Wireless Sensor Network Deployment Using a Variable ... - IEEE Xplore

IEEE WCNC'14 Track 3 (Mobile and Wireless Networks)

Wireless Sensor Network Deployment Using a Variable-Length Genetic Algorithm Dina S. Deif and Yasser Gadallah Department of Electronics Engineering The American University in Cairo Cairo, Egypt [email protected], [email protected] Abstract— The Sensor Deployment Problem (SDP) is one of the most studied problems in the field of Wireless Sensor Networks (WSNs). It can generally be defined as selecting the sensors locations in a specified Region of Interest (RoI) to achieve one or more design objectives of the WSN. Two of the commonly required design objectives are maximizing coverage and minimizing the deployment cost of the WSN. In this paper, we address the SDP of covering a finite set of target locations in a specified RoI using non-homogenous, non-isotropic sensors with minimum sensor deployment cost. We propose a novel approach for solving the SDP using a Variable-Length Genetic Algorithm (VLGA). We apply our proposed algorithm on a WSN surveillance case-study to evaluate its performance. Based on the experimental results, we show that our proposed algorithm outperforms an existing approach which uses a Fixed-Length GA (FLGA) in terms of the quality of obtained solutions, speed of convergence and scalability. Keywords—Wireless sensor networks; sensor deployment; genetic algorithm; variable-length chromosome.

I.

INTRODUCTION

In the fast evolving field of Wireless Sensor Networks (WSNs), a considerable body of research has been dedicated to addressing the Sensor Deployment Problem (SDP) [1]. The SDP is concerned with selectively deciding the locations of the sensors to optimize one or more design objectives of the WSN. Common design objectives include maximizing coverage, minimizing deployment cost and maximizing the network’s lifetime. A large number of studies have proposed algorithms for solving the SDP. These algorithms are developed using different mathematical techniques. They are also heavily influenced by the requirements of specific applications and the characteristics of the used sensors, such as their wireless communication ranges and sensing profiles. The followed mathematical techniques can be classified into deterministic and stochastic techniques. Deterministic techniques include optimization techniques such as Integer Linear Programming (ILP) [2]. Stochastic techniques include different types of Evolutionary Algorithms (EAs) such as Genetic Algorithms (GAs) [3]-[5], Particle Swarm Optimization [6] and Ant Colony Optimization [7]. The choice between deterministic and stochastic techniques depends on the complexity of the SDP as an optimization problem. Hence, the defining elements of the SDP, including the objective

function, constraints and set of feasible solutions, determine which class of techniques should be used. In general, practical SDPs are characterized by multiple design objectives, heterogeneous sensors and large search spaces. Such problems are classified as NP-hard or NP-complete problems, for which there are no polynomial-time algorithms that are capable of finding exact solutions [8]. Hence, they cannot be solved using deterministic optimization techniques. Since EAs are known to be capable of finding near-optimal solutions relatively fast for a wide range of combinatorial and multi-objective optimization problems [9], they offer an attractive alternative for solving SDPs. One of the most well-known stochastic EAs is GA. GAs are search and optimization algorithms based on the mechanics of natural selection and genetics. GAs became popular through the work of John Holland in the early 1970s [10], and have since been used for solving optimization problems in various fields such as computer networking, industrial engineering and machine learning. Several studies have proposed the use of GAs for solving complex, multi-objective SDPs for various WSNs applications. The study in [3] presents a Multi Objective GA (MOGA) for optimal deployment of ݊ static sensors in a two dimensional RoI, with two competing deployment objectives: maximizing the area coverage and maximizing the networks’ lifetime. Candidate solutions are real number encoded and are represented by a deployment vector, which contains the Cartesian coordinates of each sensor inside the RoI. The authors apply their proposed algorithm to three specific surveillance scenarios in [4]. Each scenario has its own set of competing design objectives depending on the nature of the surveillance required. However, the proposed MOGA in [3] and [4] has three limitations. The first limitation stems from the assumption that deployed sensors have isotropic coverage, also known in the literature as the disk coverage model [11]. This assumption does not apply to many types of sensors, such as image and video sensors [12]. The second limitation is that MOGA works on the assumption that the RoI has a flat terrain i.e. has no obstacles, which is an unrealistic assumption for many WSN surveillance applications. The third limitation is that MOGA cannot be used for the optimal deployment of non-homogenous sensors, i.e. heterogeneous deployment.

978-1-4799-3083-8/14/$31.00 ©2014 IEEE. 978-1-4799-3083-8/14/$31.00 ©2014IEEE

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Authors in [13] propose using an integer-encoded GA to find the minimum number and deployment positions of static homogeneous sensors to achieve complete area coverage in a grid-based RoI. The proposed GA is designed on the assumptions that sensors have a probabilistic sensing profile and that the RoI can have obstacles and/or preferential areas with known locations prior to the deployment. Although the proposed algorithm in [13] addresses the first two limitations of MOGA in [3] and [4], it still cannot be used for heterogeneous deployment. All three limitations are addressed in the study presented in [14], in which the authors address a specific class of SDPs. This class is modeled as an optimization problem in which a finite set of target points in a specified RoI is to be covered with minimum sensor deployment cost. Target points can assume both regular patterns (e.g. in a grid-based RoI) or irregular patterns depending on the application. The authors model the SDP such that the coverage calculations are decoupled from the actual deployment algorithm. This serves the original purpose of considering a wide variety of sensors with different sensing profiles without the need to make changes in the deployment algorithm itself. The authors propose solving the SDP using an integer-encoded FixedLength GA (FLGA). In an FLGA, all encoded candidate solutions in the algorithm are equal in length [9]. The authors test the validity of their deployment algorithm using a simplified SDP consisting of six target points and five deployment points, forming a security sensor fence. The output of their proposed algorithm matched the optimal solutions derived deterministically. However, the performance of the proposed algorithm was not evaluated for practical case studies that involve SDPs with larger coverage requirements. This is of paramount importance because the different performance metrics of GAs are known to change dramatically as the size of the optimization problem increases [15]. In this paper, we address the same class of SDPs as in [14]. We consider different types of sensors, i.e. a heterogeneous deployment, which differ in their cost and sensing profile. We propose a novel approach for solving this class of SDPs using a Variable-Length GA (VLGA). To the best of our knowledge, this is the first time a VLGA is used in the context of WSN deployment. We apply our proposed algorithm on a WSN surveillance case-study to evaluate its performance. Performance is evaluated in terms of convergence speed and quality of obtained solutions to the SDP. We also investigate the scalability of our proposed algorithm by considering several instances of the SDP with varying sizes. We compare our algorithm with the existing Fixed-Length GA (FLGA) proposed in [14] for solving this class of SDPs. The rest of this paper is organized as follows: In Section II, we define the SDP, in terms of its assumptions, objectives and mathematical formulation as an optimization problem. The different components of the proposed VLGA are discussed in details in Section III. We present our case-study and experimental results in Section IV. Finally, the paper is concluded in Section V.

II.

SENSOR DEPLOYMENT PROBLEM

Before presenting our proposed algorithm, we first need to define the SDP model, in terms of its assumptions, objectives and mathematical formulation. We assume a two-dimensional RoI, in which there is a finite set of locations that require some form of surveillance (e.g. sound, image, video...etc.) using static sensors. These locations are given the name target points, and they do not necessarily assume any regular pattern. The assumption that target points to be covered in a WSN form a regular pattern, such as square grids, has been widely used in WSN deployment literature [2], [13], [16], [17] . However, this is not a realistic assumption for many WSN surveillance applications, in which the target points represent the vital locations or assets that require monitoring, such as paintings in a museum hall or arrival/departure checkpoints in an airport terminal. We denote the set of target points ሼ‫ݐ‬௜ ሽ for݅ ൌ ͳǡ ǥ ǡ ݉. We also assume that there is a finite set of possible deployment locations, which are given the name deployment points, at which sensors may be deployed. This assumption is valid for WSN surveillance applications where the topology or layout of the RoI is known prior to the sensor deployment. Hence, careful examination of the RoI yields a finite set of feasible sensor deployment locations. We denote the set of possible deployment pointsሼ݀௞ ሽ for݇ ൌ ͳǡ ǥ ǡ ݊. Since image/video sensors are usually used for surveillance applications, we assume a sector coverage model [11]. This coverage model defines the coverage/sensing profile of a sensor in terms of three parameters: a sensing range‫ݎ‬௦ , a Field of View (FoV) angleߠ௦ and an orientation angle߶௦ . The model can hence be expressed in the following equation: ݂ሺ݀ሺ‫ݏ‬ǡ ‫݌‬ሻǡ ߠሺ‫ݏ‬ǡ ‫݌‬ሻሻ ൌ ൜

ͳ݂݅݀ሺ‫ݏ‬ǡ ‫݌‬ሻ ൑ ‫ݎ‬௦ ܽ݊݀ߠ௦ ൑ ߠሺ‫ݏ‬ǡ ‫݌‬ሻ ൑ ߠ௦ ൅ ߶௦ Ͳ‫݁ݏ݅ݓݎ݄݁ݐ݋‬ǡ

(1)

where ݀ሺ‫ݏ‬ǡ ‫݌‬ሻ is the Euclidean distance between the sensor and a point ‫ ݌‬in the RoI and ߠሺ‫ݏ‬ǡ ‫݌‬ሻ is the angular spacing between them. We assume that different kinds of sensors, with different sensing ranges, FoV angles and hence prices can be deployed in each deployment point݀௞ . Hence, to fully define a deployment combination or deployment-tuple, three parameters are specified: the sensor type, the deployment point ݀௞ and the orientation angle߶௦ . We denote the set of all possible deployment-tuples ൛݀‫ݐ‬௝ ൟ for݆ ൌ ͳǡ ǥ ǡ ‫ܮ‬, where ‫ܮ‬is the total number of all possible deployment tuples in the SDP. The value of ‫ ܮ‬depends on the number of deployment points݊, the number of available sensors types and the number of possible orientations. We have two design objectives for this surveillance WSN. The first objective is to cover all the target points inሼ‫ݐ‬௜ ሽ by, at least, one sensor. The second objective is to minimize the total deployment cost of the WSN. Hence, candidate solutions for the SDP are subsets of ൛݀‫ݐ‬௝ ൟ that cover all points inሼ‫ݐ‬௜ ሽ. The optimal solution(s) is a candidate solution(s) with the minimum deployment cost. To formulate the SDP as an optimization problem, we define a binary coverage matrix ࡯ for any given pair of ሼ‫ݐ‬௜ ሽ

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and൛݀‫ݐ‬௝ ൟ. The coverage matrix ࡯ is structured such that each of its rows represents a target point ‫ݐ‬௜ and each of its columns represents one of the possible deployment tuples݀‫ݐ‬௝ . Hence, ࡯ has a size of݊ ൈ ‫ܮ‬. If a deployment-tuple݀‫ݐ‬௝ covers a target point‫ݐ‬௜ , then the value of ܿ௜௝ will be set to one, otherwise it will be set to zero. Similar to [14], the SDP is expressed as follows: ‫݊݅ܯ‬൛σ௅௝ୀଵ ‫݌‬௝ ݀‫ݐ‬௝ ൟ Subject to

σ௅௝ୀଵ ܿ௜௝ ݀‫ݐ‬௝ ൒ ͳǡ݅ ൌ ͳǡ ǥ ǡ ݉

݀‫ݐ‬௝ ‫ א‬ሼͲǡͳሽǡ݆ ൌ ͳǡ ǥ ǡ ‫ܮ‬,

(2) (3) (4)

where ݀‫ݐ‬௝ is equal to 1 if it is included in a candidate solution and 0 otherwise. The price of deploying݀‫ݐ‬௝ is‫݌‬௝ . The constraint in (3) ensures that the first design objective is met. III.

VARIABLE-LENGTH GENETIC ALGORITHM

There are several GA implementations which can be categorized in a variety of ways (generational vs. steady state, single vs. multi-objective, fixed vs. variable-length, etc.) [18]. In this section, we discuss the limitations of the FLGA proposed in [14] in solving the SDP expressed in (2) and present the components of our proposed VLGA. A. Limitations of Fixed-length Genetic Algorithm As mentioned in Section I, the authors in [14] propose using an FLGA to solve the SDP expressed in (2). They use binary encoding, i.e. an encoded candidate solution, or a chromosome in the GA nomenclature, is represented by a binary string of elements, or genes. The chromosome length is equal to the number of all possible deployment-tuples ‫ ܮ‬in a specific scenario of the SDP. A gene will be set to 1 if the corresponding deployment tuple ݀௝ is included in the candidate solution/chromosome and zero otherwise. The main limitation of the binary-encoded FLGA proposed in [14] is that it would not scale well to SDPs with larger sizes, i.e. larger numbers of deployment and target points. This scalability issue stems from the choice of the binary encoding scheme, where the length of the chromosome grows with the size of the SDP. For example, if we are attempting to solve an SDP scenario with 100 possible deployment points, 4 different types of sensors to choose from and 4 possible orientations, then each chromosome will contain a massive number of 1600 genes. Since only one sensor may be deployed at a given deployment point, chromosomes corresponding to good solutions will be mostly composed of redundant zeros. We believe a more efficient method for chromosome encoding is integer encoding, where chromosomes would have a variable length and would contain only the integers corresponding to the deployed tuples. For example, assuming‫ ܮ‬ൌ ͳͲ, a binary encoded chromosome given by would be represented by using integer encoding. VLGAs were proposed in several studies [19] - [21] for solving a wide variety of optimization problems, especially those where the number of elements in an optimum solution is unknown a priori. An example of this type of

optimization problems is the famous Knapsack problem, which is similar to the SDP in (2) [18], [22]. The use of integerencoded VLGA to solve (2) has two advantages. The first advantage is that the GA becomes more computationally efficient since it eliminates the presence of the redundant zeros, especially as the SDP is scaled up. This in turn will significantly increase the efficiency of the GA in terms of speed, measured in CPU computation time. Second, the use of genetic operators designed for VLGA can enhance the ability of the GA to converge to near-optimal solutions [22]. B. Components of Varibale-Length Genetic Algorithm We now discuss the various components of our proposed VLGA, namely the encoding scheme, fitness evaluation, genetic operators, selection schemes and convergence/termination conditions. Table I shows the pseudocode of our proposed VLGA. 1) Encoding Sceheme: As discussed earlier, each possible deployment-tuple in the SDP is assigned an integer. A chromosome representing a subset of all possible deploymenttuples will hence be a combination of different (i.e. without repetitions) integers. Since the number of all possible deployment tuples is‫ܮ‬, the chromosome length, denoted by݈, can take values between ͳ and‫ܮ‬. 2) Fitness Function: We use a similar fitness function to the one proposed in [9]. The fitness function used to evaluate a chromosome of length݈,ܿሺ݈ሻǡ is expressed in the following equation: ݂൫ܿሺ݈ሻ൯ ൌ െሺσ௟௜ୀଵ ‫݌‬௜ ൅ ‫ݓ‬ሺ݉ െ ܿ‫ݒ݋‬ሻሻ, (5) where σ௟௜ୀଵ ‫݌‬௜ is the total deployment cost of the ݈ deployment tuples included in ܿሺ݈ሻ, ܿ‫ ݒ݋‬is the number of target points which fall in the coverage of the deployment tuples included in ܿሺ݈ሻ using (1) and ‫ ݓ‬is the penalty imposed on the fitness for failing to cover a single target point. The negative sign is added so that the maximum fitness would correspond to achieving both total coverage of the target points and minimum deployment cost. 3) Genetic Operators: Genetic operators in a GA, namely crossover and mutation, are the driving force behind its evolution. There are different crossover operators, but the most widely used one is the single-point crossover [15], [18]. In a VLGA, single-point crossover can be used. However, the method by which it is performed is modified to factor in the fact that parent chromosomes will almost always have different lengths. There are two different crossover events that can take place in the reproduction cycle of the VLGA [22]. The first event, called equal crossover, produces offspring that have the same lengths of the parents. This occurs when the crossover point is within the length boundaries of both parents. The second event, called inside crossover, produces offspring that are shorter than the longest parent and longer than the shortest parent. This occurs when the crossover point is outside the length boundary of the shortest parent but inside that of the longest parent. These two crossover events are illustrated in Fig.1. As for the mutation operator, there are a number of different mutation operators suitable for integer-

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encoded GAs [15][18]. Since there is no obvious method for choosing the most suitable among them, we experimented with different options and settled on randomly changing the vaue of a gene in a chromosome within the range ͳ and ‫ܮ‬. Any repetitions of integers inside offspring chromosomes after applying both operators are discarded. 4) Selection Schemes: Two types of selections schemes are needed in GA. The first one is the parent selection scheme, which determines the method by which parent chromosomes are chosen to undergo crossover. In our algorithm, we use the widely-known Roullette Wheel parent selection scheme [15]. The second one is the survivor selection scheme, which determines which chromosomes in both the parents and offspring populations survive to the next generation of the GA. We use a combination of elitism and fitness-based survivor selection as outlined in [15]. This is achieved by allowing the best half the population in terms of fitness to mate and survive to the next generation while their offspring replace the other half. 5) Convergence/Termination conditions: Three possible convergence criteria can be used to terminate a GA. The first one is reaching an optimal or acceptable solution to the optimization problem. This criterion is used for testing the validity of the GA in simple problems for which optimal solution(s) can be derived using exhaustive search or when we are only trying to reach an acceptable solution. For the SDP at hand, we seek near-optimal solutions. Hence, we must use the other two termination criteria, which are either reaching a predetermined maximum number of generations or going through a predetermined number of generations with no increase in the value of the maximum fitness. IV.

Fig. 1. Crossover events in the proposed VLGA TABLE I. Step

PERFORMANCE EVALUATION

In this section, we evaluate the performance of our proposed VLGA versus that of the FLGA of [14] in solving SDPs as expressed in (2). We first describe the case-study used for performance evaluation and the parameters used in both algorithms. We then present the obtained experimental results and compare the two algorithms in terms of specific performance metrics. In our case-study, we seek to deploy a surveillance WSN at the terminal of an international airport. Fig. 2 shows the layout of the airport terminal [23], which comprises the RoI of the SDP. Target points, marked on the figure as triangles, represent the vital locations which need to be placed under image/video surveillance, such as arrival checkpoints, entrances and staircases. Possible deployment points, marked as asterisks on the figure, are chosen such that a line of sight between the deployed image/video sensors and the target points can be maintained. We assume two types of image/video sensors are available for the deployment:

PSEUDO-CODE OF THE PROPOSED VLGA

VLGA for Solving SDP in (2)

1.

Set ‫ ݐ‬՚ Ͳ

2.

Initialize ܲሺ‫ݐ‬ሻ, such that ‫ܿ ׊‬௜ ሺ݈ሻ ‫ܲ א‬ሺ‫ݐ‬ሻǡ ݈ ൌ ͳ ՜ ݊ǡ ݅ ൌ ͳ ՜ ‫݁ݖ݅ݏ݌݋݌‬

3.

Evaluate ܲሺ‫ݐ‬ሻ using (5)

4.

While (termination condition not met)

5.

Recombine fitter half of ܲሺ‫ݐ‬ሻ to yield ‫ܥ‬ሺ‫ݐ‬ሻ

6.

Mutate ‫ܥ‬ሺ‫ݐ‬ሻ

7.

Replace less fit half of ܲሺ‫ݐ‬ሻ with ‫ܥ‬ሺ‫ݐ‬ሻ to form ܲሺ‫ ݐ‬൅ ͳሻ

8.

Evaluate ܲሺ‫ ݐ‬൅ ͳሻ using (5)

9.

‫ ݐ‬՚‫ݐ‬൅ͳ

9.

End While

10.

Output: ܿ௜ ሺ݈ሻ such that ݂൫ܿ௜ ሺ݈ሻ൯ ൌ ݂௠௜௡

•

A wide-angle image/video sensor of 90 degree FoV, 30 meters sensing range and a price of $150.

•

A standard image/video sensor of 60 degree FoV, 40 meters sensing range and a price of $100.

Each deployed sensor may have four different orientations of north, east, south and west, i.e. ߶௦ = 0, 90, 270 and 360 degree respectively. In order to find the subset of deployment tuples that provides full coverage for the specified target locations with minimum deployment cost, we apply our proposed VLGA and the FLGA in [14] to the SDP. The parameters used for both algorithms are listed in Table II. Convergence occurs when the maximum number of generations is reached or when the algorithms go through 30 generations with no increase in the value of the maximum fitness. Two performance metrics are used:

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Fig. 2. Layout of the first floor of an international airport terminal [23]

enhancing the algorithm’s ability in searching promising regions in the search space of the SDP. As the search space size increases with the dimensions of the SDP, this feature becomes more crucial and results in increasing the difference in the minimum deployment cost that is obtained by the two algorithms. Fig.4 illustrates the performance of the two algorithms in terms of speed of convergence, measured in CPU runtime. From the figure, it is clear that the VLGA exhibits a significant advantage in speed over the FLGA. Savings in CPU runtime increases steadily with the increase of the SDP dimensions and reaches a value of 50% for the SDP sets of our experiments. This behavior is attributed to the choice of integer-encoding in VLGA, which eliminates the redundant zeros from chromosome representation in the FLGA. This significantly decreases memory assignment time in VLGA execution, which in turn increases its computational efficiency. Since the chromosome length in the FLGA grows with the increase in SDP dimensions, specifically݊ , the speed advantage of the proposed VLGA shows the trend of gradual increase and is expected to become even more significant for higher SDP dimensions. From the obtained results, it is evident that the proposed VLGA outperforms FLGA in terms of speed of convergence and quality of obtained solutions. Enhancements in both parameters increase as the SDP dimensions increases, suggesting that VLGA is also more scalable than FLGA.

• The quality of obtained solutions to the SDP at convergence, which is equivalent to the deployment cost. • The speed by which the algorithms converge, measured in CPU run-time. We also investigate how these performance metrics are affected by changing the size of the SDP i.e. algorithm scalability. This is achieved by applying both algorithms on several instances of the SDP with varying sizes, i.e. SDP dimensions specified by the numbers of target and deployment points, ݉ and ݊respectively. The algorithms were run on an Intel processor Core i7-3621QM CPU, 2.1 GHz and 6 GB of RAM. Each reported result point is the average of 15 runs. To account for the inherent randomness in GAs, we calculated the 95% confidence interval for each set of runs. Results are shown in Fig. 3 and Fig. 4. Each result point consists of the average, upper and lower limits of the 95% confidence interval of the corresponding set of runs. In both figures, the dimensions of the SDP, specified by ݉ and݊, are increased gradually, ranging between 15 to 30 and 30 to 60 respectively. Fig.3 illustrates the performances of the two algorithms in terms of quality of obtained solutions. The quality of obtained solutions corresponds to the absolute value of the maximum fitness the algorithms have converged to. In our experiments, we set the penalty parameter ‫ ݓ‬in the fitness function expressed in (5) in the following manner. The chromosome which includes all possible deployment tuples in an SDP instance has a higher fitness than all the chromosomes that fail to cover one or more of the target points. This ensures that both algorithms will be able to evolve quickly to good solutions that provide full coverage. Hence, the differentiating factor in the solution quality between the two algorithms is the minimum deployment cost obtained, as shown in Fig. 3. The figure shows that the VLGA converges to solutions with smaller deployment cost than those obtained by the FLGA, for all tested instances of the SDP. This enhancement in the minimum deployment cost increases progressively as the dimensions of the SDP increase. This can be attributed to the fact that the genetic operators used in the VLGA introduce more fitness diversity in the evolving generations in the proposed VLGA as compared to the FLGA, thus

V.

CONCLUSION

In this paper, we addressed the Sensor Deployment Problem (SDP) of covering a finite set of target locations with minimum deployment cost in WSN surveillance applications. We defined the SDP in terms of its assumptions, objectives and formulation as an optimization TABLE II.

GA PARAMETERS AND THEIR VALUES

Parameter Maximum number of generations Population size Selection method Selection rate Crossover type Crossover rate Mutation rate

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Value 1000 100 roulette-wheel 0.5 single-point 1.0 0.05


3500

[3]

Minimum deployment cost ($)

3000

FLGA in [14]

[4]

Proposed VLGA

2500

[5] 2000

[6] 1500

1000

30,15

35,15 40,20 45,20 50,25 55,25 Sensor deployment problem dimensions (n,m)

[7]

60,30

Fig. 3. Comparison between the proposed VLGA and FLGA in terms of quality of obtained solutions to the SDP [8] 350

[9] 300

FLGA in [14]

[10]

Proposed VLGA

CPU runtime (s)

250

[11] 200

[12] 150

[13]

100

50

[14] 0

30,15

35,15 40,20 45,20 50,25 55,25 Sensor deployment problem dimensions (n,m)

60,30

[15]

Fig. 4. Comparison between the proposed VLGA and FLGA in terms of speed of convergence

[16]

problem. We proposed an efficient algorithm based on Variable-Length Genetic Algorithm (VLGA) for solving the SDP. Our proposed technique is the first to use VLGA methodology for WSN deployment. We evaluated the performance of our proposed algorithm and compared it to an existing algorithm that is based on Fixed-Length GA (FLGA) for different SDP instances with varying dimensions. Experimental results show that the VLGA outperforms the FLGA in terms of both the quality of obtained solutions and the speed of convergence for all tested SDP instances. Results also suggest that in terms of scalability, VLGA performs progressively better than FLGA as the problem size increases.

[17]

[18] [19]

[20]

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