I am trying to model an optimisation problem. Here is the setting:
The driver has to stop at each of the seven stations of the trip (They are ordered linearly so he first comes across station 1, then station 2 and so on). At each station, he will have to wait a certain time (this is the control variable) until he reaches a threshold passenger capacity (that will satisfy his minimum expected income restriction).
During each station some people enter the bus and some other leave it. This looks similar to a Dynamic Linear Programming Problem, as some of the passengers who enter at station 1 will leave at the station 2, hence the driver , as he notices some seats are empty, will choose to stay some time at station 2 until he reaches the minimum income restriction (say 30 seats full per station). The problem is that, not everybody who entered at station 1 chooses to leave the bus at station 2: some people could wait until station 5 for instance. At station 7, everybody will have to leave.
In summary:
Control variable: time spent at each station (chosen by the driver)
State variable: Passengers
- Restriction: Minimum income (given by minimum full seats) at each station
- The price is the same for all passengers and for all trips (whether one chooses to go from Station 1 to Station 2 or to Station 7)
- As stations increase, people could choose to leave in less station. (If somebody enters the bus at station 3, he can only leave at station 4,5,6 or 7, but if he enters the bus at station 6, he can only leave at station 7)
How could I model this? Could I use historical percentage of flows in each station to know how many people will leave at Station i, i:2,3,4,5,6,7?
Thanks in advance