# Transitioning to Society 5.0 in Africa: Tools to Support ICT Infrastructure Sharing
## 3 Mathematical Modelling and Problem Definition
### Optimization Problem  
The following nonlinear mixed integer mast colocation optimization problem is presented:  
Minimize:  
\[C=\sum_{i=1}^{N}f_{i}\left(\sum_{j=1}^{T}\left(x_{i,j}l_{j}\right)\right) \tag{5}\]Subject to:  
\[\begin{split}&\forall j=1,2,\ldots T:\sum\limits_{i=1}^{N}x_{i,j}=1\\ &\forall i,j:x_{i,j}\in\{0,1\}\\ &\forall i=1,2,\ldots N:\sum\limits_{j=1}^{T}x_{i,j}\cdot l_{j} \leq l_{i,\max}\\ &\forall k=1,2,\ldots R:\\ &\arg\max_{i=1,2,\ldots N}\left\{\left(\prod\limits_{j=1}^{T}x_{i, j}\right)\cdot r_{k,j}\right\}\geq r_{k,\min}\end{split} \tag{6}\]  
where \(x_{i,j}\) are introduced as binary decision variables to indicate an assignment of transmitter \(j\) with load \(l_{j}\) to mast location \(i\), and Equation (5) specifies the total costs for a given assignment of transmitters to masts through these variables. Constraint (6) ensures that a transmitter is assigned to one mast and one mast only, constraint (7) enforces binary behavior in the decision variables and constraint (8) ensures that each individual mast is not overloaded in terms of transmitter loading factor. Constraint (9) requires that the signal strength at each of the receiver locations, for the active masts given by the transmitter assignment specified by the decision variables, achieves the minimum required strength. When each cost function is of the form (3), and the signal strengths \(r_{k,j}\) are precomputed and stored in a \(T\)-by-\(N\) matrix, this can be specified and solved as a mixed integer linear program (MILP). In such a situation, the number of binary decision variables is fixed by the number of transmitters and masts and not directly impacted by the number of receiver locations. However, the number of constraints over the binary decision variables increases with the number of receiver locations. Thus, the problem complexity is not dominated by this measure (number of receivers) but is certainly influenced by it. Modification of constraint (9) to a case in which, say, a certain number of receivers have a signal strength above a certain threshold (coverage metric) is also relatively straightforward. As the dimensions of \(N\), \(R\) and \(T\) are normally of manageable size in many real-world applications (e.g., 10 masts, 200 receivers, 10 transmitters) this poses no real problems for modern commercial MILP solver packages. In the case of larger problem sizes, the area may potentially be subdivided or the problem instance sub-constrained to reduce combinatorial problem dimensions by splitting areas in which masts cannot feasibly cover cells (e.g., due to excessive distance or unviable path coefficient), either by application of Equation (4) or by expert inspection of signal strength analysis. The simplest way to achieve this splitting algorithmically is to limit the number of feasible transmitter assignments considered by a specific receiver location to those mast locations within a specific radius \(R\) of its location, with \(R\) set using Equation (4) to upper bound the achievable approximate signal strength for locations outside this distance in ideal path conditions. Exploration of the computational properties of the algorithm and extensions such as these is beyond the scope of the current work and will be considered as future work.