# On Mathematical Modelling of Automated Coverage Optimization in Wireless 5G and beyond Deployments
## 1 Introduction
### Problem Formulation and Related Work  
According to our previous work [65], the network infrastructure can be defined as the following graph. Assume that the network infrastructure contains \(m\) vertices (service centres), and \(n\) vertices (customer locations), and for each pair of vertices \(i\) (considered as service centres) and \(j\) (customer locations) their signal strength \(p_{ij}\) is given using a suitable propagation model (see more detailed discussion on the Equation (42)). In addition, \(P_{opt}\) is defined as the minimum signal strength which will be regarded as sufficient to establish the communication link between the service centres and customer location (these variables are described in more detail in Section 3).  
Let us consider two finite sets \(I\) and \(J\), where:  
* \(I\) is the set of service centres \(1,2,\ldots,m\),
* \(J\) is the set of customer locations \(1,2,\ldots,n\).  
The aim was to determine which vertices must be used as service centres for each customer location to be covered by at least one of the centres and to minimize the number of operating centres. In other words, with respect to the targeted real-life scenario, we want to define a minimum number of gNodeB nodes while still providing coverage for all users in a given area.  
**Remark 1**.:
1. _A condition necessary to solve the task is that all of the customer locations are reachable from at least one location where an operating service centre is considered._
2. _Customer location_ \(j\) _is reachable from vertex_ \(i\)_, which is designated as an operating service centre if_ \(p_{ij}\geq P_{opt}\)_. If this inequality is not satisfied, vertex_ \(j\) _is unreachable from_ \(i\)_._  
Here, \(a_{ij}=1\) means that vertex \(j\) has a signal with sufficient strength from the service centre \(i\), \(a_{ij}=0\) means the opposite, and \(w_{i}\) expresses the weights of service centres (since it is the minimization problem, the greater the weights are, the smaller the coefficient must be). If all the weights are set to \(1\) (i.e., all the centres are equally important), then we get a basic version of the set covering problem.  
Similarly for decision variables \(x_{i}\), \(x_{i}=1\) means that the service centre \(i\) is selected, while \(x_{i}=0\) means that it is not selected. Then, the set covering problem can be described by the following mathematical model [38; 65; 66; 67; 68; 69; 70].  
Minimise  
\[z=\sum_{i\in I}w_{i}x_{i} \tag{1}\]  
Subject to  
\[\forall j\in J:\ \sum_{i\in I}a_{ij}x_{i}\geq 1 \tag{2}\]  
\[\forall i\in I:\ x_{i}\in\{0,1\} \tag{3}\]  
The objective function (1) represents the number of operating centres, constraint (2) means that each customer location is assigned to at least one operating service centre. Following that, the \(P_{opt}\) represents a threshold of service reachability.  
To solve the above-described model, we have utilized an enhanced genetic algorithm as one of the possible solutions [65]. This was used for minimizing the number of schools in the selected area in Central Europe, but the theoretical concept was the same. However, in that work, we did not consider the capacity of service centres and the capacities potentially requested from customer locations. Therefore, in this paper, we propose the model and its implementation applied to optimizing wireless networks infrastructure while taking into account the capacity features mentioned.