T-spherical fuzzy set is a recently developed model that copes with imprecise and uncertain events of real-life with the help of four functions having no restrictions. This article’s aim is to define some improved algebraic operations for T-SFSs known as Einstein sum, Einstein product and Einstein scalar multiplication based on Einstein t-norms and t-conorms. Then some geometric and averaging aggregation operators have been established based on defined Einstein operations. The validity of the defined aggregation operators has been investigated thoroughly. The multi-attribute decision-making method is described in the environment of T-SFSs and is supported by a comprehensive numerical example using the proposed Einstein aggregation tools. As consequences of the defined aggregation operators, the same concept of Einstein aggregation operators has been proposed for q-rung orthopair fuzzy sets, spherical fuzzy sets, Pythagorean fuzzy sets, picture fuzzy sets, and intuitionistic fuzzy sets. To signify the importance of proposed operators, a comparative analysis of proposed and existing studies is developed, and the results are analyzed numerically. The advantages of the proposed study are demonstrated numerically over the existing literature with the help of examples.

The concept of linear Diophantine fuzzy sets (LDFSs) is a new approach for modeling uncertainties in decision analysis. Due to the addition of reference or control parameters with membership and non-membership grades, LDFS is more flexible and reliable than existing concepts of intuitionistic fuzzy sets (IFSs), Pythagorean fuzzy sets (PFSs), and q-rung orthopair fuzzy sets (q-ROFSs). In this paper, the notions of linear Diophantine fuzzy soft rough sets (LDFSRSs) and soft rough linear Diophantine fuzzy sets (SRLDFSs) are proposed as new hybrid models of soft sets, rough sets, and LDFS. The suggested models of LDFSRSs and SRLDFSs are more flexible to discuss fuzziness and roughness in terms of upper and lower approximation operators. Certain operations on LDFSRSs and SRLDFSs have been established to discuss robust multi-criteria decision making (MCDM) for the selection of sustainable material handling equipment. For these objectives, some algorithms are developed for the ranking of feasible alternatives and deriving an optimal decision. Meanwhile, the ideas of the upper reduct, lower reduct, and core set are defined as key factors in the proposed MCDM technique. An application of MCDM is illustrated by a numerical example, and the final ranking in the selection of sustainable material handling equipment is computed by the proposed algorithms. Finally, a comparison analysis is given to justify the feasibility, reliability, and superiority of the proposed models.

In this article, we define a new fractional technique which is known as generalized proportional fractional (GPF) integral in the sense of another function $\Psi .$ The authors prove several inequalities for newly defined GPF-integral with respect to another function $\Psi .$ Our consequences will give noted outcomes for a suitable variation to the GPF-integral in the sense of another function $\Psi$ and the proportionality index $\varsigma$ . Furthermore, we present the application of the novel operator with several integral inequalities. A few new properties are exhibited, and the numerical approximation of these new operators is introduced with certain utilities to real-world problems.

A q-rung orthopair fuzzy set (q-ROFS) provides a significant mechanism for managing symmetrical aspects in real life circumstances. The renowned distinguishing feature of q-ROFS is that the sum of the qth powers to each membership degree (MD) and non-membership degree (NMD) is less than or equal 1, and therefore the comprehensive uncertain space for q-ROF information is broader. Numerous researchers have suggested several aggregation operators based on q-ROFSs. In order to discuss prioritized relationship in the criterion and a smooth approximation of q-ROF information, we introduced q-rung orthopair fuzzy Einstein prioritized weighted averaging (q-ROFEPWA) operator and q-rung orthopair fuzzy Einstein prioritized weighted geometric (q-ROFEPWG) operator. Additionally, we presented a multi-criteria group decision making (MCGDM) technique based on q-rung orthopair fuzzy Einstein prioritized aggregation operators. These operators can evaluate the possible symmetric roles of the criterion that express the real phenomena of the problem. In order to investigate characteristic of suggested operators regarding the symmetry of attributes and their symmetrical roles under q-ROF information, we presented an application of Einstein prioritized aggregation operators. Finally, by comparing it with some other established representative MCGDM models, an illustrative example is provided to check the feasibility, efficiency and supremacy of the proposed technique.

In the present paper, we aim to prove a new lemma and quantum Simpson’s type inequalities for functions of two variables having convexity on co-ordinates over $[\alpha ,\beta ]\times [\psi ,\varphi ]$ . Moreover, our deduction introduce new direction as well as validate the previous results.

In this investigation, for convex functions, some new $(p,q)$–Hermite–Hadamard-type inequalities using the notions of ${(p,q)}^{{\pi }_2}$ derivative and ${(p,q)}^{{\pi }_2}$ integral are obtained. Furthermore, for ${(p,q)}^{{\pi }_2}$-differentiable convex functions, some new ($p,q$) estimates for midpoint and trapezoidal-type inequalities using the notions of ${(p,q)}^{{\pi }_2}$ integral are offered. It is also shown that the newly proved results for $p=1$ and $q\to 1^{-}$ can be converted into some existing results. Finally, we discuss how the special means can be used to address newly discovered inequalities.