  We present new integer linear programming (ILP) models for NP-hard
optimisation problems in instances of the Stable Marriage problem with Ties and
Incomplete lists (SMTI) and its many-to-one generalisation, the Hospitals /
Residents problem with Ties (HRT). These models can be used to efficiently
solve these optimisation problems when applied to (i) instances derived from
real-world applications, and (ii) larger instances that are randomly-generated.
In the case of SMTI, we consider instances arising from the pairing of children
with adoptive families, where preferences are obtained from a quality measure
of each possible pairing of child to family. In this case we seek a maximum
weight stable matching. We present new algorithms for preprocessing instances
of SMTI with ties on both sides, as well as new ILP models. Algorithms based on
existing state-of-the-art models only solve 6 of our 22 real-world instances
within an hour per instance, and our new models solve all 22 instances within a
mean runtime of 60 seconds. For HRT, we consider instances derived from the
problem of assigning junior doctors to foundation posts in Scottish hospitals.
Here we seek a maximum size stable matching. We show how to extend our models
for SMTI to the HRT case. For the real instances, we reduce the mean runtime
from an average of 144 seconds when using state-of-the-art methods, to 3
seconds when using our new ILP-based algorithms. We also show that our models
outperform considerably state-of-the-art models on larger randomly-generated
instances of SMTI and HRT.
