# Review of Traffic Assignment and Future Challenges
## 3 Traffic Assignment: Theoretical Background
### Dynamic Traffic Assignment (DTA)
#### 3.2.2 Resolving Approaches
**Repeat the following operations multiple times**:  
* Evaluation: Assess the \"score\" of each candidate solution. * Selection: Reduce the number of \"poor\" solutions (there are various methods to achieve this). * Construction of new solutions: Build new solutions and add them to the collection of candidate solutions. **Algorithm 2** Population-Based Search  
**4. Extended Traffic Assignment**  
Traffic assignment has been widely implemented in the context of road infrastructure design. It has been extensively used to assess various road investment scenarios and lane directions. Thus, the previous section specifically focused on travel times, vehicle traffic, and trip balance. In this section, we will discuss the expansion of the application of traffic assignment, particularly to enhance environmental conditions, likely a reduction in pollution or an increase in environmental sustainability. This section explores contributions related to alternative modes of transportation and road toll policies. In addition, it shows how agent-based modeling allowed a paradigm shift. **4.1. Traffic Assignment for Alternative Modes**  
Alternative modes of transportation are of particular interest to transportation authorities, aiming not only to provide transport options suitable for all socioeconomic backgrounds of inhabitants but also to enhance environmental conditions and the quality of services for tourists and impaired people. Traffic assignment for alternative modes is vital for facilitating economic evaluations that assess the sustainability of projects related to innovative transportation systems [100; 101; 102; 103]. Following the formulation of the traffic assignment problem, there has been a specific focus on the transit assignment. However, assigning public transport users requires some adjustments to the initial traffic assignment formulations. The first pertains to the criterion motivating users. Travel time cannot be considered the predominant criterion. Users' perceived quality of service is not limited to the travel time. It depends on vehicle and transport line capacity as well as frequency [104; 105; 106; 107; 108; 109]. These factors determine access queue lengths and transfer times [110; 111]. Additionally, the number of connections may discourage certain route choices, even if the travel time is shorter. From the supply modeling standpoint, there are two approaches to modeling transit. The first one is based on line frequencies [112; 113; 114], where specific waiting time models have been proposed, either with or without consideration of the vehicle capacities. The second one considers the transit timetables (schedule) [115; 116; 117; 118; 119; 120]. From the demand modeling standpoint, the notion of equilibrium [104; 106; 107; 121; 122; 123], including the stochastic one [124; 120; 125; 126; 127], is widely deployed in the context of transit assignment. However, the assumption of a strongly monotonic generalized cost function increasing as a flow function is not necessarily verified. First, travel time is no longer the most important criterion. Second, even if it were, the function does not benefit from the same characteristics as the functions used in road traffic. More precisely, the traveling time is not necessarily monotone, and \"an increase in flow does not deteriorate the performance of the system\" [123]. For the reasons presented previously, the static assignment of transit users has also benefited from its theoretical developments, not only for formulating the problem but also for calculating the solution. As far as the formulation of the problem is concerned, a hyperpath model is used and assumptions have also been relaxed in [128] to ensure the uniqueness of the solution. For the calculation of the solution, the contribution of [128] reformulates the problem in the form of a nonlinear complementary problem, and [129] proposes an approach for its resolution. In the latter paper, the authors highlight the convergence issue of classical gradient-based approaches.