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13 Last Step of Four Step Modeling (Trip Assignment Models)

Chapter overview.

Chapter 13 presents trip assignment, the last step of the Four-Step travel demand Model (FSM). This step determines which paths travelers choose for moving between each pair of zones. Additionally, this step can yield numerous results, such as traffic volumes in different transportation corridors, the patterns of vehicular movements, total vehicle miles traveled (VMT) and vehicle travel time (VTT) in the network, and zone-to-zone travel costs. Identification of the heavily congested links is crucial for transportation planning and engineering practitioners. This chapter begins with some fundamental concepts, such as the link cost functions. Next, it presents some common and useful trip assignment methods with relevant examples. The methods covered in this chapter include all-or-nothing (AON), user equilibrium (UE), system optimum (SO), feedback loop between distribution and assignment (LDA),  incremental increase assignment, capacity restrained assignment, and stochastic user equilibrium assignment.

Learning Objectives

•  Describe the reasons for performing trip assignment models in FSM and relate these models’ foundation through the cost-function concept.
• Compare static and dynamic trip assignment models and infer the appropriateness of each model for different situations.
• Explain Wardrop principles and relate them to traffic assignment algorithms.
• Complete simple network traffic assignment models using static models such as the all-or-nothing and user equilibrium models.
• Solve modal split analyses manually for small samples using the discrete choice modeling framework and multinominal logit models.

Introduction

In this chapter, we continue the discussion about FSM and elaborate on different methods of traffic assignment, the last step in the FSM model after trip generation, trip distribution, and modal split. The traffic assignment step, which is also called route assignment or route choice , simulates the choice of route selection from a set of alternatives between the origin and the destination zones (Levinson et al., 2014). The first three FSM steps determine the number of trips produced between each zone and the proportion completed by different transportation modes. The purpose of the final step is to determine the routes or links in the study area that are likely to be used. For example, when updating a Regional Transportation Plan (RTP), traffic assignment is helpful in determining how much shift or diversion in daily traffic happens with the introduction an additional transit line or extension a highway corridor (Levinson et al., 2014). The output from the last step can provide modelers with numerous valuable results. By analyzing the results, the planner can gain insight into the strengths and weaknesses of different transportation plans. The results of trip assignment analysis can be:

• The traffic flows in the transportation system and the pattern of vehicular movements.
• The volume of traffic on network links.
• Travel costs between trip origins and destinations (O-D).
• Aggregated network metrics such as total vehicle flow, vehicle miles traveled (VMT) , and vehicle travel time (VTT).
• Zone-to-zone travel costs (travel time) for a given level of demand.
• Modeled link flows highlighting congested corridors.
• Analysis of turning movements for future intersection design.
• Determining the Origin-Destination (O-D) pairs using a specific link or path.
• Simulation of the individual choice for each pair of origins and destinations (Mathew & Rao, 2006).

Link Performance Function

Building a link performance function is one of the most important and fundamental concepts of the traffic assignment process. This function is usually used for estimating travel time, travel cost, and speed on the network based on the relationship between speed and travel flow. While this function can take different forms, such as linear, polynomial , exponential , and hyperbolic , one of the most common functions is the link performance function which represents generalized travel costs (United States Bureau of Public Roads, 1964). This equation estimates travel time on a free-flow road (travel with speed limit) adding a function that exponentially increases travel time as the road gets more congested. The road volume-to-capacity ratio can represent congestion (Meyer, 2016).

While transportation planners now recognize that intersection delays contribute to link delays, the following sections will focus on the traditional function. Equation (1) is the most common and general formula for the link performance function.

• t and x are the travel time and vehicle flow;
• t 0 is the link free flow travel time;
• k is the link capacity;
• α and β are parameters for specific type of links and calibrated using the field data. In the absence of any field data, it is usually assumed = 0.15, and β= 4.0.

α and β are the coefficients for this formula and can take different values (model parameters). However, most studies and planning practices use the same value for them. These values can be locally calibrated for the most efficient results.

Figure 13.1 demonstrates capacity as the relationship between flow and travel time. In this plot, the travel time remains constant as vehicle volumes increase until the turning point , which indicates that the link’s volume is approaching its capacity.

The following example shows how the link performance function helps us to determine the travel time according to flow and capacity.

Performance Function Example

Assume the traffic volume on a path between zone i and j was 525. The travel time recorded on this path is 15 minutes. If the capacity of this path would be 550, then calculate the new travel time for future iteration of the model.

Based on the link performance function, we have:

Now we have to plug in the numbers into the formula to determine the new travel time:

Traffic Assignment Models

Typically, traffic assignment is calculated for private cars and transit systems independently. Recall that the impedance function differs for drivers and riders, and thus simulating utility maximization behavior should be approached differently. For public transit assignment, variables such as fare, stop or transfer, waiting time, and trip times define the utility (equilibrium) (Sheffi, 1985). For private car assignment, however, in some cases, the two networks are related when public buses share highways with cars, and congestion can also affect the performance.

Typically, private car traffic assignment models the path choice of trip makers using:

• algorithms like all-or-nothing
• user equilibrium
• system optimum assignment

Of the assignment models listed above, user equilibrium is widely adopted in the U.S. (Meyer, 2016). User equilibrium relies on the premise that travelers aim to minimize their travel costs. This algorithm achieves equilibrium when no user can decrease their travel time or cost by altering their travel path.

• incremental
• capacity-restrained
• iterative feedback loop
• Stochastic user equilibrium assignment
• Dynamic traffic assignment

All-or-nothing Model

Through the all-or-nothing (AON) assignment, it is assumed that the impedance of a road or path between each origin and destination is constant and equal to the free-flow level of service. This means that the traffic time is not affected by the traffic flow on the path. The only logic behind this model is that each traveler uses the shortest path from his or her origin to the destination, and no vehicle is assigned to other paths (Hui, 2014). This method is called the all-or-nothing assignment model and is the simplest one among all assignment models. This method is also called the 0-1 assignment model, and its advantage is its simple procedure and calculation. The assumptions of this method are:

• Congestion does not affect travel time or cost, meaning that no matter how much traffic is loaded on the route, congestion does not take place.
• Since the method assigns one route to any travel between each pair of OD, all travelers traveling from a particular zone to another particular zone choose the same route (Hui, 2014).

To run the AON model, the following process can be followed:

• Step 0: Initialization. Use free flow travel costs Ca=Ca(0) , for each link a on the empty network. Ɐ
• Step 1: Path finding. Find the shortest path P for each zonal pair.
• Step 2: Path flows assigning. Assign both passenger trips (hppod) and freight trips (hfpod) in PCEs from zonal o to d to path P.
• Step 3: Link flows computing. Sum the flows on all paths going through a link as total flows of this link.

Example 2 illustrates the above-mentioned process for the AON model

All-or-nothing Example

Table 13.1 shows a trip distribution matrix with 4 zones. Using the travel costs between each pair of them shown in Figure 13.2, assign the traffic to the network. Load the vehicle trips from the trip distribution table shown below using the AON technique. After assigning the traffic, illustrate the links and the traffic volume on each on them.

Table 13.1 Trip Distribution Results.

To solve this problem, we need to find the shortest path among all alternatives for each pair of zones. The result of this procedure would be 10 routes in total, each of which bears a specific amount of travels. For instance, the shortest path between zone 1 and 2 is the straight line with 5 min travel time. All other routes like 1 to 4 to 2 or 1 to 4 to 3 to 2 would be empty from travelers going from zone 1 to zone 2. The results are shown in Table 13.2.

As you can see, some of the routes remained unused. This is because in all-or-nothing if a route has longer travel time or higher costs, then it is assumed it would not be used at all.

User Equilibrium

The next method for traffic assignment is called User Equilibrium (UE). The rule or algorithm is adapted from the well-known Wardrop equilibrium (1952) conditions (Correa & Stier-Moses, 2011). In this algorithm, it is assumed that travelers will always choose the shortest path, and equilibrium conditions are realized when no traveler is able to decrease their travel impedance by changing paths (Levinson et al., 2014).

As we discussed, the UE method is based on the first principle of Wardrop : “for each origin- destination (OD) pair, with UE, the travel time on all used paths is equal and less than or equally to the travel time that would be experienced by a single vehicle on any unused path”( Jeihani Koohbanani, 2004, p. 10). The mathematical format of this principle is shown in equation (3):

For a given OD pair, the UE condition can be expressed in equation (3):

This model assumes that all paths have equal travel time. Additionally, the model includes the following general assumptions:

• The users possess all the knowledge needed about different paths.
• The users have perfect knowledge of the path cost.
• Travel time in a route is subject to change only by the cost flow function of that route.
• Travel times increases as we load travel into the network (Mathew & Rao, 2006).

Hence, the UE assignment comes to an optimization problem that can be formulated using equation (4):

k  is the path x a equilibrium flow in link a t a  travel time on link a f k rs  flow on path  connecting OD pairs q rs  trip rate between  and δ a, k rs is constraint function defined as 1 if link a belongs to path k and 0 otherwise

Example 3 shows how the UE method can be applied for the traffic assignment step. This example is a very simple network consisting of two zones with two possible paths between them.

UE Example 

In this example, t 1 and t 2 are travel times measured by min on each route, and x 1 and x 2 are traffic flows on each route measured by (Veh/Hour).

Using the UE method, assign 4,500 Veh/Hour to the network and calculate travel time on each route after assignment, traffic volume, and system total travel time.

According to the information provided, total flow (X 1 +X 2 ) is equal to 4,500 (4.5).

First, we need to check, with all traffic assigned to one route, whether that route is still the shortest path. Thus we have:

T 1 (4.5)=23min

T 2 (0)=3min

if all traffic is assigned to route 2:

T 1 (0)=3min

T 2 (4.5)=43.5 min

Step 2: Wardrope equilibrium rule: t 1 =t 2        5+4x 1 =3+ 2x 2 2         and we have x 1 =4.5-x 2

Now the equilibrium equation can be written as: 6 + 4(4.5 − x2)=4+ x222

x 1 = 4.5 − x 2 = 1.58

Now the updated average travel times are: t 1 =5+4(1.58)=11.3min and T 2 =3+2(2.92)2=20.05min

Now the total system travel time is:

Z(x)=X 1 T 1 (X 1 )+X 2 T 2 (X 2 )=2920 veh/hr(11.32)+1585 veh/hr(20.05)=33054+31779=64833 min

System Optimum Assignment

One traffic assignment model is similar to the previous one and is called system optimum (SO). The second principle of the Wardrop defines the model’s logic. Based on this principle, drivers’ rationale for choosing a path is to minimize total system costs with one another to minimize total system travel time (Mathew & Rao, 2006). Using the SO traffic assignment, one can solve various problems, such as optimizing the departure time for a single commuting route, minimizing the total travel time from multiple origins to a single destination, or minimizing travel time in stochastic time-dependent O-D flows from several origins to a single destination ( Jeihani & Koohbanani, 2004).

One other traffic assignment model similar to the previous one is called system optimum (SO) in which the second principle of the Wardrop defines the logic of the model. Based on this principle, drivers’ rationale for choosing a path is to minimize total system costs with one another in order to minimize total system travel time (Mathew & Rao, 2006). Using the SO traffic assignment, problems like optimizing departure time for a single commuting route, minimizing total travels from multiple origins to one destination, or minimizing travel time in stochastic time-dependent OD flows from several origins to a single destination can be solved (Jeihani Koohbanani, 2004).

The basic mathematical formula for this model that satisfies the principle of the model is shown in equation (5):

In example 4, we will use the same network we described in the UE example in order to compare the results for the two models.

In that simple two-zone network, we had:

T 1 =5+4X 1    T2=3+2X 2 2

Now, based on the principle of the model we have:

Z(x)=x 1 t 1 (x 1 )+x 2 t 2 (x 2 )

Z(x)=x 1 (5+4x 1 )+x 2 (3+2x 2 2 )

Z(x)=5x 1 +4x 1 2 +3x 2 +2x 2 3

From the flow conservation. we have: x 1 +x 2 =4.5     x 1 =4.5-x 2

Z(x)=5(4.5-x 2 )+4(4.5-x 2 )2+4x 2 +x 2 3

Z(x)=x 3 2 +4x 2 2 -27x 2 +103.5

In order to minimize the above equation, we have to take derivatives and equate it to zero. After doing the calculations, we have:

Based on our finding, the system travel time would be:

T 1 =5+4*1.94=12.76min     T 2 =3+ 2(2.56)2=10.52 min

And the total travel time of the system would be:

Z(x)=X 1 T 1 (X 1 )+X 2 T 2 (X 2 )=1940 veh/hr(12.76)+2560 veh/hr(10.52)=24754+26931=51685 min

Incremental Increase model

Incremental increase is based on the logic of the AON model and models a process designed with multiple steps. In each step or level, a fraction of the total traffic volume is assigned, and travel time is calculated based on the allocated traffic volume. Through this incremental addition of traffic, the travel time of each route in step (n) is the updated travel time from the previous step (n-1) (Rojo, 2020).

The steps for the incremental increase traffic assignment model are:

• Finding the shortest path between each pair of O-Ds (Origin Destination).
• Assigning a portion of the trips according to the matrix (usually 40, 30, 20 and 10 percent to the shortest path).
• Updating the travel time after each iteration (each incremental increase).
• Continuing until all trips are assigned.
• Summing the results.

The example below illustrates the implementation process of this method.

A hypothetical network accommodates two zones with three possible links between them. Perform an incremental increase traffic assignment model for assigning 200 trips between the two zones with increments of: 30%, 30%, 20%, 20%. (The capacity is 50 trips.)

Incremental Increase Example

Step 1 (first iteration): Using the method of AON, we now assign the flow to the network using the function below:

Since the first route has the shortest travel time, the first 30% of the trips will be assigned to route 1. The updated travel time for this path would be:

And the remaining route will be empty, and thus their travel times are unchanged.

Step 2 (second iteration): Now, we can see that the second route has the shortest travel time, with 30% of the trips being assigned to this route, and the new travel time would be:

Step 3 (third iteration): In the third step, the 20% of the remaining trips will be assigned to the shortest path, which in this case is the first route again. The updated travel time for this route is:

Step 4 (fourth iteration): In the last iteration, the remaining 10% would be assigned to first route, and the time is:

Finally, we can see that route 1 has a total of 140 trips with a 8.85 travel time, the second route has a total of 60 trips with a 9.17 travel time, and the third route was never used.

Capacity Restraint Assignment

So far, all the presented algorithms or rules have considered the model’s link capacity. The flow is assigned to a link based on travel time as the only factor. In this model, after each iteration, the total number of trips is compared with the capacity to observe how much increase in travel time was realized by the added volume. In this model, the iteration stops if the added volume in step (n) does not change the travel time updated in step (n-1). With the incorporation of such a constraint, the cost or performance function would be different from the cost functions discussed in previous algorithms (Mathew & Rao, 2006). Figure 13.6 visualizes the relationship between flow and travel time with a capacity constraint.

Based on this capacity constraint specific to each link, the α, β can be readjusted for different links such as highways, freeways, and other roads.

Feedback Loop Model (Combined Traffic Assignment and Trip Distribution)

The feedback loop model defines an interaction between the trip distribution route choice step with several iterations. The model allows travelers to change their destination if a route is congested. For example, the feedback loop models that the traveler has a choice of similar destinations, such as shopping malls, in the area. In other words, in a real-world situation, travelers usually simultaneously decide about their travel characteristics (Qasim, 2012).

The chart below shows how the combination of these two modes can take place:

Equation (6), shown below for this model, ensures convergence at the end of the model is:

where C a (t) is the same as previous

P a   is total personal trip flows on link a,

f a   is total freight trip flows on link a,

T od is the total flow from node o to node d,

p od is personal trip from node o to node d,

F od is freight trip from node o to node d,

ζ is a parameter estimated from empirical data,

K is a parameter depending on the type of gravity model used to calculate T od , Evans (1976) proved that K’ equals to 1 for distribution using doubly constrained gravity model and it equals to 1 plus attractiveness for distribution using singly constrained model. Florian et al. (1975) ignored K for distribution using a doubly constrained gravity model because it is a constant.

Stochastic User Equilibrium Traffic Assignment

Stochastic user equilibrium traffic assignment is a sophisticated and more realistic model in which the level of uncertainty regarding which link should be used based on a measurement of utility function is introduced. This model performs a discrete choice analysis through a logistic model. Based on the first Wardrop principle, this model assumes that all drivers perceive the costs of traveling in each link identically and choose the route with minimum cost. In stochastic UE, however, the model allows different individuals to have different perceptions about the costs, and thus, they may choose non-minimum cost routes (Mathew & Rao, 2006). In this model, flow is assigned to all links from the beginning, unlike previous models, which is closer to reality. The probability of using each path is calculated with the following logit formula shown in equation (7):

P i is the probability of using path i

U i is the utility function for path i

In the following, an example of a simple network is presented.

Stochastic User Equilibrium Example

There is a flow of 200 trips between two points and their possible path, each of which has a travel time specified in Figure 13.7.

Using the mentioned logit formula for these paths, we have:

Based on the calculated probabilities, the distribution of the traffic flow would be:

Q 1 =175 trips

Q 2 =24 trips

Q 3 =1 trips

Dynamic Traffic Assignment

Recall the first Wardrop principle, in which travelers are believed to choose their routes with the minimum cost. Dynamic traffic assignment is based on the same rule, but the difference is that delays result from congestion. In this way, not only travelers’ route choice affects the network’s level of service, but also the network’s level of service affects travelers’ choice. However, it is not theoretically proven that an equilibrium would result under such conditions (Mathew & Rao, 2006).

Today, various algorithms are developed to solve traffic assignment problems. In any urban transportation system, travelers’ route choice and different links’ level of service have a dynamic feedback loop and affect each other simultaneously. However, a lot of these rules are not present in the models presented here. In real world cases, there can be more than thousands of nodes and links in the network, and therefore more sensitivity to dynamic changes is required for a realistic traffic assignment (Meyer, 2016). Also, the travel demand model applies a linear sequence of the four steps, which is unlike reality. Additionally, travelers may have only a limited knowledge of all possible paths, modes, and opportunities and may not make rational decisions.

In this last chapter of land-use/transportation modeling book, we reviewed the basic concepts and principles of traffic assignment models as the last step in travel demand modeling. Modeling the route choice and other components of travel behavior and demand for transportation proven to be very challenging and can incorporate multiple factors. For instance, going from AON to incremental increase assignment, we factor in the capacity and volume (and resulting delays) relationship in the assignment to make more realistic models.  Multiple-time-period assignments for multiple classes, separate specification of facilities like high-occupancy vehicle (HOV) and high-occupancy toll (HOT) lanes; and, independent transit assignment using congested highway travel times to estimate a bus ridership assignment, are some of the new extensions and variation of algorithms that take into account more realities within transportation network. A new prospect in traffic assignment models that adds several capabilities for such efforts is emergence of ITS such as data that can be collected from connected vehicles or autonomous vehicles. Using these data, perceived utility or impedances of different modes or infrastructure from individuals perspective can be modeled accurately, leading to more accurate assignment models, which are crucial planning studies such as growth and land use control efforts, environmental studies, transportation economies, etc.

Route choice is the process of choosing a certain path for a trip from a very large choice sets.

Regional Transportation Plan is long term planning document for a region’s transportation usually updated every five years.

• Vehicles (VMT) is the aggregate number of miles driven from in an area in particular time of day.
• Total vehicle travel time is the aggregate amount of time spent in transportation usually in minutes.

Link performance function is function used for estimating travel time, travel cost, and speed on the network based on the relationship between speed and travel flow.

Hyperbolic function is a function used for linear differential equations like calculating distances and angels in hyperbolic geometry.

Free-flow road is situation where vehicles can travel with the maximum allowed travel speed.

• Algorithms like all-or-nothing an assignment model where we assume that the impedance of a road or path between each origin and destination is constant and is equal to free-flow level of service, meaning that the traffic time is not affected by the traffic flow on the path.

Capacity-restrained is a model which takes into account the capacity of a road compared to volume and updates travel times.

User equilibrium is a traffic assignment model where we assume that travelers will always choose the shortest path and equilibrium condition would be realized when no traveler is able to decrease their travel impedance by changing paths.

System optimum assignment is an assignment model based on the principle that drivers’ rationale for choosing a path is to minimize total system costs with one another in order to minimize total system travel time.

• Static user-equilibrium assignment algorithm is an iterative traffic assignment process which assumes that travelers chooses the travel path with minimum travel time subject to constraints.
• Iterative feedback loop is a model that iterates between trip distribution and route choice step based on the rational that if a path gets too congested, the travel may alter travel destination.

First principle of Wardrop is the assumption that for each origin-destination (OD) pair, with UE, the travel time on all used paths is equal and less than or equally to the travel time that would be experienced by a single vehicle on any unused path.

System optimum (SO) is a condition in trip assignment model where total travel time for the whole area is at a minimum.

• Stochastic time-dependent OD is a modeling framework where generation and distribution of trips are randomly assigned to the area.

Incremental increase is AON-based model with multiple steps in each of which, a fraction of the total traffic volume is assigned, and travel time is calculated based on the allocated traffic volume.

Stochastic user equilibrium traffic assignment employs a probability distribution function that controls for uncertainties when drivers compare alternative routes and make decisions.

Dynamic traffic assignment is a model based on Wardrop first principle in which delays resulted from congestion is incorporated in the algorithm.

Key Takeaways

In this chapter, we covered:

• Traffic assignment is the last step of FSM, and the link cost function is a fundamental concept for traffic assignment.
• Different static and dynamic assignments and how to perform them using a simplistic transportation network.
• Incorporating stochastic decision-making about route choice and how to solve assignment problems with regard to this feature.

Prep/quiz/assessments

• Explain what the link performance function is in trip assignment models and how it is related to link capacity.
• Name a few static and dynamic traffic assignment models and discuss how different their rules or algorithms are.
• How does stochastic decision-making on route choice affect the transportation level of service, and how it is incorporated into traffic assignment problems?
• Name one extension of the all-or-nothing assignment model and explain how this extension improves the model results.

Correa, J.R., & Stier-Moses, N.E.(2010).Wardrope equilibria. In J.J. Cochran( Ed.), Wiley encyclopedia of operations research and management science (pp.1–12). Hoboken, NJ: John Wiley & Sons. http://dii.uchile.cl/~jcorrea/papers/Chapters/CS2010.pdf

Hui, C. (2014). Application study of all-or-nothing assignment method for determination of logistic transport route in urban planning. Computer Modelling & New Technologies , 18 , 932–937. http://www.cmnt.lv/upload-files/ns_25crt_170vr.pdf

Jeihani Koohbanani, M. (2004).  Enhancements to transportation analysis and simulation systems (Unpublished Doctoral dissertation, Virginia Tech). https://vtechworks.lib.vt.edu/bitstream/handle/10919/30092/dissertation-final.pdf?sequence=1&isAllowed=y

Levinson, D., Liu, H., Garrison, W., Hickman, M., Danczyk, A., Corbett, M., & Dixon, K. (2014). Fundamentals of transportation . Wikimedia. https://upload.wikimedia.org/wikipedia/commons/7/79/Fundamentals_of_Transportation.pdf

Mathew, T. V., & Rao, K. K. (2006). Introduction to transportation engineering. Civil engineering–Transportation engineering. IIT Bombay, NPTEL ONLINE, Http://Www. Cdeep. Iitb. Ac. in/Nptel/Civil% 20Engineering .

Meyer, M. D. (2016). Transportation planning handbook . John Wiley & Sons.

Qasim, G. (2015). Travel demand modeling: AL-Amarah city as a case study . [Unpublished Doctoral dissertation , the Engineering College University of Baghdad]

Rojo, M. (2020). Evaluation of traffic assignment models through simulation. Sustainability , 12 (14), 5536. https://doi.org/10.3390/su12145536

Sheffi, Y. (1985). Urban transportation networks: Equilibrium analysis with mathematical programming method . Prentice-Hall. http://web.mit.edu/sheffi/www/selectedMedia/sheffi_urban_trans_networks.pdf

US Bureau of Public Roads.  (1964). Traffic assignment manual for application with a large, high speed computer . U.S. Department of Commerce, Bureau of Public Roads, Office of Planning, Urban Planning Division.

https://books.google.com/books/about/Traffic_Assignment_Manual_for_Applicatio.html?id=gkNZAAAAMAAJ

Wang, X., & Hofe, R. (2008). Research methods in urban and regional planning . Springer Science & Business Media.

Vehicles (VMT) is the aggregate number of miles deriven from in an area in particular time of day.

Polynomial is distribution that involves the non-negative integer powers of a variable.

Hyperbolic function is a function that the uses the variable values as the power to the constant of e.

A point on the curve where the derivation of the function becomes either maximum or minimum.

all-or-nothing is an assignment model where we assume that the impedance of a road or path between each origin and destination is constant and is equal to free-flow level

Incremental model is a model that the predictions or estimates or fed into the model for forecasting incrementally to account for changes that may occur during each increment.

Iterative feedback loop is a model that iterates between trip distribution and route choice step based on the rational that if a path gets too congested, the travel may alter travel destination

Wardrop equilibrium is a state in traffic assignment model where are drivers are reluctant to change their path because the average travel time is at a minimum.

second principle of the Wardrop is a principle that assumes drivers’ rationale for choosing a path is to minimize total system costs with one another in order to minimize total system travel time

Stochastic time-dependent OD is a modeling framework where generation and distribution of trips are randomly assigned to the area

feedback loop model is type of dynamic traffic assignment model where an iteration between route choice and traffic assignment step is peformed, based on the assumption that if a particular route gets heavily congested, the travel may change the destination (like another shopping center).

Transportation Land-Use Modeling & Policy Copyright © by Qisheng Pan and Soheil Sharifi is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Evaluation of traffic assignment models through simulation.

1. Introduction

2. methods for traffic assignment.

• Zone division. First, the entire study area to be analyzed must be divided into zones of more or less homogeneous characteristics.
• Origin/destination matrix construction. Once the study area has been divided into zones, we need to construct a matrix that lists the movements of each origin/destination pair. We should also distinguish between light and heavy vehicles, since the conditions for the assignment of each type of vehicle are different.
• Definition of road networks. Each defined zone is represented by its centroid, which is the fictional place that generates or attracts all trips in the area. All zones must be joined to one another through the existing transportation system and, if applicable, through the future one (constituting two separate transport networks: a current one and a future one).
• Calculation of travel costs and/or times. For each of the sections of the considered network (current and future), the generalized cost of travelling along it under certain traffic conditions must be determined. It is desirable to calculate the costs for various traffic levels in order to take into account the capacity of each segment, or even do it dynamically at each iteration of the process, depending on the traffic supported by each part of the network.
• Assignment. Using the origin and destination data, it is advisable to calculate the traffic intensities for the existing network in the current situation, and then compare the results obtained with the traffic accounts that have been made. Once this check has been carried out, each of the movements in the origin/destination matrix is assigned to the current and future networks. A separate assignment should be performed for light and heavy vehicles and for various driving conditions. The application of computer tools to traffic assignment models has resulted in the creation of new methods that introduce a greater number of theoretical complications but are perfectly applicable nowadays.
• all-or-nothing assignment
• simulation methodologies
• proportion-based methodologies
• wardrop equilibrium
• speed adaptation
• incremental assignment
• the successive averages method

2.1. All-or-Nothing Assignment

2.2. stochastic assignment.

• For each segment, it is necessary to differentiate the objective costs, which are measured or estimated by an observer (modeler), from the subjective costs, which are perceived by each user. It is accepted that the target cost roughly matches the average of the subjective costs, which are randomly distributed according to a given probability function. By analyzing the literature on this topic, we can find different assumptions regarding the distribution of these subjective costs. For example, Burrel [ 53 ] adopts a uniform distribution, while other authors adopt a normal distribution.
• The distributions of costs perceived by users are independent.
• Users choose the route that minimizes their perceived travel cost: i.e., the sum of the costs of each used section.

2.3. Assignment with Congestion

• Since we do not know the optimal downward direction in global terms, we start by descending to where it looks optimal at that point.
• We continue descending until the slope starts to rise again.
• We stop at that point, look again for a new downward direction, and take it. We continue and repeat the previous step.
• We continue like this until there are no downward directions. By that time, we will have reached the bottom of the valley.

3. Case Study

• Phase A. This phase included a comprehensive background study and the collection of numerous previous traffic, mobility, urban planning, and socio-economic data.
• Phase B. This phase contained the traffic study itself, including the complete modeling of the network, the network’s simulation via a computer, and the final results and conclusions.

3.1. Zonification

• The internal area, which incorporated the areas on which the new infrastructure had a direct impact. This area included the municipalities directly affected by the new infrastructure and almost the entire Metropolitan Area of Barcelona.
• The external area, which incorporated some areas that supported penetration or crossing trips and usually contained more than one municipality. In the external area, we had large areas that modeled long-distance relationships in the corridor, such as trips from the Mediterranean area or Portugal to Europe or Girona.

3.2. Determining the Basic Origin/Destination Matrix

4. methodological application, 4.1. network model, 4.2. traffic assignment model.

• Regression analysis. A scatter plot should be developed containing the pairs of traffic volume values obtained in each section by the model (vertical axis) and by the actual observations in metering stations (horizontal axis). Above it, a regression line must be adjusted, where the slope value should be close to 1, the intercept value on the vertical axis should be close to 0 (compared with the analyzed traffic volume ranges), and the coefficient of determination R 2 should be greater than 0.7.
• Root–mean–square error ( RMSE ) index. The total set of observations should be divided into two groups: a “contrast” sample containing at least 10% of the values and a sample containing the rest of the values. For each group, the following indicator should be calculated, and the values in all cases should be less than 30%: % R M S E = 100 ∑ ( E i − O i ) 2 N − 1 ∑ O i N (7) where E i is the estimated flow by the model, O i is the observed flow in the metering station, and N is the number of observations. In our case study, the data were divided into a sample containing 29 observations (representing 28% of the total data) and a sample containing the remaining 73 observations. These proportions were used in all considered models.

5.1. All-or-Nothing Assignment

5.2. stochastic assignment with a simulation-based method, 5.3. incremental assignment, 5.4. method of successive averages (msa), 5.5. user equilibrium using the frank and wolfe algorithm, 6. discussion and conclusions.

• The all-or-nothing and stochastic assignment methods are inadmissible in this particular case study. In our study, we did not obtain the same outcome as Chen and Pan [ 69 ], who concluded that stochastic algorithms can produce similar results to the UE and MSA. This may be due to the size of the simulated network; the examples they used were small in size, while we analyzed a large area. Further research is needed to verify this point.
• The incremental assignment algorithms are valid only when they have four or more demand partitions of equal size; thus, no individual partition can exceed 25% of the total.
• The method of successive averages (MSA) algorithm is valid only when working with more than 10 iterations. This result is consistent with those of Ameli et al. [ 1 ], who found that this method is good for small- and medium-sized networks, but is not the fastest one when estimating the traffic flow in a large network.
• Finally, the user equilibrium methods (approximated by the Frank and Wolfe algorithm) were found to be valid in all of the considered cases (five or more iterations).

Conflicts of Interest

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Rojo, M. Evaluation of Traffic Assignment Models through Simulation. Sustainability 2020 , 12 , 5536. https://doi.org/10.3390/su12145536

Rojo M. Evaluation of Traffic Assignment Models through Simulation. Sustainability . 2020; 12(14):5536. https://doi.org/10.3390/su12145536

Rojo, Marta. 2020. "Evaluation of Traffic Assignment Models through Simulation" Sustainability 12, no. 14: 5536. https://doi.org/10.3390/su12145536

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Network assignment

What is Network Assignment?

Role of Network Assignment in Travel Forecasting

Overview of Methods for Traffic Assignment for Highways

All-or-nothing Assignments

Incremental assignment

Brief History of Traffic Equilibrium Concepts

Calculating Generalized Costs from Delays

Challenges for Highway Traffic Assignment

Transit Assignment

Latest Developments

Page categories

Topic Circles

Trip Based Models

More pages in this category:

# what is network assignment.

In the metropolitan transportation planning and analysis, the network assignment specifically involves estimating travelers’ route choice behavior when travel destinations and mode of travel are known. Origin-destination travel demand are assigned to a transportation network in order to estimate traffic flows and network travel conditions such as travel time. These estimated outputs from network assignment are compared against observed data such as traffic counts for model validation .

Network assignment is a mathematical problem which is solved by a solution algorithm through the use of computer. It is usually resolved as a travel cost optimization problem for each origin-destination pair on a model network. For every origin-destination pair, a path is selected that typically minimizes travel costs. The simplest kind of travel cost is travel time from beginning to end of the trip. A more complex form of travel cost, called generalized cost, may include combinations of other costs of travel such as toll cost and auto operating cost on highway networks. Transit networks may include within generalized cost weights to emphasize out-of-vehicle time and penalties to represent onerous tasks. Usually, monetary costs of travel, such as tolls and fares, are converted to time equivalent based on an estimated value of time. The shortest path is found using a path finding algorithm .

The surface transportation network can include the auto network, bus network, passenger rail network, bicycle network, pedestrian network, freight rail network, and truck network. Traditionally, passenger modes are handled separately from vehicular modes. For example, trucks and passenger cars may be assigned to the same network, but bus riders often are assigned to a separate transit network, even though buses travel over roads. Computing traffic volume on any of these networks first requires estimating network specific origin-destination demand. In metropolitan transportation planning practice in the United States, the most common network assignments employed are automobile, truck, bus, and passenger rail. Bicycle, pedestrian, and freight rail network assignments are not as frequently practiced.

# Role of Network Assignment in Travel Forecasting

The urban travel forecasting process is analyzed within the context of four decision choices:

• Personal Daily Activity
• Locations to Perform those Activities
• Mode of Travel to Activity Locations, and
• Travel Route to the Activity Locations.

Usually, these four decision choices are named as Trip Generation , Trip Distribution , Mode Choice , and Traffic Assignment. There are variations in techniques on how these travel decision choices are modeled both in practice and in research. Generalized cost, which is typically in units of time and is an output of the path-choice step of the network assignment process, is the single most important travel input to other travel decision choices, such as where to travel and by which mode. Thus, the whole urban travel forecasting process relies heavily on network assignment. Generalized cost is also a major factor in predicting socio-demographic and spatial changes. To ensure consistency in generalized cost between all travel model components in a congested network, travel cost may be fed back to the earlier steps in the model chain. Such feedback is considered “best practice” for urban regional models. Outputs from network assignment are also inputs for estimating mobile source emissions as part of a review of metropolitan area transportation plans, a requirement under the Clean Air Act Amendments of 1990 for areas not in attainment of the National Ambient Air Quality Standard.

# Overview of Methods for Traffic Assignment for Highways

This topic deals principally with an overview of static traffic assignment. The dynamic traffic assignment is discussed elsewhere.

There are a large number of traffic assignment methods, but they all have at their core a procedure called “all-or-nothing” (AON) traffic assignment. All-or-nothing traffic assignment places all trips between an origin and destination on the shortest path between that origin and destination and no trips on any other possible path (compare path finding algorithm for a step-by-step introduction). Shortest paths may be determined by a well-known algorithm by Dijkstra; however, when there are turn penalties in the network a different algorithm, called Vine building , must be used instead.

# All-or-nothing Assignments

The simplest assignment algorithm is the all-or-nothing traffic assignment. In this algorithm, flows from every origin to every destination are assigned using the path finding algorithm , and travel time remains unchanged regardless of travel volumes.

All-or-nothing traffic assignment may be used when delays are unimportant for a network. Another alternative to the user-equilibrium technique is the stochastic traffic assignment technique, which assumes variation in link level travel time.

One of the earliest, computationally efficient stochastic traffic assignment algorithms was developed by Robert Dial. [1] More recently the k-shortest paths algorithm has gained popularity.

The biggest disadvantage of the all-or-nothing assignment and the stochastic assignment is that congestion cannot be considered. In uncongested networks, these algorithms are very useful. In congested conditions, however, these algorithm miss that some travelers would change routes to avoid congestion.

# Incremental assignment

The incremental assignment method is the simplest way to (somewhat rudimentary) consider congestion. In this method, a certain share of all trips (such as half of all trips) is assigned to the network. Then, travel times are recalculated using a volume-delay function , or VDF. Next, a smaller share (such as 25% of all trips) is assigned based using the revised travel times. Using the demand of 50% + 25%, travel times are recalculated again. Next, another smaller share of trips (such as 10% of all trips) is assigned using the latest travel times.

A large benefit of the incremental assignment is model runtime. Usually, flows are assigned within 5 to 10 iterations. Most user-equilibrium assignment methods (see below) require dozens of iterations, which increases the runtime proportionally.

In the incremental assignment, the first share of trips is assigned based on free-flow conditions. Following iterations see some congestion, on only the very last trip to be assigned will consider true congestion levels. This is reasonable for lightly congested networks, as a large number of travelers could travel at free-flow speed.

The incremental assignment works unsatisfactorily in heavily congested networks, as even 50% of the travel demand may lead to congestion on selected roads. The incremental assignment will miss the fact that a portion of the 50% is likely to select different routes.

# Brief History of Traffic Equilibrium Concepts

Traffic assignment theory today largely traces its origins to a single principle of “user equilibrium” by Wardrop [2] in 1952. Wardrop’s “first” principle simply states (slightly paraphrased) that at equilibrium not a single driver may change paths without incurring a greater travel impedance . That is, any used path between an origin and destination must have a shortest travel time between the origin and destination, and all other paths must have a greater travel impedance. There may be multiple paths between an origin and destination with the same shortest travel impedance, and all of these paths may be used.

Prior to the early 1970’s there were many algorithms that attempted to solve for Wardrop’s user equilibrium on large networks. All of these algorithms failed because they either did not converge properly or they were too slow computationally. The first algorithm to be able to consistently find a correct user equilibrium on a large traffic network was conceived by a research group at Northwestern University (LeBlanc, Morlok and Pierskalla) in 1973. [3] This algorithm was called “Frank-Wolfe decomposition” after the name of a more general optimization technique that was adapted, and it found the minimum of an “objective function” that came directly from theory attributed to Beckmann from 1956. [4] The Frank-Wolfe decomposition formulation was extended to the combined distribution/assignment problem by Evans in 1974. [5]

A lack of extensibility of these algorithms to more realistic traffic assignments prompted model developers to seek more general methods of traffic assignment. A major development of the 1980s was a realization that user equilibrium traffic assignment is a “variational inequality” and not a minimization problem. [6] An algorithm called the method of successive averages (MSA) has become a popular replacement for Frank-Wolfe decomposition because of MSA’s ability to handle very complicated relations between speed and volume and to handle the combined distribution/mode-split/assignment problem. The convergence properties of MSA were proven for elementary traffic assignments by Powell and Sheffi and in 1982. [7] MSA is known to be slower on elementary traffic assignment problems than Frank-Wolfe decomposition, although MSA can solve a wider range of traffic assignment formulations allowing for greater realism.

A number of enhancements to the overall theme of Wardop’s first principle have been implemented in various software packages. These enhancements include: faster algorithms for elementary traffic assignments, stochastic multiple paths, OD table spatial disaggregation and multiple vehicle classes.

# Calculating Generalized Costs from Delays

Equilibrium traffic assignment needs a method (or series of methods) for calculating impedances (which is another term for generalized costs) on all links (and nodes) of the network, considering how those links (and nodes) were loaded with traffic. Elementary traffic assignments rely on volume-delay functions (VDFs), such as the well-known “BPR curve” (see NCHRP Report 365), [8] that expressed travel time as a function of link volume and link capacity. The 1985 US Highway Capacity Manual (and later editions through 2010) made it clear to transportation planners that delays on large portions of urban networks occur mainly at intersections, which are nodes on a network, and that the delay on any given intersection approach relates to what is happening on all other approaches. VDFs are not suitable for situations where there is conflicting and opposing traffic that affects delays. Software for implementing trip-based models are now incorporating more sophisticated delay relationships from the Highway Capacity Manual and other sources, although many MPO forecasting models still use VDFs, exclusively.

# Challenges for Highway Traffic Assignment

Numerous practical and theoretical inadequacies pertaining to Static User Equilibrium network assignment technique are reported in the literature. Among them, most widely noted concerns and challenges are:

• Inadequate network convergence;
• Continued use of legacy slow convergent network algorithm, despite availability of faster solution methods and computers;
• Non-unique route flows and link flows for multi-class assignments and for assignment on networks that include delays from opposing and conflicting traffic;
• Continued use of VDFs , when superior delay estimation techniques are available;
• Unlikeness of a steady-state network condition;
• Impractical assumption that all drivers have flawless route information and are acting without bias;
• Every driver travels at the same congested speed, no vehicle traveling on the same link overtakes another vehicle;
• Oncoming traffic does not affect traffic flows;
• Interruptions, such as accidents or inclement weather, are not represented;
• Traffic does not form queues;
• Continued use of multi-hour time periods, when finer temporal detail gives better estimates of delay and path choice.

# Transit Assignment

Most transit network assignment in implementation is allocation of known transit network specific demand based on routes, vehicle frequency, stop location, transfer point location and running times. Transit assignments are not equilibrium, but can be either all-or-nothing or stochastic. Algorithms often use complicated expressions of generalized cost which include the different effects of waiting time, transfer time, walking time (for both access and egress), riding time and fare structures. Estimated transit travel time is not directly dependent on transit passenger volume on routes and at stations (unlike estimated highway travel times, which are dependent on vehicular volumes on roads and at intersection). The possibility of many choices available to riders, such as modes of access to transit and overlaps in services between transit lines for a portion of trip segments, add further complexity to these problems.

# Latest Developments

With the increased emphasis on assessment of travel demand management strategies in the US, there have been some notable increases in the implementation of disaggregated modeling of individual travel demand behavior. Similar efforts to simulate travel route choice on dynamic transportation network have been proposed, primarily to support the much needed realistic representation of time and duration of roadway congestion. Successful examples of a shift in the network assignment paradigm to include dynamic traffic assignment on a larger network have emerged in practice. Dynamic traffic assignments are able to follow UE principles. An even newer topic is the incorporation of travel time reliability into path building.

# References

Dial , Robert Barkley, Probabilistic Assignment; a Multipath Traffic Assignment Model Which Obviates Path Enumeration, Thesis (Ph.D.), University of Washington, 1971. ↩︎

Wardrop, J. C., Some Theoretical Aspects of Road Traffic Research, Proceedings, Institution of Civil Engineers Part 2, 9, pp. 325–378. 1952. ↩︎

LeBlanc, Larry J., Morlok, Edward K., Pierskalla, William P., An Efficient Approach to Solving the Road Network Equilibrium Traffic Assignment Problem, Transportation Research 9, 1975, 9, 309–318. ↩︎

(opens new window) ) ↩︎

Evans, Suzanne P., Derivation and Analysis of Some Models for Combining Trip Distribution and Assignment, Transportation Research, Vol 10, pp 37–57 1976. ↩︎

Dafermos, S.C., Traffic Equilibrium and Variational Inequalities, Transportation Science 14, 1980, pp. 42-54. ↩︎

Powell, Warren B. and Sheffi, Yosef, The Convergence of Equilibrium Algorithms with Predetermined Step Sizes, Transportation Science, February 1, 1982, pp. 45-55. ↩︎

(opens new window) ). ↩︎

← Mode choice Dynamic Traffic Assignment →

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• DOI: 10.1007/s13676-019-00147-4
• Corpus ID: 208125471

An assignment model for public transport networks with both schedule- and frequency-based services

• M. Eltved , O. A. Nielsen , T. K. Rasmussen
• Published in EURO Journal on… 1 December 2019
• Engineering, Business, Computer Science

5 Citations

Integrated optimization of transit networks with schedule- and frequency-based services subject to the bounded stochastic user equilibrium, calculating conditional passenger travel time distributions in mixed schedule- and frequency-based public transport networks using markov chains, evaluation of traffic assignment models through simulation, sustainable urban delivery: the learning process of path costs enhanced by information and communication technologies, commuting preferences in eastern europe: case study in town of šiauliai, 24 references, optimisation of timetable-based, stochastic transit assignment models based on msa, choice models in frequency-based transit assignment, a stochastic traffic assignment model considering differences in passengers utility functions, generation and calibration of transit hyperpaths, multimodal route choice models of public transport passengers in the greater copenhagen area, finding shortest time-dependent paths in schedule-based transit networks: a label setting algorithm, transit assignment for congested public transport systems: an equilibrium model, common bus lines, assessment of schedule-based and frequency-based assignment models for strategic and operational planning of high-speed rail services, stochastic user equilibrium with a bounded choice model, related papers.

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OPTIMAL STRATEGIES: A NEW ASSIGNMENT MODEL FOR TRANSIT NETWORKS

We describe a model for the transit assignment problem with a fixed set of transit lines. The traveler chooses the strategy that allows him or her to reach his or her destination at minimum expected cost. First we consider the case in which no congestion effects occur. For the special case in which the waiting time at a stop depends only on the combined frequency, the problem is formulated as a linear programming problem of a size that increases linearly with the network size. A label-setting algorithm is developed that solves the latter problem in polynomial time. Nonlinear cost extensions of the model are considered as well.

• Find a library where document is available. Order URL: http://worldcat.org/issn/01912615

Pergamon Press, Incorporated

• Publication Date: 1989-4
• Features: Figures; References; Tables;
• Pagination: p. 83-102
• Transportation Research Part B: Methodological
• Volume: 23B
• Issue Number: 2
• Publisher: Elsevier
• ISSN: 0191-2615
• Serial URL: http://www.sciencedirect.com/science/journal/01912615

Subject/Index Terms

• TRT Terms: Algorithms ; Mathematical models ; Optimization ; Origin and destination ; Public transit ; Travel time
• Uncontrolled Terms: Models ; Transit services
• Subject Areas: Planning and Forecasting; Public Transportation;

Filing Info

• Accession Number: 00484187
• Record Type: Publication
• Files: TRIS, ATRI
• Created Date: May 31 1990 12:00AM

Dynamic Transportation Network Modeling, Analysis, Simulation, and Control (DTN-MASC)

DTN-MASC is to understand the behavior and interactions of major components/players in a multi-modal transportation network system, develop mathematical paradigms and tools to model such interactions and behavior, and design efficient solution methods. The main purpose is to develop novel management policies to manage the multi-modal transportation system more effectively and efficiently. Important applications include emergency evacuation planning and modeling, congestion pricing, urban traffic control, among others. This is particularly critical now in the era of connected / automated vehicles (CAVs), electric vehicles / buses, new mobility systems (ridehailing, ridesharing, carsharing, bike/scooter sharing) , and AI / big data analytics in transportation. The topics we are currently investigating include the network effect of new mobility services, the transit first-mile/last-mile problem using new mobility services, urban traffic control with CAVs, and optimizing the charging infrastructure of electric buses.

Below are some specific examples of DTN-MAS related research topics and results iUTS has conducted in the past.

Dynamic User Equilibrium (DUE) and applications

Our iUTS team has worked on developing dynamic user equilibrium (DUE) methods, models, and algorithms for both the continuous-time problems and the discrete-time problems. By collaborating with optimization experts and mathematicians, we were able to apply a new mathematical paradigm, called differential variational inequality (DVI) and its special form of differential complementarity systems (DCS), to model DUE to capture its two major and distinct components, i.e., (i) drivers’ choice behavior (such as departure time, route, and mode choices); and (ii) the network traffic flow dynamics. The former is usually formulated as an optimization problem or an equilibrium problem, while the latter is often formulated as an ordinary differential equation (ODE) or a partial differential equation (PDE). DVI and DCS integrates these two (mathematically very different) components into one consistent and coherent mathematical framework and gives them a formal treatment in terms of solution existence, uniqueness, stability, and solution methods. In particular, we reformulated the Vickrey-type point queue model as a DCS and applied/extended the double queue model for link level traffic flow dynamics. However, DUE problems are inherently more challenging than regular DVI and DCS, mainly due to the time-delayed terms in some of the major components (such as route choice and flow propagations). This requires new developments of the standard DVI/DCS theories, which are challenging yet very interesting problems both mathematically and practically.

Dynamic System Optimum (DSO) and Applications

Dynamic system optimum (DSO) describes and predicts the dynamic traffic network flow from the system perspective by assuming that all drivers are fully cooperative to minimize the total cost (e.g., travel time) of the system. It was widely believed in the past that DSO is less realistic than DUE, which however could provide a benchmark to develop and compare network-wide transportation management strategies such as congestion pricing. The iUTS team has developed continuous-time DSO models that integrates the link-based double queue model. In particular, we showed the solution properties and existence of free-flow DSO models in which all drivers would wait at the origins instead of in the network. Such free-flow DSO solutions are easier to compute and may play some important role in developing future dynamic traffic network management strategies especially when connected and automated vehicles (CAVs) are widely deployed. We also applied the DSO models to network-wide emission pricing and control, and the emergency evaluation planning for Lower Manhattan of the New York City.

Modeling the Network Effect of Shared Mobility

Emerging shared mobility services such as e-hailing, transportation network companies (TNCs), and ridesourcing and ridesharing, are rapidly changing the way how people travel in urban areas. There are however uncertainty regarding how these new services will impact congestion, energy use, and emissions at a network level, as well as how they will compete or complement the public transportation system. The iUTS team has worked on developing network models to capture and quantify those network effects of shared mobility services, including network-level congestion and how they may cooperate with transit to better solve the first-mile and last-mile transit problems.

Traffic Signal Control and Optimization with Connected and Automated Vehicles

The wide deployment of CAVs may profoundly change the look of urban traffic and the way how it is controlled. There are also uncertainties about how CAV technology may evolve and deployed, as well as challenges on understanding the dynamics/interactions of CAVs and human driven vehicles (HDVs). iUTS recently investigated the modeling of CAVs and HDVs on transportation networks, developed reinforcement-learning based eco-driving methods for a single CAV to save energy, and synthesized recent CAV-based urban traffic control studies and summarized the most critical gaps for future research.

By collaborating with vehicle control experts, iUTS developed CAV-based methods to optimize traffic signal timing plans by considering the driving and fuel consumption characteristics of individual vehicles. Some key features of the study include: (i) fixed cycle length so that signal coordination can be done readily for multiple intersection; (ii) different types of vehicles (such as gasoline cars and trucks, electric vehicles, buses, etc.) with their distinct fuel consumption characteristics; (iii) an optimization model that can be approximated as a dynamic programming (DP) problem, with a two-step method to guarantee the fixed-cycle-length solution.

When all vehicles are automated vehicles, the iUTS team, via collaborations with vehicle experts from Tsinghua University, worked on developing a cooperative method for the simultaneous optimization of traffic signal timing (macro level) and vehicle control (micro level), by considering two objectives: transportation efficiency and vehicle fuel consumption. We consider transportation efficiency at the macro signal timing control level and fuel economy at the micro vehicle control level. This also implies that the primary goal of the proposed method is to ensure the efficiency of all vehicles, while at the same time to minimize vehicle fuel consumption. Such consideration helps decompose the method into two interactive components, which makes the cooperative method easier to construct and solve. The results of this research received the Best Paper Award (2nd Prize) from the IEEE Intelligent Vehicles Symposium 2017. The paper is titled “V2I Based Cooperation between Traffic Signal and Approaching Automated Vehicles”, and is one of the two papers selected from over 300 papers submitted to the Symposium.

Publications

• Guo, Q * , Ban, X. , Aziz, H.M.A., 2021. Mixed traffic flow of human driven vehicles and connected/automated vehicles on a dynamic transportation network. Transportation Research Part C 128, 103159 ( https://doi.org/10.1016/j.trc.2021.103159 ); To be presented at the International Symposium on Transportation and Traffic Theory (ISTTT) 2022.
• Yang, X. , Ma, R., Yang. P., Ban, X. , 2021. Link Travel Time Estimation in Double-Queue-Based Traffic Models. Journal Promet – Traffic&Transportation , accepted.
• Guo, Q. * , Angah, O. 1 , Liu, Z. 2 , Ban, X. , 2021. Hybrid deep reinforcement learning based eco-driving for low-level connected and automated vehicles. Transportation Research Part C 124, 102980.
• Guo, Q. * , Ban, X. , 2020. Macroscopic fundamental diagram based perimeter control considering dynamic user equilibrium. Transportation Research Part B 136, 87-109.
• Li, W. 1 , Ban, X . , Zheng, J., Liu, H., Cheng, G., Li, Y., 2020. A deep learning approach for real-time traffic volume prediction at signalized intersection. Journal of Transportation Engineering , 146(8): 04020081.
• Li, W. * , Ban, X. , 2020. Connected vehicle based traffic signal coordination. Engineering 6(12), 1463-1472.
• Li, W. * , Wang, J. * , Fan, R. * , Guo, Q. * , Zhang, Y. * , Siddique, N. * , Ban, X., 2020. Short-term traffic state prediction from latent structures: accuracy vs. efficiency. Transportation Research Part C 111, 72-90.
• Ban, X. , Dessouky, M., Pang, J.S., Fan, R. * , 2019. A general equilibrium model for transportation systems with e-hailing services and flow congestion. Transportation Research Part B 129, 273-304.
• Di, X. , Ban, X. , 2019. A mixed link-node and path formulation for equilibrium of new mobility systems. Transportation Research Part B 129, 50-78.
• Wang, J.P. * , Huang, H.J. , Ban, X. , 2019. Optimal capacity allocation for high occupancy vehicle (HOV) lane in morning commute. Physica A 524, 354-361.
• Guo, Q. * , Li, L. , Ban, X. , 2019. Urban traffic signal control with connected and automated vehicles: A survey. Transportation Research Part C 101, 313-334.
• Wang, J.P.*, Ban, X. , Huang, H.J., 2019. Dynamic ridesharing with variable-ratio charging-compensation scheme for morning commute. Transportation Research Part B 122, 390-415.
• Li, W. * , Ban, X. , 2019. Connected vehicle based traffic signal timing optimization. IEEE Transactions on Intelligent Transportation Systems 20(12), 4354-4366.
• Xu, B.*, Li, S.E., Bian, Y., Li, S., Ban, X. , Wang, J., Li, K. , 2018. Distributed conflict-free cooperation for multiple connected vehicles at unsignalized intersections. Transportation Research Part C 93, 322-334 .
• Xu, B. * , Ban, X. , Bian, Y., Li, W.*, Wang, J., Li, K. , 2018. Cooperative method of traffic signal optimization and speed control of connected vehicles at isolated intersections. I EEE Transactions on Intelligent Transportation Systems 20 (4), 1390-1403.
• Di., X. , Ma, R., Liu, X., Ban, X. , Yang, H., 2018. Network design for ridesharing user equilibrium. Transportation Research Part B 112, 230-255.
• Yang, X. * , Ban, X. , Mitchell, J., 2018. Modeling multimodal transportation network emergency evacuation considering evacuees’ cooperative behavior. Transportation Research Part A 114(B), 380-397.
• Ma, R. * , Ban, X. , Pang, J.S., 2018. A link-based dynamic complementarity system formulation for continuous-time dynamic user equilibria with queue spillbacks. Transportation Science 52(3), 564-592.
• Ji. X.F.*,  Ban, X. , Zhang, J., Ran, B., 2017. Subjective-utility travel time budget modeling in the stochastic traffic network assignment.  Journal of Intelligent Transportation Systems , Accepted.
• Yang, X.*,  Ban, X. , Mitchell, J., 2017. Modeling multimodal transportation network emergency evacuation considering evacuees’ cooperative behavior.  Transportation Research, Part A , Accepted.
• Ma, R.*,  Ban, X. , Pang, J.S., 2017. A Link-Based Dynamic Complementarity System Formulation for Continuous-time Dynamic User Equilibria with Queue Spillbacks.  Transportation Science , accepted.
• Di, X., Liu, H,  Ban, X. , Yang, H., 2017. Ridersharing user equilibrium and its implications for High-Occupancy-Toll lane pricing.  Transportation Research Record , accepted.
• Ji. X.F.*,  Ban, X. , Li, M., Zhang, J., Ran, B., 2017. Non-expected route choice model under risk on stochastic traffic networks.  Networks and Spatial Economics , in press. DOI: 10.1007/s11067-017-9344-3.
• Yang, X.*,  Ban, X. , Ma, R.*, 2017. Mixed equilibria with common constraints on transportation networks.  Networks and Spatial Economics  17(2), 547-579.
• Ma, R.*,  Ban, X. , Szeto, W.Y., 2017. Emission modeling and pricing on single-destination dynamic traffic networks.  Transportation Research Part B  100, 255-283.
• Luo, L., Ge, Y., Zhang, F.,  Ban, X. , 2016. Real-Time Route Diversion Control in a Model Predictive Control Framework with Multiple Objectives: Traffic Efficiency, Emission Reduction and Fuel Economy.  Transportation Research Part D  48, 332-356.
• Di, X., Liu, H.,  Ban, X. , 2016. Second best toll pricing within the framework of bounded rationality.  Transportation Research Part B  83, 74-90.
• Zhao, J., Li, W.*, Wang, J.,  Ban, X. , 2016. Dynamic Traffic Signal Timing Optimization Strategy Incorporating Various Vehicle Fuel Consumption Characteristics.  IEEE Transactions on Vehicular Technology  65 (6), 3874-3887.
• Sánchez-Díaz, I., Holguin-Veras, J.,  Ban, X. , 2015. A time-dependent freight tour synthesis model.  Transportation Research Part B , 78, 144-168.
• Sun, Z.*,  Ban, X. , Hao, P.*, Yang, D., 2015. Trajectory-based vehicle energy/emission estimation for signalized arterials using mobile sensing data.  Transportation Research Part D  34, 27-40.
• Ma, R.*,  Ban, X. , Pang, J.S., Liu, X., 2015. Approximating time delays in solving continuous-time dynamic user equilibria.  Networks and Spatial Economics  15(3), 443-463.
• Ma, R.*,  Ban, X. , Pang, J.S., Liu, X., 2015. Time discretization of continuous-time dynamic network loading models.  Networks and Spatial Economics  15(3), 419-441.
• Yushimito, W.*,  Ban, X. , Holguin-Veras, J., 2015. Correcting the market failure in work trips with work rescheduling: an analysis using bi-level models for the firm-workers interplay,  Networks and Spatial Economics  15(3), 883-915.
• Ge, Y.E., Stewart, K., Sun, B.,  Ban, X. , Zhang, S., 2014. Investigating undesired spatial and temporal boundary effects of congestion charging.  Transportmetrica B: Dynamics , in press.
• Ma, R.*,  Ban, X. , Pang, J.S., 2014. Continuous-time dynamic system optimal for single-destination traffic networks with queue spillbacks.  Transportation Research Part B  68, 98-122.
• Yushimito, W.*,  Ban, X. , and Holguin-Veras, J., 2014. A two stage optimization model for staggered work hours.  Journal of Intelligent Transportation Systems  18(4), 410-425.
• Sanchez, I., Holguin-Veras, J., and  Ban, X. , 2014. A time-dependent freight tour synthesis model.  In Proceedings of the 93rd Annual Meeting of Transportation Research Board, Washington, DC.
• Yushimito, W.*,  Ban, X. , Holguin-Veras, J., 2013. Correcting the market failure in work trips with work rescheduling: an analysis using bi-level models for the firm-workers interplay,  Networks and Spatial Economics , in press. DOI: 10.1007/s11067-013-9213-7.
• Di, X., Liu, H.,  Ban, X ., and Yu, J.W., 2013. One the stability of a boundedly rational day to day dynamic.  Networks and Spatial Economics , in press. DOI: 10.1007/s11067-014-9233-y.
• Di, X., Liu, H., Pang, J.S., and  Ban, X. , 2013. Boundedly rational user equilibria (BRUE): Mathematical formulation and solution sets.  Transportation Research Part B  57, 300-313.
• Ban, X. , Ferris, M.C., Tang, L., and  Lu, S. , 2013. Risk-neutral second best toll pricing.  Transportation Research Part B , 48(2), 67-87.
• Ban, X. , Pang, J.S., Liu, X., and Ma, R.*, 2012. Continuous-time Point-Queue Models in Dynamic Network Loading.  Transportation Research Part B , 46(3), 360-380.
• Ban, X. ,  Pang, J.S ., Liu, X., and Ma, R.*, 2012. Modeling and Solution of Continuous-Time Instantaneous Dynamic User Equilibria: A Differential Complementarity Systems Approach.  Transportation Research Part B , 46(3), 389-408.
• Ban, X.,  Ferris, M., Liu, H., 2010. Numerical studies on reformulation techniques for continuous network design problems with asymmetric user equilibrium.  International Journal of Operations Research and Information Systems , 1(1), 52-72.
• Yushimito , W.F.,  Ban, X. , and  Holguin-Veras, J. , 2010. Staggered work hours: a bi-level model and the role of incentives. In Proceedings of the 3rd International Symposium on Dynamic Traffic Assignment.
• Ban, X. , and Liu, H., 2009. A link-node discrete-time dynamic second best toll pricing model with a relaxation solution algorithm.  Networks and Spatial Economics  9(2), 243-267.
• Ban, X.,  Liu, H.,  Ferris, M.C. , and Ran, B. ,  2008. A link-node complementarity model and solution algorithm for dynamic user equilibria with exact flow propagations.  Transportation Research, part B , 42(9), 823-842.
• Ban, X. , and Liu, H., 2007. A link-node discrete-time dynamic second best toll pricing model with a relaxation solution algorithm. Presented at  the 86th Transportation Research Board Annual Meeting  and submitted for publication .
• Ban, X. , Liu, H., and Ran, B., 2005. A link based quasi-variational inequality model for dynamic user equilibria, towards real time traffic operations. In  Proceedings of the   8th IEEE International Conference on Intelligent Transportation Systems (CD-ROM) .
• Yang, F., Liu, H., H, R.,  Ban, X , and Ran, B., 2003. Bi-level formulation for optimal traffic information dissemination.  Transportation Research Record  1836, 21-28.
• Liu, H.,  Ban, X. , Ran, B., and Mirchandani, P. ,  2003. Formulation and solution algorithm for fuzzy dynamic traffic assignment model.  Transportation Research Record  1854, 114-123.
• Liu, H. ,  Ban, X. ,  Ran, B. , and  Mirchandani , P., 2002. Analytical dynamic traffic assignment model with probabilistic network and travelers’ perceptions.  Transportation Research Record  1783, 125-133.

Notes: * indicates graduate students Dr. Ban has advised or visiting students he has supervised.

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This nifty assignment is designed to introduce students to fundamental networking concepts, such as the client—server model, sockets, and network protocols, through hands-on experience with cross-language programming and code refactoring. The assignment targets students without a prior background in computer science. By engaging students with starter code in C, Python, and Java, the assignment facilitates the understanding of protocols across different programming languages and emphasizes the importance of code reusability and refactoring. Students are tasked with extending server functionality to include custom commands and are encouraged to use AI tools for code development. This approach aims to prepare students for the evolving pedagogical landscape where AI-assisted development plays a significant role in software engineering practices.

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• Biophysical models
• Motor control
• Network models

Animals have exquisite control of their bodies, allowing them to perform a diverse range of behaviors. How such control is implemented by the brain, however, remains unclear. Advancing our understanding requires models that can relate principles of control to the structure of neural activity in behaving animals. To facilitate this, we built a ‘virtual rodent’, in which an artificial neural network actuates a biomechanically realistic model of the rat 1 in a physics simulator 2 . We used deep reinforcement learning 3–5 to train the virtual agent to imitate the behavior of freely-moving rats, thus allowing us to compare neural activity recorded in real rats to the network activity of a virtual rodent mimicking their behavior. We found that neural activity in the sensorimotor striatum and motor cortex was better predicted by the virtual rodent’s network activity than by any features of the real rat’s movements, consistent with both regions implementing inverse dynamics 6 . Furthermore, the network’s latent variability predicted the structure of neural variability across behaviors and afforded robustness in a way consistent with the minimal intervention principle of optimal feedback control 7 . These results demonstrate how physical simulation of biomechanically realistic virtual animals can help interpret the structure of neural activity across behavior and relate it to theoretical principles of motor control.

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Experiment-free exoskeleton assistance via learning in simulation

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A shifting role of thalamocortical connectivity in the emergence of cortical functional organization

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Diego Aldarondo & Josh Merel

Present address: Fauna Robotics, 900 Broadway, Suite 903, New York, NY, USA

Jesse D. Marshall

Present address: Reality Labs, Meta, 421 8th Ave, New York, NY, USA

Authors and Affiliations

Department of Organismic and Evolutionary Biology and Center for Brain Science, Harvard University, Cambridge, MA, USA

Diego Aldarondo, Jesse D. Marshall, Ugne Klibaite, Amanda Gellis & Bence P. Ölveczky

DeepMind, Google, 5 New Street Square, London, UK

Diego Aldarondo, Josh Merel, Leonard Hasenclever, Yuval Tassa, Greg Wayne & Matthew Botvinick

Gatsby Computational Neuroscience Unit, University College London, 46 Cleveland Street, London, UK

Matthew Botvinick

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Correspondence to Diego Aldarondo or Bence P. Ölveczky .

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Supplementary information.

This file contains Supplementary Discussion and Supplementary Tables 1-3.

Reporting Summary

Supplementary video 1.

Overview of the MIMIC pipeline. The MIMIC pipeline consists of multi-camera video acquisition.

Supplementary Video 2

Accurate 3D pose estimation with DANNCE. We used DANNCE to estimate the 3D pose of freely moving rats from multi-camera recordings. This video depicts the DANNCE keypoint estimates overlain atop the original video recordings from all six cameras. Keypoint estimates are accurate across a wide range of behaviors.

Supplementary Video 3

Accurate skeletal registration with STAC. We used a custom implementation of simultaneous tracking and calibration (STAC).

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Aldarondo, D., Merel, J., Marshall, J.D. et al. A virtual rodent predicts the structure of neural activity across behaviors. Nature (2024). https://doi.org/10.1038/s41586-024-07633-4

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DOI : https://doi.org/10.1038/s41586-024-07633-4

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An eight-neuron network for quadruped locomotion with hip-knee joint control

• Wang, Dongqi
• Qu, shaoxing

The gait generator, which is capable of producing rhythmic signals for coordinating multiple joints, is an essential component in the quadruped robot locomotion control framework. The biological counterpart of the gait generator is the Central Pattern Generator (abbreviated as CPG), a small neural network consisting of interacting neurons. Inspired by this architecture, researchers have designed artificial neural networks composed of simulated neurons or oscillator equations. Despite the widespread application of these designed CPGs in various robot locomotion controls, some issues remain unaddressed, including: (1) Simplistic network designs often overlook the symmetry between signal and network structure, resulting in fewer gait patterns than those found in nature. (2) Due to minimal architectural consideration, quadruped control CPGs typically consist of only four neurons, which restricts the network's direct control to leg phases rather than joint coordination. (3) Gait changes are achieved by varying the neuron couplings or the assignment between neurons and legs, rather than through external stimulation. We apply symmetry theory to design an eight-neuron network, composed of Stein neuronal models, capable of achieving five gaits and coordinated control of the hip-knee joints. We validate the signal stability of this network as a gait generator through numerical simulations, which reveal various results and patterns encountered during gait transitions using neuronal stimulation. Based on these findings, we have developed several successful gait transition strategies through neuronal stimulations. Using a commercial quadruped robot model, we demonstrate the usability and feasibility of this network by implementing motion control and gait transitions.

• Computer Science - Robotics

Models for Traffic Assignment to Transportation Networks

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• Ennio Cascetta 3  

Part of the book series: Applied Optimization ((APOP,volume 49))

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Models for traffic assignment to transportation networks simulate how demand and supply interact in transportation systems. These models allow the calculation of performance measures and user flows for each supply element (network link), resulting from origin-destination demand flows, path choice behavior, and the reciprocal interactions between supply and demand. Assignment models combine the supply and demand models described in the previous chapters; for this reason they are also referred to as demand-supply interaction models. In fact, as seen in Chapter 4, path choices and flows depend on path generalized costs, futhermore demand flows are generally influenced by path costs in choice dimensions such as mode and destination. Also, as seen in Chapter 2, link and path performance measures and costs may depend on flows due to congestion. There is therefore a circular dependence between demand, flows, and costs, which is represented in assignment models as can be seen in Fig. 5.1.1.

Giulio Erberto Cantarella is co-author of this Chapter.

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Cascetta, E. (2001). Models for Traffic Assignment to Transportation Networks. In: Transportation Systems Engineering: Theory and Methods. Applied Optimization, vol 49. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6873-2_5

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2024 Travelers Championship odds, picks, field: Surprising predictions by golf model that's called 13 majors

Sportsline's proven model simulated the travelers championship 2024 10,000 times and revealed its surprising pga tour golf picks.

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