Neutrosophic and Information Fusion

Supplier selection is a recurrent multi-criteria decision-making (MCDM) problem in which expert judgements are rarely crisp: procurement managers routinely hesitate, disagree, and abstain. Classical fuzzy models capture membership and (at most) non-membership, but they cannot separately encode the indeterminacy that pervades real committee evaluations. This paper develops a supplier-selection model built on single-valued neutrosophic sets (SVNSs), where every judgement is represented by an independent triple of truth, indeterminacy and falsity degrees. Group opinions are aggregated by a single-valued neutrosophic weighted averaging operator, providing a transparent information-fusion step, after which an extended TOPSIS procedure ranks the alternatives by their relative closeness to neutrosophic ideal solutions. A worked case with five suppliers and six criteria illustrates the pipeline end to end, and a sensitivity study over the score function and criteria weights confirms that the top-ranked supplier is stable across 200 weight perturbations. Benchmarked against fuzzy-TOPSIS and intuitionistic-fuzzy TOPSIS baselines, the neutrosophic model separates genuinely ambiguous suppliers from clearly dominated ones more reliably, preserving a wider spread of closeness coefficients in the contested middle of the ranking.

Reliable multi-source AI requires a distinction between two non-equivalent failure modes: within-source ambiguity and between-source contradiction. This paper formulates dual-indeterminacy neutrosophic fusion (DINF), in which calibrated posteriors ps ∈ ΔK−1 are transformed into class-wise triples Nk = ⟨Tk, Ik,Fk⟩. Source weights combine chance-corrected validation reliability, normalized entropy, and a robust Jensen–Shannon consensus discount. Intrinsic ambiguity U and class-specific contradiction Ck are retained separately and joined by Ik =U +Ck −UCk. The decision distribution qk ∝ Tk exp(−λIk) therefore penalizes conflict without collapsing neutrosophic evidence into a probability simplex. Boundedness, permutation invariance, consensus preservation, conflict monotonicity, log-odds sensitivity, and missing-source neutrality are established. Repeated experiments on three public benchmarks examine clean data, 30% source dropout, 30% confident contradiction, and 40% mixed failure. Under contradiction, DINF achieves macro-F1 0.910, NLL 0.445, and selective accuracy 0.939 at approximately 90% coverage; the corresponding logarithmic-pool values are 0.896, 0.469, and 0.929. The results identify robust agreement discounting and explicit contradiction-sensitive indeterminacy as complementary mechanisms for calibrated failure-aware fusion.

Neutrosophic set theory, which explicitly models truth (T ), indeterminacy (I), and falsity (F) as independent membership components, has emerged as one of the most active mathematical frameworks for uncertain information fusion over the 2020–2025 period. This comprehensive survey reviews, synthesises, and critically analyses more than 200 research contributions spanning single-valued neutrosophic sets (SVNS), interval neutrosophic sets (INS), neutrosophic cubic sets (NCS), neutrosophic Z-numbers, linguistic neutrosophic sets, and their integration with Dempster-Shafer evidence theory. We organise the literature across four interlocking axes— mathematical foundations, aggregation operators, information measures, and decision-support methods—and map these onto seven application domains including medical diagnosis, supply chain management, environmental assessment, and engineering fault diagnosis. Three representative algorithms are formally presented with pseudocode, complexity analysis, and mathematical justifications: (i) the SVNWA entropy weighted aggregation framework, (ii) the Neutrosophic Dempster-Shafer Evidence Theory (N-DSET) fusion pipeline with conflict r edistribution, a nd (iii) the Neutrosophic TOPSIS multi-criteria d ecision-making a lgorithm. A comparative performance analysis shows that neutrosophic methods achieve mean AUC improvements of +4.2% to +7.1% over intuitionistic fuzzy set baselines across reported experimental studies. Six precisely formulated open problems are identified, and a five-horizon research roadmap from 2025 to 2030 is proposed, covering mathematical completeness, computational scalability, hybrid deep-learning architectures, domain expansion to quantum and large language model settings, and the long-term vision of a unified neutrosophic information quality standard.

Decision environments that combine laboratory indicators, expert warnings, chemical descriptors, and regulatory traces rarely produce a single consistent description of risk. Classical aggregation rules usually collapse incomplete, contradictory, and partially reliable evidence into one scalar before the contradiction itself has been modelled. This paper develops an indeterminacy-balanced neutrosophic granulation method for prioritization problems in which truth, falsity, and hesitation must remain simultaneously visible during fusion. Each alternative is represented by a single-valued neutrosophic profile, criterion weights are obtained from a contrast-sensitive entropy functional, and the final ranking is produced by an indeterminacy-penalized evidence score. The mathematical contribution is a bounded fusion operator that separates positive support, negative pressure, and contradiction-induced hesitation. A numerical study reports detailed intermediate matrices, criterion weights, fused memberships, ranking stability, sensitivity to the indeterminacy penalty, ablation results, and computational complexity. The findings show that retaining indeterminacy during fusion changes the ordering of borderline alternatives and makes the decision trace easier to audit than scalar aggregation alone.

Neutrosophic information fusion has become a rigorous computational approach for modeling evidence that is simultaneously supportive, opposing, and unresolved. This review synthesizes recent studies published from 2020 to 2025 and organizes the field around the operational semantics of truth, indeterminacy, and falsity. Rather than presenting neutrosophic sets only as an extension of fuzzy sets, the paper analyzes neutrosophic fusion as a mathematical problem of evidence representation, operator design, source weighting, contradiction control, and decision reduction. The review covers single-valued neutrosophic similarity measures, EDAS and TOPSIS extensions, neutrosophic Z-number aggregation, Einstein and Aczel–Alsina operators, trigonometric credibility operators, dynamic aggregation, divergence measures, uncertainty-aware multi-source information fusion, and evidence theoretic comparisons. A relatedwork section of more than twenty verified 2020–2025 studies is added, followed by a selection protocol, formal definitions, propositions, algorithms, operator-property analysis, and research directions. The paper concludes with a research agenda for benchmark construction, data-driven membership learn-ing, explainable indeterminacy, scalable dynamic fusion, and trustworthy integration of neutrosophic logic with intelligent decision-support systems.