Nowadays, several real-world decision-making problems concerning falling economies, power crises, depleting resources, etc., require efficient decision-making. To solve such problems, researchers proposed several hybrid models by combining the spherical fuzzy sets with other theories, such as spherical fuzzy soft sets, which is an efficient tool to deal with the uncertainties concerning positive, neutral, and negative memberships in the soft environment. However, all the existing hybridizations of spherical fuzzy sets fail to deal with information symmetrically in a bipolar soft environment. Accordingly, this paper presents a novel hybrid model called spherical fuzzy bipolar soft sets (SFBSSs) by fusing spherical fuzzy sets and bipolar soft sets, considering the opposite sets of parameters in symmetry. An example considering the selection of a chief management officer (CMO) for a multi-national company illustrates the proposed model in detail. In addition, some symmetric properties and algebraic operations of the initiated model, including subset, complement, relative null SFBSSs, relative absolute SFBSSs, extended union, extended intersection, restricted union, restricted intersection, AND, and OR operations, are discussed and illustrated via numerical examples. Further, some fundamental results, namely, commutativity, associativity, distribution, and De Morgan’s laws are presented for SFBSSs. Moreover, by considering the massive impact of hydropower in combating the energy crisis and possible dangers to fish migration, a multi-attribute decision-making problem concerning the ranking of downstream fish passage designs for a hydroelectric project is modeled and solved under the developed algorithm based on SFBSSs. Finally, a comparative analysis discusses the supremacy of the initiated model over its building blocks.

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A computational model is presented to find the q -Bernstein quasi-minimal Bézier surfaces as the extremal of Dirichlet functional, and the Bézier surfaces are used quite frequently in the literature of computer science for computer graphics and the related disciplines. The recent work [1–5] on q -Bernstein–Bézier surfaces leads the way to the new generalizations of q -Bernstein polynomial Bézier surfaces for the related Plateau–Bézier problem. The q -Bernstein polynomial-based Plateau–Bézier problem is the minimal area surface amongst all the q -Bernstein polynomial-based Bézier surfaces, spanned by the prescribed boundary. Instead of usual area functional that depends on square root of its integrand, we choose the Dirichlet functional. Related Euler–Lagrange equation is a partial differential equation, for which solutions are known for a few special cases to obtain the corresponding minimal surface. Instead of solving the partial differential equation, we can find the optimal conditions for which the surface is the extremal of the Dirichlet functional. We workout the minimal Bézier surface based on the q -Bernstein polynomials as the extremal of Dirichlet functional by determining the vanishing condition for the gradient of the Dirichlet functional for prescribed boundary. The vanishing condition is reduced to a system of algebraic constraints, which can then be solved for unknown control points in terms of known boundary control points. The resulting Bézier surface is q -Bernstein–Bézier minimal surface.

The term metric or distance of a graph plays a vital role in the study to check the structural properties of the networks such as complexity, modularity, centrality, accessibility, connectivity, robustness, clustering, and vulnerability. In particular, various metrics or distance-based dimensions of different kinds of networks are used to resolve the problems in different strata such as in security to find a suitable place for fixing sensors for security purposes. In the field of computer science, metric dimensions are most useful in aspects such as image processing, navigation, pattern recognition, and integer programming problem. Also, metric dimensions play a vital role in the field of chemical engineering, for example, the problem of drug discovery and the formation of different chemical compounds are resolved by means of some suitable metric dimension algorithm. In this paper, we take rotationally symmetric and hexagonal planar networks with all possible faces. We find the sequences of the local fractional metric dimensions of proposed rotationally symmetric and planar networks. Also, we discuss the boundedness of sequences of local fractional metric dimensions. Moreover, we summarize the sequences of local fractional metric dimension by means of their graphs.

A complex picture fuzzy set (Com-PFS) is a motivating tool for more precisely interpreting fuzzy notions. Recently, all extensions of complex fuzzy graphs (Com-FGs) have become a growing research topic as they handle ambiguous situations more explicitly than complex intuitionistic fuzzy graphs (Comp-IFG) and picture fuzzy graphs (PFG). The primary goal of this study is to demonstrate the foundation of Com-PFG due to the drawback of the complex neutral membership function in Com-IFG. This paper introduces the concept of Com-PFGs and explains many of the approaches to their development. A Com-PFG has three complex membership functions. We define the order, size, degree of vertex, and total degree of vertex in Com-PFGs. We elaborate on primary operations including: complement, join, union, Cartesian product, direct product, and composition of Com-PFG. Finally, we discuss its use in decision-making problems.