## 3 Transport System Modelling and Optimization  
To analyse systems and optimize traffic flow distribution, transport engineering commonly uses mathematical models of transport systems. They classically focus on one of the following four elements [56], namely:  
* structure--covering linear and point elements,
* characteristics--corresponding to structural properties,
* traffic flow--depicting the demand for transport services,
* traffic organization--characterizes the distribution of traffic flow in a given transport network. The mathematical notation of a transport system model is presented by Equation (1):  
\[MST=\{G,F,P,O\}, \tag{1}\]  
where: \(MST\)--transport system model, \(G\)--system structure graph, \(F\)--set of structure graph node and (or) link functions (characteristics), \(P\)--traffic flow, \(O\)--traffic organization of the transport system. A transport system graph is made up of a set of vertices that represent point elements of the transport infrastructure and a set of links that represent linear elements. In mathematical notation, the structure graph is presented by the relationship (2) [11]:  
\[G=\{W,L\}, \tag{2}\]  
where: \(W\)--set of graph nodes \(G\), \(L\)--set of graph links \(G\), which is a Cartesian product of the nodes. \begin{table}
\begin{tabular}{c c c c} \hline
**Paper** & **Case Study** & **Approaches** & **Criteria** \\ \hline
[44] & Selected road networks in India (data from Google Maps) & Deep Learning Technique (hybrid Xception support vector machine) & Traffic congestion \\
[45] & Road network in urban area of Guiyang city & Self-Organizing Feature Mapping (SOFM) & Traffic congestion \\
[46] & Traffic light junction & Ant Colony Optimization & Traffic flow \\
[46] & (Ranchi, India) & and Genetic Algorithm (GA) & management \\
[47] & Road network in New York & Bayesian optimization (BO) & Traffic flow \\  & & and Gaussian process (GP) & management \\
[48] & Motorway & Numerical optimization with & Stop-and-go waves \\ \hline \end{tabular}
\end{table} 
Table 1: \(Cont.\)The set of all nodes in the transport network can be presented in the form of a mathematical relationship (3):  
\[W=\{w:w=1,\ldots,\overline{W}\}, \tag{3}\]  
where: \(\overline{W}\) is the population of the set \(W\). The location of a demand for transport services determines the decomposition of the node set into three subsets:  
* \(A=\{a:a\in W\}\)--origin nodes,
* \(V=\{v:v\in W\}\)--intermediate nodes,
* \(B=\{b:b\in W\}\)--destination nodes. In mathematical notation, it is represented by the following relationship (4):  
\[W=A\cup V\cup B. \tag{4}\]  
According to the assumption adopted when modelling transport networks, all three subsets \(A\), \(V\) and \(B\) are disjoint pairwise, which is expressed as a relationship (5):  
\[ \begin{cases}A\cap B=\varnothing\\ A\cap V=\varnothing.\\ B\cap V=\varnothing\end{cases}  \tag{5}\]  
Assuming that the element \(l_{ij}\) presents a transport connection between nodes \(w_{i}\) and \(w_{j}\), it is assigned the following conventional values: 1--when there is a connection or 0--in the absence of a connection. Therefore, a set of relationships between transport network structure graph nodes can be presented as a binary matrix in the following form:  
\[L= \begin{bmatrix}0&\cdots&l_{1N}\\ \vdots&\ddots&\vdots\\ l_{N1}&\cdots&0\end{bmatrix} . \tag{6}\]  
Points of the transport infrastructure and linear or nonlinear characteristics are described in the model of the transport system model as node and link functions. They are both determined by properties, i.e., node and link capacity, movement distance, time, and cost of movement over given links, etc. The structure graphs of the system, together with the node and link functions form a transport network (\(S\)), described with Equation (7):  
\[S=\{G,F_{W},F_{L}\}, \tag{7}\]  
where: \(G\)--transport system structure graph, \(F_{W}\)--set of functions defined on a set of nodes \(W\), \(F_{L}\)--set of functions defined on a set of links \(L\).