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Adaptive design focuses on creating flexible products that meet evolving demands and enhance sustainability. However, evaluating adaptive design concepts poses significant challenges due to the dynamic nature of product features over time and the inherent uncertainty in decision-makers' (DMs') evaluations. Most traditional frameworks rely on static models that fail to capture the temporal evolution of attributes and often overlook decision-makers' (DMs') confidence levels, resulting in incomplete or unreliable evaluations. To bridge these gaps, we propose the m-polar linguistic Z-number (mLZN) to address these issues. This framework uses the dynamic representation capabilities of m-polar fuzzy sets (mFSs) and the symmetrical structure of linguistic Z-numbers (LZNs), which effectively integrate linguistic evaluations with corresponding confidence levels, providing a balanced and robust approach to handling uncertainty. This approach models design characteristics across multiple periods while accounting for DMs' confidence levels. Based on this framework, we develop mLZN weighted and geometric aggregation operators, computation rules, and ranking methods to support dynamic multi-attribute group decision-making (MAGDM). The proposed framework's effectiveness is demonstrated through a case study on adaptive furniture design for children, which showcases its ability to dynamically evaluate key attributes, including safety, ease of use, fun, and comfort. Furthermore, we validate its robustness and feasibility through comprehensive sensitivity and comparative analyses. [ABSTRACT FROM AUTHOR]

Aggregation is a very efficient indispensable tool in which several input values are transformed into a single output value that further supports dealing with different decision-making situations. Additionally, note that the theory of m-polar fuzzy (mF) sets is proposed to tackle multipolar information in decision-making problems. To date, several aggregation tools have been widely investigated to tackle multiple criteria decision-making (MCDM) problems in an m-polar fuzzy environment, including m-polar fuzzy Dombi and Hamacher aggregation operators (AOs). However, the aggregation tool to deal with m-polar information under Yager's operations (that is, Yager's t-norm and t-conorm) is missing in the literature. Due to these reasons, this study is devoted to investigating some novel averaging and geometric AOs in an mF information environment through the use of Yager's operations. Our proposed AOs are named as the mF Yager weighted averaging (mFYWA) operator, mF Yager ordered weighted averaging operator, mF Yager hybrid averaging operator, mF Yager weighted geometric (mFYWG) operator, mF Yager ordered weighted geometric operator and mF Yager hybrid geometric operator. The initiated averaging and geometric AOs are explained via illustrative examples and some of their basic properties, including boundedness, monotonicity, idempotency and commutativity are also studied. Further, to deal with different MCDM situations containing mF information, an innovative algorithm for MCDM is established under the under the condition of mFYWA and mFYWG operators. After that, a real-life application (that is, selecting a suitable site for an oil refinery) is explored under the conditions of developed AOs. Moreover, the initiated mF Yager AOs are compared with existing mF Hamacher and Dombi AOs through a numerical example. Finally, the effectiveness and reliability of the presented AOs are checked with the help of some existing validity tests.

In the fuzzy multicriteria decision-making approach, a committee of decision-makers is usually involved in the assessment of the suitability of different alternatives based on the evaluation criteria by using linguistic terms and their equivalent fuzzy numbers. In this context, researchers have developed the Pythagorean fuzzy set (PFS) to overcome the limitation of intuitionistic fuzzy set in the description of decision-maker information such as imposing restrictions on the representation of membership and nonmembership grades. On the one hand, PFS still does not have sufficient ability and flexibility to deal with such issues. On the other hand, multipolar technology is used to operate large-scale systems in real-life situations, especially in dealing with dissatisfaction and indeterminacy grades for the alternatives of the reference set. Thus, m-polar fuzzy set is utilized and applied with other fuzzy sets because of its remarkable ability as a tool for depicting fuzziness and uncertainty under multipolar information in many circumstances. With the practical features of m-polar fuzzy set in combination with PFS, this paper employs it to extend two considerable MCDM methods, namely, fuzzy decision by opinion score method and fuzzy-weighted zero inconsistency. Such extensions, called Pythagorean m-polar fuzzy-weighted zero-inconsistency (Pm-PFWZIC) method and Pythagorean m-polar fuzzy decision by opinion score method (Pm-PFDOSM), are formulated to weight the evaluation criteria followed by alternative ranking progressively. The research methodology is presented as follows. Firstly, the mechanisms of Pm-PFWZIC and Pm-PFDOSM are formulated and integrated into the development phase. Secondly, the description of the real-world case study of the evaluation and benchmarking of the sign language recognition systems is adapted and presented. The result of Pm-PFWZIC shows that the criterion of 'finger movements' has the highest weight amongst the rest of the criteria, whereas 'misclassification error' has the lowest weight. In the ranking results, a variation of ranking is scored by each expert, and group decision-making is applied to solve the individual ranking variety. The robustness of the formulated methods is evaluated using systematic ranking, sensitivity analysis and comparison analysis. [ABSTRACT FROM AUTHOR]

Theory of m-polar fuzzy set deals with multi-polar information. It is used when data comes from m factors (m≥2). The primary objective of this work is to explore a generalized form of m-polar fuzzy subsemigroups, which is m-polar fuzzy ternary subsemigroups. There are many algebraic structures which are not closed under binary multiplication that is a reason to study ternary operation of multiplication such as the set of negative integer is closed under the operation of ternary multiplication but not closed for the binary multiplication. This paper, presents several significant results related to the notions of m-polar fuzzy ternary subsemigroups, m-polar fuzzy ideals, m-polar fuzzy generalized bi-ideals, m-polar fuzzy bi-ideals, m-polar fuzzy quasi-ideals and m-polar fuzzy interior ideals in ternary semigroups. Also, it is proved that every m- polar fuzzy bi-ideal of ternary semigroup is an m-polar fuzzy generalized bi-ideal of ternary semigroup but converse is not true in general. Moreover, this paper characterizes regular and intra-regular ternary semigroups by the properties of m-polar fuzzy ideals, m-polar fuzzy bi-ideals.

This article presents the idea of an m-polar fuzzy KU-ideal and investigates its characteristics. The relationship between an m-polar fuzzy KU-subalgebra and an m-polar fuzzy KU-ideal is presented in the discussion. [ABSTRACT FROM AUTHOR]

An multi-polar fuzzy set is a robust mathematical model to examine multipolar, multiattribute, and multi-index data. The multi-polar fuzzy sets was created as a useful mechanism to portray uncertainty in multiattribute decision making. In this article, we consider the theoretical applications of multi-polar fuzzy sets. We present the notion of multi-polar fuzzy sets in ordered semihypergroups and define multi-polar fuzzy hyperideals (bi-hyperideals, quasi hyperideals) in an ordered semihypergroup. Relations between multi-polar fuzzy hyperideals, multi-polar fuzzy bi-hyperideals and multi-polar fuzzy quasi hyperideals are discussed. [ABSTRACT FROM AUTHOR]

The notion of a q-rung orthopair fuzzy soft rough set ( $^{q}ROFSRS$ ) appeared as an extension of q-rung orthopair fuzzy set ( $^{q}ROFS$ ) and q-rung orthopair fuzzy soft set ( $^{q}ROFSS$ ) with the aid of rough set (RS) definition. Thus, $^{q}ROFSRS$ and m-polar fuzzy set ( $_{m}PFS$ ) are convenient to deal with uncertain knowledge which helps us to solve many problems in decision making. In this paper, we define the soft rough q-rung orthopair m-polar fuzzy sets ( $^{\text {q}}RO_{\text {m}}PFS$ ) and q-rung orthopair m-polar fuzzy soft rough sets ( $^{q}RO_{\text {m}}PFSRS$ ) through crisp soft and q- rung orthopair (q-RO) m-polar fuzzy soft approximation space. The related characteristics of these models are also studied. Then, we construct two new algorithms for these models to solve MADM issues. The successful application and corresponding comparative analyses proves that our proposed models are rational and effective.

Hesitant fuzzy sets (HFSs) present a general structure to express the uncertain concepts and data that have been served as in most of the generalizations of fuzzy sets. In this research article, we introduce a novel hybrid model called hesitant m-polar fuzzy sets (HmF-sets), which is a reasonable combination of HFSs with m-polar fuzzy sets (mF sets). It is the generalization of the concept HFSs, in which the membership degrees of an element of given set deals the m different numeric and fuzzy values that enables to deal the hesitancy of multipolar information. Hesitancy integrates the conformity for the analysis of given data, and an mF format concedes to severalize the sources of multi-polar information. We highlight and explore some useful properties, construct fundamental operations and investigate comparison laws of HmF-sets. Moreover, we develop the hesitant m-polar fuzzy TOPSIS approach for multi-criteria group decision-making (MCGDM), which is the natural extension of TOPSIS method and used to rank and choose the best alternative under HmF positive and negative ideal solutions to this framework. We describe applications of HmF-sets in group decision-making and apply our proposed method in real life examples to show its efficiency. Finally, we develop an algorithm that implements our decision-making procedure by using computer programming. [ABSTRACT FROM AUTHOR]

The notions of m -polar fuzzy subalgebras and m -polar fuzzy (closed, commutative) ideals are introduced, and related properties are investigated. Characterizations of m -polar fuzzy subalgebras and m -polar fuzzy (commutative) ideals are considered. Relations between m -polar fuzzy subalgebras, m -polar fuzzy ideals and m -polar fuzzy commutative ideals are discussed. [ABSTRACT FROM AUTHOR]

The major concern of this paper is to highlight the notion of possibility m-polar fuzzy soft set (because it is useful in decision-making and other similar problems). For convenience of practical applications, several operations (such as subset, equal, complement, union, intersection, inf product, and sup product) over the possibility m-polar fuzzy soft sets are introduced. We present two algorithms by using inf product or sup product operations of possibility m-polar fuzzy soft sets for fuzzy decision-making problem. Finally, we construct an algorithm using a possibility m-polar fuzzy soft set to solve the decision-making problems and illustrate its applicability through a numerical example. From the study, we conclude that the proposed approach is viable in order to handle the uncertainties during the decision-making problems. [ABSTRACT FROM AUTHOR]

In recent year, the mathematics of three-way fuzzy concept lattice is introduced to characterize the attributes based on its acceptation, rejection and uncertain part. One of the suitable example is descriptive analysis of opinion of people in a democratic country. This became complex for the country like India where opinion (i.e. vote) of people to choose the particular leader is based on 29 independent states and their distinct issues. Adequate analysis of these type of 29-valued data based on its acceptation, rejection and uncertain part is major issue for the government and private agencies. To resolve this issue current paper introduces n-valued neutrosophic context and its graphical structure visualization for descriptive analysis. In the same time an another method is proposed to some of the similar three-way n-valued concepts. To zoom in and zoom out the n-valued neutrosophic context at user required information granules with an illustrative example. [ABSTRACT FROM AUTHOR]

An m-polar fuzzy model is a generalization of the fuzzy model. This model is useful for multi-polar information, multi-agent, multi-attribute and multi-object network models. In this research study, we introduce the notion of m-polar fuzzy directed hypergraphs, and present some properties. We depict certain operations on m-polar fuzzy directed hypergraphs. We also consider an application of m-polar fuzzy directed hypergraphs in business strategy company. [ABSTRACT FROM AUTHOR]

The notion of an m-polar fuzzy set is a generalization of a bipolar fuzzy set. We apply the concept of m-polar fuzzy sets to graphs. We introduce certain types of irregular m-polar fuzzy graphs and investigate some of their properties. We describe the concepts of types of irregular m-polar fuzzy graphs with several examples. We also present applications of m-polar fuzzy graphs in decision making and social network as examples. [ABSTRACT FROM AUTHOR]

The primary goal of this research article is to apply ELECTRE I, a fundamental multi-criteria group decision-making technique, in an m -polar fuzzy N -soft environment. This new methodology helps us to pinpoint the best alternative(s) in the presence of multi-polar options with N -graded qualities. Its basic operational idea entails the comparison between any two alternatives by the assessment of score degrees. Concordance and discordance indices are then calculated to evaluate the alternatives' superiority and inferiority. We may disqualify the incompetent alternatives using concordance and discordance levels. An m -polar fuzzy N -soft dominance matrix can represent the combined effect of concordance and discordance dominance matrices. The steps of this new multi-criteria group decision making technique are summarized in a flowchart. In order to demonstrate its authenticity and applicability, we employ a case study involving the establishment of a rehabilitation facility for drug abusers. A comparison with the m -polar fuzzy PROMETHEE and m -polar fuzzy ELECTRE I methodologies establishes its validity. Finally, we conclude our study of the methodology proposed in this paper with a critical analysis of its benefits and drawbacks. • We develop the implementation of ELECTRE I with m -polar fuzzy N -soft data. • The steps of this MCGDM technique are presented in a comprehensive flowchart. • We apply the methodology to the optimal selection of a rehabilitation center. • Comparative and critical analyses are provided. [ABSTRACT FROM AUTHOR]

Aggregation is a very efficient indispensable tool in which several input values are transformed into a single output value that further supports dealing with different decision-making situations. Additionally, note that the theory of m-polar fuzzy (mF) sets is proposed to tackle multipolar information in decision-making problems. To date, several aggregation tools have been widely investigated to tackle multiple criteria decision-making (MCDM) problems in an m-polar fuzzy environment, including m-polar fuzzy Dombi and Hamacher aggregation operators (AOs). However, the aggregation tool to deal with m-polar information under Yager's operations (that is, Yager's t-norm and t-conorm) is missing in the literature. Due to these reasons, this study is devoted to investigating some novel averaging and geometric AOs in an mF information environment through the use of Yager's operations. Our proposed AOs are named as the mF Yager weighted averaging (mFYWA) operator, mF Yager ordered weighted averaging operator, mF Yager hybrid averaging operator, mF Yager weighted geometric (mFYWG) operator, mF Yager ordered weighted geometric operator and mF Yager hybrid geometric operator. The initiated averaging and geometric AOs are explained via illustrative examples and some of their basic properties, including boundedness, monotonicity, idempotency and commutativity are also studied. Further, to deal with different MCDM situations containing mF information, an innovative algorithm for MCDM is established under the under the condition of mFYWA and mFYWG operators. After that, a real-life application (that is, selecting a suitable site for an oil refinery) is explored under the conditions of developed AOs. Moreover, the initiated mF Yager AOs are compared with existing mF Hamacher and Dombi AOs through a numerical example. Finally, the effectiveness and reliability of the presented AOs are checked with the help of some existing validity tests.