4.1. Effective Travel Cost
Due to the demand for interval travels, the travel cost of a route becomes an interval variable. If only the expected value of the interval travel cost in travel mode and travel route decision is considered similarly to the traditional equilibrium model, it is difficult to illustrate the travellers’ real choices comprehensively. Therefore, a new choice criterion needs to be defined and a corresponding equilibrium model needs to be formulated to capture travellers’ choices on travel modes and routes under the interval travel cost.
By analyzing the characteristic of interval travel cost, the concept of ETC, denoted as , is proposed, which explicitly considers both reliability and unreliability aspects of the interval travel cost in the travel mode and route choosing process. First, a predicted travel cost value is considered from the origin to the destination, which is defined as the TCB.
Definition 1. The TCB is defined as an interval variable , of which the mean value is calculated as follows (Moore (1996, 1979) [20,21]):where is the expected free-flow travel cost of routes in travel mode m for pair ω; the deviation value of TCB is calculated as follows:where ω is defined as an acceptable deviation value, which reflects the traveller’s tolerance for variability of the interval travel cost. The value is assumed to be 0.1. Comparing the upper and lower bounds of the budgeted travel cost with the actual travel cost, the actual travel cost can be divided into multiple subintervals. If the value in the sub-interval is likely to be less than the budgeted travel cost, the sub-interval is called the reliability part of the actual travel cost. If the value in the subinterval is likely to be greater than the budgeted trip cost, the subinterval is said to be the unreliable part of the actual trip cost, The definition of reliability and unreliability aspects of the actual travel costs is shown in
Figure 3. The effective trip cost is defined as the product of the mean of the reliable subinterval and the probability that the subinterval is less than the budgeted trip cost + the product of the mean of the unreliable subinterval and the probability that the subinterval is greater than the budgeted trip cost. The first group reflects the reliability aspect, under which travellers travel from their origin to their destination with an acceptable level deviation of
from their TCB. It is defined as the subintervals of the actual interval travel cost, which has a certain probability of being less than the TCB. The second group reflects the unreliability aspect, under which travellers travel from their origin to their destination but exceed their TCB by a value greater than
. It is defined as subintervals of the actual interval travel costs, which have a certain probability of exceeding the TCB.
Definition 2. The ETC for each route is defined as a weighted sum of the mean values of the reliability and unreliability aspects of the actual travel costs. The respective weights are equal to the corresponding probabilities. Therefore, the ETC equation is presented for the following six cases:
Case 1: When the actual interval travel cost and the TCB satisfy , the reliability aspect is given by the interval , and the probability of not exceeding the TCB is equal to 1.
Case 2: When the actual interval travel cost and the TCB satisfy , the reliability aspects are given by the intervals and , and their probabilities of not exceeding the TCB are, respectively, given by and ; the ATC’s unreliability aspect is , and the probability of exceeding the TCB is .
Case 3: When the actual interval travel cost and the travel cost budget satisfy , the reliability aspect of ACT is given by the interval , and the probability of not exceeding the TCB is ; the ATC’s unreliability is , and the probability of exceeding the TCB is .
Case 4: When the actual interval travel cost and the TCB satisfy , the ATC’s reliability aspect is given by , and the probability of not exceeding the TCB is ; the ATC’s unreliability aspects are given by and , and the corresponding probabilities of exceeding the TCB are and .
Case 5: When the actual interval travel cost and the TCB satisfy , the ATC’s reliability aspects are given by and , and the corresponding probabilities of not exceeding the TCB are and ; the ATC’s unreliability aspects are and , and the corresponding probabilities of exceeding the TCB are and .
Case 6: When the actual interval travel cost and the TCB satisfy , then the ATC’s unreliability aspect is , and the probability of exceeding the travel cost budget is 1.
Therefore, the ETC cases can be is formulated as follows:
4.2. Variational Inequality Formulation
The equilibrium conditions are given by the intersection of the following two subsets of conditions:
The choice of travel mode: For each
pair
, the proportion of users in every travel mode is given as the logit model function, which is defined as follows:
where
is the proportion of passengers using travel mode
m for
pair
w;
is the travel demands for
pair
w;
is the ETC of the minimum route in travel mode
m for
pair
w;
is a parameter of travel utility perception variation.
The choice of route: It is assumed that each hyper-path on the multi-modal network has an ETC that models these implicit choices. The users’ behaviors are modelled through a version of Wardrop’s user-optimal principle. Therefore, in the equilibrium state, no traveller can unilaterally change the travel path to reduce the ETC; that is, all the used paths have the same ETC, which is less than or equal to the ETC of the unused paths. This condition can be formulated as follows:
Then, the feasible set can be described as follows:
where (29) and (30) are the travel demand conservation constraints, (31) is a definitional constraint that summarizes all the route flows that pass through a given link, and (32) and (33) are non-negativity constraints on the route flows.
Then, the equilibrium model can be formulated as a variational inequality problem
, as follows:
where
represents constraints (29)–(33).
The following two propositions state the equivalence of the VI formulation and the equilibrium model, as well as the existence of an equilibrium solution.
Proposition 1. If the effective route travel time function is positive, and the solution of the VI problem (34) is equivalent to the equilibrium solution of the ETC model.
Proof 1. By considering the Karush–Kuhn–Tucker conditions of model (34), we have
Then, from (33), we obtain
which can be written in the following form:
Combined with (29), this yields
When (38) is substituted into (37), we obtain
The final equation is the logit model (34). □
Proposition 2. If the route ETC function is positive and continuous, the ETC model has at least one solution.
Proof 2. Based on Proposition 1, solving the equivalent VI formulation is sufficient. Note that the feasible set is nonempty and convex. Furthermore, according to the assumption, the route ETC is continuous. Thus, the VI problem (21) has at least one solution. □