Review of Traffic Assignment and Future Challenges
Abstract
:1. Introduction
2. Preview
2.1. Problem Statement
 Traffic supply: Road network and/or public transportation services as well as their corresponding behaviors.
 Traffic demand: A flow matrix indicating the demand volume between each origin–destination (o–d) pair.
2.2. User Criteria
 The monetary costs, such as the costs of fuel consumption, vehicle’s depreciation and maintenance, toll, and transport fare.
 The nonmonetary costs, such as the travel time and user’s perception of comfort and convenience.
2.3. Mode of Transport
2.4. Illustrative Example
 In road segment ${a}_{1}$, the capacity, ${Q}_{max}^{1}=1800$ vph, and the travel time behave as follows:$${c}_{1}\left({r}_{1}\right)=10\left(1+2{\left(\frac{{r}_{1}}{1800}\right)}^{2}\right).$$
 In road segment ${a}_{2}$, the capacity, ${Q}_{max}^{2}=3600$ vph, and the travel time behave as follows:$${c}_{2}\left({r}_{2}\right)=20\left(1+2{\left(\frac{{r}_{2}}{3600}\right)}^{2}\right).$$
 Shortest path (Allornothing): Without considering Equations (1) and (2) as though both roads are devoid of traffic, all vehicles follow road ${a}_{1}$ because it is the shortest path. However, since travel time increases with traffic flow, an allornothing assignment cannot be adopted to predict road use. In this figure, the solution of the allornothing assignment is presented by a green circle. According to Equation (1), the users spend more than an hour to go from o to d.
 Optimal assignment (Centralized): If we consider that energy consumption is proportional to travel time, the manager of a fleet of the 3000 vehicles has a strong incentive to minimize the average travel time. The red curve in Figure 2 presents the average travel time. The optimal solution is represented by the red circles. One can note that, with this solution, the vehicles taking road ${a}_{1}$ spend less travel time than the ones using road ${a}_{2}$.
 Fair assignment (Decentralized): In this configuration, each vehicle optimizes its itinerary, so that there is no better solution. The fair assignment is represented by the yellow circle.
3. Traffic Assignment: Theoretical Background
3.1. Static Traffic Assignment ($STA$)
3.1.1. User Equilibrium ($UE$): Definition and Formulation of the Problem
 For each origin–destination pair, o–d, the generalized costs of each utilized route are less than or equal to those of alternative (unused) routes.
 If multiple routes are used for an o–d pair, their generalized costs are equal.
3.1.2. Stochastic User Equilibrium ($SUE$)
 All reasonable options can be chosen, even if their probability of selection is very low (In [20], the author describes the concept)
 If two options have the same cost, the probability of selection is the same.
 The probability of choosing options depends on their costs: a route with a higher cost has a lower probability of being chosen.
 The user of the $SUE$ model must have some control over the probability of diverting routes.
3.1.3. Approaches to Solving Static Assignment
Algorithm 1 Static Traffic Assignment 

3.2. Dynamic Traffic Assignment ($DTA$)
 A traffic model in which congestion (travel time) varies over time.
 A timevarying demand.
 An equilibrium that is based on experienced travel cost, not instantaneous travel cost.
3.2.1. TimeDependent User Equilibrium
 For each o–d pair and for each departure time interval, the routes taken by users exhibit a generalized cost (experienced travel time) that is both equal and minimized to the extent possible.
 No user can unilaterally reduce their experienced generalized cost (travel time).
3.2.2. Resolving Approaches
 Initialization: In the majority of studies, the network is initialized using the allornothing assignment based on the computation of the shortest path. In the context of $STA$, the shortest path is computed with an empty network. Consequently, the allornothing assignment associates a route with each origin–destination (o–d) pair throughout the entire temporal horizon of the study. In the case of dynamic assignment, the travel time (generalized cost) for a given route, ${k}^{od}$, varies based on the departure time intervals of vehicles. Indeed, with each new time interval, it is necessary to update the generalized costs induced by vehicles already assigned in previous time intervals. Thus, the search for the shortest path occurs progressively as the network fills up.
 Iteration: Recall that, at each iteration, a flow of vehicles is shifted from a costly route to a less costly one until a convergence criterion is met. In the context of $DTA$, this flow shift occurs for all time intervals, T, within the study interval at each iteration. Several approaches exist in the literature for determining the direction and quantity of the traffic shift. The most classical approach is based on the Frank–Wolfe algorithm [30,83]. Other more efficient algorithms have been proposed since, including the gradient projection algorithm [41] and the method of successive averages ($MSA$) [79,84,85] to compute the generalized cost. Approaches based on metaheuristics have also been proposed.
 Evaluation of Travel Times: The time it takes to travel a road segment can vary from when a vehicle starts its journey to when it is moving on that segment. Drivers base their route decisions on the time spent actively driving on the segment. Thus, it is crucial to predict this travel time accurately. In advanced methods, travel times are assessed by adding up the times for each segment, forming a travel time chain. The estimated travel time for each segment is based on when the vehicle is expected to reach that specific segment [78].
 Stopping Criterion: Recall that, in $STA$, there is a unique equilibrium solution in terms of traffic flow for each segment. This solution is reached when Algorithm 1 converges. Unlike $STA$, the convergence of vehicle routes in $DTA$ does not imply that the network has reached a dynamic equilibrium. The convergence problem becomes more complex when the granularity of the traffic model is high. It is widely reported that microscopic models are intractable.
Algorithm 2 PopulationBased Search 

4. Extended Traffic Assignment
4.1. Traffic Assignment for Alternative Modes
4.2. Environmental Concerns and Traffic Control
4.3. Paradigm Shift through MultiagentBased Approaches
5. Discussion
5.1. Intelligent Transportation Systems
5.2. Promising Directions
 Mixed traffic digital twin: The cohabitation of a multitude of systems would require either new macroscopic and mesoscopic traffic models or the use of microscopic models coupled with an agentbased approach [76,169]. The latter option appears better suited to accommodate the diverse objectives of road users, especially in the context of mixed traffic: human drivers with navigation systems, human drivers familiar with the road, autonomous shuttles, optimized freight transport, etc. Furthermore, employing microscopic models could enable us to address the challenge of finding an equilibrium between diverse equitable routes and the costs induced by conflicts. However, it is worth noting that the use of microscopic traffic models for computing the travel times is resourceintensive in terms of computation resources and time. Another option is to investigate the implementation of digital twins [170,171] with different levels of granularity of the traffic entities. This digital twin should be capable of assimilating data from the cloud, ranging from macroscopic quantities such as road flow and occupancy rates to more detailed user profiles at the vehicle level, including departure times and itineraries. Harnessing $AI$ techniques is crucial [172] for generating a comprehensible map of road usage for decisionmakers (e.g., identifying saturated intersections, assessing environmental impact, improving safety, and offering the forecasts needed by cities). More importantly, $AI$ can also be used to propose solutions (e.g., the necessity for new transit services and a new toll policy). As emphasized throughout this article, the problem is undeniably intricate. Nevertheless, the substantial advancements in machine learning technologies, data analysis techniques, and cloud computing hold the promise of highly innovative approaches.The data has previously been employed for a range of tasks related to traffic assignment, including estimating the origin–destination (o–d) matrix [173,174,175], evaluating travel times [176], and calibrating models and traffic assignment results [174,177,178,179,180]. This utilization encompasses the data of cell phones, information systems, and magnetic loops. The development of the mixed traffic digital twin aims to transcend these contributions, representing a significant stride towards exploring advanced artificialintelligence ($AI$) techniques for traffic management. An illustrative foundation for datadriven traffic assignment is established in the study presented in [181]. Instead of assuming user behavior, the authors showcased the feasibility of precisely estimating road demand by directly learning flow patterns from the available data. This innovative approach lays the groundwork for incorporating $AI$ into traffic forecasting. The extension introduces additional parameters, such as traffic control strategies and new transportation scenarios. Moreover, the mixed traffic digital twin can be designed to be proactive, not only highlighting problems, such as those proposed in ref. [182], but also proposing solutions to the decisionmakers. To this end, the mixed traffic digital twin may be trained by using “classical traffic” assignment approaches.
 New generation of navigation systems: The evolving uses of $CAV$ and associated regulatory systems will likely challenge current traffic assignment approaches. These vehicles will need to plan their routes in real time based on received demands and communicate their estimated arrival times as accurately as possible. Although Wardrop’s equilibrium assumption does not allow for the perfect rationalization of vehicle usage, it remains desirable for fairness reasons. Indeed, it seems evident that no user would want to take a shuttle service that significantly takes longer than other shuttles without financial compensation. However, current traffic assignment techniques do not allow for realtime route planning. Computation times increase when considering vehicle behavior and traffic regulation details with high resolution. Currently, traditional traffic assignment approaches fall short in delivering realtime itineraries. Simultaneously, there exists a lack of consensus in the literature regarding the impact of contemporary navigation systems. Several authors have underscored adverse effects, as indicated in refs. [183,184,185,186].This issue pertains to the challenge of efficiently assigning $CAV$ on a constantly evolving road network, taking into consideration realtime traffic conditions. One crucial aspect of the problem is determining optimal routes for vehicles based on realtime traffic information. The commonly used shortest path search method, which calculates the fastest routes between two points on a road network, does not effectively solve the realtime traffic assignment problem. The main drawback of this method is its lack of responsiveness to constantly changing traffic conditions. Figure 4 and Figure 5 depict the adverse effects of such an approach on route selection. The first figure illustrates how vehicles can become trapped when new vehicles are rerouted to an alternative path with smoother traffic, highlighting the necessity for a thorough evaluation of intersection times. Meanwhile, the second figure demonstrates how a vehicle may be misdirected due to the absence of traffic that is not yet present, emphasizing the critical importance of accurately estimating upcoming traffic conditions.New strategies with simple rules must be defined to guide vehicles in real time. These rules should be capable of providing both efficient and fair routes while fostering smooth traffic flow. Some studies already address these issues by proposing itinerary reservations [187,188]. Among these studies, some focus on road booking in order to not exceed their capacity [189,190,191,192]. Others are inclined towards intersection reservations [149,193,194,195] to alleviate costs associated with conflicts arising from diversified routes. However, these approaches are relatively recent and deserve to garner broader attention within the community to receive more feedback on microscopic models of large cities incorporating innovative strategies for sharing road infrastructures with more transparency.
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
$AI$  Artificial Intelligence 
$BPR$  Bureau of Public Roads 
$CAV$  Connected and Autonomous Vehicle 
$DTA$  Dynamic Traffic Assignment 
$DUE$  Dynamic User Equilibrium 
$MSA$  Method of Successive Averages 
$STA$  Static Traffic Assignment 
$SUE$  Stochastic User Equilibrium 
$UE$  User Equilibrium 
vph  vehicle per hour 
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Elimadi, M.; AbbasTurki, A.; Koukam, A.; Dridi, M.; Mualla, Y. Review of Traffic Assignment and Future Challenges. Appl. Sci. 2024, 14, 683. https://doi.org/10.3390/app14020683
Elimadi M, AbbasTurki A, Koukam A, Dridi M, Mualla Y. Review of Traffic Assignment and Future Challenges. Applied Sciences. 2024; 14(2):683. https://doi.org/10.3390/app14020683
Chicago/Turabian StyleElimadi, Manal, Abdeljalil AbbasTurki, Abder Koukam, Mahjoub Dridi, and Yazan Mualla. 2024. "Review of Traffic Assignment and Future Challenges" Applied Sciences 14, no. 2: 683. https://doi.org/10.3390/app14020683