#### Abstract

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.

#### 1. Introduction

The 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 processing to efficiently serve society, network security risks also present a threat to enterprises and the entire society. In the information era, data losses are certainly fatal. Many noncyber security organizations find it impractical to maintain a long-term professional network security technology staff due to the extremely professional aspects of network security technology. As a result, many enterprises want to entrust network security defense to professional network security service providers for technical help. This means that choosing the proper network security service 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 . 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. The limitation of MD and NMD is that the sum, square sum, and cube sum of both would belong to . 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 sum of MD and NMD lies between . If , , and , and then, the *q*-ROFS is reduced into the IFS, PFS, and FFS, respectively.

The *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. The 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 C*q*-ROFSs and introduced the C*q*-ROF weighted average operator and C*q*-ROF weighted geometric operator. To aggregate complex *q*-rung orthopair fuzzy numbers, Liu et al. [11] extended the Einstein operations to C*q*-ROFSs and proposed a family of C*q*-ROF Einstein averaging operators, such as the C*q*-ROF Einstein weighted averaging, the C*q*-ROF Einstein ordered weighted averaging, the generalized C*q*-ROF Einstein weighted averaging, and the generalized C*q*-ROF Einstein ordered weighted averaging operators. The newly proposed C*q*-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. The amplitude term implies the extent to which an object belongs in a C*q*-ROFS, while the phase terms are frequently associated with periodicity. The C*q*-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].

The 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 [23–27] have developed several DM methods under generalized fuzzy scenario.

Later, many researchers integrated the 2TL model with several FSs and proposed 2TLIFS [28], 2TLPFS [29], and so forth. These extensions can effectively describe uncertain fuzzy information in addressing DM problems. The C*q*-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 2TLC*q*-ROFS. The 2TLC*q*-ROFS is the more universal than existing FSs because we can obtain multiple specific examples by considering some particular circumstances. In the context of 2TLC*q*-ROFS, the parameters and degenerate into the 2TLCIFS and the 2TLCPFS, respectively. Furthermore, if the imaginary part of 2TLC*q*-ROFS is set to zero, it is reduced to a 2TL*q*-ROFS. From the previous linguistic set research, the 2TLC*q*-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; 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 2TLC*q*-ROF-AOs, there is limited research by using Hamacher operations to propose new operators. Therefore, it is necessary to research AOs utilizing Hamacher operations with 2TLC*q*-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 2TLC*q*-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. The 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. The 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. This paper extends the COPRAS method to the 2TLC*q*-ROF environment, considering the flexibility of 2TLC*q*-ROFS and the quality of the COPRAS method. The 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. Thus, 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 2TLC*q*-ROFS and some new information measures.

The 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. The 2TLC*q*-ROFS and related fundamental theories are developed to enhance CFS theories and provide a reliable tool for experts to express assessment information. Using the 2TLC*q*-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 2TLC*q*-ROFH operators are presented to address two-dimensional fuzzy information in the light of the excellent superiority of the Hamacher operator. The COPRAS method establishes to rank the given 2TLC*q*-ROFNs, to develop two algorithms based on COPRAS and AOs to understand DM problems. The approach is described with a numerical illustration to examine the research study.

The main contributions of this research work are as follows:(i)We introduce the 2TL terms into the complex *q*-rung orthopair fuzzy environment and propose the construction process of 2TLC*q*-ROFNs.(ii)The 2TLC*q*-ROFHWA and 2TLC*q*-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 2TLC*q*-ROF Hamacher AOs.(iv)Based on 2TLC*q*-ROFNs, we improve the COPRAS method and develop a 2TLC*q*-ROF-COPRAS method to solve the MAGDM problem for ranking of alternatives.(v)We apply the 2TLC*q*-ROF-COPRAS method to the assessment of the network security service provider. This 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, C*q*-ROFS, and Hamacher operator. Section 4 presents some new 2TLC*q*-ROFH aggregation operators, that is, 2TLC*q*-ROFHWA operator, 2TLC*q*-ROFHOWA operator, 2TLC*q*-ROFHHA operator, 2TLC*q*-ROFHWG operator, 2TLC*q*-ROFHOWG operator, and 2TLC*q*-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 2TLC*q*-ROFHWA and 2TLC*q*-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.

#### 2. Preliminaries

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

##### 2.1. 2TL Representation Model and C*q*-ROFS

*Definition 1 (see [43]). *Let there exist a linguistic term set (LTS) with odd cardinality, where indicates a possible linguistic term for a linguistic variable. For instance, an LTS having seven terms can be described as follows:*S* = { = no influence, = very low influence, = low influence, = same influence, = high influence, = very high influence, = very high influence}.

If , then the LTS meets the following characteristics:(i)The set is ordered: , iff (ii)Max operator: , iff (iii)Min operator: , iff (iv)Negative operator: Neg such that The 2TL representation model based on the idea of symbolic translation, introduced by Herrera and Martinez [18, 44], is useful for representing the linguistic assessment information by means of a 2-tuple , where is a linguistic label from predefined LTS and is the value of symbolic translation, and .

*Definition 2 (see [18, 44]). *Let be the result of an aggregation of the indices of a set of labels assessed in an LTS , that is, the result of a symbolic aggregation operation, , where is the cardinality of . Let and be two values, such that and , and then, is called a symbolic translation.

*Definition 3 (see [18, 44]). *Let be an LTS and is a number value representing the aggregation result of linguistic symbolic. Then, the function used to obtain the 2TL information equivalent to is defined as

*Definition 4 (see [18, 44]). *Let be an LTS and be a 2-tuple; there exists a function that restore the 2-tuple to its equivalent numerical value , where

*Definition 5 (see [10]). *A C*q*-ROFS is defined aswhere are the complex-valued membership and nonmembership functions, respectively, and are defined aswhere and . Furthermore, and are complex hesitancy degree of . For simplicity, the pair is called the C*q*-ROF number (C*q*-ROFN), where , and .

##### 2.2. 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:

Clearly, when , the Hamacher t-norm and t-conorm change into the algebraic t-norm and t-conorm as follows:

Again, when , the Hamacher t-norm and t-conorm reduce to the Einstein t-norm and t-conorm [45] as follows:

#### 3. 2-Tuple Linguistic Complex *q*-Rung Orthopair Fuzzy Set

*Definition 6. *Let be a LTS with odd cardinality . The 2TLC*q*-ROFS is defined aswhere are termed as 2TL complex-valued membership and nonmembership functions, respectively, and are defined aswhereFor simplicity, the pair is called the 2TLC*q*-ROF number and is defined for and , whereIn order to compare any two 2TLC*q*-ROFNs, their score value and accuracy value are defined as follows:

*Definition 7. *Let be a 2TLC*q*-ROFN. Then, the score value of a 2TLC*q*-ROFN , can be represented as, and its accuracy function is defined as

*Definition 8. *Let , and be two 2TLC*q*-ROFNs; then, these two 2TLC*q*-ROFNs can be compared according to the following rules:(1)If , then (2)If , then(i)If , then (ii)If , then

##### 3.1. Operational Laws for 2TLC*q*-ROFNs

Some operational laws are put forward to compute the 2TLC*q*-ROFNs like complex numbers:

*Definition 9. *Let , and be three 2TLC*q*-ROFNs, then(1)(2)(3)(4)

#### 4. Some 2TLC*q*-ROFH Aggregation Operators

In this section, we present Hamacher operational laws of 2TLC*q*-ROFS, and based on these Hamacher operational laws, we propose some 2TLC*q*-ROFH AOs by using weighted average and weighted geometric operators.

*Definition 10. *Let , and be two 2TLC*q*-ROFNs, with ; then, the basic Hamacher operations between and are given as follows:

*Definition 11. *Let be a collection of 2TLC*q*-ROFNs; then, the 2TLC*q*-ROFHWA operator is defined aswhere be the weight vector of , and , .

We derive the following theorem from Def. 11 using the 2TLC*q*-ROFH operations.

Theorem 1. *Let be a collection of 2TLCq-ROFNs, where. Then, for any, the aggregated value by utilizing 2TLCq-ROFHWA operator is also a 2TLCq-ROFN, andwhere be the weight vector of , and , .*

*Proof. *We use the mathematical induction principle, to prove (20). For , utilizing the operational laws (1) and (3) of Def. 10, we obtain the following result: