# Cooperative Opportunistic Pressure Based Routing for Underwater Wireless Sensor Networks
## 3 Our Proposed Work
### Minimization of Energy Consumption  
In order to design an energy efficient routing protocol, linear programming is one of the most commonly used mathematical techniques. It begins with a problem formulation of an objective function that has some linear constraints. In this subsection, we will discuss how linear programming based model reduces the energy consumption resulting in maximum network lifetime. In our proposed schemes, sensor nodes consume their energy in data communication between the source and the destination while receiving and transmitting a data packet. Therefore, based on this assumption, we have formulated the objective function that is given in Equation (25). \[Min\sum_{r=1}^{r=max}E_{consumed}(r)\;\;\forall\;r\in R \tag{25}\]  
where \(E_{consumed}\) is the energy consumed by the sensor node during the data communication that is computed as given in Equation (26). \[E_{consumed}=\sum_{s=1}^{d=u}D(i)\times(E_{tx}+E_{rx}) \tag{26}\]  
\(E_{tx}\) is a maximum transmission energy over distance D, \(E_{rx}\) shows a receiving energy over D. These are calculated as given in Equations (27) and (28). \[E_{max-tx}=P_{tx}(DPS/DR) \tag{27}\]  
where \(DPS\) denotes data packet size, \(DR\) is a data rate, \(P_{tx}\) represents a transmission power and \(P_{rx}\) is a receiving power. \[E_{max-rx}=P_{rx}(DPS/DR) \tag{28}\]  
such as,  
\[E_{tx,rx}\leq E_{i}^{max}\;\;\forall\;i\in N \tag{29}\]  
\[E_{tx,rx}\geq E_{i}^{min}\;\;\forall\;i\in N \tag{30}\]  
\[SNR\geq SNR_{th}\;\;\forall\;SNR\in N \tag{31}\]  
\[D_{s}^{d}\leq D_{s}^{dmax}\;\;\forall\;s,d\in N \tag{32}\]  
\[D_{s}^{d}\geq D_{s}^{dmin}\;\;\forall\;s,d\in N \tag{33}\]  
According to Equation (25), our objective is to minimize the energy consumption in each round. Such that the summation of all rounds resulting in high network lifetime. Equation (26) gives the explanation about the type of energy consumed in each round and Equations (27) and (28) explain how the transmission and the receiving energy in calculated at each hop between the source and the destination. Figure 9: Data packet transmission with cooperation. Constraints defined in Equations (29) and (30) ensure that energy must be within these constraints to receive and transmit a data packet in any round successfully. Equation (31) constraints ensure that each packet received at the destination must satisfy the predefined SNR threshold otherwise it will perform the cooperation to enhance the SNR to increase the reliability of the data packet. Similarly, Equations (32) and (33) are used to restrict the transmission of data communication over too long or too small distances in order to keep data packets within the transmission range of a node and reduce the number of hops for conserving nodes energy for a long time. Suppose the total available energy is \((EN_{total})\), that is computed as given below:  
\[E_{consumed}=\sum_{s=1}^{d=u}D(i)\times(E_{tx}+E_{rx}) \tag{26}\]0  
\(EN_{initial}\) denotes the initial energy of a single nod at the time of deployment and \(N\) shows the number of nodes in the network. The total \(E_{consumed}\) after round (r) is calculated as shown in Equation (35). \[E_{consumed}=\sum_{s=1}^{d=u}D(i)\times(E_{tx}+E_{rx}) \tag{26}\]1  
Let D be divided in to K number of levels then the minimum energy consumption over K be computed as given in Equations (36) and (37). \[E_{consumed}=\sum_{s=1}^{d=u}D(i)\times(E_{tx}+E_{rx}) \tag{26}\]2 \[E_{consumed}=\sum_{s=1}^{d=u}D(i)\times(E_{tx}+E_{rx}) \tag{26}\]3  
Here, it can be noted that the energy utilization during data transmission and data collection over D provides the encouraging results in response to the objective function given in Equation (25). **Graphical analysis:** Let we have a scenario in which transmission range (250 m) be divided in to \(K\) levels i.e., \(K=1,2,3,4\), and 5. Where \(N=100\), \(DPS=200B\), \(DR=10\) Kbps, \((P_{tx}\), and \(P_{rx})\) is 2 \(watt\) (W) and 0.1 W, respectively.