Trade-off principle for standard shadowed sets and its generalization to five-regions

Abstract

A shadowed set S interprets and makes decision with a fuzzy set F from its approximation regions and a tri-valued mapping μS:F⟶{0,0.5,1} on fuzzy membership grades. The original idea of shadowed sets is to balance the uncertainty of F in S by a principle of uncertainty relocation. This paper proposes a principle of making a trade-off between uncertainty for certainty to minimize the amount of unclassified data and maximize the number of items for which crisp decisions are made. We provide detailed derivations for determining the optimum partition threshold for a trade-off three-region shadowed set and generalized it to five-region model S5. We investigate the existence and uniqueness of the optimum partition threshold of S5. Also, we outline some application examples where five-region shadowed sets can be reasonably exploited, including fuzzy clustering, decision-making, etc. To deliver guidance on how to construct S5, several detailed numeric examples are also provided.