Swarm and Evolutionary Computation 13 (2013) 74–84

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Swarm and Evolutionary Computation

2210-65http://d

n CorrE-m

pmkhila

journal homepage: www.elsevier.com/locate/swevo

Regular Paper

Detection and diagnosis of node failure in wireless sensor networks:A multiobjective optimization approach

Arunanshu Mahapatro a,n, Pabitra Mohan Khilar b

a National Institute of Science and Technology, Berhampur, Indiab National Institute of Technology Rourkela, Rourkela, India

a r t i c l e i n f o

Article history:Received 25 December 2011Received in revised form22 February 2013Accepted 10 May 2013Available online 22 May 2013

Keywords:Fault detectionIntermittent faultMultiobjective optimizationWSNs

02/$ - see front matter & 2013 Elsevier B.V. Ax.doi.org/10.1016/j.swevo.2013.05.004

esponding author. Tel.: +91 9437438294.ail addresses: arun227@gmail.com (A. Mahapar@nitrkl.ac.in (P. Mohan Khilar).

a b s t r a c t

Detection of intermittent faults in sensor nodes is an important issue in sensor networks. This requiresrepeated application of test since an intermittent fault will not occur consistently. Optimization of intertest interval and maximum number of tests required is crucial. In this paper, the intermittent faultdetection in wireless sensor networks is formulated as an optimization problem. The two objectives, i.e.,detection latency and energy overhead are taken into consideration. Tuning of detection parametersbased on two-lbests based multiobjective particle swarm optimization (2LB-MOPSO) algorithm isproposed here and compared with that of non-dominated sorting genetic algorithm (NSGA-II) andmultiobjective evolutionary algorithm based on decomposition (MOEA/D). A comparative study of theperformance of the three algorithms is carried out, which show that the 2LB-MOPSO is a better candidatefor solving the multiobjective problem of intermittent fault detection. A fuzzy logic based strategy is alsoused to select the best compromised solution on the Pareto front.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Wireless sensor network (WSN) is a special kind of networkcomposed of hundreds or even thousands of autonomous sensornodes. The nodes can perform sensing, processing, and wirelesscommunication tasks [1,2]. Experimental studies have shown thatmore than 80% of the faults that occur in real systems like WSNsare intermittent faults [3,4]. An intermittent fault originates frominside the system when software or hardware is faulty. By itsnature, an intermittent fault will not occur consistently, whichmakes its diagnosis a probabilistic event over time [5]. Since theeffect of fault is not always present, detection of intermittent faultrequires repetitive testing at a discrete time kT ðk¼ 1;2;…Þ incontrast to single test for detection of permanent fault. Intuitivelythis implies that to detect an intermittent fault the issues likenumber of test required and inter test interval (T) are crucial. If T istoo large, then probability that the error appears after kth test anddisappears before kþ 1 th test increases and thus detectionaccuracy decreases. Diagnostic latency is expected to be more forlarger value of T which might not be acceptable for applicationwith short mission time. Improvement in both detection accuracyand latency can be achieved with smaller value of T. However, if Tis too small, then frequent exchange of sensor measurements is

ll rights reserved.

tro),

required as message exchange is the only means to detect faults.This in turn increases the energy overhead. Thus, the followingquestions might be of interest:

�

What should be the value of T? � How many tests required to detect an intermittent fault?

These issues motivate to find an trade-off between detectionaccuracy, detection latency and energy overhead. As it can beperceived, finding a good trade-off can be formulated in severalpossible ways, and with emphasis on various aspects of the finaloutput expected. Thus, there may not exist a single optimalsolution but rather a whole set of possible solutions of equivalentquality. This motivates to use Multiobjective Optimization algo-rithm that deals with such simultaneous optimization of multiple,possibly conflicting, objective functions. This work introduces thetwo-lbests based multiobjective particle swarm optimization(2LB-MOPSO) [6] algorithm as a tool in finding trade-offs account-ing for the relative importance of detection accuracy, latency ofisolation of unhealthy nodes and energy overhead. As suggested in[7] a fuzzy based mechanism is employed to extract the best trade-off solution from the Pareto optimal solutions provided by2LB-MOPSO.

The specific contributions of this paper are listed below:

�

Proposes a generic parameterize diagnosis scheme that identi-fies permanent and intermittent faults with high accuracywhile maintaining low time, message and energy overhead.

A. Mahapatro, P. Mohan Khilar / Swarm and Evolutionary Computation 13 (2013) 74–84 75

�

Formulate intermittent fault detection as a multiobjectiveoptimization problem.

�

Tuning of detection parameters like T and kmax based on the2LB-MOPSO algorithm is proposed and compared with that ofNSGA-II [8] and MOEA/D [9].

The remainder of the paper is organized as follows: Section 2presents background of related works. The system model isdescribed in Section 3. Section 4 presents the fault detectionalgorithms. Section 5 presents formulation of fault detectionproblem. The multiobjective optimization problem is discussedin Section 6. In Section 7 presents performance metrics and besttrade-off solution. Simulation experiments are described inSection 8. Finally, Section 9 deals with conclusions.

2. Related research

The classical model for considering system-level faults is thatintroduced by Preparata, Metze, and Chien in [10]. This so-calledPMC model is intended to diagnose permanent faults in a wiredinter connected system. The problem of permanent fault detectionand diagnosis in wireless sensor networks is extensively studied inliteratures [11–15]. Luo et al. [16] proposed a fault-tolerant detec-tion scheme that explicitly introduces the sensor fault probabilityinto the optimal event detection process where the optimaldetection error decreases exponentially with the increase of theneighborhood size. In [13] the authors present a distributed faultdetection model for wireless sensor networks where each sensornode identifies its own state based on local comparisons of senseddata against some thresholds and dissemination of the test results.Krishnamachari et al. have presented a Bayesian fault recognitionmodel to solve the fault-event disambiguation problem in sensornetworks [14]. In [12], the authors have proposed time redun-dancy to diagnose the intermittent faults in sensing and commu-nication in a sensor network. They assume that each sensor has atleast three neighboring nodes, which may not be always possiblefor sparse networks.

The evolutionary approach for fault detection was introducedin [17] and a comparison of evolutionary algorithms for system-level diagnosis can be found in [18]. The genetic approaches havebeen used previously for fault identification in [19–21]. GeneticAlgorithms (GAs) offer several advantages over traditional optimi-zation techniques. These techniques require either gradient des-cent information or any other internal knowledge. On theopposite, GAs require only fitness information, which makes themvery suitable for fault identification. In addition, GAs are designedto search highly nonlinear spaces for global optima. In [20], aparallel evolutionary approach for identifying faults in diagnosablesystems is proposed. The new parallel version considerablyimproves the efficiency of the serial genetic approach, making asignificant contribution to the state of the art on fault diagnosisalgorithms. An ant-colony based fault diagnosis algorithm isproposed in [21]. Experimental results are presented for both thetraditional GA and specialized versions of the GA in [22].

In summary, most of the existing evolutionary approaches forfault detection focus on wired inter connected system. Further,most cited works on distributed diagnosis for WSNs work with theassumption that sensors are either permanent faulty or fault-free.This assumption may not be true in real time applications. Thispaper introduces and examines a generic detection scheme, whichcan detect both permanent and intermittent faults in WSNs and toestablish a good trade-off between detection latency and energyoverhead.

3. System model

3.1. Network model

The proposed algorithm considers a network with n sensornodes non-uniformly distributed in a square area of side L, whichis much larger than the communication range (rtx) of the sensornodes. Every node maintains a neighbor table Nð�Þ. Each sensorperiodically produces information as it monitors its vicinity.Similar to [13], nodes with malfunctioning sensors are allowedto act as a communication node for routing. However, these nodesare asked to switch off their sensors. Only those sensor nodes witha permanent fault in the transceiver and power supply are to beremoved from the network.

3.2. Fault model

The proposed algorithm considers both hard and soft faults [5].A hard faulty node is unable to communicate with other nodes inthe network, whereas a node with soft-fault continues to operateand communicate with altered behavior. These malfunctioning(soft faulty) sensors could participate in the network activitiessince still they are capable of routing information. The proposedalgorithm assumes that the sensor fault probability p is uncorre-lated and symmetric, i.e.,

PðS¼ xjA¼ pxÞ ¼ PðS¼ pxjA¼ xÞ ¼ p ð1Þwhere S is the sensor measurement (say temperature) and A is theactual ambient temperature.

3.3. Energy consumption model

Similar to [23], this work assumes a simple model for the radiohardware energy dissipation. The transmitter dissipates energy torun the radio electronics and the power amplifier. The receiverdissipates energy to run the radio electronics. Both the free space(D2 power loss) and the multipath fading (D4 power loss) channelmodels are used, depending on the distance between the trans-mitter and receiver. The energy spent for transmission of an r-bitpackets over distance D is

ETxðr;DÞ ¼ rEelec þ rϵDα ¼rEelec þ rϵfsD

2; DoD0

rEelec þ rϵampD4; D≥D0

8<: ð2Þ

The electronics energy, Eelec, depends on factors such as thedigital coding, and modulation. The amplifier energy, ϵfsD

2 orϵampD

4, depends on the transmission distance and the acceptablebit-error rate. To receive this message, the radio expends energy:

ERxðrÞ ¼ rEelec ð3Þ

4. Fault detection

4.1. Permanent fault detection

This section introduces a detection algorithm which follows thegeneral principle, where working nodes perform their own indepen-dent diagnosis of the system. The detection algorithm uses timeoutmechanism to detect hard faulty nodes. At each detection round eachnode broadcasts its own sensor reading. The node vi detects nodevj∈NðviÞ as hard faulty, if vi does not receive the sensor reading fromvj before Tout. Tout should be chosen carefully so that all the fault-freenodes vj∈NðviÞ must report node vi before Tout.

The soft faults can be detected as follows. This approachexploits the fact that sensor faults are likely to be stochasticallyunrelated, while sensor measurements are likely to be spatially

Fig. 1. Flow diagram to detect intermittent fault.

Fig. 2. Appearance and disappearance of fault.

Fig. 3. Analytical model for the occurrence of intermittent fault.

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correlated. In WSNs, sensors from the same region should haverecorded similar sensor reading [24]. Let vi be a neighbor of vj, xiand xj are the sensor reading of vi and vj respectively. In this workxi is similar to xj when jxi−xjjoδ where δ is application dependent.An arbitrary node vi receives the sensor reading from neighboringnodes and generates a set (fEg⊂fNðviÞg) of nodes with similarreading S. The node vi is detected fault-free if reading Si agreeswith S and the cardinality of set fEg is greater than the threshold(θ) else vi is marked as possibly soft faulty. The optimal value for θis 0:5ðN−1Þ (see Appendix) where N is the number of neighbors.This decision is then broadcasted. A final decision on a nodemarked as possibly soft faulty is taken as follows. A node viidentified as possibly soft faulty, first checks for a node vq∈fNigsuch that the qth entry in its fault table is fault-free. If such vqexists and vq∈fEg then vi is detected as fault-free or else faulty. Thisfinal decision is next broadcasted.

4.2. Intermittent fault detection

To test for permanent faults, any particular test need only beapplied once as these faults are software or hardware faults thatalways produce errors when they are fully exercised. In contrast,the only approach to test for intermittent faults is throughrepeated application of tests. Thus to detect intermittent faults,each node executes the algorithm discussed in Section 4.1 atdiscrete times kT ðk¼ 1;2;3;…Þ. The operation of the algorithmis described by the flow diagram in Fig. 1. The conditional blocklabeled “Faulty?” represents the snapshot view of the currentdiagnostic round. The algorithm loops as long as no errors froma node are detected. The node is isolated when a fault is observed.

5. Problem formulation

5.1. Stochastic model for intermittent fault

Once intermittent fault is activated in a sensor node, faults areobservable for a duration FAD (fault appearance duration) beforethey disappear. Eventually, errors will reappear after FDD (faultdisappearance duration) either because of permanent faults orcorrelated intermittent faults. This is depicted in Fig. 2. Thebehavior of the intermittent fault can be characterized by measur-ing or estimating the probabilities of error disappearance andreappearance in discrete time kT.

The state of a sensor node is modeled as four-state Markovmodel. Fig. 3 depicts this model where the transition probabilitiesbetween different states of the sensor node are shown. Accordingto the proposed model, the node can be in either one of the fourstates—fault-free (FF), permanent faulty (PF), intermittent faultyand fault is active (FA), and intermittent faulty but fault is inactive(FD). The sensor node in FF state can make a transition to either PFstate or FA state with a rate γ. From FD state, it can either go to PFstate or to FA state or stay in FD state. In order to analyzeintermittent fault in more details we focus on FA and FD states,which can be visualized as a two-state Markov model. The state FA(1) corresponds to fault exits and appears at the scheduled time of

test and state FD (0) corresponds to fault exits but does not appearat the scheduled time of test. The probabilities for going from onestate at time kT to either state FA or FD at time ðkþ 1ÞT depend onFDD and FAD respectively. The FDD for intermittent faults insensor node is system and deployment specific, thus, unpredict-able in most practical scenarios. Intermittent faults usually exhibita relatively high occurrence rate after its first appearance andeventually tends to become permanent. Therefore, as suggested in[25] a Weibull distribution is considered for FDD with shapeparameter β41 and failure rate λk. An exponential distributionis considered for FAD with a constant failure rateμ¼ ð1=mean time in FA stateÞ [26,25]. A similar distribution isconsidered for time to failure of a fault-free node with theconstant failure rate γ ¼ ð1=mean time in the fault� free stateÞ. Inpractice μ⪢λk⪢γ.

In order to devise such a model, let fFjg be the state spacewhere F0 denotes that node is fault-free and F1 denotes that nodeis intermittent faulty. Let ftkg is the test pattern where tk is the kthtest performed by the sensor node by using algorithm discussed inSection 4.1 at time kT ðk¼ 1;2;…Þ. The outcome of the kth test is0 if the node is either fault-free or node is intermittent faulty butthe fault does not appear during the test. Since effect of fault is notalways present, deriving an optimal test pattern which cancertainly detect the intermittent fault is hard to realize. In orderto get a near optimal test pattern, we consider an inequality wherethe probability that an intermittent fault exists and is not detectedmust be smaller than the error threshold θ1. Using the fact that thenetwork is sampled with sampling period T, the following inequal-ity is obtained:

PðF1jtk ¼ 0Þ≤θ1 ð4Þ

For k¼1, using Baye's rule we can write

Pðt1 ¼ 0jF1Þ � PðF1ÞPðt1 ¼ 0jF0Þ � PðF0Þ þ Pðt1 ¼ 0jF1Þ � PðF1Þ

≤θ1 ð5Þ

For kmax number of tests the above equation can be rewritten as

∏kmaxk ¼ 1Pðtk ¼ 0jF1Þ � p

ð1−pÞ þ∏kmaxk ¼ 1Pðtk ¼ 0jF1Þ � p

≤θ1 ð6Þ

The term ∏kmaxk ¼ 1Pðtk ¼ 0jF1Þ of (6) defines the probability that the

fault remains inactive at time instants kT where k¼ 1;2;…; kmax.Thus, the inequality can be rewritten as

∏kmaxk ¼ 1P00ðkTÞ � p

ð1−pÞ þ∏kmaxk ¼ 1P00ðkTÞ � p

≤θ1 ð7Þ

where P00ðkTÞ is called the state transition probability, which is theconditional probability that the sensor node will be in state FD(0) at time kT immediately after the next transition, given that it

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was in state FD (0) at time ðk−1ÞT . This probability is [27,26]

P00ðkTÞ ¼μ

μþ λkþ λk

μþ λke−ðλkþμÞT ð8Þ

Eq. (7) is derived under perfect test condition, i.e., a fault isalways detected by a test when it occurs. Since we adopt neighborcoordination as a test to detect faults, thus, a fault is detected by atest with probability 1−Pe and is not detected with probability Pe.The probability Pe is (A.9)

Pe ¼ f 1 � 1−p− ∑N

l ¼ 0:5ðN−1Þð1−pÞf l þ pf N−l

!

where fl is the probability that l out of N 1-hop neighbors of a nodeare fault-free. For imperfect test condition, Eq. (7) can be rewritten as

∏kmaxk ¼ 1P00ðkTÞ � p � ð1−PeÞ

ð1−pÞ þ∏kmaxk ¼ 1P00ðkTÞ � p � ð1−PeÞ

≤θ1 ð9Þ

As comprehended from (9), a better trade-off between detec-tion accuracy, detection latency and energy overhead can beachieved by properly tuning the detection parameters kmax and T.

5.2. Impact of design parameters on fault detection

The modeling framework discussed in earlier sections allows usto highlight detection accuracy, detection latency and energyoverhead trade-offs in detecting an intermittent fault. To evaluatethe impact of the design parameters on these trade-offs we havefirst used (9) to find out the number of tests required to detectfaults and the detection latency at varying values of T and θ1. Thesetheoretical results are shown in Fig. 4(a) and (b) respectively.Second, we have conducted a simulation on a simple network tofind the impact of these design parameters. This simple networkwe considered has one intermittent faulty node surrounded byfour fault-free one-hop neighbors. For simulation, the mean valueof FAD is considered 50 ms where FAD is exponentially distributed.

1 10 20 30 40 50 600

2000

4000

6000

8000

10000

12000

14000

T (sec)

Num

ber o

f tes

ts

θ1=10−2

θ1=10−4

θ1=10−6

θ1=10−8

θ1=10−10

θ1=10−12

1 10 20 30 40 50 604000

7000

9000

12000

15000

T(sec)

Num

ber o

f tes

ts

Fig. 4. Impact of design

The FDD is assumed to follow a Weibull distribution with increas-ing failure rate ðβ¼ 1:5Þ and expected value of 1 h. We run theexperiment until the fault is detected, and the results are shown inFig. 4(c) and (d). As discussed earlier and shown in Fig. 4(a) and (c),the number of tests required and thus the number of messagesexchanged to detect the intermittent faults decreases for anincrease in T. Fig. 4(b) and (d) shows the latency in detecting theintermittent fault. It is observed that the latency tends to increasewith T. As comprehended from Fig. 4(a) and (b), better detectionaccuracy, i.e., extremely small value for θ1 can be achieved at thecost of the number of messages to be exchanged and detectionlatency.

5.3. Calculation of objectives

From the above discussions, it can be concluded that theobjectives are conflicting. These two conflicting objectives are:(1) to minimize the detection latency and (2) at the same time, tominimize energy overhead (energy overhead is proportional tonumber of tests), while satisfying detection error constraints. Thisproblem is formulated, mathematically, in this section.

5.3.1. Energy overheadIn an n-node WSN, each node has a unique identifier which can

be encoded with log2 n bits. As discussed in Section 4.1, a singletest requires exchange of three diagnostic messages. The firstdiagnostic message is the sensor reading which is represented by znumber of bits. The energy dissipated in exchanging first diag-nostic message is nðz þ log2 nÞðETx þ ERxÞ. The second diagnosticmessage is the initial decision taken at each node. The correcteddecisions about the nodes detected as possibly soft faulty areexchanged as third diagnostic message. Since the state of eachnode is identified with a single bit (0: fault-free, 1: faulty), theenergy dissipated in exchanging the initial and correct decisions is2nðlog2 nþ 1ÞðETx þ ERxÞ. Thus, the total energy dissipated per test

1 10 20 30 40 50 600

10

20

30

40

50

60

T (sec)

Det

ectio

n la

tenc

y (h

ours

)

θ1=10−2

θ1=10−4

θ1=10−6

θ1=10−8

θ1=10−10

θ1=10−12

0 10 20 30 40 50 600

10

20

30

40

50

60

T (sec)

Det

ectio

n la

tenc

y (h

ours

)

parameters.

A. Mahapatro, P. Mohan Khilar / Swarm and Evolutionary Computation 13 (2013) 74–8478

is nð3 log2 nþ z þ 2ÞðETx þ ERxÞ and the energy dissipated indetecting intermittent faults is nkmaxð3 log2 nþ z þ 2ÞðETx þ ERxÞ.Thus, the first objective function can be given by

F1 ¼ nkmaxð3 log2 nþ z þ 2ÞðETx þ ERxÞ ð10Þ

5.3.2. Detection latencyDetection latency is the time elapsed between the first occur-

rence of the fault and the fault detected. Thus, the detectionlatency is a function of kmax and T. As discussed earlier and shownin Fig. 4(b) and (d), large detection latency increases with T andmight be undesirable for critical applications with short missiontime. The detection latency can be expressed as

F2 ¼ kmax � T ð11Þ

5.4. Constraint function (CF)

There is mainly one constraint corresponding detection errorthat should be satisfied, which is given as (9)

∏kmaxk ¼ 1P00ðkTÞ � p � ð1−PeÞ

ð1−pÞ þ∏kmaxk ¼ 1P00ðkTÞ � p � ð1−PeÞ

≤θ1 ð12Þ

6. Multiobjective optimization problem

Multiobjective optimization is the process of simultaneouslyoptimizing two or more conflicting objectives subject to certainconstraints. Since many conflicting objectives to be optimizedsimultaneously, there is a set of possible solutions of equivalentquality. Most real-world problems employ the optimization ofseveral objectives, which are often conflicting in nature. A multi-objective optimization problem with M conflicting objectives canbe defined as in [28]:

Maximize/minimize

y¼ f ðxÞ¼ ðf 1ðxÞ; f 2ðxÞ;…; f MðxÞÞ; x∈½Xmin;Xmax�

subject to:

gjðxÞ≤0; j¼ 1;…; J

hkðxÞ ¼ 0; k¼ 1;…;K

where x and y are the decision vector and the objective vectorrespectively. Different from the single objective optimization,there are two spaces to be considered. One is the decision spacedenoted as x and the other is the objective space denoted as y.

Definition 1 (Zhao and Suganthan [6]). Let wi and wj are twosolutions to a multiobjective problem. wi dominates wj if wi

performs at least as good as wj with respect to all the objectivesand performs strictly better than wj in at least one objective.

Definition 2 (Zhao and Suganthan [6]). Among a set of solutionsW, the non-dominated set of solutions W ′ are those that are notdominated by any member of the set W.

Definition 3 (Zhao and Suganthan [6]). When the set W is theentire feasible search space, the resulting non-dominated set W ′ iscalled the Pareto-optimal solution set.

6.1. Finding Pareto optimal solution

Numerical techniques can be adopted to find the set ofsolutions for a multiobjective optimization problem. In this workwe describe how predictor–corrector method like algorithm

CONT-Recover [29] is used for the numerical treatment of ourmultiobjective optimization problem. Starting with a given Kar-ush–Kuhn–Tucker point (KKTpoint) ~x of an multiobjective optimi-zation problem, algorithm CONT-Recover is applied to detectfurther KKT-points in the neighborhood of ~x. In the subsequentsteps, further points are computed starting with these new-foundKKT-points. To maintain a good spread of these solutions, algo-rithm CONT-Recover uses boxes for the representation of thecomputed parts of the solution set. Though predictor–correctormethods are quite effective, they are, however, based on someassumptions. First, an initial solution has to be computed beforethe process can start. Due to their local nature, predictor–correctormethods are restricted to the connected component that containsthe given initial solution [30]. Further, the Pareto set may fall intoseveral connected components.

Evolutionary algorithms are correctly fitted to multiobjectiveoptimization problems as they are essentially based on biologicalprocesses, which are inherently multiobjective. An extensivesurvey on multiobjective evolutionary algorithms is well pre-sented in [31]. Central to these articles, considering superiorperformance for solving multiobjective problems, the 2LB-MOPSO[6], MOEA/D [9] and NSGA-II [8] algorithms have been used inthis study.

In NSGA-II, initially a random population of size H, which issorted based on the non-domination, is created. This populationsubsequently undergoes selection, crossover and mutation pro-cesses to produce an offspring population of size H. A combinedpopulation of size 2H is formed from the parent and offspringpopulation. Next, the population is sorted according to the non-domination relation. This in turn classifies the complete popula-tion into several non-dominated fronts based on the values of theobjective functions. Until each member of the population falls intoone front the other fronts are determined. The new parentpopulation is generated by adding the solutions from the firstfront. Several non-dominated fronts are discarded as the popula-tion size is predefined. The required numbers of members for thenew population are selected using a new parameter calledcrowding distance. The crowding distance describes how closean individual is to its neighbors.

Similar to GA, the PSO algorithm has been successfullyextended to multiobjective optimization problems. Different fromother variants of MOPSO algorithms, 2LB-MOPSO uses two localbests instead of one personal best and one global best to lead eachparticle. The two local bests are selected to be close to each otherin order to enhance the local search ability of the algorithm.Compared to the other variants of MOPSO algorithms, 2LB-MOPSOshows great advantages in maintaining a good diversity of thesolutions, convergence speed and fine-searching ability.

In 2LB-MOPSO, the initialized archive includes all initializedsolutions at iteration 1. In every iteration, all new positions Q(t)generated in iteration t is combined with the members in thearchive A(t) to obtain the mixed-temporary external archive. Thesorted archive R(t) is obtained by applying the non-dominationsorting to this mixed-temporary archive. During this process, allthe sorted solutions retain two indicators, namely, the front rankand crowding distance value. The sorted solution with the lowestfront rank is first included in the archive Aðt þ 1Þ. When the size ofthe archive equals to the permitted maximum size of an archiveðAðt þ 1Þj ¼ jAðtÞjÞ, the crowding distance is applied to select therequired number of members to be included in Aðt þ 1Þ from thelowest front that still remains unselected in the archive R(t). Thepseudo-code of the 2LB-MOPSO algorithm is presented in Fig. 5.

In 2LB-MOPSO, each objective function range in the externalarchive is divided into a number of bins. The two lbests are chosenfrom the external archive members located in two neighboringbins, so that they are near each other in the parameter space. In

Fig. 5. The pseudo-code of the 2LB-MOPSO.

A. Mahapatro, P. Mohan Khilar / Swarm and Evolutionary Computation 13 (2013) 74–84 79

order to select the first lbests for a particle, an objective is firstrandomly selected followed by a random selection of a non-emptybin of the chosen objective. Within this bin, the archived memberwith the lowest front number and among these with the highestcrowding distance is selected as the first lbests. The second lbests isselected from a neighboring non-empty bin with the lowest frontnumber and the smallest Euclidean distance in the parameterspace to the first lbests. As velocity of each particle is adjusted bythe two lbests from two neighboring bins, the flight of eachparticle will be in the direction between the positions of twolbests and oriented to improve upon the current solutions.

Upon assigning a pair of lbests to a particle, the number ofiterations the particle fails to contribute a solution to the archive A(t) is counted. The particle is reassigned with another pair of lbestswhen the count exceeds a pre-specified threshold. During theinitialization stage and when the count is larger than the pre-specified threshold during the iterative optimization stage, thefirst lbests for a particle is chosen randomly by selecting anobjective and one bin of the objective. The second lbests is chosenfrom the neighborhood of the first lbests in the parameter space.When the count is less than or equal to the pre-specified thresholdduring the iterative optimization stage, two lbests are chosen fromthe same assignment of the objective and the bin as used in thelast iteration. The particle will accelerate potentially in a directionbetween the two lbests and hence may explore the region of thetwo lbests.

Unlike NSGA-II and 2LB-MOPSO algorithm the MOEA/D algo-rithm uses weight vectors to decompose a multiobjective optimi-zation problem into a number of single objective optimization sub-problems and optimizes them simultaneously. Each sub-problemis optimized by sharing information between its neighboring sub-problems with similar weight values. Tchebycheff approach isemployed to convert the problem of approximating the Pareto

front into a number of scalar optimization problems. Let ς1;…; ς2be a set of even spread weight vectors and zn be the referencepoint. The problem of approximation of the Pareto front can bedecomposed into N scalar optimization subproblems by using theTchebycheff approach. The objective function of the jth subpro-blem is given by

gtcðxjςj; znÞ min1 ≤ i ≤M

fςjijf iðx−zni jÞg ð13Þ

where ςj ¼ ðςj1;…; ςjmÞT . MOEA/D minimizes all these objectivefunctions simultaneously in a single run.

In MOEA/D, a neighborhood of weight vector ςi is defined as aset of its several closest weight vectors in ς1;…; ς2. The neighbor-hood of the ith subproblem consists of all the subproblems withthe weight vectors from the neighborhood of ςi. The population iscomposed of the best solution found so far for each subproblem.Only the current solutions to its neighboring subproblems areexploited for optimizing a subproblem.

7. Performance metrics and best trade-off solution

All the existing multiobjective optimization algorithms aim tofind solutions as close as possible to the Pareto optimal front andas diverse as possible in the non-dominated front. Differentperformance metrics to measure these two objectives have beensuggested in the literature. Since the true Pareto-optimal front forthe proposed application is unknown, for performance analysis,we consider coverage of the Pareto front [32], and spacing of thePareto front [33]. The first metric measures the convergence of thePareto front, while the second metric measures the distribution ofsolutions along the Pareto front.

7.1. Coverage of the Pareto front

Let A and B are two Pareto-optimal sets. This metric measuresthe relative spread of solutions between two non-dominated sets.The function C maps the ordered pair (A, B) to the interval [0, 1]and is given by

CðA;BÞ ¼ jfb∈Bj∃a∈A : a≽bgjjBj ð14Þ

where jBj represents the number of solutions in the set B, and a≽bimplies that solution a weakly dominates solution b. The valueCðA;BÞ ¼ 1 implies that all decision vectors in B are weaklydominated by A. In contrary, CðA;BÞ ¼ 0 implies that none of thepoints in B are weakly dominated by A. If CðA;BÞ4CðB;AÞ, then theset A has better solutions than the set B.

7.1.1. SpacingSchott [33] introduced a metric namely Spacing that measures

the distribution of the solutions over the non-dominated front.Spacing between solutions is computed as

Spacing ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

Q−1∑Q

i ¼ 1

did

� �2s

ð15Þ

where

di ¼minj

∑M

m ¼ 1jFim−Fjmj for j¼ 1;…;Q and i≠j: ð16Þ

Q is the number of solutions in the non-dominated set, M is thetotal number of objectives to be optimized and d is the mean of allthe di. The nearer the value of Spacing to zero, the more uniformlydistributed the solutions found over the Pareto optimal front.

A. Mahapatro, P. Mohan Khilar / Swarm and Evolutionary Computation 13 (2013) 74–8480

7.2. Fuzzy decision making

Upon obtaining a set of Pareto optimal solutions using2LB-MOPSO, we need to find a best optimum trade-off. Assuggested in [7], the fuzzy membership functions that representthe goals of each objective function are used. The fuzzy sets aredefined by these membership functions. These functions representthe degree of membership in certain fuzzy sets using values from 0 to1. The membership functions for both objectives are defined as

μi ¼

1; Fi ≤Fmini

Fmaxi −Fi

Fmaxi −Fmin

i

; Fmini oFioFmax

i

0; Fi≥Fmaxi

8>>>><>>>>:

ð17Þ

where Fmini and Fmax

i are the minimum and maximum values fromnon-dominated solutions of each objective function, respectively. Foreach non-dominated solution, the normalized membership functioncan be calculated as

μr ¼ ∑1i ¼ 1μ

ri

∑Rr ¼ 1∑

2i ¼ 1μ

ri

ð18Þ

where R is the number of non-dominated solutions. The solution thatattains the maximum membership μr in the fuzzy set can be chosenas the best solution:

Best solution¼maxfμr : r¼ 1;…;Rg

8. Simulation results and analysis

8.1. Tuning of detection parameters

This section is primarily meant to study how the designparameters namely kmax and T affect detection of intermittentfaults in terms of two important figures of merit: the detectionlatency and energy overhead while maintaining low detectionerror. In this section, we compare the design results obtained with2LB-MOPSO, algorithm CONT-Recover [29], MOEA/D [9], NSGA-II[8] and three single-objective optimization algorithms namely GA,DE and PSO. We use MATLAB as a simulation tool for tuning thedetection parameters. The mean value of FAD is considered 50 mswhere FAD is exponentially distributed. The FDD is assumed tofollow a Weibull distribution with increasing failure rate ðβ¼ 1:5Þand expected value of 1 h. For 2LB-MOPSO, the parameters are setas in the [6]: count and number of bins are considered as 5 and 10respectively, population size NP¼50, inertia weight ω¼ 0:729,C1¼ C2¼ 2:05, Vmax ¼ 0:25ðXmax−XminÞ. For NSGA-II (real-coded)

0 0.005 0.01 0.015 0.020

2

4

6

8

10

12

Normalized total energy consumption (J)

Det

ectio

n la

tenc

y (h

ours

)

Best trade off

2LBMOPSO

CONT− Recover

Fig. 6. Trade-

and MOEA/D, we use a population size of 50, crossover probabilityof 0.9 and mutation probability of 0.5. As suggested in [8], thedistribution indexes for crossover, and mutation operators are setas ηc ¼ 20 and ηm ¼ 20. For MOEA/D the number of the weightvectors in the neighborhood of each weight vector is set to 20. Thedecision variables are initialized with uniformly distributedpseudo-random numbers that take the range of these variables,i.e., T ¼ rand½Tmin; Tmax� and k¼ rand½kmin; kmax�. We considerTmin ¼ 1000 ms, Tmax ¼ 60 000 ms, kmin ¼ 1 and kmax ¼ 15 000, andθ1 ¼ 10−20. The maximum function evaluations are set as 15 000.

To optimize simultaneously F1 and F2 using GA, PSO and DE, thefitness function can be defined as

Ff ¼ ω1F1 þ ω2F2 ð19ÞThe fitness function Ff is minimized using different values of ω1

and ω2 such that ω1 þ ω2 ¼ 1. The weighted sum method is,however, subjective and the solution obtained will depend onthe values (more precisely, the relative values) of the weightsspecified [34]. It is hard, if not impossible, to choose a propercombination of ω1, and ω2 to get a optimized code. As suggested in[35], for DE the parametric setup is CR¼0.7, F¼0.5. For PSO, weused swarm size¼50, acceleration coefficients C1¼ C2¼ 2:05. ForGA, we use a population size of 50, crossover probability of 0.9 andmutation probability of 0.5.

8.1.1. Performance analysisIn order to evaluate the performance, we first compare the

Pareto fronts obtained using algorithm CONT-Recover [29] andwith one of the 20 runs of 2LB-MOPSO (Fig. 6(a)). The best trade-off solution is obtained on these two solutions by using theaforementioned fuzzy logic based mechanism and is shown inFig. 6(b). Further, 20 independent runs were conducted for2LB-MOPSO, MOEA/D, NSGA-II, GA, DE, and PSO. To illustrate thedifference between the Pareto fronts obtained with 2LB-MOPSO,MOEA/D and NSGA-II, the Pareto fronts obtained with one of the20 runs of 2LB-MOPSO, MOEA/D and NSGA-II are plotted in Fig. 6(b). The best trade-off solution is obtained on these three solutionsby using the aforementioned fuzzy logic based mechanism and isshown in Fig. 6(b). Here, we consider the normalized total energywhich is the ratio between the total energy (10) and the number ofnodes participated in the detection. Table 1 shows the tuneddetection parameters obtained using the mentioned optimizationalgorithms. For GA, PSO, and DE based implementation, weprovide the best solutions found in 50 independent trials of eachalgorithm.

The quality of the Pareto-optimal solutions obtained withalgorithm CONT-Recover, 2LB-MOPSO, MOEA/D, and NSGA-II ismeasured by the two aforementioned performance metrics. The

2 4 6 8 10 12

x 10−3

0

2

4

6

8

10

12

Normalized total energy consumption (J)

Det

ectio

n la

tenc

y (h

ours

)

2LB MOPSO

NSGA−II

MOEA/D

Best trade off

off curve.

Table 2Coverage (2LB-MOPSO vs MOEA/D).

2LB-MOPSO MOEA/D

Best 0.9461 0.3631Worst 0.7129 0.0248Average 0.9011 0.2247Median 0.8172 0.0182Variance 0.0056 0.0098Std. dev. 0.0746 0.0993

Table 3Coverage (2LB-MOPSO vs NSGA-II).

2LB-MOPSO NSGA-II

Best 0.9886 0.3126Worst 0.7835 0.0192Average 0.9133 0.2013Median 0.8361 0.0133Variance 0.0067 0.0182Std. dev. 0.0821 0.1352

Table 4Coverage (2LB-MOPSO vs CONT-Recover).

2LB-MOPSO CONT-recover

Best 1 0.137Worst 0.92 0Average 0.9533 0.0625Median 0.962 0.0986Variance 0.0039 0.0669Std. dev. 0.0624 0.2586

Table 5Spacing.

2LB-MOPSO MOEA/D NSGA-II CONT-Recover

Best 0.2096 0.2439 0.3862 0.5162Worst 0.3932 0.4218 0.6696 0.8103Average 0.3206 0.3881 0.5182 0.7312Median 0.3182 0.3437 0.5021 0.7663Variance 0.0056 0.0080 0.0144 0.0492Std. dev. 0.0749 0.0897 0.1204 0.2218

Table 6Simulation parameters.

Parameter Value

Number of sensors 1000Network grid From (0, 0) to (1000, 1000) mSink At (75,150) mInitial energy 1 JEelec 50 nJ/bitϵfs 10 pJ/bit/m2

ϵamp 0.0013 pJ/bit/m4

Table 1Tuned detection parameters and their corresponding results.

T kmax EN (J) Latency (min)

2LB-MOPSO 8600 1052 0.0044 150.786MOEA/D 9362 1124 0.0046 174.998NSGA-II 12 978 1256 0.0057 271.6728CONT-Recover 14 457 1465 0.0071 353.2258GA 10 763 1867 0.0076 334.908PSO 11 602 1597 0.0065 309.498DE 13 254 1508 0.0061 333.117

A. Mahapatro, P. Mohan Khilar / Swarm and Evolutionary Computation 13 (2013) 74–84 81

best, worst, mean, median, variance and standard deviation of thetwo performance metrics are presented in Tables 2–4. The bestaverage result with respect to each metric is shown in bold font.

In Table 2, the value for Coverage¼0.9461 implies that 94.61%of the Pareto-optimal solutions obtained with MOEA/D are weaklydominated by the solutions obtained with 2LB-MOPSO. Likewise,the value for Coverage¼0.3631 means that only 36.31% of thesolutions obtained with 2LB-MOPSO are weakly dominated bythose with MOEA/D. In addition, the standard deviation of 2LB-MOPSO with respect to Coverage implies that the performance of2LB-MOPSO is more stable.

The distributions of the Pareto-optimal solutions over thenon-dominated front obtained with algorithm CONT-Recover,2LB-MOPSO, MOEA/D and NSGA-II are evaluated with metricSpacing. Since a lower value of Spacing implies uniform spread ofsolutions, as shown in Table 5 for our application 2LB-MOPSOoutperforms algorithm CONT-Recover, MOEA/D and NSGA-II.

8.2. Simulation experiments

In order to validate the obtained detection parameter T, andkmax and measure its effectiveness, we chose to conduct anextensive set of simulations using Castalia-2.3b, a state-of-the-art WSN simulator based on the OMNET++ platform. The simula-tion parameters are given in Table 6, where the values for radiomodel parameters are same as those in [36].

As discussed earlier both FAD and FDD are system specific anddepends on multiple factors. Thus, to simulate the real faultscenario FAD follows a Weibull distribution with expected valueranging from 1 min to 10 h and FAD follows an exponentialdistribution with expected value ranging from 5 ms to 50 ms. Allthe intermittent faults are activated randomly before first test, i.e.,before 8600 ms from the start of simulation. Each sensor node inthe network is scheduled to take sensor measurement at thediscrete time kT with T¼8600 ms. The data-gathering stage isscheduled at GT where G is an integer and is application specific.For instance, applications with short mission time need the data tobe gathered more frequently in contrast to applications, wherefrequency of data gathering is less. For applications with longmission time, GT is large. Thus, to detect intermittent faults, G/Tnumber of sensor measurements needs to be broadcasted by eachnode. This in turn makes the packet to grow with G. Since energyconsumed by a sensor node is directly proportional to the numberof bits it transmits or receives, the energy overhead will be morefor large value of G and may not be practically implementable. Toaddress this issue, we suggest to sample the interval GT whereeach sample constitutes of I consecutive senor measurements. Thestandard deviation of these I sensor measurements correspondingto each sample interval is calculated and broadcasted along withthe routine data. This in turn reduces the packet size and makesthe algorithm energy efficient. Each node takes the decision bycomparing the corresponding standard deviations of one-hopneighbors. Use of standard deviation instead of individual mea-surements does not affect the detection performance since rate ofchange in sensor measurements over time is very less. In addition,a sensor often reports unusually high or low sensor measurementduring FAD. Thus, the standard deviation of sensor measurementsof a sample interval with at least one incorrect measurement willbe distinguished from the corresponding standard deviations of

A. Mahapatro, P. Mohan Khilar / Swarm and Evolutionary Computation 13 (2013) 74–8482

one-hop neighbors with all true measurements. In this experi-ment, we assume temperature sensors.

8.3. Experiment 1: efficiency with regard to da and p

In this experiment, the performance of the diagnosis algorithmin regard to DA and FAR is evaluated by first considering onlyintermittent faults and then considering both intermittent andpermanent faults. In the later experiment, the number of inter-mittent and permanent faults is randomly chosen while maintain-ing the total number of faults. For performance evolution, weassume that the number of intermittent and permanent faultsdoes not change during the simulation period. Note that thisassumption does not mean that the detection algorithm is notadaptive to change in fault type and fault rate. In this simulation,sensor nodes are assumed to be faulty with probabilities of 0.05,0.10, 0.15, 0.20, 0.25, 0.30, respectively. The transmission range ischosen for the sensor network to have the desired average nodedegree da. Since a faulty node will often report unusually high orlow sensor measurements, all the nodes with malfunctioningsensors are momentarily assumed to show a match in comparison

0.1 0.15 0.2 0.25 0.30.985

0.99

0.995

1

Sensor fault probability

DA

0.05 0.15 0.2 0.25 0.30

2

4

6

FAR

DA

x 10−3

Fig. 7. DA and FAR with da≈4 and da≈12 for a network considering (a) o

0.05 0.1 0.15 0.2 0.25 0.3100

150

200

250

300

350

400

Sensor fault probability

Det

ectio

n la

tenc

y (m

in)

0.05 0.1 0.15 0.2 0.25 0.33

4

5

6

7

8

9

10x 10−3

Sensor fault probability

Nor

mal

ized

tota

l ene

rgy

con

sum

ptio

n (J

)

Fig. 8. Average detection latency and n

with a probability of 0.5 regardless of their locations. To validatethe obtained detection parameters, the experiment was conductedfor 21 epochs ðkmax=G¼ 1052=50≈21Þ.

The results shown are average of 100 experiments. Fig. 7(a) and(b) shows the average detection accuracy and average false alarmrate of the detection algorithm considering only intermittentfaults. Interestingly, an improvement in both DA and FAR isobserved. The reason is that a faulty node will be detected asfault-free only when the node has more than θ faulty neighborsand shows a match in comparison. A fault-free node detected asfaulty only when the node has more than θ faulty neighbors. In thescenario where all faults are intermittent, the probability ofmentioned neighbors at the time of test is less as compare tothe scenario where all faults are permanent. This is because theprobability that fault appears in all the faulty neighbors at the timeof test is less.

8.4. Experiment 2: time and energy efficiency

In this experiment, we attempted to illustrate the detectionlatency and normalized total energy overhead of the detection

0.1 0.15 0.2 0.25 0.30.96

0.97

0.98

0.99

1

Sensor fault probability

0.05 0.1 0.15 0.2 0.25 0.30

0.002

0.004

0.006

0.008

0.01

FAR

nly intermittent faults (b) both intermittent and permanent faults.

0.05 0.1 0.15 0.2 0.25 0.3100

150

200

250

300

350

400

Sensor fault probability

Det

ectio

n la

tenc

y (m

in)

0.05 0.1 0.15 0.2 0.25 0.33

4

5

6

7

8

9

10x 10−3

Sensor fault probability

Nor

mal

ized

tota

l ene

rgy

con

sum

ptio

n (J

)

ormalized total energy overhead.

A. Mahapatro, P. Mohan Khilar / Swarm and Evolutionary Computation 13 (2013) 74–84 83

algorithm. We use the tuned detection parameters obtainedthrough algorithm CONT-Recover, 2LB-MOPSO, MOEA/D, NSGA-II,GA, DE, and PSO (Table 1). All results are the average of resultsobtained on 100 random topologies. For better analysis, weconsider only intermittent faults. The average detection latencyand the average normalized total energy overhead is shown inFig. 8(c) and (d) for varying fault rate and da. As shown, both thedetection latency and normalized energy overhead are lessaffected by the number of faults. The reason is that the detectionof intermittent faults depends only on T and the detection latencydepends on the number of test repetitions executed to detect thefault. It is observed that the detection latency for 2LB-MOPSObased implementation outperforms algorithm CONT-Recover,MOEA/D, NSGA-II, GA, DE, and PSO based implementations.Similarly, the normalized total energy overhead is less affectedby the fault rate. This is because it depends purely on the numberof messages exchanged to detect the fault. As discussed earliermore messages need to be exchanged if nodes fail the thresholdtest. Since only intermittent faults are considered, thus as dis-cussed in Experiment 1, the number of nodes failed the thresholdtest is less. However, a minor improvement is observed for greateraverage node degree and lower fault rate. It is observed that 2LB-MOPSO based implementation outperforms algorithm CONT-Recover, MOEA/D, NSGA-II, GA, DE, and PSO based implementa-tions from normalized total energy overhead perspective.

9. Conclusions

In this paper, an efficient fault detection technique with lowdetection latency, low energy overhead and high detection accu-racy was considered. The application of two lbest multiobjectiveparticle swarm optimization to minimize detection latency andenergy overhead simultaneously was discussed. While applyingthe algorithm, the detection error was considered as a constraint.A fuzzy based mechanism is also used to find out the bestcompromised solution on the optimal Pareto front. The tuneddetection parameters were used by the detection algorithm. Theperformance difference between 2LB-MOPSO, CONT-Recover,MOEA/D and NSGA-II based parameter tuning was observed and2LB-MOPSO based approach was found more suitable for theproposed application.

Appendix A

In this section, we formulate the threshold θ.

Theorem 4. The optimum value of θ which minimizes the detectionerror is 0:5ðN−1Þ.

Proof. Proof of this theorem closely follows a similar proof in [14].The real situation at the sensor node is modeled by two variables Sand A where S represents the sensor reading and A represents theactual reading. Let Eðx; lÞ be the manifest that l out of N 1-hopneighbors of a node vi report the similar sensor reading x. Theobjective here is to determine the fault detection estimate (DE)after obtaining information about the sensor readings of neighbor-ing nodes. The possible vales of DE are fault-free (FF) and faulty (F).The probability that the detection estimate is fault-free, given thatl out of N neighboring sensors of node vi report the same reading xis defined as

Pl ¼ PðDE¼ FFjSi ¼ x; Eiðx; lÞÞ ðA:1ÞFor faulty communication channel Chi;j, vi believes that vj∈Ni isfaulty. In the presence of channel fault, let fl is the probability that lout of N 1-hop neighbors of node vi are fault-free. This probability

is determined as

f l ¼N

l

� �PðSi ¼ xjAi ¼ x;Ch¼ GÞl � PðSi ¼ xjAi ¼ px;Ch¼ GÞN−l

¼ N

l

� �PðSi ¼ xjAi ¼ x;Ch¼ GÞl � PðSi ¼ xjAi ¼ x;Ch¼ BÞN−l

¼ Nl

� �ð1−pÞlpN−l ðA:2Þ

The possible values for variables S and A are x and px where pxdefines a value which is not similar to x. Thus eight possiblecombinations exist for DE, S and A. The correctness of the proposedalgorithm can be analyzed by the conditional probabilities corre-sponding to these combinations. From these combinations we cancalculate the probability that the algorithm estimates the node isfaulty though both the sensed and actual reading are similar. Byusing marginal probability this can be derived as

PðDE¼ FjS¼ x;A¼ xÞ¼ 1−PðDE¼ FFjS¼ x;A¼ xÞ

¼ 1− ∑N

l ¼ 0PðDE¼ FF; Eðx; lÞjS¼ x;A¼ xÞ

¼ 1− ∑N

l ¼ 0PðDE¼ FFjS¼ x;A¼ x; Eðx; lÞÞ

� PðEðx; lÞjS¼ x;A¼ xÞ

¼ 1− ∑N

l ¼ 0Pl � f l ðA:3Þ

In a similar manner, we can calculate the probability that thealgorithm estimates the node is fault-free though the sensorreading does not agree with actual reading:

PðDE¼ FFjS¼ px;A¼ xÞ

¼ ∑N

l ¼ 0PðDE¼ FF; Eðx;N−lÞjS¼ px;A¼ xÞ

¼ ∑N

l ¼ 0PðDE¼ FFjS¼ px;A¼ x; Eðx;N−lÞÞ

� PðEðx;N−lÞjS¼ px;A¼ xÞ

¼ ∑N

l ¼ 0PðDE¼ FFjS¼ px;A¼ x; Eðpx; lÞÞ

� PðEðx;N−lÞjS¼ px;A¼ xÞ

¼ ∑N

l ¼ 0Pl � f N−l ðA:4Þ

As discussed earlier fault-free nodes which failed to pass thethreshold test later diagnosed as fault-free through a fault-freeneighbor. The probability that at least one out of N 1-hop neigh-bors are fault-free can be derived from Eq. (A.2) as

f 1 ¼Nð1−pÞpN ðA:5Þ

Eqs. (A.(4) and A.5) suffice to calculate the probability that thedetection algorithm declares a fault-free node as faulty. Thisprobability is given by

Pgf ¼ PðDE¼ F; S¼ xjA¼ xÞ � f 1¼ PðDE¼ FjS¼ x;A¼ xÞ � Pðs¼ xjA¼ xÞ � f 1

¼ 1− ∑N

l ¼ 0Plf l

!� ð1−pÞ � f 1 ðA:6Þ

A. Mahapatro, P. Mohan Khilar / Swarm and Evolutionary Computation 13 (2013) 74–8484

In the similar manner the probability that the detection algo-rithm declares a faulty node as fault-free can be derived as

Pfg ¼ PðDE¼ FF; S¼ pxjA¼ xÞ � f 1¼ PðDE¼ FFjS¼ px;A¼ xÞ � Pðs¼ pxjA¼ xÞ � f 1

¼ ∑N

l ¼ 0Plf N−l

!� p � f 1 ðA:7Þ

In the proposed algorithm the detection estimation is fault-freeonly when l4θ. Thus Eq. (A.1) can be rewritten as

Pl ¼1 if l4θ

0 otherwise

�ðA:8Þ

Thus, the error probability of the proposed algorithm in detect-ing the status of a node is given by

Pe ¼ Pgf þ Pfg

¼ f 1 � 1−p− ∑N

l ¼ θ

ð1−pÞf l þ pf N−l

!ðA:9Þ

Substituting fl in Eq. (A.9), the expression of summand of Eq.(A.9) can be written as

N

l

� �ðð1−pÞlþ1pN−l−plþ1ð1−pÞN−lÞ

¼ N

l

� �ðð1−pÞlþ1pNþlðpN−2l−1−ð1−pÞN−2l−1ÞÞ ðA:10Þ

For po0:5, Eq. (A.10) is negative for N42lþ 1, zero for N¼ 2lþ 1,and positive for No2lþ 1. Additional terms with negative con-tributions are produced by decreasing θ one at a time from Nwhileθ40:5ðN−1Þ and positive contributions once θo0:5ðN−1Þ. It fol-lows that pe achieves a minimum when θ¼ 0:5ðN−1Þ. □

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