# An Effective Scheduling Algorithm for Coverage Control in Underwater Acoustic Sensor Network
## 3 Energy-Efficient Coverage Control Using Memetic Algorithm
### Node Sleep-Scheduling Mechanism Based on Memetic Algorithm
#### 3.2.2 Adaptive Function  
An adaptive function is used to evaluate the degree of chromosomes. The goal of ESACC is to use the fewest nodes to achieve the best coverage. Therefore, in this study, the superiority of node scheduling is used as the adaptive function of ESACC. Using the perceptual coverage model described in Section 2.1, suppose the coverage vector of node \(s_{i}\) is \(cov_{i}=\left[\gamma_{i,1},\gamma_{i,2},\cdots,\gamma_{i,G}\right]\), and the coverage vector of the other node \(s_{j}\) is \(cov_{j}=\left[\gamma_{j,1},\gamma_{j,2},\cdots,\gamma_{j,G}\right]\), where \(i\
eq j\). To determine whether an event is covered by a node, a Boolean operation is applied to the coverage vector by calculating the composite overlay vector of \(cov_{i}\) and \(cov_{j}\):  
\[\Psi\left(s_{i},s_{j}\right)=\left[\gamma_{i,1}\vee\gamma_{j,1},\gamma_{i,2} \vee\gamma_{j,2},\cdots,\gamma_{i,G}\vee\gamma_{j,G}\right] \tag{9}\]  
where \(\Psi\) represents the composite overlay vector, which is used to determine whether each event is covered by node \(s_{i}\) and \(s_{j}\). Therefore, the composite overlay vector of chromosome \(k\) can be expressed as  
\[\Psi\left(k\right)=\left(A_{k1}\cdot cov_{1}\right)\vee\left(A_{k2}\cdot cov_{ 2}\right)\vee\cdots\vee\left(A_{kN}\cdot cov_{N}\right) \tag{10}\]  
Furthermore, the coverage of chromosome \(k\) can be expressed as  
\[\lambda^{\left(k\right)}=\frac{\left\|\Psi\left(k\right)\right\|^{2}}{M} \tag{11}\]  
where \(\left\|\Psi\left(k\right)\right\|^{2}\) represents the number of events covered by chromosome \(k\). At this point, the utilization of the nodes in the network can be expressed as  
\[\eta^{\left(k\right)}=\frac{\sum\limits_{i=1}^{N}A_{k,i}}{N} \tag{12}\]  
where \(\sum\limits_{i=1}^{N}A_{k,i}\) indicates the number of nodes in the active state at the current time. The entire coverage-assessment process can be simplified to binary operations, greatly improving the performance of ESACC. From the aspect of evaluating the coverage of node-deployment algorithms, UASNs require the coverage of events to be as high as possible, and utilization of the nodes to be as low as possible. Therefore, we define an adaptive function \(F^{k}\) to evaluate the advantages and disadvantages of chromosome \(k\), and this is expressed as  
Figure 2: Coding rule design.  
\[F^{k}=\xi_{1}\cdot\left(\lambda^{(k)}\right)^{\alpha}-\xi_{2}\cdot\left(\eta^{(k)} \right)^{\beta} \tag{13}\]  
In the equation, \(0\leq\lambda^{(k)}\leq 1\) and \(0\leq\eta^{(k)}\leq 1\). \(\xi_{1}\) and \(\xi_{2}\) are the weight coefficients, whose values depend on the network designer's comprehensive requirements for network performance indicators. \(\alpha\) and \(\beta\) are exponential factors whose role is to distinguish the size of each chromosome \(F^{k}\). The adaptive function \(F^{k}\) is defined by Equation (13). It is a key factor affecting the performance of ESACC, and it includes four parameters: \(\xi_{1}\), \(\xi_{2}\), \(\alpha\) and \(\beta\). Coverage rate \(\lambda^{(k)}\) and utilization \(\eta^{(k)}\) are equally important indicators for evaluating UASNs, so we set \(\xi_{1}=\xi_{2}=1\). \(\alpha=3\) and \(\beta=0.7\) is based on our practical experience and repeated experiments. Figure 3 shows the trend of the adaptive function \(F^{k}\) when \(\alpha=3\) and \(\beta=0.7\). The higher is the coverage rate \(\lambda^{(k)}\), the larger is the value of \(F^{k}\), and the lower is the utilization rate of nodes \(\eta^{(k)}\), the larger is the value of \(F^{k}\), which accord with the design intention.