Hajjej et al. [16] proposed a multi-objective flower pollination algorithm (MOFPA) for stochastic WSN scheduling. It aims to deploy a set of sensors in a target area under the condition of guaranteed network connectivity while optimizing the total network coverage and energy consumption. Wang et al. [17] employ a node scheduling scheme based on a multi-objective evolutionary algorithm to schedule heterogeneous nodes in shifts, thus extending the life cycle of the network. Non-dominated Sorting Genetic Algorithm-iii (NSGA-III) is used to optimize the deployment of WSN to reduce the network energy consumption and improve network coverage while ensuring network connectivity [18]. The deployment of WSNs in environments where obstacles are present is investigated, using a multi-objective optimization algorithm to deploy sensor nodes in a way that ensures network connectivity to maximize coverage and reduce deployment costs [19]. Saad et al. [20] addressed the deployment of WSN in 3D environments using NSGA-II. It is worth mentioning that they simulated a real space 3D model using the Bresenham line-of-sight 3D environment coverage model. Zaimen et al. [21] considered the problem of optimizing the deployment of sensor nodes in heterogeneous obstacle indoor environments, using the BIM database as well as other additional inputs (sensor node parameters) and a genetic algorithm to provide the optimal deployment. A new method based on Support Vector Regression and Genetic Algorithm is proposed for the deployment optimization problem of WSN nodes, which argues that the configuration and deployment of WSN affects almost all of their performance metrics, and the support vector regression model is utilized for the configuration of the node parameters of the Wireless Sensor Networks, and the configured node model is optimized for the deployment [22]. An enhanced multi-objective marine predator algorithm (CMOMPA) is proposed, which utilizes a Gaussian elite perturbation strategy and a learning strategy based on a competitive mechanism to generate progeny with enhanced diversity and distribution, and balances the deployment cost, connectivity, and coverage of heterogeneous WSN under specific coverage constraints [23]. The Voronoi diagram method is used to divide the monitoring area to obtain the appropriate sensing radius and communication radius, and the algorithm is used to solve the problem of deploying the wireless sensor network on 3D terrain and to improve the Quality of Service (QoS) of the wireless sensor network [24]. A resource scheduling algorithm for large-scale wireless sensor networks based on differential co-evolution and multi-objective decomposition is proposed, which is used to simultaneously optimize the position and dormant state of nodes [25]. In [26], three heuristic algorithms are used to optimize the deployment of sensor nodes, thereby maximizing the network lifetime and network coverage is divided into. The deployment of sensor nodes is optimized using an improved vampire bat optimizer, which obtains the optimal deployment location and the minimum number of sensors required by optimally splicing the sensing region with the cellular grid [27].

Looking at the current state of research at domestic and abroad, it can be found that most of the models currently studied in the literature for the deployment problem of wireless sensor networks are too simple, and there is a huge difference between them and practical applications. There are obvious differences between the currently studied sensor sensing models which mainly utilize the 0–1 sensing model, the probabilistic sensing model, the advanced sensing models, and information processing schemes adopted by the existing sensor networks; and there are few literature studies on the temporal coverage problem and the quality of network monitoring of the network. This study proposed a stochastic data fusion wireless sensor model as the wireless sensor coverage deployment model for the following reasons:

(1) The model adopts a random sensing model. Most of the traditional wireless sensor networks use a 0–1 sensing model or probabilistic sensing model, which does not consider the randomness of sensing, while sensors using a stochastic sensing model sense the signal energy of the target point will be affected by the external environment, which makes the sensor’s sensing with randomness and more realistic.

(2) The current study of wireless sensor networks for monitoring using an independent transmission model which is very different from the advanced data transmission model used in practice, this project uses a data fusion model that is more in line with the practical application, a certain range of sensors to fuse their monitoring results to obtain the final monitoring results.

In 2022, Sadoun et al. used DMOA to improve the Long Short-Term Memory (LSTM) model to predict the properties of composite materials [28]. The model improved by DMOA has a prediction accuracy of 99%. In 2022, Akinola et al. introduced a simulated annealing algorithm to a binary variant of DMOA [29] which balanced the detection and exploitation capabilities of the algorithm. In 2022, Agushaka et al. improved DMOA by incorporating other social behaviors of dwarf mongooses, i.e., predation, mound protection, reproduction, and group splitting behaviors [30], greater enhancement of the dwarf mongoose’s exploration and exploitation capabilities, and improved convergence speed of the algorithm. In 2022, Mehmood et al. proposed a meta-heuristic algorithm for parameter estimation of autoregressive exogenous models based on DMOA [31]. The algorithm has fast convergence speed, high estimation accuracy, and strong robustness. In 2023, Agushaka et al. introduced adaptive and stochastic factors to the DMOA, which improves the exploration capability and availability of the algorithm and improves the diversity of the solution [32]. In 2022, Akinola et al. solved the high-dimensional feature selection problem using a binary version of DMOA [33]. In 2022, Alissa et al. proposed a dwarf mongoose algorithm based on machine learning-driven ransomware detection by combining DMOA with machine learning-driven ransomware detection [34], which is effective in identifying and classifying malware or ransomware. In 2022, Alrayes et al. used improved DMOA to select cluster heads of unmanned aerial vehicles to find the optimal routes to reach their destinations [35]. In 2022, Balasubramaniam et al. embedded DMOA into a lightweight deep neural network architecture for heart disease prediction [36]. In 2023, Zare et al. used DMOA to integrate the battery life, operations and maintenance cost, fuel cost and environmental cost of microgrids to determine the optimal operating parameters and improve the load capacity of microgrids [37]. In 2023, Dora et al. merged Symbiotic Organism Search (SOS) into DMOA to enhance the algorithm’s local search capability [38], solved the reactive power scheduling problem using the proposed enhanced DMOA and found the optimal settings to minimize the real power loss, total voltage variation and L-index. In 2023, Fu et al. proposed an improved DMOA which introduced the optimal leader mechanism and proposed a novel nonlinear control strategy based on sinusoidal function, which ensured the accuracy of the algorithm and at the same time improved the convergence speed of the algorithm [39]. In 2022, Abirami et al. combined DMOA with Gaussian Convolution Deep Confidence Networks for effective classification and extraction of retinal images to examine diabetic retinopathy [40]. In 2023, Almutairi et al. introduced quantum techniques to DMOA, which accelerated the convergence of the algorithm at a later stage [41]. In 2023, Rizk-Allah et al. used a modified DMOA to identify unknown parameters of an computational and physical elements model (single-phase transformer) and to evaluate the aging trend of the transformer at the hottest temperature [42].

Throughout the research status at domestic and abroad, the research on DMOA is mainly carried out in the following two aspects: one is to improve the performance of the basic algorithm for the deficiencies that exist, and a variety of different types of improved versions of DMOA have been proposed, the other is to broaden the scope of application of DMOA. With the deepening of the research, it can be found that although there are more literature on the improvement of DMOA performance, through a large number of numerical examples of experiments and comparative analysis found that DMOA still exists in the exploration and exploitation capacity is difficult to achieve the balance; easy to fall into the local optimum and late convergence of the slow speed and other shortcomings, and single-objective algorithms of the application of the scope of a certain degree of restriction. In this study, the shortcomings of the DMOA are as follows:

(1) EDMOA is proposed, which introduces nonlinear convergence factors and Markov Chain ideas in the original DMOA to accelerate the convergence speed of the algorithm, solve the problem of imbalance between the detection and exploitation capabilities, and improve the ability of the algorithm to jump out of the local optimal solution. In order to solve more practical application problems, EDMOA is designed to EMODMOA. The performance and application range of DMOA are comprehensively improved.

(2) Evaluate the performance of the proposed algorithm. Theoretically analyze the time complexity of the algorithm and test the performance of the algorithm using the CEC2020 multi-objective multi-modal optimization benchmark function.

(3) The proposed algorithm is used for the optimal deployment of SDFWSN. The results obtained in this study are of great theoretical significance and application prospects for advancing the development of the discipline of modern intelligent optimization technology.

As mentioned above, under the constraints of complete coverage of the monitoring area and network connectivity, this article investigates the SDFWSN deployment problem by jointly optimizing the five objectives of the network lifecycle, spatiotemporal coverage, false alarm rate, and detection rate. In summary, the SDFWSN deployment problem can be formulated as follows (Eq. (14)): (14){max(LT),min(PFj),max(PDj),max(c),max(τ)}s.t.C(GSDFWSN)=1where C(GSDFWSN)=1 indicates that the network satisfies the connectivity condition.

DMOA is a population intelligence optimization algorithm proposed in 2022 by Agushaka et al. DMOA is inspired by the semi-nomadic and compensatory adaptive behavior of dwarf mongoose populations. The algorithm has a strong global search capability. The algorithm is divided into four main phases, searching for food, searching for mounds (exploitation phase), babysitter exchange, and choosing mounds (exploration phase) [13].

(1) Food search phase:

The algorithm divides the mongoose into an alpha group (scouting group) and a babysitter group based on their compensatory adaptive behavior, and the alpha group searches for the location of the optimal food during the food search phase. The alpha group leader is selected according to Eq. (15) and the individual position update formula is given by Eq. (16). (15)α=fiti∑i=1nfiti (16)Xi+1=Xi+phi∗peepwhere phi is a uniformly distributed random number [−1, 1] and peep is the sound made by the alpha female.

(2) Mounds search phase:

The scouting group is responsible for finding the sleep mounds, and evaluation formula for individual sleep mounds as Eq. (17). Evaluation formulas for population sleep mounds are given by Eq. (18). (17)smi=(fiti+1−fiti)max{fiti+1,fiti} (18)ϕ=∑i=1nsminwhere fiti+1, fiti denote the fitness values of the mongoose’s new position and the original position respectively.

(3) Babysitter exchange phase:

Once the babysitter swap conditions are met, the babysitter group and alpha (scout) group will swap identities. Initialize the mongoose’s new location as Eq. (19). (19)xi,j=unifrnd(VarMin,VarMax,VarSize)

(4) Mounds choose phase:

During this phase, an individual chooses an optimal sleep mound to perch on, and the individual position equation is as follows Eq. (20). Where M is given by Eq. (21), CF is the adaptive operator and the formula is as Eq. (22). (20)Xi+1={Xi−CF∗phi∗rand[Xi−M],ifϕi+1>ϕiXi+CF∗phi∗rand[Xi−M],otherwise (21)M=∑i=1nXi∗smiXi (22)CF=(1−iterMaXiter)(2iterMaxiter)

The DMOA pseudo-code is shown in Algorithm 1.

This subsection describes the proposed EDMOA algorithm, and the new algorithm focuses on the improvement of the original algorithm for the shortcomings that it is easy to fall into local optimal solutions, slow convergence, and imbalance between exploration and exploitation.

The design of EMODMOA refers to the algorithm framework of NSGAIII, but in the selection of reference points, the classification algorithm KNN is used to classify individuals close to the reference points, select individuals in small categories, ensure the diversity of the population, and reduce the complexity of the algorithm. The pseudo-code of EMODMOA is shown in Algorithm 3, and the algorithm flowchart is as follows:

Algorithm 3 is the pseudo-code of EMODMOA, in EMODMOA Pt is the parent of tth generation of size N and its generated child is Qt which is also of size N. Combining the child and the parent generates Rt of size 2N and selects N individuals from it. To implement this selection process, Rt is first divided into multiple non-dominated layers (F1,F2,⋯) by non-dominated ordering. The members of the populations in the non-dominated stratum rank 1 to l are put into St in order, and if |St|=N, the following operation is not necessary, and Pt+1=St directly. If |St|>N, then a part of the next generation is solved as Pt+1=∪j=1l−1Fj, and the remaining part (K=N−|Pt+1|) is chosen from Fl. The selection principle is to classify according to the KNN algorithm, selecting individuals that are far from the reference point, thereby maintaining the diversity of the solution.

The SDFWSN is deployed on a 4000 m × 4000 m static node grid. The false alarm rate of each node is lower than 0.05 detection rate is higher than 0.95. The optimization process for the deployment of a stochastic fused wireless sensor network consisting of 220 nodes using EMODMOA is as follows:

Step 1. Initialize the parent population Pt and the reference point Nc. Pt is denoted as follows:(30)Pt=[X1,X2,,XN]Twhere Xi=[x,y,Rc,n] is the ith individual in the population Pt representing a feasible deployment of a wireless sensor network. n is the number of sensors in the network, x and y are n dimensional column vectors, and Rc is the connectivity radius of the network.

Step 2. Generate child population Qt by updating iteration of parent population Pt using EDMOA and merge the parent and child populations into Rt.

Step 3. Generate a Pareto frontier by the non-dominated ordering of Rt.

Step 4. Starting from the first Pareto frontier select individuals to population St until the number of St is greater than or equal to the initial population size.

Step 5. Use KNN to select individuals in the selected last Pareto frontier to join the population St until the number of St is equal to the initial population size. The population St is the deployment scheme of SDFWSN.

The EMODMOA flowchart is shown in Fig. 2.

The HV index evaluates the convergence and diversity of the solution set by comparing the hypercube between the solution set and the reference point. The HV index is positive. In multi-objective optimization, the larger the hypercube occupied by the solution set, the better the performance of the solution set, indicating that the algorithm has found a better solution in the objective space. The Delta index is a commonly used evaluation index that can help determine the diversity and homogeneity of the solution set generated by the optimization algorithm. It measures the degree of dispersion of the solutions within the solution set and provides information about the structure of the solution set. The closer the Delta metric is to 1, the better the diversity and homogeneity of the solution set; the closer it is to 0, the worse the diversity and homogeneity of the solution set. NDS is the number of optimal solution sets obtained by the algorithm, which intuitively reflects the algorithm’s ability to find the best solution. In this paper, the HV, Delta, and NDS values of NSGAII, MOPSO, MOEA_D, and MOGWO are compared with those of the EMODMOA algorithm. The results are shown in Figs. 3–5. The results show that the super-volume values obtained by the EMODMOA algorithm are higher than those of the other algorithms under different parameter settings, which indicates that the EMODMOA algorithm has strong convergence. Moreover, the Delta and NDS values of EMODMOA are higher than those of the other algorithms, which indicates that EMODMOA can find a Pareto frontier with more diversity and richness, and thus provide decision-makers with more choices.

Fig. 6 shows the network life cycle as the number of iterations increases. It can be seen from the figure that the network life cycle is extended with optimization. Comparing the EMODMOA algorithm with algorithms such as NSGAII, MOPSO, MOEA/D, and MOGWO, our proposed algorithm obtains the maximum network life cycle after 500 iterations. As can be seen from the direction of the broken line in the figure, EMODMOA can steadily improve the network life cycle, and it has the highest stability among all algorithms.

The analysis of the spatial coverage of the network is given in Fig. 7, which improves with the number of iterations. A comparison of the EMODMOA algorithm with NSGAII, MOPSO, MOEA/D, and MOGWO reveals that our proposed algorithm gives the optimal spatial coverage of the network at the later stage of iteration. This is due to the scientific improvement of our algorithm and shows that our proposed algorithm is highly competitive among many multi-objective optimization algorithms.

Fig. 8 gives an analysis of the network time coverage, which is optimized with the increase of the number of iterations. Comparing EMODMOA with NSGAII, MOPSO, MOEA/D, and MOGWO, it is found that the final iteration result of our proposed algorithm has the best time coverage. Compared with the broken lines obtained by other algorithms, the optimized broken lines obtained by our algorithm tend to grow steadily, which indicates that our proposed algorithm has strong stability.

Figs. 9 and 10 show the detection rate and false alarm rate of the network, respectively. The detection rate indicates the probability that the network correctly detects the target, while the false alarm rate indicates the probability that the network transmits information about the existence of a target when there is no target. Therefore, as the number of iterations increases, the algorithm should improve the detection rate of the network while reducing the false alarm rate. Observing Fig. 9, it can be found that all algorithms have optimized the detection rate, and the MOGWO algorithm has achieved the optimal detection rate. Although the algorithm proposed in this paper has not achieved the optimal detection rate, the detection rate obtained is not much different from the optimal detection rate. In addition, compared with Fig. 10, it can be found that although the MOGWO algorithm achieves the optimal detection rate, the optimization effect on the false alarm rate is not ideal. In contrast, EMODMOA, MOPSO, and MOEA/D improve the detection rate while reducing the false alarm rate. EMODMOA achieves a good detection rate and an optimal false alarm rate, thereby improving network performance.

Fig. 11 shows the deployment of the sensors optimized by the EMODMOA algorithm, and after the optimized deployment, the sensor nodes can cover almost all the target areas. Our algorithm can optimize the deployment of wireless sensors very well. Improve the spatial coverage of the network.

Fig. 12 shows the data fusion graph of SDFWSN, the red circle indicates the sensors that perform data fusion, and the optimized nodes are evenly distributed, and a certain number of sensors are distributed in each data fusion range, which ensures the quality of network transmission. Since the sensors use stochastic sensing, the quality of sensing decreases as the distance between the target node and the sensor increases, the shades of blue in the graph represent the quality of the target area covered by the sensors, i.e., the PD value of the target area, which represents the accuracy degree of the network in detecting the target, and it can be seen that most of the target areas are covered by a higher quality to ensure the accuracy of the network monitoring. This provides a good guarantee for network QoS.

Figs. 13–16 represent the data fusion maps after optimization with the MOEA/D, MOGWO, MOPSO, and NSGAII algorithms, comparing Figs. 13–16 with Fig. 12, the superiority of network nodes deployed using EMODMOA can be found. First, the nodes of the network optimized using other algorithms are unevenly distributed, and all of them have different degrees of coverage gaps, which will affect the quality of the network. Second, the uneven distribution of the number of sensors within the fusion range leads to poor detection accuracy, i.e., the network cannot provide accurate detection results, which directly affects the QoS of the network. In summary, it can be found that the deployment of the network optimized by EMODMOA is much better than the deployment of network nodes optimized using the other four algorithms, which can be seen in the algorithms of the superiority of the deployment of the network nodes.

In the simulation experiment, 220 wireless sensors are required to cover an area of 16 × 10⁶ m². After the experimental deployment, it is calculated that the coverage areas of EMODMOA, MOPSO, MOGWO, MOEA/D, and NSGAII are 15.1 × 10⁶ m², 13.1 × 10⁶ m², 12.2 × 10⁶ m², MOEA/D covers an area of 12.9 × 10⁶ m², and NSGAII covers an area of 12.4 × 10⁶ m². The calculation time for each algorithm is shown in Fig. 17 below. Within the allowable time frame, EMODMOA achieved the optimal coverage.

To enhance the credibility of the algorithm, four different use cases are designed for verification. The deployment of many nodes and a small number of nodes in a small area, i.e., 1000 m × 1000 m, is discussed, as is the deployment of many nodes and a small number of nodes in a large area, i.e., 5000 × 5000.

Case 1: 300 nodes are deployed within a 1000 m × 1000 m range, and the network coverage is as Fig. 18.

Case 2: 100 nodes are deployed within a 1000 m × 1000 m range, and the network coverage is as Fig. 19.

Case 3: 300 nodes are deployed within an area of 5000 m × 5000 m, and the network coverage is as Fig. 20.

Case 4: 100 nodes are deployed within a 5000 m × 5000 m range, and the network coverage is as Fig. 21.

As can be seen from the coverage map of the wireless sensor network, the EMODMOA algorithm can optimize multi-node networks well and can ensure the uniform distribution of wireless sensor nodes, thus covering the target area well. For networks with a small number of wireless sensors, it is better than the limited number of sensors themselves, so full coverage of the target area cannot be guaranteed, but the sensors optimized by EMODMOA can also be distributed more evenly in the target area. As can be seen from the figure, both the large target area and the small target area can be well covered after optimization by the wireless sensor network.

Table 10 shows the false alarm rate and detection rate after EMODMOA optimization in different cases. For the same target range, the more sensors there are, the lower the false alarm rate and the higher the detection rate. This shows that EMODMOA is suitable for optimizing multi-node networks. When the number of nodes is the same, the false alarm rate is low, and the detection rate is high for a small target area. In summary, EMODMOA is suitable for optimizing multi-wireless sensor node networks, and there is no limit on the range of the target area.

To verify the performance of the algorithm in large-scale wireless sensor networks, we will add an experiment to deploy a large-scale wireless sensor network for verification.

The experimental parameters are set as shown in Table 11, with 400 wireless sensor network nodes deployed within a 5000 m × 5000 m area.