  Counterfactual explanations is one of the post-hoc methods used to provide
explainability to machine learning models that have been attracting attention
in recent years. Most examples in the literature, address the problem of
generating post-hoc explanations for black-box machine learning models after
the rejection of a loan application. In contrast, in this work, we investigate
mathematical programming formulations for scorecard models, a type of
interpretable model predominant within the banking industry for lending. The
proposed mixed-integer programming formulations combine objective functions to
ensure close, realistic and sparse counterfactuals using multi-objective
optimization techniques for a binary, probability or continuous outcome.
Moreover, we extend these formulations to generate multiple optimal
counterfactuals simultaneously while guaranteeing diversity. Experiments on two
real-world datasets confirm that the presented approach can generate optimal
diverse counterfactuals addressing desired properties with assumable CPU times
for practice use.
