In recent years, numerous scholars have investigated the relationship between symmetry and generalized convexity. Due to this close relationship, generalized convexity and symmetry have become new areas of study in the field of inequalities. With the help of fuzzy up and down relation, the class of up and down $\lambda$-convex fuzzy-number valued mappings is introduced in this study; and weighted Hermite-Hadamard type fuzzy inclusions are demonstrated for these functions. The product of two up and down $\lambda$-convex fuzzy-number valued mappings also has Hermite-Hadamard type fuzzy inclusions, which is another development. Additionally, by imposing some mild restrictions on up and down $\lambda$-convex ( $\lambda$-concave) fuzzy number valued mappings, we have introduced two new significant classes of fuzzy number valued up and down $\lambda$-convexity ( $\lambda$-concavity), referred to as lower up and down $\lambda$-convex (lower up and down $\lambda$-concave) and upper up and down $\lambda$-convex ( $\lambda$-concave) fuzzy number valued mappings. Using these definitions, we have amassed many classical and novel exceptional cases that implement the key findings. Our proven results expand and generalize several previous findings in the literature body. Additionally, we offer appropriate examples to corroborate our theoretical findings.

To create various kinds of inequalities, the idea of convexity is essential. Convexity and integral inequality hence have a significant link. This study's goals are to introduce a new class of generalized convex fuzzy-interval-valued functions (convex 𝘍𝘐𝘝𝘍s) which are known as $\left(\mathfrak{p},\mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s and to establish Jensen, Schur and Hermite-Hadamard type inequalities for $\left(\mathfrak{p},\mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s using fuzzy order relation. The Kulisch-Miranker order relation, which is based on interval space, is used to define this fuzzy order relation level-wise. Additionally, we have demonstrated that, as special examples, our conclusions encompass a sizable class of both new and well-known inequalities for $\left(\mathfrak{p},\mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s. We offer helpful examples that demonstrate the theory created in this study's application. These findings and various methods might point the way in new directions for modeling, interval-valued functions and fuzzy optimization issues.

This article elaborates the enormous theory of MULTIMOORA (multi-objective optimization ratio analysis plus full multiplicative form) method to build up a new outranking approach for the innovative extension of fuzzy set theory, namely, 2-tuple linguistic Fermatean fuzzy sets (2TLFFSs). The main objective of the proposed work is to expand and present the components of MULTIMOORA method in 2-tuple linguistic Fermatean fuzzy framework. The resulted technique is named as 2-tuple linguistic Fermatean fuzzy MULTIMOORA method. This technique is designed to tackle the unclear information using 2-tuple linguistic Fermatean fuzzy numbers (2TLFFNs). The proposed model is intrinsically superior to deal with one-dimensional linguistic data. The 2TLFF-MULTIMOORA method takes into account standard relative correlations. Also, it handles the rank inversion problem when changing the rank of alternatives by adding one or more alternatives. The algorithm designed for the proposed methodology is elaborated with a numerical example (to opt for the most favorable city for the selection of quality of life). The accuracy and precision of the proposed strategy is determined by narrating a comparative study. Finally, the advantages of the developed technique over existing methods are discussed briefly.

In the literature, extensions of common fuzzy sets have been proposed one after another. The recent addition is spherical fuzzy sets theory, which is based on three independent membership parameters established on a unit sphere with a restriction linked to their squared summation. This article uses the new extension that presents bigger domains for each parameter for production design. A systematic approach for determining customer demands or requirements, Quality Function Deployment (QFD) converts them into the final production to fulfill these demands in a decision-making environment. In order to prevent information loss during the decision-making process, it offers a useful technique to describe the linguistic analysis in terms of 2-tuples. This research introduces a novel decision-making method utilizing the 2-tuple linguistic $T$-spherical fuzzy numbers (2TL $T$-SFNs) in order to select the best alternative to manufacturing a linear delta robot. Taking into account the interaction between the attributes, we develop the 2TL $T$-SF Hamacher (2TL $T$-SFH) operators by using innovative operational rules. These operators include the 2TL $T$-SFH weighted average (2TL $T$-SFHWA) operator, 2TL $T$-SFH ordered weighted average (2TL $T$-SFHOWA) operator, 2TL $T$-SFH hybrid average (2TL $T$-SFHHA) operator, 2TL $T$-SFH weighted geometric (2TL $T$-SFHWG) operator, 2TL $T$-SFH ordered weighted geometric (2TL $T$-SFHOWG) operator, and 2TL $T$-SFH hybrid geometric (2TL $T$-SFHHG) operator. In addition, we discuss the properties of 2TL $T$-SFH operators such as idempotency, boundedness, and monotonicity. We develop a novel approach according to the CODAS (Combinative Distance-based Assessment) model in order to deal with the problems of the 2TL $T$-SF multi-attribute group decision-making (MAGDM) environment. Finally, to validate the feasibility of the given strategy, we employ a quantitative example to select the best alternative to manufacture a linear delta robot. The suggested information-based decision-making methodology which is more extensively adaptable than existing techniques prevents the risk of data loss and makes rational decisions.

A contemporary fuzzy technique is employed in the current study to generalize some established and recent findings. For researchers, fixed point (FP) procedures are highly advantageous and appealing mechanisms. Discovering fuzzy fixed points of fuzzy mappings (FM) meeting Nadler's type contraction in complete fuzzy metric space (FMS) and?iri? type contraction in complete metric spaces (MS) is the core objective of this research. The outcomes are backed up by example and applications that highlight these findings. There are also preceding conclusions that are given as corollaries from the relevant literature. In this mode, numerous consequences exist in the significant literature are extended and combined by our findings.

The main purpose of this study is to examine the existence of fuzzy fixed points of fuzzy mappings meeting the criteria of some generalized contractions of Nadler's type in the framework of complete b-metric spaces. From the pertinent literature, there are additional previous observations that are provided as corollaries. Our study expands and incorporates several implications that are apparent in this mode and are addressed in considerable literature.

T-spherical fuzzy sets (T-SPFSs) have gained popularity because of their ability to account for uncertainty more effectively and spanning a larger domain. The sum of the t- $th$ power of membership grades in T-SPFSs is close to a unit interval, allowing for greater uncertainty. As a result, this set outperforms traditional fuzzy structures. The "multi-criteria decision-making" (MCDM) approach is a widely used technique that requires the use of some aggregation tools, and various such aggregation operators (AOs) have been developed over the years to achieve this purpose. The purpose of this paper is to propose some new operational laws and AOs for use in a T-spherical fuzzy environment. In this regard, we presented some new neutral or fair operational rules that combine the concept of proportional distribution to provide a neutral or fair solution to the membership, abstinence, and non-membership of T-spherical fuzzy numbers (T-SPFNs). Based on the obtained operational rules, we presented the "T-spherical fuzzy fairly weighted average operator" and the "T-spherical fuzzy fairly ordered weighted averaging operator". Compared to earlier methodologies, the proposed AOs provide more generalised, reliable, and accurate information. In addition, under T-SPFSs, an MCDM approach is developed employing suggested AOs with several decision-makers (DMs) and partial weight details. Finally, to demonstrate the applicability of the innovative technique, we give an actual case study of "food waste treatment technology" (FWTT) selection under T-SPFSs scenarios. A comparison with an existing model has also been undertaken to confirm the validity and robustness of the acquired results.