A q-rung orthopair fuzzy set is one of the effective generalizations of fuzzy set for dealing with uncertainties in information. Under this environment, in this study, we define a new type of extensions of fuzzy sets called n,m-rung orthopair fuzzy sets and investigate their relationship with Fermatean fuzzy sets, Pythagorean fuzzy sets and intuitionistic fuzzy sets. The n,m-rung orthopair fuzzy sets can supply with more doubtful circumstances than Fermatean fuzzy sets, Pythagorean fuzzy sets and intuitionistic fuzzy sets because of their larger range of depicting the membership grades. There is a symmetry between the values of this membership function and non-membership function. Here, any power function scales are utilized to widen the scope of the decision-making problems. In addition, the novel notion of an n,m-rung orthopair fuzzy set through double universes is more flexible when debating the symmetry between two or more objects that are better than the diffusing concept of an n-rung orthopair fuzzy set, as well as m-rung orthopair fuzzy set. The main advantage of n,m-rung orthopair fuzzy sets is that it can describe more uncertainties than Fermatean fuzzy sets, which can be applie in many decision-making problems. Then, we discover the essential set of operations for the n,m-rung orthopair fuzzy sets along with their several properties. Finally, we introduce a new operator, namely, n,m-rung orthopair fuzzy weighted power average (n,m-ROFWPA) over n,m-rung orthopair fuzzy sets and apply this operator to the MADM problems for evaluation of alternatives with n,m-rung orthopair fuzzy information.

The concept of soft sets was proposed as an effective tool to deal with uncertainty and vagueness. Topologists employed this concept to define and study soft topological spaces. In this paper, we introduce the concepts of soft SD-continuous, soft SD-open, soft SD-closed and soft SD-homeomorphism maps by using soft somewhere dense and soft cs-dense sets. We characterize them and discuss their main properties with the help of examples. In particular, we investigate under what conditions the restriction of soft SD-continuous, soft SD-open and soft SD-closed maps are respectively soft SD-continuous, soft SD-open and soft SD-closed maps. We logically explain the reasons of adding the null and absolute soft sets to the definitions of soft SD-continuous and soft SD-closed maps, respectively, and removing the null soft set from the definition of a soft SD-open map.

Picture fuzzy nano topological spaces is an extension of intuitionistic fuzzy nano topological spaces. Every decision in life ends with an answer such as yes or no, or true or false, but we have an another component called abstain, which we have not yet considered. This work is a gateway to study such a problem. This paper motivates an enquiry of the third component—abstain—in practical problems. The aim of this paper is to investigate the contemporary notion of picture fuzzy nano topological spaces and explore some of its properties. The stated properties are quantified with numerical data. Furthermore, an algorithm for Multiple Attribute Decision-Making (MADM) with an application regarding the file selection of building material under uncertainty by using picture fuzzy nano topological spaces is developed. As a practical problem, a comparison table is presented to show the difference between the novel concept and the existing methods.

Enhancing competitive pressure is one of the most significant roles of supply chain management. The competitive environment and customer perception have shifted in favour of an ecological mentality. As a result, green supplier selection (GSS) has emerged as a critical problem. The challenge of green supplier selection striving for agility, durability, ecological sensitivity, leanness, and sustainability is tackled in this paper. In terms of recycling applications, environmental applications, carbon footprint, and water consumption, the environmental parameters evaluated in GSS and traditional supplier selection differ. Because of the form of the problem, a resolution is defined, which comprises an algorithm entrenched in the spherical linear Diophantine fuzzy sets (SLDFSs) Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) technique. Before discussing the approach of the SLDF model, some background information on SLDF sets is provided. To assure the uniqueness of this robust extension, different operations on SLDFSs are described, along with some concise interpretations to help the reader comprehend these ideas. A robust TOPSIS approach has been utilized in the issue of GSS by taking into consideration the multicriteria decision-making (MCDM) technique particularly useful in several areas, like analyzing and choosing traditional and environmental conventionalities. Due to linguistic criteria and the inability to assess all criteria, the fuzzy technique must be used with the TOPSIS method to lessen the consequences of instability and ambiguity. The spherical linear Diophantine fuzzy TOPSIS approach is employed, as it simplifies the evaluation of decision-makers and criteria. The hybrid technique resulting from integrating the SLDFS and TOPSIS is extremely successful in selecting which provider is more suited among the alternatives established on the criteria set by the order of significance, and this method may also be incorporated into similar issues.

In this paper, we focus on several ideas associated with linear Diophantine fuzzy soft sets (LDFSSs) along with its algebraic structure. We provide operations on LDFSSs and their specific features, elaborating them with real-world examples and statistical depictions to construct an inflow of linguistic variables based on linear Diophantine fuzzy soft (LDFSS) information. We offer a study of LDFSSs to the multi-criteria decision-making (MCDM) process of university determination, together with new algorithms and flowcharts. We construct LDFSS-TOPSIS, LDFSS-VIKOR and the LDFSS-AO techniques as robust extensions of TOPSIS (a technique for order preferences through the ideal solution), VIKOR (Vlse Kriterijumska Optimizacija Kompromisno Resenje) and AO (aggregation operator). We use the LDFSS-TOPSIS, LDFSS-VIKOR and LDFSS-AO techniques to solve a real-world agricultural problem. Moreover, we present a small-sized robotic agri-farming to support the proposed technique. A comparison analysis is also performed to examine the symmetry of optimal decision and to analyze the efficiency of the suggested algorithms.

In the present paper, we introduce the notion of r-fuzzy soft δ-open sets in fuzzy soft topological spaces. Furthermore, we define and characterize the notions of fuzzy soft δ-closure (δ-interior) operators. After that, we explore the notions of fuzzy soft δ-continuous (semi-continuous and pre-continuous) functions, which are weaker forms of fuzzy soft continuity. We also study some properties of these functions along with their mutual relationships with the help of some problems. Moreover, we present a decomposition of fuzzy soft semi-continuity. Lastly, as a weaker form of fuzzy soft continuity, we define and study the notions of fuzzy soft almost (weakly) continuous functions. Also, we explore the notion of continuity in a very general setting called fuzzy soft (L, M, N, O)-continuity and introduce a historical justification.