  We present a preference learning framework for multiple criteria sorting. We
consider sorting procedures applying an additive value model with diverse types
of marginal value functions (including linear, piecewise-linear, splined, and
general monotone ones) under a unified analytical framework. Differently from
the existing sorting methods that infer a preference model from crisp decision
examples, where each reference alternative is assigned to a unique class, our
framework allows to consider valued assignment examples in which a reference
alternative can be classified into multiple classes with respective credibility
degrees. We propose an optimization model for constructing a preference model
from such valued examples by maximizing the credible consistency among
reference alternatives. To improve the predictive ability of the constructed
model on new instances, we employ the regularization techniques. Moreover, to
enhance the capability of addressing large-scale datasets, we introduce a
state-of-the-art algorithm that is widely used in the machine learning
community to solve the proposed optimization model in a computationally
efficient way. Using the constructed additive value model, we determine both
crisp and valued assignments for non-reference alternatives. Moreover, we allow
the Decision Maker to prioritize importance of classes and give the method a
flexibility to adjust classification performance across classes according to
the specified priorities. The practical usefulness of the analytical framework
is demonstrated on a real-world dataset by comparing it to several existing
sorting methods.
