# Notes on Bus User Assignment Problem Using Section Network Representation Method
## 2 Background  
In the last 50 years, several publications have been written about transit assignment models based on frequency distribution. While a number of models have been presented to forecast the behavior of passengers in picking their lines, the idea of common/attractive routes at the stations continues to be the underlying concept for such models. In [6], the notion that each passenger picks a group of routes to reduce the sum of in-vehicle time and average waiting time in her/his destination plan is initially developed. In [11], the emphasis is placed on estimating the waiting time for this issue based on the probabilistic arrival distributions of users and vehicles. In the two studies [12; 13], the concept of optimum strategies is developed as a predetermined set of steps/rules the passenger follows in his/her travel journey. The CLP is summarized in a single mathematical formulation, producing a mixed-integer non-linear programming model. Interestingly, the model could be simplified into a linear form that facilitates solution discovery. In addition, the provided formulation might be expanded to include the congestion impact, much as equilibrium models of conventional traffic models do. In [14], the optimum strategies method is adopted into a graph-theoretic formulation in which the idea of hyper-paths is added. Different hyper-paths for each Origin-Destination (O/D) consisting of specific/basic pathways could be constructed. At each station, outbound connections distribute passengers throughout the basic pathways. The frequency at each stop determines the distribution components, which are then summed into one. The least-cost hyper-path for the O/D equates to the optimum strategy derived from the [12; 13] formulation. To readily modify the Bellman optimality equation [15], bus frequencies are supposed to obey an exponential distribution with stochastic passenger arrivals. Figuring ideal hyper-paths in large networks with different frequency distributions using a label-correcting or label-setting method is more complicated, according to [16]. In [1], [PERSON] and [PERSON] suggested an alternative graph representation based on what is known as line sections. In this depiction, similar lines are intrinsic to the graph architecture. The hyperbolic equation published in [7] is used at each section formulation to solve the CLP corresponding to that section. Fortunately, the methods proposed in [1] can effectively solve the hyperbolic equation. The congestion effect has been a popular area of study in frequency models. It is an essential issue in several global transit networks [17]. Users often face completely crowded stations that force them to switch lines owing to either inability to board the preferred line due to insufficient capacity (strict capacity) or an increase in travel time impediment (moderate capacity). Modeling users' preferences becomes a more complex problem when individual preferences and network congestion levels are included [18]. Adapting the Wardrop principles of equilibrium demonstrates the utility of congestion modeling [19]. In [20], for instance, the transit network assignment is handled using stochastic passenger equilibrium under the multinomial logit (MNL) presumption, while [1; 12] employed deterministic user equilibrium principles for route selection. Instead, in [21], the probit-based approach is used to represent the path choice in order to avoid the Independence of Irrelevant Alternatives (IIA) characteristic seen in the MNL. A well-defined congestion-cost formula is required to formulate a crowded equilibrium assignment model. Some formulae may result in challenging assignment models to use or analyze. For this reason, the congestion-cost formula must have desirable mathematical qualities (such as a monotonically growing function) [22]. Typically, cost is proportional to flow. Hence, arc costs are shown as flow-dependent. This illustrates the inefficiency and discomfort caused by overcrowded transportation.