Classical fuzzy hypergraphs assign one membership value to each vertex, and this value remains the same in all hyperedges containing that vertex. In many applications, however, the role or participation of a vertex may change from one group to another. To represent such situations, we introduce edge-dependent fuzzy hypergraphs, where the membership value of a vertex is specified separately for each incident hyperedge. We define edge-dependent fuzzy subhyperedges, fuzzy [Formula: see text]-subhyperedges, fuzzy subhypergraphs, fuzzy semisubhypergraphs, and fuzzy [Formula: see text]-semisubhypergraphs. We also study fuzzy order, fuzzy size, vertex degree, and weighted vertex degree and obtain some basic results concerning these quantities. Incidence, adjacency, degree, and Laplacian matrices are introduced for the proposed model. Homomorphisms and isomorphisms of edge-dependent fuzzy hypergraphs are defined, and several structural quantities are shown to be invariant under isomorphism. An optimization model for research collaboration networks is included as an application. In this model, the same researcher may have different participation levels in different projects, which illustrates the need for edge-dependent membership in higher-order interactions.
This paper focuses on developing a method for solving a nonlinear fractional integro-differential system of Volterra–Hammerstein (NFIDS–VH) type with variable coefficients. The proposed technique employs a cubic B-spline curve as the approximate solution, while a quadratic B-spline curve is used as a predictor at a single mesh point to initiate the scheme, together with a collocation approach. Furthermore, all definite integral terms are approximated using the Gauss–Legendre quadrature formula. Following these computations, the NFIDS–VH type is transformed into a nonlinear algebraic system involving unknown control points. This nonlinear system is then solved iteratively using Newton's method to obtain accurate solutions. The proposed technique is implemented and evaluated through two illustrative examples to demonstrate its effectiveness, robustness and computational efficiency, with errors decreasing systematically as the degree of Legendre polynomial increases. Four distinct algorithms are developed to implement the proposed method, and custom MATLAB (version 9.2) code is written for each algorithm to solve the illustrative examples. In addition, comparisons between the exact and approximate solutions are presented through graphs and tables. Finally, a comparative analysis with existing methods is conducted to demonstrate the accuracy of the proposed approach.
We define a union operation for fuzzy semigraphs and prove that it is well defined when the underlying crisp union is a semigraph. Under this compatibility condition, common consecutive pairs and partial edges occur precisely within shared semigraph edges, while the consecutive-pair and induced memberships on nonshared edges remain unchanged. We obtain inclusion–exclusion formulas for order and size and an exact formula for the edge-membership sum. The latter separates the overlap correction from a nonnegative shared-edge min–max gap, which measures the difference between taking the minimum after pointwise maxima and taking the maximum of the original edge minima. We characterize when this gap vanishes. For two uniform edge fuzzy semigraphs of the same level, that level is preserved exactly when every shared edge has a consecutive pair attaining the minimum in both summands.
The main goal of this paper is to propose a novel methodology that combines the spherical fuzzy rough CRITIC (Criteria Importance Through Inter Criteria Correlation) method with the TODIM (an acronym in Portuguese of interactive and multi-criteria decision making) method for supplier selection. Spherical fuzzy rough numbers are used to preserve both the hesitancy of individual experts and the disagreement among them, while TODIM grounds the ranking stage in prospect theory and thus in the bounded rationality of the decision makers. To demonstrate its practical applicability, the proposed framework is applied to a real-world case study in which three experts evaluate five equipment suppliers of a table tennis club against six criteria. Delivery performance and service emerge as the most influential criteria, and A3 is identified as the most suitable supplier. The robustness of the solution is examined by varying the attenuation factor over two orders of magnitude, across which the best alternative remains unchanged, and by comparison with spherical fuzzy rough TOPSIS, WASPAS and MULTIMOORA formulations and with a crisp CRITIC–TODIM baseline. The findings offer valuable insights for managers and policymakers in the sports industry.
Accurate and computationally efficient simulation of nonlinear Klein-Gordon and Sine-Gordon equations is important for analyzing nonlinear wave propagation, oscillatory dynamics and localized structures arising in engineering and physical systems. In this work, two differential quadrature approaches with cubic Uniformly Algebraic Trigonometric (UAT) and Uniformly Algebraic Hyperbolic (UAH) tension B-splines are proposed and compared. The spatial derivatives are discretized by using the corresponding spline-based DQM weighting coefficients. The resulting semi-discrete systems are integrated in time by the Runge-Kutta 4(3) scheme. The methods are tested on the one-dimensional Klein-Gordon and the one- and two-dimensional Sine-Gordon benchmarks. Accuracy is assessed through L ∞ , RMS, and relative errors, exact-numerical solution comparisons, grid- and time-dependent error behavior, and convergence analysis. Matrix/eigenvalue analysis is employed to investigate numerical stability, while computational complexity, CPU time, wall-clock time, and memory requirements are evaluated to assess efficiency. The UAT- and UAH-based formulations produce nearly identical numerical results under matched computational conditions. Representative errors remain bounded, ranging from approximately 10 –9 – 10 –10 for highly accurate Klein–Gordon cases 10 –4 – 10 –5 to for the considered Sine-Gordon benchmarks. The stability and computational analyses further demonstrate reliable performance with modest memory requirements. These results establish both tension B-spline DQM formulations as accurate and computationally viable alternatives for nonlinear wave problems in scientific and engineering applications.
