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Wireless Personal Communications
An International Journal
ISSN 0929-6212
Volume 81
Number 2
Wireless Pers Commun (2015)
81:819-838
DOI 10.1007/s11277-014-2159-3
A Multi-objective Disjoint Set Covers for
Reliable Lifetime Maximization of Wireless
Sensor Networks
Bara’a A.Attea, Enan A.Khalil, Suat
Özdemir & Oktay Yıldız
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Wireless Pers Commun (2015) 81:819–838
DOI 10.1007/s11277-014-2159-3
A Multi-objective Disjoint Set Covers for Reliable
Lifetime Maximization of Wireless Sensor Networks
Bara’a A. Attea ·Enan A. Khalil ·Suat Özdemir ·
Oktay Yıldız
Published online: 12 November 2014
© Springer Science+Business Media New York 2014
Abstract An important challenge facing many large-scale surveillance applications is how
to schedule sensors into disjoint subsets to maximize the coverage time span. Due to its
NP-hard complexity, the problem of finding the largest number of disjoint set covers (DSC)
of sensors has been addressed by many researchers. Majority of these studies employs the
Boolean sensing model where a sensor covers a target if it lies within its sensing range. In
reality, however, the sensing reliability may be affected by several parameters, e.g., strength
of the generated signals, environmental conditions and the sensor’s hardware. To the best of
our knowledge, improving coverage reliability of Wireless Sensor Networks (WSNs) has not
been explored while solving DSC problem. This paper addresses the problem of improving
coverage reliability of WSNs while simultaneously maximizing the number of DSC. Thus, in
the context of WSNs design problem, our main contribution is to turn the definition of single-
objective DSC problem into a multi-objective problem (MOP) by adopting an additional
conflicting objective to be optimized. Specifically, we investigate the performance of two
multi-objective evolutionary algorithms in terms of diversity and quality of the Pareto optimal
set for the modeled MOP. The simulation results indicate that multi-objective approach results
in achieving reliable coverage and large number of DSC compared to a single-objective
approach.
B. A. Attea (B)
Department of Computer Science, Baghdad University, Baghdad, Iraq
e-mail: baraaali@yahoo.com
B. A. Attea
e-mail: baraali@scbaghdad.edu.iq
E. A. Khalil ·S. Özdemir ·O. Yıldız
Department of Computer Engineering, Gazi University, Ankara, Turkey
e-mail: enanameen@yahoo.com
S. Özdemir
e-mail: suatozdemir@gazi.edu.tr
O. Yıldız
e-mail: oyildiz@gazi.edu.tr
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Keywords DSC ·MOO ·Non-dominated solution ·NP-hard ·Probabilistic coverage ·
Reliability ·WSNs
1 Introduction
The Wireless Sensor Network (WSN) paradigm is an emerging and fast growing technolog-
ical platform in myriad working environments. To name just a few, WSNs can be used in
military and agriculture applications, harsh environment monitoring, emergency search-and-
rescue operations. One of the important characteristics of WSNs is their ability to densely
deploy in ad-hoc manner. For successful operation of WSNs, sensors should effectively cover
the required region of interest for a long period of time. Energy-aware sensing and commu-
nication mechanisms have been substantially pursued by the research community in order to
prolong the lifespan of WSNs. Energy saving techniques can generally be classified in the
following categories; (1) Energy-efficient data aggregation, gathering and routing; (2) power
management by adjusting the transmission and/or sensing range of sensor nodes; and (3)
sensor wake-up scheduling to alternate between active and idle state.
In this paper, we consider the third approach to achieve energy-efficient WSN that monitors
all the targets at all times. In this approach, sensors are divided into disjoint sets of covers
(DSC) such that every set completely covera set of targets being located with known locations.
We assume that the lifetime of a WSN is divided into intervals and at each interval only one
set is active with while the remaining set covers are in the low-energy sleep mode. The sensors
from the active set are in active state, monitoring all targets within their sensing ranges. Once
the active set cover runs out of energy, another set cover will be selected to enter the active
mode and provide the functionality continuously. As all targets should be monitored by every
set cover, the goal of this approach is to determine a maximum number of DSC. The more
set covers we can find, the longer sensor network lifetime is prolonged. It has been proven
that this problem (also called the SET k-cover problem) is a generalization of the minimum
cover problem [1] and proves its NP-completeness [2,3].
Many attempts in the literature have been made to solve DSC problem in WSNs using
either heuristic or meta-heuristic methods (like genetic algorithms). Different scheduling
rules determine when sensors change to be active or sleep mode. In localized and distributed
realizations, sensors periodically investigate their neighborhood and decide whether to change
their operation modes [4–11].
In [2], a heuristic approach called the “most constrained-minimally constraining covering
(MCMCC)” is proposed to select and successively activate mutually exclusive sets of covers,
where every set completely covers the entire monitored area. This method gives priority
to sensors which cover a high number of uncovered fields, cover sparsely covered fields
and do not cover fields redundantly. This method achieves energy savings by increasing the
number of disjoint covers. Also, the DSC problem and its extension to the maximum set
cover (MSC) problem have been solved in [12]and[3] using integer programming, linear
programming, and greedy heuristics. In [12], the DSC problem is reduced to a maximum
flow problem and solved using mixed integer programming. By a branch and bound method,
the maximum covers based on mixed integer programming algorithm (MC-MIP) acts as an
implicit exhaustive search to guarantees finding the optimal solution.
The definition of DSC problem has been re-formulated in [3,13], and [14] to include
additional coverage constraints. The definition of DSC problem has been generalized in [3]
to a maximum set covers (MSC) problem and solved it using linear programming and greedy
techniques. The extended problem in MSC lets the sensors to participate in multiple sets. In
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[13], the DSC problem has been extended to include connectivity constraint as well. Then,
the Connected Set Covers (CSC) problem has as objective finding a maximum number of set
covers such that each sensor to be activated should be connected to the base station. In [14],
DSC problem has been extended to include sensor coverage-failure probability. Each sensor
is associated with sensor’s failure probability (comes from several facts, e.g., manufacture,
weather in the monitoring area, interferences to the sensors, or unexpected accidents). The
proposed Maximum Reliability Sensor Covers (MRSC) problem has been solved in [14]
using a heuristic greedy algorithm to compute the maximal number of set covers that satisfy
a user specified coverage-reliability threshold.
The works in [15–17] also provide solutions to the DSC problem in WSNs by employing
the meta-heuristic framework of evolutionary and genetic algorithms. Like the previous
mentioned heuristic methods, the single-objective genetic algorithms proposed in [15–17]
assume simple and common isotropic (i.e., disc) sensing model. Each sensor in this model is
associated with a sensing area which is represented by a circle with the same radius. According
to [18], the upper bound, ub, of the maximum number of disjoint complete set covers depends
on the size of the target area, the total number of sensors, the sensors’ locations, and their
sensing ranges.ub can be estimated as the minimum number of sensors covering the most
sparsely covered target, and thus can be computed by polynomial-time algorithms. It has
been shown that for the same deployment of sensors and targets, ub can be increased by
enlarging the sensing radius of each sensor [15–18]. In reality, however, the sensing region
of a sensor could be irregular, resulting in imprecise target coverage. The coverage in this
case could be expressed in probabilistic terms [19–21]. In probabilistic sensing model, there
is a measure of uncertainty in sensor signal-detection being expressed by a value from 0 to
1. For reliable coverage with threshold cth, the detection uncertainty of each target should
not exceed 1 −cth. This in turn should effect on the number of sensors reliably covering
the most sparsely covered target. As more reliability degree to cover a target is required,
ub becomes smaller. Thus, there should be a trade-off between the coverage reliability (also
denoted throughout this paper as covers reliability)andub.
Like many other sensor network design problems, maintaining reliable coverage while
maximizing the number of DSC (i.e., extending network lifetime) often require optimiz-
ing two conflicting decision variables. Due to the conflicting nature of many sensor net-
work design problems, multi-objective optimization (MOO) has, recently, attracted several
researchers in formulating multi-objective optimization problems (MOPs). In MOO, a set of
alternative solutions (called non-dominated solutions) can simultaneously be obtained pro-
viding the decision maker with an optimal tradeoff between the conflicting objectives. As
multi-objective evolutionary algorithms (MOEAs) possess several interesting characteristics
for tackling MOPs, there are many MOEA approaches for solving different sensor design
problems. For organizing energy efficient WSNs, the literature provides a collection of MOO
schemes in which sensing coverage and network lifetime are the main design issues to be
optimized. However, all of these methods fall in one of the first two categories; data aggre-
gation, routing, and sensor clustering [22–25], or sensors’ power management [26–29]. For
the reliable DSC design problem in WSNs, our work in [30] formulates a single objective
evolutionary algorithm to tackle it. To the best of our knowledge, however, MOO concept
has not been exploited. The contributions of this paper are as follows:
1. The paper turns the de-facto definition of single-objective DSC in WSN design problems
to a multi-objective DSC. The goal of which becomes to cope with contradictory objec-
tives. The formulated optimization problem will be directed to effectively maintaining
reliable coverage with large number of cover sets in WSNs.
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2. The paper presents two MOEAs with a new heuristic perturbation operator (instead of
the traditional bi-perturbation of crossover and mutation operators) to solve the modeled
problem. The results show that the Pareto-optimization can provide the decision maker
with a set of non-dominated solutions that effectively render the tradeoff between the
two conflicting objectives (i.e., network lifetime and coverage probability).
The rest of this paper is outlined as follows. Section 2describes the traditional formulation
of DSC problem in WSNs as well as the new formulation of multi-objective DSC by adding
probabilistic coverage means to the original problem. Section 3provides the preliminaries to
the concept of multi-objective optimization and introduces the formulation of the proposed
perturbation operator. The results of multi-objective DSC optimizations are then compared
with a related work of single-objective genetic algorithm in Sect. 4. Finally, Sect. 5concludes
the current work and hints some further ramifications.
2 Problem Definition
2.1 System Model and Assumptions
In order to model the system, we assume that the investigated WSNs have 2D sensing area
Awith known size(Xmax,Ymax ). We also assume that Ahas a set T(i.e., target set) of n
targets with known locations, i.e., T={
(xt1,yt1),(xt2,yt2),...,(xtn,ytn)}.Therearem
homogenous sensors S={
(xs1,ys1),(xs2,ys2),...,(xsm,ysm)}having the same sens-
ing range rs. All the sensors are dropped randomly in A(1≤∀i≤m|(xsi,ysi)=
([0,Xmax],[0,Ymax ])). Depending on the sensing range rs, each sensor is responsible for
sensing and covering a part of A. We consider a probabilistic sensing model [20,21]todefine
the notion of the probabilistic coverage of a target Tj=xtj,ytjby a sensor si.
Coverage si,tj=⎧
⎨
⎩
0if r
s+ru≤d(si,tj)
e−λaβif r
s−ru<d(si,tj)
1if r
s−ru≥d(si,tj)
<rs+ru(1)
where ruis a measure of the uncertainty in sensor detection. dsi,tjis the Euclidean distance
(xsi −xtj)2+(ysi −ytj)2between sensor siand target tj.a=d(si,tj)−(rs−ru),and
λand βare probabilistic detection parameters to measure detection strength when a target
point lies within the interval {rs−ru,rs+ru}. It causes coverage value to exponentially
decrease as the distance increase. All points that lie within a distance of rs−rufrom the
sensor are said to be 1-covered. Beyond the distance rs+ru, all the points have 0-coverage
by this sensor.
To save energy and prolong WSN’s lifetime, sensors in the sensor set Sshould be divided
into duty-cycling sensor cover (also called set cover) subsets, each of which can cover all
the interested targets in T. Thus, in the traditional Boolean sensing model, the definition of
the sensor cover could be formulated as:
Definition 1 (Sensor Cover). Given a WSN consists of target set Tand sensor set S,where
each sensor si∈Scan be represented as a subset Ti⊂T, such that tj ∈Tiif and only if
Coverage si,tj=1. Any subset Si⊂Sthat can completely cover all the target set Tis
termed as a sensor cover.
However, considering probabilistic sensing model, the definition of the traditional sensor
cover needs to be re-formulated as:
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Definition 2 (Reliable Sensor Cover). Given a WSN consists of target set Tand sensor set
S, where each sensor si∈Scan be represented as a subset Ti⊂T, such that tj ∈Tiif
and only if Coverag e si,tj≥cth. Any subset Si⊂Sthat can satisfy a user coverage
constraint cth to cover all the targets in Tis termed as a reliable sensor cover or reliable set
cover.
Now, the problem of finding the maximum number of disjoint set covers (DSC) could be
turned into the problem of finding the maximum number of disjoint reliable sensor covers
(DRSC), and can be formulated as:
Definition 3 (Disjoint Reliable Sensor Covers Problem—DRSC). Given a collection Sof
reliable subsets of a finite set T, find the maximum number of disjoint reliable covers for
T. Every cover Siis a subset of S,Si⊆S, such that every element tj of Tbelongs to at
least one member of Si, and for any two covers Siand Sj,Si∩S
j=∅. Now, the upper bound
ub of the maximum number of disjoint reliable covers is determined by the most sparsely
covered target and can be calculated as:
ub =min
i=1,2,...,n(|Si|)(2)
where CS =Si,|Si|=ub denotes the set of sensors (called Critical sensors Set) covering
target ti with certainty equal to or larger than a user specified threshold cth.Asthemost
sparsely covered target only covered by ub. Sensors, the maximum number of disjoint reliable
covers Nis no larger than ub i.e., N≤ub).
2.2 Problem Formulation
In this section, we formulate the problem of getting high coverage percentage with maximum
number of DSC as multi-objective Disjoint Reliable Sensor Covers Problem:
Given:
A: 2-D plane area with size(Xmax,Ymax).
T:setofntargets being uniformly distributed in A.
S:setofmsensors with probabilistic sensing capability, being uniformly distributed in
A.
rs: sensing range.
ru,λand β: uncertainty-parameters adjusted according to the physical properties of the
sensor.
Decision (Design) variables of a solution:
The scheduling sets of sensors in S, i.e., {S1,S2,...,Sm}|Si∈[1,ub].
Objective:
Two objectives are considered as optimization functions. The first objective is to maximize
the number of scheduling sets of the sensors in S, such that each scheduling set Sican form
a complete reliable sensor cover (see Eq. 3). Thus, Eq. 3computes the number of reliable
covers (i.e., without and coverage holes) being formed by a particular solution. Detection of
coverage holes is formulated in Eq. 4. The second objective is to maximize the reliability
of the whole disjoint sensor covers (see Eq. 5). In each sensor cover, we consider only the
highest coverage of the sensors belong to this cover to each target in set T.
max DRSC =Si∈[1,ub]Cover(Si,T)(3)
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where:
Cover (Si,T)=1if ∀t∈T→∃s∈Si|Coverage (s,t)≥cth
0otherwise (4)
max Reliability =
Si|Cover(Si,T)=1
Coverage(Si,T)(5)
where:
Coverage (Si,T)=t∈Tmax
s∈SiCoverage (s,t)
n(6)
3 Multi-objective Evolutionary Algorithms for DRSC Problem
3.1 Preliminaries
The general formalization of MOP is defined as [31,32]: finding a vector X∗=
[x∗
1,x∗
2,...,x∗
n]Toptimizing (in terms of domination) the vector function F(X)=
[f1(X),f2(X),...,fk(X)]Twhere X=[x,
1x2,...,xn]Tis the decision variables vector.
Optimizing F(X), here, considers the domination concept. Let us consider two solution U
and Vfrom the solution space P. Then, solution Uis said to dominate Vif and only if the
following two conditions hold:
1. The solution Uis no worse than Vin all objectives, or fi(U)fi(V)for all i=
1,2,...,k. For example in maximization, the word “no worse” means fi(U)≮fi(V).
2. The solution Uis strictly better than Vin at least one objective, or fi(U)fi(V)for at
least one i∈1,2,...,k. For example in maximization, the word “strictly better” means
fi(U)>fi(V).
The notation (ij)is used to denote that solution iis better than solution jregardless of the
type of the optimization problem at hand (maximization or minimization). Also, the notation
(ij)is used in the same way to express that solution jis better than solution i. Hence, a
dominated set can be defined as: among a set of solutions P, the non-dominated solutions
set ´
Pare those that are not dominated by any member of the set P.
In this paper, we consider two multi-objective evolutionary algorithms (MOEAs). These
are: multi-objective evolutionary algorithm with Tchebycheff decomposition (MOEA/D) and
non-dominated sorting genetic algorithm (NSGA-II). In what follows, we only present these
two algorithms focusing on the proposed perturbation operator. We refer interested readers to
[31–35] for more detailed description of the general framework of MOEA/D and NSGA-II.
3.2 The Proposed Meta-heuristic Framework
As MOEA/D and NSGA-II are population-based optimization algorithms, let us consider
a population ρof Ksolutions. The choice of a good solution representation is a critical
issue for the applicability and performance of evolutionary algorithm. Solution representa-
tion is highly problem dependent and related to the evolution operations. In our algorithm
design, each individual solution P1≤i≤K∈ρis represented as a fixed-length vector of size
m, where each element (i.e., gene) value indicates the index of the set cover that the corre-
sponding sensor joins, specifying that the index does not exceed the upper limit ub. Thus,
∀i∈{1,...,N}and ∀j∈[1,m]:Pi=(Pi1,Pi2,...,Pim)s.t.Pij ∈[1,ub]. Then, each of
the implemented MOOs can be described as a process formulated in an iterative evolution
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function :{ρ,EP}→{ρ,EP}with (ρi)=ρi+1,whereiis the iteration index and
ρiis the population in iteration i.EP is an external archive for the generated non-dominated
solutions. The population starts with an initial random population ρ0and continues until a
maximum number of iterations maxihas been reached. Each iteration iconsists of a three
main operators: parents selection, heuristic perturbation, and EP update.
Based on the chromosome representation, we adopted an intuitive and fast random ini-
tialization mechanism, where each gene can take a random value that does not exceed the
upper limit ub. Formally speaking:
(∀i∈{1,...,N},∀j∈[1,m]):Pij ∼[1,ub](7)
where ∼[1,ub]means a uniform random integer number between 1 and ub.
For selection operator, we imitate similar functions to that found in the implementation
of the traditional MOEA/D and NSGA-II. For the perturbation function, we proposed one
evolution operator that provides heuristic for improving a solution coverage reliability
or number of disjoint covers. The process of the proposed heuristic perturbation operator
is presented next (see Algorithm 1). It takes as input two parent individuals P1,P2being
selected from ρby the selection operator and perturbs them with probability pper =1.0to
return one child Cwith its decoded set covers C1,C2,...,Ck. For each child, the proposed
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heuristic will be directed (with equal probability) either towards maximizing the reliability of
the generated set covers or towards maximizing the number of sensor covers k. Thus, for each
child Ci,1≤i≤K, the heuristic procedure also takes as input a flag Findicating whether to
prefer maximizing reliability or maximizing number of sensor covers. The proposed heuristic
also takes as input the set of the critical sensors CS.
The proposed heuristic recursively builds sensor covers in the child. For each cover i,
the set S(in line 6) collects sensors from both parents that have an index value equal to i.
The sensors in Scan then participate in creating the cover iin the child. When sensors in
Scan cover all the targets without any coverage hole (in line 8), then (and according to the
flag F), the procedure will select from Sthose sensors having the highest contribution to the
heuristic (line 12–18). In order to maximize number of sensor covers towards its upper limit
ub, a heuristic process is maintained in line 15. Note that this process is guided to restrict
the choice of sensors, for the current sensor cover i , to those that do not belong to the set of
critical sensors (i.e., s /∈CS) or at least bias the choice in their support. The biasing depends
on the number of targets being covered by each sensor s/∈CS. However, if there is still
coverage hole in the current cover i, the heuristic process turns the selection to the critical
sensors s∈CS. The remaining sensors can then contribute in the next covers (line 20). They
will migrate to the last set cover kand used again (line 23) when there is a coverage hole
(line 21) while forming a new cover i. Finally, the last cover (line 25) may form a complete
set cover or incomplete one. The generated child Cmay modify the contents of the external
archive EP according to the non-domination condition described previously.
4 Simulation Results and Discussion
In this section, we evaluate the performance of the proposed MOO algorithms in terms of
number of set covers obtained and coverage reliability. The results are obtained after setting
WSNs and algorithm parameters into the most commonly used in the related literature. The
simulation area is square-shaped with side length Xmax = 500 m of 10 randomly distributed
targets. Unless otherwise stated, the simulation is mainly divided into five groups according
to five different settings of sensor density: 50, 60, 70, 80, and 90. For each group, we vary the
sensing range of the sensor nodes Rsto three different values {100,200,300},togetadifferent
test instance. Each test instance includes 10 random WSNs with different configurations. Thus
the overall simulation examines a total of 150 random networks. As we have random WSN
configurations, we may get different value for the upper bound of set covers ub,forthe
same parameter settings. Thus in each test instance, the presented results also indicate the
average number of upper bound of set covers (denoted by ub). Uncertainty level Ruis set
to Rs∗0.5 units, both λand βare set to 0.5, and cth is set to 0.001. The setting of the
probabilistic coverage parameters also influences the overall network’s coverage reliability.
Since studying the impact of varying these parameters is out of the scope of this paper, we
fixed these parameters to one setting and evaluate the average coverage reliability (denoted
by ¯r) of all network configurations in each test instance. Population size is set to 50 and will
be allowed to evolve 500 times. Perturbation probability is fixed to 1.0. For MOEA/D, the
size of neighboring solutions that participate in the evolution of each individual solution is
fixed to 5.
Both MOEA/D and NSGA-II have been compared with Genetic Algorithm for Maximum
Disjoint Set Covers (GAMDSC). GAMDSC has been proposed in [15] to extend the lifetime
of WSNs. GAMDSC is single-objective genetic algorithm that use integer representation
for chromosome representation, traditional uniform crossover, creep mutation, and a scatter
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operator to evolve their chromosome solutions. The proposed fitness function in GAMDSC
has been devoted to maximize number of complete set covers in each chromosome. The per-
formance of GAMDSC has been compared in [15] against the most constrained-minimum
constraining heuristic (MCMCC) [2] and the exhaustive heuristic of Maximum Covers using
Mixed Integer Programming (MCMIP) [12] for solving the DSC problem in WSNs. Simu-
lation results in [15] show that GAMDSC can get near-optimal solutions and improve the
performance of MCMCC by 16% in terms of the obtained sensor covers. As compared with
the exhaustive search of MCMIP, GAMDSC can get results with acceptable computation
time.
4.1 Comparison of Multi-objective Versus Single-objective DSC
First, results are presented in Tables 1and 2to compare the quality of results obtained by
GAMDSC, MOEA/D, and NSGA-II. Table 1presents number of set covers being obtained
by the three algorithms. Table 2presents the coverage reliability being obtained by the
algorithms. Also, to examine the quality of solutions provided by the three algorithms, we
present ub and ¯ras reference values in Tables 1and 2, respectively.
For both MOEA/D and NSGA-II, Tables 1and 2present the two extreme non-dominated
solutions (in terms of number of set covers and reliability, respectively). The solution pro-
viding the closest value to ub in each test instance in Table 1has been presented in bold.
Also, the solution providing the maximum reliability in each test instance in Table 2has been
bolded. Note that in Table 1, closest solutions to ub do not necessary imply best solutions. In
terms of the algorithm’s independent single-objective measure of solution’s quality, closest
solutions to ub should result in high quality solutions. That is GAMDSC provides more
sensor covers in four test instances while NSGA-II provides more sensor covers in almost all
of the remaining test instances. However, seeking for tradeoff solutions that balance between
the sensor covers and coverage reliability (see Table 2), the performance of GAMDSC in
these four test instances is not good enough. The coverage reliability of GAMDSC gives the
lowest score while both MOEA/D and NSGA-II outperform GAMDSC in all test instances.
Similarly, in the third and tenth test instances (referred with asterisk), GAMDSC gives equal
values of sensor covers to NSGA-II or MOEA/D, respectively (as presented in Table 1), how-
ever, both NSGA-II MOEA/D outperform GAMDSC in terms of reliability (as presented in
Tabl e 2). Thus, the combined results of Tables 1and 2signify the importance of utilizing
the two objectives as optimizing criteria rather than maximizing only the number of sensor
covers. Moreover, in all test instances, Table 2presents that NSGA-II performs better than
MOEA/D even in the case of providing an equal number of set covers as in the sixth test
instance, referred with asterisk in Table 1.
Figure 1compares the performance of GAMDSC, MOEA/D, and NSGA-II in terms
of average number of disjoint sensor covers obtained while fixing sensing radius. In this
simulation, we fixed sensing range Rsto 500 and vary number of sensors from 50 to 90 with an
incremental value of 5. Each test instance of this simulation includes 10 random networks, and
thus the total simulation is experimented under test-bed of 90 different network topologies.
This simulation clarifies the impact of varying sensing ranges, of a given fixed number and
fixed locations of sensors, on the total number of disjoint sensor covers. The results of this
simulation summarize that for a fixed sensing radius, the number of sensor covers increases
with increasing number of sensors. Also, from the figure we can that NSGA-II provides more
number of sensor covers than both MOEA/D and GAMDSC.
On the other hand, Fig. 2clarifies the impact of varying sensing range Rswhile fixing
number of sensors. In this simulation, the number of sensors is set to 70 and let Rsto vary
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Tab l e 1 Comparison of the maximum number of the generated set covers for 10 WSNs in each test case
Test instance No. of sensors Rsub GAMDSC MOEA/D NSGA-II
1 50 100 5.00 4.80 4.70 4.60
1.10 1.30
2 200 17.30 15.80 15.10 15.00
6.80 5.90
3 300 23.10 21.20 21.10 21.20∗
9.50 8.20
4 60 100 6.50 5.50 5.60 5.60
1.30 1.80
5 200 22.30 18.90 19.20 19.60
8.90 7.30
6 300 30.00 25.80 26.30 26.30∗
13.40 9.90
7 70 100 7.70 6.70 6.70 6.80
1.80 1.80
8 200 23.80 20.80 20.60 20.90
10.10 8.30
9 300 32.10 28.10 29.00 29.40
14.60 11.10
10 80 100 9.50 8.30 8.30∗8.20
2.10 1.80
11 200 31.20 25.80 25.90 26.50
13.70 10.60
12 300 41.00 34.10 36.10 36.40
18.50 14.50
13 90 100 9.10 8.50 8.40 8.30
3.00 3.40
14 200 30.80 27.10 26.50 26.40
14.80 11.40
15 300 42.10 36.30 36.70 37.90
20.40 16.20
For MOEA/D and NSGA-II, results are presented in terms of max and min DRSC solutions in EP
from 100 to 500 with an incremental value of 50. Also, in this simulation, each test instance
includes 10 random networks with a total of 90 different network topologies. The results of
the figure indicates that increasing sensing radius have a positive impact on the number of
sensor covers, even when fixing the number of sensors to a certain value. Also, we can see
that NSGS-II provides more number of sensor covers than both MOEA/D and GAMDSC.
4.2 Quality and Diversity of Pareto Front Solutions
Let us now turn to quantitatively compare the performance of MOEA/D and NSGA-II.
Here, the comparison will be according to the quality of the non-dominated solutions set
being obtained by each algorithm. The performance of MOEA/D and NSGA-II can also be
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Tab l e 2 Comparison of the maximum covers reliability provided for 10 WSNs in each test case
Test instance No. of sensors Rs¯rGAMDSC MOEA/D NSGA-II
1 50 100 0.1867 0.3932 0.6971 0.7027
0.4038 0.4052
2 200 0.2114 0.3365 0.6447 0.7542
0.3463 0.3489
3 300 0.3216 0.4805 0.8049 0.9000
0.4783 0.4933
4 60 100 0.1731 0.3295 0.6066 0.6168
0.3661 0.3584
5 200 0.2000 0.3175 0.6069 0.7062
0.3273 0.3187
6 300 0.3206 0.4533 0.7501 0.8784
0.4545 0.4708
7 70 100 0.1754 0.3020 0.6275 0.6531
0.3502 0.3482
8 200 0.2090 0.3367 0.6265 0.7340
0.3357 0.3422
9 300 0.3156 0.4644 0.7706 0.8932
0.4701 0.4768
10 80 100 0.1888 0.3245 0.6221 0.6753
0.3620 0.3711
11 200 0.2072 0.3220 0.5968 0.7229
0.3422 0.3365
12 300 0.3267 0.4799 0.7665 0.8895
0.4764 0.4857
13 90 100 0.1972 0.3796 0.6549 0.6787
0.4263 0.4336
14 200 0.2214 0.3683 0.6163 0.7521
0.3695 0.3910
15 300 0.3384 0.5168 0.7822 0.8772
0.4999 0.5176
For MOEA/D and NSGA-II, results are presented in terms of max and min Reliability extreme solutions in EP
verified visually by depicting the Pareto Front EP. According to [36], if Aand Bare two
non-dominated sets resulted from two MOEAs Aand B, respectively, then, the set coverage
quality metric (C-metric) will express the fraction of solutions in one non-dominated solution
set that are dominated by those solutions of the second algorithm. The C-metric is defined
as:
C(A,B)=|b∈B|∃a∈A:ab|/|B|(8)
where reads dominate. Note that:
•C(A,B)= 1−C(B,A), and
•Ais better than Bif C(A,B)is found to be higher than C(B,A)over many trials.
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50 55 60 65 70 75 80 85 90
0
10
20
30
40
50
60
70
Number of Sensors
Avg. Number of Sensor Covers
NSGA-II
MOEA/D
GAMDSC
Fig. 1 Average number of sensor covers for 90 networks with Rs=500 and number of sensors=
{50,55,...,90}. In each group of bars:left,middle,andright bar corresponds to NSGA-II, MOEA/D,
and GAMDSC results, respectively
100 150 200 250 300 350 400 450 500
0
5
10
15
20
25
30
35
40
45
50
Sensing Redius
Avg. Number of Sensor Covers
NSGA-II
MOEA/D
GAMDSC
Fig. 2 Average number of sensor covers for 90 networks with number of sensors =70andRs=
{100,150,...,500}. In each group of bars:left,middle,andright bar corresponds to NSGA-II, MOEA/D,
and GAMDSC results, respectively
The second measure is the domination metric Dom which is defined as:
Dom (A,B)=d(A,B)
d(A,B)+d(B,A)(9)
where d(X,Y)=x∈X|{y∈Y|xy}|, is known as the domination factor. Mutually
non-dominating pairs are ignored in calculating the dominance factor d(A,B). If each solu-
tion in Adominates every solution in B,then Dom (A,B)=1. Thus, Dom (A,B)=
1−Dom(B,A).Table3presents these two metrics when comparing the quality of non-
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Tab l e 3 Comparison of coverage set and domination metric of MOEA/Dversus NSGA-II for 10 WSNs in
each test case
Test instance No. of sensors RsC(A,B)C(B,A)Dom(A.B)Dom(B,A)
1 50 100 0.4897 0.4100 0.5168 0.4832
2 200 0.3895 0.4998 0.4466 0.5534
3 300 0.2609 0.6408 0.3059 0.6941
4 60 100 0.4450 0.3226 0.5386 0.4614
5 200 0.3137 0.5704 0.3595 0.6405
6 300 0.2330 0.6198 0.2844 0.7156
7 70 100 0.3185 0.5356 0.4022 0.5978
8 200 0.2277 0.6786 0.2632 0.7368
9 300 0.2327 0.5813 0.3387 0.6613
10 80 100 0.3829 0.4071 0.4981 0.5019
11 200 0.3152 0.4757 0.4468 0.5532
12 300 0.1732 0.6750 0.2125 0.7875
13 90 100 0.2883 0.6171 0.2994 0.7006
14 200 0.3394 0.4477 0.4597 0.5403
15 300 0.2394 0.5576 0.2664 0.7336
2 2.5 3 3.5 4 4.5 55.5 6
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
DRSC
Reliability
Fig. 3 Pareto solutions of MOEA/D(black ball)andNSGA-II (+), for one network with number of sensors
=50 and Rs=100
dominated set obtained by MOEA/D (denoted as A)against those of NSGA-II (denoted
as B). Also, here best results are given in bold.
From the results, we can consider that on overall, NSGA-II performs better than MOEA/D,
except for two test instances (i.e., test instance 1 and 4) where there is a slight difference
between the solutions quality attained by the two algorithms.
For the purpose of illustration, Figs. 3and 4depict the non-dominated solution sets
provided by the both algorithms for two WSNs belong to the two extreme test instances
experimented in our simulations (i.e. test instance 1 and 15, respectively). x,y−coordinates
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15 20 25 30 35 40 45
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
DRSC
Reliability
Fig. 4 Pareto solutions of MOEA/D(black ball)andNSGA-II (+), for one network with number of sensors
=90andRs=300
in the figures correspond to number of set covers (DRSC) and total coverage reliability. In the
first test instance, the average number of upper bound of sensor covers ub is 5.00. Projection
in Fig. 3shows the cardinality and quality of both MOEA/D’s and NSGA-II’s Pareto Fronts.
The number of non-dominated solutions being obtained by both algorithms is similar (i.e.,
|A|=|B|=5). Also, we can see that both Aand Bhave nearly equal quality.
However, the dissimilarity in the quality and number of solutions in Aand Bbecomes
more apparent while increasing ub to its extreme in the last test instance (i.e., 42.10). For
this test instance, Fig. 4clarifies that NSGA-II discovers more non-dominated solutions
(|B|=24 against |A|=12). Also, we can see that more than 70 % of solutions in Bare
with better quality than those in A. On the other hand, about 25 % of MOEA/D solutions are
with better quality than those obtianed by NSGA-II.
Figures 5and 6also compare the quality and diversity (or width) of Pareto Front solutions
of both NSGA-II and MOEA/D in terms of the extremes solutions found there (i.e., minimum
and maximum values). Here, 9 simulations examine 90 different WSNs (each simulation with
10 networks) with the same sensing radius setting (Rs=500), but with different number
of sensor nodes ({50,55,...,90}). Thus, each simulation carries out a couple of extremes
averaged over 10 different networks with certain number of sensors and fixed value of Rs.
Figure 5depicts the extremes in terms of number of sensor covers, while Fig. 6depicts the
corresponding extremes’ coverage reliability (covers reliability). For complete illustration,
we also present the best results delivered by GAMDSC in both figures. Moreover, for more
clarity, the overall statistics of NSGA-II, MOEA/D and GAMDSC are presented in Table 4.
According to the tradeoff between the two optimization objectives (i.e., number of sensor
covers and covers reliability), an extreme solution (in Fig. 5) which is dedicated on maxi-
mizing the number of sensor covers is connected to the extreme solution (in Fig. 6)thatis
dedicated to the minimum coverage reliability and vice versa.
From the results of both figures, we can see that NSGA-II provides a wider Pareto Front
than MOEA/D and with better quality. The detailed results of Fig. 5clarifies that the aver-
age distance between the extremes of NSGA-II is about 28.26 sensor covers, while for the
extremes of MOEA/D is 22.68 sensor covers. Also, the results reveals that the maximum
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50 55 60 65 70 75 80 85 90
15
20
25
30
35
40
45
50
55
60
65
Number of Sensor Nodes
Number of Sensor Covers
GAMDSC
NSGA-II (Max)
MOEA/D (Max)
NSGA-II (Min)
MOEA/D (Min)
Fig. 5 Pareto extreme solutions (in terms of number of sensor covers) of MOEA/D(+)andNSGA-II (filled
circle). GAMDSC’sbest solutions (empty circle) are also included in the figure. The simulation is experimented
under 90 networks with Rs=500 and number of sensors ={50,55,...,90}
50 55 60 65 70 75 80 85 90
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Numebr of Sensors
Covers Reliability
GAMDSC
MOEA/D (Max)
NSGA-II (Max)
MOEA/D (Min)
NSGA-II (Min)
Fig. 6 Pareto extreme solutions (in terms of covers reliability) of MOEA/D(+)andNSGA-II (filled circle).
GAMDSC’s best solutions (empty circle) are also included in the figure. The simulation is experimented under
90 networks with Rs=500 and number of sensors ={50,55,...,90}
number of sensor covers provided by MOEA/D ranges from 34.8to61.8 with an average
of 49.12. On the other hand, NSGA-II provides more number of sensor covers ranges from
35.2to63.9 with average of 50.34. Both MOEA/D and NSGA-II gain more sensor covers
than GAMDSC. In Fig. 5, we can see that GAMDSC can provide from 33.8to60.6 with
average of 47.91 sensor covers.
Similarly, Fig. 6shows that NSGA-II provides wider Pareto Front than MOEA/D. Also, in
Fig. 6we can see that all algorithms have nearly equal reliability performance represented by
min extremes of MOEA/D and NSGA-II and by the best results of GAMDSC. Knowing that
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Tab l e 4 Overall statistics of Pareto extremes of MOEA/D and NSGA-II and the best solutions of GAMDSC
giveninFigs.5and 6
Objective Algorithm Extreme max min average
No. of sensor covers GAMDSC – 60.633.847.91
MOEA/Dmax 61.834.849.12
min 34.518.126.44
NSGA-II max 63.935.250.34
min 28.716 22.08
Covers relaibility GAMDSC – 0.6737 0.6217 0.644
MOEA/Dmax 0.8913 0.853 0.8726
min 0.6602 0.6077 0.6364
NSGA-II max 0.949 0.9211 0.9403
min 0.6699 0.6156 0.6456
these reliability results are associated with the max extremes (in terms of number of sensor
covers) of MOEA/D and NSGA-II and with the best results of GAMDSC in Fig. 5,wecan
then realize that NSGA-II facilitate more qualitative sensor covers. In contrast, GAMDSC is
incurred by the lowest number of sensor covers. This reflects the positive impact of the pro-
posed heuristic perturbation operator over the combined collaboration of crossover, mutation,
and the scattering operators of GAMDSC algorithm.
4.3 Comparison of Worst Case Time Complexity
We also compare the worst-case computational complexity of the single-objective GAMDSC
algorithm and the multi-objective MOEA/D and NSGA-II algorithms for solving Disjoint
Reliable Sensor Covers Problem (DRSC). Due to optimizing multiple objectives instead of
only one objective, multi-objective evolutionary algorithms, in general, are more expensive
than single-objective algorithms. Without loss of generality, the computational complexity of
single-objective genetic algorithms, such as GAMDSC is O(maxi×K)where Kis the size
of solutions to be evolved, via certain evolution operators, for maxiiterations. On the other
hand, the general computational complexity of an MOEA/D is O(maxi×2×K×Kneighbor)
(where 2 means number of objective functions and Kneighbor <Kmeans size of neighboring
solutions for each individual solution) [35]. The computational complexity of NSGA-II is
O(maxi×2×K2)[32,33].
Now, let us consider the computation time needed for the most critical parts of the above
algorithms when applied for solving DRSC problem. It is stated in [15] that in each generation
of GAMDSC, the fitness function is the most critical part which represents the computation
time bottleneck. The evaluation of the fitness function is linearly related to the number of
sensors and the number of targets. Thus, the worst-case time complexity of GAMDSC is:
O(GAMDSC)=maxi×K×|n|×|m|(10)
where |n|is the number of targets and |m|is the number of sensors. On the other hand, for
MOEA/D and NSGA-II, the proposed heuristic perturbation operator (Algorithm 1) costs
the most computation time and in the worst-case time it linearly related to the maximum
number of sensors and targets, and upper bound of sensor covers ub. Thus, the worst-case
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time complexity of MOEA/D and NSGA-II are, respectively, as:
O(MOEA/D)=maxi×2×K×Kneighbor ×|n|×|m|×|ub|(11)
O(NSGA-II)=maxi×2×K2×|n|×|m|×|ub|(12)
5 Conclusions
Maximum DSC is a well-known WSN design which is NP-hard. Unlike all of the existing
work that rely on the traditional single-objective definition, this paper introduces a multi-
objective Disjoint Reliable Sensor Covers Problem (DRSC). The formulated problem is then
tackled by two well-known and competitor multi-objective evolutionary algorithms, namely
MOEA/D and NSGA-II. Instead of the traditional use of two perturbation operators (i.e.,
crossover and mutation), one heuristic operator is proposed that involves the incorporation
of problem-specific knowledge in its mechanism. The proposed heuristic operator is proved
to be of positive effect on the overall performance of the implemented MOEAs. The results
over 150 different WSN topologies signifies the importance of utilizing the two contradictory
criteria of getting large number of disjoint sensor covers and more coverage reliability and
optimizing them simultaneously rather than optimizing only number of disjoint sensor covers.
Good overall performance of the multi-objective algorithms for the modeled DRSC highlights
that it may be motivated to consider another WSN’s design criterion while solving disjoint
sensor covers problem. As an example, we can include communication constraints (e.g.,
sensor-sink connectivity) to the multi-objective definition of disjoint sensor covers. Also, by
satisfying the sensor energy constraints, a sensor can participate in multiple sensor covers
and the definition of DRSC could be reformulated to satisfy a multi-objective non-disjoint
sensor covers (nDRSC) problem.
Acknowledgments This work is partially supported by TUBITAK under Grant No. 113E328.
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Doctoral dissertation, Zurich: Swiss Federal Institute of Technology.
Bara’a A. Attea (a.k.a. Baraa A. Atiyah) received the BS and MS
degrees in computer science from University of Baghdad, Baghdad,
Iraq in 1993 and 1996, respectively, and the PhD degree in computer
science from University of Technology, Baghdad in 2002. She is an
associate professor in Computer Science Department - University of
Baghdad. She was a visiting researcher in Gazi University, Ankara,
Turkey from 2011 to 2012. Her main research interests are computa-
tional intelligence, evolutionary algorithms, data mining, and applica-
tions of bio-inspired algorithms in wireless sensor networks.
Enan A. Khalil received the BS, and MS in computer science from
University of Baghdad, Baghdad, Iraq in 2008 and 2011, respectively.
He is currently a PhD student in Computer Engineering Department -
Gazi University, Ankara, Turkey. His main interests are computational
intelligence, evolutionary algorithms, and wireless sensor networks.
Suat Özdemir has been with the Computer Engineering Department
in Gazi University, Ankara, Turkey. He received his MSc degree from
Syracuse University and PhD degree from Arizona State University
in 2001 and 2006, respectively. Dr. Ozdemir’s research areas mainly
include wireless and sensor networks, network security, and data min-
ing. He is a member of IEEE and currently serving as editorial board
or TPC member for various leading IEEE and ACM journals and con-
ferences.
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Oktay Yıldız received his M.Sc. degree in Institute of Science from
Gazi University, in 2004 and Ph.D. degree in Institute of Information
Sciences from Gazi University, in 2012. He has been with the Com-
puter Engineering Department at Gazi University, Ankara, Turkey since
2009. His research interests include machine learning, data mining and
bioinformatics.
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