Wireless Sensor Networks (WSN) is one among the predominant domain where ‘n’ number of problems grasps the researchers for achieving reliability, accuracy, the efficiency of the network. This contribution exponentially increased when the applications of WSN are expanded in the environments such as health care, disaster tracking system, military-based applications, etc. In recent years a significant number of optimization problems are solved using nature inspired algorithms [1]. IoT based Pay-AsYou-Go Smart Parking System is proposed which uses the unused garbage spaces [2]. Another recent model, which uses digital contact tracing to handle the epidemic situations like Covid-19 using IoT [3]. In WSN, the sensor nodes are used to sense the data surrounds it and report it to the base station either using a single hop or multi-hop communication process. Sensor node deployment is of two different types, namely ad-hoc and pre-determined manner [4]. The ad-hoc type deployment plays a significant role in regions like a deep forest, aquatic nature, etc. Usage of ad-hoc in such regions requires many sensor nodes for effective coverage of the respective region and proper connectivity between deployed sensor nodes. Manual deployment (pre-determined) of sensor nodes is easy to access and requires fewer sensors. However, coverage and connectivity issues are common in both types since the node’s transmission range is less than its restricted power source. The use of wireless connection gets disrupted due to bad weather, etc. Thus, covering targets and maintaining connectivity between sensors that enhance the network lifetime has been considered one of the significant aspects of WSN.

This paper addresses k-coverage m-connected augmented node deployment model in WSN as a multiobjective problem where Non-dominated sorting Genetic Algorithm-II is used to optimize it. The problem is described in this section. Given k target regions and P available positions for node deployment, the objective is to cover k targets using m sensor nodes where m⊆P. As it is given in Fig. 1, Let T={λ1,λ2,…,λ10} be the available target regions to be sensed using S={s1,s2,…,sm} deployed nodes. As per Fig. 1 number of sensor nodes in S is six out of 10 available positions of P. Target λ2 is covered by a Deployed sensor s₁ since the target is within the sensing range of the deployed sensor s₂ which is deployed in the available position p₂ and the deployed node can transfer the data to a base station through the available positions p₃ and p₄. By allocating the sensor nodes concerning its coverage and connectivity range, the minimum objective number of deployed nodes to cover all targets are efficiently achieved. This problem is considered an NP-complete problem [5]. Achieving optimization in this problem is highly preferable since exact methods take more high time complexity when compared to the optimization approach.

In the past few decades, research contributions on optimizing WSN problems using heuristic and metaheuristics are higher than earlier research exposure [5–9]. In this literature, coverage and connectivity related algorithms are described with their predominant features.

A detailed Survey has been recorded on different methods for effective node placement in WSN by [10]. In [5] a contradictory set of objectives are taken for effective node placement in WSN, namely energy cost, sensible area, and network reliability. The flaws in their approach are the inadequate coverage of complete targets in the given region. The use of meta-heuristics failed to achieve this particular issue which made the algorithm an ineffective one. A new approach for solving the WSN issue is dividing a region into a disjoint sub-region and imposed active and passive mode of operations in it by [11]. The proposed methodology of Cardie was later redefined in [5] to enhance the lifetime of the network by imposing redundant nodes. Irrespective of coverage and connectivity, this approach achieves better results along with improved disjoint sets.

Research on the optimization of resource allocation algorithm in WSN is discussed by [12]. The lifetime of network and connectivity are the objectives considered in this paper, which takes the sensor positions and transmission power level as their attributes to solve the problem. MOEA/D is used to solve the given problem to achieve the objectives with fewer deployed sensor nodes. Coverage was not handled in this paper. Author Harizan et al. [13] proposed an NSGA-II with modified dominance has been proposed for the scheduling problem. The multiple parameters as coverage, connectivity and residual energy of the sensor nodes are considered. A probabilistic sensing model (PSM) and harmony search algorithm (HSA) approach are used in the same type of problem in WSN to achieve the balance between the network coverage performance and the network cost [14].

In 2012, a multi coverage based WSN problem is handled in [15], three different coverage, namely simple coverage, k and Q coverage. For the optimization process, an exhaustive cover set defining process is used in this approach. For the inclusion of connectivity in the proposed algorithm, an M-connected approach is included for optimization. The algorithm’s time complexity is higher since the algorithm flow included each node one by one and check for all possible combinations in the set for adaptive placement of nodes.

A simulation-based approach for effective node placement is proposed by [16], where several available positions to place the sensor nodes are given. Without colliding the connectivity between sensor nodes, the nodes should be placed with the minimal number–-another approach based on GA proposed in [17] as relay sensor node placement algorithm. This paper’s defined objective is to efficiently place the nodes in the given positions, thus minimizing the number of sensors without colliding its connectivity between relay nodes. Using GA, another approach for attacking coverage problem in WSN is proposed to enhance its lifetime. In this approach, sensing targets are not used for simulation. Only the sensors nodes are placed to cover the simulation region without any targets in it. In our proposed method, this problem is addressed. Another GA based approach for addressing the same issue as our proposed method has been made in [4]. However, this method reproduces offspring so that it becomes invalid, and a solution repair process is needed here. In our proposed algorithm, a multiobjective optimization algorithm NSGA-II is used to address this issue and Pareto set solutions.

This section comprises the prescribed definition of the multiobjective optimization problem [18] and Pareto dominance operators, which helps to understand the proposed work better.

Given a problem with solution set P=[C1,C2,…,Cn] with n individuals where each individual Ci=[Gi,1,Gi,2,…,Gi,j,…,Gi,m] with m decision variables intents to find a vector Ci*→=[Gi,1*,Gi,2*,…,Gi,m*] which satisfies p equality constraints ui(C→)=0,i=1,2,…,p, q inequality constraints vi(C→)=0,i=1,2,…,q and finds a minimized vector function f(C→)=[f1(C→),f2(C→),…,fk(C→)] where k is the number of objectives.

From the solution space given in this definition P=[C1,C2,…,Cn], an individual C1=[G1,1,G1,2,…,G1,m] is said to dominate C2=[G2,1,G2,2,…,G2,m] iff fi(C1→)≤fi(C2→)∀iϵk and fi(C1→)<fi(C2→)∃iϵk and this can be represented as C1→≺C2→. Two individuals said to be non-dominated to each other if neither C1→≺C2→ nor C2→≺C1→ and this can be represented as C1→≼C2→.

Fig. 2 shows an example for dominated and non-dominated solutions. Let us consider a multiobjective problem that consists of two objectives f1() and f2(). Three individuals C1→,C2→andC3→ are plotted in Fig. 1 based on their objective values. From the given definition, we can claim that individual C1→ dominates individual C2→ and C3→ and this can be represented as C1→≺{C2→,C3→}. And individual C2→ and C3→ are non-dominated to each other since it satisfies the given definition of the non-dominated solution set and this can be represented as C2→≼C3→.

In this section the the formulation of t-WSN problem is defined with an example along with the objectives.

This section holds a four-folded algorithm for optimizing coverage-connected problem using NSGA II [19]. In this section, a discussion on the Non-dominated sorting method followed by preservation of diversity will be explained in the second fold, and reproduction operators are discussed. In the final fold, a complete algorithm for solving the coverage-connected problem using NSGA-II is described. NSGA-II is chosen to solve m −connected k −coverge problem over other multiobjective optimization techniques due to its following advantages.The advantages are listed as follows: Non-Dominated sorting techniques are utilized to enhance the solution search towards pareto-optimal solutions. The use of crowding distance improvises the diversity of search. The usage of Elitist technique preserves the optimal solution in every iteration [20].

The experimental setup for evaluating the proposed algorithm’s performance on the k-coverage m-connected problem, NSGA-II, is implemented on the problem using MATLAB version 8.4 in the system configured with Intel Core i7 processor, 3.4 GHz processor speed, and 4 GB RAM. The designed testbed for this paper’s described problem, two different grid-based scenarios are generated, namely WSN1 with a 300-m × 300-m grid and WSN2 with a 500-m × 500-m grid. The base station for WSN1 and WSN2 were set in the position coordinates of (300, 150) and (500, 250), respectively. The available positions for sensor nodes and targets are randomly placed in the grid. The parameters are given in Tab. 1.

In this paper, the NSGA-II optimization algorithm is used to handle the multiobjective coverage connectivity problem in WSN. The described objectives were efficiently handled using NSGA-II to obtain a high number of Pareto front set solutions. The proposed NSGA-II on coverage connectivity was tested with different grid size, communication and sensing range. In terms of F-Value, the proposed method performance with superior values for most of the simulations. Also, in terms of computational time, the proposed algorithm outperforms when compared with the existing algorithms. For justifying the proposed algorithm, five existing techniques were simulated with the same grid, and the values were tabulated in terms of F-Value and Computational time. For comparing the results in terms of multiobjective performance indicators, the final iteration individuals are taken to count with respect to its Euclidian distance.

The future directions of the proposed model can be enhanced by solving more number of objectives. On the other hand, improvising heterogeneous algorithms for solving m-connected k-coverge problem is also an another scope of the research.