Bipolar Complex Fuzzy Hamacher Aggregation Operators and Their Applications in Multi-Attribute Decision Making

On the basis of Hamacher operations, in this manuscript, we interpret bipolar complex fuzzy Hamacher weighted average (BCFHWA) operator, bipolar complex fuzzy Hamacher ordered weighted average (BCFHOWA) operator, bipolar complex fuzzy Hamacher hybrid average (BCFHHA) operator, bipolar complex fuzzy Hamacher weighted geometric (BCFHWG) operator, bipolar complex fuzzy Hamacher ordered weighted geometric (BCFHOWG) operator, and bipolar complex fuzzy Hamacher hybrid geometric (BCFHHG) operator. We present the features and particular cases of the above-mentioned operators. Subsequently, we use these operators for methods that can resolve bipolar complex fuzzy multiple attribute decision making (MADM) issues. We provide a numerical example to authenticate the interpreted methods. In the end, we compare our approach with existing methods in order to show its effectiveness and practicality.


Introduction
In classical set theory (CST), we have only two possibilities, yes or no, i.e., an item either belongs to a set or does not. This means that the characteristic function of an item can give values 0 or 1. The CST fails in many situations such as age, intelligence, and height. To overcome this issue, Zadeh [1] presented the idea of the fuzzy set (FS). In FS theory, the membership degree (MD) belongs to the closed interval [0, 1] instead of {0, 1}. Mardani et al. [2] presented decision-making (DM) methods based on fuzzy aggregation operators. Merigó and Casanovas [3] described fuzzy generalized hybrid aggregation operators and their application in fuzzy DM. FS theory only considers MD, but in various circumstances, we need a non-membership degree (NMD). To handle such problems, Atanassov [4] provided the notion of intuitionistic FS (IFS) denoted by an MD and NMD with the condition that the sum of MD and NMD belongs to the closed interval [0, 1]. Xu [5] introduced intuitionistic fuzzy (IF) aggregation operators. The generalized IF aggregation operators are based on confidence levels for group DM given by Rahman et al. [6]. Verma and Merigó [7] presented multiple-attribute group DM (MAGDM) based on two-dimensional linguistic IF aggregation operators. Huang [8] defined the IF Hamacher aggregation operators and their application to MADM. Garg [9] presented IF Hamacher aggregation operators' entropy weight and their applications to multi-criteria DM (MCDM) issues.

Theorem 2.
The BCFHWA operator gives a BCFN and BCFHWA (J 1 , J 2 , J 3 , . . . , J n ) = The proof of this theorem is presented in Appendix A.
It can be easily demonstrated that the BCFHWA operator satisfies the following three properties.
Idempotency is an extremely helpful property in various circumstances, as it implies that an operation can be rehashed or revised as frequently as vital without initiating accidental impacts. In the following theorem, we invent the idempotency for the BCFHWA operator.
The proof of this theorem is presented in Appendix A.
The invented BCFHWA operator satisfies the boundedness property, which is interpreted as below.
Monotonicity is a significant trademark in numerous applications. The term comes from monotonic mathematical operations, also called non-decreasing function. In the following theorem, we invent monotonicity for the BCFHWA operator.

1.
When we take α = 1, then the BCFHWA operator transforms into the bipolar complex fuzzy weighted average (BCFWA) operator When we take α = 2 , then the BCFHWA operator transforms into the bipolar complex fuzzy Einstein weighted average (BCFEWA) operator In the following Definition 12, we invent the BCFHOWA operator.

Theorem 6.
The BCFHOWA operator gives a BCFN and Proof. The proof is similar to that of Theorem 2.
One can easily prove that the BCFHOWA operator satisfies the following three properties. Idempotency is an extremely helpful property in various circumstances, since it implies that an operation can be rehashed or revised as frequently as vital without initiating accidental impacts. In the following theorem, we invent the idempotency for the BCFHOWA operator.
The invented BCFHOWA operator satisfies the boundedness property, which is interpreted as below.
Monotonicity is a significant trademark in numerous applications. The term comes from monotonic mathematical operations, also called the non-decreasing function. In the following theorem, we invent monotonicity for the BCFHWA operator.
Particular Cases 2. We examine two particular cases of the BCFHOWA operator as follows: 1.
When we take α = 1 , then the BCFHOWA operator transforms into the bipolar complex fuzzy ordered weighted average (BCFOWA) operator

2.
When we take α = 2 , then the BCFHOWA operator transforms into the bipolar complex fuzzy Einstein ordered weighted average (BCFEOWA) operator In the following Definition 13, we invent the BCFHHA operator.
Proof. It is similar to the proof of Theorem 2.

Particular Cases 3.
We examine two particular cases of the BCFHH A operator as follows: 1.
When we take α = 1 , then the BCFHHA operator transforms into the bipolar complex fuzzy hybrid average (BCFHA) operator When we take α = 2 , then the BCFHHA operator transforms into the bipolar complex fuzzy Einstein hybrid average (BCFEHA) operator

Bipolar Complex Fuzzy Hamacher Geometric Aggregation Operators
In this subsection, we interpret the bipolar complex fuzzy Hamacher weighted geometric (BCFHWG) operator and bipolar complex fuzzy Hamacher ordered weighted geometric (BCFHOWG) operator.
In the following Definition 14, we invent the BCFHWG operator.

Theorem 11. The BCFHWG operator gives a BCFN and
The proof of this theorem is presented in Appendix A.
One can easily prove that the BCFHWG operator satisfies the following three properties. Idempotency is an extremely helpful property in various circumstances, as it implies that an operation can be rehashed or revised as frequently as vital without initi-ating accidental impacts. In the following theorem, we invent the idempotency for the BCFHWA operator.
The proof is presented in Appendix A.
The invented BCFHWG operator satisfies the boundedness property, which is interpreted as below.
Monotonicity is a significant trademark in numerous applications. The term comes from monotonic mathematical operations, also called the non-decreasing function. In the following theorem, we invent monotonicity for the BCFHWA operator.

1.
When we take α = 1 , then the BCFHWG operator transforms into the bipolar complex fuzzy weighted geometric (BCFWG) operator When we take α = 2 , then the BCFHWG operator transforms into the bipolar complex fuzzy Einstein weighted geometric (BCFEWG) operator In the following Definition 15, we invent the BCFHOWG operator.

Theorem 15.
The BCFHOWG operator gives a BCFN and Proof. The proof is similar to that of Theorem 11.
One can easily prove that the BCFHOWG operator satisfies the following three properties. Idempotency is an extremely helpful property in various circumstances, as it implies that an operation can be rehashed or revised as often as necessary without initiating accidental impacts. In the following theorem, we invent the idempotency for the BCFHOWG operator.
The invented BCFHOWG operator satisfies the boundedness property, which is interpreted as below.
Monotonicity is a significant trademark in numerous applications. The term comes from monotonic mathematical operations, otherwise called the non-decreasing function. In the following theorem, we invent monotonicity for the BCFHWA operator.
Particular Cases 5. Now we interpret two particular cases of the BCFHOWG operator as follows: 1.
When we take α = 1, then the BCFHOWG operator transforms into the bipolar complex fuzzy ordered weighted geometric (BCFOWG) operator fuzzy Einstein ordered weighted geometric (BCFEOWG) operator In the following Definition 16, we invent the BCFHHG operator.
Proof. The proof is similar to the proof of Theorem 11. Particular Cases 6. Now we interpret two particular cases of the BCFHHG operator as follows:

1.
When we take α = 1, then the BCFHHG operator transforms into the bipolar complex fuzzy geometric (BCFG) operator

2.
When we take α = 2, then the BCFHHG operator transforms into the bipolar complex fuzzy Einstein hybrid geometric (BCFEHG) operator

An Approach to MADM with Bipolar Complex Fuzzy Information
In this part of the article, we show a MADM technique based on the interpreted bipolar complex Hamacher aggregation operators in Section 4 under the BCFS environment.
is the bipolar complex fuzzy decision matrix, where F + qp denotes the PD for which the alternative A p satisfies attribute B p provided by the decision-maker, and F − qp denotes the ND for which the alternative A p does not satisfy attribute B p provided by the decision-maker.

Algorithm
We interpret the algorithm to solve MADM issues in the environment of BCFSs by utilizing BCFHWA and BCFHWG operators as follows: Step I: By employing the BCFHWA operator to the decision information provided in the matrix X, derive all the values x q (q = 1, 2, 3, . . . , m) of the alternative A q .
Step III. Rank all the alternatives A q (q = 1, 2, 3, . . . , m) in terms of S B x q (q = 1, 2, 3, . . . , m). If the two scores functions S B x q and S B x p have same values, then we use the accuracy function H B x q and H B x p to rank the alternatives A q and A p .
Step IV. Choose the best alternative.
Step V. End.

Numerical Example
In this segment, we use a practical MADM example to explain the application of interpreted operators. Consider A to be the universal set in this example and each A p ∈ A to be given in the setting of BCFN, i.e., Recognizing, assessing, and gauging the applicants against job necessities can be accepted as a capacity of the employees' selection. Employees' capabilities such as competence, knowledge, and experience perform an essential part of an organization's achievement. It is hard to assess the consequences of the incorrect recruiting decisions of an individual. One of the fundamental goals of an enterprise is to find effective methods of evaluating and positioning a number of employees who have been assessed for various capabilities. In the literature, the selection of an appropriate individual from among various applicants is an important aspect. When the policies of employee selection are accepted by the enterprise, they imply an improvement of the enterprise's performance. Enterprises invest energy in recruiting people. The employers' costs are increased by an excess of time and costs spent on engaging, training, and firing inefficient and frustrated employees. These costs increase if the employers take a longer time to realize the employee's deficiencies. Our proposed algorithm in Section 5 is the appropriate method for the selection of employees that covers all the enterprises' requirements.
Suppose an enterprise is recruiting an employee for the post of an assistant director. Firstly, the enterprise forms a selection board formed by a CEO and three other senior representatives. There are four applicants, A q (q = 1, 2, 3, 4), who applied for that post. The selection board choose four attributes to assess the applicants i.e., B 1 = qualification, B 2 = experience, B 3 = organizational skills, and B 4 = professionalism. The four applicants A q (q = 1, 2, 3, 4) are assessed by using BCFN by the decision-makers, factoring in the above-mentioned four attributes, whose weighting vector is = (0.2, 0.25, 0.15, 0.4). The decision matrix X = x qp 4×4 is given in Table 1.  For the selection of the employees, we use the BCFHWA (BCFHWG) operator with an MADM approach and bipolar complex fuzzy data, which is given below: Step I: For α = 3, employ the BCFHWA operator to determine all preference values x q of the applicants A q (q = 1, 2, 3, 4). Step II: Determine the score values S B x q (q = 1, 2, 3, 4) of the overall BCFNs x q (q = 1, 2, 3, 4).
Step III. Rank all the applicants A q (q = 1, 2, 3, 4) following score values S B x q (q = 1, 2, 3, 4) of the overall BCFNs: Step IV. A 3 is selected as the best applicant.
Step V. End. If we apply the BCFHWG operator instead of BCFHWA, then the above problem will solve similarly: Step I: For α = 3, employ the BCFHWG operator to determine all preference values x q of the applicants A q (q = 1, 2, 3, 4). Step II: Determine the score values S B x q (q = 1, 2, 3, 4) of the overall BCFNs x q (q = 1, 2, 3, 4).
Step III. Rank all the applicants A q (q = 1, 2, 3, 4) following score values S B x q (q = 1, 2, 3, 4) of the overall BCFNs: Step IV. A 3 is selected as the best applicant.
Step V. End. We observe that all rating values of alternatives are different when we use two different operators, but their ranking order is similar. The best alternative (applicant) is U 3 for both BCFHWA and BCFHWG operators.

Comparative Analysis
This section develops a comparative analysis of the aggregation operators so as to demonstrate the authenticity and dominance of our proposed methods and operators.
Consider the data given in Table 1. The data of Table 1 is two-dimensional (i.e., real part and imaginary part) along with PD and ND. We know that the work of Wei et al. [24], Jana et al. [25], and Jana et al. [26] can only operate with one-dimensional information with positive and negative aspects (i.e., positive degree and negative degree), but are incapable of accounting for the second dimension or imaginary part. From the above discussion, it is clear that the work of Wei et al. [24], Jana et al. [25], and Jana et al. [26] are unable to solve the MADM issues related to data in the environment of BCFSs. Moreover, the work of Huang [8] only can cope with one-dimensional information along with membership and non-membership grade where both membership and non-membership grade belongs to the [0, 1]. Huang [8] does not provide us with any information about the negative aspect. From this, we observe that the work of Huang [8] is also unable to solve the MADM aspects involving the data in the environment of BCFSs. Only the interpreted work can solve such type of MADM cases. This shows that our approach is superior to the existing methods. BCFS amplifies the existing methods: when the imaginary part equals zero in both PD and ND, it transforms into BFSs, and if the imaginary part equals zero in PD and the ND part is neglected, its converts into FSs. The score values and ranking results of the interpreted and existing methods are given in Table 2. Figure 1 provides a graphic of the score values of existing and interpreted methods. Table 2. Score values and ranking results of interpreted and existing work.

Conclusions
In this manuscript, we established various operations, the score function, and the accuracy function for BCFS. Furthermore, inspired by Hamacher operations, we interpreted BCFHWA operator, BCFHOWA operator, BCFHHA operator, BCFHWG operator, BCFHOWG operator, and BCFHHG operator. We described the features and the particular cases of the above operators such as BCFWA operator, BCFOWA operator, BCFHA operator, BCFWG operator, BCFOWG operator, and BCFHG operator by taking the parameter equal to 1. By taking the parameter equal to 2, we obtained BCFEWA operator, BCFEOWA operator, BCFEHA operator, BCFEWG operator, BCFEOWG operator, and BCFEHG operator. Subsequently, we used these operators to generate methods to resolve the bipolar complex fuzzy MADM issues. In order to authenticate the

Conclusions
In this manuscript, we established various operations, the score function, and the accuracy function for BCFS. Furthermore, inspired by Hamacher operations, we interpreted BCFHWA operator, BCFHOWA operator, BCFHHA operator, BCFHWG operator, BCFHOWG operator, and BCFHHG operator. We described the features and the particular cases of the above operators such as BCFWA operator, BCFOWA operator, BCFHA operator, BCFWG operator, BCFOWG operator, and BCFHG operator by taking the parameter equal to 1. By taking the parameter equal to 2, we obtained BCFEWA operator, BCFEOWA operator, BCFEHA operator, BCFEWG operator, BCFEOWG operator, and BCFEHG operator. Subsequently, we used these operators to generate methods to resolve the bipolar complex fuzzy MADM issues. In order to authenticate the interpreted methods, we provided a numerical example for a company that has recruited the best employee for the position of assistant director.
Finally, in order to show the effectiveness and practicality of our approach, we compared our results with the existing operators.
The Hamacher operators based on BCFS generalize Hamacher operators for FS, BFS, and CFS. By obtaining the unreal part equal to zero in both PD and ND, we found Hamacher operators for the data in the structure of BFS; by neglecting the ND, we acquired Hamacher operators for CFS; and by obtaining the unreal part zero in PD and neglecting the ND, we acquired Hamacher operators for FS. However, our proposed approach presents some limitations, since the invented operators cannot manage the information in the structure of bipolar complex intuitionistic FS, bipolar complex fuzzy soft set, etc.

Data Availability Statement:
The data utilized in this manuscript are hypothetical and artificial, and one can use these data before prior permission by simply citing this manuscript.

Conflicts of Interest:
Regarding the publication of this manuscript, the authors declare that they have no conflict of interest. Ethics Declaration Statement: The authors state that this is their original work, and it is neither submitted nor under consideration in any other journal simultaneously.

Appendix A
Proof of Theorem 2. We prove it through mathematical induction.
Proof of Theorem 11. We prove it through mathematical induction.