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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 6174 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. After
90 the information exchange among the sensor nodes in the subregion, each node
91 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 large enough to prevent overcoverage and thus to activate a
145 minimum number of sensors. The values of $\alpha_{i}^{j}$ can be identical
146 for all coverage intervals $i$ of one sensor $j$ in order to express that
147 the perimeter of each sensor should be uniformly covered, but
148 $\alpha_{i}^{j}$ values can be differentiated between sensors to force some
149 regions to be better covered than others. The choice of $\beta \gg \alpha$
150 prevents the overcoverage, and so limit the activation of a large number of
151 sensors, but as $\alpha$ is low, some areas may be poorly covered. This
152 explains the results obtained for $Lifetime_{50}$ with $\beta \gg \alpha$: a
153 large number of periods with low coverage ratio. With $\alpha \gg \beta$,
154 we favor the coverage even if some areas may be overcovered, so a high
155 coverage ratio is reached, but a large number of sensors are activated to
156 achieve this goal. Therefore the network lifetime is reduced. The choice
157 $\alpha=0.6$ and $\beta=0.4$ seems to achieve the best compromise between
158 lifetime and coverage ratio.}}\\
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
173 paragraph about our previous work on DiLCO protocol to explain the
174 difference with PeCO. }}\\
176 \noindent {\bf 7.} The methodology is implemented in OMNeT++ (network simulator)
177 and tested against 2 existing algorithms and a previously developed method by
178 the authors. The simulation results are thorough and show that the proposed
179 method improves the coverage and network lifetime compared with the 3 existing
180 methods. The results are similar to previous work done by their team.\\
182 \textcolor{blue}{\textbf{\textsc{Answer:} Although the study conducted in this
183 paper reuses the same protocol presented in our previous work, we focus in
184 this paper on the mathematical optimization model developed to schedule
185 nodes activities. We deliberately chose to keep the same performance
186 indicators to compare the results obtained with this new formulation with
187 other existing algorithms.}}\\
189 \noindent {\bf 8.} Since this paper is attacking the coverage problem, I would
190 like to see more information on the amount of coverage the algorithm is
191 achieving. It seems that there is a tradeoff in this algorithm that allows the
192 network to increase its lifetime but does not improve the coverage ratio. This
193 may be an issue if this approach is used in an application that requires high
196 \textcolor{blue}{\textbf{\textsc{Answer:} Your remark is very interesting. Indeed,
197 Figures 8(a) and (b) highlight this result. The PeCO protocol allows to achieve
198 a coverage ratio greater than $50\%$ for far more periods than the others
199 three methods, but for applications requiring a high level of coverage
200 (greater than $95\%$), the DiLCO method is more efficient. It is explained at
201 the end of Section 5.2.4.}}\\
203 %%%%%%%%%%%%%%%%%%%%%% ENGLISH and GRAMMAR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
205 \noindent\textcolor{black}{\textbf{\Large English and Grammar:}}\\
207 \noindent {\ding{90} The first paragraph of every Section is not indented.}\\
209 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed. The first paragraph of
210 every Section is indented in the new version. }}\\
212 \noindent {\ding{90} You seem to be writing in the first person. I suggest
213 rewriting sentences that include ``we'' ``our'' or ``I'' in the third person. (There
214 are too many instances to list them all. They are easily found using the find
217 \textcolor{blue}{\textbf{\textsc{Answer:} It is very common to find sentences
218 with ``we'' and ``our'' in scientific papers to explain the work made by the
219 authors. Nevertheless we agree with the reviewer and we reformulated some
220 sentences in the paper to avoid too many uses of the first person. }}\\
222 \noindent {\ding{90} Run-on sentence: Page 2 lines 43-48} \\
224 \textcolor{blue}{\textbf{\textsc{Answer:} We rewrote this sentence in two
225 separated sentences. }}\\
227 \noindent {\ding{90} Add an “and” after the comma on page 3 line 34.} \\
229 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
231 \noindent {\ding{90} “model as” instead of “Than” on page 10 line 12.} \\
233 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
235 \noindent {\ding{90} “no longer” instead of “no more” on page 10 line 31.} \\
237 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
239 \noindent {\ding{90} “in the active state” add the on page 10 line 34. } \\
241 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
243 \noindent { \ding{90} Lots of English and grammar mistakes. I recommend
244 rereading the paper line by line and adjusting the sentences that do not make
247 \textcolor{blue}{\textbf{\textsc{Answer:} The English of the paper has been
248 carefully revised and the readability improved. The new version has been
249 checked by an English teacher.}}\\
251 \section*{Response to Reviewer No. 2 Comments}
253 The paper entitled ``Perimeter-based Coverage Optimization to Improve Lifetime
254 in Wireless Sensor Networks'', by Ali Kadhum Idrees, Karine Deschinkel, Michel
255 Salomon, and Rapha\"el Couturier proposes a new protocol for Wireless Sensor
256 Networks called PeCO (Perimeter-based Coverage Optimization protocol) that aims
257 at optimizing the use of energy by conjointly exploiting a spatial and temporal
258 subdivision. The protocol is based on solving a Mixed Integer Linear Program at
259 each leader node, and at each iteration of the protocol. The results obtained by
260 PeCO are compared with three other competitors.\\
262 \noindent\textcolor{black}{\textbf{MAJOR COMMENTS:}} \\
264 \noindent {\bf 1.} The protocol framework is not described in details. In
265 particular, the spatial and temporal subdivision (page 2, line 11) that is at
266 the core of PeCO, is not described nor justified in much detail. How to
267 implement an efficient spatial subdivision ? On page 10, line 11, the number of
268 subdivisions is said to be equal to 16, but the clustering algorithm used is not
269 mentioned. Is this number dependent of the size of the sensing area? Of the
270 number of sensors? Of the sensing range? The proposed protocol cannot be adopted
271 by practitioners if such an important step is not documented. Temporal
272 subdivision suffers from the same lack of description and justification: why
273 should time intervals have the same duration? If they have the same duration,
274 how should this common duration should be chosen?\\
276 \textcolor{blue}{\textbf{\textsc{Answer:} Spatial and temporal choices of
277 subdivision are not the topics of the paper. In the study, we assume that
278 the deployment of sensors is almost uniform over the region. So we only
279 need to fix a regular division of the region into subregions to make the
280 problem tractable. The subdivision is made such that the number of hops
281 between any pairs of sensors inside a subregion is less than or equal
282 to~3. Concerning the choice of the sensing period duration, it is correlated
283 with the types of applications, with the amount of initial energy in sensors
284 batteries, and also with the duration of the exchange phase. All
285 applications do not have the same Quality of Service requirements. In our
286 case, information exchange is executed every hour, but the length of the
287 sensing period could be reduced and adapted dynamically. On the one hand, a
288 small sensing period would allow the network to be more reliable but would
289 have higher communication costs. On the other hand, the choice of a long
290 duration may cause problems in case of nodes failure during the sensing
291 period. Several explanations on these points are given throughout the
292 paper. In particular, we discuss the number of subregions in Section 5.2 and
293 the sensing duration in the second paragraph of Section 5.1.}}\\
295 \noindent {\bf 2.}Page 9, Section 4, is the Perimeter-based coverage problem
296 NP-hard? This question is important for justifying the use of a Mixed Integer
297 Linear Programming model.\\
299 \textcolor{blue}{\textbf{\textsc{Answer:} The perimeter scheduling coverage
300 problem is NP-hard in general, it has been proved in the paper entitled
301 ``Perimeter Coverage Scheduling in Wireless Sensor Networks Using Sensors
302 with a Single Continuous Cover Range'' from Ka-Shun Hung and King-Shan Lui
303 (EURASIP Journal on Wireless Communications and Networking 2010, 2010:926075
304 doi:10.1155/2010/926075). In this paper, authors study the coverage of the
305 perimeter of a large object requiring to be monitored. In our study, the
306 large object to be monitored is the sensor itself (or more precisely its
307 sensing area). This point has been highlighted at the beginning of
310 \noindent {\bf 3.} Page 9, the major problem with the present paper is, in my
311 opinion, the objective function of the Mixed Integer Linear Program (2). It is
312 not described in the paper, and looks like an attempt to address a
313 multiobjective problem (like minimizing overcoverage and
314 undercoverage). However, using a weighted sum is well known not to be an
315 efficient way to address biobjective problems. The introduction of various
316 performance metrics in Section 5.1 also suggests that the authors have not
317 decided exactly which objective function to use, and compare their protocols
318 against competitors without mentioning the exact purpose of each of them. If the
319 performance metrics list given in Section 5.1 is exhaustive, then the authors
320 should mention at the beginning of the paper what are the aims of the protocol,
321 and explain how the protocol is built to optimize these objectives. \\
323 \textcolor{blue}{\textbf{\textsc{Answer:} Right. The Mixed Integer Linear
324 Program adresses a multiobjective problem, where the goal is to minimize
325 overcoverage and undercoverage for each coverage interval of a sensor. To
326 the best of our knowledge, representing the objective function as a weighted
327 sum of criteria to be minimized in case of multicriteria optimization is a
328 classical method. In Section 5, the comparison of protocols with a large
329 variety of performance metrics allows to select the most appropriate method
330 according to the QoS requirement of the application.}}\\
332 \noindent {\bf 4.} Page 11 Section 5.2, the sensor nodes are said to be based on
333 Atmels AVR ATmega103L microcontroller. If I am not mistaken, these devices have
334 128 KBytes of memory, and I didn't find any clue that they can run an operating
335 system like Linux. This point is of primary importance for the proposed
336 protocol, since GLPK (a C API) is supposed to be executed by the cluster
337 leader. In addition to that, GLPK requires a non negligible amount of memory to
338 run properly, and the Atmels AVR ATmega103L microcontroller might be
339 insufficient for that purpose. The authors are urged to provide references of
340 previous works showing that these technical constraints are not preventing their
341 protocol to be implemented on the aforementioned microcontroller. Then, on page
342 13, in Section "5.2.3 Energy Consumption", the estimation of $E_p^{com}$ for the
343 considered microcontroller seems quite challenging and should be carefully
344 documented. Indeed, this is a key point in providing a fair comparison of PeCO
345 with its competitors.\\
347 \textcolor{blue}{\textbf{\textsc{Answer 1:}
348 To implement PeCO on real sensors nodes with limited memories capacities, we can act on :
350 \item the solver : GLPK is memory consuming for the resolution of integer
351 programming (IP) compared with other commercial solvers like
352 CPLEX\textregistered. Commercial solvers generally outperform open source
353 solvers (See the report : ``Analysis of commercial and free and open source
354 solvers for linear optimization problems" by B. Meindl and M. Templ from
355 Vienna University of Technology). Memory use depends on the number of
356 variables and number of constraints. For linear programs (LP), a reasonable
357 estimate of memory use with CPLEX \textregistered~is to allow one megabyte per
358 thousand constraints. For integer programs, no simple formula exists since
359 memory use depends so heavily on the size of the branch and bound tree (B \& B
360 tree). But, the estimate for linear programs still provides a lower bound. In
361 our case, the characteristics of the integer programming (2) are the
364 \item number of variables : $S* (2*I+1)$
365 \item number of constraints : $2* I *S$
366 \item number of non-zero coefficients : $2* I *S * B$
367 \item number of parameters (in the objective function) : $2* I *S$
369 where $S$ denotes the number of sensors in the subregion, $I$ the average number
370 of cover intervals per sensor, $B$ the average number of sensors involved in a
371 coverage interval. The following table gives the memory use with GLPK to solve
372 the integer program (column 3) and its LP-relaxation (column 4) for different
373 problem sizes. The sixth column gives an estimate of the memory use with
374 CPLEX\textregistered to solve the LP-relaxation according to the number of
377 \begin{tabular}{|c|c|c|c|c|c|r|}
379 Total number & S & I & GLPK IP & GLPK LP & nodes&CPLEX\\
380 of nodes &&&&relaxation &B\&B tree &\\
382 100 & 6.25& 5&0.2 MB & 0.2 MB &1 & 64 KB\\
384 200 & 12.5& 11&1.7 MB & 1.6 MB &1 & 281 KB\\
386 300 &18.5 & 17&3.6 MB & 3.5 MB & 3 &644 KB\\
389 \medskip \\ It is noteworthy that the difference of memory used with GLPK
390 between the resolution of the IP and its LP-relaxation is very weak (not more
391 than 0.1 MB). The size of the branch and bound tree does not exceed 3
392 nodes. This result leads one to believe that the memory use with
393 CPLEX\textregistered for solving the IP would be very close to that for the
394 LP-relaxation, that is to say less than 300 KB for a subregion containing $S=12$
395 sensors. Moreover the IP seems to have some specifities that encourage us to
396 develop our own solver (coefficents matrix is very sparse) or to use an existing
397 heuristic to find good approximate solutions (Reference : ``A feasibility pump
398 heuristic for general mixed-integer problems", Livio Bertacco and Matteo
399 Fischetti and Andrea Lodi, Discrete Optimization, issn 1572-5286).
400 \item the subdivision of the region of interest. To make the resolution of
401 integer programming tractable by a leader sensor, we need to limit the number
402 of nodes in each subregion (the number of variables and constraints of the
403 integer programming directly depends on the number of nodes and
404 neigbors). It is therefore necessary to adapt the subdvision according to the
405 number of sensors deployed in the area and their sensing range (impact on the
406 number of coverage intervals).
408 A discussion about memory consumption has been added in Section 5.2}}
410 \indent \textcolor{blue}{\textbf{\textsc{Answer 2:} In Section 5.2 we give a table with
411 the power consumption values which are used to compute the energy
412 consumption. These ones are based on the energy model of (Vu et al. 2006).
417 \noindent\textcolor{black}{\textbf{MINOR COMMENTS:}} \\
419 \noindent {\ding{90} Page 12, lines 7-15, the authors mention that DiLCO
420 protocol is close to PeCO. This should be mentioned earlier in the paper,
421 ideally in Section 2 (Related Literature), along with the detailed description
422 of DESK and GAF, the competitors of the proposed protocol, PeCO. } \\
424 \textcolor{blue}{\textbf{\textsc{Answer:} Right. This observation has been added
425 at the end of the introduction.}}\\
427 \noindent {\ding{90} Page 2, line 20, ``An optimal scheduling" should be
428 replaced with ``An optimal schedule" } \\
430 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
432 \noindent {\ding{90} Page 4, we first read (line 23) ``we assume that each
433 sensor node can directly transmit its measurements to a mobile sink", then on
434 line 30, "We also assume that the communication range $Rc$ satisfies $Rc
435 >=2Rs$. In fact, Zhang and Hou (2005) proved that if the transmission range
436 fulfills the previous hypothesis, the complete coverage of a convex area
437 implies connectivity among active nodes.". These two assumptions seems
440 \textcolor{blue}{\textbf{\textsc{Answer:} Yes, you are right and we removed
441 sentences about the sink. Indeed we consider multi-hop communication.}}\\
443 \noindent {\ding{90} Page 4, line 37, a definition for k-covered is missing (the
444 sentence is an equivalence property).} \\
446 \textcolor{blue}{\textbf{\textsc{Answer:} Right. A network area is said to be
447 $k$-covered if every point in the area is covered by at least k sensors. We
448 added this definition in the paper.}}\\
450 \noindent {\ding{90} Page 5, lines 34 and 37, replace [0, $2\pi$] with [0,
453 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
455 \noindent {\ding{90} Page 5, line 36 and 43, replace ``figure 2" with ``Figure 2"
458 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
460 \noindent {\ding{90} Page 5, line 50, replace ``section 4" with ``Section 4" } \\
462 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
464 \noindent {\ding{90} Page 5, line 51, replace ``figure 3" with ``Figure 3"} \\
466 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
468 \noindent {\ding{90} Page 7, line 20 ``regular homogeneous subregions" is too vague. } \\
470 \textcolor{blue}{\textbf{\textsc{Answer:} As mentioned in the previous remark,
471 the spatial subdivision was not clearly explained in the paper. We added a
472 discussion about this question in the article. Thank you for highlighting
475 \noindent {\ding{90} Page 7, line 24, replace ``figure 4" with ``Figure 4"} \\
477 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
480 \noindent {\ding{90} Page 7, line 47, replace ``Five status" with ``Five statuses" } \\
482 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
484 \noindent {\ding{90} Page 9, the constraints of the Mixed Integer Linear Program
485 (2) are not numbered. There are two inequalities for overcoverage and
486 undercoverage that are used to define Mij and Vij. Why not using replacing
487 these inequalities by equalities? } \\
489 \textcolor{blue}{\textbf{\textsc{Answer:} For minimizing the objective function,
490 $M_{i}^{j}$ and $V_{i}^{j}$ should be set to the smallest possible value
491 such that the inequalities are satisfied. It is explained in answer 4
492 for reviewer 1. But, at optimality, constraints are not necessary
493 satisfied with equality. For instance, if a sensor $j$ is overcovered, there
494 exists at least one of its coverage interval (say $i$) for which the number
495 of active sensors (denoted by $l^{i}$) covering this part of the perimeter
496 is greater than $l$. In this case, $M_{i}^{j}=0$, $V_{i}^{j}=l^{i}-l$, the
497 corresponding inequality $\sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i \geq l$
498 is a strict inequality since $\sum_{k \in A} ( a^j_{ik} ~ X_{k})=l^{i} >
501 \noindent {\ding{90} Page 10, line 50, ``or if the network is no more
502 connected". In order to assess this, the communication range should be known,
503 but it is not given in Table 2. } \\
505 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
507 \noindent {\ding{90} Page 10, line 53, the ``Coverage ratio" definition is
508 provided for a given period p? Then in the formula on top of page 11, N is set
509 to 51 times 26, why? Is it somehow related to the sensing area having size 50
512 \textcolor{blue}{\textbf{\textsc{Answer:} Yes, the ``Coverage ratio" definition
513 is provided for a given period p. N is set to 51 times 26 = 1326 grid points
514 because we discretized the sensing field as a regular grid, a point on the
515 contour and a point every meter. Yes, it is related to the sensing area
516 having size 50 meters times 25 meters.}}\\
518 \noindent {\ding{90} Page 11, line 17 in the formula of ASR, |S| should be
519 replaced with J (where J is defined page 4 line 16). } \\
521 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
523 \noindent {\ding{90} Page 13, line 41 and 43, replace ``figure 8" with ``Figure 8"
526 \textcolor{blue}{\textbf{\textsc{Answer:} Right, fixed.}}\\
528 We are very grateful to the reviewers who, by their recommendations, allowed us
529 to improve the quality of our article.
