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Exact Credibilistic Expectations of Nonmonotone Fuzzy Transformations with Applications to Higher-Order Moments

Year: 2026
Type: article
Abstract: Nonmonotone transformations of fuzzy numbers arise in loss functions, dispersion measures, and higher-order risk criteria, but inverse-credibility laws apply directly only to monotone maps. Existing methods typically either compute transformed alpha-cuts without a credibilistic scalar functional or use endpoint formulas whose validity depends on monotonicity. We develop a calculus that preserves interior extrema before integration. For any continuous transformation of a fuzzy number whose transformed output is integrable, the expected value admits an extrema-based alpha-cut representation that requires neither regularity nor an inverse credibility distribution. For continuous nondecreasing radial losses, we derive the exact risk, prove a sharp factor-two bound for the conservative endpoint proxy, characterize both equality regimes, and localize the proxy excess to center-containing cuts. We also establish Lipschitz perturbation and certified midpoint-quadrature bounds. At the decision level, proxy constraints are conservative and proxy minimization has a factor-two guarantee. A reproducible experiment further shows that endpoint substitution can reverse the optimizer and contract the feasible set. The framework is further specialized to higher-order credibilistic moments: for regular fuzzy numbers, signed odd moments retain inverse-distribution formulas, whereas absolute and even moments are obtained from the extrema-based formulation without requiring regularity. Triangular and Gaussian closed forms, standardized coefficients, a scale-free bias function, and deterministic audits demonstrate the resulting error, stability, computation, and decision guarantees.
Source: Mathematics
DOI: 10.3390/math14203768
Language: English
Open Access status: gold
Estimated APC paid: $3,137
OpenAlex ID: W7222963287
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Cites: 23
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Related to: 10