs with different alphas and betas.\\

\textcolor{blue}{\textbf{\textsc{Answer:}  To  discuss   this  point,  we  added
    Section  5.2.5  in which  we  study  the protocol  performance,  considering
    $Lifetime_{50}$ and $Lifetime_{95}$ metrics, for different couples of values
    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
    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 $\alpha_{i}^{j}$
    values can  be differentiated between  sensors to  force some regions  to be
    better covered  than others. The choice  of $\beta \gg \alpha$  prevents the
    overcoverage, and so limit 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 $Lifetime_{50}$ 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
    reached, but a  large number of sensors are activated  to achieve this goal.
    Therefore  network  lifetime  is   reduced.   The  choice  $\alpha=0.6$  and
    $\beta=0.4$  seems  to achieve  the  best  compromise between  lifetime  and
    coverage ratio.}}\\

\noindent {\bf  6.} The  authors have  performed a  thorough review  of existing
coverage  methodologies.  However,  the clarity  in the  literature review  is a
little off. Some of the descriptions of the  method s used are very vague and do
not bring out their key contributions.  Some references are not consistent and I
suggest using the journals template to adjust them for overall consistency.\\

\textcolor{blue}{\textbf{\textsc{Answer:} References have been carefully checked
    and seem to be consistent with the journal template. In Section~2, ``Related
    literature'',  we refer  to papers  dealing  with coverage  and lifetime  in
    WSN. Each  paragraph of this Section  discusses the literature related  to a
    particular aspect of the problem : 1. types of coverage, 2. types of scheme,
    3. centralized versus distributed protocols,  4. optimization method. At the
    end of each paragraph we position our approach. We have also added a last paragraph about our previous work on DilCO protocol to explain the difference with PeCO. }}\\

\noindent {\bf 7.} The methodology is implemented in OMNeT++ (network simulator)
and tested  against 2 existing algorithms  and a previously developed  method by
the authors.   The simulation results  are thorough  and show that  the proposed
method improves the  coverage and network lifetime compared with  the 3 existing
methods. The results are similar to previous work done by their team.\\

\textcolor{blue}{\textbf{\textsc{Answer:} Although  the study conducted  in this
    paper reuses the  same protocol presented in our previous  work, we focus in
    this  paper on  the mathematical  optimization model  developed to  schedule
    nodes  activities.   We deliberately  chose  to  keep the  same  performance
    indicators to  compare the results  obtained with this new  formulation with
    other existing algorithms.}}\\

\noindent {\bf 8.}  Since this paper  is attacking the coverage problem, I would
like  to  see more  information  on  the amount  of  coverage  the algorithm  is
achieving. It seems that  there is a tradeoff in this  algorithm that allows the
network to increase  its lifetime but does not improve  the coverage ratio. This
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
    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
    the end of Section 5.2.4.}}\\

%%%%%%%%%%%%%%%%%%%%%%  ENGLISH and GRAMMAR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent\textcolor{black}{\textbf{\Large English and Grammar:}}\\

\noindent {\ding{90} The first paragraph of every Section is not indented.}\\

\textcolor{blue}{\textbf{\textsc{Answer:} Right,  fixed. The first  paragraph of
    every Section is indented in the new version. }}\\

\noindent  {\ding{90} You  seem to  be writing  in the  first person.  I suggest
  rewriting sentences that include “we” “our” or “I” in the third person. (There
  are too many instances to list them  all. They are easily found using the find
  tool.)  }  \\

\textcolor{blue}{\textbf{\textsc{Answer:} It  is very  common to  find sentences
    with "we"  and "our" in  scientific papers to explain  the work made  by the
    authors. Nevertheless  we agree with  the reviewer and we  reformulated some
    sentences in the paper to avoid too many uses of the first person. }}\\

\noindent {\ding{90} Run-on sentence: Page 2 lines 43-48} \\

\textcolor{blue}{\textbf{\textsc{Answer:}  We  rewrote   this  sentence  in  two
    separated sentences.  }}\\

\noindent {\ding{90} Add an “and” after the comma on page 3 line 34.}  \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}\\

\noindent {\ding{90} “model as” instead of “Than” on page 10 line 12.}  \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}\\

\noindent {\ding{90} “no longer” instead of “no more” on page 10 line 31.}  \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}\\

\noindent {\ding{90} “in the active state” add the on page 10 line 34. }  \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}\\

\noindent  {  \ding{90}  Lots  of  English and  grammar  mistakes.  I  recommend
  rereading the paper line by line and  adjusting the sentences that do not make
  sense.} \\

\textcolor{blue}{\textbf{\textsc{Answer:}  The English  of  the  paper has  been
    carefully revised  and the readability  improved.  The new version  has been
    checked by an English teacher.}}\\

\section*{Response to Reviewer No. 2 Comments}

The paper  entitled ``Perimeter-based Coverage Optimization  to Improve Lifetime
in Wireless Sensor  Networks'', by Ali Kadhum Idrees,  Karine Deschinkel, Michel
Salomon, and  Rapha\"el Couturier  proposes a new  protocol for  Wireless Sensor
Networks called PeCO (Perimeter-based  Coverage Optimization protocol) that aims
at optimizing the use of energy  by conjointly exploiting a spatial and temporal
subdivision. The protocol is based on  solving a Mixed Integer Linear Program at
each leader node, and at each iteration of the protocol. The results obtained by
PeCO are compared with three other competitors.\\

\noindent\textcolor{black}{\textbf{MAJOR COMMENTS:}} \\

\noindent {\bf  1.}  The  protocol framework  is not  described in  details.  In
particular, the  spatial and temporal subdivision  (page 2, line 11)  that is at
the  core of  PeCO,  is not  described  nor  justified in  much  detail. How  to
implement an efficient spatial subdivision ? On  page 10, line 11, the number of
subdivisions is said to be equal to 16, but the clustering algorithm used is not
mentioned. Is  this number dependent  of the size of  the sensing area?   Of the
number of sensors? Of the sensing range? The proposed protocol cannot be adopted
by  practitioners  if  such  an  important step  is  not  documented.   Temporal
subdivision suffers  from the  same lack of  description and  justification: why
should time  intervals have the same  duration? If they have  the same duration,
how should this common duration should be chosen?\\

\textcolor{blue}{\textbf{\textsc{Answer:}  Spatial   and  temporal   choices  of
    subdivision are not  the topics of the  paper. In the study,  we assume that
    the deployment  of sensors is  almost uniform over  the region.  So  we only
    need to  fix a regular  division of the region  into subregions to  make the
    problem tractable.   The subdivision is  made such  that the number  of hops
    between  any pairs  of sensors  inside  a subregion  is less  than or  equal
    to~3. Concerning the choice of the sensing period duration, it is correlated
    with the types of applications, with the amount of initial energy in sensors
    batteries and also with the duration of the exchange phase. All applications
    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
    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
    particular,  we discuss  the number  of subregions  in Section  5.2 and  the
    sensing duration in the second paragraph of Section 5.1.}}\\

\noindent {\bf  2.}Page 9,  Section 4, is  the Perimeter-based  coverage problem
NP-hard? This  question is important for  justifying the use of  a Mixed Integer
Linear Programming model.\\

\textcolor{blue}{\textbf{\textsc{Answer:}  The   perimeter  scheduling  coverage
    problem is  NP-hard in  general, it  has been proved  in the  paper entitled
    "Perimeter  Coverage Scheduling  in Wireless  Sensor Networks  Using Sensors
    with a  Single Continuous Cover Range"  from Ka-Shun Hung and  King-Shan Lui
    (EURASIP Journal on Wireless Communications and Networking 2010, 2010:926075
    doi:10.1155/2010/926075). In this  paper, authors study the  coverage of the
    perimeter of  a large object  requiring to be  monitored. In our  study, the
    large object  to be monitored  is the sensor  itself (or more  precisely its
    sensing  area).  This  point  has  been  highlighted  at  the  beginning  of
    Section~4.}}\\

\noindent {\bf 3.}  Page 9, the major  problem with the present paper  is, in my
opinion, the objective  function of the Mixed Integer Linear  Program (2). It is
not  described  in   the  paper,  and  looks  like  an   attempt  to  address  a
multiobjective      problem      (like     minimizing      overcoverage      and
undercoverage).  However, using  a  weighted sum  is  well known  not  to be  an
efficient  way to  address  biobjective problems.  The  introduction of  various
performance  metrics in  Section 5.1  also suggests  that the  authors have  not
decided exactly  which objective  function to use,  and compare  their protocols
against competitors without mentioning the exact purpose of each of them. If the
performance metrics  list given in Section  5.1 is exhaustive, then  the authors
should mention at the beginning of the  paper what are the aims of the protocol,
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
    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
    according to the QoS requirement of the application.}}\\

\noindent {\bf 4.} Page 11 Section 5.2, the sensor nodes are said to be based on
Atmels AVR ATmega103L microcontroller. If I  am not mistaken, these devices have
128 KBytes of memory, and I didn't find  any clue that they can run an operating
system  like  Linux. This  point  is  of  primary  importance for  the  proposed
protocol,  since GLPK  (a C  API)  is supposed  to  be executed  by the  cluster
leader. In addition to that, GLPK requires  a non negligible amount of memory to
run  properly,   and  the  Atmels   AVR  ATmega103L  microcontroller   might  be
insufficient for  that purpose. The authors  are urged to provide  references of
previous works showing that these technical constraints are not preventing their
protocol to be implemented on  the aforementioned microcontroller. Then, on page
13, in Section "5.2.3 Energy Consumption", the estimation of $E_p^{com}$ for the
considered  microcontroller  seems quite  challenging  and  should be  carefully
documented. Indeed, this is  a key point in providing a  fair comparison of PeCO
with its competitors.\\

\textcolor{blue}{\textbf{\textsc{Answer 1:}
To implement PeCO on real sensors nodes with limited memories capacities, we can act on :
\begin{itemize}
\item  the solver  : GLPK  is  memory consuming  for the  resolution of  integer
  programming    (IP)   compared    with   other    commercial   solvers    like
  CPLEX\textregistered.  Commercial  solvers  generally outperform  open  source
  solvers (See  the report : ``Analysis of commercial  and free and  open source
  solvers  for linear  optimization problems"  by B.  Meindl and  M. Templ  from
  Vienna  University  of  Technology).  Memory  use depends  on  the  number  of
  variables and number  of constraints.  For linear programs  (LP), a reasonable
  estimate of memory use with CPLEX \textregistered~is to allow one megabyte per
  thousand constraints.  For integer  programs, no  simple formula  exists since
  memory use depends so heavily on the size of the branch and bound tree (B \& B
  tree). But, the estimate for linear  programs still provides a lower bound. In
  our  case,  the  characteristics  of  the  integer  programming  (2)  are  the
  following:
\begin{itemize}
\item number of variables : $S* (2*I+1)$
\item number of constraints : $2* I *S$
\item number of non-zero coefficients : $2* I *S * B$
\item number of parameters (in the objective function) : $2* I *S$
\end{itemize}
where $S$ denotes the number of sensors in the subregion, $I$ the average number
of cover intervals per  sensor, $B$ the average number of  sensors involved in a
coverage interval. The  following table gives the memory use  with GLPK to solve
the integer  program (column 3) and  its LP-relaxation (column 4)  for different
problem  sizes. The  sixth  column gives  an  estimate of  the  memory use  with
CPLEX\textregistered  to solve  the  LP-relaxation according  to  the number  of
constraints.
\medskip \\
\begin{tabular}{|c|c|c|c|c|c|r|}
\hline
Total number & S & I & GLPK IP & GLPK LP & nodes&CPLEX\\
of nodes &&&&relaxation &B\&B tree &\\
\hline
100 & 6.25& 5&0.2 MB & 0.2 Mb &1 &  64 kB\\
\hline
200 & 12.5& 11&1.7 MB & 1.6 Mb &1 & 281 kB\\
\hline
300 &18.5 & 17&3.6 MB & 3.5 Mb & 3 &644 kB\\
\hline
\end{tabular}
\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
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
some specifities that encourage us to develop our own solver (coefficents matrix
is  very sparse)  or  to use  an  existing heuristic  to  find good  approximate
solutions (Reference : ``A feasibility pump heuristic  for general mixed-integer
problems",  Livio  Bertacco  and  Matteo Fischetti  and  Andrea  Lodi,  Discrete
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
  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).
\end{itemize}
A discussion about memory consumption has been added in Section 5.2}}
\bigskip \\
\indent \textcolor{blue}{\textbf{\textsc{Answer 2:} In Section 5.2  we give a table with
    the  power  consumption  values  which   are  used  to  compute  the  energy
    consumption. These ones are  based on the energy model of  (Vu et al. 2006).
}}

\bigskip

\noindent\textcolor{black}{\textbf{MINOR COMMENTS:}} \\

\noindent  {\ding{90}  Page 12,  lines  7-15,  the  authors mention  that  DiLCO
  protocol is  close to  PeCO. This  should be mentioned  earlier in  the paper,
  ideally in Section 2 (Related Literature), along with the detailed description
  of DESK and GAF, the competitors of the proposed protocol, PeCO.  }  \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right. This observation has been added
    at the end of the introduction.}}\\

\noindent  {\ding{90}  Page 2,  line  20,  ``An  optimal scheduling"  should  be
  replaced with ``An optimal schedule" } \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\

\noindent  {\ding{90} Page  4, we  first read  (line 23)  ``we assume  that each
  sensor node can directly transmit its  measurements to a mobile sink", then on
  line  30, "We  also assume  that the  communication range  $Rc$ satisfies  $Rc
  >=2Rs$. In  fact, Zhang and Hou  (2005) proved that if  the transmission range
  fulfills  the previous  hypothesis, the  complete  coverage of  a convex  area
  implies  connectivity  among  active  nodes.".  These  two  assumptions  seems
  redundant.}  \\

\textcolor{blue}{\textbf{\textsc{Answer:}  Yes, you  are  right  and we  removed
    sentences about the sink. Indeed we consider multi-hop communication.}}\\

\noindent {\ding{90} Page 4, line 37, a definition for k-covered is missing (the
  sentence is an equivalence property).}  \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right.   A network area is  said to be
    $k$-covered if every point in the area  is covered by at least k sensors. We
    added this definition in the paper.}}\\

\noindent  {\ding{90} Page  5, lines  34 and  37, replace  [0, $2\pi$]  with [0,
    $2\pi$) } \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\

\noindent {\ding{90} Page 5, line 36 and  43, replace ``figure 2" with ``Figure 2"
} \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\

\noindent {\ding{90}  Page 5, line 50, replace ``section 4" with ``Section 4" }  \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\

\noindent {\ding{90}   Page 5, line 51, replace ``figure 3" with ``Figure 3"}  \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\

\noindent {\ding{90}  Page 7, line 20 ``regular homogeneous subregions" is too vague. }  \\

\textcolor{blue}{\textbf{\textsc{Answer:} As  mentioned in the  previous remark,
    the spatial subdivision  was not clearly explained in the  paper. We added a
    discussion about  this question in  the article. Thank you  for highlighting
    it. }}.\\

\noindent {\ding{90} Page 7, line 24, replace ``figure 4" with ``Figure 4"}  \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\


\noindent {\ding{90}  Page 7, line 47, replace ``Five status" with ``Five statuses" }  \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\

\noindent {\ding{90} Page 9, the constraints of the Mixed Integer Linear Program
  (2)  are  not  numbered.  There  are two  inequalities  for  overcoverage  and
  undercoverage that  are used to  define Mij and  Vij. Why not  using replacing
  these inequalities by equalities? }  \\

\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
    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
    is greater than  $l$. In this case,  $M_{i}^{j}=0$, $V_{i}^{j}=l^{i}-l$, the
    corresponding inequality $\sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i \geq l$
    is a  strict inequality since  $\sum_{k \in A}  ( a^j_{ik} ~  X_{k})=l^{i} >
    l$. }}\\

\noindent  {\ding{90}  Page  10,  line  50,  ``or if  the  network  is  no  more
  connected". In order to assess this,  the communication range should be known,
  but it is not given in Table 2. }  \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed}}.\\

\noindent  {\ding{90} Page  10, line  53,  the ``Coverage  ratio" definition  is
  provided for a given period p? Then in the formula on top of page 11, N is set
  to 51 times 26, why? Is it somehow  related to the sensing area having size 50
  times 25? } \\

\textcolor{blue}{\textbf{\textsc{Answer:} Yes, the  ``Coverage ratio" definition
    is provided for a given period p. N is set to 51 times 26 = 1326 grid points
    because we discretized the  sensing field as a regular grid,  a point on the
    contour and  a point  every meter. Yes,  it is related  to the  sensing area
    having size 50 meters times 25 meters.}}\\

\noindent {\ding{90}  Page 11,  line 17  in the  formula of  ASR, |S|  should be
  replaced with J (where J is defined page 4 line 16). }  \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\

\noindent {\ding{90} Page 13, line 41 and 43, replace ``figure 8" with ``Figure 8"
} \\

\textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\

We are very grateful to the  reviewers who, by their recommendations, allowed us
to improve the quality of our article.

\begin{flushright}
Best regards\\
The authors
\end{flushright} 
 


\end{document}
