Amuwa's Kingdom

Operations Research, Optimization, Scheduling, Vehicle Routing

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[amuwa-blog]

Free Bus Problems: Batching Decisions over Finite Horizons

Problem

Consider the following situation:

> A free shuttle is provided to ferry customers waiting at an assemble station to a supermarket in weekends. The driver is asked to bring up as many customers to the supermarket as he can. Customers arrive at the station gradually. Every time the bus get to the station, there may be waiting customers less than its capacity. If the driver decides to set off immediately, capacity loss occurs in the single trip. But if he decide to wait for more customers to come, there is also a capacity loss in terms of total number of trips. How can he make the decision in such cases?

This problem is named the free bus problem (FBS).

Model

Given that the capacity of the bus is $B$, it spends $\rho$ time units in travelling to the supermarket from the assemble station, and cost $\zeta$ time units to travel back to the station. Customers arrive at $t_{1},t_{2},\ldots,t_{N}$, where $$t_{1}\leq t_{2}\leq\ldots\leq t_{N}\leq T.$$

Assume the bus get to the station at time $0$.

Suppose the bus will depart at time points $\tau_{1}, \tau_{2}, \ldots, \tau_{M}$,where $\tau_{1}\lt \tau_{2}\lt \ldots \lt \tau_{M}$ then we have $$ \tau_{i+1}-\tau_{i}\geq \rho+\zeta,\; \forall i\lt M. $$

Let $N(t)=| \lbrace t_{i}|t_{i}\leq t\rbrace|$,$Q_{t}$ be the number of customers waiting at the station at time $t$, and $r_{t}$ be the number of customers aboard the bus at time $t$.

......

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