## 4 Results
### Case Study  
The presented optimization approaches have been applied to a solution to an actual road transport problem, occurring in the north-eastern part of the Warsaw agglomeration. It is an area that often experiences the phenomenon of traffic congestion caused by heavy traffic flows during the morning and afternoon rush hours. The lack of a well-developed road network in relation to the number of traffic users implies the need to search for new and better solutions, assuming appropriate limiting conditions. Figure 3 shows a road map with the main nodes plotted of the analyzed transport system. Figure 2: Flowchart of the research. Figure 4 shows a graph of the analysed transport network. The transport network consists of eight nodes, three of which are the origin nodes, and one is the traffic flow destination node. Three traffic flows marked as \(x_{1}\), \(x_{2}\) and \(x_{3}\) are found at the network entrance. Vectors were used to determine the direction of movement of allowed traffic flows along nine transport links. Traffic flow marked as \(x_{8}\), which is the sum of flows in all origin nodes, is headed for the destination node. Figure 4: Transport network graph. Figure 3: Road map with road nodes plotted. A detailed description of the transport network can be found in Table 2. There are three movement O-D pairs within the analyzed system: (1,8), (2,8) and (3,8). The transport network graph for each O-D pair was used to determine the sets of possible routes composed of links \(L_{ij}\). Each link in the transport network has been assigned its time characteristics, which determine the travel time through a given section of the road, depending on the flow of traffic. Increased traffic volume leads to an elevated average travel time through a transport link and a significant reduction in the average travel speed (especially in bottlenecks), which in consequence may result in congestion. Equation (19) was used to present the characteristic time matrix of the analyzed road network. \[T= \begin{bmatrix}-&-&-&f_{14}(x_{14})&-&-&-&-\\ -&-&-&-&f_{25}(x_{25})&-&-&-\\ -&-&-&-&-&f_{36}(x_{36})&-&-\\ -&-&-&-&-&-&f_{47}(x_{47})&t_{48}(x_{48})\\ -&-&-&f_{54}(x_{54})&-&f_{56}(x_{56})&-&-\\ -&-&-&-&-&-&f_{67}(x_{67})&-\\ -&-&-&-&-&-&-&f_{78}(x_{78})\\ -&-&-&-&-&-&-&-\end{bmatrix} . \tag{19}\]  
For the traffic flow \(x_{ij}\) expressed in thousands of vehicles, the characteristics of individual links in the transport network are shown in the form of a system of Equations (20):  
\[ \begin{cases}t_{14}(x_{14})=5+0.05x_{14}+0.025x_{14}^{2}\\ t_{25}(x_{25})=10+0.04x_{25}+0.065x_{25}^{2}\\ t_{36}(x_{36})=6+0.03x_{36}+0.025x_{36}^{2}\\ t_{47}(x_{47})=4+0.04x_{47}+0.04x_{47}^{2}\\ t_{48}(x_{48})=5+0.08x_{48}+0.125x_{48}^{2}\\ t_{54}(x_{54})=4+0.035x_{54}+0.02x_{54}^{2}\\ t_{56}(x_{56})=5+0.035x_{56}+0.04x_{56}^{2}\\ t_{67}(x_{67})=1+0.035x_{67}+0.03x_{67}^{2}\\ t_{78}(x_{78})=2+0.025x_{78}+0.035x_{78}^{2}\end{cases} . \tag{20}\]  
The set of permissible solutions is determined by the conditions to be satisfied by the distribution of the traffic flow within a transport network, which were presented through systems of equations and inequalities (21)-(24):  
* condition of satisfying transport demand (21), assuming that the total traffic flow within a given O-D pair moves entirely along identified transport network roads:  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Element of Model** & **Mathematical Notation** \\ \hline Structure & \(G=<W,L>\) \\ Set of nodes & \(W=\{1,\,2,\,3,\,4,\,5,\,6,\,7,\,8\}\) \\ Set of origin nodes & \(A=\{1,\,2,\,3\}\) \\ Set of intermediate nodes & \(V=\{4,\,5,\,6,\,7\}\) \\ Set of destination nodes & \(B=\{8\}\) \\ Set of O-D pairs & \(E=\{(1,8),\,28), (3,8)\  
\[\left\{ \begin{array}{c}x_{1}=x_{1,18}+x_{2,18}\\ x_{2}=x_{3,28}+x_{4,28}+x_{5,28},\\ x_{3}=x_{6,38}\end{array} \right., \tag{21}\]
* condition of traffic flow non-negativity (22): \[\left\{