An Extension of Bonferroni Mean under Cubic Pythagorean Fuzzy Environment and Its Applications in Selection-Based Problems
Cubic Pythagorean fuzzy (CPF) set (CPFS) is a hybrid set that can describe both interval-valued Pythagorean fuzzy (IVPF) sets (IVPFSs) and Pythagorean fuzzy (PF) sets (PFSs) simultaneously for addressing information ambiguities. Since an aggregation operator (AO) is a significant mathematical approach in decision-making (DM) problems, this article presents some novel Bonferroni mean (BM) and weighted Bonferroni mean averaging operators between CPF-numbers (CPFNs) for aggregating the different preferences of the decision-makers. Then, using the proposed AOs, we develop a DM approach under the CPF environment and demonstrated it with a numerical example. Furthermore, a comparative study between the proposed and existing methods has been performed to demonstrate the practicality and efficiency of the proposed DM approach.
Decision-making is a significant process in order to pick the best-suited alternative from among those available. In it, a number of scholars provided a variety of theories to make the best judgments. In the past, decisions were made based on crisp numbered data sets, but this resulted in insufficient outcomes that were less applicable to real-life operational scenarios. However, as time passes and the complications of the system increase, it becomes more difficult for the decision-maker to handle the inconsistencies in the data, and thus the traditional technique is unable to find the optimum alternative. Therefore, the scholars used fuzzy set (FS) theory , interval-valued fuzzy sets (IVFSs) , intuitionistic fuzzy sets (IFSs) , interval-valued intuitionistic fuzzy sets (IVIFSs) , PFS [5, 6], and IVPFS  to describe the information. Scholars have paid increasing attention to these ideas in recent decades and have efficiently implemented them in a variety of scenarios in the DM process. An aggregation operator, which generally takes the form of a mathematical formalism to accumulate all of the individual input data into a single one, is an important part of the DM process. For example, Xu and Yager  introduced certain geometric AOs to integrate various preferences of the decision-makers into intuitionistic fuzzy numbers (IFNs). Later, Wang and Liu  utilized Einstein norm operations to generalize these operators. For aggregating different intuitionistic fuzzy information, Garg  introduced generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein norm operations. Garg has presented a series of interactive AOs for IFNs in . Garg  implemented the IFS concept to PFS and proposed generalized averaging AOs. The symmetric Pythagorean fuzzy AOs were proposed by Ma and Xu . To solve DM problems, Garg  presented some improved interactive aggregation operators, while Wang and Liu  presented some hybrid weighted aggregation operators using Einstein norm operators. Apart from that, several other authors have given other approaches to solve DM problems, such as ranking functions  and AOs (see ).
As the above, AOs have been considered by many researchers during the DM process in which they have highlighted the contribution of each factor or its ordered position but cannot represent the interrelationships of the individual information. In our real-life situation, a relationship of different criteria such as importance, support, and impact on each other constantly plays a significant role throughout the aggregation process. To deal with it, Yager  developed the power average (PA) AOs to address this issue and implement it into DM analysis. Xu and Yager  and Yu  proposed the prioritized averaging and geometric AOs in an IFS environment. Further, Yager  presented the idea of the BM aggregation operators, which has the potential to represent the interrelationship between the input arguments. To alleviate the limitation of BM, Beliakov and James  introduced the generalized BM. To aggregate the intuitionistic fuzzy information, Xu and Yager  developed an intuitionistic fuzzy BM. These BM operators were generalized to the interval-valued IFSs environment by Xu and Chen . The generalized intuitionistic fuzzy BMs were presented by Xia et al. . The partitioned BM operators were described by Liu et al.  in an IFSs environment. Shi and He  discussed how to optimize BMs by applying them to different DM processes. In an intuitionistic fuzzy soft set environment, Garg and Arora  proposed the BM aggregation operator. The Pythagorean fuzzy Bonferroni mean (PFBM) is developed by Liang et al. , and several specific properties and cases are described. Wang and Li proposed a Pythagorean fuzzy interaction PFBM and weighted PFBM operators . Nie et al.  proposed a PF partitioned normalized weighted BM operator with Shapley fuzzy measure.
All of the existing research and their respective applications are mostly focused on the FS, interval-valued FS (IVFS), IFS, PFS, IVIFS, and IVPFS. Then, Jun et al.  developed several logic operations of the cubic sets and familiarized the theory of cubic set (CS) and their operational laws such as P-union, P-intersection, R-union, and R-intersection of CS and investigated several related properties. CS has been employed in a variety of real-world applications. They use highly interconnected distinctive to solve complex issues in engineering, economics, and the environment. Because of the many uncertainty models for such situations, it is not always simple to apply standard approaches to obtain good results. Therefore, Khalil and Hassan  introduced the class of cubic soft algebras and their basic characteristics. Shi and Ye  proposed Dombi Aggregation Operators of Neutrosophic Cubic Sets for Multiple Attribute Decision-Making. Ye et al.  presented Multi-fuzzy Cubic Sets and Their Correlation Coefficients for multi-criteria Group Decision-Making. Garg et al.  proposed Correlation Measures for Cubic m-Polar Fuzzy Sets with Applications. Khan et al. [37, 38] presented some cubic AOs under this set, while Mahmood et al.  presented the concepts of cubic hesitant fuzzy sets and their AOs. The above concepts only provide information in the form of membership intervals and ignore the nonmembership section of the data entities, which also perform an important role in evaluating the alternative in the DM process. In the real world, expressing the value of a membership degree by an exact value in a fuzzy set is typically challenging. In such situations, an interval value and an exact value, rather than unique interval/exact values, may be convenient to communicate uncertainty and ambiguity in the real world. Hence, the hybrid form of an interval and an exact number may be a very useful term for a person to express certainty and uncertainty as a result of his or her uncertain assessment in multifaceted DM problems. Kaur and Garg [40, 41] introduced the concept of the cubic intuitionistic fuzzy set CIFS, which is characterized by two portions at the same time, one of which reflects membership degrees by an IVIF value and the other by an intuitionistic fuzzy value. Each CIFS component is represented as that satisfies the conditions and 1. However, in some actual cases, the sum of membership and non-membership grades may be greater than 1, but their square sum is less than or equal to 1. Therefore, Abbas et al.  presented the concept of the CPFS, which is the generalization of CIFS. At the same time, CPFS has two phases, one of which reflects the degree of membership by an IVPFS and the other by a PFS that satisfies the condition and . Therefore, a CPFS is a hybrid set that includes both an IVPFS and a PFS. Obviously, the CIFS has the advantage of being able to contain a lot more information in order to express both the IVPFN and the PFN at the same time. Hence, CPFS has a high level of efficiency and importance when used to evaluate alternatives during the DM process because the general DM process may use IVPFS or PFS data, which may loss of some important evaluation information. There is currently no research on AOs that reflects the interrelationship between the multiple criteria of a DM process having CPF information.
According to the current communication, inspired by the BM notion and by utilizing the CPFS advantages, we suggest some new AOs called the CPF Bonferroni mean (CPFBM) and weighted cubic Pythagorean fuzzy Bonferroni mean (WCPFBM) to aggregate the preferences of decision-makers. We have also investigated various desirable properties of these operators in detail. The main advantage of the proposed AOs is that the interrelationships between aggregated values are taken into account. We also examine the characteristics of the proposed work and design some specific scenarios. The proposed AOs have been deduced from several previous studies, demonstrating that the proposed AOs are more flexible than the others. Finally, a DM method for rating the various alternatives based on the proposed AOs has been presented. Finally, a DM method for rating the various alternatives based on the proposed AOs has been presented. The list of abbreviations used in this article is provided in Table 1.
The rest of the article is organized as follows. The basic concepts are briefly discussed in Section 2. Section 3 presents AOs called CPF Bonferroni mean (CPFBM) and weighted CPF Bonferroni mean (WCPFBM), as well as their applications. To address multi-criteria DM (MCDM) problems, a DM approach based on proposed operators has been developed in Section 4. In Section 5, a numerical model is provided to illustrate the proposed approach and demonstrate its practicality and applicability. Section 6 presents the conclusion including closing remarks.
Some main theories associated with PFSs, IVPFSs, CSs, CIFSs and CPFSs are briefly addressed in this section.
2.1. Pythagorean Fuzzy Set
Definition 2 (see ). Let , and be three PFNs and then(i);(ii);(iii);(iv).
2.2. Interval-Valued Pythagorean Fuzzy Set
Definition 3 (see ). Let be a fixed set. An IVPFS over is defined aswhere , such that . For the sake of simplicity, we denote this pair as and called as IVPF number (IVPFN).
Definition 4 (see ). Let , and be three IVPFNs, then(i);(ii);(iii);(iv).
2.3. Cubic Set
Definition 5 (see ). Let be a universal set. A CS over is defined aswhere is IVF set in and is a FS. A CS is said to be an internal cubic set if and a CS an external cubic set if . A CS is simply denoted by .
Definition 6. (see ). Let and be CSs in . Then,(i)Equality. If and then ;(ii)P-Order. If and then ;(iii)R-Order. If and then .
Definition 7 (see ). Let be a collection CSs where , then(i)P-Union.;(ii)P-Intersection. ;(iii)R-Union. ;(iv)R-Intersection. .
2.4. Cubic Intuitionistic Fuzzy Set
Definition 8 (see [40, 41]). Let be a non-empty set. A CIFS over is defined as follows:where is IVIFS while represents IFS such that , and . Also, and . In order to keep it simple, the pair , where and and called as CIF number (CIFN).
Definition 9 (see ). For a family of CIFS , then(i)P-Union. .(ii)P-Intersection. .(iii)R-Union. .(iv)R-Intersection. .
Definition 10 (see ). Let and be two CIFSs in . Then,(i)Equality. , if and only if , , and .(ii)P-Order. if , , and .(iii)R-Order. if , , and .
Definition 11 (see ). Let , be the collections of CIFNs, and be a real number, then
2.5. Cubic Pythagorean Fuzzy Set
Definition 12. (see ). Let be a non-empty set. A CPFS over is defined as follows:where represents IVPFS while represents PFS for all such that , and . Also, and . In order to keep it simple, the pair , where and and called as CPF number (CPFN).
Definition 13 (see ). Let be a CPFN then score and accuracy function is defined as follows:where .where .
Definition 14 (see ). Let , ) be the collections of CPFNs, and be a real number, then
2.6. Bonferroni Mean Operators
Definition 16 (see ). Let be the collection of PFNs. For any , the PF Bonferroni mean (PFBM) is defined as
Definition 17 (see ). Let be the collection of PFNs and . is the weight vector of such that and , then weighted PF Bonferroni mean (WPFBM) is defined as
3. CPF-Bonferroni Mean Aggregation Operators
In this section, a series of aggregation operators, namely, CPFBM and WCPFBM are presented.
3.1. CPF Bonferroni Mean Operators
Definition 18. A CPF Bonferroni mean (CPFBM) operator is a mapping defined on a set of CPFNs and is given bywhere are CPFNs, and and are positive real numbers.
Example 1. Let , and be two CPFNs. Then, use CPFBM to aggregate these three CPFNs. The steps are outlined below. (Supposes and ).
By using (11), we getThus,
Proposition 1. Let ) be the collections of CPFNs and . For any , and , we have
Proof. By Definition 14, we haveandThen,
Proposition 2. Let ) be the collections of CPFNs and . For any and , we have
Proposition 3. Let ) be the collection of CPFNs and . We can calculate the following by using the value of k.where .
Proof. Mathematical induction on .
Step 1. By adopting Proposition 1, we obtain the following result when k = 2:Thus, we have
Step 2. We can obtain the following result. Eq. (14) holds for ,
Step 3. When , then we have ,,,