Loss Function Information Fusion and Decision Rule Deduction of Three-Way Decision by Constructing Interval-Valued $q$-Rung Orthopair Fuzzy Integral

Abstract

Three-way decision is consistent with human cognitive processes and applied to different information systems. However, since the real world is more imprecision, hesitation, and ambiguity, experts can select interval fuzzy information to represent their evaluations. In the evaluation process, experts sometimes may not guarantee the interval fuzzy information satisfy the constraint between membership and nonmembership, which further impacts make decisions. Interval-valued <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -rung orthopair fuzzy set (IVq-ROFS) are proposed to improve the fault-tolerant ability and ensure the experts’ evaluation make sense. In this case, this article generalizes IVq-ROFS to three-way decision, which consists of the construction of three-way decision and the aggregation of interval continuity of IVq-ROFS. For considering the aggregation of interval continuity of IVq-ROFS, we first investigate the IVq-ROFS integral to aggregate continuous IVq-ROFSs by developing simplified interval-valued <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -rung orthopair fuzzy set. Particularly, in order to obtain integrable region, we design <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> -amendment to amend integral region of IVq-ROFS for removing unmeaning point in advance. To simplify the calculation, we further utilize Taylor series of the integral function to replace the complicated integral function. Second, a new equivalence class method is constructed to support three-way decision. Third, with the aid of the integral of IVq-ROFS and equivalence class method, we availably fuse the loss functions of the equivalence class and obtain the classifying rules of three-way decision by comparing the expected losses. Finally, we elaborate three-way decision by using a case of medical diagnosis about nephropathy and prove the validity of the investigated model.