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When information is incomplete or uncertain, Fermatean hesitant fuzzy sets (FHFSs) can provide more information to help decision-makers deal with more complex problems. Typically, determining attribute weights assumes that each attribute has a fixed influence. Introducing probability information can enable one to consider the stochastic nature of evaluation data and better quantify the importance of the attributes. To aggregate data by considering the location and importance degrees of each attribute, this paper develops a Fermatean hesitant fuzzy multi-attribute decision-making (MADM) method with probabilistic information and an ordered weighted averaging (OWA) method. The OWA method combines the concepts of weights and sorting to sort and weigh average property values based on those weights. Therefore, this novel approach assigns weights based on the decision-maker's preferences and introduces probabilities to assess attribute importance under specific circumstances, thereby broadening the scope of information expression. Then, this paper presents four probabilistic aggregation operators under the Fermatean hesitant fuzzy environment, including the Fermatean hesitant fuzzy probabilistic ordered weighted averaging/geometric (FHFPOWA/FHFPOWG) operators and the generalized Fermatean hesitant fuzzy probabilistic ordered weighted averaging/geometric (GFHFPOWA/GFHFPOWG) operators. These new operators are designed to quantify the importance of attributes and characterize the attitudes of decision-makers using a probabilistic and weighted vector. Then, a MADM method based on these proposed operators is developed. Finally, an illustrative example of selecting the best new retail enterprise demonstrates the effectiveness and practicality of the method. [ABSTRACT FROM AUTHOR]

This paper reflects the results of research analyzing models of multi-attribute decision-making based on fuzzy preference relations. Questions of constructing the corresponding multi-attribute models to deal with quantitative information concomitantly with qualitative information based on experts' knowledge are considered. Human preferences may be represented within the fuzzy preference relations and by applying diverse other preference formats. Considering this, so-called transformation functions reduce any preference format to fuzzy preference relations. This paper's results can be applied independently or as part of a general approach to solving a wide class of problems with fuzzy coefficients, as well as within the framework of a general scheme of multi-criteria decision-making under conditions of uncertainty. The considered techniques for fuzzy preference modeling are directed at assessing, comparing, choosing, prioritizing, and/or ordering alternatives. These techniques have served to develop a computing system for multi-attribute decision-making. It has been implemented in the C# programming language, utilizing the ".NET" framework. The computing system allows one to represent decision-makers' preferences in one of five preference formats. These formats and quantitative estimates are reduced to nonreciprocal fuzzy preference relations, providing homogeneous preference information for decision procedures. This paper's results have a general character and were applied to analyze power engineering problems. [ABSTRACT FROM AUTHOR]

During the development process of complex products, selecting the best desirable alternative supplier is a challenge since an improperly selected alternative may cause losing capacity and increasing the cycle time and cost of development for a company. For this multiple-attribute decision-making problem of supplier selection, in this paper, a supplier selection problem in which the decision data are hesitant fuzzy information and the attribute weight is unknown in complex product development is investigated, and a supplier selection decision-making approach based on hesitant fuzzy information is proposed. Firstly, a bidirectional projection based on hesitant fuzzy information is established, and then the measurement equation for the degree of closeness is improved. Further, an attribute weight determination model which minimizes the projection total deviation for the hesitant fuzzy elements is constructed. By solving this model, scientific and reasonable attribute weights are provided. Subsequently, an illustrative example is employed to not only give the ranking result of alternative suppliers but also demonstrate the validity and feasibility of the developed approach. Meanwhile, sensitivity analysis and comparative analysis are put forward to illustrate the stability of the given final ranking result and the advantages and reliability of the constructed method. For alternative or strategy selection, this proposed approach can be used as a decision-making means when uncertainties are inherent. [ABSTRACT FROM AUTHOR]

It is difficult to describe the hesitation and uncertainty of experts by single-valued information, and the differences in the importance of attributes are often ignored during the decision-making process. This paper introduces the probability and interval values into Fermatean hesitant fuzzy set (FHFS) and creatively proposes the probabilistic interval-valued Fermatean hesitant fuzzy set (PIVFHFS) to deal with information loss. This new fuzzy set allows decision makers to use interval-valued information with probability to express their quantitative evaluation, which broadens the range of information expression, effectively reflects the important degree of different membership degrees, and can describe uncertain information more completely and accurately. Under the probabilistic interval-valued Fermatean hesitant fuzzy environment, several new aggregation operators based on Hamacher operation are proposed, including the probabilistic interval-valued Fermatean hesitant fuzzy Hamacher weighted averaging (PIVFHFHWA) operator and geometric (PIVFHFHWG) operator, and their basic properties and particular forms are studied. Then, considering the general correlation between different attributes, this paper defines the probabilistic interval-valued Fermatean hesitant fuzzy Hamacher Choquet integral averaging (PIVFHFHCIA) operator and geometric (PIVFHFHCIG) operator and discusses related properties. Finally, a multi-attribute decision-making (MADM) method is presented and applied to the decision-making problem of reducing carbon emissions of manufacturers in the supply chain. The stability and feasibility of this method are demonstrated by sensitivity analysis and comparative analysis. The proposed new operators can not only consider the correlation between various factors but also express the preference information of decision makers more effectively by using probability, thus avoiding information loss in decision-making progress to some extent. [ABSTRACT FROM AUTHOR]

The process of evaluating and ranking alternatives, including the aggregation of various qualitative and quantitative criteria and weights of criteria, can be recognized as a fuzzy multiple criteria decision-making (MCDM) problem. In fuzzy MCDM problems, qualitative criteria and criteria weights are usually indicated in linguistic values expressed in terms of fuzzy numbers, and values under quantitative criteria are usually crisp numbers. How to properly aggregate them for evaluating and selecting alternatives has been an important research issue. To help decision-makers make the most suitable selection, this paper proposes a fuzzy preference relation-based fuzzy VIKOR method. VIKOR is a compromise ranking method to solve discrete MCDM problems in complex systems. In this study, the F-preference relation is applied to compare fuzzy numbers with their means to produce a single index of a dominance level while still maintaining fuzzy meaning of the original linguistic values. The inverse function is applied to obtain the defuzzification values of Beta 1–4 to assist in the completion of the proposed method, and formulas can be clearly derived to facilitate the ranking procedure. Introducing fuzzy preference relation into fuzzy VIKOR can simplify the calculation procedure for more efficient decision-making. The proposed method is new and has never been seen before. A numerical example and a comparison of the proposed method are conducted to show and verify its expedience and advantage. [ABSTRACT FROM AUTHOR]

Many fuzzy concepts have been researched and described with uncertain information. Collecting data under uncertain information is a difficult task, especially when there is a difference between the opinions of experts. To deal with such situations, different types of operators have been introduced. This paper aims to develop the Maclaurin symmetric mean (MSM) operator for the information in the shape of the interval-valued spherical fuzzy set (IVSFS). In this article, a family of aggregation operators (AOs) is proposed which consists of interval valued spherical fuzzy Maclaurin symmetric mean operator (IVSFMSM), interval valued spherical fuzzy weighted Maclaurin symmetric mean (IVSFWMSM), interval valued spherical fuzzy dual Maclaurin symmetric mean (IVSFDMSM), and interval valued spherical fuzzy dual weighted Maclaurin symmetric mean (IVSFDWMSM) operators. In this paper, we studied an elucidative example to discuss the evaluation of multi-national companies for the application of the proposed operator. Then the obtained results from the proposed operators are compared. The results obtained are graphed and tabulated for a better understanding. [ABSTRACT FROM AUTHOR]

Recently, the notion of a three-way fuzzy set is presented, inspired by the basic ideas of three-way decision and various generalized fuzzy sets, including lattice-valued fuzzy sets, partial fuzzy sets, intuitionistic fuzzy sets, etc. As the new theory of uncertainty, it has been used in attribute reduction and as a new control method for the water level. However, as an extension of a three-way decision, this new theory has not been used in multi-criteria decision making (MCDM for short). Based on the previous work, in this paper, we present rough set models based on three-way fuzzy sets, which extend the existing fuzzy rough set models in both complete and incomplete information systems. Furthermore, the new models are used to solve the issue of MCDM. Firstly, three-way fuzzy relation rough set and three-way fuzzy covering rough set models are presented for complete and incomplete information systems. Because almost all existing fuzzy rough set models are proposed under complete information, the new proposed models can be seen as a supplement to these existing models. Then, a relationship between the three-way fuzzy relation rough set and the three-way fuzzy covering rough set is presented. Finally, a novel method for the issue of MCDM is presented under the novel three-way fuzzy rough set models, which is used in paper defect diagnosis. [ABSTRACT FROM AUTHOR]

The overlap function is an important class of aggregation function that is closely related to the continuous triangular norm. It has important applications in information fusion, image processing, information classification, intelligent decision-making, etc. The usual multi-attribute decision-making (MADM) is to select the decision object that performs well on all attributes (indicators), which is quite demanding. The MADM based on fuzzy quantifiers is to select the decision object that performs well on a certain proportion or quantification (such as most, many, more than half, etc.) of attributes. Therefore, it is necessary to study how to express and calculate fuzzy quantifiers such as most, many, etc. In this paper, the Sugeno integral based on the overlap function (called the O-Sugeno integral) is used as a new information fusion tool, and some related properties are studied. Then, the truth value of a linguistic quantified proposition can be estimated by using the O-Sugeno integral, and the O-Sugeno integral semantics of fuzzy quantifiers is proposed. Finally, the MADM method based on the O-Sugeno integral semantics of fuzzy quantifiers is proposed and the feasibility of our method is verified by several illustrative examples such as the logistics park location problem. [ABSTRACT FROM AUTHOR]

It is well known that the complexity of the decision-making environment frequently coexists with the diversity of linguistic information in the decision-making process. In order to solve this kind of uncertain multi-criteria decision-making problem, reasonable measures and integrals should be established. In this paper, the discrete expression of the C T -integral on the interval-valued Sugeno probability measure is proposed. The C T -integral is the Choquet integral when the t-norm is T (x , y) = x y in the C T -integral and is a pre-aggregation function. Then, the C T -integral on interval-valued Sugeno probability measure is applied to solve end-of-life (EOL) strategy in order to determine multi-criteria decision-making problems. Compared with the general Choquet integral, the method proposed in this paper significantly improves the calculation process, that is, the calculation is simpler and the amount of calculation is smaller. A case study was performed in order to validate the effectiveness of this conclusion. [ABSTRACT FROM AUTHOR]

The integration of key indicators from the results of the analysis of time series represents a constant challenge within organizations; this could be mainly due to the need to establish the belonging of each indicator within a process, geographic region or category. This paper thus illustrates how both primary and secondary indicators are relevant for decision making, and why they need to be integrated by making new final fuzzy indicators. Thus, our proposal consists of a type-2 fuzzy integration of multivariate time series, such as OECD country risk classification, inflation, population and gross national income (GNI) by using multiple type-1 fuzzy inference systems to perform time series classification tasks. Our contribution consists of the proposal to integrate multiple nested type-1 fuzzy inference systems using a type-2 fuzzy integration. Simulation results show the advantages of using the proposed method for the fuzzy classification of multiple time series. This is done in order so the user can have tools that allow them to understand the environment and generate comparative analyses of multiple information sources, and finally use it during the process prior to decision making considering the main advantage of modeling the inherent uncertainty. [ABSTRACT FROM AUTHOR]

Computation intelligence techniques are important for making decisions in an agile-based offshore software development paradigm. Offshore development faces additional challenges, such as trust, communication and coordination, and socio-cultural and knowledge transfer. There is a need to determine the rankings of challenges considering their criticality concerning practitioners working in agile-based offshore software development. This paper aims to identify and rank agile challenges in offshore software development by applying computational intelligence techniques. From the systematic literature review, we identified 30 communication and coordination challenges. The distributed agile pattern catalog consists of 15 patterns, from which eight were used to solve communication and collaboration challenges. Many researchers have used fuzzy logic to quantify their results. We further applied the fuzzy analytical technique to determine the priority order concerning the criticality of the identified agile pattern catalog. The results showed that Central Code Repository Pattern ranked the most significant for solving communication and coordination challenges. Global Scrum Board Pattern and Synchronous Communication Pattern ranked second. [ABSTRACT FROM AUTHOR]

The crypto and digital assets ecosystems have attracted investment, regulators, and speculators to their environment. As the blockchain-based framework can reduce transaction costs, generate distributed trust, and enable decentralized platforms, it has become a potential new base for decentralized business models. Previous studies have highlighted the advantages and drawbacks of each platform, such as interest rates, cost concerns, transparency issues, hacking issues, and hazards. Consequently, it is challenging for investors to evaluate the cryptocurrency trading system which determines the optimum exchanges and crucial aspects. Therefore, in order to rank the optimal digital token trading system, this paper develops an evaluation architecture to determine the various token trading systems. The developed architecture integrates fuzzy theory and the best-worst method (BWM) into the decision-making process to assess decision behaviors regarding preference for digital token trading systems in investors in Taiwan. First, this work establishes the views and parameters by modifying the Delphi method based on a literature review and survey. Second, the fuzzy-BWM is applied to obtain the fuzzy weights of the views and parameters. Then, defuzzification and BWM are used to rank the optimal alternatives of the digital token trading systems for investors. The results indicate that the optimal digital token trading system is the decentralized platform, and the critical parameters are gas fees, interest rates, and the mechanism of savings under fuzzy uncertain scenarios. This means that when considering the uncertain and ambiguous characteristics of the expert decision process in digital token trading systems, the evaluation is decentralized and the gas fees are the most important parameter in the digital token investment platform. Academically, the fuzzy BWM-based decision-making architecture can provide corporations and investors with valuable guidance to rank the optimal digital token trading systems based on fuzzy uncertain scenarios. Commercially, the proposed architecture could provide corporations and investors with a useful model to measure the optimal digital token trading system. [ABSTRACT FROM AUTHOR]

This paper analyzes the various properties of the positively correlated weights related to the subset of finite criteria in a multi-attribute decision-making problem. Given a finite number of criteria, the exact constraints of the positively correlated weights related to the subset of criteria are presented. Introducing the non-Archimedean number, the associated bounded polyhedral-set is shown. The number of the extreme points in the bounded polyhedral-set will increase as the number of criteria increase. Applying the proposed efficient extreme-point method, the pre-emptive-weights-goal-programming optimal solution is shown. These theoretical global-maximum values of the positively correlated weights related to the subset of finite criteria are useful for practical applications. [ABSTRACT FROM AUTHOR]

Rough set (RS) and fuzzy set (FS) theories were developed to account for ambiguity in the data processing. The most persuasive and modernist abstraction of an FS is the linear Diophantine FS (LD-FS). This paper introduces a resilient hybrid linear Diophantine fuzzy RS model (LDF-RS) on paired universes based on a linear Diophantine fuzzy relation (LDF-R). This is a typical method of fuzzy RS (F-RS) and bipolar FRS (BF-RS) on two universes that are more appropriate and customizable. By using an LDF-level cut relation, the notions of lower approximation (L-A) and upper approximation (U-A) are defined. While this is going on, certain fundamental structural aspects of LD-FAs are thoroughly investigated, with some instances to back them up. This cutting-edge LDF-RS technique is crucial from both a theoretical and practical perspective in the field of medical assessment. [ABSTRACT FROM AUTHOR]

The business process modeling tool selection problem has a significant impact on the overall performance of enterprise business process modeling, which will directly affect the development of enterprise information systems. Apart from that, the process to select the business process modeling tool from all alternatives is a Multi-Criteria Decision Making (MCDM) problem. This paper develops a methodology based on the hybrid fuzzy Decision-Making Trial and Evaluation Laboratory (DEMATEL) and Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) method to help companies select the optimal business process modeling tool, where the business process modeling process is more efficient, economic and safe. The proposed method has the following state-of-the-art contributions and features: (1) the latest application of the MCDM methodology to the field of BPM tool selection, (2) addressing the direct and indirect impact between criteria in the selection of BPM tools, and (3) considering the hybrid fuzzy (uncertainty) decision-making issue in the BPM tool selection process. Meanwhile, the mathematical formula in TOPSIS can be regarded as a formula for solving a symmetric problem. The hybrid fuzzy DEMATEL method is used to obtain the weight for the criteria to be considered in the BPM tool selection process, and the TOPSIS method is used to obtain the final business process modeling tool. [ABSTRACT FROM AUTHOR]

Overlap and grouping functions are important aggregation operators, especially in information fusion, classification and decision-making problems. However, when we do more in-depth application research (for example, non-commutative fuzzy reasoning, complex multi-attribute decision making and image processing), we find overlap functions as well as grouping functions are required to be commutative (or symmetric), which limit their wide applications. For the above reasons, this paper expands the original notions of overlap functions and grouping functions, and the new concepts of pseudo overlap functions and pseudo grouping functions are proposed on the basis of removing the commutativity of the original functions. Some examples and construction methods of pseudo overlap functions and pseudo grouping functions are presented, and the residuated implication (co-implication) operators derived from them are investigated. Not only that, some applications of pseudo overlap (grouping) functions in multi-attribute (group) decision-making, fuzzy mathematical morphology and image processing are discussed. Experimental results show that, in many application fields, pseudo overlap functions and pseudo grouping functions have greater flexibility and practicability. [ABSTRACT FROM AUTHOR]

The Decision-Theoretic Rough Set model stands as a compelling advancement in the realm of rough sets, offering a broader scope of applicability. This approach, deeply rooted in Bayesian theory, contributes significantly to delineating regions of minimal risk. Within the Decision-Theoretic Rough Set paradigm, the universal set undergoes a tripartite division, where distinct regions emerge and losses are intelligently distributed through the utilization of membership functions. This research endeavors to present an enhanced and more encompassing iteration of the Decision-Theoretic Rough Set framework. Our work culminates in the creation of the Generalized Intuitionistic Decision-Theoretic Rough Set (GI-DTRS), a fusion that melds the principles of Decision-Theoretic Rough Sets and intuitionistic fuzzy sets. Notably, this synthesis bridges the gaps that exist within the conventional approach. The innovation lies in the incorporation of an error function tailored to the hesitancy grade inherent in intuitionistic fuzzy sets. This integration harmonizes seamlessly with the contours of the membership function. Furthermore, our methodology deviates from established norms by constructing similarity classes based on similarity measures, as opposed to relying on equivalence classes. This shift holds particular relevance in the context of aggregating information systems, effectively circumventing the challenges associated with the process. To demonstrate the practical efficacy of our proposed approach, we delve into a concrete experiment within the information technology domain. Through this empirical exploration, the real-world utility of our approach becomes vividly apparent. Additionally, a comprehensive comparative analysis is undertaken, juxtaposing our approach against existing techniques for aggregation and decision modeling. The culmination of our efforts is a well-rounded article, punctuated by the insights, recommendations, and future directions delineated by the authors. [ABSTRACT FROM AUTHOR]

In the pursuance of engineering excellence and sustainable practices, the optimization of material selection processes plays a crucial role. Using Fermatean fuzzy aggregation Operators (AOs), this study introduces an innovative method for improving material selection procedures. Combining the advantages of Fermatean fuzzy set (FrFS) and AOs, the proposed method enables a comprehensive evaluation of materials based on multiple criteria. The authors propose two operators: the "Fermatean fuzzy hybrid weighted arithmetic geometric aggregation (FrFHWAGA) operator" and the "Fermatean fuzzy hybrid ordered weighted arithmetic geometric aggregation (FrFHOWAGA) operator". This method facilitates informed decision making in a number of industries by taking into account factors such as cost, durability, environmental impact, and availability. This research enables engineers, designers, and decision makers to optimize material selection, resulting in more efficient, cost-effective, and sustainable solutions across multiple domains. [ABSTRACT FROM AUTHOR]

A Fermatean fuzzy set (FRFS) is the extension of a fuzzy set, an intuitionistic fuzzy set, and a Pythagorean fuzzy set, and is used in different fields. Unlike other fuzzy structures, the sum of cubes of membership grades in FRFSs approximates a unit interval, increasing uncertainty. In this study, we intend to provide unique operational rules and aggregation operators (AOs) inside a Fermatean fuzzy environment. To develop a fair remedy for the membership degree and non-membership degree features of "Fermatean fuzzy numbers (FRFNs)", our solution introduces new neutral or fair operating principles, which include the concept of proportional distribution. Based on the suggested operating principles, we provide the "Fermatean fuzzy fairly weighted average operator and the Fermatean fuzzy fairly ordered weighted averaging operator". Our suggested AOs provide more generalized, reliable, and exact data than previous techniques. Combining the recommended AOs with multiple decision-makers and partial weight information under FRFSs, we also devised a technique for "multi-criteria decision-making". To illustrate the application of our novel method, we provide an example of the algorithm's effectiveness in addressing decision-making challenges. [ABSTRACT FROM AUTHOR]

Complex fuzzy sets (CFSs) are a suitable tool to manage spatial directional information which includes distance and direction. However, spatial directional information is given by linguistic values. It is very awkward for the CFS to describe this type of spatial directional information. To overcome this limitation, we first propose a novel concept called a linguistic complex fuzzy set (LCFS) to serve as an extension of the CFS. Then we put forward some basic operational laws for LCFSs. After that, we define three operators for LCFSs: the linguistic complex fuzzy weighted averaging (LCFWA) operator, the linguistic amplitude max (Amax) operator and the linguistic amplitude min (Amin) operator. In actual application, we use the LCFWA operator to deal with group decision making when the importance weights of experts are known. For the situation in which the weights of experts are unknown, we develop an Amax-Amin method for group decision making. [ABSTRACT FROM AUTHOR]

A T-spherical fuzzy set is a more powerful mathematical tool to handle uncertain and vague information than several fuzzy sets, such as fuzzy set, intuitionistic fuzzy set, Pythagorean fuzzy set, q-rung orthopair fuzzy set, and picture fuzzy set. The Aczel–Alsina t-norm and s-norm are significant mathematical operations with a high premium on affectability with parameter activity, which are extremely conducive to handling imprecise and undetermined data. On the other hand, the Hamy mean operator is able to catch the interconnection among multiple input data and achieve great results in the fusion process of evaluation information. Based on the above advantages, the purpose of this study is to propose some novel aggregation operators (AOs) integrated by the Hamy mean and Aczel–Alsina operations to settle T-spherical fuzzy multi-criteria decision-making (MCDM) issues. First, a series of T-spherical fuzzy Aczel–Alsina Hamy mean AOs are advanced, including the T-spherical fuzzy Aczel–Alsina Hamy mean (TSFAAHM) operator, T-spherical fuzzy Aczel–Alsina dual Hamy mean (TSFAADHM) operator, and their weighted forms, i.e., the T-spherical fuzzy Aczel–Alsina-weighted Hamy mean (TSFAAWHM) and T-spherical fuzzy Aczel–Alsina-weighted dual Hamy mean (TSFAAWDHM) operators. Moreover, some related properties are discussed. Then, a MCDM model based on the proposed AOs is built. Lastly, a numerical example is provided to show the applicability and feasibility of the developed AOs, and the effectiveness of this study is verified by the implementation of a parameters influence test and comparison with available methods. [ABSTRACT FROM AUTHOR]

The collection of Hamacher t-norms was created by Hamacher in 1970, which played a critical and significant role in computing aggregation operators. All aggregation operators that are derived based on Hamacher norms are very powerful and are beneficial because of the parameter 0 ≤ ζ ≤ + ∞ . Choquet first posited the theory of the Choquet integral (CI) in 1953, which is used for evaluating awkward and unreliable information to address real-life problems. In this manuscript, we analyze several aggregation operators based on CI, aggregation operators, the Hamacher t-norm and t-conorm, and Atanassov intuitionistic fuzzy (A-IF) information. These are called A-IF Hamacher CI averaging (A-IFHCIA), A-IF Hamacher CI ordered averaging (A-IFHCIOA), A-IF Hamacher CI geometric (A-IFHCIG), and A-IF Hamacher CI ordered geometric (A-IFHCIOG) operators; herein, we identify their most beneficial and valuable results according to their main properties. Working continuously, we developed a multi-attribute decision-making (MADM) procedure for evaluating awkward and unreliable information, with the help of the TOPSIS technique for order performance by similarity to the ideal solution, and derive operators to enhance the worth and value of the present information. Finally, by comparing the pioneering information with some of the existing operators, we illustrate some examples for evaluating the real-life problems related to enterprises, wherein the owner of a company appointed four senior board members of the enterprise to decide what was the best Asian company in which to invest money, to show the supremacy and superiority of the invented approaches. [ABSTRACT FROM AUTHOR]

The purpose of this article is to introduce to the literature a new extension of the Simple WISP method adapted for utilizing the triangular fuzzy numbers. This extension is proposed to allow the use of the Simple WISP method for addressing decision-making problems related to uncertainties and inaccuracies, as well as for solving problems related to predictions. In addition, this article also discusses the use of linguistic variables to collect the attitudes of the respondents, as well as their transformation into appropriate triangular fuzzy numbers. The article discusses the use of two defuzzification procedures. The first normalization procedure is easy to use, while the second procedure uses the advantages that the application of asymmetric fuzzy numbers gives in terms of analysis. The usability of the proposed extension is presented through two examples. [ABSTRACT FROM AUTHOR]

The T-spherical fuzzy set (TSFS) is a modification of the fuzzy set (FS), intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PyFS), q-rung orthopair fuzzy set (q-ROFS), and picture fuzzy set (PFS), with three characteristic functions: the membership degree (MD) denoted by S , the nonmembership degree (NMD) denoted by D , and the abstinence degree (AD) denoted by I. It can be used to solve problems of uncertain information with no restrictions. The distance measure (DM) is a tool that sums up the difference between points, while the similarity measure (SM) is a method applied to calculate the similarity between objects within an interval of [ 0 , 1 ] . The current work aims to introduce novel DMs and SMs in the environment of TSFSs to show the limitations of the previously defined DMs and SMs. The suggested DMs and SMs provide more room for all three degrees to be selected without restriction. We investigated the effectiveness of the proposed DMs and SMs by applying a pattern-recognition technique, and we determined their applicability for multicriteria decision making (MCDM) using numerical examples. The newly proposed DMs and SMs are briefly compared to existing DMs and SMs, and appropriate conclusions are drawn. [ABSTRACT FROM AUTHOR]

The bipolar neutrosophic set is an important extension of the bipolar fuzzy set. The bipolar neutrosophic set is a hybridization of the bipolar fuzzy set and neutrosophic set. Every element of a bipolar neutrosophic set consists of three independent positive membership functions and three independent negative membership functions. In this paper, we develop cross entropy measures of bipolar neutrosophic sets and prove their basic properties. We also define cross entropy measures of interval bipolar neutrosophic sets and prove their basic properties. Thereafter, we develop two novel multi-attribute decision-making strategies based on the proposed cross entropy measures. In the decision-making framework, we calculate the weighted cross entropy measures between each alternative and the ideal alternative to rank the alternatives and choose the best one. We solve two illustrative examples of multi-attribute decision-making problems and compare the obtained result with the results of other existing strategies to show the applicability and effectiveness of the developed strategies. At the end, the main conclusion and future scope of research are summarized. [ABSTRACT FROM AUTHOR]

Modeling uncertainties with spherical linear Diophantine fuzzy sets (SLDFSs) is a robust approach towards engineering, information management, medicine, multi-criteria decision-making (MCDM) applications. The existing concepts of neutrosophic sets (NSs), picture fuzzy sets (PFSs), and spherical fuzzy sets (SFSs) are strong models for MCDM. Nevertheless, these models have certain limitations for three indexes, satisfaction (membership), dissatisfaction (non-membership), refusal/abstain (indeterminacy) grades. A SLDFS with the use of reference parameters becomes an advanced approach to deal with uncertainties in MCDM and to remove strict limitations of above grades. In this approach the decision makers (DMs) have the freedom for the selection of above three indexes in [ 0 , 1 ] . The addition of reference parameters with three index/grades is a more effective approach to analyze DMs opinion. We discuss the concept of spherical linear Diophantine fuzzy numbers (SLDFNs) and certain properties of SLDFSs and SLDFNs. These concepts are illustrated by examples and graphical representation. Some score functions for comparison of LDFNs are developed. We introduce the novel concepts of spherical linear Diophantine fuzzy soft rough set (SLDFSRS) and spherical linear Diophantine fuzzy soft approximation space. The proposed model of SLDFSRS is a robust hybrid model of SLDFS, soft set, and rough set. We develop new algorithms for MCDM of suitable clean energy technology. We use the concepts of score functions, reduct, and core for the optimal decision. A brief comparative analysis of the proposed approach with some existing techniques is established to indicate the validity, flexibility, and superiority of the suggested MCDM approach. [ABSTRACT FROM AUTHOR]

Interval-valued T-spherical fuzzy set (IVTSFS) handles uncertain and vague information by discussing their membership degree (MD), abstinence degree (AD), non-membership degree (NMD), and refusal degree (RD). MD, AD, NMD, and RD are defined in terms of closed subintervals of [ 0 ,   1 ] that reduce information loss compared to the T-spherical fuzzy set (TSFS), which takes crisp values from [ 0 ,   1 ] intervals; hence, some information may be lost. The purpose of this manuscript is to develop some Hamacher aggregation operators (HAOs) in the environment of IVTSFSs. To do so, some Hamacher operational laws based on Hamacher t-norms (HTNs) and Hamacher t-conorms (HTCNs) are introduced. Using Hamacher operational laws, we develop some aggregation operators (AOs), including an interval-valued T-spherical fuzzy Hamacher (IVTSFH) weighted averaging (IVTSFHWA) operator, an IVTSFH-ordered weighted averaging (IVTSFHOWA) operator, an IVTSFH hybrid averaging (IVTSFHHA) operator, an IVTSFH-weighted geometric (IVTSFHWG) operator, an IVTSFH-ordered weighted geometric (IVTSFHOWG) operator, and an IVTSFH hybrid geometric (IVTSFHHG) operator. The validation of the newly developed HAOs is investigated, and their basic properties are examined. In view of some restrictions, the generalization and proposed HAOs are shown, and a multi-attribute decision-making (MADM) procedure is explored based on the HAOs, which are further exemplified. Finally, a comparative analysis of the proposed work is also discussed with previous literature to show the superiority of our work. [ABSTRACT FROM AUTHOR]