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.
Human opinion cannot be restricted to yes or no as depicted by conventional fuzzy set (FS) and intuitionistic fuzzy set (IFS) but it can be yes, abstain, no and refusal as explained by picture fuzzy ...set (PFS). In this article, the concept of spherical fuzzy set (SFS) and T-spherical fuzzy set (T-SFS) is introduced as a generalization of FS, IFS and PFS. The novelty of SFS and T-SFS is shown by examples and graphical comparison with early established concepts. Some operations of SFSs and T-SFSs along with spherical fuzzy relations are defined, and related results are conferred. Medical diagnostics and decision-making problem are discussed in the environment of SFSs and T-SFSs as practical applications.
In this manuscript, the notions of q-rung orthopair fuzzy sets (q-ROFSs) and complex fuzzy sets (CFSs) are combined is to propose the complex q-rung orthopair fuzzy sets (Cq-ROFSs) and their ...fundamental laws. The Cq-ROFSs are an important way to express uncertain information, and they are superior to the complex intuitionistic fuzzy sets and the complex Pythagorean fuzzy sets. Their eminent characteristic is that the sum of the qth power of the real part (similarly for imaginary part) of complex-valued membership degree and the qth power of the real part (similarly for imaginary part) of complex-valued non‐membership degree is equal to or less than 1, so the space of uncertain information they can describe is broader. Under these environments, we develop the score function, accuracy function and comparison method for two Cq-ROFNs. Based on Cq-ROFSs, some new aggregation operators are called complex q-rung orthopair fuzzy weighted averaging (Cq-ROFWA) and complex q-rung orthopair fuzzy weighted geometric (Cq-ROFWG) operators are investigated, and their properties are described. Further, based on proposed operators, we present a new method to deal with the multi‐attribute group decision making (MAGDM) problems under the environment of fuzzy set theory. Finally, we use some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods.
•Various higher-order fuzzy set theories (HO-FSTs) have been developed since 1960s.•Painstaking efforts have been devoted to devising basic measures for these HO-FSTs.•We present a unified ...representation for several of these HO-FSTs, based on u-map.•This effectively dissolves the artificial boundaries between different HO-FSTs.•Theoretical developments in HO-FSTs can now be done in a much more unified manner.
Since its creation by Zadeh in 1965, fuzzy set theory (FST) has been continuously advanced in various fronts during the past five decades. Along with Zadeh's classical FST (also termed type-1 fuzzy set theory), a number of higher-order FSTs, including e.g. type-2 fuzzy sets, intuitionistic fuzzy sets, typical hesitant fuzzy sets, and generalized hesitant fuzzy sets, have been proposed, constructed, and applied. A quick survey of the literature leads one to observe that a large amount of remarkable researches has been performed on developing theories of these higher-order fuzzy sets – a significant portion of these researches have involved painstaking efforts in devising suitable fundamental operators (e.g. the basic set operators and the aggregation operators) and measures (e.g. the similarity, subsethood, and entropy measures) under these various higher-order settings. At the same time, one also observes that the somewhat disparate frameworks assumed under these various higher-order settings have led to a highly complex landscape of the whole FST field. Arguably, this complexity poses significant barriers for any non-expert to try to apply these latest developments to his/her application domain. In this article, based on a so-called u-map representation that we have developed, we propose a very simple framework, via suitably adapting voting scenarios, that gives a unified description of several types of higher-order fuzzy sets. We further demonstrate that this framework enables us to develop fundamental measures for these higher-order fuzzy sets in an extremely streamlined and unified manner (e.g. as a result, by proving something once, you have essentially proved it for all). Thus, we believe that such a framework would be useful for the non-experts to understand and use higher-order FSTs in their application domains, and for the experts to further develop higher-order FSTs in an efficient manner.
In this paper, we will present a wider view on the relationship between interval-valued fuzzy sets and interval type-2 fuzzy sets, where we will show that interval-valued fuzzy sets are a particular ...case of the interval type-2 fuzzy sets. For this reason, both concepts should be treated in a different way. In addition, the view presented in this paper will allow a more general perspective of interval type-2 fuzzy sets, which will allow representing concepts that could not be presented by interval-valued fuzzy sets.
The aim of this paper is to present the novel concept of Complex q-rung orthopair fuzzy set (Cq-ROFS) which is a useful tool to cope with unresolved and complicated information. It is characterized ...by a complex-valued membership grade and a complex-valued non-membership grade, the distinction of which is that the sum of q-powers of the real parts (imaginary parts) of the membership and non-membership grades is less than or equal to one. To explore the study, we present some basic operational laws, score and accuracy functions and investigate their properties. Further, to aggregate the given information of Cq-ROFS, we present several weighted averaging and geometric power aggregation operators named as complex q-rung orthopair fuzzy (Cq-ROF) power averaging operator, Cq-ROF power geometric operator, Cq-ROF power weighted averaging operator, Cq-ROF power weighted geometric operator, Cq-ROF hybrid averaging operator and Cq-ROF power hybrid geometric operator. Properties and special cases of the proposed approaches are discussed in detail. Moreover, the VIKOR (“VIseKriterijumska Optimizacija I Kompromisno Resenje”) method for Cq-ROFSs is introduced and its aspects discussed. Furthermore, the above mentioned approaches apply to multi-attribute decision-making problems and VIKOR methods, in which experts state their preferences in the Cq-ROF environment to demonstrate the feasibility, reliability and effectiveness of the proposed approaches. Finally, the proposed approach is compared with existing methods through numerical examples.
•Interval- valued intuitionistic hesitant fuzzy entropy and VIKOR techniques are proposed.•The preferred alternative by IVIHF- VIKOR method is closer to the ideal solution.•The IVIHF entropy method ...utilized to calculate the importance of the criteria.
In this paper, we proposed interval valued intuitionistic hesitant fuzzy entropy for determine the importance of the criteria and interval valued intuitionistic hesitant fuzzy VIKOR method for ranking the alternatives. Industrial Robots are utilized to perform complicated and hazardous tasks accurately and also used to enhance the quality and efficiency of the work. Selecting an industrial robot for performing a particular task depends on the work and the associated criteria of the robot. The materials handled by the robots are different like powdered, adhesive, bulky, brittle etc. In this manner, choosing a suitable robot from the set of available industrial robots to handle a particular material is a challenging task. To get a more conscionable decision result, a decision organization contains a lot of decision makers. The interval- valued intuitionistic hesitant fuzzy set is utilized as a competent mathematical tool for enunciate individuals hesitant thinking. An interval- valued intuitionistic hesitant fuzzy set (IVIHFS) concedes a set of several possible interval- valued intuitionistic fuzzy membership and non- membership values. Finally, the proposed interval- valued intuitionistic hesitant fuzzy entropy and VIKOR techniques utilized for industrial robot selection.
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