# Notes on Bus User Assignment Problem Using Section Network Representation Method
## 1 Introduction  
Transit assignment models are essential assessment tools for the bus network design problem (BNDP), which is considered one of the most difficult issues to address in transportation science. Due to its great degree of intricacy, this is the case. Non-convexity and non-linearity, NP-hard/combinatorial complexity, bi-level problem formulation, and the multi-objective structure of the challenge often impede the search for an optimum solution [1]. The employment of these models determines the quality of a BNDP solution forecasting how each passenger chooses a path/route from his/her starting location to the destination location [2; 3; 4]. The BNDP might also be designed as a bi-level programming paradigm, with an assignment method serving as the low-level model. The final BNDP solution requires a recursive calling to the transit assignment stage [5]. In addition to the amount of aggregation attempted by each model, there are several assumptions about passengers' information and tactics for bus line selection at the station level, where people and buses have distinct probability distributions of their arrivals. The primary distinction between transit assignment models and ordinary traffic models is that, in virtually all circumstances, bus passengers are more likely to navigate overlapping bus lines in which some lines share common stations and sections (see Figure 1). Intriguingly, this transforms the issue, even in its uncongested situations, into a multi-path assignment [1]. The example of a simple bus network in Figure 1 illustrates the core principle influencing the results of an assignment model directed to the bus networks. In actual practice, a transportation passenger always finds the most economical route (i.e., the least travel time) to their destination. Passengers boarding at station A and traveling to station B would rationally consider line 1 (\(L_{1}\)) in their path selection as it is the direct service to the desired destination from the originating bus stop. From another perspective, \(L_{2}\) (with transfer at y or x) could be included in their desirable options if they were aware that it decreases overall journey time. Then, users would take the first vehicle to arrive on the two routes. In another situation, \(L_{1}\) might become unattractive to passengers if intelligent information on the residual wait time for \(L_{2}\) and the probable wait time at stations y and x is made accessible. All users would board \(L_{2}\) and then make transfers, even if \(L_{1}\) has the first arriving vehicle. This issue is referred to in the literature as the common lines problem (CLP) [6], in which customers would always opt to board the first coming bus among specified alternative lines if the target/main line is unavailable. The CLP is concerned with whether the user boards the approaching vehicle of a line, waits at the stop for the following vehicle of a different line, or goes to a different stop in search of other options. The whole CLP solution relies on the amount of data accessible at each stage and the choice made at that stage [7]. When line capacity is concerned, more dimensions are introduced. Even if \(L_{1}\) were the sole desirable option, some users might be unable to take the first inbound vehicle owing to the unavailability of seats. The options are to either wait for a vacant place in \(L_{1}\), switch entirely to the other line, or consider both options. Additionally, \(L_{2}\) users might not be able to determine the exact decision cost (i.e., travel time). When riding aboard, passengers must choose whether to alight to transfer at station x or remain on board until station y. All transit assignment models are regarded to be based on these reasonable scenarios. Each model defines the strategies (i.e., a set of rules for sequential line selection) that the users might take to minimize the total waiting and in-vehicle periods.