Maximizing Lifetime of Range-Adjustable Wireless Sensor Networks: A Neighborhood-Based Estimation of Distribution Algorithm

—Sensor activity scheduling is critical for prolonging the lifetime of wireless sensor networks (WSNs). However, most existing methods assume sensors to have one ﬁxed sensing range. Prevalence of sensors with adjustable sensing ranges posts two new challenges to the topic: 1) expanded search space, due to the rise in the number of possible activation modes and 2) more complex energy allocation, as the sensors differ in the energy consumption rate when using different sensing ranges. These two challenges make it hard to directly solve the lifetime maximization problem of WSNs with range-adjustable sensors (LM-RASs). This article proposes a neighborhood-based estimation of distribution algorithm (NEDA) to address it in a recursive manner. In NEDA, each individual represents a coverage scheme in which the sensors are selectively activated to monitor all the targets. A linear programming (LP) model is built to assign activation time to the schemes in the population so that their sum, the network lifetime, can be maximized conditioned on the current population. Using the activation time derived from LP as individual ﬁtness, the NEDA is driven to seek coverage schemes promising for prolonging the network lifetime. The network life-time is thus optimized by repeating the steps of the coverage scheme evolution and LP model solving. To encourage the search for diverse coverage schemes, a neighborhood sampling strategy is introduced. Besides, a heuristic repair strategy is designed to ﬁne-tune the existing schemes for further improving the search efﬁciency. Experimental results on WSNs of different scales show that NEDA outperforms state-of-the-art approaches. It is also expected that NEDA can serve as a potential framework for solving other ﬂexible LP problems that share the same structure with LM-RAS.


I. INTRODUCTION
W IRELESS sensor networks (WSNs) have been deployed for automation of various tasks, such as environmental monitoring, intrusion detection, and intelligent farming [1]- [3]. The lifetime of a WSN is the total amount of time that the WSN can stay functional and fulfill certain objectives [4]- [6]. Many functions of WSNs rely on full coverage of specified objects, which can be an area [7]- [9]; a barrier [10], [11]; or a set of targets [12]- [14]. In many scenarios, batteries of sensors have limited energy and recharging the batteries is impractical. Therefore, how to utilize sensor energy efficiently for guaranteeing full coverage becomes the key to prolong the lifetime of WSNs.
The sensor activity scheduling is an effective way to prolong the lifetime of WSNs. In order to provide high-quality service, sensors are usually densely deployed. Activating a subset of sensors is already sufficient for achieving full coverage of specified objects (called a coverage scheme), and the other sensors can switch to the sleep mode for saving energy. Therefore, the network lifetime can be prolonged by finding a set of disjoint coverage schemes that use different sensors and activating them in sequence. Following this idea, a wealth of sensor activity scheduling approaches has been proposed. Liao and Ting [15] proposed a memetic algorithm based on a genetic algorithm and a local search operator. Wan et al. [16] proposed a scheduling algorithm for the roadside sensor network, where the road should be monitored by multiple sensors simultaneously. Zhang et al. [17] proposed a Kuhn-Munkres parallel genetic algorithm that used a divide-and-conquer strategy for the large-scale WSNs. Chen et al. [18] proposed a hybrid memetic framework that considered the dynamic coverage maintenance.
All the above approaches are subject to the assumption that sensors have a fixed sensing range. However, in recent years, the usage of sensors with adjustable sensing ranges has become more prevalent [19], [20]. This posts new challenges to the sensor activity scheduling task. With adjustable sensing ranges, the sensor activation mode is no longer binary (sleep/activated) but multilevel. The search space for finding coverage schemes thus becomes larger. In addition, sensors activated at different sensing ranges consume different amounts of energy for the same length of time. Thus, sensors activated in one coverage scheme may have This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ different remaining energy left after activation. The sensors with abundant remaining energy can continue to monitor targets, which should be considered in the formation of other coverage schemes for the sake of efficient energy utilization. However, the fact that one sensor can be used in multiple coverage schemes makes energy allocation more complex. To deal with these new challenges, a few new scheduling approaches have been proposed. Chang et al. [21] proposed a sensing radius adaptation approach based on a weighted Voronoi diagram for area coverage. Zhang et al. [22] proposed a multiobjective algorithm for barrier coverage to minimize the total power consumption, the number of active sensors, and the maximum sensing range of sensors. Since area and barrier coverage can both be converted into the special cases of target coverage, we focus on solving the lifetime maximization problem of WSNs with range-adjustable sensors (LM-RASs) in a target-coverage scenario. In this regard, Rossi et al. [23] proposed a column generation approach assisted by a genetic algorithm. However, it suffers from poor efficiency due to the time-consuming steps in generating new columns, particularly when facing large-scale WSNs. Cerulli et al. [24] proposed a heuristic approach based on a greedy algorithm and a local search procedure, but it is easy to get trapped into local optima. Despite these pioneering efforts, an effective and efficient algorithm for LM-RAS is still lacking.
This article proposes a novel neighborhood-based estimation of distribution algorithm (NEDA) to address the LM-RAS problem. Instead of solving LM-RAS directly, the NEDA divides LM-RAS into two subproblems: 1) finding a set of coverage schemes and 2) determining the activation time of each coverage scheme so that sensors' energy is best allocated to maximize the network lifetime. Clearly, solutions to the first subproblem compose the premise for the second one. In fact, supposing that all the possible coverage schemes in a WSN have been found, the second subproblem can be modeled as a linear programming (LP) problem, where the activation time of each scheme can be optimally assigned so that their sum, the network lifetime, is maximized. However, due to the large search space of LM-RAS, it is difficult or even impossible to enumerate all the coverage schemes. The essential idea behind NEDA is to address LM-RAS by solving the two subproblems recursively. In NEDA, each individual in the population represents a coverage scheme. An LP model is built to assign activation time to each coverage scheme in such a way that the network lifetime is maximized conditioned on the current population. The activation time is used as the individual's fitness, which drives NEDA to generate coverage schemes that are promising for prolonging the network lifetime. The new schemes, which form a new population, are again fed into the LP model for activation time allocation that may further prolong the network lifetime. As such, by repeating the steps of the coverage scheme evolution and LP model solving, NEDA can gradually approach the globally optimal network lifetime.
Maintaining high-population diversity is the key for NEDA to achieve good performance. This is because using similar coverage schemes that have many common sensors can induce imbalanced energy consumption across the WSN, resulting in a short lifetime. Therefore, to provide a set of diverse coverage schemes for efficiently utilizing each sensor, a neighborhood sampling strategy (NSS) is developed, which generates new coverage schemes based on the probabilistic models derived from neighborhoods. Moreover, a heuristic repair strategy (HRS) is also introduced, which uses the remaining energy of sensors as heuristic information to finetune the coverage schemes and thus enhances the search efficiency. Experiments are performed on WSN instances of different scales. The experimental results are compared with state-of-the-art approaches for validating the effectiveness and efficiency of the proposed NEDA.
In real world, many problems share the same structure with the LM-RAS problem: they all contain two interdependent subproblems and the first one is the premise for optimally solving the second one in an LP manner. We name them as flexible LP (fLP) problems (refer to Section II-B for details). It is also expected that NEDA can serve as a potential framework for solving the other fLP problems.
The remainder of this article is organized as follows. Section II presents the formulation of the LM-RAS problem and its generalization as fLP problems. Section III introduces the proposed NEDA approach and its novelties in detail. Section IV presents the experimental study. Finally, Section V draws a conclusion to this article.

A. Lifetime Maximization Problem of WSNs With Range-Adjustable Sensors
The optimization objective of the LM-RAS problem is to maximize the lifetime of WSN while satisfying two constraints: one is to guarantee that all targets are monitored simultaneously and consecutively; and the other is that the sensor activation time is limited by the battery energy.

1) Sensor Features:
We define the set of sensors in WSN as S = {s 1 , s 2 , s 3 , . . . , s |S| }. All the sensors are of the same type. Each sensor's battery has an initial energy b ini and cannot be recharged. As mentioned above, the sensing range of a sensor is adjustable. The set of selectable sensing ranges is a finite set of discrete values, defined as R = {r 0 ,r 1 , r 2 , r 3 , . . . , r K }, in which the elements, without loss of generality, satisfy r i < r i+1 (i = 0, 1, . . . ,K − 1). r 0 indicates inactivation (i.e., r 0 = 0). A tuple (s j , r i ) represents the sensor s j is activated at the sensing range r i . The energy consumption rate corresponding to each sensing range, defined as E = {e 0 ,e 1 , e 2 , e 3 , . . . , e K }, is computed according to the quadratic model [23], as where e K , the energy consumption rate of the largest sensing range, is given in advance.
2) Coverage Scheme: A coverage scheme is a subset of sensors with selected sensing ranges that can achieve full coverage over all the targets. For example, in Fig. 1, there are three sensors s 1 , s 2 , and s 3 and three targets t 1 , t 2 , and t 3 . {(s 1 , r 2 ), (s 2 , r 1 ), (s 3 , r 2 )} is a coverage scheme. However, this coverage scheme is not considered energy efficient due to coverage redundancy. An improved version would be to reduce the sensing range of s 1 from r 2 to r 1 for energy saving. In the sense of energy efficiency, we define the nondominated coverage scheme (ndCS) as a coverage scheme whose sensing range reduction or inactivation of any involved sensors will cause the coverage scheme to be infeasible, that is, losing full target coverage. In the example of Fig. 1, {(s 1 , r 1 ), (s 2 , r 1 ), (s 3 , r 1 )} is an ndCS. It is obvious that using ndCSs is always more energy efficient than using simply coverage schemes.
3) LP Model for Maximizing Network Lifetime: Given a WSN with fixed sensors and targets, we could maximize its lifetime if we can identify a specific set of ndCSs and activate them one by one in a way that utilizes energy of each sensor to the best. Considering the instance in Fig. 1, assume that each sensor can be activated for 2 h with the sensing range set at r 1 , but for only 1 h at r 2 . If we activate the ndCS {(s 1 , r 1 ), (s 2 , r 1 ), (s 3 , r 1 )} till all the sensors run out of energy, the network lifetime is 2 h. However, there are also three other ndCSs: 1) {(s 1 , r 2 )}; 2) {(s 2 , r 2 )}; and 3) {(s 3 , r 2 )}. If we activate these three ndCSs in turn, the network lifetime is prolonged to 3 h.
Following the above idea, the LM-RAS problem can be divided into two interdependent subproblems: 1) identify a proper set of ndCSs and 2) determine the activation time of each identified ndCS. Let CSS = {ndCS 1 , ndCS 2 , . . . , ndCS M } denote the set of ndCSs, and ε i be the activation time of the ith ndCS in CSS, that is, ndCS i (i = 1, 2, . . . , M). The second subproblem can be formulated by an LP model for maximizing the network lifetime, as In this formulation, (2) describes the optimization objective, where the network lifetime NL is calculated as the sum of activation time of ndCSs in CSS because the ndCSs will be activated sequentially. Equation (3) reflects the constraint of battery energy for each sensor, where ndCS i,j represents the index of the selected sensing range for the sensor s j in ndCS i . Note that a sensor may be activated in multiple ndCSs, and the energy it consumes during the activation of each ndCS must be taken into account. Equation (4) is a boundary constraint by definition, which requires the activation time of each ndCS to be non-negative. Given that the first subproblem is solved, that is, CSS is determined, the second subproblem (i.e., the LP model) can be addressed by a variety of matured techniques in polynomial time [25], [26]. Therefore, the key to solve the entire problem lies in the first subproblem. A naïve solution is to enumerate all ndCSs in the considered WSN. However, considering the numerous combinations of different sensors and different sensing ranges, ndCS enumeration is too time consuming to be practical. Moreover, a large CSS may lead to poor efficiency in solving the LP model. Therefore, we propose the NEDA to find a proper ndCS set, which will be introduced in Section III.

B. Flexible Linear Programming Problems: Generalization of LM-RAS
The LM-RAS problem can be generalized as a kind of problems that share the same structure. That is, they can be formulated as where the decision variable vector β and the pattern set α have the same number of elements (denoted as NE). Each decision variable β i in β is related to the pattern α i in α (i = 1, 2, . . . , NE). g(β|α) is a group of linear functions that represents problem constraints. For example, in the LM-RAS problem, α is the set of ndCSs and β is the vector composed of their activation time. A salient feature of such problems is that once α is determined, the problems are reduced to classical LP problems, and the exact optimal solution conditioned on α can be deterministically obtained by LP techniques. We identify this kind of problems as fLP problems, where the term "flexible" reflects uncertainty in the size and content of the pattern sets, that is, α in (5). As mentioned above, the key to solve an fLP problem lies in the formation of a proper pattern set. To approximate the global optimal solution efficiently, it is expected that the set includes all the patterns used in the global optimal solution, but excludes unrelated patterns (i.e., those that are not used in the global optimal solution) as many as possible. Missing one or more patterns used in the global optimal solution will deviate the search procedure from the global optimum while including too many unrelated patterns will slow down the search procedure.
This article proposes the NEDA to address the pattern set formation in the LM-RAS problem. It can also be extended to other fLP problems for pattern set formation. Thus, it arises the main contribution of this article: the identification of fLP problems and a potential NEDA-based framework for solving them.

III. NEDA
EDA is a kind of evolutionary computation techniques [27]- [30]. It evolves the population by sampling the probabilistic model(s) constructed by promising individuals. The basic framework of EDA is shown in Algorithm 1. As can be seen, after generating and evaluating the initial population (steps 1 and 2), a set of promising individuals (usually those with good fitness) is selected to construct a probabilistic model (steps 3 and 4), according to which new individuals are sampled (step 5). Then, the new population is formed by selecting from the union of the individuals in the current population and the new individuals sampled by the probabilistic model (step 6). By repeating the steps of model construction and sampling until the termination criterion is met (step 7), EDA can drive the population toward the optimum of the problem. EDAs have been successfully applied to a number of real-world optimization problems [31]- [33]. In this article, following the basic framework of EDA, the proposed NEDA utilizes a problem-dependent encoding scheme and incorporates the following three novel strategies to solve the LM-RAS problem: 1) an LP-based fitness evaluation (LPFE) strategy; 2) an NSS; and 3) an HRS.

A. Encoding Scheme
Each individual in the population of NEDA, denoted by X i (i = 1, 2, . . . , N with N as the population size), is a sequence that represents an ndCS. Each element X i,j in X i denotes the index of the sensing range at which the sensor s j is activated (j = 1, 2, . . . , |S|). The number of dimensions is thus the number of sensors, that is, |S|. As mentioned in Section II-A, r 0 represents inactivation. Therefore, s j is not activated if X i,j equals zero. An example of the above encoding scheme is shown in Fig. 2. As can be seen, s 2 and s 5 are not activated; s 4 , s 8 , and s 10 are activated at r 1 ; s 1 and s 7 are activated at r 2 ; and s 3 , s 6 , and s 9 are activated at r 3 .
In this article, we assume that the selectable sensing ranges of each sensor are a finite set of discrete values, defined as R = {r 0 , r 1 , r 2 , r 3 , . . . , r K }. However, not all the sensing ranges are energy-efficient choices for each sensor. In some cases, activation at a larger sensing range, which consumes the sensor more energy, cannot bring more gain in target coverage. Therefore, a preprocessing procedure is performed before population initialization to find out the energy-efficient sensing ranges for each sensor according to the target distribution. If a sensing range allows the sensor to monitor more targets than its next smaller sensing range, it is considered an energyefficient sensing range, otherwise, it is an energy-inefficient sensing range. Take sensor s 1 in Fig. 3 as an example. r 3 is an energy-efficient sensing range for s 1 because it allows s 1 to monitor two more targets (t 2 and t 3 ) compared to r 2 . However, r 2 is an energy-inefficient sensing range for s 1 because it cannot allow s 1 to monitor more targets compared to r 1 . Note that after conducting the preprocessing procedure, the energyefficient sensing ranges of different sensors may be different. The individuals can only choose each sensor's energy-efficient sensing ranges in the corresponding dimension. By doing so, the search space of the problem can be reduced while all ndCSs are still preserved.

B. LP-Based Fitness Evaluation
NEDA defines the fitness of an individual as the activation time of its corresponding ndCS, which is derived from the solution of the LP model formulated in (2)-(4) with the current population fed as the ndCS set in (3). Longer activation time of an ndCS indicates a greater contribution to the network lifetime, and thus is a suitable fitness metric to guide the generation of promising ndCSs. Consider the example in Fig. 1. Assume that the population has four individuals: 1) X 1 = {(s 1 , r 1 ), (s 2 , r 1 ), (s 3 , r 1 )}; 2) X 2 = {(s 1 , r 2 )}; 3) X 3 = {(s 2 , r 2 )}; and 4) X 4 = {(s 3 , r 2 )}. Solving the LP model we obtain ε 1 = 0, ε 2 = 1, ε 3 = 1, and ε 4 = 1, which represents that {(s 1 , r 1 ), (s 2 , r 1 ), (s 3 , r 1 )} is not activated while the other three are activated for 1 h, respectively. Therefore, the fitness values of the four individuals are f (X 1 ) = 0 and Note that in NEDA, all the individuals act as the input to the LP model and after solving the LP model, the fitness of each individual is obtained. That is, the LPFE only needs to be carried out once to obtain all the individuals' fitness, which is distinct from the traditional EDAs.

C. Neighborhood Sampling Strategy
A promising ndCS set to prolong the network lifetime should make use of each sensor as fully as possible. Moreover, the overlap of the activated sensors across different ndCSs should be minimized to achieve relatively balanced energy utilization among the sensors. Herein, the word "overlap" indicates that one sensor can participate in multiple ndCSs simultaneously, using either the same or different sensing Randomly select an individual X i ∈ SI; 4: Find the n − 1 individuals nearest to X i in SI, incorporate them with X i to construct X i 's neighborhood; 5: Construct a univariate probabilistic model based on X i 's neighborhood; 6: Sample one new individual from the above probabilistic model, repair it by HRS if it is not an ndCS; 7: If the new individual is not the same as the individuals in SI and SP, add it to SP, otherwise discard it; 8: End While ranges. To achieve these two goals, the key is to enhance the population diversity of NEDA. The NSS is thus proposed.
Typically, EDAs select a proportion of the population to construct a probabilistic model from which new individuals are sampled. In NEDA, we select the individuals whose fitness is larger than zero, that is, those ndCSs that are used (activated) in the solution of the LP model. The set of selected individuals is denoted as where P represents the population. Note that EDAs usually require a parameter to determine how many individuals are selected to construct the probabilistic model, while the proposed NEDA does not require such a parameter and determines adaptively based on the solution of the LP model. NSS introduces the idea of neighborhood into the process of constructing the probabilistic models to enhance the population diversity. The procedure of NSS in one generation is shown in Algorithm 2. Let SP denote the set to store the new individuals sampled by NSS. At the beginning of the NSS procedure, SP is initialized as an empty set (step 1). Then, the new individuals are generated as follows. First, an individual is randomly selected from SI (step 3). Second, the neighborhood of the selected individual is formed, including itself as well as the other (n-1) individuals nearest to it according to the Manhattan distance in the search space (step 4). In NEDA, n is set to 5. A univariate probabilistic model is constructed based on the distribution of activation modes of each sensor across the individuals in the neighborhood (step 5). Finally, a new individual is sampled from this probabilistic model (step 6). Fig. 4 exemplifies the above process of generating a new individual. Assume that the WSN has five sensors with two sensing range options. The left grid in Fig. 4 is a neighborhood that contains five individuals. The middle grid contains five subgrids, each of which shows the selection probabilities of the activation modes for one sensor. For example, three individuals in the neighborhood select r 2 for sensor s 1 while the other two select r 0 and r 1 , respectively. Therefore, as shown in the first subgrid, the probabilities for s 1 to be inactivated, be activated at r 1 , and be activated at r 2 are p(0) = 0.2, p(1) = 0.2, and p(2) = 0.6, respectively. With the probabilities of the three activation modes calculated in the same way for the other four sensors (as tabulated in the other four subgrids), a probabilistic model is built. A new individual can then be generated by choosing an activation mode for each sensor based on the above probabilistic model. To be more exact, for each sensor s j , a random number o j is first derived from the uniform distribution in (0, 1). Then, o j is compared with the cumulative probability of each activation mode, and the first mode whose cumulative probability goes beyond o j is chosen for the sensor. Still take Fig. 4 as an example. For sensor s 1 , the cumulative probabilities of r 0 , r 1 , and r 2 are 0.2, 0.4, and 1, respectively. Given that o 1 = 0.7, sensor s 1 will be activated at r 2 since o 1 ∈ (0.4, 1]. With the activation mode for the other sensors selected in the same way, a new individual 21 201 is generated (as shown in the right grid). Such a method is better than pure random sampling as it reduces the blindness of the search process. Meanwhile, it is also better than maximum-likelihood sampling (i.e., greedily choosing the activation mode with the highest probability for each sensor) as it enables the generation of more diverse individuals and thus alleviates the risk of premature convergence caused by the loss of population diversity.
The new individual may not always be an ndCS, if it is not, repair it by HRS (refer to Section III-D for HRS). In addition, to prevent the duplicated individuals that may lead to premature convergence, the new individual is compared with the individuals in SI and the other new individuals (i.e., the individuals in SP). If no duplication is detected, add the new individual to SP (step 7). The above procedure (steps 3-7) iterates until (N − |SI|) new individuals are generated (step 2). NEDA then forms the population of the next generation by bringing these new individuals together with the original individuals in SI.

D. Heuristic Repair Strategy
In the proposed NEDA, each individual represents an ndCS, that is, a nondominated coverage scheme. However, the individuals sampled from the univariate probabilistic models are not always ndCSs. Therefore, we propose the HRS to transform an individual into an ndCS using the remaining energy of sensors as the heuristic information. The remaining energy of each sensor s j , denoted by RE j , is computed by applying the solution of the LP model in (2)-(4), that is As shown in Algorithm 3, HRS conducts two operations: 1) refinement and 2) reduction. The refinement operation tunes the coverage scheme to make it feasible. That is, it activates some sensors or increases the sensing ranges of some activated sensors to achieve the coverage over all the targets. The refinement operation deals with each target in a random order (step 3). Then, it adopts a tournament selection strategy to choose a sensor for the unmonitored target. More Algorithm 3 HRS 1: X i is the individual to be transformed into an ndCS by HRS; 2: //refinement operation 3: For Each target t k in a random order 4: If t k is not monitored 5: Randomly select two sensors from TS k , denoted by s a , s b ; 6: sel = a if RE a > RE b , otherwise sel = b; 7: X i,sel = min_r where r min_r is the minimum sensing range for s sel to monitor t k ; 8: Update the coverage information; 9: End If 10: End For 11: //reduction operation 12: For Each sensor s j in the ascending order of RE; 13: Iteratively do X i,j = X i,j −1 if X i,j >0 and the WSN can still maintain full target coverage after that; 14: End For specifically, the set of sensors that can monitor a target t k is defined as TS k . If t k is not monitored (step 4), two sensors are randomly picked from TS k and the one with higher remaining energy are selected to monitor t k (steps 5 and 6). The sensing range of the selected sensor is set at the minimum that can monitor t k (step 7). After that, the coverage information is updated immediately (step 8) because the selected sensor may also monitor some other unmonitored targets. The tournament selection strategy in the above refinement operation is intended for striking a balance between randomness and greediness. If the sensors are selected completely at random, it may improve the diversity in sensor usage but also increase the risk of selecting undesired sensors with low remaining energy. In contrast, selecting the sensors solely based on their remaining energy can avoid the energy risk, but results in a less diverse ndCS set.
After performing the refinement operation, the reduction operation is carried out to make the coverage scheme nondominated. For each activated sensor in the scheme, HRS checks whether it can reduce its sensing range or be inactivated while the WSN can still maintain full target coverage. If so, the sensing range of the sensor will be shrunk to the lowest possible level. Note that the checking order can affect the performance of the reduction operation, because several sensors may monitor the same target. Herein, we check the activated sensors in ascending order of their remaining energy. The sensor with less remaining energy thus has a higher chance to get its sensing range reduced, which eases the energy consumption of that sensor and possibly brings longer network lifetime. The reduction operation corresponds to steps 12-14 in Algorithm 3.
One note is that those randomly generated individuals in the initial population may also not ndCSs, but the remaining energy of sensors is not available since the LP model has not been solved. In this case, we simply use a random repair strategy. That is, randomly select sensors to monitor the unmonitored targets and follow a random order to check if each sensor can reduce its sensing range or be inactivated.

E. Overall Procedure of NEDA
The overall procedure of NEDA is shown in Algorithm 4. First, the population of N individuals is randomly generated. Each individual, if not an ndCS, is repaired by the random repair strategy. The next step is solving the LP model in (2)-(4) with the population as input. After that, the individuals are evaluated through LPFE and SI is obtained according to (6). (N-|SI|) new individuals are then sampled by NSS and repaired by HRS if necessary. The new individuals and those in the SI form the population of the next generation. Note that by keeping all the individuals in SI for the next generation, NEDA is able to preserve the best solution found so far. The population size N in NEDA is adaptively set to 2 × |S| according to the number of sensors in WSNs. Based on the theory of LP, if an LP model has c constraints (excluding the non-negativity constraints), the number of variables in a feasible basis will be no larger than c [34], [35]. The LP model in (2)-(4) has |S| constraints apart from the non-negativity constraints. Therefore, at most |S| ndCSs are used (activated) in the solution of the LP model. Our NEDA has a population of 2 × |S| individuals, which can provide sufficient new ndCSs to avoid premature convergence. In extreme conditions that |SI| = |S|, NEDA can still generate |S| new ndCSs.

A. Experimental Settings
Two groups of instances are used to test the performance of the proposed NEDA. The first group contains 15 types of small-scale instances, where the number of sensors |S| varies in the set of {200, 300, 400} and the number of targets |T| varies in the set of {10, 20, 30, 40, 50}. The second group contains one type of large-scale instances, where |S| = 1000 and |T| = 100. For each type of instances, ten different instances are generated with sensors and targets randomly scattered in a 100 × 100 area. We name the instances as |S|_|T|_l, where l is the index of the instance in the type specified by |S| and |T|. Each sensor has ten selectable sensing ranges, increasing from 2 to 20 with a step of 2. The initial battery energy of sensor is 1 and the energy consumption rate of the largest sensing range is also 1. That is, the sensor can be activated for 1 h at the sensing range of 20. The energy consumption rates of the other sensing ranges are computed according to (1).
We compare NEDA with three approaches, CGGA [23], AR-Iterative [24], and Greedy. Herein, CGGA and AR-Iterative are two state-of-the-art approaches proposed in [23] and [24], respectively. Greedy is a simple heuristic algorithm that acts as the baseline. It has two principles: 1) the target whose distance from its nearest available sensor is larger has higher priority to be handled and 2) the available Algorithm 5 Greedy Algorithm 1: AS = S; NL = 0; 2: While the sensors in AS can achieve full target coverage 3: Sort the targets in descending order of the distance from their nearest available sensor; 4: For each target t k in the order obtained by Step 3 5: If t k is not monitored 6: Activate t k 's nearest sensor in AS at the minimum sensing range that can monitor t k ; 7: Update the coverage information; 8: End If 9: End For 10: Calculate the maximum activation time of the current coverage scheme, denoted by CL; 11: NL = NL + CL; 12: Update AS and the remaining energy of sensors; 13: End While 14: Output NL; sensor nearest to the target is selected to monitor it. Herein, available sensors are those that have remaining energy. The detailed procedure of Greedy is shown in Algorithm 5. AS represents the set of available sensors. In steps 3-9, a coverage scheme is constructed according to the above two principles. In step 7, the selected sensor may also be able to monitor the other target(s) apart from t k , so that the coverage information is updated immediately. Since the target that has larger distance from its nearest available sensor is handled at first, the coverage scheme obtained is guaranteed to be an ndCS. After obtaining an ndCS, its activation time is set at the maximum according to the remaining energy of the sensors (step 10). Then, the network lifetime NL is updated (step 11). In addition, AS and the remaining energy of sensors are also updated (step 12). The algorithm follows an iterative procedure and terminates until the sensors in AS are unable to achieve full target coverage.
NEDA and the compared algorithms are all iterative approaches, but it may not be fair to use the same number of iterations due to the difference in their iterative mechanisms. Instead, we force the algorithms to terminate within the same execution time on identical platforms. The maximum execution time is set to 1 and 20 h for small-scale and large-scale instances, respectively. All the experiments are conducted on PCs with Ubuntu 16.04 system, Intel i7-7700 CPU and 8-GB RAM. Among the tested algorithms, NEDA and CGGA involve stochastic factors, so that these two algorithms are run for ten independent times on each instance and the average results are presented. Unless otherwise stated, the differences between the experimental results are validated by Wilcoxon rank-sum test at the significance level of 0.05. Table I tabulates the average lifetime obtained on the ten instances of each small-scale instance type. The best results on each instance type are denoted in bold. The numbers of the format (a/b/c) show that comparing NEDA with the corresponding algorithm, NEDA performs significantly better on a instances, shows no significant difference on b instances, and performs significantly worse on c instances. "Total" sums up a, b, and c across all the 15 instance types for each compared algorithm.

1) Small-Scale Instances:
Compared with AR-Iterative and Greedy, NEDA performs better on all instance types. In detail, NEDA performs significantly better than AR-Iterative on 118 out of the 150 small-scale instances, and better than Greedy on 120 instances. Moreover, AR-Iterative and Greedy do not show a significant advantage over NEDA on any instance. Comparing NEDA with CGGA, we can see that their average results are the same on all small-scale instances. That is, given a maximum execution time of 1 h, NEDA and CGGA achieve the same result on each instance of the 15 small-scale instance types. The reason for getting the same results is probably that NEDA and CGGA both have the ability to sufficiently explore the search space of small-scale instances and both find the global optimal solution.
To make a more detailed comparison between NEDA and CGGA, the "Time to Best" metric is employed and the experimental results are shown in Table II. The Time to Best metric represents the execution time consumed by the algorithm to reach its best solution. The same as in Table I, the average results on the ten instances of each instance type are presented. The unit of execution time is second. In the last row of the table, the Wilcoxon ranksum tests show that NEDA performs significantly better than CGGA on all 150 instances. The last column shows the reduced time of NEDA to reach the best solution compared  II  COMPARISON OF "TIME TO BEST" BETWEEN NEDA AND CGGA ON  SMALL-SCALE INSTANCES (THE UNIT OF EXECUTION TIME IS SECOND) with CGGA. We can observe that NEDA has comprehensively better efficiency on all 15 instance types. Particularly on 400_30, NEDA is 15 min faster than CGGA to reach the best solution. These results indicate that although CGGA can achieve competitive results, it suffers from poor time efficiency.
2) Large-Scale Instances: The average lifetime obtained by the four algorithms on the ten large-scale instances are shown in Table III, with the best results denoted in bold. The values in the parentheses show the improvement rate of NEDA compared with the other algorithms. The last row summarizes the number of instances on which NEDA performs significantly better, shows no significant difference, or performs significantly worse compared with the corresponding algorithm.
From the table, the proposed NEDA achieves the best results on eight out of the ten large-scale instances. Only on 1000_100_2 and 1000_100_7 is the performance of NEDA worse than AR-Iterative, with the differences of −3.811% and −2.227%, respectively. However, NEDA performs significantly better than AR-Iterative on seven instances, particularly on 1000_100_9 and 1000_100_10, where NEDA's improvement rates against AR-Iterative are 5.092% and 7.883%, respectively. CGGA suffers from poor search ability on large-scale WSNs. We can see that NEDA performs better than CGGA on all the instances, with an improvement rate beyond 15% on seven out of the ten instances. In particular, the improvement rate of NEDA on 1000_100_8 reaches Overall, NEDA has better performance than the other three tested algorithms on both small-scale and large-scale instances. There are mainly two reasons for this. First, NEDA benefits from the LP model in (2)-(4), which can obtain the optimal activation time of each ndCS to efficiently utilize sensor energy and maximize the network lifetime. Second, with LPFE, NSS, and HRS, NEDA can efficiently generate a high-quality set of ndCSs to prolong the network lifetime.

C. Effectiveness of LPFE
In NEDA, the LPFE evaluates the fitness of individuals based on the LP model in (2)-(4). To investigate the effectiveness of LPFE, a greedy fitness evaluation (GFE) is used for comparison. GFE defines the fitness of each individual as the maximum activation time of its corresponding ndCS. Note that in NEDA, those individuals whose fitness is larger than zero are selected to construct the SI for NSS. However, every individual's fitness is larger than zero by the GFE. In this case, |S| fittest individuals (random break in case of tie) are selected to construct the SI.
Tables IV and V show the experimental results of these two fitness evaluation strategies on small-scale and large-scale instances, respectively. The better results are bolded. The numbers of the format (a/b/c) show that compared with GFE, LPFE performs significantly better on a instances, shows no significant difference on b instances, and performs significantly worse on c instances. In Table V  LPFE's effectiveness is as follows. The LP model can obtain the optimal activation time of each ndCS to maximize the network lifetime. Using the optimal activation time as the fitness can show each ndCS's contribution to the network lifetime, which helps guide the generation of more promising ndCSs. However, GFE's greedy principle makes it easy to find a local optimum.

D. Effectiveness of NSS
In this section, we investigate the performance of NEDA with different settings of the neighborhood size n in NSS, including n = 5, 10, 20, and |SI|. Note that with the setting of n = |SI|, NEDA actually omits the use of NSS and directly use SI to construct the univariate probabilistic model from which all new individuals are sampled. Thus, we term this setting as "no NSS." The experimental results show that the lifetime obtained by these four settings are not significantly different on all small-scale instances. Due to the page limit, we only present the convergence curves of NEDA under these four settings on some typical instances, as shown in Fig. 5.  From these figures, we can see that although the differences in the obtained lifetime are not statistically significant, NSS does help accelerate the convergence. The reason is that NSS helps generate diverse ndCSs to efficiently utilize the energy of each sensor. As the neighborhood size increases, the acceleration effect is gradually compromised, implying that a larger neighborhood size may weaken the effectiveness of NSS in diversity improvement and thus reduce the efficiency for finding an ndCS set with longer lifetime.
The experimental results on large-scale instances are shown in Table VI, with the best results on each instance bolded. The  values in the parentheses show the improvement rate of the adopted setting, that is, n = 5, compared with the other tested settings. The comparison results on all ten tested instances are counted and presented in the bottom of each compared setting's column. We can see that apart from 1000_100_6 where the four settings have closed performance, the NSS with n = 5 can greatly improve the obtained results.
In conclusion, NSS can improve the search efficiency by sampling new individuals based on neighborhood. The reason is that the LM-RAS problem requires a set of diverse ndCSs to efficiently utilize sensors' energy and NSS helps conduct the distributed search to enhance the population diversity. Moreover, on both small-scale and large-scale instances, the adopted setting of n, that is, n = 5, performs the best among all tested settings.

E. Effectiveness of HRS
To validate the effectiveness of HRS, we compare the performance of HRS with the random repair strategy (denoted by RRS). Compared with HRS, RRS does not use the remaining energy of the sensors as the heuristic information, but randomly selects the sensors to monitor unmonitored targets and checks if each activated sensor can reduce its sensing range or be inactivated in a random order. The convergence curves on some small-scale instances are plotted in Fig. 6. We can see that HRS accelerates the convergence. The convergence curves on some large-scale instances are plotted in Fig. 7. We can see that HRS can not only accelerate the convergence but also improve the obtained lifetime. The above results confirm that the proposed HRS benefits from the usage of heuristic information, that is, the remaining energy of sensors, and helps improve the search efficiency of NEDA to prolong the network lifetime.

V. CONCLUSION
In this article, a novel NEDA is proposed to deal with the LM-RAS problem. Each individual in NEDA represents an ndCS. Based on the ndCS set represented by the population, an LP model is built to find the optimal activation time of each ndCS for maximizing the network lifetime. To generate more promising ndCSs for prolonging the network lifetime, three novel strategies, i.e., LPFE, NSS, HRS, are incorporated into NEDA. Individual fitness is evaluated by LPFE that shows an individual's contribution to the network lifetime. The NSS uses neighborhoods to construct the probabilistic models for sampling new individuals, which helps generate diverse ndCSs. The HRS uses the remaining energy of the sensors as heuristic information to fine-tune the coverage schemes by the refinement and reduction operations. Experimental results on WSN instances of different scales showed that the proposed NEDA outperforms the state-of-the-art approaches in terms of solution quality and efficiency. The LM-RAS problem is further generalized as a kind of problem that shares the same structure, namely, fLP problems. The NEDA is supposed to be an effective framework for solving fLP problems, which will be further examined in future work.