  In practical situations, interval-valued fuzzy sets are frequently
encountered. In this paper, firstly, we present shadowed sets for interpreting
and understanding interval fuzzy sets. We also provide an analytic solution to
computing the pair of thresholds by searching for a balance of uncertainty in
the framework of shadowed sets. Secondly, we construct errors-based three-way
approximations of interval-valued fuzzy sets. We also provide an alternative
decision-theoretic formulation for calculating the pair of thresholds by
transforming interval-valued loss functions into single-valued loss functions,
in which the required thresholds are computed by minimizing decision costs.
Thirdly, we compute errors-based three-way approximations of interval-valued
fuzzy sets by using interval-valued loss functions. Finally, we employ several
examples to illustrate that how to take an action for an object with
interval-valued membership grade by using interval-valued loss functions.
