Journal of Neutrosophic and Fuzzy Systems

Fuzzy c-means clustering assigns every data point a degree of membership to each cluster but offers no separate account of how ambiguous that assignment is. This paper builds a single-valued neutrosophic layer on top of the classical fuzzy c-means membership distribution, representing each point by a truth-membership Ti (its strongest cluster membership), an indeterminacy Ii (the normalized Shannon entropy of its full membership vector), and a falsity-membership Fi = 1−Ti. Four results are proved: the fuzzy c-means update equations are re-derived from the Lagrangian stationarity conditions of the underlying constrained optimization; the resulting (Ti, Ii,Fi) triplet is shown to be bounded and to attain its extremes exactly at crisp and maximally ambiguous membership distributions; a score function combining the three components is shown to be strictly monotone in each; the natural root-mean-square distance between two neutrosophic triplets is shown to satisfy the metric axioms; and, for the two-cluster case specifically, indeterminacy is proved to be an exact deterministic function of truth-membership, so that a third, genuinely independent source of information requires three or more clusters. Every result is checked numerically, including a direct verification of the two-cluster degeneracy result to floating-point precision. Applied to the Breast CancerFuzzy c-means clustering assigns every data point a degree of membership to each cluster but offers no separate account of how ambiguous that assignment is. This paper builds a single-valued neutrosophic layer on top of the classical fuzzy c-means membership distribution, representing each point by a truth-membership Ti (its strongest cluster membership), an indeterminacy Ii (the normalized Shannon entropy of its full membership vector), and a falsity-membership Fi = 1−Ti. Four results are proved: the fuzzy c-means update equations are re-derived from the Lagrangian stationarity conditions of the underlying constrained optimization; the resulting (Ti, Ii,Fi) triplet is shown to be bounded and to attain its extremes exactly at crisp and maximally ambiguous membership distributions; a score function combining the three components is shown to be strictly monotone in each; the natural root-mean-square distance between two neutrosophic triplets is shown to satisfy the metric axioms; and, for the two-cluster case specifically, indeterminacy is proved to be an exact deterministic function of truth-membership, so that a third, genuinely independent source of information requires three or more clusters. Every result is checked numerically, including a direct verification of the two-cluster degeneracy result to floating-point precision. Applied to the Breast CancerWisconsin Diagnostic dataset (569 cases, 30 measured features), the clustering recovers the malignant/benign partition with 91.4% accuracy and an adjusted Rand index of 0.683, matching a hard k-means baseline on point accuracy; the neutrosophic layer nonetheless adds diagnostic information the hard baseline cannot provide, since indeterminacy is significantly higher for misclassified cases than for correctly classified ones (Mann–Whitney U-test, p < 10−18), correctly flagging the cases nearest the decision boundary as the ones most likely to be wrong.Wisconsin Diagnostic dataset (569 cases, 30 measured features), the clustering recovers the malignant/benign partition with 91.4% accuracy and an adjusted Rand index of 0.683, matching a hard k-means baseline on point accuracy; the neutrosophic layer nonetheless adds diagnostic information the hard baseline cannot provide, since indeterminacy is significantly higher for misclassified cases than for correctly classified ones (Mann–Whitney U-test, p < 10−18), correctly flagging the cases nearest the decision boundary as the ones most likely to be wrong.

Label corruption is difficult for prototype classifiers because a mislabeled observation does two things at once: it perturbs the prototype vyi associated with the supplied label and obscures whether the observation is genuinely ambiguous or simply inconsistent with its assigned class. This paper develops an adaptive fuzzy–neutrosophic prototype learning algorithm that separates these effects. For each training observation, the membership vector ui = (ui1, . . . ,uiK) ∈ ΔK−1 induced by the current prototypes is converted into the evidence state zi = (Ti, Ii,Fi) ∈ [0,1]3: truth is the membership assigned to the observed class, falsity is the strongest competing membership, and indeterminacy is the normalized membership entropy. These quantities drive three coupled mechanisms: a contradiction margin ci = [Fi −Ti]+ that attenuates unreliable labels, an entropy-dependent fuzzy exponent mi ∈ [mmin,mmax] that adapts membership weighting near class overlap, and a conservative soft-label correction activated only when Fi > Ti and the hesitation Ii is sufficiently small. The resulting Adaptive Fuzzy–Neutrosophic Prototype Learning (AFNPL) algorithm remains a lightweight prototype method with linear cost in the number of observations, classes and features per iteration. A reproducible three-class study evaluates 0–40% cyclic asymmetric label corruption under low, medium and high class overlap. At medium overlap and 40% corruption, AFNPL obtains 91.54% test accuracy and 91.55% macro-F1, compared with 72.52%/72.37% for noisy class means and 80.23%/80.17% for trimmed class means. Its internal contradiction score also detects corrupted labels with mean AUC between 0.959 and 0.971 across the contaminated settings. The contribution is therefore not only a robust prototype update, but a fuzzy learning mechanism in which neutrosophic truth, indeterminacy and falsity have explicit algorithmic roles in label reliability and adaptive fuzzy weighting.

Let x = (T, I,F) ∈ [0,1]3 denote a single-valued neutrosophic assessment and let wT +wI +wF = 1 with wc > 0. This paper introduces the probability-completed embedding Φw(x) = 􀀀 wT T,wT (1−T),wI I,wI(1−I), wFF,wF (1−F) ∈ Δ5. and the pullback distance dw(x,y) = [J(Φw(x),Φw(y))/log2]1/2 , where J is Jensen–Shannon divergence. The construction yields a bounded metric, separates into three weighted Bernoulli Jensen–Shannon terms, is invariant under the neutrosophic complement xc = (F,1−I,T) when wT = wF , and has the local information metric d2w (x,x+δ) = 1 8log2 Σ c∈{T,I,F} wcδ2 c xc(1−xc) +O(∥δ∥3). On this geometry, robust consensus is posed as the bounded M-estimation problem bxτ = argmin x∈(0,1)3 mΣ r=1 ar{1−e−τd2w (x,xr)}. A majorization–minimization iteration reduces each step to three one dimensional weighted Jensen–Shannon barycenters and decreases the objective monotonically. The induced expert weight is proportional to e−τd2w, so strongly conflicting assessments are downweighted without a hard rejection threshold. In a reproducible contamination study with 15 experts, 800 replications at each of five contamination levels, and a fixed τ = 60, the proposed estimator has mean normalized Jensen–Shannon error 0.0149 at 40% oppositional contamination; the coordinate median, ordinary Jensen–Shannon barycenter, and arithmetic mean obtain 0.0380, 0.1448, and 0.1489, respectively. The contribution is therefore a metric and optimization framework for consensus itself, rather than another ranking operator: neutrosophic disagreement is represented on a common information-geometric scale and robust aggregation follows from a bounded variational principle.

Cultivar identification from viticultural chemical profiles is a multiclass recognition problem in which a hard label alone does not reveal whether global chemometric evidence agrees with the local structure of previously observed samples. This paper proposes Neutrosophic–Fuzzy Viticultural Evidence Fusion (NFVEF), an uncertainty-aware classifier that combines a global discriminant probability vector G(x) ∈ ΔK−1 with a Gaussian fuzzy-neighborhood vector L(x) ∈ ΔK−1. For every cultivar k, the two evidence views are converted into Tk = p GkLk, Fk = p (1−Gk)(1−Lk), Ik = 1−Tk −Fk. where Ik is exactly the squared Hellinger disagreement between the Bernoulli support views Gk and Lk. A logarithmic fuzzy opinion pool Hk ∝ Gηk L1−η k is then attenuated by neutrosophic disagreement, Rk ∝ Hk exp(−κIk), before classification. The winning class is accompanied by an uncertainty score U = 1−Tˆk (1−Iˆk)(1−Fˆk ), enabling uncertain chemical profiles to be flagged rather than reported with unqualified confidence. The method is evaluated on the UCI Wine cultivar dataset using 60 repeated stratified splits at four synthetic analytical-perturbation levels δ ∈ {0,0.1,0.2,0.3} measured relative to training-feature standard deviations. NFVEF obtains mean accuracies of 0.9858, 0.9836, 0.9744, and 0.9728, respectively. At δ = 0.3, its paired accuracy advantage over linear discriminant analysis is 0.00278 with a 95% bootstrap interval [0.00123,0.00463], while RBF-SVM remains slightly better in raw accuracy. The uncertainty score detects NFVEF errors with mean AUC 0.9520 at the strongest perturbation, and retaining the lowest-uncertainty 90% of cases yields 0.9922 accuracy. The contribution is therefore not universal classifier dominance, but a mathematically interpretable fuzzy–neutrosophic evidence layer for cultivar identification from ambiguous chemical measurements.