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Journal of Intelligent & Fuzzy Systems xx (20xx) x–xx
DOI:10.3233/JIFS-190605
IOS Press
1
Mathematical modelling for reducing the
sensing of redundant information in WSNs
based on biologically inspired techniques
1
2
3
Abhilash Singha,b,∗, Sandeep Sharmab, Jitendra Singhcand Rahul Kumard
4
aIndian Institute of Science Education and Research, Bhopal, India5
bDepartment of Electronics and Communication Engineering, Gautam Buddha University, Greater Noida, India6
cDepartment of Electrical Engineering, Indian Institute of Technology Kanpur, India7
dDepartment of Electrical Engineering, Indian Institute of Technology Delhi, India8
Abstract. Wireless sensor networks (WSNs) found application in many diverse fields, starting from environment monitoring
to machine health monitoring. The sensor in WSNs senses information. Sensing and transmitting this information consume
most of the energy. Also, this information requires proper processing before final usages. This paper deals with minimising
the redundant information sensed by the sensors in WSNs to reduce the unnecessary energy consumption and prolong the
network lifetime. The redundant information is expressed in terms of the overlapping sensing area of the working sensors set.
A mathematical model is proposed to find the redundant information in terms of the overlapping area. A combined heuristic
approach is used to achieve the optimal coverage, and the effect of the overlapping area is considered in the objective function
to reduce the amount of redundant information sensed by the working sensors set. Improved genetic algorithm (IGA) and
Binary ant colony algorithm (BACA) are used as heuristic tools to optimise the multi-objective function. The objective was
to find the minimum number of sensors that cover a complete scenario with minimum overlapping sensing region. The results
show that optimal coverage with the minimum working sensor set is achieved and then by incorporating the concept of
overlapping area in the objective function, sensing of redundant information is further reduced.
9
10
11
12
13
14
15
16
17
18
19
20
Keywords: Improved genetic algorithm, Binary ant colony algorithm, Redundant information, Objective function, Overlap-
ping area
21
22
1. Introduction
23
Random deployment of the wireless sensor in24
WSNs along with the sensor’s location information25
has made the localisation issue a critical challenge
26
in WSNs. One of the proficient ways to deal with
27
this type of problem is the heuristic approach based
28
on the simulation [1–4]. Several heuristic approaches
29
have been proposed in WSNs such as GA, ACO, Lion
30
∗Corresponding author. Abhilash Singh, Indian Institute of
Science Education and Research, Bhopal, India. E-mails:
abhilash.singh@ieee.org; sabhilash@iiserb.ac.in
Optimization (LO), Particle Swarm Optimization 31
(PSO), Support Vector Machines (SVM) Artificial 32
Neural Networks (ANNs) etc. [5–10]. Along with 33
these intelligent algorithms, and their modified ver- 34
sions are also used to deal with serious issues 35
associated with WSNs [11–16]. 36
Regardless of the benefits of such heuristic 37
approaches, its major weakness is the sensing of the 38
redundant information and the presence of redundant 39
sensors. Both these factors affect the lifetime of sen- 40
sors in WSNs drastically. However, the sensors have 41
a limited lifetime due to the limit of cost and volume 42
[17, 18]. Energy to the sensors cannot be supplied 43
ISSN1064-1246/19/$35.00 © 2019 – IOS Press and the authors. All rights reserved
Uncorrected Author Proof
2A. Singh et al. / Mathematical modelling for reducing the sensing of redundant information
once employed or regenerated, so the optimal cover-44
age and the energy consumed by the nodes are the
45
two critical problems in WSNs [2, 19].46
In this paper, a novel approach has been proposed47
to deal with the sensing of redundant information. A48
mathematical model is proposed to find the redundant
49
information sensed by the sensors using a combined50
heuristic approach. The contribution of the authors in51
the paper is summarised as follows.52
– A very effective combined heuristic approach,53
IGA-BACA, is applied in WSNs for maximising54
the network coverage along with reducing the55
number of sensors required.56
– A mathematical model is proposed for finding57
the redundant information sensed by the sensor
58
node having a circular sensing area. Redundant
59
information sensed by active nodes is expressed60
in terms of the overlapping area.61
– Designing of a multi-objective function that
62
incorporates the concept of the overlapping area.63
In Section 2, the theoretical background and the64
related work have been presented. In which, the basics65
of heuristics techniques used has been discussed.
66
2. Theoretical background and related work
67
2.1. GA and Adaptive GA(or IGA)
68
GA is a global heuristic random search technique,
69
proposed in the early seventies by John H. Holland70
[5]. It follows three operators, namely reproduction,71
crossover and mutation. It can be considered as a bio-
72
logical evolutionary process in which a population73
of certain solutions evolves over a series of gener-74
ations [20]. It retains a set of solutions known as75
chromosomes, which are obtained through fitness
76
function in every generation. It is then shortlisted
77
for reproduction according to their fitness value. The
78
selection process follows the principle of survival of
79
the fittest. Solutions having good fitness values are80
selected for reproduction whereas the bad solutions81
are rejected. The shortlisted solutions go through82
other two genetic operator crossover and mutation.
83
Now crossover leads to the interchange of genetic-84
based material between solutions because crossed85
solutions can produce an individual with either best or
86
worst fitness. It occurs only with some probabilityPc
87
known as crossover probability or crossover rate. The88
mutation modifies a solution with probability Pm
89
known as mutation probability. The role of mutation90
probability is to explore the unexplored genetic mate- 91
rial. GA finds a globally optimal solution or nearby 92
solution and hence, reduces the chances of getting the 93
local minimum. 94
In Adaptive Genetic algorithm (AGA), the
crossover and mutation probability changes adap-
tively according to the different condition of the
individuals which prevents the premature conver-
gence [21]. Mathematical expression for Pcand Pm
is given in equation 1 and 2.
Pc=⎧
⎨
⎩
k1(fmax−f)
fmax−favg f>f
avg
k3f<f
avg
(1)
Pm=⎧
⎨
⎩
k2(fmax−f)
fmax−favg f>f
avg
k4f<f
avg
(2)
Where Pcis known as crossover probability, Pm95
is known as mutation probability, fmax is known as 96
the highest fitness of population, favg is known as 97
the average fitness of the population, fis known as 98
the higher fitness of two strings used for crossover, f99
is known as the fitness of the individuals to mutate. 100
The constant k1,k
2,k
3and k4have to be less than 1, 101
to restrict the value of Pcand Pmto the range [0,1]. 102
Also, k1and k3are bigger than other two constants. 103
2.2. Ant Colony Optimisation (ACO) and BACA 104
ACO is a kind of simulated evolutionary algo- 105
rithm [22]. The basic concept in ACO is that an ant 106
while searching for food releases a peculiar type of 107
pheromone in its path, which makes other ants per- 108
ceive a kind of oddity and tends to lean towards the 109
route of higher intensity [20]. ACO evaluates the 110
optimal route through a continuous process of accu- 111
mulation and update of chemical released by them, 112
pheromones, and it obtains optimised value by under- 113
going appropriate times of generation. Although 114
ACO finds better solutions, gets good performance 115
on parallel computing and can be easily combined 116
with other algorithms, it still has some shortcomings. 117
The initial stages lack pheromone as a result of which 118
it searches slowly and results in the local optimal 119
solution, i.e. premature convergence [23]. 120
To overcome the above issue, BACA is used. 121
BACA network is illustrated in Fig. 1. The work of 122
ants is divided, and the same routine searches by a 123
different type of ants, and pheromone is left on each 124
edge. As an intelligent individual, each ant selects one 125
Uncorrected Author Proof
A. Singh et al. / Mathematical modelling for reducing the sensing of redundant information 3
Fig. 1. BACA network.
edge out of two. The incidence matrix traversed by126
each kind of ant, needs only (2 ×n)sspace, which,127
up to some extent, solve the problem of reduction in128
solution quality. Hence, in the next iteration or experi-129
ment, the efficiency of the algorithm increases greatly
130
[23]. The BACA make use of binary coding instead131
of decimal coding, and the binary coding has a bet-
132
ter traversal search capability as compared to decimal133
coding.134
Define a direct graph, G=(V, R), where the node
set Vis
v0(cs),v
1((c0
N)),v
2((c1
N)),v
3((c0
N−1)),v
4((c1
N−1)), ...,
v2N−3((c0
2),v
2N−2((c1
2),v
2N−1((c0
1),v
2N((c1
1)
(3)
135
Where csis the starting node, c0
jand c1
jare sepa-136
rately used to represent the value of 0 and 1 of bit bj
137
in binary-coding. For every node (j=1,2,3, ..., N),138
there are only two directed arcs pointing towards c0
j−1
139
and c1
j−1, which respectively stands for the 0 and 1140
allowed next states for all the ants and N represents141
the binary encoding length [24].142
In the initial stage, information associated to all
path are equal, let it be τi,j (0) =Cwhere Cis
a constant, τi,j (0)(i, j =1,2, ..., N ). During the
operation of move, k number of ants determines the
direction (where k=1,2, ..., m) according to the
info of each path and the probability of movement
is define as
pk
i,j =(τα
i,j (t).ηβ
i,j (t))
(s∈allowedk τα
i,j (t).ηβ
i,j (t)) (4)
Where pk
i,j represent the movement probability,
that kants move from location ito location j,mrepre-
sent the number of ants, αand βrepresent the relative
importance of moving path and visibility respec-
tively. Both αand βshould be greater than or equal
to one. τi,j is the unused information in the ij connec-
tion at tmoment. ηi,j represent the visualness in the
ij connection. Allowedk=(0,1) is the next selected
status. As pheromones evaporate as time passage so
the leaving information also starts disappearing with
passage of time. ρ(0 ≤ρ≤1) represents the perse-
verance of a moving path and information loss degree
is 1 −ρ. The ant completes one cycle after nmoment.
The τi,j is represented as
τi,j (t+1) =ρ.τi,j (t)+τi,j (5)
Where τi,j =1
fbest (S),fbest (S) is the best- 143
iteration cost. Compared with the basic ant colony, 144
the BACA differs only in the path selection of ant. 145
There are only two states 0 and 1 in a binary ant 146
colony; the ant do not need to remember that node 147
that has been travelled, it selects only from the two 148
path pheromones. 149
2.3. IGA with BACA 150
BACA finds the solution by taking the initial 151
population from the GA. The initial population is 152
generated randomly, which go through GA evolution. 153
Simultaneously, the result obtained by GA is utilised 154
to initialise the pheromones information of the net- 155
work. If the last condition of the loop is met, then the 156
algorithm will terminate; otherwise ants will traverse, 157
pheromones update and reserve of optimal solution 158
step is executed. Further, GA operators are operated 159
to obtain a set of solution and retain the optimal solu- 160
tion. And lastly, the optimal updated pheromones are 161
used to decide whether the termination is met or not. 162
A one to one correspondence between IGA-BACA 163
parameters and WSNs is given in Table 1. 164
2.4. WSN nodes optimal coverage based on 165
IGA-BACA 166
Optimal coverage of WSN node is a multiple
objective optimization problem, which can be rep-
resented as S=(s1,s
2, ..., si, ..., sN) is the set of
existing sensors, the target is to get sensor node set
S, that provides utmost coverage rate with the lowest
number of working sensors. If f1(S) represents max-
imal network coverage and f2(S), minimum working
nodes. This conflit undermines them both, multiple
objective optimisation models in WSN is changed to
maximal objective function f(S) given by
f(S)=(f1(S).f1(S)/f2(S)) (6)
The optimal coverage based on IGA-BACA imple- 167
mentation in WSN is as follows [24].
Uncorrected Author Proof
4A. Singh et al. / Mathematical modelling for reducing the sensing of redundant information
Fig. 2. Control vector of 9 working sensors out of 12 is {11101
0110111}.
2.4.1. Code168
The completion of mapping from solution space to169
coding space of optimisation problem is a key prob-170
lem. The control vector L=(l1,l
2, ..., li, ..., lN)of
171
wireless sensor node for random distribution of work-172
ing sensor is expressed in form of binary coding, that173
represent the location of all sensor nodes. Here the174
value of lican be either 0 or 1, which reflects the175
sleep or active state of the sensor node i. The initial-176
isation of a gene in a chromosome in GA has one to177
one correlation with selection of nodes.
178
2.4.2. Initial solution and adaptation function179
A total of 8 sensors are randomly distributed over a180
region, where the sensors (1,2,3,5,7,8,10,11,12)181
are working sensors nodes and the corresponding182
control vector is 111010110111which is shown183
in Fig. 2. Equation 6 is used as adaptation function;184
the larger the value of the functions, the large will be
185
selection probability of individual.
186
2.4.3. Operation operator187
It is a combination of GA and ACO opera-188
tors. A total of four operators namely reproduction,189
crossover, mutation and lastly update pheromone190
operator are there in IGA-BACA. In IGA-BACA, the
191
first operator reproduces offspring to the new colony
192
depending on probability in a fraction to their adap-
193
tive value. After the first operator, the offspring with
194
better fitness are retained, and inferior offspring are
195
rejected or removed, and hence the average fitness196
value associated with the colony is increasing, how-197
ever, in this process, the number of possible varieties198
is lost. The crossover operator selects a pair of indi-199
viduals according to probability Pc, and there may200
be a chance that crossed solutions can either produce201
ones with better or worse fitness value. The mutation202
operator modifies a solution or offspring with proba- 203
bility Pm. The role of mutation operator is to explore 204
the unexplored genetic material. 205
The update pheromone operator (Tg) repre-
sent a mapping of update pheromone for optimal
offspring selected through ant sequence. By Max-
Min rule pheromones will be only released in
the optimal ant traverse path, mathematically
τ(i, j )=1/(fbest(S)), fbest (S) is the best-iteration
cost or best global solution. The probability of update
pheromone operator is given by
PTg=xi=f(xi)
N
k=1f(xk)(7)
Where, Nis the total number of offspring. 206
A diagrammatic representation of this section is 207
illustrated in the flow chart in Fig. 8 [24]. 208
The rest of the paper is structured as follows. An 209
expression is derived for calculating the overlapping 210
area in section III. The system model for the proposed 211
approach is illustrated in section IV. The simulation 212
results are discussed in section V. Finally, in section 213
VI, the conclusion and future scope of the work is 214
highlighted. 215
3. Mathematical formulation 216
for overlapping area 217
The overlapping area between the circles repre- 218
sents the redundant information, as both the sensor 219
sensing the common information. A mathematical 220
formulation to find the common area has been given 221
below. 222
Let us assume that in the two circles, one with cen- 223
tre Aand other as centre Chave the same radius, 224
r, overlapping each other. The distance between the 225
centres, daffects the size of the overlapping area, as 226
shown in Fig. 3. But at first, without going to math- 227
ematics, one can observe some of the points. First 228
when d=0 the overlapping area is πr2and second 229
when d≥2rthen the overlapping area is 0 [25]. 230
The common area is due to the overlapping sec- 231
tors, as shown in Fig. 4. To find the overlapping area, 232
first the area of the sector for one circle is calculated, 233
then the area due to the segment of the sector is calcu- 234
lated indirectly by eliminating the area of its triangle. 235
Then, it is multiplied by two to obtain the desirable 236
overlapping area, as explained in Fig. 5. 237
The area of the sector of a circle,
Areasector =(θ/2)r2, where θis the central
Uncorrected Author Proof
A. Singh et al. / Mathematical modelling for reducing the sensing of redundant information 5
Fig. 3. Overlapping circles with centres A and C.
Fig. 4. Overlapping sectors.
Fig. 5. Segment of sector.
Fig. 6. Isosceles triangle obtained from segment of sector.
Fig. 7. Splitting of AED.
angle in radian, of the sector. AED is an isosceles
triangle with points A(centre of the first circle), E
and D(intersection point of the circle) shown in Fig.
6. The height of the isosceles triangle is d/2. To find
the central angle of the sector θ,AED is divided
into two right angle triangles of height/2, hypotenuse
rand angle between the hypotenuse and height θ/2
as shown in Fig. 7. From Fig. 7,
cos(θ/2) =(d/2)/r ⇒cos(θ/2) =d/2r(8)
solving for θwe get,
θ/2=cos−1(d/2r)⇒θ=2cos−1(d/2r)(9)
At this particular time, one can conclude that |d|≤2r
is a necessary condition for a circle to overlap. Now
the area of the sector is given by
Areasector =(θ/2)r2=(2cos−1(d/2r))r2/2
=r2cos−1(d/2r)
(10)
To obtain the area due to the sector’s segment first the 238
area of ADE needs to be evaluated, which is given 239
by Areatriangle =(1/2)bh, Where base, bis DE and 240
height, his d/2. 241
There are two methods to obtain the triangle area. 242
In the first method Pythagoras theorem is used to get 243
an expression for the base of the AED,bin term 244
of rand d.Ifbis the base of the AED then b/2is 245
the base of the right angle triangles. 246
By the Pythagoras theorem, we have,
(b/2)2+(d/2)2=r2⇒b=4r2−d2(11)
Hence the area of the triangle is given by
Areatriangle =(1/2)bh =(d/4)4r2−d2(12)
In the second method, the triangle area is expressed
in terms of rand θ. For this, first band his calculated
in terms of rand θ.
sin(θ/2) =(b/2)/r ⇒b=2rsin(θ/2) (13)
cos(θ/2) =h/r ⇒h=rcos(θ/2) (14)
Uncorrected Author Proof
6A. Singh et al. / Mathematical modelling for reducing the sensing of redundant information
Now area of the triangle can be given by
Areatriangle =r2sin(θ/2)cos(θ/2) (15)
Using double-angle formula, sin(2z)=2sinzcosz,
the area of the triangle can be reduced to
Areatriangle =(1/2)r2sin(θ)(16)
Now area due to the segment is obtained as a differ-
ence in the area between the sector and the triangle.
Area of the sector’s segment (in terms of rand θ).
Areasegment =Areasector −Areatriangle
⇒(r2/2)(θ−sinθ)
(17)
Area of the sector’s segment (in terms of rand d).
Areasegment =r2cos−1(d/2r)−(d/4)4r2−d2
(18)
Area of the overlapping region in terms of rand θ.
Areaoverlapping =2Areasegment =r2(θ−sinθ)
(19)
with 0 ≤θ≤2π.
Area of the overlapping region in terms of rand d.
Areaoverlapping =2Areasegment =2r2cos−1
(d/2r)−(d/2)4r2−d2)
(20)
with 0 ≤d≤2r.
247
4. System model
248
Network coverage is one of the critical problems249
in WSNs. The coverage should reach the maximum250
with a certain level of Quality of Service (QoS)
251
[26–28]. Maintaining the required amount of QoS252
is achieved by measuring the WSNs coverage rate to253
check whether blind areas are present or not. Blind254
areas are regions which are not sensed by any of the255
sensors. The density of sensor nodes can be changed256
based on the requirement by keeping a total number257
of sensor nodes constant, i.e. one can deploy more
258
sensors in more vital area to increase the reliabil-
259
ity. There are two types of deployment viz. random260
coverage and certainty covering. In random covering261
sensors, deployment task is random, whereas in cer-
262
tainty covering the sensors deployment task is based
263
upon specific algorithms. In the later certainty cov-
264
ering, if someones want to cover the area using a265
minimum working set of nodes, then determining the266
coverage area size is the first task that needs to be267
Table 1
Analogous parameters in IGA-BACA algorithm and sensor
deployment
IGA-BACA Algorithm Sensor deployment problem
Solution of a food source Sensor distribution
Ndimensions in each solution Nsensor coordinates
Fitness of the solution Coverage rate of interest area
Maximum fitness Optimum sensor deployment
Fig. 8. Flow chart for IGA-BACA.
done. The certainty covering technique is adopted in 268
this paper. In recent times, the method mentioned in 269
[26] is widely used to find the minimum set of nodes 270
that is enough to cover the given area and turn off the 271
remaining nodes. In WSNs, with increase in the num- 272
ber of sensing nodes, the network cost also increases. 273
The optimal set of nodes is the minimum working set 274
of nodes that will not affect the network coverage. 275
Let us suppose that the total area to be monitored 276
is Aand is a 2Dplane. Further, it is divided into mxn277
grids with each grid having unit area. A total of Nsen- 278
sor nodes are first distributed randomly in the given 279
monitoring area Aand sensor nodes set is represented 280
by S=(s1,s
2, ..., si, ..., sN). The sensor node s1has 281
Uncorrected Author Proof
A. Singh et al. / Mathematical modelling for reducing the sensing of redundant information 7
a coordinate (xi,y
i) in the region A, coordinate of282
each sensor node is known and all sensor node have
283
effective radii of rand si=xi,y
i,r
. The sensing
284
radius of node siis consider to be circular region with285
centre, (xi,y
i) and effective radius, r. Let pbe any
286
arbitrary point in the region Awhere p∈A, and dis
287
the Euclidean distance then d(p, (xi,y
i)) ≤rwhere288
i∈[1,N].289
Each grids are target point (a1,a
2, ..., aj,
290
..., am×n) and need to be sensed in the region A. Let291
the target point ajbe in the sensing region of the sen-292
sor node sithen d(aj,s
i)≤r. For the sensor node
293
sithe communication range is the circle with centre,294
(xi,y
i) and radius, r.Nowifd(si,s
j)≤rcthen there
295
will be direct communication possible between the
296
sensor node siand sj.297
An undirected communication graph G, consist-298
ing of Ssensor node is said to be connected if their
299
exit a communicating path between any two sensor300
node which means dshould be d(si,s
j)≤rcwhere301
Sis sensor node set. If any point in the region Ais302
sensed by a sensor node of S, then Sis called as the303
covering set of A, and if the communication graph G304
is connected then Sis called as the connected cover-305
ing set of A. Let Sbe the minimal subset of S, and306
if Scovers the monitoring area completely then one307
can conclude that communication diagram extracted308
from Sis connected. The goal of optimisation should309
be such that each grid in Ashould be at least sensed310
by a sensor. For connectivity to exit the communi-311
cation range rc, it should be at least twice as that of312
sensing radius r, means rc≥2r. So by maintaining313
this relation one can ensure that connectivity is there
314
and we have to only deal with the coverage problem
315
in the network [24].316
The aim of WSNs is to maximise the network317
coverage while minimising the node uses. This is a318
multi-objective optimisation problem whose cover-319
age model for WSNs is represented by equation 26.320
In a nutshell, the above problem can be represented
321
as follows:322
Input parameters
323
–N: number of sensors324
–Sj: sensor node set for j(j=1,2, ..., N) i.e.325
S=(s1,s
2, ..., si, ..., sN)326
–r: effective radius of all sensors327
–rc: communication range, so that for connectiv-
328
ity between two sensor rc≥2r329
–A: monitoring area330
–W, L: represents the width and length of the
331
monitoring area, A332
Fig. 9. Overlapping criteria.
Fig. 10. Optimal network coverage.
333
Output parameters 334
–2−Dsets of coordinates values for all Nsen- 335
sors. 336
– redundant information in terms of overlapping 337
area
Uncorrected Author Proof
8A. Singh et al. / Mathematical modelling for reducing the sensing of redundant information
Objective338
1. First objective is to maximize the coverage i.e.
maxObj1=area
m
x=1
n
y=1
cr(x, y)∩A
(21)
where the circle located at (x, y) with radius ris
339
represented by cr(x, y); the area of a region Xis340
represented by area(X).341
2. Second objective is to minimize the no. of sen-
sors i.e.
minObj2=|S|/N(22)
where the total number of the sensor node is rep-
342
resented by Nand Sis the magnitude of active343
sensors.344
– Third objective is to minimize the overlapping345
sensing area between sensors. Obviously, the346
less the overlapping coverage area, the more will347
be the coverage within A. Based on this observa-348
tion a new objective function is designed which349
is given by350
Obj3=N
i=1(N
j=i+1overlap(si,s
j)351
+4
k=1overlap(si,b
k)) (23)352
where the coverage area for two sensors located
at si,sjis represented by overlap(si,s
j) and it is
given by
overlap(si,s
j)=0if d (si,s
j)>r
c
Areaoverlapping if d (si,s
j)≤rc
(24)
with d(si,s
j) as the Euclid distance between
the two sensors siand sjand an expression
for Areaoverlapping is derived in mathemati-
cal formulation for overlapping area section.
Equation 24 is illustrated in Fig. 9. The term
overlap(si,b
k) represents the overlap between
the coverage area of sensor at siout of monitor-
ing area, A, boundary that is represented by bk.
If d(si,b
k)<rthen
overlap(si,b
k)=r(r−d(si,b
k)) (25)
where bk∈{(si.x, L),(si.x, 0),(0,s
i.y),(W, si.y)}353
In a nutshell, the complete objective is to maximize
the
maxObj =max(Obj1,1−Obj2,1−Obj3) (26)
Fig. 11. Random distribution of 100 nodes.
5. Simulation results 354
The monitoring area, Aunder considering is 100m355
times 100mwith a perception radius, r, of the sensor 356
nodes equals to 10m. A set of 42 working sensor node 357
optimally covers the monitoring area A, as shown 358
in Fig. 10. It is obtained for 300 iteration of IGA- 359
BACA. It is also used as the benchmark for further 360
comparison. The complete algorithm is implemented 361
in MATLAB 2017b. A total of 100 nodes are ran- 362
domly distributed in the monitoring target area, as 363
shown in Fig. 11. 364
Although the area is covered completely, there 365
exist a lot of redundant nodes which are sensing 366
redundant information. The uncovered areas in A367
are termed as a coverage hole and in Fig. 11, one 368
can easily detect such coverage holes or blind areas 369
(indicated in red boxes for reference). As compared 370
to the optimal coverage, randomly distributed sce- 371
nario senses a huge amount of redundant information. 372
Once, the redundant information is expressed in term 373
of the overlapping area, then the overlapping areas is 374
more for the randomly distributed case and minimum 375
for the optimal case. This is illustrated in Figs. 12 and 376
13. 377
Figure 12 indicates that the overlapping areas of 378
each sensor are equal, which is 86.7546 sq. unit. This 379
concludes that the redundant information sensed by 380
each of the sensors are equal. In contrast, Fig. 13 381
indicates that the overlapping areas are different for a 382
randomly distributed case, which means each sen- 383
sor are sensing the different amount of redundant 384
information. 385
Uncorrected Author Proof
A. Singh et al. / Mathematical modelling for reducing the sensing of redundant information 9
Fig. 12. Optimal network coverage with overlapping area.
Fig. 13. Random distribution of 100 nodes with overlapping area.
Fig. 14. 50 Generation of IGA-BACA.
Fig. 15. 100 Generation of IGA-BACA.
Fig. 16. 150 Generation of IGA-BACA.
Fig. 17. 200 Generation of IGA-BACA.
Uncorrected Author Proof
10 A. Singh et al. / Mathematical modelling for reducing the sensing of redundant information
Fig. 18. Redundant information vs iterations.
Figures 14–17 shows the coverage using the IGA-386
BACA algorithm for 50, 100, 150, 200 generations387
respectively. As the number of generation is increas-388
ing from 50 to 200, the coverage tends towards389
optimal coverage and hence, the number of redundant390
nodes decreases.
391
To quantify the redundant information relatively,392
a graph is plotted between the amount of redundant393
information sensed by the sensors to the no. of iter-
394
ations or generations, as illustrated in Fig. 18. It has
395
been noticed from Fig. 18 that as the number of
396
iterations is increased, i.e. as the network coverage397
tends towards the optimal coverage the amount of398
redundant information sense by the sensor decreases,399
and also the slope of decrements also increases for
400
higher values of iterations. To evaluate the perfor-
401
mance of the combined heuristic over WSNs, the402
relative redundant information is compared with the403
existing heuristic approaches, namely GA and ACO.404
6. Conclusion405
In this paper we propose an effective way to com-406
pute and reduce the redundant information sensed by
407
the sensors in WSNs. It is essential to reduce the over-
408
lapping sensing regions as sensing the same set of
409
information results in loss of time and limited energy
410
of the sensors. The optimal coverage is achieved411
using the combined technique of IGA-BACA. It has412
been observed that as the number of iterations is413
increased, the network coverage tend towards opti-414
mal coverage. With optimal coverage, the redundant415
information sensed by the sensors gets reduced to its416
lowest possible value. Although this approach gives417
better performance but, much is yet to be explored, 418
especially if one can use other combined heuristic 419
such as lion optimisation, ant lion optimisation etc. 420
to reduce the redundant information. 421
Acknowledgments 422
The authors would like to thanks the editor and 423
reviewers for their valuable suggestions. 424
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