Novel Dombi Aggregation Operators in Spherical Cubic Fuzzy Information with Applications in Multiple Attribute Decision-Making

,


Introduction
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 [1] 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 [2] 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. e 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. e 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. e 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][8][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.
e 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 [10] 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. [11] gave a new idea of trapezoidal cubic fuzzy numbers and their applications in multiattribute group decision making. Garg [12] presented the idea of picture fuzzy aggregation operators and further discussed its applications. Ashraf et al. [13] introduced the collection of spherical weighted aggregated operations for resolving multiattribute decision-making problems under the spherical fuzzy sets. Gündogdu and Kahraman [14] 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 [16] using IFSs to resolve the issues in multiattribute group decision-making. Liu et al. [17] presented new concept of spherical fuzzy sets for Yunnan Baiyao's R & D project selectionproblem. In [18], 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 [19] firstly defined Dombi aggregated operations and linguistic cubic sets [20][21][22][23]. In [24], 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. [25] presented the idea of picture fuzzy Dombi aggregation operators and their applications in multiattribute decisionmaking process. Rafiq et al. [26] presented the cosine similarity measures of spherical fuzzy sets and their applications in decision-making. Seikh and Mandal [27] gave the idea of intuitionistic fuzzy Dombi aggregation operators and their applications. Wei et al. [28] 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][32][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. e 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.
is model has a stronger capability than existing weighted averaging, weighted geometric, Einstein, logarithmic averaging, and logarithmic geometric aggregation operators for spherical cubic fuzzy information. is 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. e 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.
e 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.

Preliminaries
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 [30]). Let X be the universe of discourse; a fuzzy intuitionistic cubic set I x on X is presented as where [u − , u + ], α and [w − , w + ], θ〉 are known as membership and nonmembership of I x , which satisfy the condition that sup[u − , u + ] + sup[w − , w + ] ≤ 1 and α + θ ≤ 1.
Definition 2 (see [24]). Let X be the universe of discourse; a fuzzy Pythagorean cubic set P x on X is presented as where [u − , u + ], α and [w − , w + ], θ〉 are known as membership and nonmembership of I x , which satisfy the condition that (sup[u − , u + ]) 2 + (sup[w − , w + ]) 2 ≤ 1 and Definition 3 (see [10]). Let X be the universe of discourse; a spherical fuzzy set S x on X is presented as where P S x : X ⟶ θ, I S x : X ⟶ θ, and N S x : X ⟶ θ are known as membership, neutral, and nonmembership degrees, respectively, under the specific condition 0 ≤ P S x + I S x + N S x ≤ 1, and the triplet S x � < P S x , I S x , N S x > is called the spherical fuzzy numbers.

Characteristics of Spherical Cubic Fuzzy Sets
e definition of the spherical cubic fuzzy set (SCFS) and its operations are presented in this section. e extension of the spherical fuzzy set is the spherical cubic fuzzy set. e composition of the SCFS defends elements that met or did not fulfill the requirement for values to be from 0 to 1. e 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. ere is a need to describe the concept of a spherical cubic fuzzy set to solve this situation in this way. e principle of the distinction between PyCFSs and SCFSs is that we research the neutral membership in SCFSs, where it does not in PyCFSs. e 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. e 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 [34]). Let X be the universe of discourse, then the spherical cubic fuzzy set S ⌣ x in X is defined as where [u − , u + ], α represents the membership degree, We called , θ〉) which is known as spherical cubic fuzzy numbers (SCFNs).
, θ p 〉} be the collection of SCFNs where (p � 1, 2, 3, . . . , n) in X. e spherical cubic fuzzy ordered weighted average (SCFOWA) operator is defined as where c p (p � 1, 2, 3, . . . , n) are weight vectors with c p ∈ [0, 1], n p�1 c p � 1, and the p th largest value is e spherical cubic fuzzy hybrid weighted averaging (SCFHWA) operator is defined as follows: Here, the weighted vector is represented as c p (p � 1, 2, . . . . . . , n) with c p ≥ 0 and n p�1 c p � 1, and the p th largest weight value is S ), and the order is defined as S Mathematical Problems in Engineering w * n ), and n p�1 w * p � 1with w * p ≥ 0 representing the weighted vector.
e spherical cubic fuzzy weighted geometric (SCFWG) operator is where the weight vector is represented e spherical cubic fuzzy hybrid weighted geometric (SCFHWG) operator is where c p ≥ 0 represent the weight vector where (p � 1, 2, 3, . . . , n), and the p th largest weight value is S with n p�1 w p * � 1 and w p * ≥ 0.

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.
, θ 2 〉} in X and c ≥ 0. en, the operations of SCFNs on the basis of Dombi operation are presented as

Spherical Cubic Dombi Weighted Averaging Operations.
In the view of characterized Dombi operations of SCFNs, we describe the following weighted averaging aggregated operators.

Theorem 1. For any collections of SCFNs
en, the spherical cubic Dombi weighted averaging (SCFDWA) operator is defined using the operations on Dombi with some positive constant ε > 0 as follows:

Mathematical Problems in Engineering
where c p represents the weight vectors c p ≥ 0 with (p � 1, 2, . . . , n) and n p�1 c p � 1.
Proof. We will prove it by mathematical induction, so eorem 1 is true for n � 2.

Mathematical Problems in Engineering
Now, assume that equation (9) is true.
Now, we will prove equation (19) for n � k + 1, i.e., 8 Mathematical Problems in Engineering So, by the mathematical induction, it is true for all n.
Mathematical Problems in Engineering 9 which we have to prove. □ e properties of SCFDWA are as follows: (1) Idempotency. We have the collection of SCFNs , θ p 〉} (p � 1, 2, . . . , n) in X. en, the collection of SCFNs (2) Boundary. We have the collection of SCFNs us, where c p (p � 1, 2, 3, . . . , n) represent the weight vectors satisfying n p�1 c p � 1 and c p ≥ 0, and the largest p th weighted value is S ⌣ x η(p) , and the total order is S

Theorem 2. We have the collection of SCFNs
e spherical cubic fuzzy Dombi ordered weighted average (SCDOWA) operator is defined as where c p represent the weight vector where (p � 1, 2, 3, . . . , n) and n p�1 c p � 1, and the largest weight is S ⌣ x η(p) and the total order is S Proof. e proof is similar to eorem 1.
Proof. e proof is similar to eorem 1.
□ e properties of SCFDHWA are as follows: (1) Idempotency. We have the collection of SCFNs en, we say collection of SCFNs S ⌣ x p (p � 1, 2, . . . , n) is equal, i.e., (2) Boundary. We have the collection of SCFNs us, (3) Monotonicity. We have the collection of SCFNs

Spherical Cubic Dombi Weighted Geometric Operators.
On the basis of Dombi operator of SCFNs, we present the weighted geometric aggregated operations as follows.

Definition 22. We have the collection of SCFNs
e spherical cubic fuzzy Dombi weighted geometric (SCFDWG) operator is defined as where c p represent the weight vector with c p ≥ 0 and n p�1 c p � 1.

Mathematical Problems in Engineering
Step 1. We will use the concept of SCFWG operator to aggregate all spherical cubic fuzzy decision matrices that are normalized separately. e aggregated of spherical cubic fuzzy matrix is shown in Table 7.
Step 2 Case 1. We will use SCFDWA to evaluate their efficiency separately according to the weight vectors c � (0.2, 0.3, 0.5) T and ε � 0.5 > 0 shown in Table 8 Case 2. We will use SCFDOWA to evaluate their efficiency separately according to the weight vectors c � (0.2, 0.3, 0.5) T and ε � 0.5 > 0 shown in Table 9 Case 3. We will use SCFDHWA to evaluate their efficiency separately according to the weight vectors c � (0.2, 0.3, 0.5) T and ε � 0.5 > 0 shown in Table 10 Table 1: Investing capacity in a wealth administration firm D 1 .      Table 11 Case 5. We will use SCFDOWG to evaluate their efficiency separately according to the weight vectors c � (0.2, 0.3, 0.5) T and ε � 0.5 > 0 shown in Table 12 Case 6 We will use SCFDHWG to evaluate their efficiency separately according to the weight vectors c � (0.2, 0.3, 0.5) T and ε � 0.5 > 0 shown in Table 13 Step 3. e score of each alternative is shown in Table 14.     Step 4. By using the rank criteria b i (i � 1, 2, . . . , m), we will select the best one which has the largest score.

Comparative Analysis
Here, we provide two comparison analysis that show that our proposed operators reliable and effective to aggregate the spherical cubic data. In Figure 1, ranking of spherical cubic fuzzy Dombi operators is given (Tables 14-20).
In 2019, Jana et al. suggested the picture fuzzy Dombi aggregation operators; in this study, we give comparison between proposed spherical cubic Dombi aggregation operators and existed Dombi aggregation operators as follows. Now, utilizing spherical cubic fuzzy Dombi weighted averaging operator to chose the best alternative is as follows as follows.
� Q 1 is the optimal alternative. e findings are similar to those provided by Jana et al. [25]. e methodology   presented by Jana et al. [25] tackles picture fuzzy set and fails to deal spherical cubic fuzzy set. erefore, the new method proposed in this paper is more generalized to deal uncertainties in decision-making problems. erefore, novel spherical cubic Dombi aggregation operators are more efficient and accurate in solving group decision-making problems compared with existing Dombi aggregation operators.
Ashraf et al. [5] suggested the spherical fuzzy Dombi aggregated operations to aggregate the spherical fuzzy numbers; in the present section, we give the comparison between proposed and novel spherical cubic Dombi aggregated operators. We get the spherical Dombi information from Ashraf et al. [5] as follows. e best alternative is A 2 . Now, to utilize spherical cubic fuzzy Dombi weighted averaging operator to chose the best alternative is as follows. e best choice is A 4 . Results are similar to Ashraf et al. [5]. e methodology proposed by Ashraf et al. [5] dealt with spherical fuzzy set, and we extend it for spherical cubic fuzzy set to get more accurate results. e methodology presented for this article is more comprehensive to tackle the vagueness in decision-making problems. Consequently, spherical cubic Dombi aggregated operations are more reliable and effective to comprehend the decision issues when contrasted to existed Dombi aggregated operations. e obtained results by using the concept of spherical cubic fuzzy Dombi aggregated operations give nearest results of ranking of spherical Dombi aggregated operations, and more suitable and accurate results of decision-making problems are given in Figure 2.
Mathematical Problems in Engineering

Conclusion
In this paper, the concept of spherical cubic fuzzy set is introduced that is the generalization of the spherical fuzzy set. Further, some spherical cubic fuzzy operational laws are established. Moreover, score and accuracy functions are defined for the comparison of spherical cubic fuzzy numbers. In addition, the idea for Dombi aggregated operations of spherical cubic fuzzy set is presented. Furthermore, the fundamental characteristics of spherical cubic fuzzy Dombi aggregated operations are presented. For the aggregation of spherical cubic fuzzy sets, we proposed SCFDWA, SCFDOWA, SCFDHWA, SCFDWG, SCFDOWG, and SCFDHWG under the spherical cubic fuzzy information. Additionally, some properties like idempotency, boundary, and monotonicity are discussed, and a relation between these established operators is shown. Likewise, a multiattribute decision-making methodology to illustrate the efficiency of the proposed operators is suggested. In addition, we applied the developed aggregation operators to discuss the decision-making problems. A numerical illustration was proposed to demonstrate the efficiency of the suggested operators over alternate methods in decisionmaking problems. Finally, to determine the validity and efficiency of the novel approach, we carried out the comparative analysis among the existing and the proposed operators.
In future, we will be integrating other approaches with SCFSs like Einstein sum and product to develop the ideas of spherical cubic fuzzy Einstein weighted averaging (SCFEWA), spherical cubic fuzzy Einstein ordered weighted averaging (SCFEOWA), spherical cubic fuzzy Einstein hybrid weighted averaging (SCFEHWA), spherical cubic fuzzy Einstein weighted geometric (SCFEWG), spherical cubic   fuzzy Einstein ordered weighted geometric (SCFEOWG), spherical cubic fuzzy Einstein hybrid weighted geometric (SCFEHWG), and more generalized operators in multiattribute decision-making problems. We can extend these defined Dombi aggregation operators for Hamacher and Frank norms to deal with uncertainty in the data by using spherical cubic fuzzy information. We can extend this work for power aggregation operators, Dombi Bonferroni mean operators, and their application to multiattribute group decision-making. e application of our proposed model in the future may be used in decision-making theory, risk analysis, and other domains in uncertain environments.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.