Problem Summary
To match passengers (the number of passengers) to capacitated vehicles such that the profit is increased. All the vehicles have the same capacity $c$. It is not important to track which passenger is matched to which vehicle.
Problem Background
I'm trying to find a solution to the following passenger matching problem:
The network is represented by graph $G=(V,E)$. $V$ is the set of nodes/stations. $p_{ij}$ is the profit of traveling through an edge $(i,j)$. Let $N$ be the number of vehicles, all the vehicles have the same capacity $c$.
At each (discreet) time step, the passengers arrive at their origin station and need to be transported to their destination station. $d^t_{ij}$ is the demand. $f^t_{ij} \le d^t_{ij}$ is the passenger flow, i.e., the number of passengers that are traveling from $i$ to $j$ at time $t$ and are successfully matched to a vehicle. The unmatched passers will leave the system. $X^t_i$ is the total number of available vehicles at station $i$ at time $t$.
Objective
The objective is to maximize profit.
Problem formulation for single occupancy vehicle
I read a few resources and found the following formulation. However, this formulation assumes that the vehicles are single occupancy.
$$ max \sum_{i,j \in V} f^t_{ij}p^t_{ij} $$ $$ 0 \le f^t_{ij} \le d^t_{ij} \quad i,j \in V \quad (1)$$ $$ \sum_{j\in V} f^t_{ij} \le X^t_i \quad i \in V \quad \quad (2)$$
The constraint $(1)$ ensures that the passenger flow doesn't exceed the demand and is non-negative.
The constraint $(2)$ ensures that the number of vehicles doesn't exceed the number of available vehicles $X^t_i$ at time $t$ at station $i$.
As long as the demand is integral, this formulation results in integral passenger flow.
Question
I'd like to extend the above single occupancy formulation to multi-passenger vehicles. I explored some resources online but the methods focused on individual vehicles instead of stations and were using ILP.
Update
I modified the problem formulation to set an upper bound to the vehicle count.
$$ max \sum_{i,j \in V} f_{ij}p_{ij} $$ $$ 0 \le c.x_{i,j} \le d_{ij}+c \quad i,j \in V \quad (1)$$ $$ \sum_{j \in V} x_{ij} \le X_i \quad \forall i \in V \quad \quad (2)$$ $$ f_{i,j} \le c.x_{ij} \quad \forall i,j \in V \quad \quad (3)$$