# Boundedly Rational Dynamic User Equilibrium Route Choice Assignment

Boundedly rational dynamic user equilibrium route choice assignment can be considered a special case of (within-day) DTA, in which the travel choice principle or route choice principle is defined by the tolerance-based dynamic user optimal principle.

## Tolerance-Based Dynamic User Optimal Principle

The traditional DUO principle (e.g. Ran & Boyce, 1996) requires that all used paths between the same OD pair have equal and minimum travel time. As a relaxation, the tolerance-based DUO principle only requires the travel times of all used routes between the same OD pair to be similar, or within an acceptable tolerance εmax from the minimum OD route travel time, where the tolerance level is purely a function of the behaviour of the network users.

The relaxation recognises the important fact that it is physically impossible to always fulfil the requirement that all used routes on the same OD pair have exactly the same travel time, as is demonstrated in a study with the more realistic physical queue representation (Szeto, 2003). This relaxation, adapted from the bounded- rationality behavioural notion, can be expressed as:

(8.2)

(8.3)

where and are respectively the flow between OD pair *rs* entering route *p* at time *t* and the corresponding route travel time; is the shortest OD travel time between OD pair *rs* for flows departing at time *t;* εmax is the acceptable tolerance, a non-negative parameter obtained through travel behaviour surveys and experiments. In Eqs. (8.2) and (8.3), *t* is an instant of time; in this chapter, *t* is a time-slice index, as we consider a discrete-time DTA formulation. Condition (8.3) is included in this principle to ensure to be the shortest OD travel time among all the possible routes between OD pair *rs* for flows departing at time *t.*

By employing the following transformation function:

(8.4)

where *и* and *у* are independent non-negative variables, the tolerance-based DUO principle can be alternatively formulated as:

(8.5)

(8.6)

Condition (8.6) implies condition (8.3) due to the requirement of the non-negative independent variable to the transformation function. Condition (8.5) implies (8.2), meaning that the travel time of a used route is greater than the minimum route travel time by not more than an acceptable level εmaч. According to (8.5), if route *p* carries a positive flow at time *t* (i.e.), the transformation function must be equal to zero, implying due to the first condition of Eq. (8.4). In other words, if route *p* carries a positive flow at time /, the travel time of route *p* is greater than the minimum route travel time by not more than an acceptable level εmax. Note that if route *p* carries zero flow at time / (i.e.),, εmax) must be nonnegative due to Eq. (8.6) and hence the route travel timemust be greater than or equal to the minimum route travel time !!!*n*rs*(t).*

As a special case, if εmax equals zero, conditions (8.5)-(8.6) can be reduced to the following:

(8.7)

**(**8**.**8**)**

According to Eq. (8.4), we have

Therefore, Eqs. (8.7)-(8.8) are simplified to

(8.9)

**(**8**.**10**)**

or

**(**8**.**11**)**

**(**8**.**12**)**

Conditions (8.9)-(8.10) are the ideal DUO conditions. That is, if equals zero, Eqs. (8.5)-(8.6) can be reduced to the ideal DUO conditions. According to this result, the tolerance-based principle is a generalisation of the traditional DUO principle.