In practical situations, interval-valued fuzzy sets are frequentlyencountered. In this paper, firstly, we present shadowed sets for interpretingand understanding interval fuzzy sets. We also provide an analytic solution tocomputing the pair of thresholds by searching for a balance of uncertainty inthe framework of shadowed sets. Secondly, we construct errors-based three-wayapproximations of interval-valued fuzzy sets. We also provide an alternativedecision-theoretic formulation for calculating the pair of thresholds bytransforming 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-valuedfuzzy sets by using interval-valued loss functions. Finally, we employ severalexamples to illustrate that how to take an action for an object withinterval-valued membership grade by using interval-valued loss functions.
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