## 3 Multi-Objective Coverage Optimization Problem  
In this paper, we assume that sensor nodes are randomly and uniformly distributed over the target area, with the same physical structure, communication capacity, sensing range, initial limited energy and computing capacity. The sink node or the base station (sink) is assumed to have unlimited energy with plenty of power supplies. The sink or every sensor node has a unique identifier, and can get its own location information and communicate with its neighbor nodes usually using normal power or with each other through power control.  
We define the objective optimization model of the network coverage as:  
\[\begin{cases}\min\sum\limits_{i=1}^{n}\gamma_{i}\\ \max\varpi(\mathrm{T})\end{cases} \tag{8}\]  
In WSNs, due to the limited energy of sensor nodes, reducing the energy consumption of some nodes and balancing the energy consumption of the entire network can effectively save the limited energy of the network and extend the network lifetime. Under the premise of guaranteeing some certain coverage degree, we may reduce the number of sensor nodes in working and deactivate the redundant nodes to reduce the energy consumption as much as possible to increase the network lifetime.  
We define node utilization as the ratio of the number of cells having active sensor nodes to the total cell number in the target area given by:  
\[U(T)=\frac{\parallel T\parallel}{M\times N} \tag{9}\]  
where \(\parallel T\parallel\) is the number of cells having active sensor nodes and sensor nodes are deployed in the target field \(D\) which is divided into \(M\times N\) virtual cells.  
To extend the lifetime of sensor networks, we also consider the residual energy of WSNs in the target field \(D\). If \(E_{i}\) is the residual energy of the sensor node \(t_{i}\), the energy distribution of the network is expressed as:  
\[E(T)=\frac{\frac{1}{n}\sum\limits_{i=1}^{n}(E_{i}-\frac{1}{n}\sum\limits_{i=1} ^{n}E_{i})^{2}}{(\frac{1}{n}\sum\limits_{i=1}^{n}E_{i})^{2}}=\frac{n\sum \limits_{i=1}^{n}(E_{i}-\frac{1}{n}\sum\limits_{i=1}^{n}E_{i})^{2}}{(\sum \limits_{i=1}^{n}E_{i})^{2}} \tag{10}\]  
where \(E(T)\) reflects the degree of energy difference in the network; if its value is smaller, the energy difference between sensor nodes may be smaller, and the residual energy of the sensor nodes may be higher.  
To increase the network lifetime, we may not only use as few nodes as possible to reduce energy consumption, but also make sure to select the nodes with balanced energy consumption and a higher residual energy. Therefore, we define the objective optimization model of the energy consumption as:  
\[\begin{cases}\min\alpha\times U(T)+\beta\times E(T)\\ \max\varpi(\mathrm{T})\end{cases} \tag{11}\]  
where \(\alpha\) is the node utilization weighting coefficient, \(\beta\) is the energy balance weighting coefficient and \(\alpha+\beta=1\).  
According to Equations (8) and (11), in MOCOAMA, we pay attention to Pareto optimal solutions of the multi-objective coverage optimization problem of WSNs formulated as:  
\[\begin{cases}\min\sum\limits_{i=1}^{n}\gamma_{i}\\ \max\varpi(\mathrm{T})\\ \min\alpha\times U(T)+\beta\times E(T)\\ s.t.\gamma_{i}=0\text{ }or\text{ }1,\text{i}\in[1,n]\end{cases} \tag{12}\]