Evaluation of Network Security Service Provider Using 2-Tuple Linguistic Complex q -Rung Orthopair Fuzzy COPRAS Method

. In recent years, network security has become a major concern. Using the Internet to store and analyze data has become an integral aspect of the production and operation of many new and traditional enterprises. However, many enterprises lack the necessary resources to secure information security, and selecting the best network security service provider has become a real issue for many enterprises. This research introduces a novel decision-making method utilizing the 2-tuple linguistic complex q -rung orthopair fuzzy numbers (2TLC q -ROFNs) to tackle this issue. We propose the 2TLC q -ROF concept by combining the complex q -rung orthopair fuzzy set with 2-tuple linguistic terms, including the fundamental deﬁnition, operational rules, scoring, and accuracy functions. Aggregation operators are the fundamental mathematical approach used to combine various inputs into a single output. Taking into account the interaction between the attributes, we develop the 2TLC q -ROF Hamacher (2TLC q -ROFH) operators by using the innovative operational rules. These operators include the 2TLC q -ROFH weighted average (2TLC q - ROFHWA), 2TLC q -ROFH ordered weighted average (2TLC q -ROFHOWA), 2TLC q -ROFH hybrid average (2TLC q -ROFHHA), 2TLC q -ROFH weighted geometric (2TLC q -ROFHWG), 2TLC q -ROFH ordered weighted geometric (2TLC q -ROFHOWG), and 2TLC q -ROFH hybrid geometric (2TLC q -ROFHHG) operators. In addition, we talk about the properties of 2TLC q -ROFH operators such as idempotency, commutativity, monotonicity, and boundedness and also examine their spatial cases. To tackle the problems of the 2TLC q -ROF multiattribute group decision-making (MAGDM) environment, we develop a novel approach according to the COPRAS (complex proportional assessment) model. Finally, to validate the feasibility of the given strategy, we employ a quantitative example related to select the best network security service provider. In comparison with existing approaches, the developed decision-making algorithm is most extensively used and reduces the loss of information.


Introduction
e use of the computer network has spread to every industry as a result of its popularization and advancement. People in the information society are becoming increasingly reliant on the network, and as the network has grown in size and complexity, network security has become a major concern. Although the Internet and other information technologies empower businesses, financial institutions, and even government agencies with the ease of data storing and providers has much impact on ordinary enterprises. As a result, selecting the best network security company is a MADM issue. MAGDM is an integrative research field that combines MADM with the group of decision-makers, particularly analyzing different alternatives through different decision-making (DM) approaches. MAGDM usually provides structures to fuse individual preference information into group preference information. Due to the increasing complexity in economics and management, it is almost impossible for decision-makers (DMs) to collect all information about optimal alternatives associated with MAGDM problems. Hence, uncertainty and fuzziness occur in real-life issues, and how to effectively deal with such kind of fuzziness is crucial to select the best alternative. Many scholars and researchers have worked hard to develop different methods to represent fuzzy DM information in the MAGDM process. Recently, for expressing vagueness and uncertainty, various tools have been developed. For some MAGDM problems, DMs experience problems in describing attribute values of alternatives by using crisp numbers. To describe the uncertainties, Zadeh [1] introduced the fuzzy set (FS) as a generalization of the crisp set, and the value of FS lies between [0, 1]. However, FS has only a membership degree (MD) and ignores the nonmembership degree (NMD) in DM problems. Furthermore, intuitionistic FS (IFS) [2], Pythagorean FS (PFS) [3], and Fermatean FS (FFS) [4], whose elements are pairs of fuzzy numbers, have been introduced. All of the above described FSs demonstrate the MD and NMD. e limitation of MD and NMD is that the sum, square sum, and cube sum of both would belong to [0, 1]. Yager [5] realized that the current IFS, PFS, and FFS frameworks are unable to represent human opinion more realistically and developed the q-rung orthopair FS (q-ROFS), which effectively enhances the scope of information by establishing novel subjective constraints where the qth sum of MD and NMD lies between [0, 1]. If q � 1, q � 2, and q � 3, and then, the q-ROFS is reduced into the IFS, PFS, and FFS, respectively. e q-ROFS theory deals only with one dimension at a time, which sometimes destroys information. However, in real life, we encounter complex natural phenomena in which it becomes significant to integrate the second dimension for the representation of MD and NMD. e development of the second dimension allows complete information to be projected into a set, avoiding any information loss. With the unit disc, Ramot et al. [6] extended the MD range from real number to complex number and proposed the concept of a complex FS (CFS). Furthermore, representing the complex-valued NMD, Alkouri and Salleh [7,8] extended an idea of CFS to complex IFS (CIFS) and also put forward the concept of CIF relations and a distance measure in CIF circumstances. Ullah et al. [9] developed various distance measures of the complex PFS (CPFS) and an algorithm for addressing pattern recognition problems. Liu et al. [10] put forward an innovative, effective, and powerful tool to describe uncertain phenomena named Cq-ROFSs and introduced the Cq-ROF weighted average operator and Cq-ROF weighted geometric operator. To aggregate complex qrung orthopair fuzzy numbers, Liu et al. [11] extended the Einstein operations to Cq-ROFSs and proposed a family of Cq-ROF Einstein averaging operators, such as the Cq-ROF Einstein weighted averaging, the Cq-ROF Einstein ordered weighted averaging, the generalized Cq-ROF Einstein weighted averaging, and the generalized Cq-ROF Einstein ordered weighted averaging operators. e newly proposed Cq-ROFSs are incredibly flexible and efficient, as opposed to many existing FS theories, which can clearly describe the DM perspectives of experts in a complex environment. e amplitude term implies the extent to which an object belongs in a Cq-ROFS, while the phase terms are frequently associated with periodicity. e Cq-ROFS differs from typical q-ROFS theories because of these phase terms. Akram et al. developed novel decision-making methods based on complex Pythagorean fuzzy [12] and complex Fermatean fuzzy N-soft circumstances [13]. e above FSs can only represent information from a quantitative perspective. And it is difficult for DMs to provide precise numerical values to describe their point of view. As a result, Zadeh [14] developed the linguistic variable (LV) as a tool to express qualitative information in DM problems. Following that, various innovative concepts based on the LV and FS were proposed, including intuitionistic linguistic numbers [15], single-valued neutrosophic linguistic set [16], and linguistic q-ROF numbers [17]. Furthermore, Herrera and Martnez [18] introduced the concept of a 2-tuple linguistic FS (2TLFS) established by LV and numerical value to reduce information loss in the DM procedure. Zhao et al. [19] presented an advanced TODIM strategy based on 2-tuple linguistic neutrosophic sets and cumulative prospect theory as a novel approach to MAGDM problems. Based on previous research, Zhang et al. [20] improved dramatically the TODIM technique as well as the cumulative prospect theory under the 2TL Pythagorean fuzzy sets. Naz and Akram [21,22] developed a new DM approach to deal with the MADM problems based on the graph theory. Recently, many research studies [23][24][25][26][27] have developed several DM methods under generalized fuzzy scenario.
ese extensions can effectively describe uncertain fuzzy information in addressing DM problems.
e Cq-ROFS and the 2TL terms, as previously noted, are two strategies for describing the quantitative and qualitative assessment information. Motivated by the concept of a 2TLPFS, Rong et al. [30] introduced the novel concept of 2TLCq-ROFS. e 2TLCq-ROFS is the more universal than existing FSs because we can obtain multiple specific examples by considering some particular circumstances. In the context of 2TLCq-ROFS, the parameters q � 1 and q � 2 degenerate into the 2TLCIFS and the 2TLCPFS, respectively. Furthermore, if the imaginary part of 2TLCq-ROFS is set to zero, it is reduced to a 2TLq-ROFS. From the previous linguistic set research, the 2TLCq-ROFS is stronger because: (1) it can prevent information distortion throughout the linguistic information procedure; (2) it can avoid information loss by expressing assessment information through complex-valued MD and complex-valued NMD; 2 Complexity and (3) in real-life applications, it can tackle problems with two dimensions of information. An aggregation operator (AO) is a well-known approach in the field of information fusion, and it has provided lots of new research results on a variety of topics. To design the MAGDM method, Liu and Wang [31] developed a weighted average and geometric operator for q-ROFS. However, in DM problems, these operators fail to evaluate the interrelationship of attributes. Hamacher product and Hamacher sum were first presented by Hamacher [32] as part of the Hamacher operations. Furthermore, as a generalization of the algebraic and Einstein t-norm and t-conorm, the Hamacher t-norm and t-conorm are more general and flexible. According to a review of the 2TLCq-ROF-AOs, there is limited research by using Hamacher operations to propose new operators. erefore, it is necessary to research AOs utilizing Hamacher operations with 2TLCq-ROF information. Moreover, in decision analysis, selecting the appropriate alternative(s) is critical. As a result, it is crucial to use Hamacher operations to develop 2TLCq-ROF-AOs for solving MAGDM problems. Akram et al. [33] introduced the complex intuitionistic fuzzy Hamacher-weighted averaging operator, complex intuitionistic fuzzy Hamacher ordered weighted averaging operator, complex intuitionistic fuzzy Hamacher weighted geometric operator, and complex intuitionistic fuzzy Hamacher ordered weighted geometric operator. With the use of Hamacher operations and I2TL elements, Faizi et al. [34] developed the intuitionistic 2-tuple linguistic Hamacher weighted average (I2TLHWA) and intuitionistic 2-tuple linguistic Hamacher weighted geometric (I2TLHWG) operators. Rawat [35] introduced q-rung orthopair fuzzy Hamacher Muirhead mean aggregation operators and developed a decision-making approach utilizing proposed operators. Pamucar et al. [36] introduced a novel weighted aggregated sum product assessment approach for advantage prioritization of the electric ferry's sustainable supply chain based on the fuzzy Hamacher weighted averaging function and weighted geometric averaging function.
In recent years, a wide range of methods such as AHP, VIKOR, TOPSIS, and COPRAS that can effectively deal with the ranking procedure has been introduced. e basic purpose of these methods is to select the best alternative by aggregating the information and ranking the objectives according to their significance. Zavadskas et al. [37] proposed the COPRAS method, which compares each alternative and computes their priorities based on attribute weights. COPRAS method is one of the most appropriate methods for ranking the alternatives among all of these methods, and it is widely used for both quantitative and qualitative analyses. e COPRAS method considers direct and proportional reliance of the weights and the utility degree of examined adaptations on a framework of the attributes. To explain logistic regression, boosted regression trees, and random forest, Arabameri et al. [38] built three new ensemble models and assessed them using the COPRAS method. A comparative analysis of COPRAS and the other existing methods such as AHP, TOPSIS, and VIKOR is conducted by Chatterjee et al. [39] and concluded that the COPRAS method indicates good transparency, less calculation time, and a high possibility of graphical understanding of their counterpart strategies. Alipour et al. [40] provided an integrated approach for fuel cell combined with hydrogen supplier selection based on entropy, step-wise weight assessment ratio analysis, and COPRAS methods in a Pythagorean fuzzy environment. Balali et al. [41] utilized the COPRAS approach for risk assessment and the analytic network process technique for determining the weights of each risk assessment criteria. Narang et al. [42] introduced a new hybrid multicriteria decision-making method comprised of group fuzzy COPRAS and fuzzy BCM, followed by a strategy based on the combination of the fuzzy set theory and the COPRAS to rank alternatives in uncertain and ambiguous contexts.
is paper extends the COPRAS method to the 2TLCq-ROF environment, considering the flexibility of 2TLCq-ROFS and the quality of the COPRAS method. e crucial properties of the COPRAS method are (1) during the execution of the process, it evaluates the proportions of the ideal and worst solutions at the same time; (2) this method evaluates the direct and relative dependencies of the significance and the utility degree of the alternatives under the contrary attribute values; and (3) this method is designed to obtain the decision much more effective and sensible. us, considering the advantages of the AOs and the COPRAS method, this article intends to establish an innovative MAGDM approach for managing the information associated with the 2TLCq-ROFS and some new information measures. e motivation and objectives of this study are to find the best network security service provider. After conducting several experiments, the MAGDM method is applied to make the final decision. A significant component of MAGDM is the selection of attributes. Attributes are divided into two types: benefit attribute and cost attribute, to select the best alternative in the application based on whether they are beneficial or not. Existing CFS theories fail to depict uncertain information through the 2TL representation model, which has a higher capability to express linguistic information and can avoid information distortion loss while dealing with linguistic decision problems. e 2TLCq-ROFS and related fundamental theories are developed to enhance CFS theories and provide a reliable tool for experts to express assessment information. Using the 2TLCq-ROFS in this type of MAGDM method gives rise to the clear thinking of DMs who assigns value to complex membership and complex nonmembership functions. Information fusion is essential for aggregating the opinions of DMs. In addition, in a range of practical problems, the correlation of selected attributes is essentially addressed. Several 2TLCq-ROFH operators are presented to address two-dimensional fuzzy information in the light of the excellent superiority of the Hamacher operator. e COPRAS method establishes to rank the given 2TLCq-ROFNs, to develop two algorithms based on COPRAS and AOs to understand DM problems. e approach is described with a numerical illustration to examine the research study.
e main contributions of this research work are as follows: Complexity (i) We introduce the 2TL terms into the complex qrung orthopair fuzzy environment and propose the construction process of 2TLCq-ROFNs. (ii) e 2TLCq-ROFHWA and 2TLCq-ROFHWG operators are proposed combining 2TL terms with complex q-rung orthopair fuzzy set, Hamacher weighted average, and Hamacher weighted geometric operators. (iii) We propose some operational properties and special cases of 2TLCq-ROF Hamacher AOs. (iv) Based on 2TLCq-ROFNs, we improve the COPRAS method and develop a 2TLCq-ROF-COPRAS method to solve the MAGDM problem for ranking of alternatives. (v) We apply the 2TLCq-ROF-COPRAS method to the assessment of the network security service provider. is method is verified to provide a new idea for the assessment of the network security service provider.
To achieve the cognitive approach, the overall framework of this article is as follows: In Section 2, we give several fundamental concepts and definitions including 2TL term, Cq-ROFS, and Hamacher operator. Section 4 presents some new 2TLCq-ROFH aggregation operators, that is, 2TLCq-ROFHWA operator, 2TLCq-ROFHOWA operator, 2TLCq-ROFHHA operator, 2TLCq-ROFHWG operator, 2TLCq-ROFHOWG operator, and 2TLCq-ROFHHG operator, and also discussed some desirable properties and particular cases of them. In Section 5, we design an extended COPRAS method for the MAGDM problem based on the 2TLCq-ROFHWA and 2TLCq-ROFHWG operators. Section 6 employs an example of the best network security service provider to show the application of the proposed method. Some sensitive and comparative analysis is also provided. Finally, Section 7 presents the conclusions, remarks, and also future directions.

Preliminaries
In this section, some correlative basic concepts of LTS, 2TL, and Cq-ROFS, are recapped to facilitate the next sections.

2TL Representation Model and Cq-ROFS
Definition 1 (see [43]). Let there exist a linguistic term set (LTS) S � s ϵ |ϵ � 0, 1, . . . , τ with odd cardinality, where s ϵ indicates a possible linguistic term for a linguistic variable. For instance, an LTS S having seven terms can be described as follows: S � {s 0 � no influence, s 1 � very low influence, s 2 � low influence, s 3 � same influence, s 4 � high influence, s 5 � very high influence, s 6 � very high influence}.
Definition 2 (see [18,44]). Let ϑ be the result of an aggregation of the indices of a set of labels assessed in an LTS S, that is, the result of a symbolic aggregation operation, ϑ ∈ [1, τ], where τ is the cardinality of S. Let ϵ � round(ϑ) and υ � ϑ − ϵ be two values, such that ϵ ∈ [1, τ] and υ ∈ [− 0.5, 0.5), and then, υ is called a symbolic translation.

Hamacher t-Norm and Hamacher t-Conorm.
To extend the existing operations of t-norm and t-conorm, Hamacher [32] introduced the Hamacher product t-norm and Hamacher sum t-conorm as generalizations of t-norms and t-conorms, respectively, as follows: 4 Complexity Clearly, when ϱ � 1, the Hamacher t-norm and t-conorm change into the algebraic t-norm and t-conorm as follows: Again, when ϱ � 2, the Hamacher t-norm and t-conorm reduce to the Einstein t-norm and t-conorm [45] as follows:

Some 2TLCq-ROFH Aggregation Operators
In this section, we present Hamacher operational laws of 2TLCq-ROFS, and based on these Hamacher operational laws, we propose some 2TLCq-ROFH AOs by using weighted average and weighted geometric operators.
Based on Hamacher sum operations of the 2TLCq-ROF values described, we can derive eorem 4.
e 2TLCq-ROFHHA operator has the same properties as eorem 2.
Using the 2TLCq-ROFH operations, we deduce the following theorem from Def. 14.
Based on the parameter ϱ, we can derive the following special cases of eorem 5.
Based on Hamacher sum operations of the 2TLCq-ROF values described, we can derive eorem 8.

An Extended COPRAS Method for MAGDM Approach
In this section, the 2TLCq-ROFHWA and 2TLCq-ROFHWG operators are used to integrate the evaluation values of the 2TLCq-ROF-MAGDM problem and develop a ranking procedure based on the COPRAS method for the 2TLCq-ROF-MAGDM problem. Firstly, we demonstrate the MAGDM problem with the 2TLCq-ROFS. For the 2TLCq-ROF-MAGDM problem, we build an extended COPRAS method, and then, with the assistance of the 2TLCq-ROFHWA and 2TLCq-ROFHWG operators, we fuse the individual input arguments into a combined viewpoint and also describe the DM algorithm.

An Extended COPRAS Method with 2TLCq-ROFH Aggregation Operators.
In this section, we will extend COPRAS method to solve a MAGDM problem within the 2TLCq-ROF environment. According to the work of Zhang [46], group decision-making problems can be solved from two angles: (1) aggregation stage and (2) exploitation stage.
In the aggregation stage, collective evaluation values are obtained from the individual evaluation values of the alternatives. erefore, we employ the 2TLCq-ROFHWA and 2TLCq-ROFHWG operators to combine the individual decision matrices into a group decision matrix. In the exploitation stage, the best alternative(s) is selected according to the priorities of the cumulative evaluation values. We will develop an extended COPRAS method based on the 2TLCq-ROFH aggregation operator, named the 2TLCq-ROFH-COPRAS method, to tackle the information in the group decision matrix.
By utilizing the 2TLCq-ROFHWA operator, the overall value of alternative ℷ κ based on attributes Z ϵ is calculated, the result is computed as follows.
Phase 1. Establish the attributes as well as the alternatives. e goal of the MADM process is to choose the best alternative from a set of m alternatives ℷ � ℷ 1 , ℷ 2 , . . . , ℷ m under the attributes set Z � Z 1 , Z 2 , . . . , Z n . Assume a group of DEs appointed to serve on a panel E � e 1 , e 2 , . . . , e Λ , which was formulated in order to find the optimal alternative(s). Let ℷ � (ℷ λ κϵ ), κ � 1(1)m, ϵ � 1(1)n be the linguistic decision matrix provided by the DEs, where ℷ λ κϵ shows the assessed values of an alternative ℷ κ over attributes Z ϵ in the form of linguistic values for λ th the expert.
Phase 3. Calculate the assessment values of the favorabletype and nonfavorable-type attributes.
Each alternative is defined throughout the designed model in terms of its total of maxima α ⌣ κ (favorable-type) and minima β ⌣ κ (nonfavorable-type); that is, maxima and minima, respectively, produce the optimal outcomes. In such circumstances, α ⌣ κ and β ⌣ κ can be obtained as described below.
Let Δ � 1, 2, . . . , l { } be a favorable-type attribute. Afterward, for every alternative, we compute the greatest possible index value in contexts of 2TLCq-ROFNs, as follows: Let ∇ � l + 1, l + 2, . . . , n { } be a nonfavorable-type attribute. Afterward, for every alternative, we assess the index value in contexts of 2TLCq-ROFNs as follows: where l represents favorable types and n represents the attributes.
Phase 4. Furthermore, we calculate the relative degree Γ κ of each alternative ℷ κ (κ � 1(1)m). Obviously, the bigger the value of Γ κ , the higher the importance of the alternative. Γ κ can be obtained as follows: where 5(α ⌣ κ ) is the score value of α ⌣ κ and 5(β ⌣ κ ) is the score value of β ⌣ κ . Equation (46) can be simplified as Γ κ from (47) reflects the satisfaction measure of each alternative. Based on the Γ κ , maximal value 5 can be determined.
Phase 5. Calculate the summary of priority.
us, the alternative(s) with the associated maximal relative degree is selected among the possible alternatives. Moreover, we can ascertain the utility degree U κ of each alternative with the aid of the Γ κ . U κ can be determined by using the following formula: Hence, the bigger the value U κ , the higher the rank of the alternative ℷ κ .

Numerical Example
In this part, we present a numerical example to evaluate how well our strategy works. With the increased reliance on technology, it is becoming increasingly important to protect all aspects of Internet information and data. Data integrity has become one of the most critical issues for enterprises to address, as the Internet and computer networks increase over time. No matter how little or large your enterprises is, network security is one of the most crucial factors to consider while working over the Internet, LAN, or other technology. While no network is immune to cyber threats, a solid and effective network security solution is critical for securing client data. An effective network security solution minimizes the danger of data theft and tampering in the workplace. Workstations will be protected from malicious software, thanks to network security service provider. It also assures the safety of shared information. In this section, we illustrate the application of 2TLCq-ROF COPRAS method on the choice of network security service provider.
By the above analysis, let L � ℷ 1 , ℷ 2 , ℷ 3 , ℷ 4 , ℷ 5 , ℷ 6 , ℷ 7 be a set of seven network security service providers (see Table 1) and let A � Z 1 , Z 2 , Z 3 , Z 4 be a set of four attributes with weighting vector ξ � (0.17, 0.31, 0.27, 0.25) T . Suppose, seven network security service providers are evaluated by three experts E � e 1 , e 2 , e 3 , with weighting vector φ ′ � (0.2, 0.5, 0.3) T for choosing the best network security service provider. To quantify each LTS, three experts provide their opinions. Based on their experience, each decision expert has an opinion for the selection of the best network security service provider according to four attributes, including (1) Z 1 : web security (2) Z 2 : data loss prevention (3) Z 3 : antivirus and anti-malware software (4) Z 4 : mobile device security Experts should evaluate the effectiveness of network security service provider concerning all attributes in Table 1: Network security service providers (alternatives).
Tier 3 cyber security services ℷ 6 Cyber security consultancy company ℷ 7 Institute of cyber security 18 Complexity accordance with their interaction and identify the most suitable one. Each decision expert uses the 2TLCq-ROFNs to assess each network security service provider's ability to control each attribute.        Collective assessment matrix Alternatives  Tables 2-4) and Table 3 by utilizing (15), the collective 2TLCq-ROF assessing matrix is computed. e aggregated outcomes are listed in Table 5. Construct the assessing matrix (see Table 6) of favorableand nonfavorable-type attributes by utilizing the (15) and (44), and (45).

Parameter Analysis.
Surely, the parameter ϱ has a great influence on the ranking results. e influence of parameters on score values and ranking results based on 2TLCq-    Collective assessment matrix Alternatives ROFHWA and 2TLCq-ROFHWG operators are evaluated in this subsection. We fix the several values of q and evaluate the scores of the overall aggregation. Furthermore, scores are used to rank the alternatives. In Tables 11 and 12 score values are evaluated by varying q and ϱ, respectively, based on the 2TLCq-ROFHWA operator. en, these scores are used to rank the alternatives. Ranking results are used to select the best alternative, and ℷ 1 is the best alternative based on the 2TLCq-ROFHWA operator. As the values of q and ϱ vary, the scores of the seven alternatives change as well, resulting in an irregular change accordingly, based on the 2TLCq-ROFHWA operator. e change in values of q and ϱ have a significant influence on the results of the alternative ranking. Tables 11 and 12 demonstrate that when q and ϱ are changed, the ranking results are relatively stable, and the best alternative remained unchanged. e decision preference can be represented in the actual DM process by varying the values of q and ϱ to obtain the best decision result.
We fix the several values of q and ϱ and evaluate the scores of the overall aggregation. Furthermore, scores are used to rank the alternatives. In Tables 13 and 14 score values are evaluated by varying q and ϱ, respectively, based on the 2TLCq-ROFHWG operator. en, these scores are used to rank the alternatives. e ranking results are used to select the best alternative, and ℷ 1 is the best alternative based on the 2TLCq-ROFHWG operator. As the values of q and ϱ vary, the scores of the seven alternatives change as well, resulting in an irregular change accordingly, based on the 2TLCq-ROFHWG operator. e change in values of q and ϱ has a significant influence on the results of the alternative    Table 11: Score values and ranking outcomes according to the parameter q by the 2TLCq-ROFHWA operator. ranking. Tables 13 and 14 demonstrate that when q and ϱ are changed, the ranking results are relatively stable, and the best alternative remained unchanged. e decision preference can be represented in the actual DM process by varying the values of q and ϱ to obtain the best decision result.

Comparative Analysis.
In this subsection, we compare our proposed work with existing work to demonstrate its reliability and efficiency. Basically, we fused the concept of 2TLq-ROFS with the concept of the complex set to present HWA, HOWA, HHA, HWG, HOWG, and HHG in the context of 2TLCq-ROFS. In real-world problems, the 2TLCq-ROFS effectively deals with uncertain and linguistic information. Determining the weight of the criteria is seen as an important aspect of dealing with the MAGDM challenges. Different criteria may have different weights, and different weights of criteria may provide different results. It is challenging for experts to obtain accurate and objective weight values from real-world data. Because the method of obtaining weight values is complicated, the experts' knowledge and biases may impact their decisions. As a result, as an objective method, the COPRAS method is an excellent solution for addressing these problems. e CO-PRAS method has been investigated extensively in different MAGDM issues. We use various MAGDM strategies to address the network security service provider selection problem to verify the feasibility and superiority of our proposed method. To synthesize the individual assessments of the DMs, we use the 2TLCq-ROFHWA and 2TLCq-  Table 13: Score values and ranking outcomes according to the parameter q by the 2TLCq-ROFHWG operator.

Comparative Analysis with Different Fuzzy Sets.
We compared our suggested operators to the 2TLCIFH (for q � 1), 2TLCPFH (for q � 2), and 2TLCFFH (for q � 3) operators in this subsection. When we set the parameter q � 1,2,3, we can see that the 2TLCIFH, 2TLCPFH, and 2TLCFFH operators are all special cases of our approach. Clearly, our technique can represent more fuzzy information and is applicable in a wide range of real-world MAGDM situations. Furthermore, in a complicated DM environment, the DM's risk attitude is an important factor to consider; our method can accomplish this goal by changing the parameter q, whereas 2TLCIFH, 2TLCPFH, and 2TLCFFH operators cannot dynamically adjust the parameter based on the DM's risk attitude. e characteristic of the proposed set is that the sum of q th power of MD and NMD is constrained to unit disc instead of real numbers in the range [0, 1]. As a result, our proposed work is more effective in solving MAGDM problems. e comparison of the proposed work with existing work is shown in Tables 17 and 18. Utilizing 2TLCq-ROFH, 2TLCFFH, 2TLCPFH, and 2TLCIFH operators the optimal alternative is ℷ 1 based on the 2TLCq-ROFHWA (2TLCq-ROFHWG) operator.
6.4. Advantages of the Proposed Work. Different aggregation operators perform different functions, and the decision expert can select appropriate aggregation operators based on the real-world DM situation. In this subsection, we try to express how the presented approach is superior. e merits of our proposed method are summarized as follows: (i) e presented method is superior to other existing methods because they effectively handle the interdependence of the multi-input arguments. Moreover, they have monotonicity for the parameter ϱ and can impact the risk perspective of the DMs. us, we can conclude that the proposed methods  are significantly preferable and have more comprehensive applications. e extended 2TLCq-ROFH-COPRAS method utilizes 2TLCq-ROFS as the information representation, and 2TLCq-ROF can provide more comprehensive assessment details as it combines the excellent aspects of the Cq-ROFS and 2TL terms. Furthermore, 2TLCq-ROFS can tackle realistic problems both quantitatively and qualitatively. erefore, the proposed approach is clear and has less loss of data.
(ii) e 2TLCq-ROFH-COPRAS method can provide more versatile and robust information fusion and make it more feasible to tackle risk MAGDM problems. It is based on the Hamacher operator so that the attribute's interrelationship can be interpreted and it can be widely applied in various cases by assigning preference values with different parameter t. us, the 2TLCq-ROFH-COPRAS method has an excellent ability to describe the interaction among multi-input parameters and is efficiently applied in the information fusion process.

Conclusions
In this work, we contributed to the development of MAGDM by analyzing difficulties in a 2TLCq-ROF context. e theoretical basis of aggregation operators must be carefully addressed in preparation for their use in decisionmaking. e inadequacies of existing methods, together with the advantageous properties of Hamacher aggregation operators, led us to evaluate their capacity to generate optimal combinations of 2TLCq-ROFNs. e following conclusions can be drawn in summary: (1) e 2TLCq-ROFS idea has been used to describe uncertain data. e Cq-ROFS is an extended version of the CIFS and CPyFS. e Cq-ROFS is defined by two functions that express the degree of complexvalued membership and nonmembership. A flexible parameter q in C q-ROFS will influence its values to reflect information in a broader domain. (2) e 2TL terms can better reflect the human perception and Cq-ROFSs are more reliable due to the q-th power of MD and NMD. So, we developed a new concept of the 2TLCq-ROFS by incorporating the Cq-ROFS with 2TL terms. (3) We expanded the arithmetic mean, geometric mean, and hybrid operators into the 2TLCq-ROF environment and utilized Hamacher operational rules to propose six novel aggregation operators, the 2TLCq-ROFHWA, 2TLCq-ROFHOWA, 2TLCq-ROFHHA, 2TLCq-ROFHWG, 2TLCq-ROFHOWG, and 2TLCq-ROFHHG operators. Several novel characteristics of these proposed operators are being considered. (4) Additionally, we developed a novel DM technique entitled the 2TLCq-ROF-COPRAS approach for solving 2TLCq-ROF-MAGDM problems.
(5) Finally, we solved a problem for selecting the best network security service provider by using our newly developed MAGDM approach.
In a nutshell, the fundamental contribution of this research is that it consolidates both the role of Hamacher aggregation operators and the favorable properties of 2TLCq-ROFNs. is model of uncertain knowledge demonstrates its versatility in presenting vague and imprecise data in complex situations. So, our proposed approach is more generic and versatile than previous approaches.
Moreover, there are some limitations to this approach that should be taken into account in future studies. Firstly, this research focuses entirely on the aggregation of the 2TLCq-ROFNs by utilizing Hamacher AOs. Future research will include a variety of assessment knowledge, such as triangular fuzzy numbers, interval-valued IF number, trapezoidal fuzzy number, and interval-valued picture fuzzy number with the integration of Hamacher operators. erefore, this research will continue to expand. Secondly, the one parameter in our developed approach may present DMs with a quandary regarding how to choose the proper values of parameter in parameter analysis. A methodological combination with alternative techniques may therefore be justified for determining the quantities of parameters objectively.
e DM approach based on the 2TLCq-ROF-COPRAS method, which have complicated assessments due to the involvement of two-step aggregation: (1) for benefit attributes and (2) for cost attributes. erefore, the DM approach can be expanded further with other MAGDM methods, namely, MABAC, EDAS, CODAS, and TOPSIS. In addition, no details were given of the parameter estimation of the attribute weighting values. In the future, more attention may receive on the development of innovative types of AOs, DM strategies [47][48][49][50][51], and distance measures for interval-valued Cq-ROFSs. Furthermore, Hamacher weighted average and geometric operators will be extended to (1) 2TL complex interval-valued q-ROFSs; (2) 2TL complex simplified interval-valued q-ROFSs; and (3) 2TL complex hesitant q-ROFSs.

Data Availability
No data were used to support this study.

Ethical Approval
is article does not contain any studies with human participants or animals performed by any of the authors.

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