**3.1 Assignment to uncongested networks**

In uncongested networks the arc flows depend on the arc costs, through the arcflow function obtained as described below; its structure is shown in **Figure 2**.

The arc costs (**c**) may be different among the vehicle types to reflect different performances, and we assume that the arc cost per vehicle type are given by an affine transformation of the arc generic costs.

**Figure 2.** *The arc-flow function for assignment to uncongested networks.*

Then, the route costs (**w**) for each o-d pair, user category and vehicle type can be obtained from the corresponding arc total costs through an affine transformation from the arc space to the route space defined by the transpose of arc-route incidence matrix.

The utility function for each o-d pair, user category and vehicle type is almost always specified through an affine transformation of costs both in research analysis and in practical applications.

Route choice behavior for users of each o-d pair, user category and vehicle type m can be modeled by applying any discrete choice modeling theory, such the wellestablished Random Utility Theory. In this case the choice proportion of an alternative is given by the probability that its perceived utility is equal to maximum among all alternatives. When the perceived utility co-variance matrix is non-singular, probabilistic route choice functions are obtained.

Demand conservation flow relation for each o-d pair, user category, vehicle type assures that flows of all connecting routes (**h**) sum up to demand flow (**d**).

The arc flows (**f**) due to each o-d pair, user category and vehicle type can be obtained from the route flows through a linear transformation from the route space to the arc space defined by the arc-route incidence matrix. Having assumed that all arc flows are measured in TVs per time unit, the arc total flows are given by the sum over all o-d pairs, user categories and vehicle types.

Main input data of the arc flow function are arc costs, and demand flows. Vehicle types may be distinguished with respect to:


The arc flow function is monotone non-increasing with respect to arc costs under mild assumptions. It can be computed for large scale applications through algorithms derived from network theory, avoiding explicitly path enumeration.

### **3.2 Equilibrium assignment to congested networks**

In congested transportation networks arc flows depend on arc costs, and user equilibrium assignment searches for mutually consistent arc flows and costs. Arc generic costs depend on the arc total flows through the arc cost function, which models user driving behavior at macroscopic level.

Equilibrium assignment can effectively be described through fixed-point (FP) models obtained combining the arc-flow function and the arc cost function. These models can be solved for large scale applications through algorithms based on the Method of Successive Averages, which avoid the use of matrix algebra and computation of derivatives. Their structure is shown in **Figure 3**.

Existence is guaranteed if both the arc flow function and the arc cost function are continuous (and the network is connected), applying Brouwer theorem. For a monotone decreasing arc flow function, if the arc cost function is monotone strictly increasing uniqueness is guaranteed. Uniqueness of arc flows also guarantees

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*Advanced Vehicles: Challenges for Transportation Systems Engineering*

uniqueness of arc costs as well as route flows and costs, and of flows and cost per

The evolution over time of arc flows and costs can effectively be described through day-to-day dynamic assignment models. The specification of these models requires an extension of models for the equilibrium assignment by including

• User memory and learning: how users forecast the level of service that they will experience today, from experience and other sources of information, such

• User habit and inertia to change: how users make a choice today, possibly repeating yesterday choice to avoid the effort needed to take a decision, or

The arc cost updating relation, modeling user memory and learning, gives the today forecasted route costs with respect to previous day costs. It extends the arc cost function. In the simplest instance, this relation can be specified by an exponential smoothing (ES) filter, say a convex combination of yesterday route forecasted costs and yesterday actual route costs, given by an affine transformation of the

The arc flow updating relation, modeling user habit and inertia to change, gives the today arc flow with respect to forecasted costs and previous day flows. It extends the above arc flow function. In the simplest instance, this relation too can be specified by an exponential smoothing (ES) filter, say a convex combination of yesterday arc flows due to users who do not reconsider their yesterday choice and

reconsidering it according to the forecasted level of service.

today arc flows due to users who reconsider their yesterday choice.

**3.3 Day-to-day dynamic assignment to congested networks**

*Fixed point models for equilibrium assignment to congested networks.*

as informative systems, about previous days;

*DOI: http://dx.doi.org/10.5772/intechopen.94105*

o-d pair, user category, vehicle type.

sub-models of

**Figure 3.**

yesterday arc costs.

*Advanced Vehicles: Challenges for Transportation Systems Engineering DOI: http://dx.doi.org/10.5772/intechopen.94105*

*Models and Technologies for Smart, Sustainable and Safe Transportation Systems*

and in practical applications.

probabilistic route choice functions are obtained.

over all o-d pairs, user categories and vehicle types.

types may be distinguished with respect to:

• mean number of users on board

• flow equivalence

• cost equivalence

• route choice function

Then, the route costs (**w**) for each o-d pair, user category and vehicle type can be obtained from the corresponding arc total costs through an affine transformation from the arc space to the route space defined by the transpose of arc-route incidence matrix. The utility function for each o-d pair, user category and vehicle type is almost always specified through an affine transformation of costs both in research analysis

Route choice behavior for users of each o-d pair, user category and vehicle type m can be modeled by applying any discrete choice modeling theory, such the wellestablished Random Utility Theory. In this case the choice proportion of an alternative is given by the probability that its perceived utility is equal to maximum among all alternatives. When the perceived utility co-variance matrix is non-singular,

Demand conservation flow relation for each o-d pair, user category, vehicle type

Main input data of the arc flow function are arc costs, and demand flows. Vehicle

The arc flows (**f**) due to each o-d pair, user category and vehicle type can be obtained from the route flows through a linear transformation from the route space to the arc space defined by the arc-route incidence matrix. Having assumed that all arc flows are measured in TVs per time unit, the arc total flows are given by the sum

assures that flows of all connecting routes (**h**) sum up to demand flow (**d**).

• specific arc, cost, e.g. monetary cost (and VoT), access cost, …

• route utility parameter and route choice function parameters

**3.2 Equilibrium assignment to congested networks**

models user driving behavior at macroscopic level.

tation of derivatives. Their structure is shown in **Figure 3**.

The arc flow function is monotone non-increasing with respect to arc costs under mild assumptions. It can be computed for large scale applications through algorithms derived from network theory, avoiding explicitly path enumeration.

In congested transportation networks arc flows depend on arc costs, and user equilibrium assignment searches for mutually consistent arc flows and costs. Arc generic costs depend on the arc total flows through the arc cost function, which

Equilibrium assignment can effectively be described through fixed-point (FP) models obtained combining the arc-flow function and the arc cost function. These models can be solved for large scale applications through algorithms based on the Method of Successive Averages, which avoid the use of matrix algebra and compu-

Existence is guaranteed if both the arc flow function and the arc cost function are continuous (and the network is connected), applying Brouwer theorem. For a monotone decreasing arc flow function, if the arc cost function is monotone strictly increasing uniqueness is guaranteed. Uniqueness of arc flows also guarantees

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**Figure 3.** *Fixed point models for equilibrium assignment to congested networks.*

uniqueness of arc costs as well as route flows and costs, and of flows and cost per o-d pair, user category, vehicle type.
