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13 %\title{Response to the reviewers of \bf "Perimeter-based Coverage Optimization to Improve Lifetime in Wireless Sensor Networks"}
14 %\author{Ali Kadhum Idrees, Karine Deschinkela, Michel Salomon and Raphael Couturier}
22 \vspace{-0.5cm}\hspace{-2cm}FEMTO-ST Institute, UMR 6714 CNRS
24 \hspace{-2cm}University Bourgogne Franche-Comt\'e
26 \hspace{-2cm}IUT Belfort-Montb\'eliard, BP 527, 90016 Belfort Cedex, France.
31 Detailed changes and addressed issues in the revision of the article
33 ``Perimeter-based Coverage Optimization \\
34 to Improve Lifetime in Wireless Sensor Networks''\\
36 by Ali Kadhum Idrees, Karine Deschinkel, Michel Salomon, and Raph\"ael Couturier
41 Dear Editor and Reviewers,
43 First of all, we would like to thank you very much for your kind help to improve
44 our article named: ``Perimeter-based Coverage Optimization to Improve Lifetime
45 in Wireless Sensor Networks''. We highly appreciate the detailed valuable
46 comments of the reviewers on our article. The suggestions are quite helpful for
47 us and we incorporate them in the revised article. We are happy to submit to you
48 a revised version that considers most of your remarks and suggestions to improve
49 the quality of our article.
51 As below, we would like to clarify some of the points raised by the reviewers
52 and we hope the reviewers and the editors will be satisfied by our responses to
53 the comments and the revision for the original manuscript.
55 %Journal: Engineering Optimization
56 %Reviewer's Comment to the Author Manuscript id GENO-2015-0094
57 %Title: \bf "Perimeter-based Coverage Optimization to Improve Lifetime in Wireless Sensor Networks"
58 %Authors: Ali Kadhum Idrees, Karine Deschinkela, Michel Salomon and Raphael Couturier
60 \section*{Response to Reviewer No. 1 Comments}
62 This paper proposes a scheduling technique for WSN to maximize coverage and
63 network lifetime. The novelty of this paper is the integration of an existing
64 perimeter coverage measure with an existing integer linear programming
65 model. Here are few comments:\\
67 \noindent {\bf 1.} The paper makes use of the existing integer optimization
68 model to govern the state of each sensor node within the WSN to maximize
69 coverage and network lifetime. This formulation of the coverage problem is
70 different from the literature in the sense that they use the perimeter coverage
71 measures to optimize coverage as opposed to the targets/points coverage. The
72 methodology uses existing methods and the original contribution lies only in the
73 application of these methods for the coverage scheduling problem.\\
75 \textcolor{blue}{\textbf{\textsc{Answer:} To the best of our knowledge, no
76 integer linear programming based on perimeter coverage has ever been
77 proposed in the literature. As specified in the paper, in Section 4, it is
78 inspired from a model developed for brachytherapy treatment planning for
79 optimizing dose distribution. In this model the deviation between an actual
80 dose distribution and a required dose distribution in each organ is
81 minimized. In WSN the deviations between the actual level of coverage and
82 the required level are minimized. Outside this parallel between these two
83 applications the mathematical formulation is completely different.}}\\
85 \noindent {\bf 2.} The theory seems mathematically sound. However, the
86 assumption made on the selection criteria for the leader seems too vague. \\
88 \textcolor{blue}{\textbf{\textsc{Answer:} The selection criteria for the leader
89 inside each subregion is explained page~9, at the end of Section~3.3
90 After the information exchange among the sensor nodes in the subregion, each
91 node will have all the information needed to decide if it will be the leader or
92 not. The decision is based on selecting the sensor node that has the larger
93 number of one-hop neighbors. If this value is the same for many sensors, the
94 node that has the largest remaining energy will be selected as a leader. If
95 there exists sensors with the same number of neighbors and the same value
96 for the remaining energy, the sensor node that has the largest index will be
97 finally selected as a leader. }}\\
99 %{\bf In fact, we gave a high priority to the number of neighbors to reduce the communication energy consumption - PAS CLAIR }}.\\
101 \noindent {\bf 3.} The communication and information sharing required to
102 cooperate and make these decisions was not discussed.\\
104 \textcolor{blue}{\textbf{\textsc{Answer:} The communication and information
105 sharing required to cooperate and make these decisions is discussed at the
106 end of page 8. Position coordinates, remaining energy, sensor node ID and
107 number of one-hop neighbors are exchanged.}}\\
109 \noindent {\bf 4.} The definitions of the undercoverage and overcoverage
110 variables are not clear. I suggest adding some information about these values,
111 since without it, you cannot understand how M and V are computed for the
112 optimization problem.\\
114 \textcolor{blue}{\textbf{\textsc{Answer:} The perimeter of each sensor may be
115 cut in parts called coverage intervals (CI). The level of coverage of one CI
116 is defined as the number of active sensors neighbors covering this part of
117 the perimeter. If a given level of coverage $l$ is required for one sensor,
118 the sensor is said to be undercovered (respectively overcovered) if the
119 level of coverage of one of its CI is less (respectively greater) than
120 $l$. In other terms, we define undercoverage and overcoverage through the
121 use of variables $M_{i}^{j}$ and $V_{i}^{j}$ for one sensor $j$ and its
122 coverage interval $i$. If the sensor $j$ is undercovered, there exists at
123 least one of its CI (say $i$) for which the number of active sensors
124 (denoted by $l^{i}$) covering this part of the perimeter is less than $l$
125 and in this case : $M_{i}^{j}=l-l^{i}$, $V_{i}^{j}=0$. On the contrary, if
126 the sensor $j$ is overcovered, there exists at least one of its CI (say $i$)
127 for which the number of active sensors (denoted by $l^{i}$) covering this
128 part of the perimeter is greater than $l$ and in this case : $M_{i}^{j}=0$,
129 $V_{i}^{j}=l^{i}-l$. This explanation has been added in the penultimate
130 paragraph of Section~4.}}\\
132 \noindent {\bf 5.} Can you mathematically justify how you chose the values of
133 alpha and beta? This is not very clear. I would suggest possibly adding more
134 results showing how the algorithm performs with different alphas and betas.\\
136 \textcolor{blue}{\textbf{\textsc{Answer:} To discuss this point, we added
137 Section 5.2.5 in which we study the protocol performance, considering
138 $Lifetime_{50}$ and $Lifetime_{95}$ metrics, for different couples of values
139 for alpha and beta. Table 4 presents the results obtained for a WSN of
140 200~sensor nodes. It explains the value chosen for the simulation settings
141 in Table~2. \\ \indent The choice of alpha and beta should be made according
142 to the needs of the application. Alpha should be large enough to prevent
143 undercoverage and thus to reach the highest possible coverage ratio. Beta
144 should be enough large to prevent overcoverage and thus to activate a minimum
145 number of sensors. The values of $\alpha_{i}^{j}$ can be identical for all
146 coverage intervals $i$ of one sensor $j$ in order to express that the
147 perimeter of each sensor should be uniformly covered, but $\alpha_{i}^{j}$
148 values can be differentiated between sensors to force some regions to be
149 better covered than others. The choice of $\beta \gg \alpha$ prevents the
150 overcoverage, and so limit the activation of a large number of sensors, but
151 as $\alpha$ is low, some areas may be poorly covered. This explains the
152 results obtained for $Lifetime_{50}$ with $\beta \gg \alpha$: a large number
153 of periods with low coverage ratio. With $\alpha \gg \beta$, we favor the
154 coverage even if some areas may be overcovered, so a high coverage ratio is
155 reached, but a large number of sensors are activated to achieve this goal.
156 Therefore the network lifetime is reduced. The choice $\alpha=0.6$ and
157 $\beta=0.4$ seems to achieve the best compromise between lifetime and
160 \noindent {\bf 6.} The authors have performed a thorough review of existing
161 coverage methodologies. However, the clarity in the literature review is a
162 little off. Some of the descriptions of the method s used are very vague and do
163 not bring out their key contributions. Some references are not consistent and I
164 suggest using the journals template to adjust them for overall consistency.\\
166 \textcolor{blue}{\textbf{\textsc{Answer:} References have been carefully checked
167 and seem to be consistent with the journal template. In Section~2, ``Related
168 literature'', we refer to papers dealing with coverage and lifetime in
169 WSN. Each paragraph of this Section discusses the literature related to a
170 particular aspect of the problem : 1. types of coverage, 2. types of scheme,
171 3. centralized versus distributed protocols, 4. optimization method. At the
172 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. }}\\
174 \noindent {\bf 7.} The methodology is implemented in OMNeT++ (network simulator)
175 and tested against 2 existing algorithms and a previously developed method by
176 the authors. The simulation results are thorough and show that the proposed
177 method improves the coverage and network lifetime compared with the 3 existing
178 methods. The results are similar to previous work done by their team.\\
180 \textcolor{blue}{\textbf{\textsc{Answer:} Although the study conducted in this
181 paper reuses the same protocol presented in our previous work, we focus in
182 this paper on the mathematical optimization model developed to schedule
183 nodes activities. We deliberately chose to keep the same performance
184 indicators to compare the results obtained with this new formulation with
185 other existing algorithms.}}\\
187 \noindent {\bf 8.} Since this paper is attacking the coverage problem, I would
188 like to see more information on the amount of coverage the algorithm is
189 achieving. It seems that there is a tradeoff in this algorithm that allows the
190 network to increase its lifetime but does not improve the coverage ratio. This
191 may be an issue if this approach is used in an application that requires high
194 \textcolor{blue}{\textbf{\textsc{Answer:} Your remark is very interesting. Indeed,
195 Figures 8(a) and (b) highlight this result. The PeCO protocol allows to achieve
196 a coverage ratio greater than $50\%$ for far more periods than the others
197 three methods, but for applications requiring a high level of coverage
198 (greater than $95\%$), the DiLCO method is more efficient. It is explained at
199 the end of Section 5.2.4.}}\\
201 %%%%%%%%%%%%%%%%%%%%%% ENGLISH and GRAMMAR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
203 \noindent\textcolor{black}{\textbf{\Large English and Grammar:}}\\
205 \noindent {\ding{90} The first paragraph of every Section is not indented.}\\
207 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed. The first paragraph of
208 every Section is indented in the new version. }}\\
210 \noindent {\ding{90} You seem to be writing in the first person. I suggest
211 rewriting sentences that include “we” “our” or “I” in the third person. (There
212 are too many instances to list them all. They are easily found using the find
215 \textcolor{blue}{\textbf{\textsc{Answer:} It is very common to find sentences
216 with "we" and "our" in scientific papers to explain the work made by the
217 authors. Nevertheless we agree with the reviewer and we reformulated some
218 sentences in the paper to avoid too many uses of the first person. }}\\
220 \noindent {\ding{90} Run-on sentence: Page 2 lines 43-48} \\
222 \textcolor{blue}{\textbf{\textsc{Answer:} We rewrote this sentence in two
223 separated sentences. }}\\
225 \noindent {\ding{90} Add an “and” after the comma on page 3 line 34.} \\
227 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}\\
229 \noindent {\ding{90} “model as” instead of “Than” on page 10 line 12.} \\
231 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}\\
233 \noindent {\ding{90} “no longer” instead of “no more” on page 10 line 31.} \\
235 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}\\
237 \noindent {\ding{90} “in the active state” add the on page 10 line 34. } \\
239 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}\\
241 \noindent { \ding{90} Lots of English and grammar mistakes. I recommend
242 rereading the paper line by line and adjusting the sentences that do not make
245 \textcolor{blue}{\textbf{\textsc{Answer:} The English of the paper has been
246 carefully revised and the readability improved. The new version has been
247 checked by an English teacher.}}\\
249 \section*{Response to Reviewer No. 2 Comments}
251 The paper entitled ``Perimeter-based Coverage Optimization to Improve Lifetime
252 in Wireless Sensor Networks'', by Ali Kadhum Idrees, Karine Deschinkel, Michel
253 Salomon, and Rapha\"el Couturier proposes a new protocol for Wireless Sensor
254 Networks called PeCO (Perimeter-based Coverage Optimization protocol) that aims
255 at optimizing the use of energy by conjointly exploiting a spatial and temporal
256 subdivision. The protocol is based on solving a Mixed Integer Linear Program at
257 each leader node, and at each iteration of the protocol. The results obtained by
258 PeCO are compared with three other competitors.\\
260 \noindent\textcolor{black}{\textbf{MAJOR COMMENTS:}} \\
262 \noindent {\bf 1.} The protocol framework is not described in details. In
263 particular, the spatial and temporal subdivision (page 2, line 11) that is at
264 the core of PeCO, is not described nor justified in much detail. How to
265 implement an efficient spatial subdivision ? On page 10, line 11, the number of
266 subdivisions is said to be equal to 16, but the clustering algorithm used is not
267 mentioned. Is this number dependent of the size of the sensing area? Of the
268 number of sensors? Of the sensing range? The proposed protocol cannot be adopted
269 by practitioners if such an important step is not documented. Temporal
270 subdivision suffers from the same lack of description and justification: why
271 should time intervals have the same duration? If they have the same duration,
272 how should this common duration should be chosen?\\
274 \textcolor{blue}{\textbf{\textsc{Answer:} Spatial and temporal choices of
275 subdivision are not the topics of the paper. In the study, we assume that
276 the deployment of sensors is almost uniform over the region. So we only
277 need to fix a regular division of the region into subregions to make the
278 problem tractable. The subdivision is made such that the number of hops
279 between any pairs of sensors inside a subregion is less than or equal
280 to~3. Concerning the choice of the sensing period duration, it is correlated
281 with the types of applications, with the amount of initial energy in sensors
282 batteries and also with the duration of the exchange phase. All applications
283 do not have the same Quality of Service requirements. In our case,
284 information exchange is executed every hour, but the length of the sensing
285 period could be reduced and adapted dynamically. On the one hand, a small
286 sensing period would allow the network to be more reliable but would have higher
287 communication costs. On the other hand, the choice of a long duration may
288 cause problems in case of nodes failure during the sensing period.
289 Several explanations on these points are given throughout the paper. In
290 particular, we discuss the number of subregions in Section 5.2 and the
291 sensing duration in the second paragraph of Section 5.1.}}\\
293 \noindent {\bf 2.}Page 9, Section 4, is the Perimeter-based coverage problem
294 NP-hard? This question is important for justifying the use of a Mixed Integer
295 Linear Programming model.\\
297 \textcolor{blue}{\textbf{\textsc{Answer:} The perimeter scheduling coverage
298 problem is NP-hard in general, it has been proved in the paper entitled
299 "Perimeter Coverage Scheduling in Wireless Sensor Networks Using Sensors
300 with a Single Continuous Cover Range" from Ka-Shun Hung and King-Shan Lui
301 (EURASIP Journal on Wireless Communications and Networking 2010, 2010:926075
302 doi:10.1155/2010/926075). In this paper, authors study the coverage of the
303 perimeter of a large object requiring to be monitored. In our study, the
304 large object to be monitored is the sensor itself (or more precisely its
305 sensing area). This point has been highlighted at the beginning of
308 \noindent {\bf 3.} Page 9, the major problem with the present paper is, in my
309 opinion, the objective function of the Mixed Integer Linear Program (2). It is
310 not described in the paper, and looks like an attempt to address a
311 multiobjective problem (like minimizing overcoverage and
312 undercoverage). However, using a weighted sum is well known not to be an
313 efficient way to address biobjective problems. The introduction of various
314 performance metrics in Section 5.1 also suggests that the authors have not
315 decided exactly which objective function to use, and compare their protocols
316 against competitors without mentioning the exact purpose of each of them. If the
317 performance metrics list given in Section 5.1 is exhaustive, then the authors
318 should mention at the beginning of the paper what are the aims of the protocol,
319 and explain how the protocol is built to optimize these objectives. \\
321 \textcolor{blue}{\textbf{\textsc{Answer:} Right. The mixed Integer Linear
322 Program adresses a multiobjective problem, where the goal is to minimize
323 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
324 criteria to be minimized in case of multicriteria optimization is a
325 classical method. In Section 5, the comparison of protocols with a large
326 variety of performance metrics allows to select the most appropriate method
327 according to the QoS requirement of the application.}}\\
329 \noindent {\bf 4.} Page 11 Section 5.2, the sensor nodes are said to be based on
330 Atmels AVR ATmega103L microcontroller. If I am not mistaken, these devices have
331 128 KBytes of memory, and I didn't find any clue that they can run an operating
332 system like Linux. This point is of primary importance for the proposed
333 protocol, since GLPK (a C API) is supposed to be executed by the cluster
334 leader. In addition to that, GLPK requires a non negligible amount of memory to
335 run properly, and the Atmels AVR ATmega103L microcontroller might be
336 insufficient for that purpose. The authors are urged to provide references of
337 previous works showing that these technical constraints are not preventing their
338 protocol to be implemented on the aforementioned microcontroller. Then, on page
339 13, in Section "5.2.3 Energy Consumption", the estimation of $E_p^{com}$ for the
340 considered microcontroller seems quite challenging and should be carefully
341 documented. Indeed, this is a key point in providing a fair comparison of PeCO
342 with its competitors.\\
344 \textcolor{blue}{\textbf{\textsc{Answer 1:}
345 To implement PeCO on real sensors nodes with limited memories capacities, we can act on :
347 \item the solver : GLPK is memory consuming for the resolution of integer
348 programming (IP) compared with other commercial solvers like
349 CPLEX\textregistered. Commercial solvers generally outperform open source
350 solvers (See the report : ``Analysis of commercial and free and open source
351 solvers for linear optimization problems" by B. Meindl and M. Templ from
352 Vienna University of Technology). Memory use depends on the number of
353 variables and number of constraints. For linear programs (LP), a reasonable
354 estimate of memory use with CPLEX \textregistered~is to allow one megabyte per
355 thousand constraints. For integer programs, no simple formula exists since
356 memory use depends so heavily on the size of the branch and bound tree (B \& B
357 tree). But, the estimate for linear programs still provides a lower bound. In
358 our case, the characteristics of the integer programming (2) are the
361 \item number of variables : $S* (2*I+1)$
362 \item number of constraints : $2* I *S$
363 \item number of non-zero coefficients : $2* I *S * B$
364 \item number of parameters (in the objective function) : $2* I *S$
366 where $S$ denotes the number of sensors in the subregion, $I$ the average number
367 of cover intervals per sensor, $B$ the average number of sensors involved in a
368 coverage interval. The following table gives the memory use with GLPK to solve
369 the integer program (column 3) and its LP-relaxation (column 4) for different
370 problem sizes. The sixth column gives an estimate of the memory use with
371 CPLEX\textregistered to solve the LP-relaxation according to the number of
374 \begin{tabular}{|c|c|c|c|c|c|r|}
376 Total number & S & I & GLPK IP & GLPK LP & nodes&CPLEX\\
377 of nodes &&&&relaxation &B\&B tree &\\
379 100 & 6.25& 5&0.2 MB & 0.2 Mb &1 & 64 kB\\
381 200 & 12.5& 11&1.7 MB & 1.6 Mb &1 & 281 kB\\
383 300 &18.5 & 17&3.6 MB & 3.5 Mb & 3 &644 kB\\
387 It is noteworthy that the difference of memory used with GLPK between the
388 resolution of the IP and its LP-relaxation is very weak (not more than 0.1
389 MB). The size of the branch and bound tree does not exceed 3 nodes. This result
390 leads one to believe that the memory use with CPLEX\textregistered for solving
391 the IP would be very close to that for the LP-relaxation, that is to say around
392 100 Kb for a subregion containing $S=10$ sensors. Moreover the IP seems to have
393 some specifities that encourage us to develop our own solver (coefficents matrix
394 is very sparse) or to use an existing heuristic to find good approximate
395 solutions (Reference : ``A feasibility pump heuristic for general mixed-integer
396 problems", Livio Bertacco and Matteo Fischetti and Andrea Lodi, Discrete
397 Optimization, issn 1572-5286).
398 \item the subdivision of the region of interest. To make the resolution of
399 integer programming tractable by a leader sensor, we need to limit the number
400 of nodes in each subregion (the number of variables and constraints of the
401 integer programming directly depends on the number of nodes and
402 neigbors). It is therefore necessary to adapt the subdvision according to the
403 number of sensors deployed in the area and their sensing range (impact on the
404 number of coverage intervals).
406 A discussion about memory consumption has been added in Section 5.2}}
408 \indent \textcolor{blue}{\textbf{\textsc{Answer 2:} In Section 5.2 we give a table with
409 the power consumption values which are used to compute the energy
410 consumption. These ones are based on the energy model of (Vu et al. 2006).
415 \noindent\textcolor{black}{\textbf{MINOR COMMENTS:}} \\
417 \noindent {\ding{90} Page 12, lines 7-15, the authors mention that DiLCO
418 protocol is close to PeCO. This should be mentioned earlier in the paper,
419 ideally in Section 2 (Related Literature), along with the detailed description
420 of DESK and GAF, the competitors of the proposed protocol, PeCO. } \\
422 \textcolor{blue}{\textbf{\textsc{Answer:} Right. This observation has been added
423 at the end of the introduction.}}\\
425 \noindent {\ding{90} Page 2, line 20, ``An optimal scheduling" should be
426 replaced with ``An optimal schedule" } \\
428 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
430 \noindent {\ding{90} Page 4, we first read (line 23) ``we assume that each
431 sensor node can directly transmit its measurements to a mobile sink", then on
432 line 30, "We also assume that the communication range $Rc$ satisfies $Rc
433 >=2Rs$. In fact, Zhang and Hou (2005) proved that if the transmission range
434 fulfills the previous hypothesis, the complete coverage of a convex area
435 implies connectivity among active nodes.". These two assumptions seems
438 \textcolor{blue}{\textbf{\textsc{Answer:} Yes, you are right and we removed
439 sentences about the sink. Indeed we consider multi-hop communication.}}\\
441 \noindent {\ding{90} Page 4, line 37, a definition for k-covered is missing (the
442 sentence is an equivalence property).} \\
444 \textcolor{blue}{\textbf{\textsc{Answer:} Right. A network area is said to be
445 $k$-covered if every point in the area is covered by at least k sensors. We
446 added this definition in the paper.}}\\
448 \noindent {\ding{90} Page 5, lines 34 and 37, replace [0, $2\pi$] with [0,
451 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\
453 \noindent {\ding{90} Page 5, line 36 and 43, replace ``figure 2" with ``Figure 2"
456 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\
458 \noindent {\ding{90} Page 5, line 50, replace ``section 4" with ``Section 4" } \\
460 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\
462 \noindent {\ding{90} Page 5, line 51, replace ``figure 3" with ``Figure 3"} \\
464 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\
466 \noindent {\ding{90} Page 7, line 20 ``regular homogeneous subregions" is too vague. } \\
468 \textcolor{blue}{\textbf{\textsc{Answer:} As mentioned in the previous remark,
469 the spatial subdivision was not clearly explained in the paper. We added a
470 discussion about this question in the article. Thank you for highlighting
473 \noindent {\ding{90} Page 7, line 24, replace ``figure 4" with ``Figure 4"} \\
475 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\
478 \noindent {\ding{90} Page 7, line 47, replace ``Five status" with ``Five statuses" } \\
480 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\
482 \noindent {\ding{90} Page 9, the constraints of the Mixed Integer Linear Program
483 (2) are not numbered. There are two inequalities for overcoverage and
484 undercoverage that are used to define Mij and Vij. Why not using replacing
485 these inequalities by equalities? } \\
487 \textcolor{blue}{\textbf{\textsc{Answer:} For minimizing the objective function,
488 $M_{i}^{j}$ and $V_{i}^{j}$ should be set to the smallest possible value
489 such that the inequalities are satisfied. It is explained in answer 4
490 for reviewer 1. But, at optimality, constraints are not necessary
491 satisfied with equality. For instance, if a sensor $j$ is overcovered, there
492 exists at least one of its coverage interval (say $i$) for which the number
493 of active sensors (denoted by $l^{i}$) covering this part of the perimeter
494 is greater than $l$. In this case, $M_{i}^{j}=0$, $V_{i}^{j}=l^{i}-l$, the
495 corresponding inequality $\sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i \geq l$
496 is a strict inequality since $\sum_{k \in A} ( a^j_{ik} ~ X_{k})=l^{i} >
499 \noindent {\ding{90} Page 10, line 50, ``or if the network is no more
500 connected". In order to assess this, the communication range should be known,
501 but it is not given in Table 2. } \\
503 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed}}.\\
505 \noindent {\ding{90} Page 10, line 53, the ``Coverage ratio" definition is
506 provided for a given period p? Then in the formula on top of page 11, N is set
507 to 51 times 26, why? Is it somehow related to the sensing area having size 50
510 \textcolor{blue}{\textbf{\textsc{Answer:} Yes, the ``Coverage ratio" definition
511 is provided for a given period p. N is set to 51 times 26 = 1326 grid points
512 because we discretized the sensing field as a regular grid, a point on the
513 contour and a point every meter. Yes, it is related to the sensing area
514 having size 50 meters times 25 meters.}}\\
516 \noindent {\ding{90} Page 11, line 17 in the formula of ASR, |S| should be
517 replaced with J (where J is defined page 4 line 16). } \\
519 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\
521 \noindent {\ding{90} Page 13, line 41 and 43, replace ``figure 8" with ``Figure 8"
524 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed }}.\\
526 We are very grateful to the reviewers who, by their recommendations, allowed us
527 to improve the quality of our article.
