2.2. Modeling
We divided passengers into four categories on the basis of where they get on and off, namely, PD (boarding at the control station), PND (boarding at the control station, getting off at the optional station), NPD (boarding at the optional station, getting off at the control station), and NPND (get off at the optional station) [
16]. If the vehicle was parked at the control station, it was marked as
, if it was parked at the optional station, it was marked as
, where S is the total number of vehicles parked. In order to simplify the calculation results, we divided the NPD and NPND passengers into separate travel needs, that is, their travel needs did not occur at the same alternative station.
This article considered both the system vehicle operating cost and the passenger travel cost, and established a dispatch model.
- (1)
Vehicle operating cost.
This term expresses the running cost of the vehicle in terms of time cost [
17], that is, the running time cost of the vehicle, and the expression is as shown in Formula (1).
- (2)
Passenger travel time cost.
In the RDT bus system, we believed that the time for passengers to board the bus at each stop was equal to the vehicle start time, and the time to get off the bus was equal to the vehicle parking time, and thus the passenger’s travel time cost expression was as shown in Equation (2).
- (3)
Passenger walking and waiting time costs
In the RDT bus system, when a passenger sends a travel request to the dispatching platform, the dispatching platform responds to this request and sends feedback to the passenger, and the passenger goes to the station to wait; then, the passenger’s walking and waiting time cost expression is as shown in Equation (3).
Thus, the system operation scheduling model is as shown in Formula (4):
—the departure time of the vehicle; —when the vehicle arrived at the station, the first station had no arrival time; —the boarding time of passenger travel demand q; —passenger travel demand q alighting time.
represents the (0,1) variable; if the vehicle was driving on the road segment , ; otherwise, . represents the (0,1) variable, if the vehicle can provide passengers with the control station getting on and off service, ; otherwise, .
The constraints of this model are as follows:
In terms of route constraints, it was assumed that except for PD passengers, other types of passengers were independent of each other when getting on and off the bus at optional stations [
18]. Constraints (5) and (6) indicate that in each vehicle operation, except for the first and last stations, each station (including control station and optional station) had one and only one route through, that is, there was only one incident route and one exit route.
Constraint (7) is to limit the departure time of the control station. The departure time of the vehicle must be in accordance with the departure timetable of the control station. Restriction (8) means that except for PND passengers, the vehicle will leave the station immediately after other passengers get on the bus. Assuming that when the loaded vehicle arrives at the station, the passengers get off immediately. Constraint (9) indicates that the PD, PND, and NPND passengers got off the station when the vehicle arrived at the station.
Constraint (10) indicates that the boarding time of each type of passenger was no earlier than the time it took to arrive at the boarding station from the location. Constraint (12–15) indicates that each type of passenger got on and then got off.
Constraints (12) and (13) indicate that if
, the vehicle left the station immediately after each PND passenger got on the bus. Constraints (14) and (15) indicate that if
, NPD passengers got off immediately when the vehicle arrived. Among the four constraints, in order to ensure that the constraint can be invalid when
, M must be large enough [
19].
Regarding the vehicle
constraint, constraint (16) means that each passenger can only travel in a unique vehicle, and cannot change vehicles halfway through the entire RDT bus system. Constraint (17) means that the number of running vehicles was not greater than the system’s maximum number of vehicles
, which can effectively save operating costs, not only causing waste of vehicle resources, but also meeting the travel needs of passengers and improving service levels.
Constraint (19) is the core constraint of this model and also a priority constraint. Constraint (18) means that at , the time for the vehicle to arrive at stop was not less than the sum of its departure time from the stop and the travel time. Moreover, at this time, M must be large enough. This ensures that each operating line does not include an inner loop and is a one-way path from the first station to the end station. Constraint (19) represents vehicle arrival time constraints.