Journal of Neutrosophic and Fuzzy Systems

Smart irrigation decisions are often computed from multiple sensor and model channels whose evidence may be gradual, missing, or mutually contradictory. A conventional fuzzy controller compresses these conditions into a scalar membership µ ∈ [0, 1], whereas a single-valued neutrosophic representation can retain support, indeterminacy, and opposition as (T, I, F ). This paper proposes a contradiction-aware neutrosophic–fuzzy irrigation system (CNFIS) in which three evidence channels for criterion j are summarized by an availability ratio qj , a robust fuzzy center m j , and a contradiction index cj . The resulting state is(Tj , Ij , Fj) = 􀀀 qj emj , 1 − qj + qjcj , qj(1 − emj), which satisfies Tj + Ij + Fj = 1 + qjcj and therefore separates missingness from contradiction without interpreting the triple as a probability vector. A neutral-prior shrinkage rule sj = (1 − αIj)rj + αIj/2 is then fused through b d = P j wjsj to estimate normalized irrigation demand, while U = P j wjIj supplies an explicit re-sensing or abstention signal. The method is evaluated in a fully reproducible Monte Carlo benchmark containing 30 independent runs of N = 5000 cases under five evidence regimes; no field measurements are claimed. At moderate missingness/corruption (pm, pc) = (0.10, 0.10), CNFIS obtains RMSE 0.03318 ± 0.00086 compared with 0.04192 ± 0.00079 for mean fuzzy fusion, a reduction of 20.84%. The reductions remain 12.63% and 10.57% under severe and extreme regimes, respectively, while the ordinary fuzzy mean retains a small advantage in clean evidence. At 80% selective coverage, the uncertainty score decreases CNFIS RMSE from 0.03320 to 0.02825 in the moderate regime. These results support the proposed representation as a mathematically transparent robustness layer for irrigation systems in which evidence quality, not only fuzzy demand, must be modeled explicitly.

Medical diagnosis is an uncertainty-sensitive decision problem in which a patient state x = (x1, . . . , xm) is mapped to a candidate disease dk ∈ C using incomplete, gradual, and sometimes conflicting evidence. Fuzzy systems represent such evidence by a membership grade µ(x) ∈ [0, 1]; intuitionistic fuzzy systems use inde-pendently assigned (µ, ν) with the derived hesitation π = 1 − µ − ν; and single-valued neutrosophic systems use an independently specified triple (T, I, F ) ∈ [0, 1 ]3. This article provides a structured mathematical re-view of these models in medical diagnosis using literature available no later than 31 May 2024. The literature is synthesized by representation space, constraints, distance and similarity measures, aggregation operators, temporal structure, and diagnostic decision rules. A common embedding Φ places fuzzy and intuitionistic fuzzy states inside the neutrosophic cube Ω = [0, 1]3, making the geometric relation among the three model families explicit. Within this space, a weighted diagnostic distance is written as D(r) k =  Xm j=1 wj 3 (|Tpj − Tkj |r + |Ipj − Ikj |r + |Fpj − Fkj |r)   1/r, and the normalization residual κ = T + I + F − 1 is used to distinguish normalized states from under-specified or overlapping evidence without interpreting (T, I, F ) as probabilities. The review further devel-ops a synthesis-derived decision layer that combines distance, indeterminacy, and diagnostic margin through Sk = (1 + Dk)−1, Ck = Sk(1 − I¯p)γ , and ∆C = C(1) − C(2). The analysis indicates that fuzzy systems remain appropriate when uncertainty is predominantly gradual and rule interpretability is central, whereas neutrosophic systems are better justified when independent support, opposition, and indeterminacy must be retained. The resulting taxonomy clarifies when additional uncertainty dimensions are mathematically infor-mative and identifies calibration, metric stability, explainability, temporal modeling, and clinical validation as the principal unresolved problems.