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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

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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.

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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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