Generalised Fuzzy Models Applied to Logical Algebras and Intelligent Systems in EngineeringView this Special Issue
Novel Dombi Aggregation Operators in Spherical Cubic Fuzzy Information with Applications in Multiple Attribute Decision-Making
The notion of spherical fuzzy sets (SFSs) is one of the most effective ways to model the fuzzy information in decision-making processes. The sum of squares of membership, neutral, and nonmembership degrees in SFSs lies in the interval [0, 1] and accommodates more uncertainties. Henceforth, in this article, the idea of spherical cubic fuzzy sets (SCFSs) is introduced, which is the generalization of SFSs. Spherical cubic fuzzy set is the combination of spherical fuzzy sets and interval-valued spherical fuzzy sets. The membership, neutral, and nonmembership degrees in an SCFS are cubic fuzzy numbers (CFNs). Consequently, this set outperforms the pre-existing structures of fuzzy set theory. Moreover, some fundamental operations for the comparison of two spherical CFNs are defined such as score function and accuracy function. Further, several new operations through Dombi t-norm and Dombi t-conorms are characterized to get the best results during the decision criteria. Furthermore, spherical cubic fuzzy Dombi weighted averaging (SCFDWA), SCFD ordered weighted averaging (SCFDOWA), SCFD hybrid weighted averaging (SCFDHWA), SCFD weighted geometric (SCFDWG), SCFD ordered weighted geometric (SCFDOWG), and the SCFD hybrid weighted geometric (SCFDHWG) aggregated operators are discussed, and their characteristics are examined. In addition, some of the operational laws of these operators are defined. Also, a decision-making approach based on these operators is proposed. Since the proposed methods and operators are the generalizations of the existing methods and operators, therefore, these techniques produce more general, accurate, and precise results as compared with existing ones. Finally, a descriptive example is given in order to describe the validity, practicality, and effectiveness of the proposed methods.
Multiattribute group decision-making is an investigation of recognizing and choosing the alternatives depending upon the values and priorities of decision makers. Settling on a chance infers that there are alternative decisions to be considered. In such a case, we do not need to recognize the same number of these options as could be allowed to select the best possibility to attain our objectives, targets, or expectations. Zadeh in  presented the idea of fuzzy set theory and logic. In the fuzzy set theory, he just examined the membership grade, known as the membership degree. Moreover, Zadeh established the application of fuzzy sets (FSs) in various fields like design, software engineering, and information technology. After numerous uses of FSs, Atanassov  saw that there are several deficiencies in FSs so he presented the idea of intuitionistic fuzzy sets (IFSs). Each element in an IFS is described by a pair of mappings, and each of these mappings is categorized by a membership and a nonmembership grade. The IFS was further extended by Yager and Yager [3, 4] who developed the idea of Pythagorean fuzzy sets (PyFSs) by adding constraints on the membership and nonmembership grades as the aggregate of the squares of membership and nonmembership grades must not exceed by 1. In [5, 6], the new approach of decision-making using the concept of picture fuzzy numbers was presented and further it was extended for picture fuzzy linguistic sets.
An IFS and a PyFS have been effectively implemented in various fields, but in numerous circumstances, these theories fail, for instance, in vote casting, human judgements include more responses like yes, no, abstain, and refusal. In voting station, the chamber issues forms for the applicants. The voting outcomes are distributed into four categories, and the results are as follows: vote in favor, abstain or neutral votes, vote in opposition, and refuse to vote. Here, abstain means blank voting form, i.e., nobody gave the vote in favor or against. Refusal of vote means that a person refuses to give the vote. The applicant is viewed as effective in the light of the fact that the quantity of supportive papers is greater than the vote in opposition. In these sorts of situations where the abstinence and rejection occur, the concepts of FSs and IFSs fail to be applied. Hence, the concept of picture fuzzy set (PFS) is presented by Cuong et al. [7–9], that is, the expanded form of the FSs and IFSs. PFS gives three membership grades known as positive membership grade, neutral membership grade, and negative membership grade correspondingly. The issue of voting criteria successfully resolved by picture fuzzy set theory, and this theory is further applied in various fields. By making use of those operations for PFN, plenty of working has been performed by the research workers, combined with aggregated methodology, multiattribute group decision, and information measures. From this overview, it is commented that neither FS nor IFS and PFS theory are useful to tackle the vague conflicting information. For example, if an individual gives their inclination about the item as far as indeed is, no is, and abstained is, at that point, we see that accordingly, IFSs and PFSs may not be able to tackle such circumstances. To overcome these complications, many researchers  presented the notion of spherical fuzzy set (SFS) by the additional constraints, i.e., the sum of the squares must not exceed. According to above situation, we see, thus, SFS is the generalization of PFS and IFS to cope with the problems in making decision. Fahmi et al.  gave a new idea of trapezoidal cubic fuzzy numbers and their applications in multiattribute group decision making. Garg  presented the idea of picture fuzzy aggregation operators and further discussed its applications. Ashraf et al.  introduced the collection of spherical weighted aggregated operations for resolving multiattribute decision-making problems under the spherical fuzzy sets. Gündogdu and Kahraman  presented the new idea of the spherical fuzzy TOPSIS method.
Dombi operations introduced in 1982 are an important complement to the existing operations. It is characterized by good flexibility for information aggregation. Many scholars presented the t-norm and t-conorm Dombi operations in [5, 15] which have an inclination of fluctuation with the operation of parameters. For this preferred position, Lin et al. utilized the IFSs and combined them with Dombi operations and presented the concept of Dombi Bonferroni mean operator  using IFSs to resolve the issues in multiattribute group decision-making. Liu et al.  presented new concept of spherical fuzzy sets for Yunnan Baiyao’s R & D project selectionproblem. In , Shi and Ye extended Dombi operation to neutrosophic cubic sets and utilized it in travelling decision-making approaches. To resolve the various issues in multiattribute group decision makers, Lu and Ye  firstly defined Dombi aggregated operations and linguistic cubic sets [20–23]. In , Jana et al. presented several Dombi bipolar fuzzy aggregated operations under the picture fuzzy data based on averaging, geometric, and various Dombi operations. Jana et al.  presented the idea of picture fuzzy Dombi aggregation operators and their applications in multiattribute decision-making process. Rafiq et al.  presented the cosine similarity measures of spherical fuzzy sets and their applications in decision-making. Seikh and Mandal  gave the idea of intuitionistic fuzzy Dombi aggregation operators and their applications. Wei et al.  presented the similarity measure of spherical fuzzy sets using the cosine function and their applications. Muneeza et al. [29, 30] gave the new idea of intuitionistic cubic fuzzy sets and their applications in supplier’s selection and hydropower plant locations. In [31–33], many researchers introduced the idea of Pythagorean cubic fuzzy sets, which is the generalization of Pythagorean fuzzy sets and cubic sets and further discussed its application in multiattribute decision-making. Ayaz et al. [34, 35] introduced the idea of spherical cubic fuzzy sets and defined the various aggregation operators and their applications in decision-making.
In aggregated procedure, the significant method is to characterize the operational laws. The applications of SFSs require several new operations and aggregated operations to be developed. By maintaining the advantages of the SFS, we describe the collection of SFSs. Besides this, by using Dombi norms, the fundamental weighted geometric average operations have been characterized by utilizing the idea of cubic fuzzy set theory as spherical cubic fuzzy Dombi weighted average (SCFDWA), spherical cubic fuzzy Dombi ordered weighted average (SCFDOWA), spherical cubic fuzzy Dombi hybrid weighted average (SCFDHWA), spherical cubic fuzzy Dombi weighted geometric (SCFDWG), spherical cubic fuzzy Dombi ordered weighted geometric (SCFOWG), and spherical cubic fuzzy Dombi hybrid weighted geometric (SCFDHWG). We utilized these operations to propose the method for multiattribute group decision-making. At last, we enlighten the practicality of the proposed methods in the selection of spherical cubic fuzzy numbers. In order to get a fair decision during the process, some new operational laws by Dombi t-norm and t-conorm are defined in this manuscript. A new approach based on spherical cubic fuzzy set models via spherical cubic fuzzy Dombi aggregation operators is proposed. A method to deal with decision-making problems using spherical cubic Dombi weighted averaging, Dombi weighted geometric, and Dombi hybrid weighted aggregation operators is established. This model has a stronger capability than existing weighted averaging, weighted geometric, Einstein, logarithmic averaging, and logarithmic geometric aggregation operators for spherical cubic fuzzy information. This study presents the novel decision-making techniques to tackle the uncertainty in decision-making processes through proposed generalized structure of spherical cubic fuzzy set and well known Dombi norms.
The paper is designed in the following manner. In Section 2, we presented fundamental information of extended fuzzy sets. In Section 3, we have discussed the idea of spherical cubic fuzzy set and various aggregation operators. The Dombi aggregation operations are introduced in Section 4. Section 5 discusses the applications of the proposed method, and some numerical applications are given in Section 6. In Section 7, analysis with the suggested Dombi aggregated operations is carried out, and finally we conclude our work in Section 8.
We support the reader’s interpretation of the standard definitions and outcomes of the spherical fuzzy set theory, but to make this work more introspective, basic ideas used in the literature are described, and we note a portion of the idea and findings used in the rest of the work.
Definition 1. (see ). Let be the universe of discourse; a fuzzy intuitionistic cubic set on is presented aswhere and are known as membership and nonmembership of , which satisfy the condition that and .
Definition 2. (see ). Let be the universe of discourse; a fuzzy Pythagorean cubic set on is presented aswhere and are known as membership and nonmembership of , which satisfy the condition that and .
Definition 3. (see ). Let be the universe of discourse; a spherical fuzzy set on is presented aswhere , , and are known as membership, neutral, and nonmembership degrees, respectively, under the specific condition , and the triplet is called the spherical fuzzy numbers.
3. Characteristics of Spherical Cubic Fuzzy Sets
The definition of the spherical cubic fuzzy set (SCFS) and its operations are presented in this section. The extension of the spherical fuzzy set is the spherical cubic fuzzy set. The composition of the SCFS defends elements that met or did not fulfill the requirement for values to be from 0 to 1.
The spherical cubic fuzzy set is a direct generalization of the cubic fuzzy set of Pythagoras and the cubic fuzzy set of images. When the pythagorean cubic fuzzy set (PyCFS) and image cubic fuzzy set both could not cope with the situation, a fascinating situation grows. There is a need to describe the concept of a spherical cubic fuzzy set to solve this situation in this way. The principle of the distinction between PyCFSs and SCFSs is that we research the neutral membership in SCFSs, where it does not in PyCFSs. The association of positive, neutral, and negative degrees of an object is defined in the closed unit interval in image cubic fuzzy, but the sum of positive, neutral, and negative degrees of the object is greater than 1. In this situation, we used spherical cubic fuzzy sets.
Each feature of the intuitionist cubic fuzzy set (ICFS) consists of a membership function and a nonmembership function in which the function of membership is cubic fuzzy set and cubic fuzzy set is also nonmembership. We then generalize (ICFS) and describe a fresh definition of the spherical cubic fuzzy set (SCFS) consisting of membership function, neutral membership function, and nonmembership function in which the function of membership is cubic fuzzy set, cubic fuzzy set is the function of neutral membership, and cubic fuzzy set is also nonmembership. The issue is addressed by SCFS in all current systems such as ICFS, PyCFS, and PCFS. So, the SCFS is the generalization of the entire system that exists.
Definition 4. (see ). Let be the universe of discourse, then the spherical cubic fuzzy set in is defined aswhere represents the membership degree, represents neutral, and represents the nonmembership degree where and , under the conditionWe called as spherical cubic fuzzy numbers (SCFNs) denoted by . which is known as spherical cubic fuzzy numbers (SCFNs).
Definition 5. (see ). Let two SCFNs be and and be any constant. Then, the operations are defined as follows: (1) iff and (2)(3)(4)
Definition 6. (see ). Let two SCFNs be and . The operations of the SCFNS are defined as follows:(1)(2)(3)(4)
Definition 7. (see ). Let and and are any two constants, then we have the following properties:(1)(2)(3), (4), (5), (6), (7)(8)
Definition 8. (see ). Let SCFN; the score function is presented asHere, .
Definition 9. An accuracy function is presented asHere, .
Definition 10. For any SCFNs, in . Then, the comparison scheme is defined as follows:(1), then (2), then (3), then(a), then (b), then (c), then
Definition 11. Let be the collection of SCFNs where in . The spherical cubic fuzzy weighted averaging (SCFWA) operator is illustrated as are weight vectors with and.
Definition 12. Let be the collection of SCFNs where in . The spherical cubic fuzzy ordered weighted average (SCFOWA) operator is defined aswhere are weight vectors with ,, and the largest value is ; therefore, the order is .
Definition 13. Let be the collection of SCFNs where in . The spherical cubic fuzzy hybrid weighted averaging (SCFHWA) operator is defined as follows:Here, the weighted vector is represented as with and , and the largest weight value is , and the order is defined as , and , and with representing the weighted vector.
Definition 14. Let be the collection of SCFNs where in . The spherical cubic fuzzy weighted geometric (SCFWG) operator iswhere the weight vector is represented with .
Definition 15. Let be the collection of SCFNs where in . The spherical cubic fuzzy ordered weighted geometric (SCFOWG) operator isand the largest weight value is , and the order is defined as .
Definition 16. Let be the collection of SCFNs where in . The spherical cubic fuzzy hybrid weighted geometric (SCFHWG) operator iswhere represent the weight vector where , and the largest weight value is , and the order is defined as . Also, associated weights are with and .
Definition 17. Assume that be real numbers and . Then, the norms of the Dombi are determined as
Example 1. Assume , and . Then,
4. Spherical Cubic Fuzzy Dombi Aggregated Operators
Presently, we proposed the concept of spherical cubic fuzzy Dombi aggregated operations and discussed some of their characteristics in this section on the basis of Definition 7. We will introduce the spherical cubic fuzzy Dombi weighted averaging (SCFDWA) operator, spherical cubic fuzzy Dombi ordered weighted averaging (SCFDOWA) operator, spherical cubic fuzzy Dombi weighted geometric (SCFDWG) operator, spherical cubic fuzzy Dombi ordered weighted geometric (SCFDOWG) operator, and spherical cubic fuzzy Dombi hybrid weighted geometric (SCFDHWG) operator and studied its fundamental properties, i.e., boundary property, idempotency property, and monotonic property.(1)(2)(3)(4)
Definition 18. Let two SCFNs be and in and . Then, the operations of SCFNs on the basis of Dombi operation are presented as
4.1. Spherical Cubic Dombi Weighted Averaging Operations
In the view of characterized Dombi operations of SCFNs, we describe the following weighted averaging aggregated operators.
Definition 19. For any collection of SCFNs, in . Then, structure of spherical cubic fuzzy Dombi weighted averaging (SCFDWA) operator iswhere is the weight vector with and .
Theorem 1. For any collections of SCFNs, in . Then, the spherical cubic Dombi weighted averaging (SCFDWA) operator is defined using the operations on Dombi with some positive constant as follows:where represents the weight vectors with and .