#### Abstract

This article takes advantage of advancements in two different fields in order to produce a novel decision-making framework. First, we contribute to the theory of aggregation operators, which are mappings that combine large amounts of data into more advantageous forms. They are extensively used in different settings from classical to fuzzy set theory alike. Secondly, we expand the literature on complex Pythagorean fuzzy model, which has an edge over other models due to its ability to handle uncertain data of periodic nature. We propose some aggregation operators for complex Pythagorean fuzzy numbers that depend on the Hamacher t-norm and t-conorm, namely, the complex Pythagorean fuzzy Hamacher weighted average operator, the complex Pythagorean fuzzy Hamacher ordered weighted average operator, and the complex Pythagorean fuzzy Hamacher hybrid average operator. We explore some properties of these operators inclusive of idempotency, monotonicity, and boundedness. Then, the operators are applied to multicriteria decision-making problems under the complex Pythagorean fuzzy environment. Furthermore, we present an algorithm along with a flow chart, and we demonstrate their applicability with the assistance of two numerical examples (selection of most favorable country for immigrants and selection of the best programming language). We investigate the adequacy of this algorithm by conducting a comparative study with the case of complex intuitionistic fuzzy aggregation operators.

#### 1. Introduction

Aggregation operators (AOs) are the tools to convert an -tuple crisp information into a single beneficial form. The AOs find extensive applications in decision-making. Multicriteria decision-making (MCDM) refers to a procedure for choosing the most worthwhile alternative with respect to some crucial factors. In real world MCDM problems, the human decisions are usually unclear and inexact which lead us to take help from the fuzzy set theory to handle the uncertain data.

The crisp set theory was developed on the basis of crisp logic in which logical connectors conjunction and disjunction are employed to define operations. On the other hand, triangular norm (t-norm) and triangular conorm (t-conorm) are used to define the operations in fuzzy set theory as a generalization of Boolean logical connectives. The concept of t-norm and t-conorm was originally initiated by Menger [1] in the context of probabilistic metric spaces in 1942. Later, Schweizer and Sklar [2] worked for the development of t-norm and t-conorms. The notion of t-norm and t-conorm gained the attention of researchers. Many researchers contributed in this field and proposed the multipurpose t-norms, including drastic product, algebraic product, Lukasiewicz t-norm, Yager t-norm, Schweizer and Sklar t-norm and Frank t-norm, and the corresponding t-conorms. Later on, Zimmermann and Zysno [3] noticed that t-norm and t-conorm exhibit extreme behaviors. This lead to the development of compensation and average operators [4] that provide the results inside the interval. In 1978, Hamacher [5] introduced a parameterized t-norm and its dual t-conorm as a generalization of Einstein product and Einstein sum, respectively.

Many AOs, including arithmetic mean, median, minimum, maximum, weighted maximum, weighted minimum, geometric mean, harmonic mean, and quasiarithmetic mean, are used to aggregate the crisp data in classical set theory. Yager [6] proposed the notion of ordered weighted average operators which was further extended by many quality researchers in different models. The fundamental properties of AOs are monotonicity, continuity, associativity, symmetry, bisymmetry, idempotency, and invariance.

In 1965, Zadeh [7] proposed the fuzzy set (FS) as an extension of classical set and introduced several operations for FS. Zadeh’s idea made a way to solve those decision-making and MCDM problems, comprising imprecise and ambiguous data that cannot be solved by the notion of crisp set. Song et al. [8] discussed fuzzy operators and studied their properties. Merigó et al. [9] utilized the generalized AOs for decision-making in fuzzy environment. Atanassov [10] associated the nonmembership degree with the membership degree along with the condition that the sum of membership and nonmembership degree should not exceed 1 and named the new structure as intuitionistic fuzzy set (IFS). Xu [11] extended weighted averaging operators in the intuitionistic fuzzy (IF) environment. Wei [12] and Xu and Wang [13] presented induced AOs and induced generalized AOs for IF as well as interval-valued IF information and illustrate their application in group decision-making. Zhao et al. [14] introduced IF generalized weighted average operators. Wei and Zhao [15] proposed a decision-making approach on the basis of induced IF correlated averaging operators and induced IF correlated geometric operators. Huang [16] and Yang et al. [17] examined IF Hamacher AOs and dynamic intuitionistic normal fuzzy AOs, respectively. Later, Yager [18, 19] presented the notion of Pythagorean fuzzy set (PFS) which has more generalized structure and accommodates more uncertainty than IFS. Zhang [20] investigated some AOs for Pythagorean fuzzy (PF) model and presented a decision-making approach. Akram et al. [21] and Jana et al. [22] worked on the Pythagorean Dombi fuzzy AOs and utilized these operators for decision-making in textile industry and enterprise resource planning system, respectively. Wu and Wei [23] presented the PF Hamacher AOs with their properties and apply these operators to opt the most talented enterpriser. Wei et al. [24, 25] considered the Pythagorean hesitant fuzzy AOs and Pythagorean fuzzy power AOs. Shahzadi et al. [26] presented a decision-making approach on the basis of PF Yager operators. Aydin et al. [27] established harmonic AOs for trapezoidal PF numbers.

To overcome the limitations of FS, IFS, and PFS, Ramot introduced the complex fuzzy set (CFS) [28] as well as complex fuzzy logic [29]. The membership function of a CFS, restricted to complex unit circle, consists of two real valued terms, i.e., amplitude term and phase term which make it distinct and superior to all existing models. The novelty of CFS is due to the phase term associated with membership which enables it to handle periodic data. Bi et al. developed the complex fuzzy arithmetic AOs [30] and complex fuzzy geometric AOs [31]. Alkouri et al. [32, 33] extended the CFS to a superior model, namely, complex intuitionistic fuzzy set (CIFS) and introduced some basic operations along with complex intuitionistic fuzzy (CIF) relations. The nonmembership function in the CIFS distinguishes it from CFS and provides more accurate results than CFS. Garg and Rani [34] presented decision-making approaches on the basis of CIF weighted average operator and CIF weighted geometric operator. Further, Garg et al. [35, 36] put forward some new complex intuitionistic fuzzy aggregation operators (CIFAOs) on the basis of Archimedean t-norm and t-conorm. Rani and Garg [37] introduced CIF power AOs and explained their application in practical decision-making. Akram et al. [38] presented the CIF Hamacher aggregation operators with impactful applications in the decision-making scenarios.

Ullah et al. [39] proposed the complex Pythagorean fuzzy set (CPFS), having relatively relaxed conditions for amplitude and phase terms, to overcome the deficiencies of CIFS. Phase term of CPFS is of vital importance and makes it dominant to all other models due to its tendency to tackle two-dimensional vague information efficiently. Akram et al. [40–42] worked for the development of aggregation operators on the basis of Yager and Dombi operations for complex Pythagorean fuzzy (CPF) model. Tan et al. [43], Wei et al. [44], and Waseem et al. [45] contributed to literature by proposing the decision-making models based on hesitant fuzzy Hamacher AOs, bipolar fuzzy Hamacher AOs, and m-polar fuzzy Hamacher AOs, respectively. Recently, Akram et al. [46] proposed an innovative extension of the existing models, namely, complex spherical fuzzy sets.

The motivation of this article can be described as follows:(i)The models of fuzzy set theory with complex membership and nonmembership have an edge over the other existing models as they are proficient enough to deal with two-dimensional obscure data and information due to phase term. The traditional decision-making approaches of fuzzy set theory cannot be applied to periodic data because this may cause loss of important information.(ii)The constraints on the amplitude and phase terms of CPFS allow it to capture more imprecision and vagueness. Therefore, CPFS works effectively when the sum of amplitude or phase term does not fall under the conditions of CIFS.(iii)Hamacher AOs represent a generalized family of operators which provide more accurate results in decision-making. The purpose of this study is to define a MCDM approach for CPF model using the foundations of Hamacher operations to exploit the strengths of both models for more precise decisions.(iv)The complex Pythagorean fuzzy Hamacher arithmetic aggregation operators (CPFHAAOs) also overcome the deficiencies of existing operators and MCDM approaches for two-dimensional information, including complex fuzzy model and CIF model.

In this research article, we propose complex Pythagorean fuzzy Hamacher weighted average (CPFHWA) operator, complex Pythagorean fuzzy Hamacher ordered weighted average (CPFHOWA) operator, and complex Pythagorean fuzzy Hamacher hybrid average (CPFHHA) operator to aggregate the CPF data for decision-making purpose. We also present a MCDM strategy based on these operators under CPF environment. We illustrate the implementation of the proposed algorithm by two explanatory numerical examples: one for the selection of best country for immigrants and other for the selection of best programming language. We verify the results of the proposed strategy by conducting a comparative study with existing MCDM techniques using complex intuitionistic fuzzy weighted average (CIFWA) operator, complex intuitionistic fuzzy Einstein weighted average (CIFEWA) operator and complex intuitionistic fuzzy Hamacher weighted average (CIFHWA) operator, complex intuitionistic fuzzy weighted geometric (CIFWG) operator, complex intuitionistic fuzzy Einstein weighted geometric (CIFEWG) operator, and complex intuitionistic fuzzy Hamacher weighted geometric (CIFHWG) operator. The main contributions of this research article can be summarized as follows:(i)The considerable contribution of this study is to make capital of the parametric and flexible framework of Hamacher operations under the competitive and innovative model of CPFSs to accumulate the obscure periodic data for decision-making.(ii)The novelty of the proposed operators is due to their flexible structure and authentic outputs as they compile the CPF data deploying the brilliance of Hamacher operations, whereas the existing operators, developed on the basis of Hamacher norms, are not applicable for CPF data due to nonavailability or strict condition of phase terms.(iii)The main goal of this article is to employ the competency and potential of the proposed operators for the development of a MCDM approach to enhance the accuracy of decision-making results for two-dimensional information.(iv)The presented methodology is supported with the help of two rational numerical examples, one for the selection of best country for immigrants and other for the selection of best programming language.(v)The calibre of the proposed methodology is demonstrated via comparative study to prove the dominance of the proposed operators over existing operators.

The rest of this research article is organized as follows: Section 2 comprises the basic concepts and Hamacher operations of CPFNs which have been employed to define the CPFHAAOs and their properties. Section 3 presents a mathematical approach to address the MCDM problem under CPF environment along with two illustrative numerical examples. Section 4 describes the comparison of the proposed operators with well-known complex intuitionistic fuzzy AOs. Section 5 highlights the eminence and excellence of the proposed operators. Section 6 summarizes the article with concluding remarks and future directions.

#### 2. Complex Pythagorean Fuzzy Hamacher Aggregation Operators

*Definition 1. *(see [39]). Let be a universe of discourse. A complex Pythagorean fuzzy set over the universe is an object of the following form:where the membership function and nonmembership function are defined by the mapping . For every , the membership grade is of the form and nonmembership grade has the form , where , , , , , , and . The pair of membership and nonmembership is called a complex Pythagorean fuzzy number (CPFN).

*Definition 2. *(see [42]). The score function of a CPFN is defined as follows:where .

*Definition 3. *(see [42]). The accuracy function of a CPFN is defined aswhere .

*Definition 4. *(see [42]). For the comparison of any two CPFNs and ,(1)If , then ( is superior to );(2)If , then(i)If , then ( is superior to );(ii)If , then ( is equivalent to ).

*Definition 5. *(see [47]). Let , , and be three CPFNs. The operations corresponding to these three CPFNs can be defined as follows:(1);(2);(3);(4).

##### 2.1. Hamacher t-Norm and Hamacher t-Conorm

The notions of t-norm and t-conorm are the elementary tools to define operations in fuzzy set theory. Later, Hamacher [5] presented the more general t-norm and t-conorm, namely, Hamacher product and Hamacher sum, respectively. For all , Hamacher operations including product and sum are defined as follows:

Special cases:(i)For , Hamacher t-norm and t-conorm give algebraic t-norm and t-conorm, respectively.(ii)For , Hamacher product and Hamacher sum reduce to Einstein product and Einstein sum, respectively.

##### 2.2. Hamacher Operations of Complex Pythagorean Fuzzy Numbers

For three CPFNs , and , Hamacher operations are defined as

##### 2.3. Complex Pythagorean Fuzzy Hamacher Arithmetic Aggregation Operators

In this subsection, we present a few CPFHAAOs based on the Hamacher operations of CPFNs.

*Definition 6. *For any collection () of CPFNs and the weight vector , the complex Pythagorean fuzzy Hamacher weighted average (CPFHWA) operator is defined as follows:where , representing the weight of , belongs to and satisfies the condition .

Theorem 1. *For any collection () of CPFNs and the weight vector , the accumulated value by deploying the CPFHWA operator is also a CPFN which is given aswhere . , representing the weight of , belongs to and satisfies the condition .*

*Proof. *We prove the theorem with the help of mathematical induction. *Case 1.* When , the CPFHWA operator given in equation (9) gives Thus, aggregated value is the same CPFN and equation (9) holds for . *Case 2.* We now assume that equation (9) is true for , where *r* denotes a natural number. Then, equation (9) becomesNow, for ,Thus, equation (9) holds for . Hence, it is proved that equation (9) is true for all *n* (natural numbers).

Special cases:(i)For , CPFHWA operator becomes which represents the complex Pythagorean fuzzy weighted averaging (CPFWA) operator.(ii)For , CPFHWA operator becomes which represents the complex Pythagorean fuzzy Einstein weighted averaging (CPFEWA) operator.(iii)When , CPFHWA operator becomeswhich represents the Pythagorean fuzzy Hamacher weighted averaging (PFHWA) operator.

*Example 1. *Let , , and be three CPFNs. Let be the associated weight vector. Then, for ,

Theorem 2. *(idempotency property). Let be a family of CPFNs. If , then*

*Proof. *Since , then equation (9) becomesHence, it is proved that .

Theorem 3. *(boundedness property). Let () be a family of CPFNs. Letthen*

Theorem 4. *(monotonicity property). Let and () be two families of CPFNs. If , , and , then*

*Definition 7. *For any collection () of CPFNs and the weight vector , the complex Pythagorean fuzzy Hamacher ordered weighted average (CPFHOWA) operator is defined as follows:where is a permutation of , such that . , representing the weight of , belongs to and satisfies the condition .

Theorem 5. *For any collection () of CPFNs and the weight vector , the accumulated value by deploying the CPFHOWA operator is also a CPFN which is given aswhere and is a permutation of , such that . , representing the weight of , belongs to and satisfies the condition .*

*Example 2. *Let , , and be three CPFNs. Let be the associated weight vector. The scores of these CPFNs can be evaluated by equation (2).Since, , therefore,Then, for ,

Special cases:(i)For , CPFHOWA operator becomes which represents the complex Pythagorean fuzzy ordered weighted averaging (CPFOWA) operator.(ii)For , CPFHOWA operator becomes which represents the complex Pythagorean fuzzy Einstein ordered weighted averaging (CPFEOWA) operator.(iii)When , CPFHOWA operator becomeswhich represents the Pythagorean fuzzy Hamacher ordered weighted averaging (PFHOWA) operator.

Theorem 6. *(idempotency property). Let be a family of CPFNs. If , then*