Multi-objective optimization in wireless sensorsnetworks

Matthieu Le BerreLM2S, UMR CNRS 6279 STMR

University of Technology of TroyesTroyes, France

Email: matthieu.le berre@utt.fr

Faicel HnaienLOSI, UMR CNRS 6279 STMR

University of Technology of TroyesTroyes, France

Email: faicel.hnaien@utt.fr

Hichem SnoussiLM2S, UMR CNRS 6279 STMR

University of Technology of TroyesTroyes, France

Email: hichem.snoussi@utt.fr

Abstract—In most wireless sensor network (WSN), energy isa limited resource. Indeed, a sensor has a limited power source(like a battery). When this power source is empty, the sensorstops working. In this paper, we will present the modeling of amulti-objective problem. The first objective is the maximizationof the coverage of the area under time, the second objective isthe maximization of the lifetime of the network depending oncoverage, and finally the minimization of the financial cost (i.e.the number of sensors). The resolution of this problem will bedone by multi-objective algorithm NSGA-II [6], SPEA-II [5] andmulti-objective ant colony optimization (MOACO) [8].

I. INTRODUCTION

Wireless Sensor Network (WSN) are used in various fieldlike fire detection, wildlife monitoring, security monitoring[11]. A WSN is a set of sensors communicating with eachother, to share information like collected data, positions, orother. A sensor is a device constituted by four key features:(1) a sensor unit which captures information, (2) a radiounit which allows communication with other sensors, (3) aCPU for the preprocess of the information and (4) a powersource [7]. The radio unit allows communications with thestatic node (a unit with no limited energy, communicatingwith the user). All sensors have to be able to communicate,directly or indirectly, with this static node to send capturedinformation to the user. Many works have been done on WSN.In many papers, authors try to optimize either the coverage orlife, or both simultaneously. Optimizing coverage of a WSNrepresents the maximization of the proportion of the monitoredarea relative to the total area. Better is the coverage, greateris the probability of detecting an event. Another importantpoint in WSN is the lifetime. Indeed the sensor’s power source(usually a battery) is generally limited, and the radio unitand sensor unit are both energy efficient. Optimize energyconsumption is really important because of cost objective, andecologic impact (used battery are highly polluting). Authorsusually formulate this lifetime by the moment when one ofthe sensors stops working. In this paper, we will try to placesensors and to adjust their sensing radius for maximize twoobjectives: (1) the coverage (formulate by the ratio of thecovered positions by the total number of positions) and (2) thelifetime in the WSN while minimizing the number of sensors.However, we will introduce a new formulation of this problem,considering the evolution of the coverage under time, even

if one or more sensors stops working. We will compare theresults of the resolution of our problem by NSGA-II, SPEA-IIand MOACO.

II. STATE OF ART

There are some works about the application of multi-objective optimization on WSN. Usually, authors try to max-imize coverage and lifetime, or minimize number of sensorsindividually or both of these objectives.

A. Calculation of coverage

Given a set of sensors, find the optimal positions of thesensors is a NP-Complete problem [12]. There are manydifferent ways to calculate the coverage, in [2] and [1], theauthor defines the deployment and power assignment problem(DPAP). He use the following formulation to calculate thecoverage:

Let X a set of N sensors,∀si ∈ X, si = {xi, yi} (1)

Cv(X) =[∑xx′=0

∑yy′=0 g(x

′, y′)]

(x× y)(2)

where x× y is the total grids of the area and

g(x′, y′) =

{1 if ∃j ∈ {1, ..., N}, d(xj ,yj),(x′,y′) ≤ Rs0 otherwise (3)

where Rs is the range of a sensor. This formulation assumesthat if a event takes place within range of a sensor, this onewill inevitably detects the event. In [10], the author definesa problem where a lot of sensors have been deployed in aarea. He tries to maximize coverage and minimize the numberof active sensors. To calculate coverage, the author use aprobabilistic model. The coverage Cxy(si) of a grid point Pby a sensor si is expressed by:

Cxy(si) =

0 if r + re ≤ d(si, P )exp(−λaβ) if r − re < d(si, P ) < r + re1 if r − re ≥ d(si, P )

(4)

with α = d(si, P ) − (r − re) and re is a measure ofuncertainty in the sensor detection.

978-1-4577-2209-7/11/$26.00 ©2011 IEEE

In [9], the author use a significantly different formulation.Indeed, there are three objectives: the coverage rate expressedby the probability to sense an event, the financial cost charac-terized by the number of active sensors, and the minimizationof energy consumption considering the adjustable sensingradius of sensors. In our paper, we will try to consideratethese objectives differently.

B. Calculation of lifetime

To calculate the lifetime, the author calculates the minimumlifetime of sensors in the configuration, considering sensorunit and transmission consumption. The second objective(minimization of the number of active sensors) is similar tothe maximization of lifetime. Indeed, when the first set ofactive sensors stops working, a second set takes over. Sominimization of the set’s size allows a better lifetime in thenetwork. Similar works on lifetime can be found, like [3],where the author extends the problem of activation to the SETK-COVER problem.

III. PROBLEM FORMULATION

The problem takes place in a rectangular plane area. Wehave to place and set sensors to maximize three objectives:the coverage, the lifetime and the financial cost consideringthe number of deployed sensors.

H Height of the areaW Width of the areaRmin minimal radius of sensorsRmax maximal radius of sensorsNmin minimal number of sensors in a setNmax maximal number of sensors in a setSN the static nodeDmax maximal range of communication|.| the number of elements in the set

We have to define the formulation of a configuration (i.e. aset of deployed sensors). This formulation have to admit theuser can place the sensors in the area, and is be able to settheir radius.

Let S a set of sensors,

|S| ∈ {Nmin, ..., Nmax}∀si ∈ S, si = {xi, yi, ri}xi ∈ {1, ...,W}yi ∈ {1, ...,H}ri ∈ {Rmin, ..., Rmax}

In this formulation, the decision variables are the size ofthe set S (i.e. the number of deployed sensors), the positionsof sensors and their radius. Larger is the radius, better is thecoverage of the sensor, but shorter is its lifetime. We use thefollowing formulation :

Max f1(S(t)) = cov(S(t)) (5)Max f2(S) =t as (cov(S(t)) < ϕcov)

∧ (cov(S(t− 1)) ≥ ϕcov)(6)

Min f3(S) = |S| (7)

S.t. :∀si ∈ S(0), ∃S′ ⊆ S as S′ = {si, .., SN}∀sj ∈ S′/{SN}, dist(sj , sj+1) ≤ Dmax

(8)

Where :

cov(S(t)) =|positons covered(t)|

H ∗Wϕcov ∈ [0, .., 1]

dist(sl, sk) =√(xl − xk)2 + (yl − yk)2

The first objective function (see Equation 5) representsthe coverage over time : we have to maximize the coverageat every moment t. The second objective (see Equation 6)is the lifetime of the network. We admit the network isoperational while it ensures a coverage greater than or equalto a threshold ϕcov determined by the user. The third objective(see Equation 7) is the financial cost of the network, in otherwords the number of deployed sensors. The main constraintin our problem (see Equation 8) is the connectivity on thenetwork. Indeed, all sensors have to be able to communicatewith the static node SN in the initial state, which assuresthe connection with the user, and allows sending informationoutside the network. The communication can be direct (thedistance between the sensor and the static node is less orequal to Dmax) or indirect (data has to transit across thenetwork). Each sensor has an initial energy. When its energyis gone, the sensor is no more considered like an active nodein the network, so it is not considered in calculation of thecoverage. Furthermore, it can happen a disabled node provokesthe isolation of some sensors in the network (they are no longerable to communicate with the static node). In that case, allthese sensors are no longer considered by the network.

IV. IMPLEMENTATION

The resolution of this problem will be done by multi-objective genetic algorithms NSGA-II, SPEA-II and the multi-objective evolutionary algorithm MOACO.

A. NSGA-II/SPEA-II adaptation

The modeling of a set of sensors in a chromosome istrivial: for |S| sensors, the corresponding chromosome is avector of integers, with the first integer corresponds to thenumber of sensors |S| and each triplet represents the position(in two dimensions) and the radius of a sensor. The specificcrossover operator is an adaptation of the classic one-pointcrossover operator. Chromosomes have various size, so theclassic crossover is ineffective. To avoid this difficulty, weconvert chromosomes to a boolean representation of the area.

The crossover doesn’t be applied to the chromosome butthe area.

B. MOACO adaptation

The MOACO is an adaptation of the ant colony optimizationfor multi-objective optimization. To adapt our problem to theMOACO, we have to express the construction of configura-tions as a graph problem. The algorithm is : there are antcolonies, each colony is in a start node, and each ants ofa colony has to cross the graph to build a configuration,considering the objective of the colony (one of the objectivefunction). After the build of the configuration, the ant layspheromone on the ground, considering the evaluation of theconfiguration. In our formulation, each node n represents atriplet with positions and radius, and successors nodes to nare in communication range of n.

Let n a node, and succ(n) successors of n

∀n′ ∈ succ(n), dist(n, n′) ≤ Dmax

Start nodes represents positions in communication range ofSN , to avoid the constraint of connectivity.

V. EXPERIMENTATION

Algorithms have been implemented in the C++ language,run on a Core I5-2520M CPU,and compared to complete enu-meration for small instance problem.. Parameters of NSGA-IIand SPEA-II are following: population size is equal to 120, thecrossover probability is equal to 0.9, the mutation probabilityis equal to 1

|S| and the number of generations is set to 250.For MOACO algorithm, the number of ants is equal to 200and the number of cycles is set to 100. Instances are describedin following:

H W Nmin Nmax Rmin Rmax Dmax

Set1 10 10 4 7 1 8 3Set2 15 15 4 7 1 8 4Set3 20 20 5 10 5 10 7

The comparison between NSGA-II, SPEA-II and MOACOalgorithms will be done by the C measure introduced in [4].The following table displays the minimum, maximum and av-erage values of C(MOACO,NSGA− II) and C(NSGA−II,MOACO) over ten runs on each set.

Set1min avg max

C(NSGAII,MOACO) 0.90 0.96 0.99C(MOACO,NSGAII) 0.07 0.28 0.75C(SPEAII,MOACO) 0 0.70 0.96C(MOACO,SPEAII) 0 0.20 1C(NSGAII, SPEAII) 0 0.40 1C(SPEAII,NSGAII) 0 0.06 0.29

Set2min avg max

C(NSGAII,MOACO) 0.81 0.92 0.98C(MOACO,NSGAII) 0.02 0.27 0.39C(SPEAII,MOACO) 0 0.68 0.73C(MOACO,SPEAII) 0 0.21 1C(NSGAII, SPEAII) 0 0.71 1C(SPEAII,NSGAII) 0 0.02 0.16

Set3min avg max

C(NSGAII,MOACO) 0.92 0.97 1C(MOACO,NSGAII) 0 0.05 0.14C(SPEAII,MOACO) 0 0.63 0.92C(MOACO,SPEAII) 0 0.30 1C(NSGAII, SPEAII) 0 0.60 1C(SPEAII,NSGAII) 0 0.04 0.24

Results show NSGA-II outperforms SPEA-II and MOACOon the resolution of our problem.

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