Archimedean T-norm Research Articles

Archimedean [formula omitted]-norm and [formula omitted]-conorm coupled q-rung orthopair fuzzy TOPSIS method for unknown criteria weighting information

To evaluate some real-life multi-criteria decision-making (MCDM) problems, the decision-makers (DMs) need an interactive and flexible MCDM approach that works well for extended decision-making space and has the ability to apply any suitable operational laws to fuse the information under incomplete criteria weighting information. The objective of this study is to propose a multi-criteria group decision-making (MCGDM) approach to evaluate any real-life MCDM problem under incomplete criteria weighting information. To achieve the objective, the paper proposes an integrated MCGDM approach in which the q-rung orthopair fuzzy set (q-ROFS) is used to provide the extended decision-making space to the DMs, a transformative approach is applied to find the DMs’ weights, the maximizing deviation method is implemented to find the criteria weights, Archimedean t-norm and t-conorm (ATU)-based operational laws are adopted to fuse the information, and the TOPSIS method is applied to rank the alternatives. The proposed integrated MCGDM approach has been applied to a real-life MCDM problem associated with the evaluation of different strategies implemented by governments to contain the COVID-19 pandemic. A sensitivity analysis is conducted to examine the impact of changes in the parameters involved in the proposed MCGDM approach on the results. A comparative analysis is also performed with some existing classical and aggregation operator-based MCDM methods to demonstrate the consistency of the proposed approach. Findings suggest that the proposed MCGDM approach is general and flexible enough to evaluate any real-life MCDM problem under unknown criteria weighting conditions.

Retraction Note: Some new dual hesitant fuzzy linguistic operators based on Archimedean t-norm and t-conorm

Retraction Note: Some new dual hesitant fuzzy linguistic operators based on Archimedean t-norm and t-conorm

A multi-period intuitionistic fuzzy consensus reaching model for group decision making problem in social network

A new intuitionistic fuzzy consensus reaching model is developed with multi-period public opinions and expert evaluation values in social network environment. First, the public opinions are obtained by using the crawler software and sentiment analysis technology is used to transform public opinions into intuitionistic fuzzy decision matrix in each period. Attribute weights are calculated by using the time attenuation factor and changes in public opinion. Second, the social trust relationship is modeled and incomplete social trust relationships are completed by using Archimedean t-norm. The expert weights are calculated by using the dynamic trust degree and similarity degree. Third, a consensus framework is proposed for multiple-period decision making problem, which coordinates conflicts between experts through dual feedback paths. The collective opinion scores are calculated by using weights of periods and attribute weights obtained from the word frequency of public opinions. The tourism attraction recommendation method is used to illustrate the proposed method.

The weak dominance between t-norms and t-conorms revisited

The weak dominance relation between two binary operations was introduced as an extension of the dominance relation and the modularity equation. This paper continues the study of the weak dominance between t-norms and t-conorms. First, we present the characterization of the weak dominance of a nilpotent t-norm over a continuous Archimedean t-norm. Second, we establish the generalized sufficient and necessary conditions for the weak dominance of continuous t-norms over continuous t-norms. Finally, we obtain a generalized characterization of a continuous t-conorm weakly dominating a continuous t-norm by showing that SM weakly dominates any t-norm.

Separable programming problem with bipolar max-T fuzzy relation equation constraints

One of the most important classes of nonlinear programming problems is separable programming problem due to its applications in theory and practice. This paper studies this class of the problems subject to a system of bipolar fuzzy relation equations using the max-T composition operator, where T is a continuous and Archimedean t-norm. Its feasible solution set structure is determined by two vectors called lower and upper bound vector of its feasible domain. It is shown that there exists an optimal solution for the problem with max-continuous and Archimedean t-norm such that its components are the corresponding components of either the lower or upper bound vector. Based on the interesting property, some sufficient conditions are proposed to detect some of its optimal components or one of its optimal solutions. These conditions can reduce the dimensions of the original problem when some its optimal components are determined. The objective function of the reduced problem is equivalently rewritten in a specific form. This form guides us to design a value matrix based on the characteristic matrix and ascending or descending of univariate functions constituting the objective function. An approach is extended on the matrix to find the optimal solution of the (reduced) problem. It is important to note that a number of problems can completely be solved using the sufficient conditions.

New consensus reaching process with minimum adjustment and feedback mechanism for large-scale group decision making problems under social trust networks

Large-scale group decision making problems in social networks environment have become an important research topic in decision making science. In this paper, we propose a new consensus reaching process with minimum adjustment and feedback mechanism considering public opinions for large-scale group decision making problems under social trust networks. The trust propagation operator based on Archimedean t-norm is used to calculate incomplete trust values first. The fast unfolding algorithm is used to divide experts into subgroups. The opinion evolution model is presented to calculate experts’ weights based on dynamic trust values and expert influence. Public opinions are used to determine consensus thresholds of attributes. A dual-path adjustment model is developed for group consensus in social network. Divide experts’ evaluation values into three different types: ‘conflict behavior’, ‘tolerant behavior’, ‘trust behavior’. Minimum cost adjustment models have been developed considering adjustment directions for the ‘conflict behavior’ and interactive path is activated for the ‘tolerant behavior’. The drugs selection from E-commerce platform for therapy of influenza at home is used to illustrate the new method and comparisons have also been conducted.

Fermatean fuzzy Archimedean Heronian Mean-Based Model for estimating sustainable urban transport solutions

Public transportation frameworks assume a critical role in the metropolitan region, especially in huge urban communities, where they offer a feasible answer for easing gridlock, moderating commotion contamination, and diminishing CO2 discharges. This paper presents some novel Fermatean fuzzy Heronian mean operators based on Archimedean t-norms, specifically the generalized Fermatean fuzzy Archimedean Heronian mean (GFFAHM) and the Fermatean fuzzy Archimedean geometric Heronian mean (FFAGHM). The study investigates different unique instances of these operators while exploring their essential properties. Besides, a powerful multiattribute decision-making (MADM) method is developed using the proposed operators. This approach offers a key asset for handling complex decision-making problems. Overcoming the capabilities of conventional BM operators, the GFFAHM and FFAGHM operators effectively minimize the potential redundancy in interrelationships during the decision-making process. The inclusion of the flexible parameters ρ and ϕ, which have a big impact on the decision-making process’ outcomes, increases the adaptability and resilience of the Archimedean t-based operators. To depict the feasibility of the proposed MADM method, a thorough quantitative model is introduced, outlining its useful execution. Through an exhaustive case study, the predominance of the proposed approach over existing strategies is experimentally established. Remarkably, the study uncovers that reducing fares is the most compelling factor in expanding the public transport framework, subsequently advancing sustainable urban transport. This study’s findings provide useful insights into decision-making processes and practical implications for policymakers, allowing them to make informed and impactful changes to the public transportation system. The proposed MADM method can play a vital role in upgrading the public transport framework, making way for remarkable strategy interventions in urban transport sustainability.

Q-Rung Orthopair Fuzzy Archimedean Aggregation Operators: Application in the Site Selection for Software Operating Units

The q-rung orthopair fuzzy (q-ROF) set is an efficient tool for dealing with uncertain and inaccurate data in real-world multi-attribute decision-making (MADM). In MADM, aggregation operators play a significant role. The majority of well-known aggregation operators are formed using algebraic, Einstein, Hamacher, Frank, and Yager t-conorms and t-norms. These existing t-conorms and t-norms are some special cases of Archimedean t-conorms (ATCNs) and Archimedean t-norms (ATNs). Therefore, this article aims to extend the ATCN and ATN operations under the q-ROF environment. In this paper, firstly, we present some new operations for q-ROF sets based on ATCN and ATN. After that, we explore a few desirable characteristics of the suggested operational laws. Then, using these operational laws, we develop q-ROF Archimedean weighted averaging (geometric) operators, q-ROF Archimedean order weighted averaging (geometric) operators, and q-ROF Archimedean hybrid averaging (geometric) operators. Next, we develop a model based on the proposed aggregation operators to handle MADM issues. Finally, we elaborate on a numerical problem about site selection for software operating units to highlight the adaptability and dependability of the developed model.

Generating methods of some classes of uninorms and associated structures

Uninorms are an important class of fuzzy logic connectives that have, so far, neither generating methods nor algebraic structures. In this work, we attempt to propose some novel generating methods of some well established classes of uninorms (with a fixed but arbitrary neutral element e) and investigate the algebraic structures that these generating methods can impose on those specific classes of uninorms. Towards this, we show that the operations point-wise maximum (∨) and the point-wise minimum (∧) on the set of idempotent uninorms are closed and make it a distributive lattice. Also, we show that the operations ∧ and ∨ on some sub-classes of idempotent uninorms with other classes of uninorms generate uninorms. Since the operations ∨ and ∧ on the set of uninorms are not closed in general, we propose some novel generating methods of uninorms that come from various classes such as representable uninorms, pseudo-continuous uninorms, uninorms continuous in the open unit square, uninorms with underlying continuous Archimedean t-norms and t-conorms and for each class of these uninorms, we investigate the algebraic structure imposed on it by the proposed generating methods with the help of results obtained recently in Vemuri et al. (2022) [51].

Archimedean [formula omitted]-norm and [formula omitted]-conorm based intuitionistic fuzzy WASPAS method to evaluate health-care waste disposal alternatives with unknown weight information

Health-care waste (HCW) contains numerous hazardous, infectious, and radioactive contents that may cause serious health issues for society and drastic environmental problems. Therefore, it is crucial to dispose of HCW safely and appropriately. Four commonly used HCW disposal methods are incineration, steam sterilization, microwave, and landfill disposal. To select an appropriate HCW disposal method, there are several conflicting criteria. Furthermore, when information is lacking due to uncertainty and criteria weighing conditions are also unknown, the decision to choose an appropriate HCW disposal method becomes more challenging. So, the objective of this paper is to introduce a novel integrated multi-criteria group decision-making (MCGDM) approach to evaluate HCW disposal methods in which incomplete information is quantified using intuitionistic fuzzy sets (IFSs), unknown criteria weights are evaluated using the maximizing deviation method, and the evaluation of HCW disposal methods is done by applying the Archimedean t-norm and t-conorm-based interactive intuitionistic fuzzy weighted aggregated sum product assessment (WASPAS) method. Herein, the Archimedean t-norm and t-conorm cover a wide range of t-norms and t-conorms, including the Algebraic, Einstein, Hamacher, and Frank, etc., by adopting prescribed combinations of additive generators. The proposed MCGDM approach has been applied to evaluate the above-listed four HCW disposal methods for a case study conducted by Mishra et al. (2020) in the context of India. A sensitivity analysis is performed to examine the impact of changes in the parameters involved in the proposed MCGDM approach. A comparative study is also performed with some well-known multi-criteria decision making(MCDM) and MCGDM methods to demonstrate the consistency and stability of the proposed approach. Findings suggest that the newly developed MCGDM approach is more general, flexible, and provides realistic results to evaluate HCW disposal methods under uncertainty and unknown criteria weight information.

On metrization of fuzzy metrics and application to fixed point theory

It is a well-known fact that the topology induced by a fuzzy metric is metrizable. Nevertheless, the problem of how to obtain a classical metric from a fuzzy one in such a way that both induce the same topology is not solved completely. A new method to construct a classical metric from a fuzzy metric, whenever it is defined by means of an Archimedean t-norm, has recently been introduced in the literature. Motivated by this fact, we focus our efforts on such a method in this paper. We prove that the topology induced by a given fuzzy metric M and the topology induced by the metric constructed from M by means of such a method coincide. Besides, we prove that the completeness of the fuzzy metric space is equivalent to the completeness of the associated classical metric obtained by the aforementioned method. Moreover, such results are applied to obtain fuzzy versions of two well-known classical fixed point theorems in metric spaces, one due to Matkowski and the other one proved by Meir and Keeler. Although such theorems have already been adapted to the fuzzy context in the literature, we show an inconvenience on their applicability which motivates the introduction of these two new fuzzy versions.

Generalized q-rung picture linguistic aggregation operators and their application in decision making

The q-rung picture linguistic set (q-RPLS) is an effective tool for managing complex and unpredictable information by changing the parameter ‘q’ regarding hesitancy degree. In this article, we devise some generalized operational laws of q-RPLS in terms of the Archimedean t-norm and t-conorm. Based on the proposed generalized operations, we define two types of generalized aggregation operators, namely the q-rung picture linguistic averaging operator and the q-rung picture linguistic geometric operator, and study their relevant characteristics in-depth. With a view toward applications, we discuss certain specific cases of the proposed generalized aggregation operators with a range of parameter values. Furthermore, we explore q-rung picture linguistic distance measure and its required axioms. Then we put forward a technique for q-RPLSs based on the proposed aggregation operators and distance measure to solve multi-attribute decision-making (MADM) challenges with unknown weight information. At last, a practical example is presented to demonstrate the suggested approaches’ viability and to perform the sensitivity and comparison analysis.

Two new methods to construct fuzzy metrics from metrics

In the last years, the interest in the notion of fuzzy metric has been growing in such a way that many works have focused their efforts on the study of their topological properties and their applications to Engineering problems. However, the applicability of fuzzy metrics is limited due to lack of examples in the literature. Motivated, on the one hand, by these facts and, on the other hand, by the fact that most of the instances of fuzzy metrics in the literature are constructed from classical metrics, in this paper we introduce two new techniques which allow us to construct systematically fuzzy metrics from metrics in such a way that the celebrated classical method for constructing indistinguishability operators from metrics is retrieved as a particular case. Hence, we construct strong fuzzy metrics from a given classical one considering continuous Archimedean t-norms and the pseudo-inverse of their additive generators acting on the metric modified by a positive real function. Moreover, we extend this technique tackling the particular case of the minimum t-norm, which is continuous but non-Archimedean. In such a construction, two non-negative real functions are now involved in order to modify the classical metric and one of them must be superadditive. In this case, the fuzzy metric obtained is not strong in general. Furthermore, the new methods are illustrated by means of different examples which, in addition, show that some celebrated examples of fuzzy metrics can be retrieved as a particular case through them. Finally, in the light of the developed theory, an open problem about strong fuzzy metrics is solved completing the partial solutions that can be found in the literature.

Probabilistic linguistic multi-attribute decision making approach based upon novel GMSM operators

<abstract><p>Probabilistic linguistic terms set (PLTS), a new tool for expressing uncertain decision information, is composed of all possible linguistic terms (LTs) and their related probabilities. It also increases the corresponding probability of LTs in hesitant fuzzy linguistic term set (HFLTS). On the other hand, aggregation operator is an important information fusion tool, the Maclaurin symmetric mean (MSM) operator can provide more flexibility and robustness in information fusion, and make it more suitable for solving MADM problems with independent attributes. This current study adopts the merits of PLTS and MSM operator, and then a novel probabilistic linguistic decision making approach are targeted. Firstly, the operations of two PLTSs are redefined based upon Archimedean t-norm (ATN) and Archimedean t-conorm (ATC); Secondly, the probabilistic linguistic generalized MSM operator (PLGMSM) is proposed based on ATN and ATC, some properties of PLGMSM are investigated, then some special PLGMSM operators have been studied in detail when the parameters take different values and the generator of ATN takes different functions. Thirdly, the weighted probabilistic linguistic generalized MSM operator (WPLGMSM) is studied along with some properties of PLGMSM, some special WPLGMSM operators have been also investigated in detail when the parameters take different values and the generator of ATN takes different functions. Finally, on the basis of our proposed aggregation operators, the aggregated-based decision making approach is designed and an example is supplied to manifest the effectiveness of the proposed approach. Furthermore, some comparison analyses with extant decision approaches are carried out to illustrate the validity and feasibility of the proposed approach.</p></abstract>

A Novel Linguistic Interval-Valued Pythagorean Fuzzy Multi-Attribute Group Decision-Making for Sustainable Building Materials Selection

The linguistic interval-valued Pythagorean fuzzy (LIVPF) sets, which absorb the advantages of linguistic terms set and interval-valued Pythagorean fuzzy sets, can efficiently describe decision makers’ evaluation information in multi-attribute group decision-making (MAGDM) problems. When investigating aggregation operators of linguistic interval-valued Pythagorean fuzzy (LIVPF) information, we have to consider two important issues, viz. the operational rules of LIVPF numbers and aggregation functions. The classical Archimedean t-norm and t-conorm (ATT) are a famous t-norm and t-conorm, which can produce some special cases. Recently, ATT has been widely applied in different fuzzy decision-making information. Hence, in this paper, for the first issue, we propose some novel operational rules of LIVPF numbers based on ATT. The new operational laws are flexible and can generate some useful operations. For the second issue, we choose a powerful function, i.e., the extended power average (EPA) operator as the aggregation function. The prominent advantages of EPA are that it not only considers the relationship among input arguments, but also dynamically changes the weights of input arguments by employing a parameter. Hence, our proposed novel aggregation operators for LIVPFNs are flexible and is suitable to handle MAGDM problems in actual life. Afterward, we further present a novel MAGDM method under LIVPF conditions. The main finding of our study is a new MAGDM method, which is more powerful and flexible than existing ones. Finally, we apply the method in a sustainable building materials selection to show its effectiveness. Additionally, comparison analysis is provided to demonstrate the advantages and superiorities of the proposed method.

On interval fuzzy implications derived from interval additive generators of interval t-norms

Yager's f-generated implication is one important class of fuzzy implication, which is based on additive generators of continuous Archimedean t-norms. Considering the related interval extension, this paper presents an analysis of interval fuzzy implications derived from interval additive generators of interval t-norms, called F-generated interval fuzzy implications. The relationship between Yager's f-generated implications and F-generated interval fuzzy implications is discussed. Then We study the basic properties of F-generated interval fuzzy implications and discuss some general forms of classical logic tautologies (i.e. law of importation, contrapositive symmetry, and distributivity over interval t-norms or interval t-conorms). Meanwhile, the relationships between F-generated interval fuzzy implications and several families of interval fuzzy implications are obtained.

Archimedean Aggregation Operators Based on Complex Pythagorean Fuzzy Sets Using Confidence Levels and Their Application in Decision Making

The diagnosed complex Pythagorean fuzzy (CPF) set is a more valuable and dominant tool than the Pythagorean and intuitionistic fuzzy sets to describe awkward and unreliable information more effectively. Further, Archimedean t-norm and t-conorm have a significant influence to depict the relation among aggregated values. To take advantage of the CPF set and Archimedean t-norm and t-conorm, and assume the relation among Archimedean norms and algebraic, Einstein, Hamacher, and frank norms at the same time, in this analysis, first, we proposed the fundamental Archimedean operational laws. Second, based on these laws, we developed confidence CPF Archimedean-weighted averaging (CCPFSAWA), confidence CPF Archimedean-ordered weighted averaging (CCPFSAOWA), confidence CPF Archimedean-weighted geometric (CCPFSAWG), confidence CPF Archimedean-ordered weighted geometric (CCPFSAOWG) operators and implemented their valuable results and properties. We know that Archimedean t-norm and t-conorm are the general form of the all-aggregation operators, so by using different values of t-norm and t-conorm, we explored the confidence CPF-weighted averaging (CCPFWA), confidence CPF-ordered weighted averaging (CCPFOWA), confidence CPF Einstein-weighted averaging (CCPFEWA), confidence CPF Einstein-ordered weighted averaging (CCPFEOWA), confidence CPF Hamacher-weighted averaging (CCPFHWA), confidence CPF Hamacher-ordered weighted averaging (CCPFHOWA), confidence CPF frank-weighted averaging (CCPFFWA), confidence CPF frank-ordered weighted averaging (CCPFFOWA), confidence CPF-weighted geometric (CCPFWG), confidence CPF-ordered weighted geometric (CCPFOWG), confidence CPF Einstein-weighted geometric (CCPFEWG), confidence CPF Einstein-ordered weighted geometric (CCPFEOWG), confidence CPF Hamacher-weighted geometric (CCPFHWG), confidence CPF Hamacher-ordered weighted geometric (CCPFHOWG), confidence CPF frank-weighted geometric (CCPFFWG), and confidence CPF frank-ordered weighted geometric (CCPFFOWG) operators. Then, we developed a multi-attribute decision-making (MADM) method based on the proposed operators. Finally, many examples are used to do comparative analysis among proposed and existing methods to show the validation of the new approaches.

Various aggregation operators of the generalized hesitant fuzzy numbers based on Archimedean t-norm and t-conorm functions.

This paper intends to introduce mathematical tools for aggregation of the generalized hesitant fuzzy numbers in order to increase the use of them in the real world. The proposed operators, are based on general form of t-norm and t-conorm functions, enable us to do some mathematical computations and aggregate the given generalized hesitant fuzzy numbers. At first, some famous Archimedean t-norms and t-conorms, i.e., Algebraic, Einstein, Hamacher, and Frank t-norms and t-conorms, and their properties, have been developed to be employed with generalized hesitant fuzzy numbers. Then, several averaging and geometric-based aggregation operators for generalized hesitant fuzzy numbers have been proposed. Later on, a decision-making algorithm has been defined based on such operators to address the problems. The necessity and application of the proposed concepts have been explained by some numerical examples.

The Fibonacci sequence in the description of maximal discrete Archimedean t-norms

There are many arguments for counting with more than two “truth values”; this allows to imitate human reasoning of facts which are not binary. For theoretical reasons, it is natural to use the whole real interval as the scale. However, this brings practical problems: it is difficult, and even impossible, to represent exact values. Often only a small scale of values suffices to express what we need. Therefore, finite chains are frequently used as domains of fuzzy logical operations. Their representation and manipulation are easy. In this paper, we focus on triangular norms (t-norms). The choice of a finite domain admits some operations (Gödel, Łukasiewicz), while it excludes others (all strict ones, including the product). A disadvantage of the Gödel (minimum) t-norm is that repetition of arguments does not change their meaning. This is often desirable to emphasize the statement. (“Words, words, words!”) Thus we do not consider the Gödel operations sufficient for representation of all fuzzy logical statements in human reasoning. Together with them, we discard all operations with idempotent elements other than 0, 1, thus we restrict attention to Archimedean t-norms (and t-conorms as their duals, not treated explicitly in the sequel). A disadvantage of many Archimedean t-norms is that, when applied to several arguments, the result is very often zero. Then, it gives no clue in comparing the outcomes, e.g., in giving priorities to alternatives evaluated by fuzzy rules. For instance, when the continuous Łukasiewicz t-norm is applied to 5 entries, it is nonzero only on 1/5!=1/120 of the volume of its 5-dimensional domain; in its discrete versions, nonzero results are even more rare. Thus we are interested in Archimedean t-norms with values “as large as possible”, here in the maximal Archimedean t-norms (=those which are not majorized by other Archimedean t-norms). It was shown in previous works that there is an abundance of discrete t-norms; their number grows fast with the number of elements of the underlying chain (no exponential bound seems to be known). There is also an abundance of Archimedean t-norms. In contrast to that, when we counted the number of maximal Archimedean t-norms, it grows asymptotically exponentially with a mild base. What is more interesting, these numbers follow the Fibonacci sequence. We have found a description of maximal Archimedean t-norms and explained also the processes behind their construction and the role of the Fibonacci sequence. These results link the t-norms to a construction using a kind of additive generators, although this could not be used without modifications. We do not describe all Archimedean t-norms and their numbers, but we have completed at least a significant step towards this goal, too.

Generation of continuous T-norms through latticial operations

It is well known that the usual point-wise ordering over the set T of t-norms makes it a poset but not a lattice, i.e., the point-wise maximum or minimum of two t-norms need not always be a t-norm again. In this work, we propose, two binary operations ▪ on the set TCA of continuous Archimedean t-norms and obtain, via these binary operations, a partial order relation ⊑, different from the usual point-wise order ≤, on the set TCA. As an interesting outcome of this structure, some stronger versions of some existing results dealing with the upper and lower bounds of two continuous Archimedean t-norms with respect to the point-wise order ≤ are also obtained. Finally, with the help of the operations ▪ on the set TCA, two binary operations ⊕,⊗ on the set TC of continuous t-norms are proposed and showed that (TC,⊕,⊗) is a lattice. Thus we have both a way of generating continuous t-norms from continuous t-norms and also obtain an order on them.