  This paper introduces the concept of task-splitting into home healthcare
(HHC) routing and scheduling. It focuses on the design of routes and timetables
for caregivers providing services at patients' homes. Task-splitting is the
division of a (lengthy) patient visit into separate visits that can be
performed by different caregivers at different times. The resulting split parts
may have reduced caregiver qualification requirements, relaxed visiting time
windows, or a shorter/longer combined duration. However, additional temporal
dependencies can arise between them. To incorporate task-splitting decisions
into the planning process, we introduce two different mixed integer linear
programming formulations, a Miller-Tucker-Zemlin and a time-indexed variant.
These formulations aim to minimize operational costs while simultaneously
deciding which visits to split and imposing a potentially wide range of
temporal dependencies. We also propose pre-processing routines for the
time-indexed formulation and two heuristic procedures. These methods are
embedded into the branch-and-bound approach as primal and improvement
heuristics. The results of our computational study demonstrate the additional
computational difficulty introduced by task-splitting and the associated
additional synchronization, and the usefulness of the proposed heuristic
procedures. From a planning perspective, our results indicate that introducing
task-splitting reduces staff requirements, decreases HHC operational costs, and
allows caregivers to spend relatively more time on tasks aligned with their
qualifications. Moreover, we observe that the potential of task-splitting is
not specific to the chosen planning objective; it can also be beneficial when
minimizing travel time instead.
