An advanced multi-attribute decision-making model for Urban transportation planning based on complex intuitionistic fuzzy sets with hierarchical parameters

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.

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>

Susceptibility analysis is an intelligent technique that not only assists decision makers in assessing the suspected severity of any sort of brain tumour in a patient but also helps them diagnose and cure these tumours. This technique has been proven more useful in those developing countries where the available health-based and funding-based resources are limited. By employing set-based operations of an arithmetical model, namely fuzzy parameterised complex intuitionistic fuzzy hypersoft set (FPCIFHSS), this study seeks to develop a robust multi-attribute decision support mechanism for appraising patients’ susceptibility to brain tumours. The FPCIFHSS is regarded as more reliable and generalised for handling information-based uncertainties because its complex components and fuzzy parameterisation are designed to deal with the periodic nature of the data and dubious parameters (sub-parameters), respectively. In the proposed FPCIFHSS-susceptibility model, some suitable types of brain tumours are approximated with respect to the most relevant symptoms (parameters) based on the expert opinions of decision makers in terms of complex intuitionistic fuzzy numbers (CIFNs). After determining the fuzzy parameterised values of multi-argument-based tuples and converting the CIFNs into fuzzy values, the scores for such types of tumours are computed based on a core matrix which relates them with fuzzy parameterised multi-argument-based tuples. The sub-intervals within [0, 1] denote the susceptibility degrees of patients corresponding to these types of brain tumours. The susceptibility of patients is examined by observing the membership of score values in the sub-intervals.

Representing complex intuitionistic fuzzy set by quaternion numbers and applications to decision making

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.

<abstract><p>The study on N-soft sets (NSSs) has been significantly developed recently. Hybrid models such as fuzzy N-soft sets, Intuitionistic fuzzy N-soft sets, and hesitant fuzzy N-soft sets were introduced to combine fuzzy sets, intuitionistic fuzzy sets and hesitant fuzzy sets with NSSs. Related to the hybrid models, it was also constructed some complements, operations and related properties. This article aims to construct a new hybrid model called hesitant intuitionistic fuzzy N-soft sets (HIFNSSs) to combine intuitionistic fuzzy N-soft sets and hesitant fuzzy N-soft sets. Moreover, we generalise HIFNSSs to generalized hesitant intuitionistic fuzzy N-soft sets (GHIFNSSs) as a hybrid model between generalized hesitant intuitionistic fuzzy sets and N-soft sets. It was also defined some complements of GHIFNSSs, intersection and union operations between GHIFNSSs, and proved that the operations between some particular complements hold De Morgan Law. In applying a GHIFNSS, we provide an algorithm for decision-making problems and its numerical illustration.</p></abstract>

The paper aims are to propose a new concept called as complex picture fuzzy set (CPFS), as an extension of complex intuitionistic fuzzy set (CIFS). The addition of a neutral membership degree to the definition of CIFS makes CPFS a generalized form of CIFS. The uniqueness of this new theory lies in the capability to attain the wider range with the help of degree of neutral membership, non-membership, and membership. The range of values of membership degrees is broaden to the unit disk in a complex plane. We define elementary operations and properties of CPFSs and explore the MCDM issues with the help of CPFSs, based on Hamacher operations and some aggregation methods. Then, we introduce some operators to aggregate the CPF data, namely complex picture fuzzy Hamacher weighted averaging, ordered weighted averaging, hybrid averaging and complex picture fuzzy Hamacher weighted geometric, ordered weighted geometric and hybrid geometric operators, benefited from the basic Hamacher operations, and averaging, geometric aggregation techniques. We also construct MCDM problem using these operators and perform a calculation for the selection of best ERP system, to demonstrate the authenticity and efficiency of this manuscript. Moreover, we study a comparison to validate the consistency and superiority of our techniques.

<abstract><p>Soft set has limitation for the consideration of disjoint attribute-valued sets corresponding to distinct attributes whereas hypersoft set, an extension of soft set, fully addresses this scarcity by replacing the approximate function of soft sets with multi-argument approximate function. Some structures (i.e., possibility fuzzy soft set, possibility intuitionistic fuzzy soft set) exist in literature in which a possibility of each element in the universe is attached with the parameterization of fuzzy sets and intuitionistic fuzzy sets while defining fuzzy soft set and intuitionistic fuzzy soft set respectively. This study aims to generalize the existing structure (i.e., possibility intuitionistic fuzzy soft set) and to make it adequate for multi-argument approximate function. Therefore, firstly, the elementary notion of possibility intuitionistic fuzzy hypersoft set is developed and some of its elementary properties i.e., subset, null set, absolute set and complement, are discussed with numerical examples. Secondly, its set-theoretic operations i.e., union, intersection, AND, OR and relevant laws are investigated with the help of numerical examples, matrix and graphical representations. Moreover, algorithms based on AND/OR operations are proposed and are elaborated with illustrative examples. Lastly, similarity measure between two possibility intuitionistic fuzzy hypersoft sets is characterized with the help of example. This concept of similarity measure is successfully applied in decision making to judge the eligibility of a candidate for an appropriate job. The proposed similarity formulation is compared with the relevant existing models and validity of the generalization of the proposed structure is discussed.</p></abstract>

In this paper we introduce the concentration and dilation operators on interval valued intuitionistic fuzzy soft sets of root type which is a generalization of fuzzy set, intuitionistic fuzzy set, interval valued intuitionistic fuzzy set and intuitionistic fuzzy set of root type and establish some properties of these operators. We define Hamming distance between two interval valued intuitionistic fuzzy soft sets of root type and it is shown that it is a metric. A similarity measure based on this Hamming distance is defined and some properties are established. We also develop a decision making method based on the similarity measure of Hamming distance between interval valued intuitionistic fuzzy soft sets of root type. We develop an algorithm for the decision making problem and illustrate its working by means of examples.

<abstract><p>The main influence of this analysis is to derive two different types of aggregation operators under the consideration of algebraic <italic>t</italic>-norm and <italic>t</italic>-conorm and Einstein <italic>t</italic>-norm and <italic>t</italic>-conorm for CIF set theory. Because these operators are very effective for evaluating the collection of information into a singleton preference. For this, first, we discover the Algebraic and Einstein operational laws for CIF sets. Then, we aim to discover the theory of CCIFWA, CCIFOWA, CCIFWG, CCIFOWG operators and their valuable properties "idempotency, monotonicity and boundedness" and results. Furthermore, we also derive the theory of CCIFEWA, CCIFEOWA, CCIFEWG, CCIFEOWG operators and their valuable properties "idempotency, monotonicity, and boundedness" and results. Some special cases of the derived work are also described in detail. Finally, we illustrate a MADM procedure under the consideration of derived operators to enhance the worth of the presented information. Finally, we compare the presented operators with various existing operators with the help of various suitable examples for showing the reliability and stability of the derived approaches.</p></abstract>

Complex evidence theory is regarded as the extension of traditional evidence theory to address uncertainty problem. The evidence evaluation for complex basic belief assignments (CBBAs) is still an open issue. This research defines a novel distance to quantify similarity degree in the framework of complex intuitionistic fuzzy set (CIFS). Base on the newly defined distance, a new evidence evaluation method is proposed. Then, a novel combination method for CBBAs is presented. A numerical example is introduced to verify the performance of the proposed method.

The basic notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation. This paper proposes an intuitionistic fuzzy rough set (IFRS) model which is a combination of intuitionistic fuzzy set (IFS) and rough set. We approximate an IFS by using soft binary relations instead of fixed binary relations. By using this technique, we get two pairs of intuitionistic fuzzy (IF) soft sets, called the upper approximation and lower approximation with respect to foresets and aftersets. Properties of newly defined rough set model (IFRS) are studied. Similarity relations between IFS with respect to this rough set model (IFRS) are also studied. Finally, an algorithm is constructed depending on these approximations of IFSs and score function for decision-making problems, although a method of decision-making algorithm has been introduced for fuzzy sets already. But, this new IFRS model is more accurate to solve the problem because IFS has degree of nonmembership and degree of hesitant.