# Node Deployment of Marine Monitoring Networks: A Multiobjective Optimization Scheme
## 3 Network Model and Problem Formulation
### Problem Formulation
#### 3.3.2 Maximization of the Network Lifetime  
Besides the network deployment cost, it is also critical to ensure that the HMMN-EC provisions quality marine monitoring services as long as possible. The network lifetime is defined as the operation time of the network until the first battery-powered node exhausts the energy supply and become out \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbols** & **Definition** \\ \hline \(g_{i}\) & The amount of data of MP,(per unit time). \\ \(\text{R}_{\text{MP}}\) & The perceived radius of MPs. \\ \(\text{D}^{\text{ARN}}\) & The communication distance of ARNs. \\ \(\text{D}^{\text{ECN}}\) & The communication distance of ECNs. \\ \(\text{D}^{\text{SSN}}\) & The communication distance of SSNs. \\ \(\text{D}^{\text{URN}}\) & The communication distance of URNs. \\ \(\text{C}^{\text{ARN}}\) & The cost of ARNs. \\ \(\text{C}^{\text{ECN}}\) & The cost of ECNs. \\ \(\text{C}^{\text{SSN}}\) & The cost of SSNs. \\ \(\text{C}^{\text{URN}}\) & The cost of URNs. \\ \(\text{E}_{\text{elec}}\) & Energy consumption for sending and receiving data per bit. \\ \(\text{P}_{\text{UR}}\) & The reception power of URNs. \\ \(\text{P}_{\text{UT}}\) & The transmission power of URNs. \\ \(\text{E}I_{i}\) & The initial energy of node \(i\). \\ \(\text{E}c_{i}\) & The energy consumption of node \(i\) per unit time. \\ \(\text{K}\) & K-coverage: each MP must be covered by K URNs. \\ \(\overset{\rightarrow}{\text{E}}=\{e_{ij}\}_{|\mathbb{V}|\times|\mathbb{V}|^{ \prime}}\forall i,j\in\mathbb{V}\) & The matrix of edge variables, where \(e_{ij}\in\{0,1\}\) is a binary variable and \(e_{ij}=1\) denotes node i can directly communicate with node j; and vice versa. The location incidence vector of nodes, where \(a_{m}\in\{0,1\}\) is a binary variable and \(a_{m}=1\) denotes that the candidate location m is selected to deploy a URN; and vice versa. \\ \(\text{B}=\{b_{n}\}_{1\times|\mathbb{V}_{\text{SSN}}|},\forall n\in\mathbb{V}_{ \text{SSN}}\) & The location incidence vector of nodes, where \(b_{n}\in\{0,1\}\) is a binary variable and \(b_{n}=1\) denotes that the candidate location n is selected to deploy a SSN; and vice versa. \\ \(\text{H}=\{h_{l}\}_{1\times|\mathbb{V}_{\text{ECN}}|},\forall l\in\mathbb{V}_{ \text{ECN}}\) & The location incidence vector of nodes, where \(h_{l}\in\{0,1\}\) is a binary variable and \(h_{l}=1\) denotes that the candidate location l is selected to deploy an ECN; and vice versa. \\ \(\text{Z}=\{z_{t}\}_{1\times|\mathbb{V}_{\text{ARN}}|},\forall t\in\mathbb{V}_{ \text{ARN}}\) & The location incidence vector of nodes, where \(z_{t}\in\{0,1\}\) is a binary variable and \(z_{t}=1\) denotes that the candidate location \(t\) is selected to deploy an ARN; and vice versa. \\ \(\text{f}_{ij},\forall i,j\in\mathbb{V}\) & The data flow from node \(i\) to node \(j\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Table of Notations. of service. Given the initial battery of a battery-powered node i, \(\text{EI}_{\text{i}}\), and the energy consumption of node i, \(\text{E}_{\text{i}}\) per unit time, the lifetime of node i is then given by  
\[\text{T}_{i}=\frac{\text{EI}_{i}}{\text{E}_{i}},\forall i\in\text{V}_{\text{URN}} \cup\text{V}_{\text{SSN}}. \tag{6}\]  
Therefore, the network lifetime \(\text{T}_{net}\) is defined as  
\[\text{T}_{net}=\min_{\forall i\in\text{V}_{\text{URN}}\cup\text{V}_{\text{SSN }}}\text{T}_{\text{i}}=\min_{\forall i\in\text{V}_{\text{URN}}\cup\text{V}_{ \text{SSN}}}\frac{\text{EI}_{i}}{\text{E}_{i}}. \tag{7}\]  
Notice that \(\text{E}_{\text{i}}\) is dependent on the communication distance and the communication data volume shown as (1)-(4). Thus, the network lifetime is determined by \(\text{E}_{\text{ci}}\) of the first energy-exhausted node i. Accordingly, to maximize the network lifetime, it is equivalent to minimize the energy consumption of the first energy-exhausted node.