Analysis of Einstein aggregation operators based on complex intuitionistic fuzzy sets and their applications in multi-attribute decision-making

Aggregation operators are the standard mathematical tool for the combination of several inputs into one unique output. This paper presents some aggregation operators for complex intuitionistic fuzzy (CIF) information. Complex intuitionistic fuzzy set (CIFS) theory has a large ability to capture vagueness since it can represent the complete specifications of problems with both intuitionistic uncertainty and periodicity. We introduce the CIF Hamacher weighted averaging (CIFHWA) operator, CIF Hamacher ordered weighted averaging (CIFHOWA) operator, CIF Hamacher weighted geometric (CIFHWG) operator and CIF Hamacher ordered weighted geometric (CIFHOWG) operator. We bring to light some remarkable properties of these operators and explicitly state some noteworthy special cases. Moreover, we contribute to the advancement of multi-attribute decision-making (MADM) with a novel proposal of an algorithm in the CIF environment. Furthermore, we consider and solve a MADM problem that finds the best source for generation of electricity with the aid of the proposed operators, thus proving their usefulness for the purpose of decision-making. Finally, we check the effectiveness of these operators by a validity test.

Since traffic congestion is a major issue worldwide, particularly in urban areas due to the increasing population on a daily basis, it negatively impacts human life, leading to time wastage, health problems, and economic repercussions. Therefore, establishing a sustainable transportation system in densely populated urban areas is crucial. To overcome this challenge, various factors on which a sustainable transportation system depends must be deeply analyzed. However, this analysis may involve fuzziness and vagueness due to imprecise data and incomplete information. In this regard, a mathematical framework called the Complex Intuitionistic Fuzzy Hypersoft Set (CIFHSS) is introduced in this research. This article presents and thoroughly investigates essential set-theoretical operations based on the complex intuitionistic fuzzy hypersoft set, using appropriate examples. Furthermore a novel Multi-Attribute Decision Making approach, based on the CIFHSS is developed. This approach incorporates decision-valued matrices (both maximum and minimum), a scoring framework for CIFHSS, and matrix-based aggregations, such as the core matrix. Moreover, the algorithm is implemented in a real-world scenario to solve traffic congestion in urban areas, demonstrating the versatility of the proposed algorithm. The model assigns scores of 0.6161 to the Electric Bus Rapid Transit system, 0.6089 to the Expanded Bicycle Lane Network, 0.6571 to Light Rail Transit, and 0.7627 to the Hybrid Car-Sharing Program, identifying the latter as the most suitable option. A sensitivity analysis of the proposed model is also presented using statistical tools. Finally, a detailed conclusion and future directions for the application of this research are provided.

Complex q-rung orthopair fuzzy sets (CQROFSs) are proposed to convey vague material in decision-making problems. The CQROFSs can enthusiastically modify the region of proof by altering the factor q ≥ 1 for real and imaginary parts based on the variation degree and, therefore, favor further uncountable options. Consequently, this set reverses over the existing theories, such as complex intuitionistic fuzzy sets (CIFSs) and complex Pythagorean fuzzy sets (CPFS). In everyday life, there are repeated situations that can occur, which involve an impartial assertiveness of the decision-makers. To determine the best decision to handle such situations, in this study, we propose modern operational laws by joining the characteristics of the truth factor sum and collaboration between the truth degrees into the analysis for CQROFSs. Based on these principles, we determined several weighted averaging neutral aggregation operators (AOs) to collect the CQROF knowledge. Subsequently, we established an original multiattribute decision-making (MADM) procedure by using the demonstrated AOs based on CQROFS. To evaluate the effectiveness, in terms of reliability and consistency, of the proposed operators, they were applied to some numerical examples. A comparative analysis of the investigated operators and other existing operators was also performed to find the dominance and validity of the introduced MADM method.

Complex q-rung orthopair fuzzy sets (Cq-ROFS) become one of the effective tools to deal with unresolved and complex information. In recent years, a variety of multi-attribute decision making(MADM) methods have been developed in the Cq-ROFS environment, such as complex intuitionistic fuzzy set (CIFS) and complex Pythagorean fuzzy sets(CPFS). Cq-ROFS is superior to CIFs and CPFS, which can describe a wider range of uncertain information spaces. Therefore, this paper studies MADM based on Cq-ROFS. However, decision making experts may have personal biases. In order to reduce the influence of individual preference on decision-making, the dependent Hamacher aggregation operators for Complex q-rung orthopair fuzzy sets(Cq-ROFDHA) are proposed based on the Hamacher operation and dependent method. The operator can improve personal bias by assigning lower weights to biased evaluations (unduly high or unduly low values) and higher weights to mid values. Firstly, the basic operation rules, score function, distance and properties of Cq-ROFDHA are described. We then applied Cq-ROFDHA to service utility decisions for online health communities. Finally, we compare it with other aggregation factors. The results show that the Cq-ROFDHA operator has the advantages of consistency and superiority.

As a generalization of complex fuzzy set (CFS), complex hesitation fuzzy set (CHFS) is a powerful tool to express two-dimensional fuzzy information, but it ignores the randomness that exists widely in real life. Considering the randomness and ambiguity often simultaneously occur in real-world problems, we come up with an innovative fuzzy set called complex probabilistic hesitant fuzzy set (CPHFS) in this article, which is a joint extension of CFS and probabilistic hesitant fuzzy set (PHFS). Moreover, we establish four basic operations for complex probabilistic hesitant fuzzy element(CPHFE) and a series of useful operational laws are also developed. Secondly, several aggregation operators including complex probabilistic hesitant fuzzy weighted averaging operator(CPHFWA), complex probabilistic hesitant fuzzy weighted geometric operator(CPHFWG), complex probabilistic hesitant fuzzy Heronian Mean operator(CPHFHM) and complex probabilistic hesitant fuzzy Geometric Heronian Mean operator(CPHFGHM) as well as their weighted forms that can reflect the interrelationship among all decision attributes are proposed, which is of great use in multi-attribute decision making(MADM) or in multi-attribute group decision making(MAGDM) issues when we want to synthesize multiple evaluation values into one. Besides, we also present a number of valuable properties of these provided aggregation operators. Thirdly, a MADM process is built in the CPHF environment step by step based on the aggregation operators we have just established. Furthermore, we provide an example about purchasing a most suitable car to demonstrate the effectiveness and practicability of the proposed decision process. Finally, we point out the superiorities of MADM in working under the CPHF environment through comparing it with the existing method and also offer some possible prospects for further research. The method displayed in this work can effectively solve the MADM problems where the decision-making information is described by CPHFEs and the attributes are interactive.

A complex intuitionistic fuzzy (CIF) set contains the membership and non-membership in the shape of a complex number whose amplitude term and phase term are covered in the unit interval. Moreover, Hamacher interaction aggregation operators are the combination of two major operators, called Hamacher aggregation operators and interaction aggregation operators, and they are used to aggregate the collection of information into one value. In this manuscript, we present the concept of Hamacher interaction operational laws for CIF sets (CIFSs). Further, we develop the CIF Hamacher interaction weighted averaging (CIFHIWA) operator, CIF Hamacher interaction ordered weighted averaging (CIFHIOWA) operator, CIF Hamacher interaction weighted geometric (CIFHIWG) operator, and CIF Hamacher interaction ordered weighted geometric (CIFHIOWG) operator. For these operators, we also discuss some basic properties, such as idempotency, monotonicity, and boundedness. Additionally, we develop a MADM method based on the developed operators and apply it to solve the green supply chain management problems, which can implement environmentally friendly practices to minimize carbon emissions, resource consumption, and waste generation while promoting long-term sustainability. Finally, we verify the superiority and effectiveness of the proposed method based on a comparative analysis between the proposed techniques and existing methods.

Complex intuitionistic fuzzy set (CIFS) is a special intuitionistic fuzzy set where the membership and non-membership degrees are expressed by a complex-valued membership degree and can more easily describe the vagueness and uncertainty in the real world. Archimedean t-conorm and t-norm (ATT), as an important class of the t-norm (TN) and t-conorm (TC), have greater flexibility in the information fusion process. In this paper, we extend the ATT to CIFSs and present some generalized geometric aggregation operators, which can be used to handle the multiple criteria decision-making (MCDM) problems. For it, we firstly define some new operational laws of the CIFSs based on ATT, then some weighted geometric aggregation operators based on proposed operations are proposed. Further, some desirable properties and special cases of them are studied. Finally, a decision-making approach is developed for the MCDM problem with complex intuitionistic fuzzy information. A practical example is given to show the availability and advantages of the proposed method by comparison with some existing methods. The proposed aggregation operators are more generalized than the existing ones to utilize the uncertain and imprecise information. Several existing operators are considered as special cases of the proposed one. Finally, the proposed method will offer various choices to the decision-maker to access the finest alternatives.

<abstract> <p>A complex intuitionistic fuzzy set is a generalization framework to characterize several applications in decision making, pattern recognition, engineering, and other fields. This set is considered more fitting and coverable to Intuitionistic Fuzzy Sets (IDS) and complex fuzzy sets. In this paper, the abstraction of (${{\alpha _{1, 2}}, {\beta _{1, 2}}}$) complex intuitionistic fuzzy sets and (${{\alpha _{1, 2}}, {\beta _{1, 2}}}$)-complex intuitionistic fuzzy subgroups were introduced regarding to the concept of complex intuitionistic fuzzy sets. Besides, we show that (${{\alpha _{1, 2}}, {\beta _{1, 2}}}$)-complex intuitionistic fuzzy subgroup is a general form of every complex intuitionistic fuzzy subgroup. Also, each of (${{\alpha _{1, 2}}, {\beta _{1, 2}}}$)-complex intuitionistic fuzzy normal subgroups and cosets are defined and studied their relationship in the sense of the commutator of groups and the conjugate classes of group, respectively. Furthermore, some theorems connected the (${{\alpha _{1, 2}}, {\beta _{1, 2}}}$)-complex intuitionistic fuzzy subgroup of the classical quotient group and the set of all (${{\alpha _{1, 2}}, {\beta _{1, 2}}}$)-complex intuitionistic fuzzy cosets were studied and proved. Additionally, we expand the index and Lagrange's theorem to be suitable under (${{\alpha _{1, 2}}, {\beta _{1, 2}}}$)-complex intuitionistic fuzzy subgroups.</p> </abstract>

Intuitionistic fuzzy geometric aggregation operators in the framework of Aczel-Alsina triangular norms and their application to multiple attribute decision making

In this study, the authors explore the complex q-rung orthopair fuzzy sets, which are preferred to be enhanced of the complex intuitionistic fuzzy sets and the complex Pythagorean fuzzy sets, individually. The intention is regarding the buildup of certain innovative operational laws and their related weighted aggregation operators based on the complex q-rung orthopair fuzzy (CQROF) information. In this regard, they characterize certain original neutral or fair operational laws that involve the model of proportional distribution to accomplish a neutral or fair usage to the truth and falsity functions of CQROFSs. Consequently, with these operations, they acquire CQROF weighted fairly aggregation (CQROFWFA) and CQROF ordered weighted fairly aggregation (CQROFOWFA) operators which can neutrally or fairly provide the truth and falsity degrees. They implement an MADM (multi-attribute decision-making) methodology with multiple decision makers and partial weight knowledge in the structure of CQROFSs.

In some multi-attribute decision-making (MADM) models studying attributes’ interactive phenomena is very important for the minimizing decision risks. Usually, the Choquet integral type aggregations are considered in such problems. However, the Choquet integral aggregations do not consider all attributes’ interactions; therefore, in many cases, when these interactions are revealed in less degree, they do not perceive these interactions and their utility in MADM problems is less useful. For the decision of this problem, we create the Choquet integral-based new aggregation operators’ family which considers all pair interactions between attributes. The problem under the discrimination q-rung picture linguistic and q-rung orthopair fuzzy environments is considered. Construction of a 2-order additive fuzzy measure (TOAFM) involves pair interaction indices and importance values of attributes of a MADM model. Based on the attributes’ pair interactions for the identification of associated probabilities of a 2-order additive fuzzy measure, the Shapley entropy maximum principle is used. The associated probabilities q-rung picture linguistic weighted averaging (APs-q-RPLWA) and the associated probabilities q-rung picture linguistic weighted geometric (APs-q-RPLWG) aggregation operators are constructed with respect to TOAFM. For an uncertainty pole of experts’ evaluations on attributes regarding the possible alternatives, the associated probabilities of a fuzzy measure are used. The second pole of experts’ evaluations as arguments of the aggregation operators by discrimination q-rung picture linguistic values is presented. Discrimination q-rung picture linguistic evaluations specify the attribute’s dominant, neutral and non-dominant impacts on the selection of concrete alternative from all alternatives. Constructed operators consider the all relatedness between attributes in any consonant attribute structure. Main properties on the rightness of extensions are showed: APs-q-RPLWA and APs-q-RPLWG operators match with q-rung picture linguistic Choquet integral averaging and geometric operators for the lower and upper capacities of order two. The conjugation among the constructed operators is also considered. Connections between the new operators and the compositions of dual triangular norms (Tp,Spq) and (Tmin,Smax) are also constructed. Constructed operators are used in evaluation of a selection reliability index (SRI) of candidate service centers in the facility location selection problem, when small degree interactions are observed between attributes. In example MADM, the difference in optimal solutions is observed between the Choquet integral aggregation operators and their new extensions. The difference, however, is due to the need to use indices of all interactions between attributes.

In this study, in view of expressing the uncertain information more elegantly, we shall enlighten the q-rung orthopair fuzzy sets (q-ROFSs) and the q-rung orthopair fuzzy numbers (q-ROFNs) which are considered to be superior of the intuitionistic fuzzy sets and the Pythagorean fuzzy sets, respectively. Here our aim is towards the development of some new operational laws and their corresponding weighted aggregation operators under the q-rung orthopair fuzzy environment. In this regard, at the very beginning, we define some new neutral or fair operational laws that include the concept of proportional distribution to achieve a neutral or fair treatment to the membership and non-membership functions of q-ROFN. Subsequently, with these operations, we develop q-rung orthopair fuzzy weighted fairly aggregation operator (qROFWFA) and q-rung orthopair fuzzy ordered weighted fairly aggregation operator (qROFOWFA) which can neutrally or fairly serve the membership and non-membership degrees. We observe the noteworthy features of these proposed aggregation operators. Furthermore, we exercise also an MADM (multi-attribute decision-making) approach with multiple decision makers and partial weight information in the framework of q-rung orthopair fuzzy sets. At the end of this study, we provide an illustrative example to highlight the feasibility and a practical look of the approach proposed herein.

<abstract><p>The fractional orthotriple fuzzy sets (FOFSs) are a generalized fuzzy set model that is more accurate, practical, and realistic. It is a more advanced version of the present fuzzy set models that can be used to identify false data in real-world scenarios. Compared to the picture fuzzy set and Spherical fuzzy set, the fractional orthotriple fuzzy set (FOFS) is a powerful tool. Additionally, aggregation operators are effective mathematical tools for condensing a set of finite values into one value that assist us in decision making (DM) challenges. Due to the generality of FOFS and the benefits of aggregation operators, we established two new aggregation operators in this article using the Frank t-norm and conorm operation, which we have renamed the fractional orthotriple fuzzy Choquet-Frank averaging (FOFCFA) and fractional orthotriple fuzzy Choquet-Frank geometric (FOFCFG) operators. A few of these aggregation operators' characteristics are also discussed. To demonstrate the efficacy of the introduced work, the multi-attribute decision making (MADM) algorithm is discussed along with applications. To demonstrate the validity and value of the suggested work, a comparison of the proposed work has also been provided.</p></abstract>

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

The interval‐valued q‐rung orthopair fuzzy set (IVq‐ROFS) and complex fuzzy set (CFS) are two generalizations of the fuzzy set (FS) to cope with uncertain information in real decision making problems. The aim of the present work is to develop the concept of complex interval‐valued q‐rung orthopair fuzzy set (CIVq‐ROFS) as a generalization of interval‐valued complex fuzzy set (IVCFS) and q‐rung orthopair fuzzy set (q‐ROFS), which can better express the time‐periodic problems and two‐dimensional information in a single set. In this article not only basic properties of CIVq‐ROFSs are discussed but also averaging aggregation operator (AAO) and geometric aggregation operator (GAO) with some desirable properties and operations on CIVq‐ROFSs are discussed. The proposed operations are the extension of the operations of IVq‐ROFS, q‐ROFS, interval‐valued Pythagorean fuzzy, Pythagorean fuzzy (PF), interval‐valued intuitionistic fuzzy, intuitionistic fuzzy, complex q‐ROFS, complex PF, and complex intuitionistic fuzzy theories. Further, the Analytic hierarchy process (AHP) and technique for order preference by similarity to ideal solution (TOPSIS) method are also examine based on CIVq‐ROFS to explore the reliability and proficiency of the work. Moreover, we discussed the advantages of CIVq‐ROFS and showed that the concepts of IVCFS and q‐ROFS are the special cases of CIVq‐ROFS. Moreover, the flexibility of proposed averaging aggregation operator and geometric aggregation operator in a multi‐attribute decision making (MADM) problem are also discussed. Finally, a comparative study of CIVq‐ROFSs with pre‐existing work is discussed in detail.