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A boundedly rational user equilibrium model with restricted unused routes (R-BRUE) considering the restrictions of both used route cost and unused route cost is proposed. The proposed model hypothesizes that for each OD pair no traveler can reduce his/her travel time by an indifference band by unilaterally changing route. Meanwhile, no route is unutilized if its travel time is lower than sum of indifference band and the shortest route cost. The largest and smallest used route sets are defined using mathematical expression. We also show that, with the increase of the indifference band, the largest and smallest used route sets will be augmented, and the critical values of indifference band to augment these two path sets are identified by solving the mathematical programs with equilibrium constraints. Based on the largest and smallest used route sets, the R-BRUE route set without paradoxical route is generated. The R-BRUE solution set can then be obtained by assigning all traffic demands to the corresponding generated route set. Various numerical examples are also provided to illustrate the essential ideas of the proposed model and structure of R-BRUE route flow solution set.

Perfect rationality is widely used in studying traditional transportation network models in which traveler always chooses the shortest (i.e., least utility) route, such as user equilibrium (UE [

In the literature of evaluating habitual routes in route choice behavior, only 30% of respondents from Boston [

It is more practical that traveler is boundedly rational (BR); traveler will not change his/her route if his/her travel time is a little longer than the shortest route. A series of experiments were conducted to empirically validate bounded rationality [

Simon, in 1957 [

Boundedly rational user equilibrium (BRUE) is a network state such that travelers can take any route whose travel time is within a threshold of the shortest route time [

However, unlike the conventional UE model, the traffic flow under BRUE may not utilize any shortest or least-cost route; in another word the unused route cost may be lower than the used one. For example, the route flows are 0, 5, and 7 for three different routes on one OD pair, and the travel times are 10, 12, and 13, respectively. If the indifference band is 3, the above route flow solution is a BRUE solution. From the behavioral point of view, one might question the plausibility of this that the least travel time route has no traffic on it. Therefore, to make the model become behavioral more defensible, not only the used route cost should be restricted, but also the unused route cost should be restricted.

Uncertainties are unavoidable in transportation systems and make people become boundedly rational. Travelers do not know exactly the time that they arrive at the destination due to the travel time variability which is made by uncertainties. However, in many cases (such as going to work, having a meeting, and catching the train), travelers care more about arrive time than travel time; no one wants to be late; thus the largest and smallest route sets exist in travels’ trip process.

This paper makes contributions in three major areas:

The remainder of the paper is organized as follows. In Section

In this section, we propose

Consider a transportation network

We should point out that boundedly rational user equilibrium (BRUE) model only considers

Equation (

In other words, a used route has lower cost than an unused one, which is the same as that in the UE (user equilibrium) setting. When

Any

let

For the converse situation, suppose that a flow allocation satisfies

Usually

If the link cost function is continuous,

First, Patriksson [

We use slack variables

In this section, we give the definition of the largest and smallest

Here we give three definitions of used route set, largest

Given

A largest

A smallest

In the following, we will discuss the impact of the value of

If

Let

We will give the definition of paradoxical route first.

With the increasing of OD demand, the traffic flow on route

This phenomenon contradicts our intuition. Following that, we use an example to illustrate this paradox.

The network topology of the test network, link, and traffic demand characteristics are depicted in Figure

Test network and routes characteristics.

The equilibrium route flow

The UE traffic flow of route 1 under different demand.

When

Test network and routes characteristics.

Solving the UE where

If

Assume that there exist two bands

When

Given

The route set is finite, while

It is assumed that there are

The largest/smallest

We give the physical meaning of

We use Figure

Monotonically nondecreasing property of critical points.

Clearly, from Figure

Definition

For any route

Equations (

The network topology of the test network, travel time functions, and traffic demand are depicted in Figure

Test network and routes characteristics.

Solving (

And there are three cases for the smallest

All largest and smallest

For a network with total

For one OD pair

Then the same approach can be used to generate the

By far we have proposed how to solve

We have already analyzed the interior structure of

Define a sequence of sets

We use a bridge network to illustrate (

A bridge network and routes characteristics.

The UE is

And there are three cases for the smallest

Then we assign all the traffic demands with diffident

When

When

When

When

When

For convenience, we only show the R-BRUE solution sets with

R-BRUE solution sets illustration.

R-BRUE solution set with

R-BRUE solution set with

R-BRUE solution set with

From Figure

In the last section, we obtain

In this paper, a boundedly rational user equilibrium model with restricted unused routes (R-BRUE) was proposed. This new model assumes that travel cost of all used route is less than or equal to the sum of given indifference band and the shortest route cost; meanwhile, the travel cost of the unused route is greater than or equal to the sum of indifference band and the shortest route cost. The mathematical formulation of the proposed model was then established.

Before constructing the R-BRUE flow set, largest and smallest used route sets were explored first. As the value of the indifference band increases, some routes which were not utilized before will be taken, and thus the route set that contains the equilibrium flow was named as the largest used route set. As the value of the indifference band increases, some routes which must be utilized for all R-BRUE flow patterns will be taken, and this route set was defined as the smallest used route set. Paradoxical route is defined as that with the increasing of OD demand; the UE flow on the paradoxical route increases first and then decreases. The monotonically nondecreasing property of largest and smallest used route sets without paradoxical route is proved.

The critical values of the indifference band to augment the largest and smallest used route sets can be identified by sequentially solving a class of mathematical programs. After the largest and smallest used route sets are obtained, the whole R-BRUE flow set can be obtained by assigning all traffic demands to the corresponding generated route set.

The proposed model is appealing in modeling realistic travel behavior. But due to the nonconvexity of the feasible region, it is difficult to get the solution using mathematical programming method, and our method which is separation of the solution sets takes heavy computational burdens. In future research, we will study the solution algorithm and then apply the built model to real traffic network, such as the autonomous vehicles. Also, it is worthwhile to extend the proposed model to the network design problem, such as enhancing capacities of the established links, congestion pricing, and adding new links to an existing road network.

The authors declare that they have no competing interests.

This research is supported by the National Natural Science Foundation of China (no. 51578150 and no. 51378119), the Scientific Research Foundation of Graduate School of Southeast University (no. YBJJ1679), the Fundamental Research Funds for the Central Universities and the Research Innovation Program for College Graduates of Jiangsu Province (no. KYLX15_0150), and the China Scholarship Council (CSC) Program sponsored by the Ministry of Education in China.