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Part III: Travel Demand Modeling

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 landuse/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 deriven 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.

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).

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3.6: 3-6 Route Choice

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Route assignment , route choice , or traffic assignment concerns the selection of routes (alternative called paths) between origins and destinations in transportation networks. It is the fourth step in the conventional transportation forecasting model, following Trip Generation, Destination Choice, and Mode Choice. The zonal interchange analysis of trip distribution provides origin-destination trip tables. Mode choice analysis tells which travelers will use which mode. To determine facility needs and costs and benefits, we need to know the number of travelers on each route and link of the network (a route is simply a chain of links between an origin and destination). We need to undertake traffic (or trip) assignment. Suppose there is a network of highways and transit systems and a proposed addition. We first want to know the present pattern of travel times and flows and then what would happen if the addition were made.

Link Performance Function

The cost that a driver imposes on others is called the marginal cost. However, when making decisions, a driver only faces his own cost (the average cost) and ignores any costs imposed on others (the marginal cost).

• \[AverageCost=\dfrac{S_T}{Q}\]
• \[MarginalCost=\dfrac{\delta S_T}{\delta Q}\]

where \(S_T\) is the total cost, and \(Q\) is the flow.

BPR Link Performance Function

Suppose we are considering a highway network. For each link there is a function stating the relationship between resistance and volume of traffic. The Bureau of Public Roads (BPR) developed a link (arc) congestion (or volume-delay, or link performance) function, which we will term S a (Q a )

\[S_a(Q_a)=t_a(1+0.15\dfrac ({Q_a}{c_a})^4)\]

t a = free-flow travel time on link a per unit of time

Q a = flow (or volume) of traffic on link a per unit of time (somewhat more accurately: flow attempting to use link a )

c a = capacity of link a per unit of time

S a (Q a ) is the average travel time for a vehicle on link a

There are other congestion functions. The CATS has long used a function different from that used by the BPR, but there seems to be little difference between results when the CATS and BPR functions are compared.

Can Flow Exceed Capacity?

On a link, the capacity is thought of as “outflow.” Demand is inflow.

If inflow > outflow for a period of time, there is queueing (and delay).

For Example, for a 1 hour period, if 2100 cars arrive and 2000 depart, 100 are still there. The link performance function tries to represent that phenomenon in a simple way.

Wardrop's Principles of Equilibrium

User Equilibrium

Each user acts to minimize his/her own cost, subject to every other user doing the same. Travel times are equal on all used routes and lower than on any unused route.

• System optimal

Each user acts to minimize the total travel time on the system.

Price of Anarchy

The reason we have congestion is that people are selfish. The cost of that selfishness (when people behave according to their own interest rather than society's) is the price of anarchy .

The ratio of system-wide travel time under User Equilibrium and System Optimal conditions.

For a two-link network with linear link performance functions (latency functions), Price of Anarchy is < 4/3.

Is this too much? Should something be done, or is 33% waste acceptable? [The loss may be larger/smaller in other cases, under different assumptions, etc.]

Conservation of Flow

An important factor in road assignment is the conservation of flow. This means that the number of vehicles entering the intersection (link segment) equals the number of vehicles exiting the intersection for a given period of time (except for sources and sinks).

Similarly, the number of vehicles entering the back of the link equals the number exiting the front (over a long period of time).

Auto assignment

Long-standing techniques.

The above examples are adequate for a problem of two links, however real networks are much more complicated. The problem of estimating how many users are on each route is long standing. Planners started looking hard at it as freeways and expressways (motorways) began to be developed. The freeway offered a superior level of service over the local street system and diverted traffic from the local system. At first, diversion was the technique. Ratios of travel time were used, tempered by considerations of costs, comfort, and level of service.

The Chicago Area Transportation Study (CATS) researchers developed diversion curves for freeways versus local streets. There was much work in California also, for California had early experiences with freeway planning. In addition to work of a diversion sort, the CATS attacked some technical problems that arise when one works with complex networks. One result was the Moore algorithm for finding shortest paths on networks.

The issue the diversion approach didn’t handle was the feedback from the quantity of traffic on links and routes. If a lot of vehicles try to use a facility, the facility becomes congested and travel time increases. Absent some way to consider feedback, early planning studies (actually, most in the period 1960-1975) ignored feedback. They used the Moore algorithm to determine shortest paths and assigned all traffic to shortest paths. That’s called all or nothing assignment because either all of the traffic from i to j moves along a route or it does not.

The all-or-nothing or shortest path assignment is not trivial from a technical-computational view. Each traffic zone is connected to n - 1 zones, so there are numerous paths to be considered. In addition, we are ultimately interested in traffic on links. A link may be a part of several paths, and traffic along paths has to be summed link by link.

An argument can be made favoring the all-or-nothing approach. It goes this way: The planning study is to support investments so that a good level of service is available on all links. Using the travel times associated with the planned level of service, calculations indicate how traffic will flow once improvements are in place. Knowing the quantities of traffic on links, the capacity to be supplied to meet the desired level of service can be calculated.

Heuristic procedures

To take account of the affect of traffic loading on travel times and traffic equilibria, several heuristic calculation procedures were developed. One heuristic proceeds incrementally. The traffic to be assigned is divided into parts (usually 4). Assign the first part of the traffic. Compute new travel times and assign the next part of the traffic. The last step is repeated until all the traffic is assigned. The CATS used a variation on this; it assigned row by row in the O-D table.

The heuristic included in the FHWA collection of computer programs proceeds another way.

• Step 0: Start by loading all traffic using an all or nothing procedure.
• Step 1: Compute the resulting travel times and reassign traffic.
• Step 2: Now, begin to reassign using weights. Compute the weighted travel times in the previous two loadings and use those for the next assignment. The latest iteration gets a weight of 0.25 and the previous gets a weight of 0.75.
• Step 3. Continue.

These procedures seem to work “pretty well,” but they are not exact.

Frank-Wolfe algorithm

Dafermos (1968) applied the Frank-Wolfe algorithm (1956, Florian 1976), which can be used to deal with the traffic equilibrium problem.

Equilibrium Assignment

To assign traffic to paths and links we have to have rules, and there are the well-known Wardrop equilibrium (1952) conditions. The essence of these is that travelers will strive to find the shortest (least resistance) path from origin to destination, and network equilibrium occurs when no traveler can decrease travel effort by shifting to a new path. These are termed user optimal conditions, for no user will gain from changing travel paths once the system is in equilibrium.

The user optimum equilibrium can be found by solving the following nonlinear programming problem

\[min \displaystyle \sum_{a} \displaystyle\int\limits_{0}^{v_a}S_a(Q_a)\, dx\]

subject to:

\[Q_a=\displaystyle\sum_{i}\displaystyle\sum_{j}\displaystyle\sum_{r}\alpha_{ij}^{ar}Q_{ij}^r\]

\[sum_{r}Q_{ij}^r=Q_{ij}\]

\[Q_a\ge 0, Q_{ij}^r\ge 0\]

where \(Q_{ij}^r\) is the number of vehicles on path r from origin i to destination j . So constraint (2) says that all travel must take place: i = 1 ... n; j = 1 ... n

\(\alpha_{ij}^{ar}\)= 1 if link a is on path r from i to j ; zero otherwise.

So constraint (1) sums traffic on each link. There is a constraint for each link on the network. Constraint (3) assures no negative traffic.

Transit assignment

There are also methods that have been developed to assign passengers to transit vehicles. In an effort to increase the accuracy of transit assignment estimates, a number of assumptions are generally made. Examples of these include the following:

• All transit trips are run on a set and predefined schedule that is known or readily available to the users.
• There is a fixed capacity associated with the transit service (car/trolley/bus capacity).

Solve for the flows on Links a and b in the Simple Network of two parallel links just shown if the link performance function on link a :

\(S_a=5+2*Q_a\)

and the function on link b :

\(S_b=10+Q_b\)

where total flow between the origin and destination is 1000 trips.

Time (Cost) is equal on all used routes so \(S_a=S_b\)

And we have Conservation of flow so, \(Q_a+Q_b=Q_o=Q_d=1000\)

\(5+2*(1000-Q_b)=10+Q_b\)

\(1995=3Q_b\)

\(Q_b=665;Q_a=335\)

An example from Eash, Janson, and Boyce (1979) will illustrate the solution to the nonlinear program problem. There are two links from node 1 to node 2, and there is a resistance function for each link (see Figure 1). Areas under the curves in Figure 2 correspond to the integration from 0 to a in equation 1, they sum to 220,674. Note that the function for link b is plotted in the reverse direction.

\(S_a=15(1+0.15(\dfrac{Q_a}{1000})^4)\)

\(S_b=20(1+0.15(\dfrac{Q_a}{3000})^4)\)

\(Q_a+Q_b=8000\)

Show graphically the equilibrium result.

At equilibrium there are 2,152 vehicles on link a and 5,847 on link b . Travel time is the same on each route: about 63.

Figure 3 illustrates an allocation of vehicles that is not consistent with the equilibrium solution. The curves are unchanged, but with the new allocation of vehicles to routes the shaded area has to be included in the solution, so the Figure 3 solution is larger than the solution in Figure 2 by the area of the shaded area.

Assume the traffic flow from Milwaukee to Chicago, is 15000 vehicles per hour. The flow is divided between two parallel facilities, a freeway and an arterial. Flow on the freeway is denoted \(Q_f\), and flow on the two-lane arterial is denoted \(Q_a\).

The travel time (in minutes) on the freeway (\(C_f\)) is given by:

\(C_f=10+Q_f/1500\)

\(C_a=15+Q_a/1000\)

Apply Wardrop's User Equilibrium Principle, and determine the flow and travel time on both routes.

The travel times are set equal to one another

\(C_f=C_a\)

\(10+Q_f/1500=15+Q_a/1000\)

The total traffic flow is equal to 15000

\(Q_f+Q_a=15000\)

\(Q_a=15000-Q_f\)

\(10+Q_f/1500=15+(15000-Q_f)/1000\)

Solve for \(Q_f\)

\(Q_f=60000/5=12000\)

\(Q_a=15000-Q_f=3000\)

Thought Questions

• How can we get drivers to consider their marginal cost?
• Alternatively: How can we get drivers to behave in a “System Optimal” way?

Sample Problems

Given a flow of six (6) units from origin “o” to destination “r”. Flow on each route ab is designated with Qab in the Time Function. Apply Wardrop's Network Equilibrium Principle (Users Equalize Travel Times on all used routes)

A. What is the flow and travel time on each link? (complete the table below) for Network A

Link Attributes

B. What is the system optimal assignment?

C. What is the Price of Anarchy?

What is the flow and travel time on each link? Complete the table below for Network A:

These four links are really 2 links O-P-R and O-Q-R, because by conservation of flow Qop = Qpr and Qoq = Qqr.

By Wardrop's Equilibrium Principle, the travel time (cost) on each used route must be equal. Therefore \(C_{opr}=C_{oqr}\)

OR \(25+6*Q_{opr}=20+7*Q_{oqr}\)

\(5+6*Q_{opr}=7*Q_{oqr}\)

\(Q_{oqr}=5/7+6*Q_{opr}/7\)

By the conservation of flow principle

\(Q_{oqr}+Q_{opr}=6\)

\(Q_{opr}=6-Q_{oqr}\)

By substitution

\Q_{oqr}=5/7+6/7(6-Q_{oqr})=41/7-6*Q_{oqr}/7\)

\(13*Q_{oqr}=41\)

\(Q_{oqr}=41/13=3.15\)

\(Q_{opr}=2.84\)

\(42.01=25+6(2.84)\)

\(42.05=20+7(3.15)\)

Check (within rounding error)

or expanding back to the original table:

User Equilibrium: Total Delay = 42.01 * 6 = 252.06

What is the system optimal assignment?

Conservation of Flow:

\(Q_{opr}+Q_{oqr}=6\)

\(TotalDelay=Q_{opr}(25+6*Q_{oqr})+Q_{oqr}(20+7*Q_{oqr})\)

\(25Q_{opr}+6Q_{opr}^2+(6_Q_{opr})(20+7(6-Q_{opr}))\)

\(25Q_{opr}+6Q_{opr}^2+(6_Q_{opr})(62-7Q_{opr}))\)

\(25Q_{opr}+6Q_{opr}^2+372-62Q_{opr}-42Q_{opr}+7Q_{opr}^2\)

\(13Q_{opr}^2-79Q_{opr}+372\)

Analytic Solution requires minimizing total delay

\(\deltaC/\deltaQ=26Q_{opr}-79=0\)

\(Q_{opr}=79/26-3.04\)

\(Q_{oqr}=6-Q_{opr}=2.96\)

And we can compute the SO travel times on each path

\(C_{opr,SO}=25+6*3.04=43.24\)

\(C_{opr,SO}=20+7*2.96=40.72\)

Note that unlike the UE solution, \(C_{opr,SO}\g C_{oqr,SO}\)

Total Delay = 3.04(25+ 6*3.04) + 2.96(20+7*2.96) = 131.45+120.53= 251.98

Note: one could also use software such as a "Solver" algorithm to find this solution.

What is the Price of Anarchy?

User Equilibrium: Total Delay =252.06 System Optimal: Total Delay = 251.98

Price of Anarchy = 252.06/251.98 = 1.0003 < 4/3

The Marcytown - Rivertown corridor was served by 3 bridges, according to the attached map. The bridge over the River on the route directly connecting Marcytown and Citytown collapsed, leaving two alternatives, via Donkeytown and a direct. Assume the travel time functions Cij in minutes, Qij in vehicles/hour, on the five links routes are as given.

Marcytown - Rivertown Cmr = 5 + Qmr/1000

Marcytown - Citytown (prior to collapse) Cmc = 5 + Qmc/1000

Marcytown - Citytown (after collapse) Cmr = ∞

Citytown - Rivertown Ccr = 1 + Qcr/500

Marcytown - Donkeytown Cmd = 7 + Qmd/500

Donkeytown - Rivertown Cdr = 9 + Qdr/1000

Also assume there are 10000 vehicles per hour that want to make the trip. If travelers behave according to Wardrops user equilibrium principle.

A) Prior to the collapse, how many vehicles used each route?

Route A (Marcytown-Rivertown) = Ca = 5 + Qa/1000

Route B (Marcytown-Citytown-Rivertown) = Cb = 5 + Qb/1000 + 1 + Qb/500 = 6 + 3Qb/1000

Route C (Marcytown-Donkeytown-Rivertown)= Cc = 7 + Qc/500 + 9 + Qc/1000 = 16 + 3Qc/1000

At equilibrium the travel time on all three used routes will be the same: Ca = Cb = Cc

We also know that Qa + Qb + Qc = 10000

Solving the above set of equations will provide the following results:

Qa = 8467;Qb = 2267;Qc = −867

We know that flow cannot be negative. By looking at the travel time equations we can see a pattern.

Even with a flow of 0 vehicles the travel time on route C(16 minutes) is higher than A or B. This indicates that vehicles will choose route A or B and we can ignore Route C.

Solving the following equations:

Route A (Marcytown-Rivertown) = Ca = 5 + Qa /1000

Route B (Marcytown-Citytown-Rivertown) = Cb = 6 + 3Qb /1000

Qa + Qb = 10000

We can the following values:

Qa = 7750; Qb = 2250; Qc = 0

B) After the collapse, how many vehicles used each route?

We now have only two routes, route A and C since Route B is no longer possible. We could solve the following equations:

Route C (Marcytown- Donkeytown-Rivertown) = Cc = 16 + 3Qc /1000

Qa+ Qc= 10000

But we know from above table that Route C is going to be more expensive in terms of travel time even with zero vehicles using that route. We can therefore assume that Route A is the only option and allocate all the 10,000 vehicles to Route A.

If we actually solve the problem using the above set of equations, you will get the following results:

Qa = 10250; Qc = -250

which again indicates that route C is not an option since flow cannot be negative.

C) After the collapse, public officials want to reduce inefficiencies in the system, how many vehicles would have to be shifted between routes? What is the “price of anarchy” in this case?

TotalDelayUE =(15)(10,000)=150,000

System Optimal

TotalDelaySO =(Qa)(5+Qa/1000)+(Qc)(16+3Qc/1000)

Using Qa + Qc = 10,000

TotalDelaySO =(Qa2)/250−71Qa+460000

Minimize total delay ∂((Qa2)/250 − 71Qa + 460000)/∂Qa = 0

Qa/125−7 → Qa = 8875 Qc = 1125 Ca = 13,875 Cc = 19,375

TotalDelaySO =144938

Price of Anarchy = 150,000/144,938 = 1.035

• \(C_T\) - total cost
• \(C_k\) - travel cost on link \(k\)
• \(Q_k\) - flow (volume) on link \(k\)

Abbreviations

• VDF - Volume Delay Function
• LPF - Link Performance Function
• BPR - Bureau of Public Roads
• UE - User Equilbrium
• SO - System Optimal
• DTA - Dynamic Traffic Assignment
• DUE - Deterministic User Equilibrium
• SUE - Stochastic User Equilibrium
• AC - Average Cost
• MC - Marginal Cost
• Route assignment, route choice, auto assignment
• Volume-delay function, link performance function
• User equilibrium
• Conservation of flow
• Average cost
• Marginal cost

External Exercises

Use the ADAM software at the STREET website and try Assignment #3 to learn how changes in network characteristics impact route choice.

Additional Questions

1. If trip distribution depends on travel times, and travel times depend on the trip table (resulting from trip distribution) that is assigned to the road network, how do we solve this problem (conceptually)?

2. Do drivers behave in a system optimal or a user optimal way? How can you get them to move from one to the other.

3. Identify a mechanism that can ensure the system optimal outcome is achieved in route assignment, rather than the user equilibrium. Why would we want such an outcome? What are the drawbacks to the mechanism you identified?

4. Assume the flow from Dakotopolis to New Fargo, is 5300 vehicles per hour. The flow is divided between two parallel facilities, a freeway and an arterial. Flow on the freeway is denoted \(Q_f\), and flow on the two-lane arterial is denoted \(Q_r\). The travel time on the freeway \(C_f\) is given by:

\(C_f=5+Q_f/1000\)

The travel time on the arterial (Cr) is given by

\(C_r=7+Q_r/500\)

(a) Apply Wardrop's User Equilibrium Principle, and determine the flow and travel time on both routes from Dakotopolis to New Fargo.

(b) Solve for the System Optimal Solution and determine the flow and travel time on both routes.

5. Given a flow of 10,000 vehicles from origin to destination traveling on three parallel routes. Flow on each route A, B, or C is designated with \(Q_a\), \(Q_b\), \(Q_c\) in the Time Function Respectively. Apply Wardrop's Network Equilibrium Principle (Users Equalize Travel Times on all used routes), and determine the flow on each route.

\(T_A=500+20Q_A\)

\(T_B=1000+10Q_B\)

\(T_C=2000+30Q_C\)

• How does average cost differ from marginal cost?
• How do System Optimal and User Equilibrium travel time differ?
• Why do we want people to behave in an SO way?
• How can you get people to behave in an SO way?
• Who was John Glen Wardrop?
• What are Wardrop’s Two Principles?
• What does conservation of flow require in route assignment?
• Can Variable Message Signs be used to encourage System Optimal behavior?
• What is freeflow travel time?
• If a problem has more than two routes, where does the extra equation come from?
• How can you determine if a route is unused?
• What is the difference between capacity and flow
• Draw a typical volume-delay function for a deterministic, static user equilibrium assignment.
• Can Q be negative?
• What is route assignment?
• Is it important that the output travel times from route choice be consistent with the input travel times for destination choice and mode choice? Why?

Travel Demand Forecasting: Parameters and Techniques (2012)

Chapter: chapter 1 - introduction.

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

1 1.1 Background In 1978, the Transportation Research Board (TRB) published NCHRP Report 187: Quick-Response Urban Travel Estimation Techniques and Transferable Parameters (Sosslau et al., 1978). This report described default parameters, factors, and manual techniques for doing planning analysis. The report and its default data were used widely by the transportation planning profession for almost 20 years. In 1998, drawing on several newer data sources, including the 1990 Census and Nation- wide Personal Transportation Survey, an update to NCHRP Report 187 was published in the form of NCHRP Report 365: Travel Estimation Techniques for Urban Planning (Martin and McGuckin, 1998). Since NCHRP Report 365 was published, significant changes have occurred affecting the complexity, scope, and context of transportation planning. Transportation planning tools have evolved and proliferated, enabling improved and more flexible analyses to support decisions. The demands on trans- portation planning have expanded into special populations and broader issues (e.g., safety, congestion, pricing, air quality, environment, climate change, and freight). In addition, the default data and parameters in NCHRP Report 365 need to be updated to reflect the planning requirements of today and the next 10 years. The objective of this report is to revise and update NCHRP Report 365 to reflect current travel characteristics and to pro- vide guidance on travel demand forecasting procedures and their application for solving common transportation problems. It is written for “modeling practitioners,” who are the public agency and private-sector planners with responsibility for devel- oping, overseeing the development of, evaluating, validating, and implementing travel demand models. This updated report includes the optional use of default parameters and appropriate references to other more sophisticated techniques. The report is intended to allow practitioners to use travel demand fore- casting methods to address the full range of transportation planning issues (e.g., environmental, air quality, freight, multimodal, and other critical concerns). One of the features of this report is the provision of trans- ferable parameters for use when locally specific data are not available for use in model estimation. The parameters pre- sented in this report are also useful to practitioners who are modeling urban areas that have local data but wish to check the reasonableness of model parameters estimated from such data. Additionally, key travel measures, such as average travel times by trip purpose, are provided for use in checking model results. Both the transferable parameters and the travel measures come from two main sources: the 2009 National Household Travel Survey (NHTS) and a database of model documentation for 69 metropolitan planning organizations (MPOs) assembled for the development of this report. There are two primary ways in which planners can make use of this information: 1. Using transferable parameters in the development of travel model components when local data suitable for model development are insufficient or unavailable; and 2. Checking the reasonableness of model outputs. This report is written at a time of exciting change in the field of travel demand forecasting. The four-step modeling process that has been the paradigm for decades is no longer the only approach used in urban area modeling. Tour- and activity-based models have been and are being developed in several urban areas, including a sizable percentage of the largest areas in the United States. This change has the potential to significantly improve the accuracy and analytical capability of travel demand models. At the same time, the four-step process will continue to be used for many years, especially in the smaller- and medium- sized urban areas for which this report will remain a valuable resource. With that in mind, this report provides information on parameters and modeling techniques consistent with the C h a p t e r 1 Introduction

2four-step process and Chapter 4, which contains the key information on parameters and techniques, is organized con- sistent with the four-step approach. Chapter 6 of this report presents information relevant to advanced modeling practices, including activity-based models and traffic simulation. This report is organized as follows: • Chapter 1—Introduction; • Chapter 2—Planning Applications Context; • Chapter 3—Data Needed for Modeling; • Chapter 4—Model Components: – Vehicle Availability, – Trip Generation, – Trip Distribution, – External Travel, – Mode Choice, – Automobile Occupancy, – Time-of-Day, – Freight/Truck Modeling, – Highway Assignment, and – Transit Assignment; • Chapter 5—Model Validation and Reasonableness Checking; • Chapter 6—Emerging Modeling Practices; and • Chapter 7—Case Studies. This report is not intended to be a comprehensive primer for persons developing a travel model. For more complete information on model development, readers may wish to consult the following sources: • “Introduction to Urban Travel Demand Forecasting” (Federal Highway Administration, 2008); • “Introduction to Travel Demand Forecasting Self- Instructional CD-ROM” (Federal Highway Administra- tion, 2002); • NCHRP Report 365: Travel Estimation Techniques for Urban Planning (Martin and McGuckin, 1998); • An Introduction to Urban Travel Demand Forecasting— A Self-Instructional Text (Federal Highway Administration and Urban Mass Transit Administration, 1977); • FSUTMS Comprehensive Modeling Online Training Workshop (http://www.fsutmsonline.net/online_training/ index.html#w1l3e3); and • Modeling Transport (Ortuzar and Willumsen, 2001). 1.2 Travel Demand Forecasting: Trends and Issues While there are other methods used to estimate travel demand in urban areas, travel demand forecasting and mod- eling remain important tools in the analysis of transportation plans, projects, and policies. Modeling results are useful to those making transportation decisions (and analysts assisting in the decision-making process) in system and facility design and operations and to those developing transportation policy. NCHRP Report 365 (Martin and McGuckin, 1998) pro- vides a brief history of travel demand forecasting through its publication year of 1998; notably, the evolution of the use of models from the evaluation of long-range plans and major transportation investments to a variety of ongoing, every- day transportation planning analyses. Since the publication of NCHRP Report 365, several areas have experienced rapid advances in travel modeling: • The four-step modeling process has seen a number of enhancements. These include the more widespread incor- poration of time-of-day modeling into what had been a process for modeling entire average weekdays; common use of supplementary model steps, such as vehicle availability models; the inclusion of nonmotorized travel in models; and enhancements to procedures for the four main model components (e.g., the use of logit destination choice models for trip distribution). • Data collection techniques have advanced, particularly in the use of new technology such as global positioning systems (GPS) as well as improvements to procedures for performing household travel and transit rider surveys and traffic counts. • A new generation of travel demand modeling software has been developed, which not only takes advantage of modern computing environments but also includes, to various degrees, integration with geographic information systems (GIS). • There has been an increased use of integrated land use- transportation models, in contrast to the use of static land use allocation models. • Tour- and activity-based modeling has been introduced and implemented. • Increasingly, travel demand models have been more directly integrated with traffic simulation models. Most travel demand modeling software vendors have developed traffic simulation packages. At the same time, new transportation planning require- ments have contributed to a number of new uses for models, including: • The analysis of a variety of road pricing options, including toll roads, high-occupancy toll (HOT) lanes, cordon pricing, and congestion pricing that varies by time of day; • The Federal Transit Administration’s (FTA’s) user benefits measure for the Section 5309 New Starts program of transit projects, which has led to an increased awareness of model properties that can inadvertently affect ridership forecasts;

3 • The evaluation of alternative land use patterns and their effects on travel demand; and • The need to evaluate (1) the impacts of climate change on transportation supply and demand, (2) the effects of travel on climate and the environment, and (3) energy and air quality impacts. These types of analyses are in addition to several traditional types of analyses for which travel models are still regularly used: • Development of long-range transportation plans; • Highway and transit project evaluation; • Air quality conformity (recently including greenhouse gas emissions analysis); and • Site impact studies for developments. 1.3 Overview of the Four-Step Travel Modeling Process The methods presented in this report follow the conven- tional sequential process for estimating transportation demand that is often called the “four-step” process: • Step 1—Trip Generation (discussed in Section 4.4), • Step 2—Trip Distribution (discussed in Section 4.5), • Step 3—Mode Choice (discussed in Section 4.7), and • Step 4—Assignment (discussed in Sections 4.11 and 4.12). There are other components commonly included in the four-step process, as shown in Figure 1.1 and described in the following paragraphs. The serial nature of the process is not meant to imply that the decisions made by travelers are actually made sequentially rather than simultaneously, nor that the decisions are made in exactly the order implied by the four-step process. For example, the decision of the destination for the trip may follow or be made simultaneously with the choice of mode. Nor is the four-step process meant to imply that the decisions for each trip are made independently of the decisions for other trips. For example, the choice of a mode for a given trip may depend on the choice of mode in the preceding trip. In four-step travel models, the unit of travel is the “trip,” defined as a person or vehicle traveling from an origin to a destination with no intermediate stops. Since people traveling for different reasons behave differently, four-step models segment trips by trip purpose. The number and definition of trip purposes in a model depend on the types of information the model needs to provide for planning analyses, the char- acteristics of the region being modeled, and the availability of data with which to obtain model parameters and the inputs to the model. The minimum number of trip purposes in most models is three: home-based work, home-based nonwork, and nonhome based. In this report, these three trip purposes are referred to as the “classic three” purposes. The purpose of trip generation is to estimate the num- ber of trips of each type that begin or end in each location, based on the amount of activity in an analysis area. In most models, trips are aggregated to a specific unit of geography (e.g., a traffic analysis zone). The estimated number of daily trips will be in the flow unit that is used by the model, which is usually one of the following: vehicle trips; person trips in motorized modes (auto and transit); or person trips by all modes, including both motorized and nonmotorized (walking, bicycling) modes. Trip generation models require some explanatory variables that are related to trip-making behavior and some functions that estimate the number of trips based on these explanatory variables. Typical variables include the number of households classified by characteristics such as number of persons, number of workers, vehicle availability, income level, and employment by type. The output of trip generation is trip productions and attractions by traffic analysis zone and by purpose. Trip distribution addresses the question of how many trips travel between units of geography (e.g., traffic analysis zones). In effect, it links the trip productions and attractions from the trip generation step. Trip distribution requires explanatory variables that are related to the cost (including time) of travel between zones, as well as the amount of trip-making activity in both the origin zone and the destination zone. The outputs of trip distribution are production-attraction zonal trip tables by purpose. Models of external travel estimate the trips that originate or are destined outside the model’s geographic region (the model area). These models include elements of trip generation and distribution, and so the outputs are trip tables represent- ing external travel. Mode choice is the third step in the four-step process. In this step, the trips in the tables output by the trip distri- bution step are split into trips by travel mode. The mode definitions vary depending on the types of transportation options offered in the model’s geographic region and the types of planning analyses required, but they can be generally grouped into auto mobile, transit, and nonmotorized modes. Transit modes may be defined by access mode (walk, auto) and/or by service type (local bus, express bus, heavy rail, light rail, commuter rail, etc.). Nonmotorized modes, which are not yet included in some models, especially in smaller urban areas, include walking and bicycling. Auto modes are often defined by occupancy levels (drive alone, shared ride with two occupants, etc.). When auto modes are not modeled separately, automobile occupancy factors are used to convert the auto person trips to vehicle trips prior to assignment. The outputs of the mode choice process include person trip tables by mode and purpose and auto vehicle trip tables.

4Time-of-day modeling is used to divide the daily trips into trips for various time periods, such as morning and afternoon peak periods, mid-day, and evening. This division may occur at any point between trip generation and trip assignment. Most four-step models that include the time-of-day step use fixed factors applied to daily trips by purpose, although more sophisticated time-of-day choice models are sometimes used. While the four-step process focuses on personal travel, commercial vehicle/freight travel is a significant component of travel in most urban areas and must also be considered in the model. While simple factoring methods applied to per- sonal travel trip tables are sometimes used, a better approach is to model such travel separately, creating truck/commercial vehicle trip tables. The final step in the four-step process is trip assignment. This step consists of separate highway and transit assignment processes. The highway assignment process routes vehicle trips from the origin-destination trip tables onto paths along Forecast Year Highway Network Forecast Year Transit Network Forecast Year Socioeconomic DataTrip Generation Model Internal Productions and Attractions by Purpose Trip Distribution Model Mode Choice Model Person and Vehicle Trip Tables by Purpose/Time Period Time of Day Model Person and Vehicle Trip Tables by Mode/Purpose/Time Period Highway Assignment CHECK: Input and output times consistent? Transit Assignment Highway Volumes/ Times by Time Period Transit Volumes/ Times by Time Period Input Data Model Output Model Component Decision Feedback Loop Yes No Truck Trip Generation and Distribution Models Production/Attraction Person Trip Tables by Purpose Truck Vehicle Trip Tables by Purpose Truck Time of Day Model Truck Vehicle Trip Tables by Time Period External Trip Generation and Distribution Models External Vehicle Trip Tables by Time Period Figure 1.1. Four-step modeling process.

5 the highway network, resulting in traffic volumes on network links by time of day and, perhaps, vehicle type. Speed and travel time estimates, which reflect the levels of congestion indicated by link volumes, are also output. The transit assignment process routes trips from the transit trip tables onto individual transit routes and links, resulting in transit line volumes and station/ stop boardings and alightings. Because of the simplification associated with and the resul- tant error introduced by the sequential process, there is some- times “feedback” introduced into the process, as indicated by the upward arrows in Figure 1.1 (Travel Model Improvement Program, 2009). Feedback of travel times is often required, particularly in congested areas (usually these are larger urban areas), where the levels of congestion, especially for forecast scenarios, may be unknown at the beginning of the process. An iterative process using output travel times is used to rerun the input steps until a convergence is reached between input and output times. Because simple iteration (using travel time outputs from one iteration directly as inputs into the next iteration) may not converge quickly (or at all), averaging of results among iterations is often employed. Alternative approaches include the method of successive averages, constant weights applied to each iteration, and the Evans algorithm (Evans, 1976). Although there are a few different methods for implement- ing the iterative feedback process, they do not employ param- eters that are transferable, and so feedback methods are not discussed in this report. However, analysts should be aware that many of the analysis procedures discussed in the report that use travel times as inputs (for example, trip distribution and mode choice) are affected by changes in travel times that may result from the use of feedback methods. 1.4 Summary of Techniques and Parameters Chapter 4 presents information on (1) the analytical tech- niques used in the various components of conventional travel demand models and (2) parameters for these mod- els obtained from typical models around the United States and from the 2009 NHTS. These parameters can be used by analysts for urban areas without sufficient local data to use in estimating model parameters and for areas that have already developed model parameters for reasonableness checking. While it is preferable to use model parameters that are based on local data, this may be impossible due to data or other resource limitations. In such cases, it is common practice to transfer parameters from other applicable models or data sets. Chapter 4 presents parameters that may be used in these cases, along with information about how these parameters can be used, and their limitations. 1.5 Model Validation and Reasonableness Checking Another important use of the information in this report will be for model validation and reasonableness checking. There are other recent sources for information on how the general process of model validation can be done. Chapter 5 provides basic guidance on model validation and reasonable- ness checking, with a specific focus on how to use the informa- tion in the report, particularly the information in Chapter 4. It is not intended to duplicate other reference material on validation but, rather, provide an overview on validation consistent with the other sources. 1.6 Advanced Travel Analysis Procedures The techniques and parameters discussed in this report focus on conventional modeling procedures (the four-step process). However, there have been many recent advances in travel modeling methods, and some urban areas, especially larger areas, have started to use more advanced approaches to modeling. Chapter 6 introduces concepts of advanced model- ing procedures, such as activity-based models, dynamic traffic assignment models, and traffic simulation models. It is not intended to provide comprehensive documentation of these advanced models but rather to describe how they work and how they differ from the conventional models discussed in the rest of the report. 1.7 Case Study Applications One of the valuable features in NCHRP Report 365 was the inclusion of a case study to illustrate the application of the parameters and techniques contained in it. In this report, two case studies are presented to illustrate the use of the information in two contexts: one for a smaller urban area and one for a larger urban area with a multimodal travel model. These case studies are presented in Chapter 7. 1.8 Glossary of Terms Used in This Report MPO—Metropolitan Planning Organization, the federally designated entity for transportation planning in an urban area. In most areas, the MPO is responsible for maintaining and running the travel model, although in some places, other agencies, such as the state department of transportation, may have that responsibility. In this report, the term “MPO” is sometimes used to refer to the agency responsible for the model, although it is recognized that, in some areas, this agency is not officially the MPO.

6Model area—The area covered by the travel demand model being referred to. Often, but not always, this is the area under the jurisdiction of the MPO. The boundary of the model area is referred to as the cordon. Trips that cross the cordon are called external trips; modeling of external trips is discussed in Section 4.6. Person trip—A one-way trip made by a person by any mode from an origin to a destination, usually assumed to be without stops. In many models, person trips are the units used in all model steps through mode choice. Person trips are the usual units in transit assignment, but person trips are converted to vehicle trips for highway assignment. Trip attraction—In four-step models, the trip end of a home-based trip that occurs at the nonhome location, or the destination end of a nonhome-based trip. Trip production—In four-step models, the trip end of a home-based trip that occurs at the home, or the origin end of a nonhome-based trip. Vehicle trip—A trip made by a motorized vehicle from an origin to a destination, usually assumed to be without stops. It may be associated with a more-than-one-person trip (for example, in a carpool). Vehicle trips are the usual units in highway assignment, sometimes categorized by the number of passengers per vehicle. In some models, vehicle trips are used as the units of travel throughout the modeling process. Motorized and nonmotorized trips—Motorized trips are the subset of person trips that are made by auto or transit, as opposed to walking or bicycling trips, which are referred to as nonmotorized trips. In-vehicle time—The total time on a person trip that is spent in a vehicle. For auto trips, this is the time spent in the auto and does not include walk access/egress time. For transit trips, this is the time spent in the transit vehicle and does not include walk access/egress time, wait time, or time spent transferring between vehicles. Usually, transit auto access/ egress time is considered in-vehicle time. Out-of-vehicle time—The total time on a person trip that is not spent in a vehicle. For auto trips, this is usually the walk access/egress time. For transit trips, this is the walk access/ egress time, wait time, and time spent transferring between vehicles. In some models, components of out-of-vehicle time are considered separately, while in others, a single out-of- vehicle time variable is used.

TRB’s National Cooperative Highway Research Program (NCHRP) Report 716: Travel Demand Forecasting: Parameters and Techniques provides guidelines on travel demand forecasting procedures and their application for helping to solve common transportation problems.

The report presents a range of approaches that are designed to allow users to determine the level of detail and sophistication in selecting modeling and analysis techniques based on their situations. The report addresses techniques, optional use of default parameters, and includes references to other more sophisticated techniques.

Errata: Table C.4, Coefficients for Four U.S. Logit Vehicle Availability Models in the print and electronic versions of the publications of NCHRP Report 716 should be replaced with the revised Table C.4 .

NCHRP Report 716 is an update to NCHRP Report 365 : Travel Estimation Techniques for Urban Planning .

In January 2014 TRB released NCHRP Report 735 : Long-Distance and Rural Travel Transferable Parameters for Statewide Travel Forecasting Models , which supplements NCHRP Report 716.

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Traffic Assignment: A Survey of Mathematical Models and Techniques

• First Online: 17 May 2018

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• Pushkin Kachroo 14 &
• Kaan M. A. Özbay 15  

Part of the book series: Advances in Industrial Control ((AIC))

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This chapter presents the fundamentals of the theory and techniques of traffic assignment problem. It first presents the steady-state traffic assignment problem formulation which is also called static assignment, followed by Dynamic Traffic Assignment (DTA), where the traffic demand on the network is time varying. The static assignment problem is shown in a mathematical programming setting for two different objectives to be satisfied. The first one where all users experience same travel times in alternate used routes is called user-equilibrium and another setting called system optimum in which the assignment attempts to minimize the total travel time. The alternate formulation uses variational inequality method which is also presented. Dynamic travel routing problem is also reviewed in the variational inequality setting. DTA problem is shown in discrete and continuous time in terms of lumped parameters as well as in a macroscopic setting, where partial differential equations are used for the link traffic dynamics. A Hamilton–Jacobi- based travel time dynamics model is also presented for the links and routes, which is integrated with the macroscopic traffic dynamics. Simulation-based DTA method is also very briefly reviewed. This chapter is taken from the following Springer publication and is reproduced here, with permission and with minor changes: Pushkin Kachroo, and Neveen Shlayan, “Dynamic traffic assignment: A survey of mathematical models and technique,” Advances in Dynamic Network Modeling in Complex Transportation Systems (Editor: Satish V. Ukkusuri and Kaan Özbay) Springer New York, 2013. 1-25.

This chapter is taken from the following Springer publication and is reproduced here, with permission and with minor changes: Pushkin Kachroo, and Neveen Shlayan, “Dynamic traffic assignment: A survey of mathematical models and techniques,” Advances in Dynamic Network Modeling in Complex Transportation Systems (Editor: Satish V. Ukkusuri and Kaan Özbay) Springer New York, 2013. 1–25.

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Kachroo, P., Özbay, K.M.A. (2018). Traffic Assignment: A Survey of Mathematical Models and Techniques. In: Feedback Control Theory for Dynamic Traffic Assignment. Advances in Industrial Control. Springer, Cham. https://doi.org/10.1007/978-3-319-69231-9_2

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Trip Assignment

Consider you have the following traffic network.

The O-D traffic volumes are as follows:

Link attributes are as follows:

The volume-delay function is as follows (where t_0 is the free-flow travel time and t is the adjusted travel time): \[t=\frac{t_0}{1-v/c}\]

Calculate the all-or-nothing assignment results.

Calculate the incremental assignment results. Assume link assignment ordering is 1-2, 1-4, 2-3, 3-4,2-4, 1-3.

Calculate the DUE assignment results. Use lambda equal to 0, 1/3, 2/3, and 1 rather than Excel Solver. Use the minimum cost result in each iteration.

• View Record

TRID the TRIS and ITRD database

EQUILIBRIUM TRIP ASSIGNMENT: ADVANTAGES AND IMPLICATIONS FOR PRACTICE

During the past 10 years the problem of assignment of vehicles to large, congested urban transportation networks according to the principle of equal travel times has been solved and an efficient, convergent computer algorithm devised. Although the algorithm is available in the Urban Transportation Planning System, many practitioners continue to use the heuristic trip-assignment algorithms devised in the early 1960s. As in many other cases, this slow implementation of a new, improved algorithm appears to come from (a) a lack of understanding of its basic concepts, (b) an unfamiliarity with the computer program for applying the algorithm, and (c) a lack of evidence concerning the new algorithm's performance in large-scale applications. These three issues are addressed in this paper. Based on the experience with its implementation on a large network, it is recommended that equilibrium trip assignment should always be used instead of iterative assignment. Better results, as judged by the criterion of equalizing travel times for alternative paths between each origin-destination pair, will always be obtained with the equilibrium algorithm for any given amount of computational effort. Which method best replicates the observed vehicle flows may depend on the detail of the network, the adequacy of the capacity-restraint functions, and the time period of the assignment (24 h or peak period). (Author)

• Record URL: http://onlinepubs.trb.org/Onlinepubs/trr/1979/728/728-001.pdf
• Record URL: https://scholar.google.com/scholar_lookup?title=EQUILIBRIUM+TRIP+ASSIGNMENT%3A+ADVANTAGES+AND+IMPLICATIONS+FOR+PRACTICE&author=R.+EASH&author=B.+Janson&author=D.+Boyce&publication_year=1979
• Find a library where document is available. Order URL: http://worldcat.org/isbn/0309029813
• Publication of this paper sponsored by Committee on Passenger Travel Demand Forecasting. Distribution, posting, or copying of this PDF is strictly prohibited without written permission of the Transportation Research Board of the National Academy of Sciences. Unless otherwise indicated, all materials in this PDF are copyrighted by the National Academy of Sciences. Copyright © National Academy of Sciences. All rights reserved
• Janson, B N
• Publication Date: 1979
• Media Type: Print
• Features: Figures; References; Tables;
• Pagination: pp 1-8
• Monograph Title: Passenger travel forecasting
• Transportation Research Record
• Issue Number: 728
• Publisher: Transportation Research Board
• ISSN: 0361-1981

Subject/Index Terms

• TRT Terms: Algorithms ; Computer programs ; Origin and destination ; Traffic assignment ; Traffic congestion ; Traffic flow ; Transportation planning ; Travel ; Travel time ; Urban transportation
• Uncontrolled Terms: Trip
• Subject Areas: Highways; Planning and Forecasting; Public Transportation;

Filing Info

• Accession Number: 00310695
• Record Type: Publication
• ISBN: 0309029813
• Files: TRIS, TRB
• Created Date: May 21 1981 12:00AM

Lessons Learned From a Two-Week Business Trip

My first trip lasted one week, and I worked almost nonstop the whole time I was away. I’m not a spring chicken and a 75-hour week surrounded by two travel days was enough for me. Here are five things I learned from that experience .

This time, there was an opportunity for a two-week assignment in Brooklyn, NY. The work schedule was more favorable, requiring me to work only a 40-hour week. After getting approval from Sharon, I applied for it and was accepted.

Then, the realization that I was going to be away from home for two weeks began to set in. What had I signed myself up for?

Hotel bars and restaurants are the last resort (and they know it)

We left work late for several nights of our stay and didn’t return to the hotel until after 10 PM. Going out to eat was not an option, and delivery would take too long, so we ended up eating at the lobby restaurant. Let me say that I’m willing to eat an $18 burger if it’s worth it. This burger definitely wasn’t worth it. Neither was the $15 bruschetta or the $12 French Onion soup.

These places exist to feed the likes of us. You know, people who are forced to eat there because of work and on an expense account, so it doesn’t matter if the food is bland and overpriced. The only other people eating there were those who were staying for an early morning flight and had no transportation to get anywhere else.

All of my meals outside the hotel were memorable. None of the hotel meals were. The only memorable thing was the round of Lemon Drop shots we ordered one night that cost $14 each because the bar policy is to use Grey Goose unless you specify otherwise.

Hotel breakfasts change from day to day, and some days are better than others

I never thought I’d become a connoisseur of the Hampton Inn hotel breakfast, but I developed my favorites after two weeks of rotating through the selections. The egg white frittata was light and flavorful, and the applewood-smoked chicken sausage had the right amount of smoke and sweetness. The prime rib hash was good but had me thinking I was eating the Hilton’s restaurant leftovers from the night before ( Note from Sharon: of course you were! ).

I tried the make-your-own waffle station one morning, but the waffle was all crust and no inside, so it lost its heat immediately and didn’t even melt the butter I put on it. Between that and the artificial maple-flavored syrup, I skipped them for the rest of the stay.

I did see something I never experienced before. A guest removed an English muffin from the case and put it in the toaster without splitting it in half first, like a psychopath.

The breakfast bagels need to go because, as one of my co-workers said, “I’m not a poor college student anymore, so don’t feel me pizza bagels and call it breakfast.” ( Note from Sharon: I tasted these when I visited. They were NASTY! )

I don’t need housekeeping every day

I stayed here at a time when housekeeping was offered every day. After the first few days, I discovered that I didn’t need housekeeping services every day. I knew this because we don’t get any housekeeping services when we stay at Candlewood Suites for four nights, but I still let housekeeping into the room every day for most of the first week. At that point, I realized I could straighten my own bed, hang my towel on the hook to dry and rinse my own glasses for the next day ( Note from Sharon: Dude, people are gonna start thinking you don’t do this at home on a regular basis ).

Since I was in New York, I refilled my water bottles with tap water and put them into the fridge to get cold for the evening because NYC tap water is better than most bottled water.

Two weeks is a long time to be away from home

While my time in NY went by quickly because I was busy all the time, either at work or sightseeing, it was still a long time to be away from home. I made myself as comfortable as possible in my hotel room, but it wasn’t home. After a while, I just wanted to be home with Sharon and Dobby, our toy poodle, sleeping in my own bed.

I understand why people love hotel promotions

I earned 91,980 Hilton points for my stay. It took me several years to use those points, but they paid for two nights in Manhattan, which isn’t a bad deal.

I don’t want to work on the road

I know now, more than ever, that my life is at home. I like to work at my desk (the same one I’ve had since I was 12) while sitting in my office chair with my dog in my lap, occasionally turning around to bother Sharon while she’s working ( Note from Sharon: true story. #rolleyes ). It’s just not the same as sending her a text. After a while, Sharon couldn’t even say my name in the house while I was gone because our dog would go to the door and wait for me to come home.

I respect those of you who live on the road more than ever. It’s a life that, once upon a time, I dreamed about having. But now that I realize all of the things it entails, I’m more than happy to stay at home and just travel for vacations. There’s more for me at home than there is on the road.

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How Duty of Care Can Help Support and Retain Talent

As the world of work continues to evolve, global mobility has experienced a substantial shift, especially after the pandemic. One aspect of global mobility in particular, duty of care, has had to adjust accordingly.

Recently, Sirva’s Jennifer Rowe, senior manager, intercultural services, participated on a panel of Mobility and Human Resources experts at a NYC SHRM event discussing duty of care and the evolution of this important aspect of mobility. Among the topics discussed were:

As the world has faced challenging situations over the past few years, it is important that global mobility programs adapt to accommodate the evolving needs of a changing workforce. Looking at ways to support employees with robust duty of care support can ensure the right candidates are moving to the right locations, their families are supported, and the company’s goals are met.

For more information on the ways you can incorporate duty of care into your relocation program, please contact your Sirva representative or email us at [email protected] .

Smart. Helpful. Human. Responsible.