## 4 Problem Formulations and Optimization Models
For this purpose the connectivity between nodes \(s_{i\to 1}\) and \(s_{(i\to 1)k},\ \forall k\in K\) on the one hand, and the connectivity between nodes \(s_{i\to 2}\) and \(s_{i+1}\) on the other hand should be maintained. The optimal location for the node \(s_{i\to 2}\) is obtained solving the **P1** optimization problem already introduced in Case 1. We should now obtain the optimal placement for the node \(s_{i\to 1}\). We define:  
* \(u_{i\to 1}=(x_{i},\ y_{i},\ z_{i\to 1})\) is the new location of node \(s_{i\to 1}\). * \(u_{(i\to 1)k}=(x_{(i\to 1)k},\ y_{(i\to 1)k},\ z_{(i\to 1)k}),\ \forall k\in K\) is the location of node \(s_{(i\to 1)k}\). * \(x_{i},x_{(i\to 1)},y_{i},y_{(i\to 1)k},\ z_{(i\to 1)k},H\,,R_{M\omega X},TL_{T_{R}},\)
* \(\alpha,\chi,A,z_{z\to start},z_{z\to end},E_{Bw},E_{T_{R}}\)  
_Case 3: The uplink \(l_{i\,i+1}\) between \(s_{i}\) and \(s_{i+1}\) and the uplinks \(l_{(i-1)k\,i}\) between \(s_{(i-1)k}\), \(\forall k\in K\) and \(s_{i}\)\(l_{(i-2)k\,i-1}\) between \(s_{(i-2)k}\), \(\forall k\in K\) and a particular \(s_{(i-1)k}\),..., \(l_{(i-x)k\,i-x+1}\) between \(s_{(i-x)k}\), \(\forall k\in K\), \(\forall x\in N\) and a particular \(s_{(i-x+1)k}\) are located in the shadow zone._  
In this case the sensor node located at the lowest depth in the shadow zone \(s_{i}\) is uncoupled into two sensor nodes \(s_{i\to 1}\) and \(s_{i\to 2}\) as shown in \[Min:\quad TL_{i\to 2~{}i+1}=\chi\log r_{i\to 2~{}i+1}+\alpha r_{i\to 2~{}i+1}.10^{-3}+A\]3  
The optimal placement for node \(s_{i\to 2}\) is obtained solving the **P1** optimization problem already introduced in Case 1. The node \(s_{i\to 1}\) should move to (\(x_{i}\), \(y_{i}\), \(z_{zz\gets end}+\varepsilon\)), where \(\varepsilon\in R^{+}\) is a very small value, that is, \(s_{i\to 1}\) moves vertically downwards outside the shadow zone. In this case, the connectivity with the nodes \(s_{(i-1)k}\)\(\forall k\in K\) through the uplinks can only be maintained if they also move outside the shadow zone. Therefore, the sensor nodes \(s_{(i-1)k}\), \(\forall k\in K\) are uncoupled into two sensor nodes \(s_{(i-1)k\to 1}\) and \(s_{(i-1)k\to 2}\). The node \(s_{(i-1)k\to 2}\) remains in the same location as node \(s_{(i-1)k}\), \(\forall k\in K\) was and continues sensing data at the required depth level. We should obtain the optimal placement for the node \(s_{(i-1)k\to 1}\). The data sensed by \(s_{(i-1)k\to 2}\) is sent through a wire between \(s_{(i-1)k\to 2}\) and \(s_{(i-1)k\to 1}\) to maintain communication inside the shadow zone. Generally speaking, the same process should be repeated for all the nodes down in the hierarchy \(s_{(i-x)k}\), \(\forall k\in K\), \(\forall x\in N\) located in the shadow zone. They are uncoupled into two sensor nodes \(s_{(i-x)k\to 1}\) and \(s_{(i-x)k\to 2}\). The node \(s_{(i-x)k\to 2}\) remains in the same location as node \(s_{(i-x)k}\)\(\forall k\in K\), \(\forall x\in N\) was and continues sensing data at the required depth level. We should obtain the optimal placement for the node \(s_{(i-x)k\to 1}\). The data sensed by \(s_{(i-x)k\to 2}\) is sent through a wire between \(s_{(i-x)k\to 2}\) and \(s_{(i-x)k\to 1}\) to maintain communication inside the shadow zone. If the node \(s_{(i-x+1)k\to 1}\) has already moved to a new location outside of the shadow zone and node \(s_{(i-x)k\to 1}\)does not know its depth \(z_{(i-x+1)k\to 1}\), node \(s_{(i-x)k\to 1}\) can move to (\(x_{(i-x)k}\), \(y_{(i-x)k}\), \(z_{zz\gets end}+\varepsilon\)) and once  
\[Min:\quad TL_{i\to 2~{}i+1}=\chi\log r_{i\to 2~{}i+1}+\alpha r_{i\to 2~{}i+1}.10^{-3}+A\]4 Scenario in the 3D UWSN with a shadow zone (Case 3). there establish connection with node \(s_{(i\cdot x+1)k\to 1}\) to find out the depth and afterwards compute its optimal location using this information.