## 4 Methods
### Modelling Framework
\[\sum_{k}x_{k,j,i}\leq 1\ ;\ \forall j,i \tag{2}\]  
\[\sum_{i}x_{k,j,i}=1\ ;\ \forall k,j \tag{3}\]  
In addition to the above, it was necessary to define additional conditions for the traffic modelling that describes reality and validates safety aspects. These took into account the limitations of speed, acceleration and deceleration. To avoid collisions, the crossing vehicle movements at the same time steps were prohibited by the following criterion (Equation (4)). \[ \begin{array}{c}\left(x_{k,j,i}+x_{k,(j+1),q}+x_{p,j,r}+x_{p,(j+1),s}\right)*r_ {cross_{(i-1)*o+q,(r-1)*o+s}}\leq 3;\\ \hskip 142.26378 pt\forall\;k,p,j,i,q,r,s\end{array}  \tag{4}\]  
The expression ensures that two vehicles (\(k\) and \(p\)) cannot travel routes (\(i\)-\(q\) and \(r\)-\(s\)) with common elements at the same time step (\(j,j+1\)), by assigning the appropriate element of \(R\_CROSS\) matrix [46]. As the main limitation of the concept, it has been established that the discretization of the time and space domains and the high number of vehicles encountered in real-world applications may significantly increase the number of model variables. This could result in increasing computational complexity and processing time. To alleviate this problem, authors have developed new methods to reduce the complexity of the presented transport management process [47]. The basic principle of the developed simplification methods was the exclusion of different possible values of the variable from the investigation in the case of the introduced constraints. Based on the specific characteristics of the predefined origin and destination locations (e.g., vehicles could not enter any origin or leave any destination location during their travel), their examination was also omitted for some of the constraining expressions. To highlight an example, we present the approach of excluding variables from the expression prohibiting the crossing movements of the vehicles (see Equation (4)). According to the approach, route pairs that do not have any common elements can be directly excluded from the investigation, resulting in the investigation of only the following cases (Equation (5)) instead of \(\forall k,p,j,i,q,r,s\):  
\[\forall k,p,j,i,q,r,s;\;where\;r_{cross_{iq,rs}}=1 \tag{5}\]  
When implementing the elaborated simplifications, it was crucial not to compromise the feasibility of the solution. Since these methods were intended to modify the model, system safety procedures were applied to mitigate the potential safety risks associated with them [47]. Focusing on the safety aspects of traffic flows in the managed autonomous transport system, indicators were developed characterizing the level of road safety for the obtained results. The different indicators examined different types of risks, considering the spatial and temporal characteristics of the travel processes and the speed data of the vehicles [47]):  
* \(CM\): the indicator expressed the risks arising from crossing vehicle movements, calculating the number and temporal distance of these manoeuvres, as shown in Equation (6). \[ \begin{array}{c}M=\sum_{k,h,j,u,i,q,r,s=1}^{m,n,(t-1),(t-1),o,o,o,o}(r_{ cross_{iq,rs}}*(x_{k,j,i}*x_{k,j+1,q}*x_{h,u,r}*x_{h,u+1,s})*\frac{1}{(u-j)^{2}})\\ \hskip 142.26378 ptk\
eq h\\ \hskip 142.26378 ptu\
eq j\end{array}  \tag{6}\]  
The value of \(CM\) is higher if there are more crossing vehicle movements (meaning more summands in the summation), or if the crossing movements take place in shorter time intervals. Accordingly, a lower value of a \(CM\) safety indicator implies a safer flow of traffic [47]. A higher \(CM\) value indicates that more vehicle routes are assigned that intersect with each other, and it may also suggest that the intersecting trajectories are traversed by the respective vehicles closer in time, which increases the likelihood of side collisions. * \(\overline{VN}\): the indicator implied the level of safety by the average speed at the network level, calculated as the mean of the average speeds of vehicles in the network.