Differential Evolution for Lifetime Maximization of Heterogeneous Wireless Sensor Networks

Abstract

Maximizing the lifetime of wireless sensor networks (WSNs) is a hot and significant issue. However, using differential evolution (DE) to research this problem has not appeared so far. This paper proposes a DE-based approach that can maximize the lifetime of WSN through finding the largest number of disjoint sets of sensors, with every set being able to completely cover the target. Different from other methods in the literature, firstly we introduce a common method to generate test data set and then propose an algorithm using differential evolution to solve disjoint set covers (DEDSC) problems. The proposed algorithm includes a recombining operation, which performs after initialization and guarantees at least one critical target’s sensor is divided into different disjoint sets. Moreover, the fitness computation in DEDSC contains both the number of complete cover subsets and the coverage percent of incomplete cover subsets. Applications for sensing a number of target points, named point-coverage, have been used for evaluating the effectiveness of algorithm. Results show that the proposed algorithm DEDSC is promising and simple; its performance outperforms or is similar to other existing excellent approaches in both optimization speed and solution quality.

1. Introduction

The past two decades have witnessed the boom of wireless sensor networks (WSNs) and their applications, such as battlefield surveillance, environment supervision, traffic control, animal tracking, and home applications [1–5]. A key technology for various applications that involve long-term and low-cost monitoring, in other words, the fundamental criterion for evaluating a WSN, is the network lifetime [6], which is defined as the period that the network satisfies the application requirements. Since most devices of WSNs are powered by nonrenewable batteries, studies for prolonging the network lifetime have become one of the most significant and challenging issues in WSNs. Thus, how to maximize the network’s lifetime is a critical research topic in WSNs.

Various methods have been proposed for prolonging the lifetime of WSNs focusing on the issues of data processing [7], routing [8–11], device placement [12], topology management [13], and device control [14–16]. In a WSN where devices are densely deployed, a subset of the devices can already address the coverage and connectivity issues. Since it has been proven that the network connectivity of active sensors in complete coverage is guaranteed by having the communication range of each sensor at least twice of its sensing range [17, 18], thus, only the sensing coverage problem is researched in this paper.

In a WSN, coverage problem is an important issue, which determines how well an area of interest is monitored or tracked by sensors, and the device control approach that schedules the devices’ sleep/wakeup activities has shown to be promising [19, 20]. In general, random deployment is performed to place the sensors (dropping from a plane) to the target area. And a sensor generally has two operation modes, active mode and sleep mode [4]. When in active mode, a sensor can carry out its full operations, such as sensing, computation, and communication. To maintain those operations, sensors need to consume a relatively large amount of energy. In contrast, a sensor in a sleep mode uses only a small amount of energy and can be awoken in a scheduled working interval for full operations. When a subset of sensors in the area can already cover the target area completely, the other sensors can be scheduled to be in the sleep mode to save energy; thus, if the number of subset is maxed, the lifetime of WSN is prolonged. In other words, maximizing the number of completely cover subset is a more direct way to maximize the network lifetime [21]. Note that the target includes point target and area target; so the coverage problem is divided into point-coverage and area-coverage, respectively. Because the point target is the most common phenomenon in practical application, in this paper, the point-coverage problem is considered to maximize the lifetime of WSN by maximizing the number of completely cover subsets.

The problem of finding the maximum number of complete cover subsets is difficult because each subset must fulfill complete coverage to the target area, and the WSN can satisfy the surveillance task with only one subset of sensors active at any time. This problem in a WSN is called the disjoint set covers problem or the SET 𝑘-cover problem, which has been proven to be nondeterministic polynomial complete (NPC) complex problem [22]. In [22], a maximum cover using mixed integer programming (MC-MIP) algorithm is proposed to find the maximum number of complete cover subsets. In [23], Slijepcevic and Potkonjak proposed a greedy algorithm, which is named the most constrained minimally constraining covering (MCMCC) heuristic to completely cover the target area. In [24], Lai et al. firstly introduce gene algorithm to solve the point-coverage problems for WSN by finding the maximum number of complete cover subsets, termed GAMDSC. Recently, a novel hybrid genetic algorithm using a forward encoding scheme (STHGA) was proposed by Hu et al. [25] for lifetime maximization of WSN, which adopts a forward encoding scheme for chromosomes and sensor schedule transition operations.

Besides, from other perspectives to research wireless sensor networks such as evolutionary algorithm (EA) [26, 27] are promising direction. Furthermore, using other gene algorithms to research this problem is worth considering. Among the popular gene algorithms and their application in the recent years [28–31], differential evolution (DE) algorithm [32, 33] appears to be a competent candidate. Due to its original definition by Storn and Price [34], the DE algorithm and it’s variant are perceived as a reliable and versatile population-based heuristic optimization technique, which exhibits remarkable performance in a wide variety of application problems [35–40].

In this paper, a DE-based algorithm is presented for aiming at solving disjoint set covers problems for maximizing the WSN lifetime. The proposed algorithm, termed the differential evolution for solving disjoint set covers (DEDSC) problems, comprises the following features. Firstly, before running of DEDSC, test data set is produced by a common generation method, which includes the targets and sensors coverage information and the upper bound of disjoint set covers number (𝐶_max). And then, at initialization of DEDSC, the population is generated randomly; in each chromosome the value of gene is a random integer between [1, 𝐶_max]. After that, a recombination operation is performed to guarantee at least one critical target’s sensor is divided into different disjoint subsets. Fourthly, there are just two parameters in DEDSC, and the mutation strategy of DEDSC is “DE/best/1” differential evolution; the round number style is ceiling. Lastly, the fitness computation in our proposed algorithm considers both the number of complete cover subset and the coverage percent of incomplete cover subset. In general, these distinct features of DEDSC make it simple and effective to implement.

The rest of this paper is organized as follows: Section 2 defines the problem addressed in this paper. Section 3 describes the implementation of the proposed algorithm in detail and includes the encoding method of chromosomes, the design of the fitness function, the recombining operation, the crossover operation, and mutation operations. Experimental results and discussions are presented in Section 4. Finally, Section 5 draws a conclusion and provides guidelines for future research.

2. Disjoint Set Covers Problem

In this section, the problem of finding the maximum number of disjoint set covers in WSN is defined. And then we introduce a method for estimating an upper bound of the number of disjoint set covers.

2.1. Problem Definition

Suppose that, in an 𝐿×𝑊 area, there are have a set of targets   𝑇={𝑡₁,𝑡₂,𝑡₃,…,𝑡_𝑚}, and then randomly deploy a set of sensors 𝑆={𝑠₁,𝑠₂,𝑠₃,…,𝑠_𝑛} in this area to monitor the targets. All of the sensors have sleep mode and active mode: in the active mode sensors can sense information of target, and assume sensors have the same sensing region, but in sleep mode they cannot sense due to saving energy. A target is said to be covered by a sensor if it lies within the sensing region of the sensor. In order to prolong the lifetime of WSN, we need find the maximum number of disjoint sensor covers. The problem can be solved via transformation to the DSC problem [22], which is defined that to find the maximum number 𝐶 of disjoint complete cover sets, and the corresponding cover set 𝐶_𝑖 is satisfied [24, 25].

Every cover 𝐶_𝑖 is a subset of 𝑆,𝑖∈[1,𝐶_max], where the 𝐶_max is the upper bound of disjoint set covers number 𝐶. Namely, 𝐶_𝑖∈𝑆, and each 𝐶_𝑖 can complete coverage to all of the targets. Beyond that, for every of 𝑡_𝑖 belongs to at least one member of 𝐶_𝑖, and for any two different covers 𝐶_𝑖 and 𝐶_𝑗, 𝐶_𝑖∩𝐶_𝑗 = 𝜙. Take Figure 1 as an example; there are five sensors S1, S2, S3, S4, and S5 and four targets t1, t2, t3, and t4; every sensor has the same circular sensing region. Note that in the real applications, sensing region of a sensor can be an irregular shape [41].

Figure 1 

A randomly deployed figure of WSNs by five sensors and four targets.

According to Figure 1 it can be transformed into bipartite graph of Figure 2. The link line between sensor 𝑆_𝑖 and target 𝑡_𝑗 means 𝑆_𝑖 can sense 𝑡_𝑗. For each sensor, which can sense targets set 𝑆₁={𝑡₁,𝑡₃}, 𝑆₂={𝑡₁,𝑡₂}, 𝑆₃={𝑡₂,𝑡₄}, 𝑆₄={𝑡₃,𝑡₄}, and 𝑆₅={𝑡₁}.

Figure 2 

The bipartite graph of Figure 1.

2.2. Upper Bound of Disjoint Set Covers Number 𝐶

In WSNs, the maximum number of disjoint set covers cannot exceed the maximum number of full cover subsets that satisfy the coverage constraint. Thus, the maximum number of full cover subsets (𝐶_max) can be used as the upper limit of the number of disjoint set covers. It has been proven that finding the maximum number of full cover subsets is an NPC problem [21, 42], but we can compute the upper bound of the number of full cover subsets with the following method in test data sets.

When all the deployed sensors are active, all the targets are covered. For each target, it can be covered by at least one sensor. The targets which are covered by the minimum number of sensors is named critical targets, and their corresponding sensors are named critical sensors. It is worth noting that in the WSNs there maybe exists one or many critical targets, and each critical target has its own corresponding critical sensors. In the literature [24], the authors proposed term critical sensors which is just a special case; namely, there is just one critical target in WSNs. The number of sensors covering a critical target can be estimated by the upper bound of the number of full cover subsets [23, 25]. Therefore, the minimum number of sensors covered, which is denoted by 𝐶_max, can be used as the upper bound of 𝐶.

Take the case in Figures 1 and 2 as an example; for these targets, the covered sets by sensors are 𝑡₁∈(𝑆₁,𝑆₂,𝑆₅),𝑡₂∈(𝑆₂,𝑆₃), 𝑡₃∈(𝑆₁,𝑆₄), and 𝑡₄∈(𝑆₃,𝑆₄). According to above analysis, there are three critical targets (𝑡₂, 𝑡₃, and 𝑡₄), and each critical target is covered by 2 sensors; thus the upper of the 𝐶 is 2 (𝐶_max=2).

3. Differential Evolution for Maximizing the Disjoint Set Covers Problems

In this section, we will describe the DEDSC approach for maximizing the number of disjoint set covers problems in point-coverage targets WSN. Firstly, the test set generation algorithm is given. Then the implementation of the proposed DEDSC is presented step by step, including the initialization, the evaluation of the population, the recombination operations, and the crossover, mutation, and selection operations. For easy understanding, the notations used in this paper are shown uniform in Table 1.

Table 1 

Notations.

3.1. The Test Data Set Generation Algorithm

According to the application of WSN in the real-world, sensors and targets are randomly deployed in the specific area. Thus the test data set needs to be randomly generated by algorithm; however, in all of the above literature [23–25], the test data set generation algorithm is not introduced. Therefore, a general test data set generation algorithm is given in this subsection. The pseudocode of this algorithm is shown in Pseudocode 1. Note that the code is described by Matlab language, and the bold words are inner functions or keywords in Matlab. For example, the cell is the cell structure type.

/* generate test data set algorithm */ (1) randomly generate 𝑚 sensors 𝑆𝑖, each 𝑆𝑖 has abscissa 𝑆𝑖𝑥  and ordinate 𝑆𝑖𝑦. (2) randomly generate 𝑛 targets 𝑡𝑗, each 𝑡𝑗 has abscissa 𝑡𝑗𝑥  and ordinate 𝑡𝑗𝑦. (3) 𝑆cell=  cell (𝑚,1); % save every sensor can sense  the targets (4)  for  𝑖=1: 𝑚 (5) for  𝑗= 1: 𝑛 (6)  if  (𝑆𝑖𝑥−𝑡𝑗𝑥)2+(𝑆𝑖𝑦−𝑡𝑗𝑦)2 ≤𝑅 (7)    St{𝑖}=[St{𝑖}𝑗]; % sensor 𝑖 can sense       target 𝑗 (8)  end if (9) end for (10)  end for /* compute upper of disjoint complete cover sets (𝐶max) */ (11) 𝑇cell=  cell(𝑛,1) (12)  for  𝑗= 1: 𝑛 (13) for  𝑖= 1: 𝑚 (14)  if (find (𝑆cell{𝑖}==𝑗)) (15)   count++; (16)   𝑇cell{𝑗}=[𝑇cell{𝑗}𝑖]; % target 𝑗 is covered      by sensor 𝑖 (17)  end (18) end (19) 𝑐=[𝑐count]; (20)   end (21) 𝐶max=  min (𝑐); /* count all of critical targets and corresponding sensors */ (22)  for  𝑡= 1: 𝑛 (23) if 𝐶max==  size (𝑇cell{𝑡},2) (24)  CritialTarget=[CritialTarget𝑇cell{𝑡}]; (25)  numOfCritialTarget++; (26) end (27)  end (28) CritialTargetMatrix=  reshape (CritialTarget);

Pseudocode 1 

Pseudocode of generated test data set algorithm.

The test data set is obtained after running this algorithm, which contains the sensing information of every sensor (Scell), the information of every target covered (Tcell), the matrix of all critical targets and their corresponding critical sensors (Critial Target Matrix), and the upper limit of the number of disjoint set covers (𝐶_max).

3.2. The Representation of Chromosomes

It is known that to find disjoint covers is that each sensor randomly joins a group among prescribed groups. A group forms a cover if it can cover all targets. Based on this idea, we use integer representation to encode a grouping combination of sensors. The value of a gene indicates the index of the subset that the sensor joins [24], and thus the sensors with the same index number form a disjoint cover set.

For example, suppose a chromosome is CH₁ = (2, 1, 2, 3, 1, 3, 2, and 1). It means that there are eight sensors and set 1, set 2, and set 3 three disjoint subsets in WSN. The set 1 contains sensor 𝑆₂, 𝑆₅, and 𝑆₈, set 2 contains sensor 𝑆₁, 𝑆₃, and 𝑆₇, and set 3 contains 𝑆₄, 𝑆₆. If each of the above three sets can completely cover all targets, it means that the number of disjoint set cover is 3. In application, the sensors with the gene value 1 are scheduled to be activated at first, the other sensors will be kept in a sleep mode until the second set of sensors is activated, and the sensors in set 3 are activated in the last scheduling.

3.3. Initialization and Recombination

According to the representation of chromosomes, we randomly generate an integer between 1 and 𝐶_max as each gene value (subset number) in the initialization. The sensors with same gene value mean they form a subset. Take Figure 1 as an example, there are five sensors, and 𝐶_max=2; therefore after initialization a chromosome maybe is CH₂ = (2, 1, 1, 1, 2), each gene value is random, and sensors 𝑆₂, 𝑆₃, and 𝑆₄ form subset 1; sensors 𝑆₁, 𝑆₅ form subset 2.

After initialization a recombination operation is performed, which guarantees at least one critical target’s sensor is divided into different disjoint sets (subsets). Note that the recombination operation is different from the scattering in paper [24]; in this literature the authors assume there is only one critical target and scatter the corresponding sensors into different subsets. However, there maybe exist many critical targets in WSNs, and it is very difficult to scatter all of corresponding sensors into different subsets, this is likely an NP problem. Thus, our recombination operation considers every chromosome, randomly chooses a critical target, and recombines different gene values to their corresponding sensors. Take the chromosome CH₂ as an example; after initialization the chromosome CH₂ = (2, 1, 1, 1, and 2); then doing recombination operation, according to above analysis in Section 2.2, there are three critical targets (𝑡₂, 𝑡₃, and 𝑡₄), which 𝑡₂∈(𝑆₂,𝑆₃), 𝑡₃∈(𝑆₁,𝑆₄), and 𝑡₄∈(𝑆₃,𝑆₄). Thus randomly choose one critical target from 𝑡₂, 𝑡₃, and 𝑡₄, suppose the target 𝑡₄ is chosen, and its corresponding sensors are scatted into different subset; one case is 𝑆₃ = 2, 𝑆₄ = 1. Namely, the chromosome CH₂ became CH₂ = (2, 1, 2, 1, 2); it is obvious that after recombination the CH₂ has better fitness. The pseudocode of recombination is given in Pseudocode 2. It needs to explain that the code is described by Matlab language, and the bold words are inner functions or keywords in Matlab.

/* recombination operation */ (1)  for  𝑖= 1: popsize  % for every chromosome    % randomly generates 𝐶max different integers and    each integer value belong [1,𝐶max]. (2)  sequ = randperm(𝐶max);    % randomly choose one critical target (3)   rnum = randint(1,1,[1, numOfCritialTarget]);    % find the corresponding sensors (4)   rt =  CritialTargetMatrix(rnum,:); (5)    for 𝑗= 1: 𝐶max  % all corresponding sensors (6)    pop(i,rt(j)) = sequ(j); % recombination (7)    end for (8)  end for

Pseudocode 2 

The pseudocode of recombination operation.

3.4. Objective Function

The objective fitness function of a chromosome CH_𝑖 in the population is defined as follow: 𝑓_𝑖=𝐶+𝑝, (1) where 𝐶 (𝐶⩾1) is the number of disjoint complete cover sets and 𝑝(𝑝∈(0,1)) is the average coverage percentage of incomplete cover subset.

Actually, it is not difficult to calculate the average coverage percentage of incomplete cover subset in the point-coverage problem. Assume in a chromosome, the number of disjoint complete cover is 𝐶, and then the number of incomplete cover subset is 𝐶^′(𝐶_max−𝐶). For the 𝑖th incomplete cover, if it can cover 𝑘 targets, then the average coverage of 𝑖th incomplete cove subset is 𝑘/𝑛.

3.5. Differential Evolution for DSC Problem

Combined with the above analysis, this subsection introduces the detail of using differential evolution to solve DSC problem. Differential evolution [34] is a novel evolutionary algorithm, which has the following different mutation strategies which are frequently used in the literature:(1)“DE/rand/1” 𝑣_𝑖,𝑔=𝑥_𝑟1,𝑔+𝐹_𝑖⋅𝑥_𝑟2,𝑔−𝑥_𝑟3,𝑔, (2)(2) “DE/best/1” 𝑣_𝑖,𝑔=𝑥_best,𝑔+𝐹_𝑖⋅𝑥_𝑟1,𝑔−𝑥_𝑟2,𝑔, (3)(3)“DE/current-to-best/1” 𝑣_𝑖,𝑔=𝑥_𝑖,𝑔+𝐹_𝑖⋅𝑥_best,𝑔−𝑥_𝑖,𝑔+𝐹_𝑖⋅𝑥_𝑟1,𝑔−𝑥_𝑟2,𝑔, (4)where the indices 𝑟₁, 𝑟₂, and 𝑟₃ are distinct integers uniformly chosen from the set {𝑖=1,2,…,NP}⧵{𝑖}, (𝑥_𝑟1,𝑔−𝑥_𝑟2,𝑔) is a difference vector to mutate the corresponding parent 𝑥_𝑖,𝑔, 𝑥_best,𝑔 is the best vector at the current generation 𝑔, and 𝐹_𝑖 is the mutation factor which usually ranges on the interval (0, 1). In classic DE, 𝐹_𝑖=𝐹 is a fixed parameter used to generate all mutation vectors at all generations, while in many adaptive DE algorithms each individual 𝑖 is associated with its own mutation factor 𝐹_𝑖.

Following the mutation stage, a binomial crossover operator is implemented to enhance diversity of the population. In crossover operation, the target vector 𝑥_𝑖,𝑔 is connected with its corresponding mutant vector 𝑣_𝑖,𝑔 by using binomial crossover to generate a trial vector 𝑢_𝑖,𝑔. The aforementioned scheme can be outlined in 𝑢^𝑗_𝑖,𝐺=𝑣^𝑗_𝑖,𝑔,ifrand_𝑖,𝑗[0,1]≤𝐶_𝑟or𝑗=𝑗_rand,𝑥^𝑗_𝑖,𝑔,otherwise. (5)

The rand_𝑖,𝑗[0,1] is a uniformly distributed random number with the range [0,1], different for each 𝑗. 𝐶_𝑟 denotes a user-defined parameter, called crossover probability, which determines how similar the trial vector will be with respect to the mutant vector. 𝑗_rand∈[1,𝐷] indicates a randomly selected integer which ensures at least one dimension of the trial vector 𝑢_𝑖,𝑔 will differ from its associated target vector 𝑥_𝑖,𝑔. It has been proven that large values of 𝐶_𝑟 improve variation and hence accelerate convergence speed, while small values promote exploitation.

Finally in selection operation, the selection operator is a one-to-one spawning strategy. In selection, the target vector competes against the trial vector according to their fitness values, and the better one will be selected to enter the next generation: 𝑥_𝑖,𝑔+1=𝑢_𝑖,𝑔,if𝑓𝑢_𝑖,𝑔≤𝑓𝑥_𝑖,𝑔,𝑥_𝑖,𝑔,otherwise. (6)

For using DE to solve the disjoint set cover problem systemically and concisely, we provide the flow chart and the pseudocode of basic DEDSE in Figure 3 and Pseudocode 3, respectively. Figure 3 contains all operation modules, and Pseudocode 3 shows the corresponding pseudocode.

/* proposed DEDSC algorithm */ (1) Load test data set. (2) Initialization parameters 𝐹,𝐶𝑟, et. al    % Initialization population (3) popold=  round(unifrnd(1, 𝐶max, popsize, m)); (4) Recombination operation (5) FES=0; % the generation of evolution (6) while FES<𝑚∗100 (7) pop=popold; (8) 𝑣𝑖,𝑔=𝑥𝑟1,𝑔+𝐹𝑖⋅(𝑥𝑟2,𝑔−𝑥𝑟3,𝑔)  % mutation (9) for  𝑖= 1: popsize (10)    for  𝑗= 1: 𝑚      𝑣𝑖,𝑔(𝑗)=  ceil (𝑣𝑖,𝑔(𝑗)); % round numbers (11)    if  𝑣𝑖,𝑔(𝑗)<1or𝑣𝑖,𝑔(𝑗)>𝐶max  % boundary       checkout         % regenerate a random integer         % between [1,𝐶max] to update 𝑣𝑖,𝑔(𝑗) (12)      𝑣𝑖,𝑔(𝑗)= randint(1,1,[1, 𝐶max]); (13)    end (14)   end (15) end (16) Crossover between 𝑣𝑖,𝑔 and 𝑥𝑖,𝑔 by (5)      % Evaluate fitness for 𝑥𝑖,𝑔 and 𝑢𝑖,𝑔 (17) for  𝑖= 1: popsize % for each chromosome 𝑖 (18)  for  𝑗= 1: 𝐶max (19)   find those sensors with same subset index 𝑗. (20)   for every sensors 𝑘. (21)    count each 𝑘 cover the targets and       union them. (22)   end (23)   if the number of targets 𝑧 is equal 𝑛 (24)     this subset 𝑗 can complete coverage; (25)   else (26)     this subset 𝑗 coverage percentage is 𝑧/𝑛; (27)   end (28)  end (29)  count fitness of chromosome 𝑖 by (1); (30)  end (31) Selection operation by (6); (32) Update popold; (33) FES=FES+popsize;  % next generation    % Judgment of terminal condition (34) if  FES≥𝑚∗100  or the best fitness equals 𝐶max (35)  break; (36) end (37) end

Pseudocode 3 

The pseudocode of proposed DEDSC algorithm. Note that the pseudo code is described by Matlab language, and the bold words are inner functions or keywords in Matlab.

Figure 3 

Flowchart of the proposed DEDSC.

From Figure 3 and Pseudocode 3, we summarize that the distinct features of DEDSC are as follows. Firstly, test data set is produced by generation algorithm before running DEDSC, and the overall ideal of DEDSC is differential evolution algorithm. Second is initialization, the population is generated randomly, and thus the value of genes in chromosome is a random integer between 1 and 𝐶_max. After that, a recombination operation is performed to guarantee at least one critical target’s sensor is divided into different disjoint subsets.

Fourthly, there are just two parameters in DEDSC, and the mutation strategy of DEDSC is “DE/best/1” differential evolution; the round numbers style is ceiling. In addition, the fitness computation in our proposed algorithm considers not only the number of complete cover sets but also the coverage percent of incomplete cover sets. Lastly, termination condition of DEDSC is that the best fitness in current population equals 𝐶_max or the generation of evolution greater than 𝑚∗100.

3.6. Runtime Complexity Analysis of DEDSC Algorithm

Runtimecomplexity analysis of the population-based stochastic search techniques, like DE, GA, and so forth, is a critical issue by its own right. In this section, we examine the computational complexity of the proposed DEDSC algorithm, which can be estimated based on the number of calculations of metrics and reproduction that are required.

Assuming NP is the number of individuals, namely, size of population, and 𝐷 is the dimension of problem; the serial DE model needs 𝑂(NP⋅𝐷) time algorithm for upgrading population and computing values and fitness object function. Hence, if the algorithm is terminated after a fixed number of generations 𝐺_max, the overall runtime is 𝑂(NP⋅𝐷⋅𝐺_max).

In DEDSC, besides the fundamental operations for two parts of vectors, we have to take into account both of them. In our DEDSC, a recombination operation after initialization is proposed, which just performsone time in every time run. For each chromosome in NP, the best and the worst run complexity is 𝑂(1) and 𝑂(𝐷), respectively. Thus, the average complexity of recombination is 𝑂(0.5⋅(1+𝐷)⋅NP). Therefore, the time complexity of DEDSC is 𝑂(0.5⋅(1+𝐷)⋅NP)+𝑂(NP⋅𝐷⋅𝐺_max).

The above formula reveals the proposed DEDSC is slightly higher than original algorithm on runtime complexity.

4. Simulation Experiment and Analysis

In this section, a series of experiments are performed to evaluate the performance of DEDSC. Since the proposed approach is for maximizing the number of disjoint sets cover in point-average WSNs, the state-of-the-art algorithms, that is, GAMDSC [24] and STHGA [25], are used for comparison.

4.1. Parameter and Test Environment Setting

It needs to explain that, if not specially stated, the experiments for DEDSC use the same parameters settings as the population size 𝑝𝑜𝑝𝑠𝑖𝑧𝑒=10; the rate of mutation and crossover are initialized 𝐹=0.5 and 𝐶_𝑟=0.3, respectively, in which value of 𝐹 and 𝐶 are empirical values. And in DEDSC, the mutation strategy is DE/best/1, the maximum number of evolution generation function evaluations (FES) is 𝑚×100, and if the number of disjoint complete cover sets reaches 𝐶_max, the algorithm also terminates. These parameters influence the performance of DEDSC, which will be analyzed in Sections 4.3 and 4.4.

Parameter settings of GAMDSC and STHGA can be referred in [24, 25]. For DEDSC and other referred algorithms, each case is tested 100 times independently, the sensors are deployed in a 50 × 50 rectangle area, and the coordinates of sensors’ locations are randomly generated as float-point values in [0,50]. All cases are run by a computer with a Core I3 2.8 GHz CPU. We use the proposed generated test data set algorithm to obtain the test cases 1–7 and show the details of these test cases in Table 2.

Table 2 

The generated test cases index 1–7.

4.2. Overall Test for DEDSC and Referred Algorithm

In this subsection, the overall test for DEDSC and comparison of referred optimization algorithms are provided in Table 3. The best result among those methods is shown by Boldface in the table. The “ok%” indicates the percent to find 𝐶_max in 100 runs, “Best” is the best result in algorithm, “Mean” indicates the mean value of 100 runs, and the “avgE” denotes the average of evolution generation when finding 𝐶_max.

Table 3 

Experimental results of cases 1–7, averaged over 100 independent runs.

From Table 3, it can be observed that the proposed DEDSC is significant improvement rather than GAMDSC in cases 1-2 and 4–7; in case 3 they have equal performance. Moreover, the proposed DEDSC algorithm almost has the same performance as the STHGA, and it can reach a hundred percent to find 𝐶_max in all cases except for case 6. However, the result shows that our proposed DEDSC need the least evolution generation to find 𝐶_max in cases 1–5 and case 7, which implies the DEDSC has the most fast convergence rate.

In order to analyze the performance of DEDSC detailed, the average optimization curves of DEDSC in cases 1 and 6 are provided in Figures 4 and 5, respectively. From Figures 4 and 5 we can conclude that (1) DEDSC finds the best result much faster than other two methods which benefit from differential evolution and the comprehensive evaluation of fitness; (2) due to the recombination operation, DEDSC has the best initial population. In general, the traditional intelligence method, that is, Genetic Algorithm, has better initial population than STHGA, that is because the GA randomly generates group number, but STHGA generates index increase by degrees. It is can be seen from Figures 4 and 5 STHGA has the least number of complete cover sets in initialization; (3) the performance of STHGA, stable progressive increase, and it easily finds global optima because of the forward encoding scheme. DEDSC has better convergence rate and initial population than STHGA. However, in the later stage of evolution it may fall into the local optimal solution. That is, the reason for our proposed DEDSC is not better than STHGA in case 6.

Figure 4 

Average optimization curves of DEDSC, STHGA, and GAMDSC in case 1. Each of 50 generations evaluates one time.

Figure 5 

Average optimization curves of DEDSC, STHGA, and GAMDSC in case 6. Each of 20 generations evaluates one time.

4.3. Test for the Method of Round Numbers

In this subsection, the different way of round numbers and mutation are tested. The used test data is cases 1 to 7 are shown in Table 2.

After mutation operation, the value of gene maybe is float type, but the index of subset is integer type; therefore, it needs to do round number operation for population. There are there popular ways for round number operation, ceiling, floor, and round. In order to find which method is more suitable for DEDSC, all of three ways are tested and compared in Table 4.

Table 4 

Experimental results of cases 1–7, averaged over 100 independent runs.

Table 4 reports the experimental results of the three methods for round numbers in 100 independent runs. And these tests are used in mutation strategy “DE/rand/1”; parameter value are F = 0.5, 𝐶_𝑟 = 0.3. The best result among those methods is indicated by Boldface in the table. The results show that different round number methods have different performances. Specifically, with ceiling methods the performance and convergence rate of DEDSC is better than other methods in all cases especially in cases 5 and 6; thus as a conclusion, the ceiling round number methods are applied suitably in our DEDSC algorithm.

4.4. Test for Mutation Strategy and 𝐹, 𝐶_𝑟

There are many mutation strategies in DE, such as “DE/rand/1,” “DE/best/1,” and “DE/current-to-best/1.” Meanwhile there are two important parameters 𝐹 and 𝐶_𝑟 in DE. Therefore, finding the suitable mutation strategy and 𝐹, 𝐶_𝑟 are significant to the performance of DEDSC. In this subsection, we apply many popular mutations and 𝐹, 𝐶_𝑟 for testing DEDSC. The used test data sets are cases 1 to 7, and each test averaged over 100 independent runs with 𝑚∗100 FES, and the used round numbers method is ceiling.

Here, we provide an overall performance comparison between these tests in Tables 5, 6, and 7. The better result among those is indicated by Boldface in the table. Storn and Price in [34] have indicated that a reasonable value for 𝐹 is usually between 0.4 and 1, and a good initial choice of 𝐹 was 0.5. The parameter 𝐶_𝑟 controls how many parameters in expectation are changed in a population member. Feoktistov and Janaqi [43] claimed that a plausible choice of the crossover rate 𝐶_𝑟 is between [0.3, 0.9]. Recently, the authors in [44] state that 𝐶_𝑟 should lie in (0, 0.2) when the function is separable, while in (0.9, 1) when the function’s parameters are dependent. For the disjoint set covers problem it likely belongs to separable function; so we apply 𝐶_𝑟=0.3 and 𝐶_𝑟=0.5 as comparison, respectively, and provide the detail results in Tables 5–7.

Table 5 

Test “DE/rand/1” strategy on cases 1–7.

Table 6 

Test “DE/current-to-best/1” strategy on cases 1–7.

Table 7 

Test “DE/best/1” strategy on cases 1–7.

From Tables 5–7, it can be seen that the “DE/best/1” strategy with 𝐹=0.5 and 𝐶_𝑟=0.3 provides the best performance on cases 1–7. To be specific, considering the values of 𝐹 and 𝐶_𝑟 in each mutation strategy, it is observed that using 𝐹=0.5 and 𝐶_𝑟=0.3 the results are better than that using 𝐹=0.9 and 𝐶_𝑟=0.5. In especially the cases 1, 5, and 6, the performance of using the former parameter combinations significantly improves.

On the other hand, consider the mutation strategy with different values of 𝐹 and 𝐶_𝑟. Tables 5–7 show that only using “DE/best/1” strategy can reach 100 percent success rates to find 𝐶_max in case 5. And in case 6 the performance of using “DE/best/1” strategy is 87, which is the best in the three mutation strategies.

As a conclusion, according to the experimental results in Tables 5–7 and the above analysis, the “DE/best/1” mutation strategy using 𝐹=0.5 and 𝐶_𝑟=0.3 has the best performance in DEDSC.

4.5. Test for DE Variants with Self-Adaptive Parameters

In this subsection, we study the effect of DE’s variants which with self-adaptive parameter control strategy.

Inspired by “jDE” [39] we humbly employ the self-adaptive scheme to update parameters for solving the DSC problem, which can self-update to 𝐹 and 𝐶_𝑟 among perform. Moreover, various mutation strategies are tested, and the comparison results are provided in Table 8.

Table 8 

Effects of paramaters on performance of jDE, averaged over 100 independent runs with m * 100 FES.

In this subsection test, we not only apply the idea of self-adaptive in “jDE” to update 𝐹 and 𝐶_𝑟, but also test some variants of “jDE,” such that the various mutation strategies in “jDE” are tested. Note that in original “jDE,” the initialized 𝐹=0.5, 𝐶_𝑟=0.5, and the mutation strategy is “DE/rand/1.” Therefore in this subsection test, 𝐹=0.5 and 𝐶_𝑟=0.5 are used in all “jDE” and their variants, but different mutation strategies are applied in different variants.

Table 8 shows that in the case 5, DEDSC outperforms “jDE” and its variants significantly, only DEDSC can attain 100 percent success rate to find 𝐶_max. In case 1, both DEDSC and “jDE/best/1” can achieve 100% success rate; the “jDE/current-to-best/1” and original “jDE” only reach 84% and 94%, respectively. This means that the original “jDE” maybe is not very suitable for this problem, and the mutation strategy “/best/1” is better than others. Consider case 6, all of those algorithms cannot reach 100% success rate, original “jDE” shows the best performance, and our proposed DEDSC ranks second. It implies that our algorithm maybe lost in local optimum and needs more suitable control parameters to improve its performance.

As a conclusion, DEDSC is the best among the three algorithms in comparison to cases 1, 5, and 6. However, because the performance of DEDSC is not better than STHGA in case 6, we intend to study more suitable parameters in the future work.

5. Conclusion

Considering maximizing the lifetime of wireless sensor networks, our paper here proposes using differential evolution to solve this optimization problem, in which algorithm is termed DEDSC.

In DEDSC, there are just two parameters, mutate factor rate 𝐹 and crossover rate 𝐶_𝑟. The major features of DEDSC are recombination operation and fitness computation, and the former guarantees at least one critical target’s sensor is divided into different disjoint sets. This is a very important operation; especially there are many critical targets in test data set. Our method for computing fitness not only considers the number of complete cover subset, but also contains the cover percent of those incomplete cover subsets. To verify the DEDSC’s effectiveness, an extensive performance comparison has been conducted over 7 commonly used test cases.

The experimental results suggested that its overall performance was better than GAMDSC and almost same as STHGA. Considering on the convergence rate, DEDSC needs the least evolution generations to find 𝐶_max among the above algorithms. Moreover, a scalability study test was carried out using the different parameters and mutation strategies for DEDSC. In addition, for updating the parameter self-adaptive we tentatively combine “jDE” with DEDSC for comparing, and three “jDE” mutation strategies are tested. From the above results we obtain that the DEDSC with “DE/best/1” mutation strategy and F = 0.5, 𝐶_𝑟 = 0.3 have promising performance to maximize the lifetime of WSN.

However, using other self-adaptive parameter schemes, such as “JADE” [29], to improve the performance of DEDSC is a future direction for the algorithm.

Furthermore, there are many evolutionary algorithms such as ant colony optimization (ACO) have been applied to research the problem of WSN. Therefore, our further research work is possible to study ACO [21, 26] for improving the performance of our method. Some possible experimental research is now under our investigation in algorithm design and test.

Authors’ Contribution

Y. Xu and J. Fang contributed equally to this work.

Acknowledgments

This paper was supported by the Key Creative Project of Shanghai Education Community (13ZZ050) and the Key Foundation Project of Shanghai (12JC1400400). The authors are grateful to the Editor-in-Chief, Associate Editor, and anonymous reviewers for their constructive suggestions that helped to improve the content as well as the quality of the paper.