  Inconsistency in prediction problems occurs when instances that relate in a
certain way on condition attributes, do not follow the same relation on the
decision attribute. For example, in ordinal classification with monotonicity
constraints, it occurs when an instance dominating another instance on
condition attributes has been assigned to a worse decision class. It typically
appears as a result of perturbation in data caused by incomplete knowledge
(missing attributes) or by random effects that occur during data generation
(instability in the assessment of decision attribute values). Inconsistencies
with respect to a crisp preorder relation (expressing either dominance or
indiscernibility between instances) can be handled using symbolic approaches
like rough set theory and by using statistical/machine learning approaches that
involve optimization methods. Fuzzy rough sets can also be seen as a symbolic
approach to inconsistency handling with respect to a fuzzy relation. In this
article, we introduce a new machine learning method for inconsistency handling
with respect to a fuzzy preorder relation. The novel approach is motivated by
the existing machine learning approach used for crisp relations. We provide
statistical foundations for it and develop optimization procedures that can be
used to eliminate inconsistencies. The article also proves important properties
and contains didactic examples of those procedures.
