Hamacher Operations for Complex Cubic q-Rung Orthopair Fuzzy Sets and Their Application to Multiple-Attribute Group Decision Making

Abstract

In this paper, based on the advantages of q-rung orthopair fuzzy sets (q-ROFSs), complex fuzzy sets (CFSs) and cubic sets (CSs), the concept of complex cubic q-rung orthopair fuzzy sets (CCuq-ROFSs) is introduced and their operation rules and properties are discussed. The objective of this paper was to develop some novel Maclaurin symmetric mean (MSM) operators for any complex cubic q-rung orthopair fuzzy numbers (CCuq-ROFNs) using Hamacher t-norm and t-conorm inspired arithmetic operations. The advantage of employing Hamacher t-norm and t-conorm based arithmetic operations with the MSM operator lies in their ability to take into account not only the interrelationships among multiple attributes but also to provide flexibility in the aggregation process due to the involvement of additional parameters. Also, the prominent characteristic of the MSM is that it can capture the interrelationship among the multi-input arguments and can provide more flexible and robust information fusion. Thus, based on the CCuq-ROF environment, we develop some new Hamacher operations for CCuq-ROFSs, such as the complex cubic q-rung orthopair fuzzy Hamacher average (CCuq-ROFHA) operator, the weighted complex cubic q-rung orthopair fuzzy Hamacher average (WCCuq-ROFHA) operator, the complex cubic q-rung orthopair fuzzy Hamacher Maclaurin symmetric mean (CCuq-ROFHMSM) operator and the weighted complex cubic q-rung orthopair fuzzy Hamacher Maclaurin symmetric mean (WCCuq-ROFHMSM) operator. Further, we develop a novel multi-attribute group decision-making (MAGDM) approach based on the proposed operators in a complex cubic q-rung orthopair fuzzy environment. Finally, a numerical example is provided to demonstrate the effectiveness and superiority of the proposed method through a detailed comparison with existing methods.

1. Introduction

Multi-attribute group decision making in uncertain environments (MAGDM) is an important component of modern decision-making science, and its theory and methods have been widely applied in many fields [1,2,3,4,5,6,7,8,9,10,11]. However, with the continuous expansion and application of decision theory, limited by individual cognitive level and thinking characteristics, people often make decisions in practical problems with a certain degree of ambiguity and uncertainty. In order to describe and deal with this fuzzy phenomenon, Zadeh [12] proposed the fuzzy set (FS) theory in 1965 by introducing the concept of membership to express subjective opinions. Subsequently, some extended forms of FS have been proposed, such as interval valued fuzzy set (IVFS) [13], intuitionistic fuzzy sets (IFS) [14], Pythagorean fuzzy set (PFS) [15] and the q-rung orthopair fuzzy set (q-ROFS) [16]. Among these, the q-ROFS is an extension that generalizes the PFS and IFS, which satisfies that the sum of the q-power of membership degree and the q-power of non-membership degree is restricted to [0, 1]; this solved the phenomenon that IFS and PFS are limited in the scope of information expression. Both IFS and PFS are special cases of the q-ROFS when q takes different values. We can contend that q-ROFS is more comprehensive. Consequently, q-ROFS offers decision makers a broader spectrum for articulating their uncertain information [17,18,19,20,21,22,23,24].

None of the existing models can effectively represent partial ignorance of the data or its fluctuations at specific phases of time. However, in complex data sets, uncertainty and vagueness in the data often coexist with variations in the data’s phases (periodicity). To address such problems, the concept of the complex fuzzy set (CFS) [25] was first proposed by Ramot et al. Because the CFS cannot only expand the range of membership and non-membership values from the unit interval to the unit circle of the complex plane, but also the complex membership and complex non-membership values can be expressed in polar coordinates. Therefore, it has received widespread attention from scholars both domestically and internationally. Rahman et al. [26] proposed an algorithm for a decision-making process using a complex Pythagorean fuzzy set and applied them to a hospital setting for COVID-19 patients. Mahmood et al. [27] studied Heronian mean operators for managing complex picture fuzzy uncertain linguistic settings. Azam et al. [28] developed a decision-making approach for the evaluation of information security management under a complex intuitionistic fuzzy set environment. Garg et al. [29] proposed a decision-making approach based on generalized aggregation operators with complex single-valued neutrosophic hesitant fuzzy set information. Janani et al. [30] defined complex probabilistic fuzzy set and proposed some aggregation operators in group decision making extended to TOPSIS. Mahmood et al. [31] proposed some Bonferroni mean operators based on a bipolar complex fuzzy setting and applied them in multi-attribute decision making. Harish et al. [32] developed some complex q-rung orthopair fuzzy Hamy mean operators. Also, Liu et al. [33] proposed some complex q-rung orthopair fuzzy aggregation operators and used them in multi-attribute group decision making. However, it is difficult to accurately represent the membership function values for complex fuzzy sets, and there is a lack of expression of non-membership degrees; this leads to inaccurate understanding of the degree of ambiguity, which can easily affect decision results.

In reality, decision makers may need to express both interval values and corresponding fixed-value evaluation information. This mixed fuzzy information cannot be represented by the above single form of Fuzzy set and its extended set. For this reason, Jun et al. [34] put forward the concept of the cubic set (CS), which is a new extension of FS, characterized by the mixed description of interval information and deterministic information, that is, a CS is composed of an IVFS and an FS. Compared with FS, cubic set has the advantage that a set contains different forms of fuzzy evaluation information at the same time, which can meet the requirements of decision makers to express different forms of evaluation information for an evaluation object in real situations. Shahzad et al. [35] proposed some operations and properties of the cubic intuitionistic set with application in multi-criteria decision making. Muhammad et al. [36] studied cubic bipolar fuzzy set with application to multi-criteria group decision making using geometric aggregation operators. Wang et al. [37] discussed similarity and Pythagorean reliability measures of multivalued neutrosophic cubic set. Muhammad et al. [38] defined cubic Pythagorean fuzzy soft set and applied them in multi-attribute decision making. Farhadinia [39] defined cubic hesitant fuzzy set and presented a cubic hesitant fuzzy set characterized in the form of triangular, where is known as triangular cubic hesitant fuzzy set. Zhang et al. [40] proposed some cubic q-rung orthopair fuzzy Heronian mean operators and discussed their applications in multi-attribute group decision making. However, cubic fuzzy information is unable to capture the partial uncertainty of data and its variations at specific time phases during execution. When handling substantial amounts of uncertain information with phase changes, such as in digital and image processing, cubic sets prove ineffective. At present, Ren et al. [41] proposed a new hybrid model called complex cubic q-rung orthopair fuzzy set (CCuq-ROFS) by combining complex fuzzy set and Cq-ROFS. It contains more information than complex fuzzy sets and cubic sets, so it is more suitable to deal with complex MAGDM problems.

Meantime, the Maclaurin symmetric mean (MSM) operator was originally introduced by Maclaurin [42] and then developed by Detemple and Robertson [43]. The prominent characteristic of the MSM is that it can capture the interrelationship among the multi-input arguments. The MSM operator can provide more flexible and robust information fusion and make it more adequate to solve MADM in which the attributes are independent. Moreover, for a given collection of arguments, the MSM operator is monotonically decreasing with respect to the values of the parameter, which can reflect the risk preferences of the decision makers in practical situations. In the past few years, the MSM has received more and more attention, and many important results both in theory and application have been developed [44,45,46,47,48,49]. Ali et al. [50] proposed complex intuitionistic fuzzy Maclaurin symmetric mean operators and their applications for emergency program selection. Song et al. [51] proposed some single-valued neutrosophic uncertain linguistic Maclaurin symmetric mean operators and their application to multiple-attribute decision making. Aliya et al. [52] proposed Maclaurin symmetric mean aggregation operators based on cubic Pythagorean linguistic fuzzy number. Mu et al. [53] proposed a novel approach to multi-attribute group decision making based on interval-valued Pythagorean fuzzy power Maclaurin symmetric mean operator. Liu et al. [54] propose multiple-attribute decision-making method based on power generalized Maclaurin symmetric mean operators under normal wiggly hesitant fuzzy environment. Yang et al. [55] studied three-way decisions based on q-rung orthopair fuzzy 2-tuple linguistic sets with generalized Maclaurin symmetric mean operators.

The previous discussions have shown that many of the developed operators rely on algebraic products and algebraic sums. Nonetheless, it is essential to recognize that these operations are not the sole options available for FSs. Hamacher [56] introduced the Hamacher operations, which encompass the Hamacher product and the Hamacher sum. Both the Hamacher product and Hamacher sum serve as effective alternatives to the conventional algebraic product and algebraic sum. Additionally, the Hamacher t-conorm and Hamacher t-norm are considered as more general and flexible generalizations of the algebraic and Einstein t-conorm and t-norm, respectively. Despite their potential advantages, a review of the existing literature on aggregation operators reveals a scarcity of research exploring the application of Hamacher operations in developing new operators. Therefore, it is imperative to undertake research on aggregation operators that leverage Hamacher operations in the context of CCuq-ROF information. The contributions of this paper are outlined below:

(1) We propose some new aggregation operators for CCuq-ROFS, such as the complex cubic q-rung orthopair fuzzy Hamacher average (CCuq-ROFHA) operator, the weighted complex cubic q-rung orthopair fuzzy Hamacher average (WCCuq-ROFHA) operator, the complex cubic q-rung orthopair fuzzy Hamacher Maclaurin symmetric mean (CCuq-ROFHMSM) operator and the weighted complex cubic q-rung orthopair fuzzy Hamacher Maclaurin symmetric mean (WCCuq-ROFHMSM) operator;

(2) The combination of Hamacher t-norm and t-conorm based operations with the MSM operator not only captures the interrelationships among multiple attributes but also enhances decision-making flexibility through the incorporation of additional parameters γ and k;

(3) The proposed operators are inherently more general, offering a variety of aggregation methods by substituting specific values for the parameters γ and k.

The structure of the remaining sections in this paper is as follows: Section 2 introduces the fundamental concepts of CCuq-ROFS, Hamacher operations and MSM operators. In Section 3, we focus on the development of Hamacher aggregation operators within the complex cubic q-rung orthopair fuzzy environment and discuss their properties. Section 4 presents a brief study on the MAGDM approach, incorporating the proposed Hamacher aggregation operators. In Section 5, we provide an example to demonstrate the application of the proposed method. Additionally, we conduct sensitivity analysis and comparison analysis. Finally, Section 6 provides a brief conclusion.

2. Preliminaries

2.1. q-Rung Orthopair Fuzzy Set (Q-ROFS)

In this portion, the definition of Q-ROFS and the q-rung orthopair fuzzy number (q-ROFN) were introduced as follows:

Definition 1 ([16]).  

Let $X=\left\{x_1,x_2,\dots ,x_n\right\}$  be an ordinary set, then a q-rung orthopair fuzzy set (q-ROFS) $\tilde{A}$  defined on $X$  is expressed as follows:

$$\tilde{A}=\left\{\langle x_i,\left(u_{\tilde{A}}\left(x_i\right),v_{\tilde{A}}\left(x_i\right)\right)\rangle |x_i\in X\right\}$$

(1)

where $u_{\tilde{A}}\left(x_i\right):X\to \left[0,1\right]$  and $v_{\tilde{A}}\left(x_i\right):X\to \left[0,1\right]$  denote the membership and non-membership degrees of the element $x\in X$  to the set $\tilde{A}$, respectively, satisfying $0\le {\left(u_{\tilde{A}}\left(x_i\right)\right)}^q+{\left(v_{\tilde{A}}\left(x_i\right)\right)}^q\le 1$ $\left(q\ge 1\right)$. Then the degree of indeterminacy of the element $x$  to the set $\tilde{A}$  is expressed as ${\pi }_{\tilde{A}}\left(x_i\right)={\left(1-{\left(u_{\tilde{A}}\left(x_i\right)\right)}^q-{\left(v_{\tilde{A}}\left(x_i\right)\right)}^q\right)}^{1/q}$. Obviously, $0\le {\pi }_{\tilde{A}}\left(x_i\right)\le 1$ $\left(x_i\in X\right)$. For convenience, Yager [16] called the $A=\left(u_{\tilde{A}},v_{\tilde{A}}\right)$  a q-rung orthopair fuzzy number(q-ROFN), which can be denoted as a $a=\left(u,v\right)$.

2.2. Complex q-Rung Orthopair Fuzzy Set (Cq-ROFS)

In this portion, the definition of Cq-ROFS and complex q-rung orthopair fuzzy number (Cq-ROFN) were introduced as follows:

Definition 2 ([33]). 

Let  $X=\left\{x_1,x_2,\dots ,x_n\right\}$  be an ordinary set, then a complex q-rung orthopair fuzzy set (Cq-ROFS)  $C$  defined on  $X$  is expressed as follows:

$$C=\left\{\langle x_i,\left(u_{{}_C}^{\prime }\left(x_i\right),v_{{}_C}^{\prime }\left(x_i\right)\right)\rangle |x_i\in X\right\}, q\ge 1$$

(2)

where $u_{{}_C}^{\prime }\left(x_i\right)=u\left(x\right)e^{i2\pi w_{u\left(x\right)}}$  and $v_{{}_C}^{\prime }\left(x_i\right)=v\left(x\right)e^{i2\pi w_{v\left(x\right)}}$  denote the complex-valued membership and complex-valued non-membership degrees of the element $x\in X$  to the set $C$, respectively, satisfying $0\le {\left(u\left(x\right)\right)}^q+{\left(v\left(x\right)\right)}^q\le 1$  and $0\le {\left(w_{u\left(x\right)}\right)}^q+{\left(w_{v\left(x\right)}\right)}^q\le 1$  $\left(q\ge 1\right)$. $u\left(x\right)$  and $w_{u\left(x\right)}$  represent the real and imaginary parts of the complex-valued membership degree, respectively, $v\left(x\right)$  and $w_{v\left(x\right)}$  represent the real and imaginary parts of the complex-valued non-membership degree, respectively. Then the degree of indeterminacy of the element $x$  to the set $C$  is expressed as follows:

$$\pi \left(x\right)={\left(1-{\left(u\left(x\right)\right)}^q-{\left(v\left(x\right)\right)}^q\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}{e}^{i2\pi {\left(1-{\left(w_{u\left(x\right)}\right)}^q-{\left(w_{v\left(x\right)}\right)}^q\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$q$}\right.}}$$

(3)

For convenience, Liu [33] called the $C=\left(u\left(x\right)e^{i2\pi w_{u\left(x\right)}},v\left(x\right)e^{i2\pi w_{v\left(x\right)}}\right)$  a complex q-rung orthopair fuzzy number (Cq-ROFN), which can be denoted as a $C=\left(u{e}^{i2\pi w_u},v{e}^{i2\pi w_v}\right)$.

2.3. Cubic q-Rung Orthopair Fuzzy (Cuq-ROFS)

In this portion, the definition of Cuq-ROFS and cubic q-rung orthopair fuzzy number (Cuq-ROFN) were introduced as follows:

Definition 3 ([40]). 

Let  $X=\left\{x_1,x_2,\dots ,x_n\right\}$  be an ordinary set, then a cubic q-rung orthopair fuzzy set (Cuq-ROFS)  $\tilde{C}$  defined on  $X$  is expressed as follows:

$$\tilde{C}=\left\{\langle x,\left(\zeta \left(x\right),\xi \left(x\right)\right)\rangle |x\in X\right\}, q\ge 1$$

(4)

where  $\zeta \left(x\right)=\left(\left[u_{\varsigma }^{-}\left(x\right),u_{\varsigma }^{+}\left(x\right)\right],\left[v_{\varsigma }^{-}\left(x\right),v_{\varsigma }^{+}\left(x\right)\right]\right)$  denotes the interval value q-rung orthopair fuzzy set (IVq-ROFS) on  $X$  ,  $\xi \left(x\right)=\left(u_{\xi }\left(x\right),v_{\xi }\left(x\right)\right)$  denotes the q-ROFS on  $X$  , satisfying  $0\le {\left(u_{\varsigma }^{+}\left(x\right)\right)}^q+{\left(v_{\varsigma }^{+}\left(x\right)\right)}^q\le 1$  ,  $0\le {\left(u_{\xi }\left(x\right)\right)}^q+{\left(v_{\xi }\left(x\right)\right)}^q\le 1$  , respectively. Then the degree of indeterminacy of the element  $x$  to set  $\tilde{C}$  is expressed as follows:  $\pi \left(x\right)=\left(\left[{\pi }_{\varsigma }^{-}\left(x\right),{\pi }_{\varsigma }^{+}\left(x\right)\right],{\pi }_{\xi }\left(x\right)\right)$, ${\pi }_{\varsigma }^{-}\left(x\right)={\left(1-{\left(u_{\varsigma }^{+}\left(x\right)\right)}^q-{\left(v_{\varsigma }^{+}\left(x\right)\right)}^q\right)}^{1/q}$, ${\pi }_{\varsigma }^{+}\left(x\right)={\left(1-{\left(u_{\varsigma }^{-}\left(x\right)\right)}^q-{\left(v_{\varsigma }^{-}\left(x\right)\right)}^q\right)}^{1/q}$, ${\pi }_{\xi }\left(x\right)={\left(1-{\left(u_{\xi }\left(x\right)\right)}^q-{\left(v_{\xi }\left(x\right)\right)}^q\right)}^{1/q}$.

For convenience, Zhang [40] called the $\tilde{C}=\left(\left(\left[u_{\varsigma }^{-}\left(x\right),u_{\varsigma }^{+}\left(x\right)\right],\left[v_{\varsigma }^{-}\left(x\right),v_{\varsigma }^{+}\left(x\right)\right]\right),\left(u_{\xi }\left(x\right),v_{\xi }\left(x\right)\right)\right)$  a cubic q-rung orthopair fuzzy number (Cuq-ROFN), which can be denoted as a $\tilde{C}=\left(\left(\left[u_{\varsigma }^{-},u_{\varsigma }^{+}\right],\left[v_{\varsigma }^{-},v_{\varsigma }^{+}\right]\right),\left(u_{\xi },v_{\xi }\right)\right)$.

2.4. Complex Cubic q-Rung Orthopair Fuzzy (Ccuq-ROFS)

In this portion, the definitions of Ccuq-ROFS and complex cubic q-rung orthopair fuzzy number (Ccuq-ROFN) were introduced as follows:

Definition 4 ([41]). 

Let  $X=\left\{x_1,x_2,\dots ,x_n\right\}$  be an ordinary set, then a complex cubic q-rung orthopair fuzzy set (Ccuq-ROFS) defined on is expressed as follows:

$$A=\left\{\langle x,\left(\zeta \left(x\right),\xi \left(x\right)\right)\rangle |x\in X\right\}, q\ge 1$$

(5)

where  $\zeta \left(x\right)=\left(\left[u_{\varsigma }^{-}\left(x\right),u_{\varsigma }^{+}\left(x\right)\right]e^{i2\pi \left[w_{u_{\varsigma }^{}\left(x\right)}^{-},w_{u_{\varsigma }^{}\left(x\right)}^{+}\right]},\left[v_{\varsigma }^{-}\left(x\right),v_{\varsigma }^{+}\left(x\right)\right]e^{i2\pi \left[w_{v_{\varsigma }^{}\left(x\right)}^{-},w_{v_{\varsigma }^{}\left(x\right)}^{+}\right]}\right)$  denotes the complex interval value q-rung orthopair fuzzy set (CIVq-ROFS) on  $X$  ,  $\xi \left(x\right)=\left(u_{\xi }\left(x\right)e^{i2\pi w_{u_{\xi }\left(x\right)}},v_{\xi }\left(x\right)e^{i2\pi w_{v_{\xi }\left(x\right)}}\right)$  denotes the Cq-ROFS on  $X$  , satisfying  $u_{\varsigma }^{-}\left(x\right)\le u_{\varsigma }^{+}\left(x\right)$  ,  $w_{u_{\varsigma }^{-}\left(x\right)}\le w_{u_{\varsigma }^{+}\left(x\right)}$  ,  $v_{\varsigma }^{-}\left(x\right)\le v_{\varsigma }^{+}\left(x\right)$  ,  $w_{v_{\varsigma }^{-}\left(x\right)}\le w_{v_{\varsigma }^{+}\left(x\right)},$   $0\le {\left(u_{\varsigma }^{+}\left(x\right)\right)}^q+{\left(v_{\varsigma }^{+}\left(x\right)\right)}^q\le 1$  ,  $0\le {\left(u_{\xi }\left(x\right)\right)}^q+{\left(v_{\xi }\left(x\right)\right)}^q\le 1$  ,  $0\le {\left(w_{u_{\xi }\left(x\right)}\right)}^q+{\left(w_{v_{\xi }\left(x\right)}\right)}^q\le 1$  ,  $q\ge 1$  , respectively. Then the degree of indeterminacy of the element  $x$  to the set  $A$  is expressed as follows:

$$\pi \left(x\right)=\left(\left[{\pi }_{\varsigma }^{-}\left(x\right),{\pi }_{\varsigma }^{+}\left(x\right)\right],{\pi }_{\xi }\left(x\right)\right),$$

where

$${\pi }_{\varsigma }^{-}\left(x\right)={\left(1-{\left(u_{\varsigma }^{+}\left(x\right)\right)}^q-{\left(v_{\varsigma }^{+}\left(x\right)\right)}^q\right)}^{1/q}{e}^{i2\pi {\left(1-{\left(w_{u_{\varsigma }^{}\left(x\right)}^{+}\right)}^q-{\left(w_{v_{\varsigma }^{}\left(x\right)}^{+}\right)}^q\right)}^{1/q}}$$

$${\pi }_{\varsigma }^{+}\left(x\right)={\left(1-{\left(u_{\varsigma }^{-}\left(x\right)\right)}^q-{\left(v_{\varsigma }^{-}\left(x\right)\right)}^q\right)}^{1/q}{e}^{i2\pi {\left(1-{\left(w_{{}_{u_{\varsigma }^{}\left(x\right)}}^{\_}\right)}^q-{\left(w_{v_{\varsigma }^{}\left(x\right)}^{-}\right)}^q\right)}^{1/q}}$$

$${\pi }_{\xi }\left(x\right)={\left(1-{\left(u_{\xi }\left(x\right)\right)}^q-{\left(v_{\xi }\left(x\right)\right)}^q\right)}^{1/q}{e}^{i2\pi {\left(1-{\left(w_{u_{\xi }\left(x\right)}\right)}^q-{\left(w_{v_{\xi }\left(x\right)}\right)}^q\right)}^{1/q}}$$

For convenience, we called the $A=\left(\left(\left[u_{}^{-},u_{}^{+}\right]e^{i2\pi \left[w_u^{-},w_u^{+}\right]},\left[v_{}^{-},v_{}^{+}\right]e^{i2\pi \left[w_v^{-},w_v^{+}\right]}\right),\left(u{e}^{i2\pi w_u},v{e}^{i2\pi w_v}\right)\right)$  a complex cubic q-rung orthopair fuzzy number (CCuq-ROFN).

Definition 5. 

Let  $A=\left(\left(\left[u_{}^{-},u_{}^{+}\right]e^{i2\pi \left[w_u^{-},w_u^{+}\right]},\left[v_{}^{-},v_{}^{+}\right]e^{i2\pi \left[w_v^{-},w_v^{+}\right]}\right),\left(u{e}^{i2\pi w_u},v{e}^{i2\pi w_v}\right)\right)$  ,  $A_1=\left(\left(\left[u_1^{-},u_1^{+}\right]e^{i2\pi \left[w_{u_1}^{-},w_{u_1}^{+}\right]},\left[v_1^{-},v_1^{+}\right]e^{i2\pi \left[w_{v_1}^{-},w_{v_1}^{+}\right]}\right),\left(u_1{e}^{i2\pi w_{u_1}},v_1{e}^{i2\pi w_{v_1}}\right)\right)$  and  $A_2=\left(\left(\left[u_2^{-},u_2^{+}\right]e^{i2\pi \left[w_{u_2}^{-},w_{u_2}^{+}\right]},\left[v_2^{-},v_2^{+}\right]e^{i2\pi \left[w_{v_2}^{-},w_{v_2}^{+}\right]}\right),\left(u_2{e}^{i2\pi w_{u_2}},v_2{e}^{i2\pi w_{v_2}}\right)\right)$  be three CCuq-ROFNs, then their operations are defined as follows:

(1)

$A_1\oplus A_2=\left(\begin{array}{l}\left(\begin{array}{l}\left[{\left(u_1^{q^{-}}+u_2^{q^{-}}-u_1^{q^{-}}{u}_2^{q^{-}}\right)}^{1/q},{\left(u_1^{q^{+}}+u_2^{q^{+}}-u_1^{q^{+}}{u}_2^{q^{+}}\right)}^{1/q}\right]e^{i2\pi \left[{\left(w_{u_1}^{q^{-}}+w_{u_2}^{q^{-}}-w_{u_1}^{q^{-}}{w}_{u_2}^{q^{-}}\right)}^{1/q},{\left(w_{u_1}^{q^{+}}+w_{u_2}^{q^{+}}-w_{u_1}^{q^{+}}{w}_{u_2}^{q^{+}}\right)}^{1/q}\right]},\\ \left[\left[v_1^{-}{v}_2^{-},v_1^{+}{v}_2^{+}\right]e^{i2\pi \left[w_{v_1}^{-}{w}_{v_2}^{-},w_{v_1}^{+}{w}_{v_2}^{+}\right]}\right]\end{array}\right),\\ \left({\left(u_1^q+u_2^q-u_1^q{u}_2^q\right)}^{1/q}{e}^{i2\pi {\left(w_{u_1}^q+w_{u_2}^q-w_{u_1}^q{w}_{u_2}^q\right)}^{1/q}},v_1{v}_2{e}^{i2\pi w_{v_1}{w}_{v_2}}\right)\end{array}\right)$ 

(2)

$A_1\otimes A_2=\left(\begin{array}{l}\left(\begin{array}{l}\left[\left[u_1^{-}{u}_2^{-},u_1^{+}{u}_2^{+}\right]e^{i2\pi \left[w_{u_1}^{-}{w}_{u_2}^{-},w_{u_1}^{+}{w}_{u_2}^{+}\right]}\right],\\ \left[{\left(v_1^{q^{-}}+v_2^{q^{-}}-v_1^{q^{-}}{v}_2^{q^{-}}\right)}^{1/q},{\left(v_1^{q^{+}}+v_2^{q^{+}}-v_1^{q^{+}}{v}_2^{q^{+}}\right)}^{1/q}\right]e^{i2\pi \left[{\left(w_{v_1}^{q^{-}}+w_{v_2}^{q^{-}}-w_{v_1}^{q^{-}}{w}_{v_2}^{q^{-}}\right)}^{1/q},{\left(w_{v_1}^{q^{+}}+w_{v_2}^{q^{+}}-w_{v_1}^{q^{+}}{w}_{v_2}^{q^{+}}\right)}^{1/q}\right]}\end{array}\right),\\ \left(u_1{u}_2{e}^{i2\pi w_{u_1}{w}_{u_2}},{\left(v_1^q+v_2^q-v_1^q{v}_2^q\right)}^{1/q}{e}^{i2\pi {\left(w_{v_1}^q+w_{v_2}^q-w_{v_1}^q{w}_{v_2}^q\right)}^{1/q}}\right)\end{array}\right)$ 

(3)

$\lambda A=\left(\begin{array}{l}\left(\left[{\left(1-{\left(1-u^{q^{-}}\right)}^{\lambda }\right)}^{1/q},{\left(1-{\left(1-u^{q^{+}}\right)}^{\lambda }\right)}^{1/q}\right]e^{i2\pi \left[{\left(1-{\left(1-w_u^{q^{-}}\right)}^{\lambda }\right)}^{1/q},{\left(1-{\left(1-w_u^{q^{-}}\right)}^{\lambda }\right)}^{1/q}\right]}{,}^{\left[v^{{\lambda }^{-}},v^{{\lambda }^{+}}\right]e^{i2\pi \left[w_v^{{\lambda }^{-}},w_v^{{\lambda }^{-}}\right]}}\right),\\ \left({\left(1-{\left(1-u^q\right)}^{\lambda }\right)}^{1/q}{e}^{i2\pi {\left(1-{\left(1-w_u^q\right)}^{\lambda }\right)}^{1/q}},v^{\lambda }{e}^{i2\pi w_v^{\lambda }}\right)\end{array}\right)$ 

(4)

$A^{\lambda }=\left(\left(\begin{array}{l}\left[u^{{\lambda }^{-}},u^{{\lambda }^{+}}\right]e^{i2\pi \left[w_u^{{\lambda }^{-}},w_u^{{\lambda }^{-}}\right]},\\ {}^{\begin{array}{l}\left[{\left(1-{\left(1-v^{q^{-}}\right)}^{\lambda }\right)}^{1/q},{\left(1-{\left(1-v^{q^{+}}\right)}^{\lambda }\right)}^{1/q}\right]e^{i2\pi \left[{\left(1-{\left(1-w_v^{q^{-}}\right)}^{\lambda }\right)}^{1/q},{\left(1-{\left(1-w_v^{q^{-}}\right)}^{\lambda }\right)}^{1/q}\right]}\\ \end{array}}\end{array}\right),\left(u^{\lambda }{e}^{i2\pi w_u^{\lambda }},{\left(1-{\left(1-v^q\right)}^{\lambda }\right)}^{1/q}{e}^{i2\pi {\left(1-{\left(1-w_v^q\right)}^{\lambda }\right)}^{1/q}}\right)\right)$ 

Theorem 1. 

Let  $A=\left(\left(\left[u_{}^{-},u_{}^{+}\right]e^{i2\pi \left[w_u^{-},w_u^{+}\right]},\left[v_{}^{-},v_{}^{+}\right]e^{i2\pi \left[w_v^{-},w_v^{+}\right]}\right),\left(u{e}^{i2\pi w_u},v{e}^{i2\pi w_v}\right)\right)$  ,  $A_1=\left(\left(\left[u_1^{-},u_1^{+}\right]e^{i2\pi \left[w_{u_1}^{-},w_{u_1}^{+}\right]},\left[v_1^{-},v_1^{+}\right]e^{i2\pi \left[w_{v_1}^{-},w_{v_1}^{+}\right]}\right),\left(u_1{e}^{i2\pi w_{u_1}},v_1{e}^{i2\pi w_{v_1}}\right)\right)$  and  $A_2=\left(\left(\left[u_2^{-},u_2^{+}\right]e^{i2\pi \left[w_{u_2}^{-},w_{u_2}^{+}\right]},\left[v_2^{-},v_2^{+}\right]e^{i2\pi \left[w_{v_2}^{-},w_{v_2}^{+}\right]}\right),\left(u_2{e}^{i2\pi w_{u_2}},v_2{e}^{i2\pi w_{v_2}}\right)\right)$  be three CCuq-ROFNs, then when  $\lambda ,{\lambda }_1,{\lambda }_2$  > 0, then they have the following basic operational properties:

(1)

$A_1\oplus A_2=A_2\oplus A_1$ 

(2)

$A_1\otimes A_2=A_2\otimes A_1$ 

(3)

$\left(A\oplus A_1\right)\oplus A_2=A\oplus \left(A_1\oplus A_2\right)$ 

(4)

$\left(A\otimes A_1\right)\otimes A_2=A\otimes \left(A_1\otimes A_2\right)$ 

(5)

$\lambda \left(A_1\oplus A_2\right)=\lambda A_1\oplus \lambda A_2$ 

(6)

${\lambda }_{1}A\oplus {\lambda }_{2}A=\left({\lambda }_1+{\lambda }_2\right)A$ 

(7)

$A^{{\lambda }_1}\otimes A^{{\lambda }_2}=A^{{}^{{\lambda }_1+{\lambda }_2}}$ 

(8)

${\left(A_1\otimes A_2\right)}^{\lambda }=A_1^{\lambda }\otimes A_2^{\lambda }$ 

Definition 6. 

Let $A=\left(\left(\left[u_{}^{-},u_{}^{+}\right]e^{i2\pi \left[w_{u_{}}^{-},w_{u_{}}^{+}\right]},\left[v_{}^{-},v_{}^{+}\right]e^{i2\pi \left[w_{v_{}}^{-},w_{v_{}}^{+}\right]}\right),\left(u_{}{e}^{i2\pi w_{u_{}}},v_{}{e}^{i2\pi w_{v_{}}}\right)\right)$  be a CCuq-ROFN, then the score function $S\left(A\right)$  and accurate function $H\left(A\right)$  are defined as follows:

$$S\left(A\right)=\frac{1}{4}\left(\frac{1}{2}\left(\left(\left(u_A^{q-}+u_A^{q+}\right)-\left(v_A^{q-}+v_A^{q+}\right)\right)+\left(\left(w_{u_A^{}}^{q-}+w_{u_A^{}}^{q+}\right)-\left(w_{v_A^{}}^{q-}+w_{v_A^{}}^{q+}\right)\right)\right)+\left(u_A^q-v_A^q\right)+\left(w_{u_A^{}}^q-w_{v_A^{}}^q\right)\right)$$

$$H\left(A\right)=\frac{1}{4}\left(\frac{1}{2}\left(\left(\left(u_A^{q-}+u_A^{q+}\right)+\left(v_A^{q-}+v_A^{q+}\right)\right)+\left(\left(w_{u_A^{}}^{q-}+w_{u_A^{}}^{q+}\right)+\left(w_{v_A^{}}^{q-}+w_{v_A^{}}^{q+}\right)\right)\right)+\left(u_A^q+v_A^q\right)+\left(w_{u_A^{}}^q+w_{v_A^{}}^q\right)\right)$$

Definition 7. 

Let $A_1=\left(\left(\left[u_1^{-},u_1^{+}\right]e^{i2\pi \left[w_{u_1}^{-},w_{u_1}^{+}\right]},\left[v_1^{-},v_1^{+}\right]e^{i2\pi \left[w_{v_1}^{-},w_{v_1}^{+}\right]}\right),\left(u_1{e}^{i2\pi w_{u_1}},v_1{e}^{i2\pi w_{v_1}}\right)\right)$  and $A_2=\left(\left(\left[u_2^{-},u_2^{+}\right]e^{i2\pi \left[w_{u_2}^{-},w_{u_2}^{+}\right]},\left[v_2^{-},v_2^{+}\right]e^{i2\pi \left[w_{v_2}^{-},w_{v_2}^{+}\right]}\right),\left(u_2{e}^{i2\pi w_{u_2}},v_2{e}^{i2\pi w_{v_2}}\right)\right)$  be two CCuq-ROFNs:

(1)

If  $S\left(A_1\right)$  >  $S\left(A_2\right)$, then  $A_1\succ A_2$  ;

(2)

If  $S\left(A_1\right)$  =  $S\left(A_2\right)$, then

If  $H\left(A_1\right)$  >  $H\left(A_2\right)$, then  $A_1\succ A_2$  ;

If $H\left(A_1\right)$  = $H\left(A_2\right)$, then $A_1\sim A_2$.

2.5. Hamacher Operations

In fuzzy set theory, t-norm and t-conorm are fundamental concepts used to define a generalized union and intersection of fuzzy sets. To give meaning to t-norm and t-conorm, Roychowdhury and Wang [57] proposed specific definitions and conditions. Additionally, Deschrijver et al. [58] developed a generalized union and a generalized intersection based on a t-norm (T) and t-conorm (T*) for intuitionistic fuzzy sets (IFSs). To further generalize the existing t-norm and t-conorm operations, Hamacher [56] introduced the Hamacher product and Hamacher sum as extensions of t-norms and t-conorms, respectively. The Hamacher product $\otimes$  (t-norm) and the Hamacher sum $\oplus$  (t-conorm) are defined as follows:

$$T\left(a,b\right)=a\otimes b=\frac{ab}{\gamma +\left(1-\gamma \right)\left(a+b-ab\right)}, \gamma >0$$

$$T^{*}\left(a,b\right)=a\oplus b=\frac{a+b-ab-\left(1-\gamma \right)ab}{1-\left(1-\gamma \right)ab}, \gamma >0$$

Especially, when $\gamma =1$ , then Hamacher $t$-norm and $t$-conorm will reduce to the following:

$$T\left(a,b\right)=a\otimes b=ab$$

$$T^{*}\left(a,b\right)=a\oplus b=a+b-ab$$

which are the algebraic $t$-norm and $t$-conorm, respectively; when $\gamma =2$ , then Hamacher $t$-norm and $t$-conorm will reduce to the following:

$$T\left(a,b\right)=a\otimes b=\frac{ab}{1+\left(1-a\right)\left(1-b\right)}$$

$$T^{*}\left(a,b\right)=a\oplus b=\frac{a+b}{1+ab}$$

which are called the Einstein $t$-norm and $t$-conorm, respectively.

2.6. Maclaurin Symmetric Mean Operators

The MSM operator was originally introduced by Maclaurin [42,43], which is a useful technique characterized by the ability to capture the interrelationship among the multi-input arguments. The definition of MSM is defined as follows:

Definition 8 ([42,43]). 

Let $a_j\left(j=1,2,\dots ,n\right)$be a collection of nonnegative numbers, and $k=1,2,\dots ,n$. If

$$MS{M}^{\left(k\right)}\left(a_1,a_2,\dots ,a_n\right)={\left(\frac{{\displaystyle {\sum }_{1\le i_1\le \dots \le i_k\le n}{\displaystyle {\prod }_{j=1}^k{a}_{i_j}}}}{C_n^k}\right)}^{\frac{1}{k}},$$

(6)

then  $MS{M}^{\left(k\right)}$  is called the MSM operator, where  $\left(i_1,i_2,\dots ,i_k\right)$  traverse all the  $k$-tuple combinations of  $\left(1,2,\dots ,n\right)$  , and  $C_n^k$  is the binomial coefficient.

Obviously, the MSM has the following properties:

(1)

$MS{M}^{\left(k\right)}\left(0,0,\dots ,0\right)=0$ ;

(2)

$MS{M}^{\left(k\right)}\left(a,a,\dots ,a\right)=a$ ;

(3)

$MS{M}^{\left(k\right)}\left(a_1,a_2,\dots ,a_n\right)\le MS{M}^{\left(k\right)}\left(b_1,b_2,\dots ,b_n\right)$ , if $a_i\le b_i$  for all $i$ ;

(4)

$\underset{i}{\min }\left\{a_i\right\}\le MS{M}^{\left(k\right)}\left(a_1,a_2,\dots ,a_n\right)\le \underset{i}{\max }\left\{a_i\right\}$.

3. Complex Cubic q-Rung Orthopair Fuzzy Hamacher Aggregation Operators

In this section, we propose complex cubic q-rung orthopair fuzzy Hamacher aggregation operators based on the arithmetic average operator and the Maclaurin symmetric mean operator.

3.1. Hamacher Operations of Complex Cubic q-Rung Orthopair Fuzzy Set

Motivated by the arithmetic aggregation operators, the Hamacher product $\otimes$  and the Hamacher sum $\oplus$  [56], then the generalized intersection and union on two CCuq-ROFNs $a_1$  and $a_2$  become the Hamacher product (denoted by $a_1\otimes a_2$ ) and Hamacher sum (denoted by $a_1\oplus a_2$ ) of two CCuq-ROFNs $a_1$  and $a_2$ , $\gamma >0$ , respectively, as follows:

Definition 9. 

Let  $a_{}=\left(\left(\left[u_{}^{-},u_{}^{+}\right]e^{i2\pi \left[w_{u_{}}^{-},w_{u_{}}^{+}\right]},\left[v_{}^{-},v_{}^{+}\right]e^{i2\pi \left[w_{v_{}}^{-},w_{v_{}}^{+}\right]}\right),\left(u_{}{e}^{i2\pi w_u},v_{}{e}^{i2\pi w_v}\right)\right),$   $a_1=\left(\left(\left[u_1^{-},u_1^{+}\right]e^{i2\pi \left[w_{u_1}^{-},w_{u_1}^{+}\right]},\left[v_1^{-},v_1^{+}\right]e^{i2\pi \left[w_{v_1}^{-},w_{v_1}^{+}\right]}\right),\left(u_1{e}^{i2\pi w_{u_1}},v_1{e}^{i2\pi w_{v_1}}\right)\right)$  and  $a_2=\left(\left(\left[u_2^{-},u_2^{+}\right]e^{i2\pi \left[w_{u_2}^{-},w_{u_2}^{+}\right]},\left[v_2^{-},v_2^{+}\right]e^{i2\pi \left[w_{v_2}^{-},w_{v_2}^{+}\right]}\right),\left(u_2{e}^{i2\pi w_{u_2}},v_2{e}^{i2\pi w_{v_2}}\right)\right)$  be three CCuq-ROFNs, then Hamacher operations are defined as follows:

$$(1) a_1\oplus a_2=\left(\begin{array}{l}\left(\begin{array}{l}\left[\begin{array}{l}{\left(\frac{{\left(u_1^{-}\right)}^q+{\left(u_2^{-}\right)}^q-{\left(u_1^{-}\right)}^q{\left(u_2^{-}\right)}^q-\left(1-\gamma \right){\left(u_1^{-}\right)}^q{\left(u_2^{-}\right)}^q}{1-\left(1-\gamma \right){\left(u_1^{-}\right)}^q{\left(u_2^{-}\right)}^q}\right)}^{1/q},\\ {\left(\frac{{\left(u_1^{+}\right)}^q+{\left(u_2^{+}\right)}^q-{\left(u_1^{+}\right)}^q{\left(u_2^{+}\right)}^q-\left(1-\gamma \right){\left(u_1^{+}\right)}^q{\left(u_2^{+}\right)}^q}{1-\left(1-\gamma \right){\left(u_1^{+}\right)}^q{\left(u_2^{+}\right)}^q}\right)}^{1/q}\end{array}\right]e^{i2\pi \left[\begin{array}{l}{\left(\frac{w_{u_1^{-}}^q+w_{u_2^{-}}^q-w_{u_1^{-}}^q{w}_{u_2^{-}}^q-\left(1-\gamma \right)w_{u_1^{-}}^q{w}_{u_2^{-}}^q}{1-\left(1-\gamma \right)w_{u_1^{-}}^q{w}_{u_2^{-}}^q}\right)}^{1/q},\\ {\left(\frac{w_{u_1^{+}}^q+w_{u_2^{+}}^q-w_{u_1^{+}}^q{w}_{u_2^{+}}^q-\left(1-\gamma \right)w_{u_1^{+}}^q{w}_{u_2^{+}}^q}{1-\left(1-\gamma \right)w_{u_1^{+}}^q{w}_{u_2^{+}}^q}\right)}^{1/q}\end{array}\right]},\\ \left[\begin{array}{l}{\left(\frac{v_1^{-}{v}_2^{-}}{\gamma +\left(1-\gamma \right)\left({\left(v_1^{-}\right)}^q+{\left(v_2^{-}\right)}^q-{\left(v_1^{-}\right)}^q{\left(v_2^{-}\right)}^q\right)}\right)}^{1/q},\\ {\left(\frac{v_1^{+}{v}_2^{+}}{\gamma +\left(1-\gamma \right)\left({\left(v_1^{+}\right)}^q+{\left(v_2^{+}\right)}^q-{\left(v_1^{+}\right)}^q{\left(v_2^{+}\right)}^q\right)}\right)}^{1/q}\end{array}\right]e^{i2\pi \left[\begin{array}{l}{\left(\frac{w_{v_1}^{-}{w}_{v_2}^{-}}{\gamma +\left(1-\gamma \right)\left({\left(w_{v_1}^{-}\right)}^q+{\left(w_{v_2}^{-}\right)}^q-{\left(w_{v_1}^{-}\right)}^q{\left(w_{v_2}^{-}\right)}^q\right)}\right)}^{1/q},\\ {\left(\frac{v_1^{+}{v}_2^{+}}{\gamma +\left(1-\gamma \right)\left({\left(w_{v_1}^{+}\right)}^q+{\left(w_{v_2}^{+}\right)}^q-{\left(w_{v_1}^{+}\right)}^q{\left(w_{v_2}^{+}\right)}^q\right)}\right)}^{1/q}\end{array}\right]}\end{array}\right),\\ \left({\left(\frac{u_1^q+u_2^q-u_1^q{u}_2^q-\left(1-\gamma \right)u_1^q{u}_2^q}{1-\left(1-\gamma \right)u_1^q{u}_2^q}\right)}^{1/q}{e}^{i2\pi {\left(\frac{w_{u_1}^q+w_{u_2}^q-w_{u_1}^q{w}_{u_2}^q-\left(1-\gamma \right)w_{u_1}^q{w}_{u_2}^q}{1-\left(1-\gamma \right)w_{u_1}^q{w}_{u_2}^q}\right)}^{1/q}},{\left(\frac{v_1^{}{v}_2^{}}{\gamma +\left(1-\gamma \right)\left(v_1^q+v_2^q-v_1^q{v}_2^q\right)}\right)}^{1/q}{e}^{i2\pi {\left(\frac{w_{v_1}{w}_{v_2}}{\gamma +\left(1-\gamma \right)\left(w_{v_1}^q+w_{v_2}^q-w_{v_1}^q{w}_{v_2}^q\right)}\right)}^{1/q}}\right)\end{array}\right)\gamma >0$$

$$(2) a_1\otimes a_2=\left(\begin{array}{l}\left(\begin{array}{l}\left[{\left(\frac{u_1^{-}{u}_2^{-}}{\gamma +\left(1-\gamma \right)\left({\left(u_1^{-}\right)}^q+{\left(u_2^{-}\right)}^q-{\left(u_1^{-}\right)}^q{\left(u_2^{-}\right)}^q\right)}\right)}^{1/q},{\left(\frac{u_1^{+}{u}_2^{+}}{\gamma +\left(1-\gamma \right)\left({\left(u_1^{+}\right)}^q+{\left(u_2^{+}\right)}^q-{\left(u_1^{+}\right)}^q{\left(u_2^{+}\right)}^q\right)}\right)}^{1/q}\right]e^{i2\pi \left[\begin{array}{l}{\left(\frac{w_{u_1}^{-}{w}_{u_2}^{-}}{\gamma +\left(1-\gamma \right)\left({\left(w_{u_1}^{-}\right)}^q+{\left(w_{u_2}^{-}\right)}^q-{\left(w_{u_1}^{-}\right)}^q{\left(w_{u_2}^{-}\right)}^q\right)}\right)}^{1/q},\\ {\left(\frac{u_1^{+}{u}_2^{+}}{\gamma +\left(1-\gamma \right)\left({\left(w_{u_1}^{+}\right)}^q+{\left(w_{u_2}^{+}\right)}^q-{\left(w_{u_1}^{+}\right)}^q{\left(w_{u_2}^{+}\right)}^q\right)}\right)}^{1/q}\end{array}\right]}\\ \left[{\left(\frac{{\left(v_1^{-}\right)}^q+{\left(v_2^{-}\right)}^q-{\left(v_1^{-}\right)}^q{\left(v_2^{-}\right)}^q-\left(1-\gamma \right){\left(v_1^{-}\right)}^q{\left(v_2^{-}\right)}^q}{1-\left(1-\gamma \right){\left(v_1^{-}\right)}^q{\left(v_2^{-}\right)}^q}\right)}^{1/q},{\left(\frac{{\left(v_1^{+}\right)}^q+{\left(v_2^{+}\right)}^q-{\left(v_1^{+}\right)}^q{\left(v_2^{+}\right)}^q-\left(1-\gamma \right){\left(v_1^{+}\right)}^q{\left(v_2^{+}\right)}^q}{1-\left(1-\gamma \right){\left(v_1^{+}\right)}^q{\left(v_2^{+}\right)}^q}\right)}^{1/q}\right]e^{i2\pi \left[\begin{array}{l}{\left(\frac{w_{v_1^{-}}^q+w_{v_2^{-}}^q-w_{v_1^{-}}^q{w}_{v_2^{-}}^q-\left(1-\gamma \right)w_{v_1^{-}}^q{w}_{v_2^{-}}^q}{1-\left(1-\gamma \right)w_{v_1^{-}}^q{w}_{v_2^{-}}^q}\right)}^{1/q},\\ {\left(\frac{w_{v_1^{+}}^q+w_{v_2^{+}}^q-w_{v_1^{+}}^q{w}_{v_2^{+}}^q-\left(1-\gamma \right)w_{v_1^{+}}^q{w}_{v_2^{+}}^q}{1-\left(1-\gamma \right)w_{v_1^{+}}^q{w}_{v_2^{+}}^q}\right)}^{1/q}\end{array}\right]},\end{array}\right),\\ \left({\left(\frac{u_1^{}{u}_2^{}}{\gamma +\left(1-\gamma \right)\left({\left(u_1^{}\right)}^q+{\left(u_2^{}\right)}^q-{\left(u_1^{}\right)}^q{\left(u_2^{}\right)}^q\right)}\right)}^{1/q}{e}^{i2\pi {\left(\frac{w_{u_1}^{}{w}_{u_2}^{}}{\gamma +\left(1-\gamma \right)\left({\left(w_{u_1}^{}\right)}^q+{\left(w_{u_2}^{}\right)}^q-{\left(w_{u_1}^{}\right)}^q{\left(w_{u_2}^{}\right)}^q\right)}\right)}^{1/q}},{\left(\frac{{\left(v_1^{}\right)}^q+{\left(v_2^{}\right)}^q-{\left(v_1^{}\right)}^q{\left(v_2^{}\right)}^q-\left(1-\gamma \right){\left(v_1^{}\right)}^q{\left(v_2^{}\right)}^q}{1-\left(1-\gamma \right){\left(v_1^{}\right)}^q{\left(v_2^{}\right)}^q}\right)}^{1/q}{e}^{i2\pi {\left(\frac{w_{v_1^{}}^q+w_{v_2^{}}^q-w_{v_1^{}}^q{w}_{v_2^{}}^q-\left(1-\gamma \right)w_{v_1^{}}^q{w}_{v_2^{}}^q}{1-\left(1-\gamma \right)w_{v_1^{}}^q{w}_{v_2^{}}^q}\right)}^{1/q}}\right)\end{array}\right)\gamma >0$$

$$(3) \lambda a_1=\left(\begin{array}{l}\left(\left[\begin{array}{l}{\left(\frac{{\left(1+\left(\gamma -1\right){\left(u_1^{-}\right)}^q\right)}^{\lambda }-{\left(1-{\left(u_1^{-}\right)}^q\right)}^{\lambda }}{{\left(1+\left(\gamma -1\right){\left(u_1^{-}\right)}^q\right)}^{\lambda }+\left(\gamma -1\right){\left(1-{\left(u_1^{-}\right)}^q\right)}^{\lambda }}\right)}^{1/q},\\ {\left(\frac{{\left(1+\left(\gamma -1\right){\left(u_1^{+}\right)}^q\right)}^{\lambda }-{\left(1-{\left(u_1^{+}\right)}^q\right)}^{\lambda }}{{\left(1+\left(\gamma -1\right){\left(u_1^{+}\right)}^q\right)}^{\lambda }+\left(\gamma -1\right){\left(1-{\left(u_1^{+}\right)}^q\right)}^{\lambda }}\right)}^{1/q}\end{array}\right]e^{i2\pi \left[\begin{array}{l}{\left(\frac{{\left(1+\left(\gamma -1\right)w_{{}^{u_1^{-}}}^q\right)}^{\lambda }-{\left(1-w_{{}^{u_1^{-}}}^q\right)}^{\lambda }}{{\left(1+\left(\gamma -1\right)w_{{}^{u_1^{-}}}^q\right)}^{\lambda }+\left(\gamma -1\right){\left(1-w_{{}^{u_1^{-}}}^q\right)}^{\lambda }}\right)}^{1/q},\\ {\left(\frac{{\left(1+\left(\gamma -1\right)w_{{}^{u_1^{+}}}^q\right)}^{\lambda }-{\left(1-w_{{}^{u_1^{+}}}^q\right)}^{\lambda }}{{\left(1+\left(\gamma -1\right)w_{{}^{u_1^{+}}}^q\right)}^{\lambda }+\left(\gamma -1\right){\left(1-w_{{}^{u_1^{+}}}^q\right)}^{\lambda }}\right)}^{1/q}\end{array}\right]},\left[\begin{array}{l}\frac{{\gamma }^{1/q}{\left(v_1^{-}\right)}^{\lambda }}{{\left({\left(1+\left(\gamma -1\right)\left(1-{\left(v_1^{-}\right)}^q\right)\right)}^{\lambda }+\left(\gamma -1\right){\left(v_1^{-}\right)}^{q\lambda }\right)}^{1/q}},\\ \frac{{\gamma }^{1/q}{\left(v_1^{+}\right)}^{\lambda }}{{\left({\left(1+\left(\gamma -1\right)\left(1-{\left(v_1^{+}\right)}^q\right)\right)}^{\lambda }+\left(\gamma -1\right){\left(v_1^{+}\right)}^{q\lambda }\right)}^{1/q}}\end{array}\right]e^{i2\pi \left[\begin{array}{l}\frac{{\gamma }^{1/q}{\left(w_{v_1^{-}}\right)}^{\lambda }}{{\left({\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_1^{-}}\right)}^q\right)\right)}^{\lambda }+\left(\gamma -1\right){\left(w_{v_1^{-}}\right)}^{q\lambda }\right)}^{1/q}},\\ \frac{{\gamma }^{1/q}{\left(v_1^{+}\right)}^{\lambda }}{{\left({\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_1^{+}}\right)}^q\right)\right)}^{\lambda }+\left(\gamma -1\right){\left(w_{v_1^{+}}\right)}^{q\lambda }\right)}^{1/q}}\end{array}\right]}\right),\\ \left({\left(\frac{{\left(1+\left(\gamma -1\right)u_1^q\right)}^{\lambda }-{\left(1-u_1^q\right)}^{\lambda }}{{\left(1+\left(\gamma -1\right)u_1^q\right)}^{\lambda }+\left(\gamma -1\right){\left(1-u_1^q\right)}^{\lambda }}\right)}^{1/q}{e}^{i2\pi {\left(\frac{{\left(1+\left(\gamma -1\right)w_{u_1}^q\right)}^{\lambda }-{\left(1-w_{u_1}^q\right)}^{\lambda }}{{\left(1+\left(\gamma -1\right)w_{u_1}^q\right)}^{\lambda }+\left(\gamma -1\right){\left(1-w_{u_1}^q\right)}^{\lambda }}\right)}^{1/q}},\frac{{\gamma }^{1/q}{\left(v_1^{}\right)}^{\lambda }}{{\left({\left(1+\left(\gamma -1\right)\left(1-{\left(v_1^{}\right)}^q\right)\right)}^{\lambda }+\left(\gamma -1\right){\left(v_1^{}\right)}^{q\lambda }\right)}^{1/q}}{e}^{i2\pi \frac{{\gamma }^{1/q}{\left(w_{v_1}\right)}^{\lambda }}{{\left({\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_1}\right)}^q\right)\right)}^{\lambda }+\left(\gamma -1\right){\left(w_{v_1}\right)}^{q\lambda }\right)}^{1/q}}}\right)\end{array}\right)\gamma >0$$

$$(4) a_1^{\lambda }=\left(\begin{array}{l}\left(\left[\begin{array}{l}\frac{{\gamma }^{1/q}{\left(u_1^{-}\right)}^{\lambda }}{{\left({\left(1+\left(\gamma -1\right)\left(1-{\left(u_1^{-}\right)}^q\right)\right)}^{\lambda }+\left(\gamma -1\right){\left(u_1^{-}\right)}^{q\lambda }\right)}^{1/q}},\\ \frac{{\gamma }^{1/q}{\left(u_1^{+}\right)}^{\lambda }}{{\left({\left(1+\left(\gamma -1\right)\left(1-{\left(u_1^{+}\right)}^q\right)\right)}^{\lambda }+\left(\gamma -1\right){\left(u_1^{+}\right)}^{q\lambda }\right)}^{1/q}}\end{array}\right]e^{i2\pi \left[\begin{array}{l}\frac{{\gamma }^{1/q}{\left(w_{u_1^{-}}\right)}^{\lambda }}{{\left({\left(1+\left(\gamma -1\right)\left(1-{\left(w_{u_1^{-}}\right)}^q\right)\right)}^{\lambda }+\left(\gamma -1\right){\left(w_{u_1^{-}}\right)}^{q\lambda }\right)}^{1/q}},\\ \frac{{\gamma }^{1/q}{\left(u_1^{+}\right)}^{\lambda }}{{\left({\left(1+\left(\gamma -1\right)\left(1-{\left(w_{u_1^{+}}\right)}^q\right)\right)}^{\lambda }+\left(\gamma -1\right){\left(w_{u_1^{+}}\right)}^{q\lambda }\right)}^{1/q}}\end{array}\right]},\left[\begin{array}{l}{\left(\frac{{\left(1+\left(\gamma -1\right){\left(v_1^{-}\right)}^q\right)}^{\lambda }-{\left(1-{\left(v_1^{-}\right)}^q\right)}^{\lambda }}{{\left(1+\left(\gamma -1\right){\left(v_1^{-}\right)}^q\right)}^{\lambda }+\left(\gamma -1\right){\left(1-{\left(v_1^{-}\right)}^q\right)}^{\lambda }}\right)}^{1/q},\\ {\left(\frac{{\left(1+\left(\gamma -1\right){\left(v_1^{+}\right)}^q\right)}^{\lambda }-{\left(1-{\left(v_1^{+}\right)}^q\right)}^{\lambda }}{{\left(1+\left(\gamma -1\right){\left(v_1^{+}\right)}^q\right)}^{\lambda }+\left(\gamma -1\right){\left(1-{\left(v_1^{+}\right)}^q\right)}^{\lambda }}\right)}^{1/q}\end{array}\right]e^{i2\pi \left[\begin{array}{l}{\left(\frac{{\left(1+\left(\gamma -1\right)w_{{}^{v_1^{-}}}^q\right)}^{\lambda }-{\left(1-w_{{}^{v_1^{-}}}^q\right)}^{\lambda }}{{\left(1+\left(\gamma -1\right)w_{{}^{v_1^{-}}}^q\right)}^{\lambda }+\left(\gamma -1\right){\left(1-w_{{}^{v_1^{-}}}^q\right)}^{\lambda }}\right)}^{1/q},\\ {\left(\frac{{\left(1+\left(\gamma -1\right)w_{{}^{v_1^{+}}}^q\right)}^{\lambda }-{\left(1-w_{{}^{v_1^{+}}}^q\right)}^{\lambda }}{{\left(1+\left(\gamma -1\right)w_{{}^{v_1^{+}}}^q\right)}^{\lambda }+\left(\gamma -1\right){\left(1-w_{{}^{v_1^{+}}}^q\right)}^{\lambda }}\right)}^{1/q}\end{array}\right]}\right),\\ \left(\frac{{\gamma }^{1/q}{\left(u_1^{}\right)}^{\lambda }}{{\left({\left(1+\left(\gamma -1\right)\left(1-{\left(u_1^{}\right)}^q\right)\right)}^{\lambda }+\left(\gamma -1\right){\left(u_1^{}\right)}^{q\lambda }\right)}^{1/q}}{e}^{i2\pi \frac{{\gamma }^{1/q}{\left(w_{u_1}\right)}^{\lambda }}{{\left({\left(1+\left(\gamma -1\right)\left(1-{\left(w_{u_1}\right)}^q\right)\right)}^{\lambda }+\left(\gamma -1\right){\left(w_{u_1}\right)}^{q\lambda }\right)}^{1/q}}},{\left(\frac{{\left(1+\left(\gamma -1\right)v_1^q\right)}^{\lambda }-{\left(1-v_1^q\right)}^{\lambda }}{{\left(1+\left(\gamma -1\right)v_1^q\right)}^{\lambda }+\left(\gamma -1\right){\left(1-v_1^q\right)}^{\lambda }}\right)}^{1/q}{e}^{i2\pi {\left(\frac{{\left(1+\left(\gamma -1\right)w_{v_1}^q\right)}^{\lambda }-{\left(1-w_{v_1}^q\right)}^{\lambda }}{{\left(1+\left(\gamma -1\right)w_{v_1}^q\right)}^{\lambda }+\left(\gamma -1\right){\left(1-w_{v_1}^q\right)}^{\lambda }}\right)}^{1/q}}\right)\end{array}\right)\gamma >0$$

3.2. Complex Cubic q-Rung Orthopair Fuzzy Hamacher Average Operators

In this section, we expand the application of the averaging operator to the CCuq-ROF environment and propose two new aggregation operators based on the Hamacher operations. The first operator introduced is the complex cubic q-rung orthopair fuzzy Hamacher average (CCuq-ROFHA) operator, and the second one is the weighted complex cubic q-rung orthopair fuzzy Hamacher average (WCCuq-ROFHA) operator.

3.2.1. Complex Cubic q-Rung Orthopair Fuzzy Hamacher Average (CCuq-ROFHA) Operator

Based on Definitions 4 and 9, we hereby present the definition of the complex cubic q-rung orthopair fuzzy Hamacher average (CCuq-ROFHA) operator as follows:

Definition 10. 

Let  $a_i=\left(\left(\left[u_i^{-},u_i^{+}\right]e^{i2\pi \left[w_{u_i}^{-},w_{u_i}^{+}\right]},\left[v_i^{-},v_i^{+}\right]e^{i2\pi \left[w_{v_i}^{-},w_{v_i}^{+}\right]}\right),\left(u_i{e}^{i2\pi w_{u_i}},v_i{e}^{i2\pi w_{v_i}}\right)\right)$   $\left(i=1,2,\dots ,n\right)$  be a collection of CCuq-ROFNs, then the complex cubic q-rung orthopair fuzzy Hamacher averaging (CCuq-ROFHA) operator is defined as follows:

$$CCuq-ROFHA\left(a_1,a_2,\dots ,a_n\right)=\underset{i=1}{\overset{n}{\oplus }}\left(\frac{1}{n}\left(a_i\right)\right)$$

(7)

Based on the CCuq-ROF Hamacher operations, we derive the following theorem from Definition 10.

Theorem 2. 

Let  $a_i=\left(\left(\left[u_i^{-},u_i^{+}\right]e^{i2\pi \left[w_{u_i}^{-},w_{u_i}^{+}\right]},\left[v_i^{-},v_i^{+}\right]e^{i2\pi \left[w_{v_i}^{-},w_{v_i}^{+}\right]}\right),\left(u_i{e}^{i2\pi w_{u_i}},v_i{e}^{i2\pi w_{v_i}}\right)\right)$   $\left(i=1,2,\dots ,n\right)$  be a collection of CCuq-ROFNs, Then, for any  $q\ge 1$  ,γ > 0, the aggregated value found using the CCuq-ROFHA operator is also a CCuq-ROFN, and

$$\begin{array}{l}CCuq-ROFHA\left(a_1,a_2,\dots ,a_n\right)=\underset{i=1}{\overset{n}{\oplus }}\left(\frac{1}{n}\left(a_i\right)\right)\\ =\left(\begin{array}{l}\left(\begin{array}{l}\left[\begin{array}{l}{\left(\frac{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(u_i^{-}\right)}^q\right)}^{\frac{1}{n}}}-{\displaystyle {\prod }_{i=1}^n{\left(1-{\left(u_i^{-}\right)}^q\right)}^{\frac{1}{n}}}}{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(u_i^{-}\right)}^q\right)}^{\frac{1}{n}}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(1-{\left(u_i^{-}\right)}^q\right)}^{\frac{1}{n}}}}\right)}^{1/q},\\ {\left(\frac{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(u_i^{+}\right)}^q\right)}^{\frac{1}{n}}}-{\displaystyle {\prod }_{i=1}^n{\left(1-{\left(u_i^{+}\right)}^q\right)}^{\frac{1}{n}}}}{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(u_i^{+}\right)}^q\right)}^{\frac{1}{n}}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(1-{\left(u_i^{+}\right)}^q\right)}^{\frac{1}{n}}}}\right)}^{1/q}\end{array}\right]e^{i2\pi \left[{\left(\frac{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(w_{u_i^{-}}\right)}^q\right)}^{\frac{1}{n}}}-{\displaystyle {\prod }_{i=1}^n{\left(1-{\left(w_{u_i^{-}}\right)}^q\right)}^{\frac{1}{n}}}}{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(w_{u_i^{-}}\right)}^q\right)}^{\frac{1}{n}}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(1-{\left(w_{u_i^{-}}\right)}^q\right)}^{\frac{1}{n}}}}\right)}^{1/q},{\left(\frac{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(w_{u_i^{+}}\right)}^q\right)}^{\frac{1}{n}}}-{\displaystyle {\prod }_{i=1}^n{\left(1-{\left(w_{u_i^{+}}\right)}^q\right)}^{\frac{1}{n}}}}{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(w_{u_i^{+}}\right)}^q\right)}^{\frac{1}{n}}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(1-{\left(w_{u_i^{+}}\right)}^q\right)}^{\frac{1}{n}}}}\right)}^{1/q}\right]},\\ \left[\begin{array}{l}\frac{{\gamma }^{1/q}{\displaystyle {\prod }_{i=1}^n{\left(v_i^{-}\right)}^{\frac{1}{n}}}}{{\left({\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right)\left(1-{\left(v_i^{-}\right)}^q\right)\right)}^{\frac{1}{n}}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(v_i^{-}\right)}^{q\frac{1}{n}}}\right)}^{1/q}},\\ \frac{{\gamma }^{1/q}{\displaystyle {\prod }_{i=1}^n{\left(v_i^{+}\right)}^{\frac{1}{n}}}}{{\left({\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right)\left(1-{\left(v_i^{+}\right)}^q\right)\right)}^{\frac{1}{n}}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(v_i^{+}\right)}^{q\frac{1}{n}}}\right)}^{1/q}}\end{array}\right]e^{i2\pi \left[\frac{{\gamma }^{1/q}{\displaystyle {\prod }_{i=1}^n{\left(w_{v_i^{-}}\right)}^{\frac{1}{n}}}}{{\left({\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_i^{-}}\right)}^q\right)\right)}^{\frac{1}{n}}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(w_{v_i^{-}}\right)}^{q\frac{1}{n}}}\right)}^{1/q}},\frac{{\gamma }^{1/q}{\displaystyle {\prod }_{i=1}^n{\left(w_{v_i^{+}}\right)}^{\frac{1}{n}}}}{{\left({\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_i^{+}}\right)}^q\right)\right)}^{\frac{1}{n}}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(w_{v_i^{+}}\right)}^{q\frac{1}{n}}}\right)}^{1/q}}\right]}\end{array}\right),\\ \left(\begin{array}{l}{\left(\frac{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(u_i^{}\right)}^q\right)}^{\frac{1}{n}}}-{\displaystyle {\prod }_{i=1}^n{\left(1-{\left(u_i^{}\right)}^q\right)}^{\frac{1}{n}}}}{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(u_i^{}\right)}^q\right)}^{\frac{1}{n}}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(1-{\left(u_i^{}\right)}^q\right)}^{\frac{1}{n}}}}\right)}^{1/q}{e}^{i2\pi {\left(\frac{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(w_{u_i^{}}\right)}^q\right)}^{\frac{1}{n}}}-{\displaystyle {\prod }_{i=1}^n{\left(1-{\left(w_{u_i^{}}\right)}^q\right)}^{\frac{1}{n}}}}{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(w_{u_i^{}}\right)}^q\right)}^{\frac{1}{n}}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(1-{\left(w_{u_i^{}}\right)}^q\right)}^{\frac{1}{n}}}}\right)}^{1/q}},\\ \frac{{\gamma }^{1/q}{\displaystyle {\prod }_{i=1}^n{\left(v_i^{}\right)}^{\frac{1}{n}}}}{{\left({\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right)\left(1-{\left(v_i^{}\right)}^q\right)\right)}^{\frac{1}{n}}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(v_i^{}\right)}^{q\frac{1}{n}}}\right)}^{1/q}}{e}^{i2\pi \frac{{\gamma }^{1/q}{\displaystyle {\prod }_{i=1}^n{\left(w_{v_i^{}}\right)}^{\frac{1}{n}}}}{{\left({\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_i^{}}\right)}^q\right)\right)}^{\frac{1}{n}}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(w_{v_i^{}}\right)}^{q\frac{1}{n}}}\right)}^{1/q}}}\end{array}\right)\end{array}\right)\end{array}$$

(8)

Property 1 (Idempotency). 

Let $a_i=\left(\left(\left[u_i^{-},u_i^{+}\right]e^{i2\pi \left[w_{u_i}^{-},w_{u_i}^{+}\right]},\left[v_i^{-},v_i^{+}\right]e^{i2\pi \left[w_{v_i}^{-},w_{v_i}^{+}\right]}\right),\left(u_i{e}^{i2\pi w_{u_i}},v_i{e}^{i2\pi w_{v_i}}\right)\right)$ $\left(i=1,2,\dots ,n\right)$  be a collection of all CCuq-ROFNs. If all $a_i$ $\left(i=1,2,\dots ,n\right)$  are equal, i.e., $a_i=a=\left(\left(\left[u_{}^{-},u_{}^{+}\right]e^{i2\pi \left[w_u^{-},w_u^{+}\right]},\left[v_{}^{-},v_{}^{+}\right]e^{i2\pi \left[w_v^{-},w_v^{+}\right]}\right),\left(u{e}^{i2\pi w_u},v{e}^{i2\pi w_v}\right)\right)$  for all $i$ , then

$$CCuq-ROFHA\left(a_1,a_2,\dots ,a_i\right)=a.$$

Property 2 (Commutativity). 

Let $a_i=\left(\left(\left[u_i^{-},u_i^{+}\right]e^{i2\pi \left[w_{u_i}^{-},w_{u_i}^{+}\right]},\left[v_i^{-},v_i^{+}\right]e^{i2\pi \left[w_{v_i}^{-},w_{v_i}^{+}\right]}\right),\left(u_i{e}^{i2\pi w_{u_i}},v_i{e}^{i2\pi w_{v_i}}\right)\right)$ $\left(i=1,2,\dots ,n\right)$  be a collection of all CCuq-ROFNs. If $a_i^{\prime }$  is any permutation of $a_i$  $\left(i=1,2,\dots ,n\right)$  then we have the following relationship:

$$CCuq-ROFHA\left(a_1,a_2,\dots ,a_i\right)=CCuq-ROFHA\left(a_1^{\prime },a_2^{\prime },\dots ,a_i^{\prime }\right).$$

3.2.2. Weighted Complex Cubic q-Rung Orthopair Fuzzy Hamacher Average (WCCuq-ROFHA) Operator

In order to address real-life problems and consider the importance of the aggregated inputs, based on Definition 4 and the Hamacher operational laws of Definition 11, we define the weighted complex cubic q-rung orthopair fuzzy Hamacher average (WCCuq-ROFHA) operator as follows:

Definition 11. 

Let $a_i=\left(\left(\left[u_i^{-},u_i^{+}\right]e^{i2\pi \left[w_{u_i}^{-},w_{u_i}^{+}\right]},\left[v_i^{-},v_i^{+}\right]e^{i2\pi \left[w_{v_i}^{-},w_{v_i}^{+}\right]}\right),\left(u_i{e}^{i2\pi w_{u_i}},v_i{e}^{i2\pi w_{v_i}}\right)\right)$ $\left(i=1,2,\dots ,n\right)$  be a collection of CCuq-ROFNs, $\omega =\left({\omega }_1,{\omega }_2,\dots ,{\omega }_n\right)$  is the weight vector, where ${\omega }_i$  indicates the importance degree of $a_i$, satisfying $\omega \in \left[0,1\right]$  and ${\displaystyle \sum _{\mathrm{i}}^n{\omega }_i}=1$. Then weighted q-rung orthopair fuzzy Hamacher average (WCCuq-ROFHA) operator is defined as follows:

$$WCCuq-ROFHA\left(a_1,a_2,\dots ,a_n\right)=\underset{i=1}{\overset{n}{\oplus }}\left({\omega }_i{a}_i\right)$$

(9)

With the help of the WCCuq-ROF Hamacher operations, we derive the following theorem from Definition 13:

Theorem 3. 

Let  $a_i=\left(\left(\left[u_i^{-},u_i^{+}\right]e^{i2\pi \left[w_{u_i}^{-},w_{u_i}^{+}\right]},\left[v_i^{-},v_i^{+}\right]e^{i2\pi \left[w_{v_i}^{-},w_{v_i}^{+}\right]}\right),\left(u_i{e}^{i2\pi w_{u_i}},v_i{e}^{i2\pi w_{v_i}}\right)\right)$   $\left(i=1,2,\dots ,n\right)$  be a collection of CCuq-ROFNs, Then, for any  $q\ge 1$  ,  $\gamma$  > 0, the aggregated value found using the WCCuq-ROFHA operator is also a CCuq-ROFN, and

$$\begin{array}{l}WCCuq-ROFHA\left(a_1,a_2,\dots ,a_n\right)=\underset{i=1}{\overset{n}{\oplus }}\left({\omega }_i{a}_i\right)\\ =\left(\begin{array}{l}\left(\begin{array}{l}\left[{\left(\frac{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(u_i^{-}\right)}^q\right)}^{{\omega }_i}}-{\displaystyle {\prod }_{i=1}^n{\left(1-{\left(u_i^{-}\right)}^q\right)}^{{\omega }_i}}}{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(u_i^{-}\right)}^q\right)}^{{\omega }_i}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(1-{\left(u_i^{-}\right)}^q\right)}^{{\omega }_i}}}\right)}^{1/q},{\left(\frac{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(u_i^{+}\right)}^q\right)}^{{\omega }_i}}-{\displaystyle {\prod }_{i=1}^n{\left(1-{\left(u_i^{+}\right)}^q\right)}^{{\omega }_i}}}{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(u_i^{+}\right)}^q\right)}^{{\omega }_i}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(1-{\left(u_i^{+}\right)}^q\right)}^{{\omega }_i}}}\right)}^{1/q}\right]e^{i2\pi \left[\begin{array}{l}{\left(\frac{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(w_{u_i^{-}}\right)}^q\right)}^{{\omega }_i}}-{\displaystyle {\prod }_{i=1}^n{\left(1-{\left(w_{u_i^{-}}\right)}^q\right)}^{{\omega }_i}}}{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(w_{u_i^{-}}\right)}^q\right)}^{{\omega }_i}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(1-{\left(w_{u_i^{-}}\right)}^q\right)}^{{\omega }_i}}}\right)}^{1/q},\\ {\left(\frac{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(w_{u_i^{+}}\right)}^q\right)}^{{\omega }_i}}-{\displaystyle {\prod }_{i=1}^n{\left(1-{\left(w_{u_i^{+}}\right)}^q\right)}^{{\omega }_i}}}{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(w_{u_i^{+}}\right)}^q\right)}^{{\omega }_i}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(1-{\left(w_{u_i^{+}}\right)}^q\right)}^{{\omega }_i}}}\right)}^{1/q}\end{array}\right]},\\ \left[\frac{{\gamma }^{1/q}{\displaystyle {\prod }_{i=1}^n{\left(v_i^{-}\right)}^{{\omega }_i}}}{{\left({\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right)\left(1-{\left(v_i^{-}\right)}^q\right)\right)}^{{\omega }_i}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(v_i^{-}\right)}^{q{\omega }_i}}\right)}^{1/q}},\frac{{\gamma }^{1/q}{\displaystyle {\prod }_{i=1}^n{\left(v_i^{+}\right)}^{{\omega }_i}}}{{\left({\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right)\left(1-{\left(v_i^{+}\right)}^q\right)\right)}^{{\omega }_i}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(v_i^{+}\right)}^{q{\omega }_i}}\right)}^{1/q}}\right]e^{i2\pi \left[\begin{array}{l}\frac{{\gamma }^{1/q}{\displaystyle {\prod }_{i=1}^n{\left(w_{v_i^{-}}\right)}^{{\omega }_i}}}{{\left({\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_i^{-}}\right)}^q\right)\right)}^{{\omega }_i}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(w_{v_i^{-}}\right)}^{q{\omega }_i}}\right)}^{1/q}},\\ \frac{{\gamma }^{1/q}{\displaystyle {\prod }_{i=1}^n{\left(w_{v_i^{+}}\right)}^{{\omega }_i}}}{{\left({\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_i^{+}}\right)}^q\right)\right)}^{{\omega }_i}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(w_{v_i^{+}}\right)}^{q{\omega }_i}}\right)}^{1/q}}\end{array}\right]}\end{array}\right),\\ \left({\left(\frac{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(u_i^{}\right)}^q\right)}^{{\omega }_i}}-{\displaystyle {\prod }_{i=1}^n{\left(1-{\left(u_i^{}\right)}^q\right)}^{{\omega }_i}}}{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(u_i^{}\right)}^q\right)}^{{\omega }_i}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(1-{\left(u_i^{}\right)}^q\right)}^{{\omega }_i}}}\right)}^{1/q}{e}^{i2\pi {\left(\frac{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(w_{u_i^{}}\right)}^q\right)}^{{\omega }_i}}-{\displaystyle {\prod }_{i=1}^n{\left(1-{\left(w_{u_i^{}}\right)}^q\right)}^{{\omega }_i}}}{{\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right){\left(w_{u_i^{}}\right)}^q\right)}^{{\omega }_i}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(1-{\left(w_{u_i^{}}\right)}^q\right)}^{{\omega }_i}}}\right)}^{1/q}},\frac{{\gamma }^{1/q}{\displaystyle {\prod }_{i=1}^n{\left(v_i^{}\right)}^{{\omega }_i}}}{{\left({\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right)\left(1-{\left(v_i^{}\right)}^q\right)\right)}^{{\omega }_i}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(v_i^{}\right)}^{q{\omega }_i}}\right)}^{1/q}}{e}^{i2\pi \frac{{\gamma }^{1/q}{\displaystyle {\prod }_{i=1}^n{\left(w_{v_i^{}}\right)}^{{\omega }_i}}}{{\left({\displaystyle {\prod }_{i=1}^n{\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_i^{}}\right)}^q\right)\right)}^{{\omega }_i}}+\left(\gamma -1\right){\displaystyle {\prod }_{i=1}^n{\left(w_{v_i^{}}\right)}^{q{\omega }_i}}\right)}^{1/q}}}\right)\end{array}\right)\end{array}$$

(10)

3.3. Complex Cubic q-Rung Orthopair Fuzzy Hamacher Maclaurin Symmetric Mean Operators

In this part, the MSM operator is modified to accommodate the CCuq-ROF environment according to the Hamacher operations. Two new aggregation operators which are complex cubic q-rung orthopair fuzzy Hamacher Maclaurin symmetric mean (CCuq-ROFHMSM) operator and weighted complex cubic q-rung orthopair fuzzy Hamacher Maclaurin symmetric mean (WCCuq-ROFHMSM) operator are proposed.

3.3.1. Complex Cubic q-Rung Orthopair Fuzzy Hamacher Maclaurin Symmetric Mean (CCuq-ROFHMSM) Operator

Based on Definitions 4 and 9, the definition of the complex cubic qrung orthopair fuzzy Hamacher Maclaurin symmetric mean (CCuq-ROFHMSM) is defined as follows:

Definition 12. 

Let $a_i=\left(\left(\left[u_i^{-},u_i^{+}\right]e^{i2\pi \left[w_{u_i}^{-},w_{u_i}^{+}\right]},\left[v_i^{-},v_i^{+}\right]e^{i2\pi \left[w_{v_i}^{-},w_{v_i}^{+}\right]}\right),\left(u_i{e}^{i2\pi w_{u_i}},v_i{e}^{i2\pi w_{v_i}}\right)\right)$ $\left(i=1,2,\dots ,n\right)$  be a collection of CCuq-ROFNs, then the complex cubic q-rung orthopair fuzzy Hamacher Maclaurin symmetric mean (CCuq-ROFHMSM) operator is defined as follows:

$$CCuq-ROFHMSM\left(a_1,a_2,\dots ,a_n\right)={\left(\frac{{\oplus }_{1\le i_1\le \dots \le i_k\le n}\left({\otimes }_{j=1}^k{a}_{i_j}\right)}{C_n^k}\right)}^{\frac{1}{k}}$$

(11)

where  $\left(i_1,i_2,\dots ,i_k\right)$  traverse all the  $k$-tuple combinations of  $\left(1,2,\dots ,n\right)$, and  $C_n^k$  is the binomial coefficient.

In light of the CCuq-ROFN Hamacher operations, the following propositions from Definition 12 can be deduced:

Proposition 1. 

Let $a_i=\left(\left(\left[u_i^{-},u_i^{+}\right]e^{i2\pi \left[w_{u_i}^{-},w_{u_i}^{+}\right]},\left[v_i^{-},v_i^{+}\right]e^{i2\pi \left[w_{v_i}^{-},w_{v_i}^{+}\right]}\right),\left(u_i{e}^{i2\pi w_{u_i}},v_i{e}^{i2\pi w_{v_i}}\right)\right)$ $\left(i=1,2,\dots ,n\right)$  be a collection of CCuq-ROFNs, where $k=1,2,\dots ,n$  and $\gamma$  > 0. Then, for any $q\ge 1$, we have

$$\begin{array}{l}\underset{j=1}{\overset{k}{\otimes }}{a}_{i_j}\\ =\left(\begin{array}{l}\left(\begin{array}{l}\left[\frac{\sqrt[q]{\gamma }{\displaystyle {\prod }_{j=1}^k\left(u_{i_j}^{-}\right)}}{\sqrt[q]{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right)\left(1-{\left(u_{i_j}^{-}\right)}^q\right)\right)+\left(\gamma -1\right){{\displaystyle {\prod }_{j=1}^k\left(u_{i_j}^{-}\right)}}^q}}},\frac{\sqrt[q]{\gamma }{\displaystyle {\prod }_{j=1}^k\left(u_{i_j}^{+}\right)}}{\sqrt[q]{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right)\left(1-{\left(u_{i_j}^{+}\right)}^q\right)\right)+\left(\gamma -1\right){{\displaystyle {\prod }_{j=1}^k\left(u_{i_j}^{+}\right)}}^q}}}\right]e^{i2\pi \left[\begin{array}{l}\frac{\sqrt[q]{\gamma }{\displaystyle {\prod }_{j=1}^k\left(w_{u_{i_j}}^{-}\right)}}{\sqrt[q]{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right)\left(1-{\left(w_{u_{i_j}}^{-}\right)}^q\right)\right)+\left(\gamma -1\right){{\displaystyle {\prod }_{j=1}^k\left(w_{u_{i_j}}^{-}\right)}}^q}}},\\ \frac{\sqrt[q]{\gamma }{\displaystyle {\prod }_{j=1}^k\left(w_{u_{i_j}}^{+}\right)}}{\sqrt[q]{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right)\left(1-{\left(w_{u_{i_j}}^{+}\right)}^q\right)\right)+\left(\gamma -1\right){{\displaystyle {\prod }_{j=1}^k\left(w_{u_{i_j}}^{+}\right)}}^q}}}\end{array}\right]},\\ \left[\sqrt[q]{\frac{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(v_{i_j}^{-}\right)}^q\right)-{\displaystyle {\prod }_{j=1}^k\left(1-{\left(v_{i_j}^{-}\right)}^q\right)}}}{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(v_{i_j}^{-}\right)}^q\right)+\left(\gamma -1\right){\displaystyle {\prod }_{j=1}^k\left(1-{\left(v_{i_j}^{-}\right)}^q\right)}}}},\sqrt[q]{\frac{{\displaystyle {\prod }_{j=1}^3\left(1+\left(\gamma -1\right){\left(v_{i_j}^{+}\right)}^q\right)-{\displaystyle {\prod }_{j=1}^3\left(1-{\left(v_{i_j}^{+}\right)}^q\right)}}}{{\displaystyle {\prod }_{j=1}^3\left(1+\left(\gamma -1\right){\left(v_{i_j}^{+}\right)}^q\right)+\left(\gamma -1\right){\displaystyle {\prod }_{j=1}^3\left(1-{\left(v_{i_j}^{+}\right)}^q\right)}}}}\right]e^{i2\pi \left[\begin{array}{l}\sqrt[q]{\frac{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(w_{v_{i_j}}^{-}\right)}^q\right)-{\displaystyle {\prod }_{j=1}^k\left(1-{\left(w_{v_{i_j}}^{-}\right)}^q\right)}}}{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(w_{v_{i_j}}^{-}\right)}^q\right)+\left(\gamma -1\right){\displaystyle {\prod }_{j=1}^k\left(1-{\left(w_{v_{i_j}}^{-}\right)}^q\right)}}}},\\ \sqrt[q]{\frac{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(w_{v_{i_j}}^{+}\right)}^q\right)-{\displaystyle {\prod }_{j=1}^k\left(1-{\left(w_{v_{i_j}}^{+}\right)}^q\right)}}}{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(w_{v_{i_j}}^{+}\right)}^q\right)+\left(\gamma -1\right){\displaystyle {\prod }_{j=1}^k\left(1-{\left(w_{v_{i_j}}^{+}\right)}^q\right)}}}}\end{array}\right]}\end{array}\right),\\ \left(\frac{\sqrt[q]{\gamma }{\displaystyle {\prod }_{j=1}^k\left(u_{i_j}^{}\right)}}{\sqrt[q]{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right)\left(1-{\left(u_{i_j}^{}\right)}^q\right)\right)+\left(\gamma -1\right){{\displaystyle {\prod }_{j=1}^k\left(u_{i_j}^{}\right)}}^q}}}{e}^{i2\pi \frac{\sqrt[q]{\gamma }{\displaystyle {\prod }_{j=1}^k\left(w_{u_{i_j}}^{}\right)}}{\sqrt[q]{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right)\left(1-{\left(w_{u_{i_j}}^{}\right)}^q\right)\right)+\left(\gamma -1\right){{\displaystyle {\prod }_{j=1}^k\left(w_{u_{i_j}}^{}\right)}}^q}}}},\sqrt[q]{\frac{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(v_{i_j}^{}\right)}^q\right)-{\displaystyle {\prod }_{j=1}^k\left(1-{\left(v_{i_j}^{}\right)}^q\right)}}}{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(v_{i_j}^{}\right)}^q\right)+\left(\gamma -1\right){\displaystyle {\prod }_{j=1}^k\left(1-{\left(v_{i_j}^{}\right)}^q\right)}}}}{e}^{i2\pi \sqrt[q]{\frac{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(w_{v_{i_j}}^{}\right)}^q\right)-{\displaystyle {\prod }_{j=1}^k\left(1-{\left(w_{v_{i_j}}^{}\right)}^q\right)}}}{{\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(w_{v_{i_j}}^{}\right)}^q\right)+\left(\gamma -1\right){\displaystyle {\prod }_{j=1}^k\left(1-{\left(w_{v_{i_j}}^{}\right)}^q\right)}}}}}\right)\end{array}\right)\end{array}$$

(12)

Proof. 

The proof of Proposition 1 is presented in the “Appendix A”. □

Proposition 2. 

Let  $a_i=\left(\left(\left[u_i^{-},u_i^{+}\right]e^{i2\pi \left[w_{u_i}^{-},w_{u_i}^{+}\right]},\left[v_i^{-},v_i^{+}\right]e^{i2\pi \left[w_{v_i}^{-},w_{v_i}^{+}\right]}\right),\left(u_i{e}^{i2\pi w_{u_i}},v_i{e}^{i2\pi w_{v_i}}\right)\right)$   $\left(i=1,2,\dots ,n\right)$  be a collection of CCuq-ROFNs, where  $k=1,2,\dots ,n$  and  $\gamma$  > 0. Then, for any  $q\ge 1$  , we have

$$\frac{1}{C_n^k}\left(\underset{1\le i_1\le \dots \le i_k\le n}{\oplus }\left(\underset{j=1}{\overset{k}{\otimes }}{a}_{i_j}\right)\right)=\left(\begin{array}{l}\left(\begin{array}{l}\left[\begin{array}{l}\sqrt[q]{\frac{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}+\left({\gamma }^2-1\right)f^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}-f^{-}\right)}^{\frac{1}{c_n^k}}}}}{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}+\left({\gamma }^2-1\right)f^{-}\right)}^{\frac{1}{c_n^k}}+\left(\gamma -1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}-f^{-}\right)}^{\frac{1}{c_n^k}}}}},}\\ \sqrt[q]{\frac{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}+\left({\gamma }^2-1\right)f^{+}+\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}-f^{+}\right)}^{\frac{1}{c_n^k}}}}}{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}+\left({\gamma }^2-1\right)f^{+}\right)}^{\frac{1}{c_n^k}}+\left(\gamma -1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}-f^{+}\right)}^{\frac{1}{c_n^k}}}}}}\end{array}\right]e^{i2\pi \left[\begin{array}{l}\sqrt[q]{\frac{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}+\left({\gamma }^2-1\right)w_f^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}-w_f^{-}\right)}^{\frac{1}{c_n^k}}}}}{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}+\left({\gamma }^2-1\right)w_f^{-}\right)}^{\frac{1}{c_n^k}}+\left(\gamma -1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}-w_f^{-}\right)}^{\frac{1}{c_n^k}}}}},}\\ \sqrt[q]{\frac{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}+\left({\gamma }^2-1\right)w_f^{+}+\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}-w_f^{+}\right)}^{\frac{1}{c_n^k}}}}}{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}+\left({\gamma }^2-1\right)w_f^{+}\right)}^{\frac{1}{c_n^k}}+\left(\gamma -1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}-w_f^{+}\right)}^{\frac{1}{c_n^k}}}}}}\end{array}\right]},\\ \left[\begin{array}{l}\sqrt[q]{\frac{\gamma {\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}-h^{-}\right)}^{\frac{1}{c_n^k}}}}{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}+\left({\gamma }^2-1\right)h^{-}\right)}^{\frac{1}{c_n^k}}+\left(\gamma -1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}-h^{-}\right)}^{\frac{1}{c_n^k}}}}}},\\ \sqrt[q]{\frac{\gamma {\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}-h^{+}\right)}^{\frac{1}{c_n^k}}}}{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}+\left({\gamma }^2-1\right)h^{+}\right)}^{\frac{1}{c_n^k}}+\left(\gamma -1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}-h^{+}\right)}^{\frac{1}{c_n^k}}}}}}\end{array}\right]e^{i2\pi \left[\begin{array}{l}\sqrt[q]{\frac{\gamma {\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}-w_h^{-}\right)}^{\frac{1}{c_n^k}}}}{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}+\left({\gamma }^2-1\right)w_h^{-}\right)}^{\frac{1}{c_n^k}}+\left(\gamma -1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}-w_h^{-}\right)}^{\frac{1}{c_n^k}}}}}},\\ \sqrt[q]{\frac{\gamma {\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}-w_h^{+}\right)}^{\frac{1}{c_n^k}}}}{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}+\left({\gamma }^2-1\right)w_h^{+}\right)}^{\frac{1}{c_n^k}}+\left(\gamma -1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}-w_h^{+}\right)}^{\frac{1}{c_n^k}}}}}}\end{array}\right]}\end{array}\right),\\ \left(\begin{array}{l}\sqrt[q]{\frac{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}+\left({\gamma }^2-1\right)f^{}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}-f^{}\right)}^{\frac{1}{c_n^k}}}}}{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}+\left({\gamma }^2-1\right)f\right)}^{\frac{1}{c_n^k}}+\left(\gamma -1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}-f^{}\right)}^{\frac{1}{c_n^k}}}}}}{e}^{i2\pi \left(\sqrt[q]{\frac{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}+\left({\gamma }^2-1\right)w_f^{}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}-w_f^{}\right)}^{\frac{1}{c_n^k}}}}}{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}+\left({\gamma }^2-1\right)w_f^{}\right)}^{\frac{1}{c_n^k}}+\left(\gamma -1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}-w_f^{}\right)}^{\frac{1}{c_n^k}}}}}}\right)},\\ \sqrt[q]{\frac{\gamma {\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}-h^{}\right)}^{\frac{1}{c_n^k}}}}{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}+\left({\gamma }^2-1\right)h^{}\right)}^{\frac{1}{c_n^k}}+\left(\gamma -1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}-h^{}\right)}^{\frac{1}{c_n^k}}}}}}{e}^{i2\pi \left(\sqrt[q]{\frac{\gamma {\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}-w_h^{}\right)}^{\frac{1}{c_n^k}}}}{{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}+\left({\gamma }^2-1\right)w_h^{}\right)}^{\frac{1}{c_n^k}}+\left(\gamma -1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}-w_h^{}\right)}^{\frac{1}{c_n^k}}}}}}\right)}\end{array}\right)\end{array}\right)$$

(13)

where

$$e^{-}={\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right)\left(1-{\left(u_{i_j}^{-}\right)}^q\right)\right)}, e^{+}={\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right)\left(1-{\left(u_{i_j}^{+}\right)}^q\right)\right)}, e^{}={\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right)\left(1-{\left(u_{i_j}^{}\right)}^q\right)\right)}, f^{-}={\displaystyle {\prod }_{j=1}^k{\left(u_{i_j}^{-}\right)}^q},$$

$$f^{+}={\displaystyle {\prod }_{j=1}^k{\left(u_{i_j}^{+}\right)}^q}, f^{}={\displaystyle {\prod }_{j=1}^k{\left(u_{i_j}^{}\right)}^q}, g^{-}={\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(v_{i_j}^{-}\right)}^q\right)}, g^{+}={\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(v_{i_j}^{+}\right)}^q\right)}, g^{}={\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(v_{i_j}^{}\right)}^q\right)},$$

$$h^{-}={\displaystyle {\prod }_{j=1}^k\left(1-{\left(v_{i_j}^{-}\right)}^q\right)}, h^{+}={\displaystyle {\prod }_{j=1}^k\left(1-{\left(v_{i_j}^{+}\right)}^q\right)}, h^{}={\displaystyle {\prod }_{j=1}^k\left(1-{\left(v_{i_j}^{}\right)}^q\right)}.$$

Proof .

The proof of Proposition 2 is presented in the “Appendix A”. □

Based on the operational law (4) of Definitions 9 and 12 and the results of Proposition 2, we obtain the following theorem:

Theorem 4. 

Let  $a_i=\left(\left(\left[u_i^{-},u_i^{+}\right]e^{i2\pi \left[w_{u_i}^{-},w_{u_i}^{+}\right]},\left[v_i^{-},v_i^{+}\right]e^{i2\pi \left[w_{v_i}^{-},w_{v_i}^{+}\right]}\right),\left(u_i{e}^{i2\pi w_{u_i}},v_i{e}^{i2\pi w_{v_i}}\right)\right)$   $\left(i=1,2,\dots ,n\right)$  be a collection of CCuq-ROFNs, where  $k=1,2,\dots ,n$  ,  $\gamma$  > 0, then for any  $q\ge 1$  , the aggregated value found using the CCuq-ROFHMSM operator is also a CCuq-ROFN. The complex cubicq-rung orthopair fuzzy Hamacher Maclaurin symmetric mean (CCuq-ROFHMSM) operator is denoted as follows:

$$\begin{array}{l}CCuq-ROFHMSM{\left(a_1,a_2,\dots ,a_n\right)}^{\left(k\right)}={\left(\frac{{\oplus }_{1\le i_1\le \dots \le i_k\le n}\left({\otimes }_{j=1}^k{a}_{i_j}\right)}{C_n^k}\right)}^{\frac{1}{k}}\\ =(\left(\begin{array}{l}\left[\begin{array}{l}\sqrt[q]{\frac{\gamma {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}+\left({\gamma }^2-1\right)f^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}-f^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}+\left({\gamma }^2-1\right)f^{-}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}-f^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}+\left({\gamma }^2-1\right)f^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}-f^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)},}\\ \sqrt[q]{\frac{\gamma {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}+\left({\gamma }^2-1\right)f^{+}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}-f^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}+\left({\gamma }^2-1\right)f^{+}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}-f^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}+\left({\gamma }^2-1\right)f^{+}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}-f^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)},}\end{array}\right]e^{i2\pi \left[\begin{array}{l}\sqrt[q]{\frac{\gamma {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}+\left({\gamma }^2-1\right)w_f^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}-w_f^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}+\left({\gamma }^2-1\right)w_f^{-}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}-w_f^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}+\left({\gamma }^2-1\right)w_f^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}-w_f^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)},}\\ \sqrt[q]{\frac{\gamma {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}+\left({\gamma }^2-1\right)w_f^{+}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}-w_f^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}+\left({\gamma }^2-1\right)w_f^{+}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}-w_f^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}+\left({\gamma }^2-1\right)w_f^{+}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}-w_f^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)},}\end{array}\right]},\\ \left[\begin{array}{l}\sqrt[q]{\frac{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}+\left({\gamma }^2-1\right)h^{-}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}-h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}-\\ {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}+\left({\gamma }^2-1\right)h^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}-h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)}{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}+\left({\gamma }^2-1\right)h^{-}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}-h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}+\left({\gamma }^2-1\right)h^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}-h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)}},\\ \sqrt[q]{\frac{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}+\left({\gamma }^2-1\right)h^{+}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}-h^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}-\\ {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}+\left({\gamma }^2-1\right)h^{+}\right)}^{\frac{1}{c_n^k}}+{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}-h^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)}{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}+\left({\gamma }^2-1\right)h^{+}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}-h^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}+\left({\gamma }^2-1\right)h^{+}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}-h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)}}\end{array}\right]e^{i2\pi \left[\begin{array}{l}\sqrt[q]{\frac{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}+\left({\gamma }^2-1\right)w_h^{-}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}-w_h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}-\\ {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}+\left({\gamma }^2-1\right)w_h^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}-w_h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)}{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}+\left({\gamma }^2-1\right)w_h^{-}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}-w_h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}+\left({\gamma }^2-1\right)w_h^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}-w_h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)}},\\ \sqrt[q]{\frac{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}+\left({\gamma }^2-1\right)w_h^{+}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}-w_h^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}-\\ {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}+\left({\gamma }^2-1\right)w_h^{+}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}-w_h^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)}{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}+\left({\gamma }^2-1\right)w_h^{+}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}-w_h^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}+\left({\gamma }^2-1\right)w_h^{+}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}-w_h^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)}}\end{array}\right]}\end{array}\right),\end{array}$$

$$\begin{array}{l}\\ \left(\begin{array}{l}\sqrt[q]{\frac{\gamma {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}+\left({\gamma }^2-1\right)f^{}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}-f\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}+\left({\gamma }^2-1\right)f^{}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}-f^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}+\left({\gamma }^2-1\right)f^{}\right)}^{\frac{1}{c_n^k}}-\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e-f^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)},}{e}^{i2\pi \sqrt[q]{\frac{\gamma {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}+\left({\gamma }^2-1\right)w_f^{}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}-w_f^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}+\left({\gamma }^2-1\right)w_f^{}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}-w_f^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}+\left({\gamma }^2-1\right)w_f^{}\right)}^{\frac{1}{c_n^k}}-\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}-w_f^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)},}},\\ \sqrt[q]{\frac{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}+\left({\gamma }^2-1\right)h^{}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}-h^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}-\\ {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}+\left({\gamma }^2-1\right)h^{}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}-h^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)}{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}+\left({\gamma }^2-1\right)h^{}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}-h^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}+\left({\gamma }^2-1\right)h^{}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}-h^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)}}{e}^{i2\pi \sqrt[q]{\frac{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}+\left({\gamma }^2-1\right)w_h{}^{}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}-w_h{}^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}-\\ {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}+\left({\gamma }^2-1\right)w_h\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}-w_h\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)}{\left(\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}+\left({\gamma }^2-1\right)w_h^{}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}-w_h^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}+\left({\gamma }^2-1\right)w_h^{}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}-w_h^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}\right)}}}\end{array}\right))\end{array}$$

(14)

where

$$e_{}^{-}={\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right)\left(1-{\left(u_{i_j}^{-}\right)}^q\right)\right)}, e_{}^{+}={\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right)\left(1-{\left(u_{i_j}^{+}\right)}^q\right)\right)},$$

$$e_{}^{}={\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right)\left(1-{\left(u_{i_j}^{}\right)}^q\right)\right)}, f_{}^{-}={\displaystyle {\prod }_{j=1}^k{\left(u_{i_j}^{-}\right)}^q}, f_{}^{+}={\displaystyle {\prod }_{j=1}^k{\left(u_{i_j}^{+}\right)}^q}, f_{}^{}={\displaystyle {\prod }_{j=1}^k{\left(u_{i_j}^{}\right)}^q};$$

$$g_{}^{-}={\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(v_{i_j}^{-}\right)}^q\right)}, g_{}^{+}={\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(v_{i_j}^{+}\right)}^q\right)}, g_{}^{}={\displaystyle {\prod }_{j=1}^k\left(1+\left(\gamma -1\right){\left(v_{i_j}^{}\right)}^q\right)};$$

$$h_{}^{-}={\displaystyle {\prod }_{j=1}^k\left(1-{\left(v_{i_j}^{-}\right)}^q\right)}, h_{}^{+}={\displaystyle {\prod }_{j=1}^k\left(1-{\left(v_{i_j}^{+}\right)}^q\right)}, h_{}^{}={\displaystyle {\prod }_{j=1}^k\left(1-{\left(v_{i_j}^{}\right)}^q\right)}.$$

Proof. 

The proof of Theorem 4 is presented in the “Appendix A”. □

On the basis of the results of the CCuq-ROFHMSM operator, we can discuss some properties of Theorem 4 as follows:

Property 3 (Idempotency). 

Let $a_i=\left(\left(\left[u_i^{-},u_i^{+}\right]e^{i2\pi \left[w_{u_i}^{-},w_{u_i}^{+}\right]},\left[v_i^{-},v_i^{+}\right]e^{i2\pi \left[w_{v_i}^{-},w_{v_i}^{+}\right]}\right),\left(u_i{e}^{i2\pi w_{u_i}},v_i{e}^{i2\pi w_{v_i}}\right)\right)$ $\left(i=1,2,\dots ,n\right)$  be a collection of all CCuq-ROFNs. If all $a_i$ $\left(i=1,2,\dots ,n\right)$  are equal, i.e., $a_i=a=\left(\left(\left[u_{}^{-},u_{}^{+}\right]e^{i2\pi \left[w_u^{-},w_u^{+}\right]},\left[v_{}^{-},v_{}^{+}\right]e^{i2\pi \left[w_v^{-},w_v^{+}\right]}\right),\left(u{e}^{i2\pi w_u},v{e}^{i2\pi w_v}\right)\right)$  for all $i$ , then

$$CCuq-ROFHMS{M}^{\left(k\right)}\left(a_1,a_2,\dots ,a_i\right)=CCuq-ROFHMS{M}^{\left(k\right)}\left(a_{},a,\dots ,a_{}\right)=a.$$

Property 4 (Commutativity). 

Let $a_i=\left(\left(\left[u_i^{-},u_i^{+}\right]e^{i2\pi \left[w_{u_i}^{-},w_{u_i}^{+}\right]},\left[v_i^{-},v_i^{+}\right]e^{i2\pi \left[w_{v_i}^{-},w_{v_i}^{+}\right]}\right),\left(u_i{e}^{i2\pi w_{u_i}},v_i{e}^{i2\pi w_{v_i}}\right)\right)$ $\left(i=1,2,\dots ,n\right)$  be a collection of all CCuq-ROFNs. If $a_i^{\prime }$  is any permutation of $a_i$   $\left(i=1,2,\dots ,n\right)$  then we have the following relationship:

$$CCuq-ROFHMS{M}^{\left(k\right)}\left(a_1,a_2,\dots ,a_i\right)=CCuq-ROFHMS{M}^{\left(k\right)}\left(a_1^{\prime },a_2^{\prime },\dots ,a_i^{\prime }\right).$$

3.3.2. Weighted Complex Cubic q-Rung Orthopair Fuzzy Hamacher Maclaurin Symmetric Mean (WCCuq-ROFHMSM) Operator

The significance of the input parameters significantly affects the final outcomes of the decision-making process, but the CCuq-ROFHMSM operator does not deal with the importance of the aggregated arguments. Hence, the WCCuq-ROFHMSM operator is introduced and defined as follows:

Definition 13. 

Let $a_i=\left(\left(\left[u_i^{-},u_i^{+}\right]e^{i2\pi \left[w_{u_i}^{-},w_{u_i}^{+}\right]},\left[v_i^{-},v_i^{+}\right]e^{i2\pi \left[w_{v_i}^{-},w_{v_i}^{+}\right]}\right),\left(u_i{e}^{i2\pi w_{u_i}},v_i{e}^{i2\pi w_{v_i}}\right)\right)$ $\left(i=1,2,\dots ,n\right)$  be a collection of CCuq-ROFNs, $\omega =\left({\omega }_1,{\omega }_2,\dots ,{\omega }_n\right)$  is the weight vector, where ${\omega }_i$  indicates the importance degree of $a_i$, satisfying $\omega \in \left[0,1\right]$  and ${\displaystyle \sum _{\mathrm{i}}^n{\omega }_i}=1$. Then weighted complex cubic q-rung orthopair fuzzy Hamacher maclaurin symmetric mean (WCCuq-ROFHMSM) operator is defined as follows:

$$WCCuq-ROFHMS{M}^{\left(k\right)}\left(a_1,a_2,\dots ,a_i\right)={\left(\frac{{\oplus }_{1\le i_1\le \dots \le i_k\le n}\left({\otimes }_{j=1}^k{\omega }_{i_j}{a}_{i_j}\right)}{C_n^k}\right)}^{\frac{1}{k}}$$

(15)

where  $\left(i_1,i_2,\dots ,i_k\right)$  traversal all the k-tuple combination of (1, 2, …,n) and  $C_n^k=\frac{n!}{k!\left(n-k\right)!}$  is the binomial coefficient.

On the basis of Definition 13, Theorem 4 and the CCuq-ROFN Hamacher operations, the following theorem can be obtained:

Theorem 5. 

Let  $a_i=\left(\left(\left[u_i^{-},u_i^{+}\right]e^{i2\pi \left[w_{u_i}^{-},w_{u_i}^{+}\right]},\left[v_i^{-},v_i^{+}\right]e^{i2\pi \left[w_{v_i}^{-},w_{v_i}^{+}\right]}\right),\left(u_i{e}^{i2\pi w_{u_i}},v_i{e}^{i2\pi w_{v_i}}\right)\right)$   $\left(i=1,2,\dots ,n\right)$  be a collection of CCuq-ROFNs, where  $k=1,2,\dots ,n$  and  $\gamma$  > 0, then, for any  $q\ge 1$  , the aggregated value found using the WCCuq-ROFHMSM operator is also a CCuq-ROFN, and we have

$$\begin{array}{l}WCCuq-ROFHMSM\left(a_1,a_2,\dots ,a_n\right)={\left(\frac{{\oplus }_{1\le i_1\le \dots \le i_k\le n}\left({\otimes }_{j=1}^k{\omega }_{i_j}{a}_{i_j}\right)}{C_n^k}\right)}^{\frac{1}{k}}\\ =(\left(\begin{array}{l}\left[\begin{array}{l}\sqrt[q]{\frac{\gamma {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}+\left({\gamma }^2-1\right)f^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}-f^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}+\left({\gamma }^2-1\right)f^{-}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}-f^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}+\left({\gamma }^2-1\right)f^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{-}-f^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}},}\\ \sqrt[q]{\frac{\gamma {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}+\left({\gamma }^2-1\right)f^{+}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}-f^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}+\left({\gamma }^2-1\right)f^{+}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}-f^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}+\left({\gamma }^2-1\right)f^{+}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{+}-f^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}},}\end{array}\right]e^{i2\pi \left[\begin{array}{l}\sqrt[q]{\frac{\gamma {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}+\left({\gamma }^2-1\right)w_f^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}-w_f^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}+\left({\gamma }^2-1\right)w_f^{-}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}-w_f^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}+\left({\gamma }^2-1\right)w_f^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{-}-w_f^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}},}\\ \sqrt[q]{\frac{\gamma {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}+\left({\gamma }^2-1\right)w_f^{+}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}-w_f^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}+\left({\gamma }^2-1\right)w_f^{+}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}-w_f^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}+\left({\gamma }^2-1\right)w_f^{+}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{+}-w_f^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}},}\end{array}\right]},\\ \left[\begin{array}{l}\sqrt[q]{\frac{{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}+\left({\gamma }^2-1\right)h^{-}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}-h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}-{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}+\left({\gamma }^2-1\right)h^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}-h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}+\left({\gamma }^2-1\right)h^{-}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}-h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}+\left({\gamma }^2-1\right)h^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{-}-h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}},\\ \sqrt[q]{\frac{{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}+\left({\gamma }^2-1\right)h^{+}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}-h^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}-{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}+\left({\gamma }^2-1\right)h^{+}\right)}^{\frac{1}{c_n^k}}+{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}-h^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}+\left({\gamma }^2-1\right)h^{+}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}-h^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}+\left({\gamma }^2-1\right)h^{+}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{+}-h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}}\end{array}\right]e^{i2\pi \left[\begin{array}{l}\sqrt[q]{\frac{\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}+\left({\gamma }^2-1\right)w_h^{-}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}-w_h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}-\\ {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}+\left({\gamma }^2-1\right)w_h^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}-w_h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}}{\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}+\left({\gamma }^2-1\right)w_h^{-}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}-w_h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}+\left({\gamma }^2-1\right)w_h^{-}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{-}-w_h^{-}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}}},\\ \sqrt[q]{\frac{\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}+\left({\gamma }^2-1\right)w_h^{+}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}-w_h^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}-\\ {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}+\left({\gamma }^2-1\right)w_h^{+}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}-w_h^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}}{\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}+\left({\gamma }^2-1\right)w_h^{+}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}-w_h^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}+\left({\gamma }^2-1\right)w_h^{+}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{+}-w_h^{+}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}}}\end{array}\right]}\end{array}\right),\end{array}$$

$$\begin{array}{l}\\ \left(\begin{array}{l}\sqrt[q]{\frac{\gamma {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}+\left({\gamma }^2-1\right)f^{}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}-f^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}+\left({\gamma }^2-1\right)f^{}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}-f^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}+\left({\gamma }^2-1\right)f^{}\right)}^{\frac{1}{c_n^k}}-\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(e^{}-f^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}},}{e}^{i2\pi \sqrt[q]{\frac{\gamma {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}+\left({\gamma }^2-1\right)w_f^{}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}-w_f^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}+\left({\gamma }^2-1\right)w_f^{}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}-w_f^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}+\left({\gamma }^2-1\right)w_f^{}\right)}^{\frac{1}{c_n^k}}-\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_e^{}-w_f^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}},}},\\ \sqrt[q]{\frac{{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}+\left({\gamma }^2-1\right)h^{}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}-h^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}-{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}+\left({\gamma }^2-1\right)h^{}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}-h^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}{{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}+\left({\gamma }^2-1\right)h^{}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}-h^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}+\left({\gamma }^2-1\right)h^{}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(g^{}-h^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}}}{e}^{i2\pi \sqrt[q]{\frac{\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}+\left({\gamma }^2-1\right)w_h{}^{}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}-w_h{}^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}-\\ {\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}+\left({\gamma }^2-1\right)w_h\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}-w_h\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}}{\begin{array}{l}{\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}+\left({\gamma }^2-1\right)w_h^{}\right)}^{\frac{1}{c_n^k}}+\left({\gamma }^2-1\right){\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}-w_h^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}+\\ \left(\gamma -1\right){\left({\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}+\left({\gamma }^2-1\right)w_h^{}\right)}^{\frac{1}{c_n^k}}-{\displaystyle {\prod }_{1\le i_1\le \dots \le i_k\le n}{\left(w_g^{}-w_h^{}\right)}^{\frac{1}{c_n^k}}}}\right)}^{\frac{1}{k}}\end{array}}}}\end{array}\right))\end{array}$$

(16)

where

$$e_{}^{-}={\displaystyle {\prod }_{j=1}^k\left(\left(\frac{{\left(1+\left(\gamma -1\right){\left(u_{i_j}^{-}\right)}^q\right)}^{{\omega }_{i_j}}+\left({\gamma }^2-1\right){\left(1-{\left(u_{i_j}^{-}\right)}^q\right)}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right){\left(u_{i_j}^{-}\right)}^q\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(1-{\left(u_{i_j}^{-}\right)}^q\right)}^{{\omega }_{i_j}}}\right)e^{i2\pi \frac{{\left(1+\left(\gamma -1\right){\left(w_{u_{i_j}}^{-}\right)}^q\right)}^{{\omega }_{i_j}}+\left({\gamma }^2-1\right){\left(1-{\left(w_{u_{i_j}}^{-}\right)}^q\right)}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right){\left(w_{u_{i_j}}^{-}\right)}^q\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(1-{\left(w_{u_{i_j}}^{-}\right)}^q\right)}^{{\omega }_{i_j}}}}\right)}{e}_{}^{+}={\displaystyle {\prod }_{j=1}^k\left(\left(\frac{{\left(1+\left(\gamma -1\right){\left(u_{i_j}^{+}\right)}^q\right)}^{{\omega }_{i_j}}+\left({\gamma }^2-1\right){\left(1-{\left(u_{i_j}^{+}\right)}^q\right)}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right){\left(u_{i_j}^{+}\right)}^q\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(1-{\left(u_{i_j}^{+}\right)}^q\right)}^{{\omega }_{i_j}}}\right)e^{i2\pi \left(\frac{{\left(1+\left(\gamma -1\right){\left(w_{u_{i_j}}^{+}\right)}^q\right)}^{{\omega }_{i_j}}+\left({\gamma }^2-1\right){\left(1-{\left(w_{u_{i_j}}^{+}\right)}^q\right)}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right){\left(w_{u_{i_j}}^{+}\right)}^q\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(1-{\left(w_{u_{i_j}}^{+}\right)}^q\right)}^{{\omega }_{i_j}}}\right)}\right)}$$

$$e_{}^{}={\displaystyle {\prod }_{j=1}^k\left(\left(\frac{{\left(1+\left(\gamma -1\right){\left(u_{i_j}^{}\right)}^q\right)}^{{\omega }_{i_j}}+\left({\gamma }^2-1\right){\left(1-{\left(u_{i_j}^{}\right)}^q\right)}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right){\left(u_{i_j}^{}\right)}^q\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(1-{\left(u_{i_j}^{}\right)}^q\right)}^{{\omega }_{i_j}}}\right)e^{i2\pi \left(\frac{{\left(1+\left(\gamma -1\right){\left(w_{u_{i_j}}^{}\right)}^q\right)}^{{\omega }_{i_j}}+\left({\gamma }^2-1\right){\left(1-{\left(w_{u_{i_j}}^{}\right)}^q\right)}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right){\left(w_{u_{i_j}}^{}\right)}^q\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(1-{\left(w_{u_{i_j}}^{}\right)}^q\right)}^{{\omega }_{i_j}}}\right)}\right)}{f}_{}^{-}={\displaystyle {\prod }_{j=1}^k\left(\left(\frac{{\left(1+\left(\gamma -1\right){\left(u_{i_j}^{-}\right)}^q\right)}^{{\omega }_{i_j}}-{\left(1-{\left(u_{i_j}^{-}\right)}^q\right)}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right){\left(u_{i_j}^{-}\right)}^q\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(1-{\left(u_{i_j}^{-}\right)}^q\right)}^{{\omega }_{i_j}}}\right)e^{i2\pi \left(\frac{{\left(1+\left(\gamma -1\right){\left(w_{u_{i_j}}^{-}\right)}^q\right)}^{{\omega }_{i_j}}-{\left(1-{\left(w_{u_{i_j}}^{-}\right)}^q\right)}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right){\left(w_{u_{i_j}}^{-}\right)}^q\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(1-{\left(w_{u_{i_j}}^{-}\right)}^q\right)}^{{\omega }_{i_j}}}\right)}\right)}$$

$$f_{}^{+}={\displaystyle {\prod }_{j=1}^k\left(\left(\frac{{\left(1+\left(\gamma -1\right){\left(u_{i_j}^{+}\right)}^q\right)}^{{\omega }_{i_j}}-{\left(1-{\left(u_{i_j}^{+}\right)}^q\right)}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right){\left(u_{i_j}^{+}\right)}^q\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(1-{\left(u_{i_j}^{+}\right)}^q\right)}^{{\omega }_{i_j}}}\right)e^{i2\pi \left(\frac{{\left(1+\left(\gamma -1\right){\left(w_{u_{i_j}}^{+}\right)}^q\right)}^{{\omega }_{i_j}}-{\left(1-{\left(w_{u_{i_j}}^{+}\right)}^q\right)}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right){\left(w_{u_{i_j}}^{+}\right)}^q\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(1-{\left(w_{u_{i_j}}^{+}\right)}^q\right)}^{{\omega }_{i_j}}}\right)}\right)}{f}_{}^{}={\displaystyle {\prod }_{j=1}^k\left(\left(\frac{{\left(1+\left(\gamma -1\right){\left(u_{i_j}^{}\right)}^q\right)}^{{\omega }_{i_j}}-{\left(1-{\left(u_{i_j}^{}\right)}^q\right)}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right){\left(u_{i_j}^{}\right)}^q\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(1-{\left(u_{i_j}^{}\right)}^q\right)}^{{\omega }_{i_j}}}\right)e^{i2\pi \left(\frac{{\left(1+\left(\gamma -1\right){\left(w_{u_{i_j}}^{}\right)}^q\right)}^{{\omega }_{i_j}}-{\left(1-{\left(w_{u_{i_j}}^{}\right)}^q\right)}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right){\left(w_{u_{i_j}}^{}\right)}^q\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(1-{\left(w_{u_{i_j}}^{}\right)}^q\right)}^{{\omega }_{i_j}}}\right)}\right)}$$

$$g_{}^{-}={\displaystyle {\prod }_{j=1}^k\left(\left(\frac{{\left(1+\left(\gamma -1\right)\left(1-{\left(v_{i_j}^{-}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left({\gamma }^2-1\right){\left(v_{i_j}^{-}\right)}^q{}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right)\left(1-{\left(v_{i_j}^{-}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(v_{i_j}^{-}\right)}^q{}^{{\omega }_{i_j}}}\right)e^{i2\pi \left(\frac{{\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_{i_j}}^{-}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left({\gamma }^2-1\right){\left(w_{v_{i_j}}^{-}\right)}^q{}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_{i_j}}^{-}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(w_{v_{i_j}}^{-}\right)}^q{}^{{\omega }_{i_j}}}\right)}\right)}{g}_{}^{+}={\displaystyle {\prod }_{j=1}^k\left(\left(\frac{{\left(1+\left(\gamma -1\right)\left(1-{\left(v_{i_j}^{+}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left({\gamma }^2-1\right){\left(v_{i_j}^{+}\right)}^q{}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right)\left(1-{\left(v_{i_j}^{+}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(v_{i_j}^{+}\right)}^q{}^{{\omega }_{i_j}}}\right)e^{i2\pi \left(\frac{{\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_{i_j}}^{+}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left({\gamma }^2-1\right){\left(w_{v_{i_j}}^{+}\right)}^q{}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_{i_j}}^{+}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(w_{v_{i_j}}^{+}\right)}^q{}^{{\omega }_{i_j}}}\right)}\right)}$$

$$g_{}^{}={\displaystyle {\prod }_{j=1}^k\left(\left(\frac{{\left(1+\left(\gamma -1\right)\left(1-{\left(v_{i_j}^{}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left({\gamma }^2-1\right){\left(v_{i_j}^{}\right)}^q{}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right)\left(1-{\left(v_{i_j}^{}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(v_{i_j}^{}\right)}^q{}^{{\omega }_{i_j}}}\right)e^{i2\pi \left(\frac{{\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_{i_j}}^{}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left({\gamma }^2-1\right){\left(w_{v_{i_j}}^{}\right)}^q{}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right)\left(1-{\left(w_{v_{i_j}}^{}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(w_{v_{i_j}}^{}\right)}^q{}^{{\omega }_{i_j}}}\right)}\right)}{h}_{}^{-}={\displaystyle {\prod }_{j=1}^k\left(\frac{{\left(1+\left(\gamma -1\right)\left(1-{\left(v_{i_j}^{-}\right)}^q\right)\right)}^{{\omega }_{i_j}}-{\left(v_{i_j}^{-}\right)}^q{}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right)\left(1-{\left(v_{i_j}^{-}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(v_{i_j}^{-}\right)}^q{}^{{\omega }_{i_j}}}\right)}{h}_{}^{+}={\displaystyle {\prod }_{j=1}^k\left(\frac{{\left(1+\left(\gamma -1\right)\left(1-{\left(v_{i_j}^{+}\right)}^q\right)\right)}^{{\omega }_{i_j}}-{\left(v_{i_j}^{+}\right)}^q{}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right)\left(1-{\left(v_{i_j}^{+}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(v_{i_j}^{+}\right)}^q{}^{{\omega }_{i_j}}}\right)}$$

$$h_{}^{}={\displaystyle {\prod }_{j=1}^k\left(\frac{{\left(1+\left(\gamma -1\right)\left(1-{\left(v_{i_j}^{}\right)}^q\right)\right)}^{{\omega }_{i_j}}-{\left(v_{i_j}^{}\right)}^q{}^{{\omega }_{i_j}}}{{\left(1+\left(\gamma -1\right)\left(1-{\left(v_{i_j}^{}\right)}^q\right)\right)}^{{\omega }_{i_j}}+\left(\gamma -1\right){\left(v_{i_j}^{}\right)}^q{}^{{\omega }_{i_j}}}\right)}$$

Proof. 

The proof of Theorem 5 is presented in the “Appendix A”. □

4. A Method for MAGDM with CCuq-ROF Information

In this section, a MAGDM method for complex cubic q-rung orthopair fuzzy numbers are developed using the proposed WCCuq-ROFHMSM operator.

For a MAGDM problem, suppose $A=\left\{A_1,A_2,\dots ,A_m\right\}$ $\left(i=1,2,\dots ,m\right)$  is the set of alternatives, $E=\left\{E_1,E_2,\dots ,E_l\right\}$ $\left(s=1,2,\dots ,l\right)$  is the set of experts, $C=\left\{C_1,C_2,\dots ,C_n\right\}$  $\left(j=1,2,\dots ,n\right)$  is the set of attributes. Let $\omega =\left({\omega }_1,{\omega }_2,\dots ,{\omega }_n\right)$  and $\tilde{\omega }=\left({\tilde{\omega }}_1,{\tilde{\omega }}_2,\dots ,{\tilde{\omega }}_l\right)$  be the weight vectors of attributes and experts, respectively, where ${\omega }_j\ge 0$ , ${\displaystyle {\sum }_{j=1}^n{\omega }_j}=1$ , ${\tilde{\omega }}_s\ge 0$ , ${\displaystyle {\sum }_{s=1}^l{\tilde{\omega }}_s}=1$. Suppose that for attribute $C_j$  of alternative $A_i$ , each expert $E_s$  expresses his /her assessment information by a complex cubic q-rung orthopair fuzzy decision matrix $R^s={\left(a_{ij}^s\right)}_{m\times n}={\left(\left(\left[u_{ij}^{s-},u_{ij}^{s+}\right]e^{i2\pi \left[w_{u_{ij}}^{s-},w_{u_{ij}}^{s+}\right]},\left[v_{ij}^{s-},v_{ij}^{s+}\right]e^{i2\pi \left[w_{v_{ij}}^{s-},w_{v_{ij}}^{s+}\right]}\right),\left(u_{ij}^s{e}^{i2\pi w_{u_{ij}}^s},v_{ij}^s{e}^{i2\pi w_{v_{ij}}^s}\right)\right)}_{m\times n},(i=1,2,\dots ,m,$ $j=1,2,\dots ,n,s=1,2,\dots ,l)$. In the following, the WCCuq-ROFHMSM operator is applied to the MAGDM with CCuq-ROF information.

Step 1. Normalize each decision matrix $R^s$.

The attributes present in the decision matrix can be categorized into two types: cost type and benefit type. To simultaneously account for both types of attributes, we must normalize the decision matrix using the following formula:

$$a_{ij}^s=\{\begin{array}{l}{a}_{ij}^s{ \mathrm{for} \mathrm{benefit} \mathrm{attribute} \mathrm{C}}_j\\ {\tilde{a}}_{ij}^s{ \mathrm{for} \cos \mathrm{t} \mathrm{attribute} \mathrm{C}}_j \end{array}$$

However, if all the attributes are of the benefit type, there is no need to normalize the decision matrix.

Step 2. The comprehensive complex cubic q-rung orthopair fuzzy decision matrix $R$  is evaluated.

Using the proposed WCCuq-ROFHMSM operator to aggregate all the individual complex cubic q-rung orthopair fuzzy decision matrixes $R^s$ $\left(s=1,2,\dots ,l\right)$  into the collective complex cubic q-rung orthopair fuzzy decision matrix $R$.

The comprehensive evaluation value of each attribute by experts in each alternative is computed as follows:

$$a_{ij}^{}=WCCuq-ROFHMSM\left(a_{ij}^{(1)},a_{ij}^{(2)},\dots ,a_{ij}^{(l)}\right).$$

Step 3. The comprehensive complex cubic q-rung orthopair fuzzy value is evaluated.

Using the proposed WCCuq-ROFHMSM operator to aggregate all the individual complex cubic q-rung orthopair fuzzy values $a_{ij}^{}$  into the collective complex cubic q-rung orthopair fuzzy value $a_i$ :

$$a_i=WCCuq-ROFHMSM\left(a_{i1},a_{i2},\dots ,a_{in}\right).$$

Step 4. Calculation of score function and accuracy function.

Calculate the score function $S\left(a_i\right)$  and accuracy function $H\left(a_i\right)$  for each comprehensive value $a_i$ $\left(i=1,2,\dots ,m\right)$.

Step 5. Ranking of alternatives.

The alternatives are ranked base on their score function $S\left(a_i\right)$ $\left(i=1,2,\dots ,m\right)$  and accuracy function $H\left(a_i\right)$ $\left(i=1,2,\dots ,m\right)$ , and the most suitable alternative is selected.

Step 6. End.

The flowchart of the proposed algorithm is given in Figure 1.

5. Application of the Proposed Method

5.1. A Numerical Example

In this section, a numerical example is presented to show potential evaluation of quality of digital government services with complex cubic q-rung orthopair fuzzy information in order to illustrate the method proposed in this paper.

In recent years, with “Internet plus government service” as the starting point, the “run at most once” reform as the overall traction, and the integrated data platform as the key support, Chongqing has achieved remarkable results in government digital governance, which is very representative. In addition, Chongqing has achieved significant results in economic development and political reform due to its geographical advantage and has a certain exemplary role in the country. Therefore, this study takes Chongqing as an example to comprehensively evaluate the quality of its digital government services.

There are five Chongqing government affairs WeChat official accounts $A_i\left(i=1,2,3,4,5\right)$  (Chongqing Medical, Chongqing Transportation, Chongqing Weather, Chongqing Public Security Bureau, Chongqing Taxation) to evaluate. The three experts $E_s\left(s=1,2,3\right)$  select the best account from four attributes: $C_1$  is platform design quality; $C_2$  is the information and administrative services; $C_3$  the information technology and $C_4$  is the government WeChat information environment. The weighting vectors of attributes and experts are $\omega ={\left(0.2,0.1,0.3,0.4\right)}^T$  and $\stackrel{-}{\omega }={\left(0.3,0.3,0.4\right)}^T$, respectively. The five official accounts $A_i\left(i=1,2,3,4,5\right)$  are to be evaluated using the complex cubic q-rung orthopair fuzzy information using the decision maker under the above four attributes, and the CCuq-ROF decision matrix $R^s={\left(a_{ij}^s\right)}_{5\times 4}$ $(i=1,2,3,4,5;j=1,2,3,4;s=1,2,3)$  is shown in

Table 1. Decision matrix $R^1$.

	C1	C2	C3	C4
A1	$\left(\begin{array}{l}\left(\left[0.1,0.7\right]e^{i2\pi \left[0.4,0.5\right]},\left[0.4,0.5\right]e^{i2\pi \left[0.3,0.4\right]}\right),\\ \left(0.2e^{i2\pi \left(0.3\right)},0.8e^{i2\pi \left(0.8\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.3,0.6\right]e^{i2\pi \left[0.4,0.5\right]},\left[0.2,0.3\right]e^{i2\pi \left[0.2,0.5\right]}\right),\\ \left(0.3e^{i2\pi \left(0.3\right)},0.7e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.7\right]e^{i2\pi \left[0.3,0.7\right]},\left[0.4,0.6\right]e^{i2\pi \left[0.4,0.6\right]}\right),\\ \left(0.4e^{i2\pi \left(0.3\right)},0.7e^{i2\pi \left(0.4\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.3,0.6\right]e^{i2\pi \left[0.3,0.4\right]},\left[0.3,0.5\right]e^{i2\pi \left[0.2,0.4\right]}\right),\\ \left(0.1e^{i2\pi \left(0.2\right)},0.8e^{i2\pi \left(0.5\right)}\right)\end{array}\right)$
A2	$\left(\begin{array}{l}\left(\left[0.2,0.6\right]e^{i2\pi \left[0.3,0.4\right]},\left[0.2,0.7\right]e^{i2\pi \left[0.2,0.6\right]}\right),\\ \left(0.3e^{i2\pi \left(0.2\right)},0.8e^{i2\pi \left(0.4\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.7\right]e^{i2\pi \left[0.4,0.5\right]},\left[0.1,0.7\right]e^{i2\pi \left[0.2,0.8\right]}\right),\\ \left(0.2e^{i2\pi \left(0.3\right)},0.6e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.4,0.5\right]e^{i2\pi \left[0.2,0.7\right]},\left[0.1,0.8\right]e^{i2\pi \left[0.4,0.6\right]}\right),\\ \left(0.3e^{i2\pi \left(0.1\right)},0.6e^{i2\pi \left(0.8\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.1,0.7\right]e^{i2\pi \left[0.4,0.5\right]},\left[0.2,0.4\right]e^{i2\pi \left[0.3,0.6\right]}\right),\\ \left(0.2e^{i2\pi \left(0.3\right)},0.6e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$
A3	$\left(\begin{array}{l}\left(\left[0.1,0.5\right]e^{i2\pi \left[0.2,0.5\right]},\left[0.1,0.8\right]e^{i2\pi \left[0.2,0.7\right]}\right),\\ \left(0.2e^{i2\pi \left(0.2\right)},0.7e^{i2\pi \left(0.8\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.3,0.5\right]e^{i2\pi \left[0.1,0.8\right]},\left[0.1,0.2\right]e^{i2\pi \left[0.1,0.4\right]}\right),\\ \left(0.2e^{i2\pi \left(0.2\right)},0.8e^{i2\pi \left(0.8\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.3,0.6\right]e^{i2\pi \left[0.2,0.6\right]},\left[0.4,0.5\right]e^{i2\pi \left[0.3,0.5\right]}\right),\\ \left(0.4e^{i2\pi \left(0.3\right)},0.6e^{i2\pi \left(0.4\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.4,0.6\right]e^{i2\pi \left[0.1,0.7\right]},\left[0.2,0.7\right]e^{i2\pi \left[0.3,0.6\right]}\right),\\ \left(0.3e^{i2\pi \left(0.2\right)},0.5e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$
A4	$\left(\begin{array}{l}\left(\left[0.1,0.9\right]e^{i2\pi \left[0.2,0.7\right]},\left[0.1,0.4\right]e^{i2\pi \left[0.1,0.3\right]}\right),\\ \left(0.3e^{i2\pi \left(0.2\right)},0.6e^{i2\pi \left(0.6\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.7\right]e^{i2\pi \left[0.4,0.6\right]},\left[0.1,0.4\right]e^{i2\pi \left[0.2,0.5\right]}\right),\\ \left(0.4e^{i2\pi \left(0.2\right)},0.6e^{i2\pi \left(0.8\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.3,0.4\right]e^{i2\pi \left[0.2,0.3\right]},\left[0.2,0.7\right]e^{i2\pi \left[0.1,0.7\right]}\right),\\ \left(0.1e^{i2\pi \left(0.3\right)},0.8e^{i2\pi \left(0.4\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.1,0.7\right]e^{i2\pi \left[0.1,0.8\right]},\left[0.1,0.4\right]e^{i2\pi \left[0.2,0.5\right]}\right),\\ \left(0.4e^{i2\pi \left(0.1\right)},0.6e^{i2\pi \left(0.8\right)}\right)\end{array}\right)$
A5	$\left(\begin{array}{l}\left(\left[0.2,0.7\right]e^{i2\pi \left[0.2,0.8\right]},\left[0.2,0.5\right]e^{i2\pi \left[0.1,0.4\right]}\right),\\ \left(0.1e^{i2\pi \left(0.2\right)},0.6e^{i2\pi \left(0.3\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.8\right]e^{i2\pi \left[0.1,0.8\right]},\left[0.1,0.5\right]e^{i2\pi \left[0.2,0.4\right]}\right),\\ \left(0.1e^{i2\pi \left(0.2\right)},0.8e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.1,0.8\right]e^{i2\pi \left[0.2,0.3\right]},\left[0.4,0.6\right]e^{i2\pi \left[0.3,0.5\right]}\right),\\ \left(0.4e^{i2\pi \left(0.3\right)},0.6e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.3,0.5\right]e^{i2\pi \left[0.4,0.5\right]},\left[0.2,0.6\right]e^{i2\pi \left[0.1,0.7\right]}\right),\\ \left(0.4e^{i2\pi \left(0.1\right)},0.6e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$

,

Table 2. Decision matrix $R^2$.

	C1	C2	C3	C4
A1	$\left(\begin{array}{l}\left(\left[0.1,0.2\right]e^{i2\pi \left[0.2,0.4\right]},\left[0.1,0.7\right]e^{i2\pi \left[0.2,0.6\right]}\right),\\ \left(0.2e^{i2\pi \left(0.3\right)},0.6e^{i2\pi \left(0.5\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.3\right]e^{i2\pi \left[0.2,0.5\right]},\left[0.3,0.6\right]e^{i2\pi \left[0.2,0.6\right]}\right),\\ \left(0.3e^{i2\pi \left(0.3\right)},0.7e^{i2\pi \left(0.4\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.5\right]e^{i2\pi \left[0.2,0.7\right]},\left[0.1,0.8\right]e^{i2\pi \left[0.2,0.6\right]}\right),\\ \left(0.3e^{i2\pi \left(0.2\right)},0.6e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.7\right]e^{i2\pi \left[0.4,0.5\right]},\left[0.3,0.5\right]e^{i2\pi \left[0.2,0.4\right]}\right),\\ \left(0.1e^{i2\pi \left(0.2\right)},0.8e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$
A2	$\left(\begin{array}{l}\left(\left[0.1,0.8\right]e^{i2\pi \left[0.2,0.8\right]},\left[0.1,0.4\right]e^{i2\pi \left[0.2,0.3\right]}\right),\\ \left(0.3e^{i2\pi \left(0.1\right)},0.7e^{i2\pi \left(0.6\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.7\right]e^{i2\pi \left[0.1,0.7\right]},\left[0.3,0.6\right]e^{i2\pi \left[0.4,0.5\right]}\right),\\ \left(0.4e^{i2\pi \left(0.2\right)},0.6e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.1,0.5\right]e^{i2\pi \left[0.2,0.3\right]},\left[0.1,0.7\right]e^{i2\pi \left[0.2,0.8\right]}\right),\\ \left(0.2e^{i2\pi \left(0.3\right)},0.5e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.3\right]e^{i2\pi \left[0.2,0.4\right]},\left[0.3,0.6\right]e^{i2\pi \left[0.4,0.6\right]}\right),\\ \left(0.1e^{i2\pi \left(0.2\right)},0.8e^{i2\pi \left(0.5\right)}\right)\end{array}\right)$
A3	$\left(\begin{array}{l}\left(\left[0.2,0.7\right]e^{i2\pi \left[0.4,0.5\right]},\left[0.3,0.5\right]e^{i2\pi \left[0.2,0.4\right]}\right),\\ \left(0.1e^{i2\pi \left(0.2\right)},0.8e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.6\right]e^{i2\pi \left[0.2,0.8\right]},\left[0.3,0.4\right]e^{i2\pi \left[0.1,0.4\right]}\right),\\ \left(0.1e^{i2\pi \left(0.3\right)},0.7e^{i2\pi \left(0.5\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.5\right]e^{i2\pi \left[0.5,0.7\right]},\left[0.1,0.8\right]e^{i2\pi \left[0.2,0.6\right]}\right),\\ \left(0.2e^{i2\pi \left(0.1\right)},0.6e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.1,0.2\right]e^{i2\pi \left[0.2,0.4\right]},\left[0.1,0.9\right]e^{i2\pi \left[0.2,0.6\right]}\right),\\ \left(0.2e^{i2\pi \left(0.3\right)},0.6e^{i2\pi \left(0.5\right)}\right)\end{array}\right)$
A4	$\left(\begin{array}{l}\left(\left[0.1,0.8\right]e^{i2\pi \left[0.2,0.7\right]},\left[0.3,0.4\right]e^{i2\pi \left[0.2,0.4\right]}\right),\\ \left(0.2e^{i2\pi \left(0.4\right)},0.8e^{i2\pi \left(0.5\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.1,0.5\right]e^{i2\pi \left[0.2,0.4\right]},\left[0.2,0.7\right]e^{i2\pi \left[0.5,0.6\right]}\right),\\ \left(0.2e^{i2\pi \left(0.1\right)},0.8e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.1,0.8\right]e^{i2\pi \left[0.2,0.8\right]},\left[0.1,0.4\right]e^{i2\pi \left[0.2,0.3\right]}\right),\\ \left(0.1e^{i2\pi \left(0.1\right)},0.8e^{i2\pi \left(0.6\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.6\right]e^{i2\pi \left[0.2,0.6\right]},\left[0.2,0.5\right]e^{i2\pi \left[0.4,0.6\right]}\right),\\ \left(0.3e^{i2\pi \left(0.2\right)},0.5e^{i2\pi \left(0.4\right)}\right)\end{array}\right)$
A5	$\left(\begin{array}{l}\left(\left[0.2,0.3\right]e^{i2\pi \left[0.2,0.5\right]},\left[0.4,0.5\right]e^{i2\pi \left[0.1,0.3\right]}\right),\\ \left(0.1e^{i2\pi \left(0.4\right)},0.8e^{i2\pi \left(0.5\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.3,0.6\right]e^{i2\pi \left[0.3,0.5\right]},\left[0.4,0.6\right]e^{i2\pi \left[0.4,0.6\right]}\right),\\ \left(0.2e^{i2\pi \left(0.1\right)},0.7e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.1,0.9\right]e^{i2\pi \left[0.1,0.8\right]},\left[0.3,0.4\right]e^{i2\pi \left[0.2,0.4\right]}\right),\\ \left(0.2e^{i2\pi \left(0.3\right)},0.7e^{i2\pi \left(0.4\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.6\right]e^{i2\pi \left[0.2,0.7\right]},\left[0.4,0.6\right]e^{i2\pi \left[0.4,0.6\right]}\right),\\ \left(0.1e^{i2\pi \left(0.2\right)},0.7e^{i2\pi \left(0.6\right)}\right)\end{array}\right)$

and

Table 3. Decision matrix $R^3$.

	C1	C2	C3	C4
A1	$\left(\begin{array}{l}\left(\left[0.1,0.6\right]e^{i2\pi \left[0.2,0.8\right]},\left[0.3,0.4\right]e^{i2\pi \left[0.1,0.4\right]}\right),\\ \left(0.1e^{i2\pi \left(0.3\right)},0.4e^{i2\pi \left(0.4\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.3\right]e^{i2\pi \left[0.5,0.7\right]},\left[0.2,0.7\right]e^{i2\pi \left[0.2,0.6\right]}\right),\\ \left(0.2e^{i2\pi \left(0.5\right)},0.7e^{i2\pi \left(0.6\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.4,0.6\right]e^{i2\pi \left[0.2,0.7\right]},\left[0.4,0.6\right]e^{i2\pi \left[0.4,0.6\right]}\right),\\ \left(0.1e^{i2\pi \left(0.2\right)},0.7e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.4\right]e^{i2\pi \left[0.2,0.5\right]},\left[0.3,0.6\right]e^{i2\pi \left[0.2,0.6\right]}\right),\\ \left(0.1e^{i2\pi \left(0.2\right)},0.8e^{i2\pi \left(0.5\right)}\right)\end{array}\right)$
A2	$\left(\begin{array}{l}\left(\left[0.3,0.4\right]e^{i2\pi \left[0.1,0.7\right]},\left[0.2,0.6\right]e^{i2\pi \left[0.2,0.6\right]}\right),\\ \left(0.3e^{i2\pi \left(0.2\right)},0.6e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.5\right]e^{i2\pi \left[0.2,0.3\right]},\left[0.1,0.7\right]e^{i2\pi \left[0.2,0.8\right]}\right),\\ \left(0.3e^{i2\pi \left(0.3\right)},0.5e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.8\right]e^{i2\pi \left[0.1,0.8\right]},\left[0.3,0.4\right]e^{i2\pi \left[0.1,0.4\right]}\right),\\ \left(0.2e^{i2\pi \left(0.2\right)},0.7e^{i2\pi \left(0.3\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.3,0.6\right]e^{i2\pi \left[0.2,0.6\right]},\left[0.2,0.5\right]e^{i2\pi \left[0.4,0.6\right]}\right),\\ \left(0.3e^{i2\pi \left(0.3\right)},0.5e^{i2\pi \left(0.4\right)}\right)\end{array}\right)$
A3	$\left(\begin{array}{l}\left(\left[0.2,0.7\right]e^{i2\pi \left[0.4,05\right]},\left[0.2,0.5\right]e^{i2\pi \left[0.2,0.4\right]}\right),\\ \left(0.2e^{i2\pi \left(0.2\right)},0.8e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.3,0.4\right]e^{i2\pi \left[0.3,0.4\right]},\left[0.3,0.6\right]e^{i2\pi \left[0.4,0.6\right]}\right),\\ \left(0.4e^{i2\pi \left(0.2\right)},0.5e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.1,0.8\right]e^{i2\pi \left[0.2,0.7\right]},\left[0.3,0.5\right]e^{i2\pi \left[0.3,0.4\right]}\right),\\ \left(0.2e^{i2\pi \left(0.2\right)},0.7e^{i2\pi \left(0.3\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.5\right]e^{i2\pi \left[0.2,0.4\right]},\left[0.1,0.7\right]e^{i2\pi \left[0.2,0.8\right]}\right),\\ \left(0.3e^{i2\pi \left(0.3\right)},0.5e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$
A4	$\left(\begin{array}{l}\left(\left[0.2,0.7\right]e^{i2\pi \left[0.1,0.7\right]},\left[0.3,0.5\right]e^{i2\pi \left[0.4,0.5\right]}\right),\\ \left(0.1e^{i2\pi \left(0.4\right)},0.6e^{i2\pi \left(0.6\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.6\right]e^{i2\pi \left[0.6,0.7\right]},\left[0.5,0.7\right]e^{i2\pi \left[0.3,0.6\right]}\right),\\ \left(0.2e^{i2\pi \left(0.5\right)},0.5e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.3,0.6\right]e^{i2\pi \left[0.3,0.8\right]},\left[0.3,0.5\right]e^{i2\pi \left[0.2,0.4\right]}\right),\\ \left(0.4e^{i2\pi \left(0.2\right)},0.7e^{i2\pi \left(0.3\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.3\right]e^{i2\pi \left[0.2,0.5\right]},\left[0.4,0.6\right]e^{i2\pi \left[0.5,0.6\right]}\right),\\ \left(0.5e^{i2\pi \left(0.2\right)},0.7e^{i2\pi \left(0.5\right)}\right)\end{array}\right)$
A5	$\left(\begin{array}{l}\left(\left[0.3,0.7\right]e^{i2\pi \left[0.5,0.7\right]},\left[0.5,0.6\right]e^{i2\pi \left[0.2,0.6\right]}\right),\\ \left(0.4e^{i2\pi \left(0.6\right)},0.5e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.6\right]e^{i2\pi \left[0.3,0.5\right]},\left[0.1,0.4\right]e^{i2\pi \left[0.1,0.4\right]}\right),\\ \left(0.1e^{i2\pi \left(0.1\right)},0.7e^{i2\pi \left(0.3\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.7\right]e^{i2\pi \left[0.5,0.7\right]},\left[0.1,0.6\right]e^{i2\pi \left[0.2,0.6\right]}\right),\\ \left(0.1e^{i2\pi \left(0.3\right)},0.8e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.2,0.6\right]e^{i2\pi \left[0.2,0.4\right]},\left[0.1,0.8\right]e^{i2\pi \left[0.2,0.8\right]}\right),\\ \left(0.4e^{i2\pi \left(0.4\right)},0.5e^{i2\pi \left(0.7\right)}\right)\end{array}\right)$

.

Step 1. Normalize the individual decision matrix $R^s$.

Since all attributes are of beneficial types, there is no requirement to normalize each decision matrix. Therefore, the decision matrix provided in Table 1, Table 2 and Table 3 is utilized for subsequent analysis.

Step 2. Comprehensive decision matrix of experts is evaluated.

Based on the weight of experts and decision matrixes, comprehensive decision matrix $R$  is obtained using the WCCuq-ROFHMSM operator (let $k=2$, $q=2$, $\gamma =1$). Then the comprehensive decision matrix $R$  is shown in

Table 4. Comprehensive decision matrix $R$.

	C1	C2	C3	C4
A1	$\left(\begin{array}{l}\left(\left[0.34,0.55\right]e^{i2\pi \left[0.33,0.59\right]},\left[0.35,0.51\right]e^{i2\pi \left[0.22,0.43\right]}\right),\\ \left(0.27e^{i2\pi \left(0.28\right)},0.57e^{i2\pi \left(0.52\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.22,0.47\right]e^{i2\pi \left[0.36,0.51\right]},\left[0.45,0.52\right]e^{i2\pi \left[0.19,0.43\right]}\right),\\ \left(0.39e^{i2\pi \left(0.37\right)},0.58e^{i2\pi \left(0.53\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.33,0.54\right]e^{i2\pi \left[0.31,0.62\right]},\left[0.29,0.57\right]e^{i2\pi \left[0.33,0.55\right]}\right),\\ \left(0.39e^{i2\pi \left(0.42\right)},0.61e^{i2\pi \left(0.58\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.35,0.53\right]2e^{i2\pi \left[0.27,0.45\right]},\left[0.32,0.56\right]e^{i2\pi \left[0.35,0.49\right]}\right),\\ \left(0.45e^{i2\pi \left(0.38\right)},0.58e^{i2\pi \left(0.43\right)}\right)\end{array}\right)$
A2	$\left(\begin{array}{l}\left(\left[0.24,0.56\right]e^{i2\pi \left[0.37,0.52\right]},\left[0.26,0.44\right]e^{i2\pi \left[0.37,0.56\right]}\right),\\ \left(0.22e^{i2\pi \left(0.41\right)},0.57e^{i2\pi \left(0.46\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.34,0.55\right]e^{i2\pi \left[0.32,0.47\right]},\left[0.31,0.66\right]e^{i2\pi \left[0.22,0.53\right]}\right),\\ \left(0.30e^{i2\pi \left(0.35\right)},0.48e^{i2\pi \left(0.54\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.37,0.52\right]e^{i2\pi \left[0.26,0.41\right]},\left[0.34,0.59\right]e^{i2\pi \left[0.32,0.52\right]}\right),\\ \left(0.31e^{i2\pi \left(0.29\right)},0.48e^{i2\pi \left(0.35\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.26,0.48\right]e^{i2\pi \left[0.32,0.41\right]},\left[0.33,0.49\right]e^{i2\pi \left[0.35,0.52\right]}\right),\\ \left(0.28e^{i2\pi \left(0.27\right)},0.59e^{i2\pi \left(0.45\right)}\right)\end{array}\right)$
A3	$\left(\begin{array}{l}\left(\left[0.31,0.55\right]e^{i2\pi \left[0.23,056\right]},\left[0.37,0.61\right]e^{i2\pi \left[0.32,0.62\right]}\right),\\ \left(0.29e^{i2\pi \left(0.34\right)},0.71e^{i2\pi \left(0.72\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.33,0.52\right]e^{i2\pi \left[0.32,0.58\right]},\left[0.23,0.39\right]e^{i2\pi \left[0.21,0.44\right]}\right),\\ \left(0.31e^{i2\pi \left(0.36\right)},0.58e^{i2\pi \left(0.49\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.33,0.51\right]e^{i2\pi \left[0.32,0.62\right]},\left[0.39,0.68\right]e^{i2\pi \left[0.24,0.43\right]}\right),\\ \left(0.44e^{i2\pi \left(0.45\right)},0.64e^{i2\pi \left(0.39\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.25,0.39\right]e^{i2\pi \left[0.21,0.57\right]},\left[0.34,0.61\right]e^{i2\pi \left[0.23,0.64\right]}\right),\\ \left(0.22e^{i2\pi \left(0.41\right)},0.60e^{i2\pi \left(0.58\right)}\right)\end{array}\right)$
A4	$\left(\begin{array}{l}\left(\left[0.36,0.45\right]e^{i2\pi \left[0.26,0.59\right]},\left[0.30,0.51\right]e^{i2\pi \left[0.24,0.39\right]}\right),\\ \left(0.38e^{i2\pi \left(0.32\right)},0.53e^{i2\pi \left(0.52\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.22,0.51\right]e^{i2\pi \left[0.32,0.53\right]},\left[0.35,0.42\right]e^{i2\pi \left[0.27,0.67\right]}\right),\\ \left(0.34e^{i2\pi \left(0.40\right)},0.43e^{i2\pi \left(0.52\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.25,0.55\right]e^{i2\pi \left[0.43,0.54\right]},\left[0.27,0.39\right]e^{i2\pi \left[0.34,0.45\right]}\right),\\ \left(0.38e^{i2\pi \left(0.23\right)},0.62e^{i2\pi \left(0.36\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.11,0.53\right]e^{i2\pi \left[0.27,0.52\right]},\left[0.11,0.43\right]e^{i2\pi \left[0.31,0.53\right]}\right),\\ \left(0.32e^{i2\pi \left(0.25\right)},0.46e^{i2\pi \left(0.45\right)}\right)\end{array}\right)$
A5	$\left(\begin{array}{l}\left(\left[0.22,0.42\right]e^{i2\pi \left[0.26,0.53\right]},\left[0.28,0.56\right]e^{i2\pi \left[0.13,0.39\right]}\right),\\ \left(0.23e^{i2\pi \left(0.32\right)},0.58e^{i2\pi \left(0.58\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.31,0.57\right]e^{i2\pi \left[0.42,0.53\right]},\left[0.19,0.45\right]e^{i2\pi \left[0.21,0.43\right]}\right),\\ \left(0.43e^{i2\pi \left(0.32\right)},0.55e^{i2\pi \left(0.58\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.28,0.66\right]e^{i2\pi \left[0.36,0.44\right]},\left[0.31,0.62\right]e^{i2\pi \left[0.23,0.45\right]}\right),\\ \left(0.42e^{i2\pi \left(0.46\right)},0.51e^{i2\pi \left(0.53\right)}\right)\end{array}\right)$	$\left(\begin{array}{l}\left(\left[0.18,0.46\right]e^{i2\pi \left[0.33,0.45\right]},\left[0.37,0.48\right]e^{i2\pi \left[0.43,0.69\right]}\right),\\ \left(0.42e^{i2\pi \left(0.43\right)},0.58e^{i2\pi \left(0.62\right)}\right)\end{array}\right)$

.

Step 3. Comprehensive value evaluation.

Similarly, there is no requirement to normalize decision matrix $R$. The WCCuq-ROCFHMSM operator is utilized to aggregate all the complex cubic q-rung orthopair cubic fuzzy information into the overall complex cubic q-rung orthopair cubic fuzzy values $a_i$ $\left(i=1,2,3,4,5\right)$. The aggregated results are provided in column 2 of

Table 5. Comprehensive values, score functions, and ranking results of alternatives.

Alternative	Comprehensive Value ${\mathit{a}}_{\mathit{i}}$	Score Value $\mathit{S}\left({\mathit{a}}_{\mathit{i}}\right)$	Rank
$A_1$	$\left(\left(\left[0.21,0.36\right]e^{i2\pi \left[0.12,0.41\right]},\left[0.22,0.41\right]e^{i2\pi \left[0.10,0.33\right]}\right),\left(0.20e^{i2\pi \left(0.22\right)},0.45e^{i2\pi \left(0.32\right)}\right)\right)$	$S\left(A_1\right)=-0.0515$	2
$A_2$	$\left(\left(\left[0.17,0.36\right]e^{i2\pi \left[0.19,0.28\right]},\left[0.20,0.39\right]e^{i2\pi \left[0.20,0.36\right]}\right),\left(0.09e^{i2\pi \left(0.14\right)},0.45e^{i2\pi \left(0.36\right)}\right)\right)$	$S\left(A_2\right)=-0.0872$	5
$A_3$	$\left(\left(\left[0.17,0.30\right]e^{i2\pi \left[0.11,0.40\right]},\left[0.26,0.41\right]e^{i2\pi \left[0.13,0.35\right]}\right),\left(0.19e^{i2\pi \left(0.23\right)},0.50e^{i2\pi \left(0.34\right)}\right)\right)$	$S\left(A_3\right)=-0.0797$	4
$A_4$	$\left(\left(\left[0.19,0.43\right]e^{i2\pi \left[0.20,0.30\right]},\left[0.12,0.31\right]e^{i2\pi \left[0.16,0.30\right]}\right),\left(0.30e^{i2\pi \left(0.17\right)},0.40e^{i2\pi \left(0.51\right)}\right)\right)$	$S\left(A_4\right)=-0.0422$	1
$A_5$	$\left(\left(\left[0.20,0.44\right]e^{i2\pi \left[0.15,0.36\right]},\left[0.22,0.41\right]e^{i2\pi \left[0.15,0.40\right]}\right),\left(0.21e^{i2\pi \left(0.19\right)},0.45e^{i2\pi \left(0.42\right)}\right)\right)$	$S\left(A_5\right)=-0.0763$	3

.

Step 4. Calculation of score and accuracy functions.

Now, calculate the score function $S\left(a_i\right)$ $\left(i=1,2,3,4,5\right)$  for each $a_i$. The calculated score functions are given in column 3 of Table 5. Since each $S\left(a_i\right)$  is different, thus the $H\left(a_i\right)$  of any $a_i$  is no need to calculate.

Step 5. Ranking of all accounts.

Finally, the ranking results all accounts $A_i$ $\left(i=1,2,3,4,5\right)$  on the basis of $S\left(a_i\right)$  are provided in column 4 of Table 5. It can be seen from Table 5 that A₄ is the best choice among the five accounts.

The determination of the optimal alternative relies upon the values of parameters q, γ and k. Likewise, various aggregation operators might yield distinct ranking outcomes based on their aggregation characteristics. It is crucial to examine the method’s effectiveness by considering the sensitivity of parameter value selection and the applied aggregation operator. Therefore, in the subsequent sections, sensitivity and comparative analyses have been carried out.

5.2. Sensitivity Analysis

To illustrate the effectiveness of the developed MAGDM approach using the WCCuq-ROFHMSM operator, a sensitivity analysis was carried out in several stages by altering the values of parameters q and γ, and by considering various values for parameter k. The subsequent sections provide an analysis and discussion of the repercussions of these variations on the outcomes.

Firstly, by varying parameter q across integer values within the range of 2 to 10, while simultaneously maintaining the other two parameters fixed at k = 2 and γ = 1, the obtained outcomes have been condensed and presented in

Table 6. Ranking results for different values of $q$  ($k=2$, $\gamma =1$ ).

$q$	Score Values	Ranking Results
$q=2$	$S\left(A_1\right)=-0.0515$, $S\left(A_2\right)=-0.0872$, $S\left(A_3\right)=-0.0797$, $S\left(A_4\right)=-0.0422$, $S\left(A_5\right)=-0.0763$.	$A_4\succ A_1\succ A_5\succ A_3\succ A_2$
$q=3$	$S\left(A_1\right)=-0.0511$, $S\left(A_2\right)=-0.0779$, $S\left(A_3\right)=-0.0759$, $S\left(A_4\right)=-0.0415$, $S\left(A_5\right)=-0.0760$.	$A_4\succ A_1\succ A_3\succ A_5\succ A_2$
$q=4$	$S\left(A_1\right)=-0.0498$, $S\left(A_2\right)=-0.0772$, $S\left(A_3\right)=-0.0751$, $S\left(A_4\right)=-0.0386$, $S\left(A_5\right)=-0.0755$.	$A_4\succ A_1\succ A_3\succ A_5\succ A_2$
$q=5$	$S\left(A_1\right)=-0.0492$, $S\left(A_2\right)=-0.0764$, $S\left(A_3\right)=-0.0746$, $S\left(A_4\right)=-0.0352$, $S\left(A_5\right)=-0.0491$.	$A_4\succ A_5\succ A_1\succ A_3\succ A_2$
$q=6$	$S\left(A_1\right)=-0.0479$, $S\left(A_2\right)=-0.0585$, $S\left(A_3\right)=-0.0481$, $S\left(A_4\right)=-0.0299$, $S\left(A_5\right)=-0.0482$.	$A_4\succ A_1\succ A_3\succ A_5\succ A_2$
$q=7$	$S\left(A_1\right)=-0.0336$, $S\left(A_2\right)=-0.0569$, $S\left(A_3\right)=-0.0477$, $S\left(A_4\right)=-0.0274$, $S\left(A_5\right)=-0.0479$.	$A_4\succ A_1\succ A_3\succ A_5\succ A_2$
$q=8$	$S\left(A_1\right)=-0.0515$, $S\left(A_2\right)=-0.0389$, $S\left(A_3\right)=-0.0445$, $S\left(A_4\right)=-0.0256$, $S\left(A_5\right)=-0.0462$.	$A_4\succ A_2\succ A_3\succ A_5\succ A_1$
$q=9$	$S\left(A_1\right)=-0.0256$, $S\left(A_2\right)=-0.0377$, $S\left(A_3\right)=-0.0359$, $S\left(A_4\right)=-0.0187$, $S\left(A_5\right)=-0.0364$.	$A_4\succ A_1\succ A_3\succ A_5\succ A_2$
$q=10$	$S\left(A_1\right)=-0.0217$, $S\left(A_2\right)=-0.0351$, $S\left(A_3\right)=-0.0321$, $S\left(A_4\right)=-0.0109$, $S\left(A_5\right)=-0.0335$.	$A_4\succ A_1\succ A_3\succ A_5\succ A_2$

. The outcomes indicate that as parameter q undergoes alteration, there is a corresponding shift in the scores and subsequent ranking order of all available alternatives. However, it consistently remains that the optimal choice is A₄, and the least favorable choice is A₂ in the most cases with the variations in parameter q. This observation suggests that parameter q not only extends the scope of potential decisions but also exerts influence over the final determination.

Secondly, parameter γ is varied using integer values within the interval [1, 10], while parameters k = 2 and q = 2 remain constant. The computed outcomes are presented in

Table 7. Ranking results for different values of $\gamma$  ($k=2$, $q=2$ ).

$\mathit{\gamma }$	Score Values	Ranking Results
$\gamma =1$	$S\left(A_1\right)=-0.0515$, $S\left(A_2\right)=-0.0872$, $S\left(A_3\right)=-0.0797$, $S\left(A_4\right)=-0.0422$, $S\left(A_5\right)=-0.0763$.	$A_4\succ A_1\succ A_5\succ A_3\succ A_2$
$\gamma =2$	$S\left(A_1\right)=-0.0511$, $S\left(A_2\right)=-0.0869$, $S\left(A_3\right)=-0.0763$, $S\left(A_4\right)=-0.0399$, $S\left(A_5\right)=-0.0760$.	$A_4\succ A_1\succ A_5\succ A_3\succ A_2$
$\gamma =3$	$S\left(A_1\right)=-0.0486$, $S\left(A_2\right)=-0.0789$, $S\left(A_3\right)=-0.0759$, $S\left(A_4\right)=-0.0342$, $S\left(A_5\right)=-0.0754$.	$A_4\succ A_1\succ A_5\succ A_3\succ A_2$
$\gamma =4$	$S\left(A_1\right)=-0.0441$, $S\left(A_2\right)=-0.0780$, $S\left(A_3\right)=-0.0678$, $S\left(A_4\right)=-0.0309$, $S\left(A_5\right)=-0.0650$.	$A_4\succ A_1\succ A_5\succ A_3\succ A_2$
$\gamma =5$	$S\left(A_1\right)=-0.0372$, $S\left(A_2\right)=-0.0577$, $S\left(A_3\right)=-0.0664$, $S\left(A_4\right)=-0.0288$, $S\left(A_5\right)=-0.0552$.	$A_4\succ A_1\succ A_5\succ A_2\succ A_3$
$\gamma =6$	$S\left(A_1\right)=-0.0309$, $S\left(A_2\right)=-0.0469$, $S\left(A_3\right)=-0.0489$, $S\left(A_4\right)=-0.0257$, $S\left(A_5\right)=-0.0554$.	$A_4\succ A_1\succ A_2\succ A_3\succ A_5$
$\gamma =7$	$S\left(A_1\right)=-0.0398$, $S\left(A_2\right)=-0.0471$, $S\left(A_3\right)=-0.0408$, $S\left(A_4\right)=-0.0208$, $S\left(A_5\right)=-0.0371$.	$A_4\succ A_5\succ A_1\succ A_3\succ A_2$
$\gamma =8$	$S\left(A_1\right)=-0.0366$, $S\left(A_2\right)=-0.0452$, $S\left(A_3\right)=-0.0232$, $S\left(A_4\right)=-0.0192$, $S\left(A_5\right)=-0.0343$.	$A_4\succ A_3\succ A_5\succ A_1\succ A_2$
$\gamma =9$	$S\left(A_1\right)=-0.0324$, $S\left(A_2\right)=-0.0449$, $S\left(A_3\right)=-0.0290$, $S\left(A_4\right)=-0.0178$, $S\left(A_5\right)=-0.0292$.	$A_4\succ A_5\succ A_3\succ A_1\succ A_2$
$\gamma =10$	$S\left(A_1\right)=-0.0173$, $S\left(A_2\right)=-0.0397$, $S\left(A_3\right)=-0.0252$, $S\left(A_4\right)=-0.0052$, $S\left(A_5\right)=-0.0236$.	$A_4\succ A_1\succ A_5\succ A_3\succ A_2$

. Examination of the results in Table 7 reveals that as parameter γ undergoes variation, the scores and ranking order of distinct alternatives correspondingly shift. In this context, it is noteworthy that the optimal alternative consistently remains as A₄ for all considered variations in parameter γ, while the least favorable alternative changes. This observation implies that parameter γ introduces a degree of adaptability into the aggregation process and exerts an influence on the ultimate decision.

Thirdly, the impact of interrelationships among multiple attributes is assessed through the variation in parameter k, while maintaining fixed values for parameters q = 2 and γ = 1. The computed outcomes are presented in

Table 8. Ranking results for different values of $k$  ($q=2$, $\gamma =1$ ).

$\mathit{k}$	Score Values	Ranking Results
$k=1$	$S\left(A_1\right)=-0.0848$, $S\left(A_2\right)=-0.1395$, $S\left(A_3\right)=-0.1234$, $S\left(A_4\right)=-0.0410$, $S\left(A_5\right)=-0.1194$.	$A_4\succ A_1\succ A_5\succ A_3\succ A_2$
$k=2$	$S\left(A_1\right)=-0.0624$, $S\left(A_2\right)=-0.1176$, $S\left(A_3\right)=-0.1010$, $S\left(A_4\right)=-0.0195$, $S\left(A_5\right)=-0.0965$.	$A_4\succ A_1\succ A_5\succ A_3\succ A_2$
$k=3$	$S\left(A_1\right)=-0.0500$, $S\left(A_2\right)=-0.0822$, $S\left(A_3\right)=-0.0703$, $S\left(A_4\right)=-0.0289$, $S\left(A_5\right)=-0.0747$.	$A_4\succ A_1\succ A_3\succ A_5\succ A_2$
$k=4$	$S\left(A_1\right)=-0.0375$, $S\left(A_2\right)=-0.0632$, $S\left(A_3\right)=-0.0496$, $S\left(A_4\right)=-0.0147$, $S\left(A_5\right)=-0.1058$.	$A_4\succ A_1\succ A_3\succ A_2\succ A_5$
$k=5$	$S\left(A_1\right)=-0.0308$, $S\left(A_2\right)=-0.0485$, $S\left(A_3\right)=-0.0270$, $S\left(A_4\right)=-0.0120$, $S\left(A_5\right)=-0.0743$.	$A_4\succ A_3\succ A_1\succ A_2\succ A_5$

, indicating that the interplay between multiple attributes does exert a certain influence on the ultimate decision. Notably, due to the incorporation of interrelationship considerations within the proposed aggregation operators, the resultant outcomes tend to exhibit a higher degree of realism. In this specific case, it is worth highlighting that the optimal alternative consistently remains A₄ across all examined variations in parameter k.

5.3. Comparison Analysis

The theory of complex cubic q-rung orthopair fuzzy sets (CCuq-ROFSs) serves as an effective framework for describing intricate fuzzy information within the real-world context. Due to the inclusion of parameter q within Ccuq-ROFSs, which enables the adjustment of the range of complex fuzzy information expression, they exhibit superiority over both complex cubic intuitionistic and complex cubic Pythagorean fuzzy sets. Hamacher aggregation operators play a crucial role as a tool for generating operational rules based on complex cubic q-rung orthopair fuzzy numbers (Ccuq-ROFNs).

In order to provide further validation of the efficacy of the methodologies introduced in this paper, several existing approaches are employed to solve the same illustrative example, and the outcomes are juxtaposed for comparison. Within this subsection, an encompassing analysis is conducted to succinctly encapsulate the utility and preeminence of the innovative techniques. In this subsection, we compare our proposed approach with complex cubic Pythagorean fuzzy weighted average (CCPFWA) operator and complex cubic Pythagorean fuzzy weighted geometric (CCPFWG) operator proposed by Chinnadurai et al. [59], complex cubic q-rung orthopair fuzzy weighted average (Ccuq-ROFWA) operator and complex cubic q-rung orthopair fuzzy weighted geometric (Ccuq-ROFWG) operator proposed by Ren et al. [41], complex cubic intuitionistic fuzzy Bonferroni mean (CCIFBM) operator and complex cubic intuitionistic fuzzy weighted Bonferroni mean (CCIFWBM) operator and proposed by Mahmood et al. [60].

Among the above-mentioned operators, the Ccuq-ROFWA operator, Ccuq-ROFWG operator, CCPFWA operator and CCPFWG operator do not consider any interrelationship between attributes. Also, neither of these operators have any additional parameters. However, the CCIFBM operator, CCIFWBM operator and the proposed operators have different types of interrelationships between input arguments, and there are some additional parameters involved in these operators as per their basic definitions. Furthermore, the CCIFWBM operator considers the interrelationship between any two attributes, while the proposed operators can take into account not only the interrelationship among multiple attributes but can also offer flexibility in the aggregation process, due to the involvement of additional parameters.

In this subsection, for using CCIFWBM operator, and the parameters are taken as $s=1, t=1$. And for applying WCCuq-ROFHMSM aggregation operators, the selected values of their additional parameters are $q=2, k=2, \gamma =1$. The comparison results between the proposed method with existing methods are discussed in

Table 9. Comparison of sorting results of different decision methods.

Methods	Score Values of the Methods	Rankings Results
Ccuq-ROFWA operator [41]	$S\left(A_1\right)=-0.0597$, $S\left(A_2\right)=-0.0854$, $S\left(A_3\right)=-0.0718$, $S\left(A_4\right)=-0.0369$, $S\left(A_5\right)=-0.1280$	$A_4\succ A_1\succ A_3\succ A_2\succ A_5$
Ccuq-ROFWG operator [41]	$S\left(A_1\right)=-0.0959$, $S\left(A_2\right)=-0.1506$, $S\left(A_3\right)=-0.1346$, $S\left(A_4\right)=-0.0521$, $S\left(A_5\right)=-0.1305$.	$A_4\succ A_1\succ A_5\succ A_3\succ A_2$
CCPFWA operator [59]	$S\left(A_1\right)=0.7165$, $S\left(A_2\right)=0.5986$, $S\left(A_3\right)=0.6639$, $S\left(A_4\right)=0.7726$, $S\left(A_5\right)=0.6125$.	$A_4\succ A_1\succ A_3\succ A_5\succ A_2$
CCPFWG operator [59]	$S\left(A_1\right)=0.4246$, $S\left(A_2\right)=0.5952$, $S\left(A_3\right)=0.4219$, $S\left(A_4\right)=0.6176$, $S\left(A_5\right)=0.3179$.	$A_4\succ A_2\succ A_1\succ A_3\succ A_5$
CCIFBM operator [60]	$S\left(A_1\right)=0.1886$, $S\left(A_2\right)=0.1346$, $S\left(A_3\right)=0.1909$, $S\left(A_4\right)=0.3843$, $S\left(A_5\right)=0.3622$.	$A_4\succ A_5\succ A_3\succ A_1\succ A_2$
CCIFWBM operator [60]	$S\left(A_1\right)=0.3120$, $S\left(A_2\right)=0.3143$, $S\left(A_3\right)=0.2580$, $S\left(A_4\right)=0.5077$, $S\left(A_5\right)=0.4856$.	$A_4\succ A_5\succ A_2\succ A_1\succ A_3$
WCCuq-ROFHMSM operator ($q=2,k=2,\gamma =1$) (The proposed operator)	$S\left(A_1\right)=-0.0515$, $S\left(A_2\right)=-0.0872$, $S\left(A_3\right)=-0.0797$, $S\left(A_4\right)=-0.0422$, $S\left(A_5\right)=-0.0763$.	$A_4\succ A_1\succ A_5\succ A_3\succ A_2$

. From Table 9, although the ranking is slightly different, it can be noted that the best alternative obtained from all the operators under the same example is almost the same.

According to comparisons with the existing methods, and compared with the help of ranking, the proposed methods based on Ccuq-ROFSs in this paper are better than the existing methods for aggregating the complex cubic fuzzy information. Moreover, in comparison with the existing approaches, our methods incorporate supplementary parameters that serve to encapsulate decision makers’ preferences. This enables decision makers to select diverse values in accordance with their risk preferences. Drawing from the aforementioned comparisons and analyses, our method emerges as a more practical, potent and logically sound approach for resolving real-world issues compared to existing methods. Meanwhile, our proposed method is superior to these methods by not only effectively capturing the interrelations among multiple input arguments but also offering decision makers the flexibility to describe their fuzzy information across a broader range. Additionally, our method empowers decision makers to select their risk preferences based on the monotonically decreasing value of the parameter. In summary, our proposed approach is versatile and well suited for dealing with MAGDM problems under Ccuq-ROF environment.

6. Conclusions

In this paper, a novel MAGDM approach based on complex cubic q-rung orthopair fuzzy set (Ccuq-ROFS) is proposed. And some properties, operational laws, score and accuracy functions of Ccuq-ROFS are discussed. Considering that the advantage of Hamacher t-norm and t-conorm provides flexibility in the aggregation process due to the parameter (γ) involved, and the use of the MSM aggregation operator provides flexibility in capturing the interactions between any number of attributes with every possible permutation, this paper developed some novel MSM aggregation operators based on Hamacher t-norm and t-conorm under Ccuq-ROF environment. The main contributions of this study are as follows:

(1)

The concept of the Ccuq-ROFS was proposed by combing the cubic q-rung orthopair fuzzy set (CqROFS) with complex fuzzy set;

(2)

Some Ccuq-ROF aggregation operators were proposed, such as the complex cubic q-rung orthopair fuzzy Hamacher average (Ccuq-ROFHA) operator, the weighted complex cubic q-rung orthopair fuzzy Hamacher average (WCCuq-ROFHA) operator, the complex cubic q-rung orthopair fuzzy Hamacher Maclaurin symmetric mean (Ccuq-ROFHMSM) operator and the weighted complex cubic q-rung orthopair fuzzy Hamacher Maclaurin symmetric mean (WCCuq-ROFHMSM) operator;

(3)

An MAGDM method based on the proposed operators was developed, and a numerical example was illustrated to demonstrate the effectiveness and superiority of the proposed method through comparison with existing methods.

The limitation of the method lies in its computational complexity. The future research work will focus on the following directions to improve the capabilities of the method and alleviate its limitations:

(1)

The method proposed in this article, as a new extended fuzzy environment, can also apply more aggregation operators and different multi-attribute decision-making methods to explore more and more effective decision-making methods;

(2)

The method proposed in this article is decision making on a two-dimensional plane. As the fuzzy environment becomes more complex, it can also be further extended to spherical fuzzy sets, that is, three-dimensional space, to further expand the fuzzy set theory;

(3)

The method proposed in this article has strong processing power in handling fuzzy information and can also be explored and studied in various fields such as scientific research, medical diagnosis, government decision making, enterprise management, commercial competition, etc., to solve more practical decision-making difficulties.