# 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
According to (3) and 4), the energy consumption of the first energy-exhausted node is given by  
\[\text{E}_{\text{max}}=\max_{i\in\text{V}_{\text{URN}}\cup\text{V}_{\text{SSN }}}(\sum_{j\in\text{V}}\text{E}_{\text{tr}}^{\text{i}}(\text{l}_{\text{ij}}, \text{d}_{\text{ij}})\text{e}_{\text{ij}}+\sum_{l\in\text{V}}\text{E}_{\text{ re}}^{\text{i}}(\text{l}_{\text{ii}})\text{e}_{\text{ii}}+\text{E}_{\text{sleep}}^{ \text{i}}). \tag{8}\]  
Without the loss of generality, the initial energy of node (EI) is regarded as 100% in the following formulation. Thus, the MO problem is formulated as follows:  
\[\text{P1}:\quad\text{\bf minimize}\quad\mathbb{C}_{net}+\omega \text{E}_{max}, \tag{9}\] \[\text{s.t.}\sum_{j\in\text{V}_{\text{URN}}}e_{ij}\geq K, \forall i\in\text{V}_{\text{MP}}, \tag{10}\]  
\[\sum_{i\in\text{V}_{\text{MP}}\cup\text{V}_{URN}}e_{im}\geq a_{m},\forall m \in\text{V}_{\text{URN}},m\
eq i, \tag{11}\]  
\[\sum_{j\in\text{V}_{\text{URN}}\cup\text{V}_{\text{SSN}}}e_{mj}=a_{m},\forall m \in\text{V}_{\text{URN}},m\
eq j, \tag{12}\]  
\[\sum_{j\in\text{V}_{\text{URN}}\cup\text{V}_{\text{SSN}}}e_{jn}\geq b_{n}, \forall n\in\text{V}_{\text{SSN}},n\
eq j, \tag{13}\]  
\[\sum_{q\in\text{V}_{\text{SSN}}\cup\text{V}_{\text{ECN}}}e_{nq}=b_{n},\forall n \in\text{V}_{\text{SSN}},n\
eq q, \tag{14}\]  
\[\text{T}_{i}=\frac{\text{EI}_{i}}{\text{E}_{i}},\forall i\in\text{V}_{\text{URN}} \cup\text{V}_{\text{SSN}}. \tag{6}\]0  
\[\text{T}_{i}=\frac{\text{EI}_{i}}{\text{E}_{i}},\forall i\in\text{V}_{\text{URN}} \cup\text{V}_{\text{SSN}}. \tag{6}\]1  
\[\text{T}_{i}=\frac{\text{EI}_{i}}{\text{E}_{i}},\forall i\in\text{V}_{\text{URN}} \cup\text{V}_{\text{SSN}}. \tag{6}\]2  
\[\text{T}_{i}=\frac{\text{EI}_{i}}{\text{E}_{i}},\forall i\in\text{V}_{\text{URN}} \cup\text{V}_{\text{SSN}}. \tag{6}\]3  
\[\text{T}_{i}=\frac{\text{EI}_{i}}{\text{E}_{i}},\forall i\in\text{V}_{\text{URN}} \cup\text{V}_{\text{SSN}}. \tag{6}\]4  
\[\text{T}_{i}=\frac{\text{EI}_{i}}{\text{E}_{i}},\forall i\in\text{V}_{\text{URN}} \cup\text{V}_{\text{SSN}}. \tag{6}\]5Equation (9) is the weighted sum of the two main objectives, where the network cost \(\mathbb{C}_{net}\) and the energy consumption of the first exhausted node \(E_{max}\) are defined in (5) and (8), respectively, and \(\omega\) is the weight to strike a balance of the two objectives. Constraint (10) specifies that each MP must be covered by at least K URNs. Different monitoring tasks have different requirements of K. Equation (11) indicates that if CL \(j\) is selected to deploy a URN, then \(a_{j}=1\), and there exists at least one link of receiving data, e.g., between a MP or URN node \(i\) and \(j\). Equation (12) indicates that when an URN \(j\) is deployed, there exists one link of forwarding data, e.g., between node \(j\) to another URN or SSN nodes \(l\). Similarly, SSNs, ECNs, and ARNs are subject to the constraints (13)-(18), respectively. Equations (13) and (14) specify that if a SSU \(n\) is deployed, i.e., \(b_{n}=1\), the SSU \(n\) has one link of forwarding data and at least one link of receiving data. Similarly, Equations (15) and (16) specify that if CL \(l\) is selected to deploy an ECN, i.e., \(h_{l}=1\), there exists one link of forwarding data and at least one link of receiving data. Equations (17) and (18) specify that if CL \(t\) is selected to deploy an ARN, i.e., \(z_{t}=1\), there exists one link of forwarding data and at least one link of receiving data. Equations (19) and (20) specify the flow constraint that the output data should be the same as the input data.