  Post-hoc explanation methods for machine learning models have been widely
used to support decision-making. One of the popular methods is Counterfactual
Explanation (CE), also known as Actionable Recourse, which provides a user with
a perturbation vector of features that alters the prediction result. Given a
perturbation vector, a user can interpret it as an "action" for obtaining one's
desired decision result. In practice, however, showing only a perturbation
vector is often insufficient for users to execute the action. The reason is
that if there is an asymmetric interaction among features, such as causality,
the total cost of the action is expected to depend on the order of changing
features. Therefore, practical CE methods are required to provide an
appropriate order of changing features in addition to a perturbation vector.
For this purpose, we propose a new framework called Ordered Counterfactual
Explanation (OrdCE). We introduce a new objective function that evaluates a
pair of an action and an order based on feature interaction. To extract an
optimal pair, we propose a mixed-integer linear optimization approach with our
objective function. Numerical experiments on real datasets demonstrated the
effectiveness of our OrdCE in comparison with unordered CE methods.
