# On Mathematical Modelling of Automated Coverage Optimization in Wireless 5G and beyond Deployments
## 2 Models Developed for Network Coverage and Capacity Problems
### Wireless Interference Considerations  
In wireless networks, it is essential to consider the signal interferences from service centres (gNodeB nodes) in the deployment. Assume that coverage cannot be provided from a single node. In our case, it is necessary to extend the model presented in Section 2.1 with additional objective and constraints to minimize the potential wireless interferences. Further in this section, we define that goal in the mathematical notation and discuss the benefits of using this extension. If \(d_{ij},\;i\in I,\;j\in I\) is the distance between centres \(i\) and \(j\), then it is possible to solve this by: (i) an additional condition that, for all pairs of selected centres, will have a distance greater or equal than a certain threshold; or (ii) extending the objective function where the sum of mutual distances of selected centres will be maximised. The constraint for the first case can be expressed by the following equation. \[(\forall i\in I)(\forall j\in I)(i\
eq j):\;d_{ij}\geq d_{\min}x_{i}x_{j} \tag{15}\]  
The product \(x_{i}x_{j}\) in Equation (16) provides that this relation will be checked only for the selected pair of centres. However, since the previous equation is nonlinear, we replace the product of binary variables with the following linear equation. \[(\forall i\in I)(\forall j\in I)(i\
eq j):\;d_{ij}\geq(x_{i}+x_{j}-1)d_{\min} \tag{16}\]  
Consider the second case. Then the problem changes to a multicriteria one. Both criteria can be aggregated by a scalarization procedure. Since the first criterion (number of selected centres) is minimizing and the second criterion (sum of mutual distances) is maximizing, the second criterion will be considered with a negative sign. In addition, it is necessary to unify both criteria into the same range of values from the interval \([0,\;1]\). Equation (8) for the objective function changes as follows:  
\[z=\left(\sum_{i\in I}w_{i}x_{i}\right)\;/\;\sum_{i\in I}w_{i}-\left(\sum_{i \in I}\sum_{j\in I}d_{ij}\min(x_{i},x_{j})\right)\;/t\;\left(\sum_{i\in I} \sum_{j\in I}d_{ij}\right) \tag{17}\]  
Instead of \(\min(x_{i},x_{j})\) in the objective function, \(h_{ij}\) and the following additional constraints may be used  
\[(\forall i\in I)(\forall j\in I):\;h_{ij}\leq x_{i} \tag{18}\] \[(\forall i\in I)(\forall j\in I):\;h_{ij}\leq x_{j}\] (19) \[(\forall i\in I)(\forall j\in I):\;h_{ij}\geq(x_{i}+x_{j}-1)\] (20) \[(\forall i\in I)(\forall j\in I):\;h_{ij}\in\{0,1\}. \tag{21}\]Of course, even here, the degree of importance of both sub-criteria can be expressed by weights, \(v_{1}\), \(v_{2}\), which leads to Equation (22). \[z=v_{1}\left(\left(\sum_{i\in I}w_{i}x_{i}\right)/\sum_{i\in I}w_{i}\right)-v_{2 }\left(\left(\sum_{i\in I}\sum_{j\in I}d_{ij}\min(x_{i},x_{j})\right)/\left( \sum_{i\in I}\sum_{j\in I}d_{ij}\right)\right) \tag{22}\]  
By including both approaches, the final model is as follows:  
Minimize  
\[z=v_{1}\left(\left(\sum_{i\in I}w_{i}x_{i}\right)/\sum_{i\in I}w_{i}\right)-v_ {2}\left(\left(\sum_{i\in I}\sum_{j\in I}d_{ij}\min(x_{i},x_{j})\right)/\left( \sum_{i\in I}\sum_{j\in I}d_{ij}\right)\right) \tag{23}\]  
subject to:  
\[\forall j\in J:\ \sum_{i\in I}a_{ij}x_{i}\geq 1 \tag{24}\]  
\[(\forall i\in I)(\forall j\in I)(i\
eq j):\;d_{ij}\geq(x_{i}+x_{j}-1)d_{\min} \tag{16}\]0  
\[(\forall i\in I)(\forall j\in I)(i\
eq j):\;d_{ij}\geq(x_{i}+x_{j}-1)d_{\min} \tag{16}\]1  
\[(\forall i\in I)(\forall j\in I)(i\
eq j):\;d_{ij}\geq(x_{i}+x_{j}-1)d_{\min} \tag{16}\]2  
\[(\forall i\in I)(\forall j\in I)(i\
eq j):\;d_{ij}\geq(x_{i}+x_{j}-1)d_{\min} \tag{16}\]3  
\[(\forall i\in I)(\forall j\in I)(i\
eq j):\;d_{ij}\geq(x_{i}+x_{j}-1)d_{\min} \tag{16}\]4  
\[(\forall i\in I)(\forall j\in I)(i\
eq j):\;d_{ij}\geq(x_{i}+x_{j}-1)d_{\min} \tag{16}\]5  
Using this model extension, we reduce the potential wireless interference by deploying service centers (gNodeB nodes) with the greatest possible distance between them.