When one adds capacity constraints to transportation vehicles
and disallows transshipments, the one-to-one PDP is
commonly referred to as the Stacker Crane Problem (SCP).
The SCP is a route optimization problem at the core of several
transportation systems, including demand-responsive transport
(DRT) systems, where users formulate requests for transportation
from a pickup point to a delivery point. Despite
its importance, current algorithms for its solution are either of
exponential complexity or come with quite poor guarantees on
their performance; furthermore, most of the literature on the
SCP does not consider the dynamic setting where pickups/deliveries
are revealed sequentially in time. Broadly speaking,
the objective of this paper is to demonstrate the existence of
simple polynomial-time algorithms for the SCP with probabilistic
optimality guarantees, and derive stability conditions
for its dynamic counterpart (where pickup/delivery requests are
generated by an exogenous Poisson process and that serves as
a model for DRT systems).