## 4 Methods
### Modelling Framework  
The referred model developed by the authors was based on a nonlinear binary integer programming approach [46; 47]. The applied 3-dimensional binary decision variable (\(x_{k,j,i}\)) described accurately the position (region \(i\)) of each vehicle (\(k\)) at each time moment (\(j\)) in the investigated timeframe. Thus, time-space was discretized, and the road network was partitioned into directed locations (regions) in the model in accordance with the occupancy grid concept [27]. The total number of locations on the network was denoted by \(o\), the total number of considered vehicles was denoted by \(m\), and the total timeframe covered \(t\) time moments. The aim of the developed model was to manage the travel demands effectively by optimizing the load of the transport network at system-level. Note that system-level processes were managed by the implemented vehicle-level transport management process in our model. The following parameters were considered as predefined constant values [46]:  
* The discretization of the road network was carried out using square shaped regions of the same size. The side length of the regions was set at 5 m (based on the size of a typical passenger vehicle). Besides the size, shape, and position of the regions, the directions of possible further travel for each elementary region were also pre-defined. * The starting and target locations of vehicles were defined in \(\textbf{ORIG}\in R^{m\times o}\) and \(\textbf{DEST}\in R^{m\times o}\) binary matrices (\(orig_{k,i}\) and \(dest_{k,q}\) are represented with a value of 1 if \(i\) is the starting and/or \(q\) is the ending location of vehicle \(k\), respectively). The origin and destination locations were connected to the network by one-way links and were interpreted as sources and sinks (an unlimited number of vehicles could use them at the same time moment). * The shortest distance between each location pair was defined in \(\textbf{D}\in R^{o\times o}\) matrix (\(d_{i,q}\) represents the value of the shortest distance between location \(i\) and \(q\), assigning infinite value for non-traversable location pairs). * \(\textbf{R\_CROSS}\in R^{o^{2}\times o^{2}}\) binary relation matrix [46] was created to define the crossing route pairs. The rows and columns of the matrix represented the location pairs (routes) in ascending order, comparing the first location against all others, then the second location against all others, and so on. The value of 1 was assigned for the elements of the matrix representing route pairs that had any common elements (excluding the starting locations). For this purpose, the shortest path between the location pairs were investigated. I.e., element \(r\_crossig,rs\) had a value of 1 if the shortest paths between the investigated location pairs (\(i\)-\(q\) and \(r\)-\(s\)) existed and had any common location excluding the starting points, otherwise 0. * The maximum allowed speed (\(c1\)), acceleration (\(c2\)) and deceleration (\(c3\)) limits were also predefined constants. In accordance with the aim of the optimization, the objective function was defined to sum up and minimize the distances between the current and the destination location of each vehicle within the investigated time frame (Equation (1)). \[y=\sum_{k}\sum_{j}\textbf{x}(\textbf{k},j)*\textbf{D}*\textbf{dest}(\textbf{ k})\to min \tag{1}\]  
where  
\(y\): is the objective function depending on \(x_{k,j,i}\);  
\(\textbf{x}(\textbf{k},j)\in R^{o}\): is a row vector identifying the location of vehicle \(k\) at the time step \(j\);  
\(\textbf{dest}(\textbf{k})\in R^{o}\): is a column vector from \(\textbf{DEST}\) matrix, identifying the destination location of vehicle \(k\). To ensure safe transport management, the appropriate assignment of time and space elements is a fundamental requirement. This ensures that a vehicle can only be at one place at a time, and only one vehicle can be at one place at a time. The conditions were described in the following expressions (Equations (2) and (3)).