# Notes on Bus User Assignment Problem Using Section Network Representation Method
## 5 Transit Assignment Algorithm  
The ultimate goal of a transit assignment model is to create \(\Omega_{R}^{w}\) and \(\alpha_{ij/R}\) variables that determine the ridership of transit lines. As noted before, transit riders' route selection is a combination of en-route and pre-trip selections. This behavior might be implicitly reflected in the outlined model by establishing route \(R\). Even when a single route \(R\) is considered for a given demand (\(d_{w}\)), multi-line selection will be involved. The recursive calculation of the shortest route \(R\) (for each s and r) may be expressed using the generalized Bellman's equation for the proposed deterministic model as follows:  
\[q_{i}=\left\{ \begin{aligned} 0\ if\ i=r\\ min_{ij\in A(i^{+})}\ \{\ q_{j}+c_{ij}\}\ if\ i\in S-\{r\}\end{aligned} \right. \tag{15}\]  
where \(q_{i}\) is the length of the shortest \(\overline{R}\) (\(\subset\)\(R\)) route from an intermediate node \(i\) to a final node r.  
It is essential to include volume-delay relationship models in the assignment process in order to simulate the consequences of growing waiting periods caused by the inconvenience of boarding (or inability to board) the first-arriving bus(es). As the suggested function may not reflect real waiting times, this assessment would aid the analyst in determining the adequacy of the analyzed design of a transit network more properly during the strategic phase. Nonetheless, it might indicate the comparative effectiveness of many examined alternatives. In this research, the impact of the higher cost will be expressed using a formula similar to the one used by the Bureau of Public Roads (BPR):  
\[c_{ij}=\frac{1+\sum_{l\in\overline{L}ij}TT_{l}\ \varphi_{l}}{\sum_{l\in \overline{L}ij}\ \varphi_{l}}+\mathcal{R}\left(\frac{f_{ij}+\sum_{j\in E}\ \overline{f}_{ij}}{\sum_{m\in \overline{L}ij}lcm}\right)^{\mathcal{P}} \tag{16}\]  
Figure 2: Section-based network graph representation for the network presented in Section 1.  
where \(\overline{f}_{ij}\) is the competitive flow of other sections that possess the CLP, with \(\delta\) and \(\mathcal{P}\) as parameters that must be calibrated to evaluate how the flow impacts journey time. The solution flows for these cost functions may exceed the physical limits of the lines (i.e., capacities). When employing volume-delay functions immediately from calibration, this may be the case. The use of these formulae may be defended in two different ways. First, the line's physical capacity might be surpassed in real life by lengthening the wait until an empty spot becomes available. Second, these formulae satisfy the necessary convergence and uniqueness conditions for the majority of assignment models [66; 67; 68; 69]. Consequently, they are the most effective means of communicating the supply-demand interplay at the strategic phase of design. The objective of the assignment procedure is to resolve the variational inequalities given in Equation (6) or (7), where they may provide a unique solution in terms of arc flows. The monotonicity assumption of the performance function in Equation (16) guarantees this uniqueness, and the assumption that the non-additive cost is flow-independent ensures that the non-additive cost is independent of the flow. The MSA might solve either fixed-point or variational inequality versions of the problem. As the segment representation, a has an inseparable cost function architecture and an asymmetric Jacobian matrix. It would need a diagonalization phase in which the Jacobian matrix would be approximated to a diagonal matrix by taking into account just the fluctuation of the link cost at each diagonal cell. In addition, the approach proposed in [70] can be deployed to minimize the number of iterations the whole algorithm needs (see Algorithm 1). For improved convergence performance, it is essential to note that in step 4, the calculation of an improving direction and optimum step length would be necessary.