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

Authors: Sumera Naz; Muhammad Akram; Mohammed M. Ali Al-Shamiri; Muhammad Ramzan Saeed
Journal: Complexity
 (2022)

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## 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 complexq-rung orthopair fuzzy numbers (2TLCq-ROFNs) to tackle this issue. We propose the 2TLCq-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 2TLCq-ROF Hamacher (2TLCq-ROFH) operators by using the innovative operational rules. These operators include the 2TLCq-ROFH weighted average (2TLCq-ROFHWA), 2TLCq-ROFH ordered weighted average (2TLCq-ROFHOWA), 2TLCq-ROFH hybrid average (2TLCq-ROFHHA), 2TLCq-ROFH weighted geometric (2TLCq-ROFHWG), 2TLCq-ROFH ordered weighted geometric (2TLCq-ROFHOWG), and 2TLCq-ROFH hybrid geometric (2TLCq-ROFHHG) operators. In addition, we talk about the properties of 2TLCq-ROFH operators such as idempotency, commutativity, monotonicity, and boundedness and also examine their spatial cases. To tackle the problems of the 2TLCq-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.


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## Body

## 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 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. The 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.Theq-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 Cq-ROFSs and introduced the Cq-ROF weighted average operator and Cq-ROF weighted geometric operator. To aggregate complex q-rung 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. The 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. The amplitude term implies the extent to which an object belongs in a Cq-ROFS, while the phase terms are frequently associated with periodicity. The 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].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 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. The 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; 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. Therefore, 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. 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 2TLCq-ROF environment, considering the flexibility of 2TLCq-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 2TLCq-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 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. The COPRAS method establishes to rank the given 2TLCq-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 complexq-rung orthopair fuzzy environment and propose the construction process of 2TLCq-ROFNs.(ii)
The 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. 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 Section2, 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.


## 2. Preliminaries

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


### 2.1. 2TL Representation Model and Cq-ROFS

Definition 1 (see [43]).
Let there exist a linguistic term set (LTS)S=sϵ|ϵ=0,1,…,τ with odd cardinality, where sϵ indicates a possible linguistic term for a linguistic variable. For instance, an LTS S having seven terms can be described as follows:
S = {s0  = no influence, s1  = very low influence, s2  = low influence, s3  = same influence, s4  = high influence, s5  = very high influence, s6  = very high influence}.
Ifsϵ,sκ∈S, then the LTS meets the following characteristics:(i)
The set is ordered:sϵ>sκ, iff ϵ>κ(ii)
Max operator:maxsϵ,sκ=sϵ, iff ϵ≥κ(iii)
Min operator:minsϵ,sκ=sϵ, iff ϵ≤κ(iv)
Negative operator: Negsϵ=sκ such that κ=τ−ϵ
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 sϵ,υϵ, where sϵ is a linguistic label from predefined LTS S and υϵ is the value of symbolic translation, and υϵ∈−0.5,0.5.Definition 2 (see [18, 44]).
Letϑ 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=roundϑ 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.Definition 3 (see [18, 44]).
LetS=sϵ|ϵ=1,…,τ be an LTS and ϑ∈1,τ is a number value representing the aggregation result of linguistic symbolic. Then, the function Δ used to obtain the 2TL information equivalent to ϑ is defined as(1)Δ:1,τ⟶S×−0.5,0.5,Δϑ=sϵ,ϵ=roundϑυ=ϑ−ϵ,υ∈−0.5,0.5.Definition 4 (see [18, 44]).
LetS=sϵ|ϵ=1,…,τ be an LTS and sϵ,υϵ be a 2-tuple; there exists a function Δ−1 that restore the 2-tuple to its equivalent numerical value ϑ∈1,τ⊂R, where(2)Δ−1:S×−0.5,0.5⟶1,τ,Δ−1sε,υ=ε+υ=ϑ.Definition 5 (see [10]).
A Cq-ROFS T is defined as(3)T=l,μTl,νTl:l∈L,where μT,νT are the complex-valued membership and nonmembership functions, respectively, and are defined as(4)μTl=ηTlei2πωηTl;νTl=γTlei2πωγTlwhere 0≤ηTl,γTl,ηTql+γTql≤1 and 0≤ωηTql,ωγTql,ωηTql+ωγTql≤1. Furthermore, πTl=1−ηTql+γTql1/q and ωπTl=1−ωηTql+ωγTql1/q are complex hesitancy degree of l. For simplicity, the pair τ=η,ωη,γ,ωγ is called the Cq-ROF number (Cq-ROFN), where 0≤η,γ,ηq+γq≤1, and 0≤ωη,ωγ,ωηq+ωγq≤1.


### 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:(5)TϱHr,s=rsϱ+1−ϱr+s−rsifϱ>0,rsr+s−rsifϱ=0.T∗ϱHr,s=r+s−rs−1−ϱrs1−1−ϱrsifϱ>0,r+s−2rs1−rsifϱ=0..Clearly, whenϱ=1, the Hamacher t-norm and t-conorm change into the algebraic t-norm and t-conorm as follows:(6)Pr,s=rs,P∗r,s=r+s−rs.Again, whenϱ=2, the Hamacher t-norm and t-conorm reduce to the Einstein t-norm and t-conorm [45] as follows:(7)Ir,s=rs1+1−r1−s,I∗r,s=r+s1+rs.


## 2.1. 2TL Representation Model and Cq-ROFS

Definition 1 (see [43]).
Let there exist a linguistic term set (LTS)S=sϵ|ϵ=0,1,…,τ with odd cardinality, where sϵ indicates a possible linguistic term for a linguistic variable. For instance, an LTS S having seven terms can be described as follows:
S = {s0  = no influence, s1  = very low influence, s2  = low influence, s3  = same influence, s4  = high influence, s5  = very high influence, s6  = very high influence}.
Ifsϵ,sκ∈S, then the LTS meets the following characteristics:(i)
The set is ordered:sϵ>sκ, iff ϵ>κ(ii)
Max operator:maxsϵ,sκ=sϵ, iff ϵ≥κ(iii)
Min operator:minsϵ,sκ=sϵ, iff ϵ≤κ(iv)
Negative operator: Negsϵ=sκ such that κ=τ−ϵ
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 sϵ,υϵ, where sϵ is a linguistic label from predefined LTS S and υϵ is the value of symbolic translation, and υϵ∈−0.5,0.5.Definition 2 (see [18, 44]).
Letϑ 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=roundϑ 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.Definition 3 (see [18, 44]).
LetS=sϵ|ϵ=1,…,τ be an LTS and ϑ∈1,τ is a number value representing the aggregation result of linguistic symbolic. Then, the function Δ used to obtain the 2TL information equivalent to ϑ is defined as(1)Δ:1,τ⟶S×−0.5,0.5,Δϑ=sϵ,ϵ=roundϑυ=ϑ−ϵ,υ∈−0.5,0.5.Definition 4 (see [18, 44]).
LetS=sϵ|ϵ=1,…,τ be an LTS and sϵ,υϵ be a 2-tuple; there exists a function Δ−1 that restore the 2-tuple to its equivalent numerical value ϑ∈1,τ⊂R, where(2)Δ−1:S×−0.5,0.5⟶1,τ,Δ−1sε,υ=ε+υ=ϑ.Definition 5 (see [10]).
A Cq-ROFS T is defined as(3)T=l,μTl,νTl:l∈L,where μT,νT are the complex-valued membership and nonmembership functions, respectively, and are defined as(4)μTl=ηTlei2πωηTl;νTl=γTlei2πωγTlwhere 0≤ηTl,γTl,ηTql+γTql≤1 and 0≤ωηTql,ωγTql,ωηTql+ωγTql≤1. Furthermore, πTl=1−ηTql+γTql1/q and ωπTl=1−ωηTql+ωγTql1/q are complex hesitancy degree of l. For simplicity, the pair τ=η,ωη,γ,ωγ is called the Cq-ROF number (Cq-ROFN), where 0≤η,γ,ηq+γq≤1, and 0≤ωη,ωγ,ωηq+ωγq≤1.


## 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:(5)TϱHr,s=rsϱ+1−ϱr+s−rsifϱ>0,rsr+s−rsifϱ=0.T∗ϱHr,s=r+s−rs−1−ϱrs1−1−ϱrsifϱ>0,r+s−2rs1−rsifϱ=0..Clearly, whenϱ=1, the Hamacher t-norm and t-conorm change into the algebraic t-norm and t-conorm as follows:(6)Pr,s=rs,P∗r,s=r+s−rs.Again, whenϱ=2, the Hamacher t-norm and t-conorm reduce to the Einstein t-norm and t-conorm [45] as follows:(7)Ir,s=rs1+1−r1−s,I∗r,s=r+s1+rs.


## 3. 2-Tuple Linguistic Complexq-Rung Orthopair Fuzzy Set

Definition 6.
LetS=s1,s2,…,sτ be a LTS with odd cardinality τ+1. The 2TLCq-ROFS is defined as(8)T=r,υTr,ϑTr:r∈R,where υT,ϑT are termed as 2TL complex-valued membership and nonmembership functions, respectively, and are defined as(9)υTr=spϵ,℘ϵTrei2πωspϵ,℘ϵTr;ϑTr=srϵ,ℜϵTrei2πωsrϵ,ℜϵTr,where(10)0≤Δ−1spϵ,℘ϵTr≤τ,0≤Δ−1srϵ,ℜϵTr≤τ,0≤Δ−1spϵ,℘ϵTqr+Δ−1srϵ,ℜϵTqr≤τq0≤ωΔ−1spϵ,℘ϵTq≤τ,0≤ωΔ−1srϵ,ℜϵTq≤τ,0≤ωΔ−1spϵ,℘ϵTqr+ωΔ−1srϵ,ℜϵTqr≤τq.
For simplicity, the pairsp,℘,ωsp,℘,sr,ℜ,ωsr,ℜ is called the 2TLCq-ROF number and is defined for sp,sr∈S and ℘,ℜ∈−0.5,0.5, where(11)0≤Δ−1spϵ,℘ϵ≤τ,0≤Δ−1srϵ,ℜϵ≤τ,0≤Δ−1spϵ,℘ϵq+Δ−1srϵ,ℜϵq≤τq0≤ωΔ−1spϵ,℘ϵTq≤τ,0≤ωΔ−1srϵ,ℜϵTq≤τ,0≤ωΔ−1spϵ,℘ϵTq+ωΔ−1srϵ,ℜϵTq≤τq.
In order to compare any two 2TLCq-ROFNs, their score value and accuracy value are defined as follows:Definition 7.
Letς˜=sp,℘,ωsp,℘,sr,ℜ,ωsr,ℜ be a 2TLCq-ROFN. Then, the score value Ϝ of a 2TLCq-ROFN ς˜, can be represented as(12)Ϝς˜=Δτ21+Δ−1sp,℘τq−Δ−1sr,ℜτq+1+ωΔ−1sp,℘τq−ωΔ−1sr,ℜτq.
Ϝς˜∈0,τ, and its accuracy function ℶ is defined as(13)ℶς˜=ΔτΔ−1sp,℘τq+Δ−1sr,ℜτq+ωΔ−1sp,℘/τq+ωΔ−1sr,ℜ/τq,ℶς˜∈0,τ.Definition 8.
Letς˜1=sp1,℘1,ωsp1,℘1,sr1,ℜ1,ωsr1,ℜ1, and ς˜2=sp2,℘2,ωs2,℘2,sr2,ℜ2,ωsr2,ℜ2 be two 2TLCq-ROFNs; then, these two 2TLCq-ROFNs can be compared according to the following rules:(1)
IfϜς˜1>Ϝς˜2, then ς˜1≻ς˜2(2)
IfϜς˜1=Ϝς˜2, then(i)
Ifℶς˜1>ℶς˜2, then ς˜1≻ς˜2(ii)
Ifℶς˜1=ℶς˜2, then ς˜1∼ς˜2


### 3.1. Operational Laws for 2TLCq-ROFNs

Some operational laws are put forward to compute the 2TLCq-ROFNs like complex numbers:Definition 9.
Letς˜=sp,℘,ωsp,℘,sr,ℜ,ωsr,ℜ,ς˜1=sp1,℘1,ωsp1,℘1,sr1,ℜ1,ωsr1,ℜ1, and ς˜2=sp2,℘2,ωsp2,℘2,sr2,ℜ2,ωsr2,ℜ2 be three 2TLCq-ROFNs, then(1)
(14)ς˜1⊕ς˜2=Δτq1−1−Δ−1sp1,℘1/τq1−Δ−1sp2,℘2/τq,Δτq1−1−ωΔ−1sp1,℘1/τq1−ωΔ−1sp2,℘2/τqΔτΔ−1sr1,ℜ1/τΔ−1sr2,ℜ2/τ,ΔτωΔ−1sr1,ℜ1/τωΔ−1sr2,ℜ2/τ.(2)
(15)ς˜1⊕ς˜2=ΔτΔ−1sp1,℘1/τΔ−1sp2,℘2/τ,ΔτωΔ−1sp1,℘1/τωΔ−1sp1,℘2/τ,Δτq1−1−Δ−1sr1,ℜ1/τq1−Δ−1sr2,ℜ2/τq,Δτq1−1−ωΔ−1sr1,ℜ1/τq1−ωΔ−1sr2,ℜ2/τq(3)
(16)λς˜=Δτq1−1−Δ−1sp,℘/τqλ,Δτq1−1−ωΔ−1sp,℘/τqλ,ΔτΔ−1sr,ℜ/τλ,ΔτωΔ−1sr,ℜ/ττ,λ0(4)
(17)ς˜λ=ΔτΔ−1sp,℘/τλ,ΔτωΔ−1sp,℘/τλ,Δτq1−1−Δ−1sr,ℜ/τqλ,Δτq1−1−ωΔ−1sr,ℜ/τqλ,,λ>0


## 3.1. Operational Laws for 2TLCq-ROFNs

Some operational laws are put forward to compute the 2TLCq-ROFNs like complex numbers:Definition 9.
Letς˜=sp,℘,ωsp,℘,sr,ℜ,ωsr,ℜ,ς˜1=sp1,℘1,ωsp1,℘1,sr1,ℜ1,ωsr1,ℜ1, and ς˜2=sp2,℘2,ωsp2,℘2,sr2,ℜ2,ωsr2,ℜ2 be three 2TLCq-ROFNs, then(1)
(14)ς˜1⊕ς˜2=Δτq1−1−Δ−1sp1,℘1/τq1−Δ−1sp2,℘2/τq,Δτq1−1−ωΔ−1sp1,℘1/τq1−ωΔ−1sp2,℘2/τqΔτΔ−1sr1,ℜ1/τΔ−1sr2,ℜ2/τ,ΔτωΔ−1sr1,ℜ1/τωΔ−1sr2,ℜ2/τ.(2)
(15)ς˜1⊕ς˜2=ΔτΔ−1sp1,℘1/τΔ−1sp2,℘2/τ,ΔτωΔ−1sp1,℘1/τωΔ−1sp1,℘2/τ,Δτq1−1−Δ−1sr1,ℜ1/τq1−Δ−1sr2,ℜ2/τq,Δτq1−1−ωΔ−1sr1,ℜ1/τq1−ωΔ−1sr2,ℜ2/τq(3)
(16)λς˜=Δτq1−1−Δ−1sp,℘/τqλ,Δτq1−1−ωΔ−1sp,℘/τqλ,ΔτΔ−1sr,ℜ/τλ,ΔτωΔ−1sr,ℜ/ττ,λ0(4)
(17)ς˜λ=ΔτΔ−1sp,℘/τλ,ΔτωΔ−1sp,℘/τλ,Δτq1−1−Δ−1sr,ℜ/τqλ,Δτq1−1−ωΔ−1sr,ℜ/τqλ,,λ>0


## 4. 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.Definition 10.
Letς˜1=sp1,℘1,ωsp1,℘1,sr1,ℜ1,ωsr1,ℜ1, and ς˜2=sp2,℘2,ωsp2,℘2,sr2,ℜ2,ωsr2,ℜ2 be two 2TLCq-ROFNs, with ϱ>0; then, the basic Hamacher operations between ς˜1 and ς˜2 are given as follows:(18)1.ς˜1⊕ς˜2=ΔτΔ−1sp1,℘1/τq+Δ−1sp2,℘2/τq−Δ−1sp1,℘1/τqΔ−1sp2,℘2/τq−1−ϱΔ−1sp1,℘1/τqΔ−1sp2,℘2/τq1−1−ϱΔ−1sp1,℘1/τqΔ−1sp2,℘2/τqqΔτωΔ−1sp1,℘1/τq+ωΔ−1sp2,℘2/τq−ωΔ−1sp1,℘1/τqωΔ−1sp2,℘2/τq−1−ϱωΔ−1sp1,℘1/τqωΔ−1sp2,℘2/τq1−1−ϱωΔ−1sp1,℘1/τqωΔ−1sp2,℘2/τqqΔτΔ−1sr1,ℜ1/τΔ−1sr2,ℜ2/τϱ+1−ϱΔ−1sr1,ℜ1/τq+Δ−1sr2,ℜ2/τq−Δ−1sr1,ℜ1/τqΔ−1sr2,ℜ2/τqqΔτωΔ−1sr1,ℜ1/τωΔ−1sr2,ℜ2/τϱ+1−ϱωΔ−1sr1,ℜ1/τq+ωΔ−1sr2,ℜ2/τq−ωΔ−1sr1,ℜ1/τqωΔ−1sr2,ℜ2/τqq,2ς1˜⊗ς2˜=ΔτΔ−1sp1,℘1/τΔ−1sp2,℘2/τϱ+1−ϱΔ−1sp1,℘1/τq+Δ−1sp2,℘2/τq−Δ−1sp1,℘1/τqΔ−1sp2,℘2/τqq,ΔτωΔ−1sp1,℘1/τωΔ−1sp2,℘2/τϱ+1−ϱωΔ−1sp1,℘1/τq+ωΔ−1sp2,℘2/τq−ωΔ−1sp1,℘1/τqωΔ−1sp2,℘2/τqqΔτΔ−1sr1,ℜ1/τq+Δ−1sr2,ℜ2/τq−Δ−1sr1,ℜ1/τqΔ−1sr2,ℜ2/τq−1−ϱΔ−1sr1,ℜ1/τqΔ−1sr2,ℜ2/τq1−1−ϱΔ−1sr1,ℜ1/τqΔ−1sr2,ℜ2/τqqΔτΔ−1sr1,ℜ1/τq+ωΔ−1sr2,ℜ2/τq−Δ−1sr1,ℜ1/τqωΔ−1sr2,ℜ2/τq−1−ϱωΔ−1sr1,ℜ1/τqωΔ−1sr2,ℜ2/τq1−1−ϱωΔ−1sr1,ℜ1/τqωΔ−1sr2,ℜ2/τqq,3.λς˜1=Δτ1+ϱ−1Δ−1sp1,℘1/τqλ−1−Δ−1sp1,℘1/τqλ1+ϱ−1Δ−1sp1,℘1/τqλ+ϱ−11−Δ−1sp1,℘1/τqλqΔτ1+ϱ−1ωΔ−1sp1,℘1/τqλ−1−ωΔ−1sp1,℘1/τqλ1+ϱ−1ωΔ−1sp1,℘1/τqλ+ϱ−11−ωΔ−1sp1,℘1/τqλqΔτϱΔ−1sr1,ℜ1/τλq1+ϱ−11−Δ−1sr1,ℜ1/τqλ+ϱ−1Δ−1sr1,ℜ1/τqλqΔτϱωΔ−1sr1,ℜ1/τλq1+ϱ−11−ωΔ−1sr1,ℜ1/τqλ+ϱ−1ωΔ−1sr1,ℜ1/τqλq;4.ς1˜λ=ΔτϱΔ−1sp1,℘1/τλq1+ϱ−11−Δ−1sp1,℘1/τqλ+ϱ−1Δ−1sp1,℘1/τqλqΔτϱωΔ−1sp1,℘1/τλq1+ϱ−11−ωΔ−1sp1,℘1/τqλ+ϱ−1ωΔ−1sp1,℘1/τqλqΔτ1+ϱ−1Δ−1sr1,ℜ1/τqλ−1−Δ−1sr1,ℜ1/τqλ1+ϱ−1Δ−1sr1,ℜ1/τqλ+ϱ−11−Δ−1sr1,ℜ1/τqλqΔτ1+ϱ−1ωΔ−1sr1,ℜ1/τqλ−1−ωΔ−1sr1,ℜ1/τqλ1+ϱ−1ωΔ−1sr1,ℜ1/τqλ+ϱ−11−ωΔ−1sr1,ℜ1/τqλq.Definition 11.
Letς˜ε=spε,℘ε,ωspε,℘ε,srε,ℜε,ωsrε,ℜεε=1,2,…,n be a collection of 2TLCq-ROFNs; then, the 2TLCq-ROFHWA operator is defined as(19)2TLCq−ROFHWAφς˜1,ς˜2,…,ς˜n=⊕ϵ=1nφϵς˜ϵ,where φ=φ1,φ2,…,φnT be the weight vector of ς˜ϵϵ=1,2,…,n, and φϵ>0, ∑ϵ=1nφϵ=1.
We derive the following theorem from Def. 11 using the 2TLCq-ROFH operations.Theorem 1.
Letς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵϵ=1,2,…,n be a collection of 2TLCq-ROFNs, whereϱ>0. Then, for anyq≥0, the aggregated value by utilizing 2TLCq-ROFHWA operator is also a 2TLCq-ROFN, and(20)2TLCq−ROFHWAφς˜1,ς˜2,…,ς˜n=⊕ϵ=1nφϵς˜ϵΔτ∏ϵ=1n1+ϱ−1Δ−1spϵ,℘ϵ/τqφϵ−∏ϵ=1n1−Δ−1spϵ,℘ϵ/τqφϵ∏ϵ=1n1+ϱ−1Δ−1spϵ,℘ϵ/τqφϵ+ϱ−1∏ϵ=1n1−Δ−1spϵ,℘ϵ/τqφϵqΔτ∏ϵ=1n1+ϱ−1ωΔ−1spϵ,℘ϵ/τqφϵ−∏ϵ=1n1−ωΔ−1spϵ,℘ϵ/τqφϵ∏ϵ=1n1+ϱ−1Δ−1spϵ,℘ϵ/τqφϵ+ϱ−1∏ϵ=1n1−Δ−1spϵ,℘ϵ/τqφϵqΔτϱq∏ϵ=1nΔ−1srϵ,ℜϵ/τφϵ∏ϵ=1n1+ϱ−1Δ−1srϵ,ℜϵ/τqφϵ+ϱ−1∏ϵ=1n1−Δ−1srϵ,ℜϵ/τqφϵqΔτϱq∏ϵ=1nωΔ−1srϵ,ℜϵ/τφϵ∏ϵ=1n1+ϱ−1Δ−1srϵ,ℜϵ/τqφϵ+ϱ−1∏ϵ=1n1−Δ−1srϵ,ℜϵ/τqφϵq,whereφ=φ1,φ2,…,φnTbe the weight vector ofς˜ϵϵ=1,2,…,n, andφϵ>0, ∑ϵ=1nφϵ=1,ϱ>0.Proof.
We use the mathematical induction principle, to prove (20). For n=2, utilizing the operational laws (1) and (3) of Def. 10, we obtain the following result:(21)φ1ς˜1⊕φ2ς˜2=Δτ1+ϱ−1Δ−1sp1,℘1/τqφ1−1−Δ−1sp1,℘1/τqφ11+ϱ−1Δ−1sp1,℘1/τqφ1+ϱ−11−Δ−1sp1,℘1/τqφ1qΔτ1+ϱ−1ωΔ−1sp1,℘1/τqφ1−1−ωΔ−1sp1,℘1/τqφ11+ϱ−1ωΔ−1sp1,℘1/τqφ1+ϱ−11−ωΔ−1sp1,℘1/τqφ1qΔτϱqΔ−1sr1,ℜ1/τφ11+ϱ−11−Δ−1sr1,ℜ1/τqφ1+ϱ−1Δ−1sr1,ℜ1/τqφ1qΔτϱq∏1=1nωΔ−1sr1,ℜ1/τφ11+ϱ−11−ωΔ−1sr1,ℜ1/τqφ1+ϱ−1ωΔ−1sr1,ℜ1/τqφ1q.⊕Δτ1+ϱ−1Δ−1sp2,℘2/τqφ2−1−Δ−1sp2,℘2/τqφ21+ϱ−1Δ−1sp2,℘2/τqφ2+ϱ−11−Δ−1sp2,℘2/τqφ2qΔτ1+ϱ−1ωΔ−1sp2,℘2/τqφ2−1−ωΔ−1sp2,℘2/τqφ21+ϱ−1ωΔ−1sp2,℘2/τqφ2+ϱ−11−ωΔ−1sp2,℘2/τqφ2qΔτϱqΔ−1sr2,ℜ2/τφ21+ϱ−11−Δ−1sr2,ℜ2/τqφ2+ϱ−1Δ−1sr2,ℜ2/τqφ2qΔτϱq∏2=1nωΔ−1sr2,ℜ2/τφ21+ϱ−11−ωΔ−1sr2,ℜ2/τqφ2+ϱ−1ωΔ−1sr2,ℜ2/τqφ2q=Δτ∏ϵ=121+ϱ−1Δ−1spϵ,℘ϵ/τqφϵ−∏ϵ=121−Δ−1spϵ,℘ϵ/τqφϵ∏ϵ=121+ϱ−1Δ−1spϵ,℘ϵ/τqφϵ+ϱ−1∏ϵ=121−Δ−1spϵ,℘ϵ/τqφϵqΔτ∏ϵ=121+ϱ−1ωΔ−1spϵ,℘ϵ/τqφϵ−∏ϵ=121−ωΔ−1spϵ,℘ϵ/τqφϵ∏ϵ=121+ϱ−1ωΔ−1spϵ,℘ϵ/τqφϵ+ϱ−1∏ϵ=121−ωΔ−1spϵ,℘ϵ/τqφϵqΔτϱq∏ϵ=12Δ−1srϵ,ℜϵ/τφϵ∏ϵ=121+ϱ−11−Δ−1srϵ,ℜϵ/τqφϵ+ϱ−1∏ϵ=12Δ−1srϵ,ℜϵ/τqφϵqΔτϱq∏ϵ=12ωΔ−1srϵ,ℜϵ/τφϵ∏ϵ=121+ϱ−11−ωΔ−1srϵ,ℜϵ/τqφϵ+ϱ−1∏ϵ=12ωΔ−1srϵ,ℜϵ/τqφϵq.
The result in (20) holds true for n=2. Suppose the result is true for n=k, and we can get the result:(22)⊕ϵ=1kφϵς˜ϵ=Δτ∏ϵ=1k1+ϱ−1Δ−1spϵ,℘ϵ/τqφϵ−∏ϵ=1k1−Δ−1spϵ,℘ϵ/τqφϵ∏ϵ=1k1+ϱ−1Δ−1spϵ,℘ϵ/τqφϵ+ϱ−1∏ϵ=1k1−Δ−1spϵ,℘ϵ/τqφϵq,Δτ∏ϵ=1k1+ϱ−1ωΔ−1spϵ,℘ϵ/τqφϵ−∏ϵ=1k1−ωΔ−1spϵ,℘ϵ/τqφϵ∏ϵ=1k∏ϵ=1k1+ϱ−1ωΔ−1spϵ,℘ϵ/τqφϵ+ϱ−1∏ϵ=1k1−ωΔ−1spϵ,℘ϵ/τqφϵqΔτϱq∏ϵ=1kΔ−1srϵ,ℜϵ/τφϵ∏ϵ=1k1+ϱ−11−Δ−1srϵ,ℜϵ/τqφϵ+ϱ−1∏ϵ=1kωΔ−1srϵ,ℜϵ/τqφϵqΔτϱq∏ϵ=1kωΔ−1srϵ,ℜϵ/τφϵ∏ϵ=1k1+ϱ−11−ωΔ−1srϵ,ℜϵ/τqφϵ+ϱ−1∏ϵ=1kωΔ−1srϵ,ℜϵ/τqφϵq.
Whenn=k+1, by using the operational laws of Def. 10, we have(23)⊕ϵ=1k+1φϵς˜ϵ=⊕ϵ=1kφϵς˜ϵ⊕φk+1ς˜k+1=Δτ∏ϵ=1k1+ϱ−1Δ−1spϵ,℘ϵ/τqφϵ−∏ϵ=1k1−Δ−1spϵ,℘ϵ/τqφϵ∏ϵ=1k1+ϱ−1Δ−1spϵ,℘ϵ/τqφϵ∏ϵ=1k+ϱ−1∏ϵ=1k1−Δ−1spϵ,℘ϵ/τqφϵqΔτ∏ϵ=1k1+ϱ−1ωΔ−1spϵ,℘ϵ/τqφϵ−∏ϵ=1k1−ωΔ−1spϵ,℘ϵ/τqφϵ∏ϵ=1k1+ϱ−1ωΔ−1spϵ,℘ϵ/τqφϵ+ϱ−1∏ϵ=1k1−ωΔ−1spϵ,℘ϵ/τqφϵqΔτϱq∏ϵ=1kΔ−1srϵ,ℜϵ/τφϵ∏ϵ=1k1+ϱ−11−Δ−1srϵ,ℜϵ/τqφϵ+ϱ−1∏ϵ=1kΔ−1srϵ,ℜϵ/τqφϵqΔτϱq∏ϵ=1kωΔ−1srϵ,ℜϵ/τφϵ∏ϵ=1k1+ϱ−11−ωΔ−1srϵ,ℜϵ/τqφϵ+ϱ−1∏ϵ=1kωΔ−1srϵ,ℜϵ/τqφϵq⊕Δτ1+ϱ−1Δ−1spk+1,℘k+1/τqφk+1−1−Δ−1spk+1,℘k+1/τqφk+11+ϱ−1Δ−1spk+1,℘k+1/τqφk+1+ϱ−1∏k+1=1n1−Δ−1spk+1,℘k+1/τqφk+1qΔτ1+ϱ−1ωΔ−1spk+1,℘k+1/τqφk+1−1−ωΔ−1spk+1,℘k+1/τqφk+11+ϱ−1ωΔ−1spk+1,℘k+1/τqφk+1+ϱ−11−ωΔ−1spk+1,℘k+1/τqφk+1qΔτϱqΔ−1srk+1,ℜk+1/τφk+11+ϱ−11−Δ−1srk+1,ℜk+1/τqφk+1+ϱ−1Δ−1srk+1,ℜk+1/τqφk+1qΔτϱqωΔ−1srk+1,ℜk+1/τφk+11+ϱ−11−ωΔ−1srk+1,ℜk+1/τqφk+1+ϱ−1ωΔ−1srk+1,ℜk+1/τqφk+1qΔτ∏ϵ=1k+11+ϱ−1Δ−1spϵ,℘ϵ/τqφϵ−∏ϵ=1k+11−Δ−1spϵ,℘ϵ/τqφϵ∏ϵ=1k+11+ϱ−1Δ−1spϵ,℘ϵ/τqφϵ+ϱ−1∏ϵ=1k+11−Δ−1spϵ,℘ϵ/τqφϵqΔτ∏ϵ=1k+11+ϱ−1ωΔ−1spϵ,℘ϵ/τqφϵ−∏ϵ=1k+11−ωΔ−1spϵ,℘ϵ/τqφϵ∏ϵ=1k+11+ϱ−1ωΔ−1spϵ,℘ϵ/τqφϵ+ϱ−1∏ϵ=1k+11−ωΔ−1spϵ,℘ϵ/τqφϵqΔτϱq∏ϵ=1k+1Δ−1srϵ,ℜϵ/τφϵ∏ϵ=1k+11+ϱ−11−Δ−1srϵ,ℜϵ/τqφϵ+ϱ−1∏ϵ=1k+1Δ−1srϵ,ℜϵ/τqφϵqΔτϱq∏ϵ=1k+1ωΔ−1srϵ,ℜϵ/τφϵ∏ϵ=1k+11+ϱ−11−ωΔ−1srϵ,ℜϵ/τqφϵ+ϱ−1∏ϵ=1k+1ωΔ−1srϵ,ℜϵ/τqφϵq.
Hence, (20) is also true for n=k+1, and therefore, (20) holds for all n.
Based on the parameterϱ, we can derive the following special cases of Theorem 1.Remark 1.
Whenϱ=1, the 2TLCq-ROFHWA operator transforms to the 2TLCq-ROF weighted average (2TLCq-ROFWA) operator as follows:(24)2TLCq−ROFWAφς˜1,ς˜2,…,ς˜n=Δτ1−∏ϵ=1n1−Δ−1spϵ,℘ϵτqφϵq,Δτ1−∏ϵ=1n1−ωΔ−1spϵ,℘ϵ/τqφϵqΔτ∏ϵ=1nΔ−1srϵ,ℜϵτφϵ,Δτ∏ϵ=1nωΔ−1srϵ,ℜϵ/τφϵ.Remark 2.
Whenϱ=2, the 2TLCq-ROFHWA operator reduces to the 2TLCq-ROF Einstein weighted average (2TLCq-ROFEWA) operator as follows:(25)2TLCq−ROFEWAφς˜1,ς˜2,…,ς˜n=Δτ∏ϵ=1n1+Δ−1spϵ,℘ϵ/τqφϵ−∏ϵ=1n1−Δ−1spϵ,℘ϵ/τqφϵ∏ϵ=1n1+Δ−1spϵ,℘ϵ/τqφϵ+∏ϵ=1n1−Δ−1spϵ,℘ϵ/τqφϵqΔτ∏ϵ=1n1+ωΔ−1spϵ,℘ϵ/τqφϵ−∏ϵ=1n1−ωΔ−1spϵ,℘ϵ/τqφϵ∏ϵ=1n1+ωΔ−1spϵ,℘ϵ/τqφϵ+∏ϵ=1n1−ωΔ−1spϵ,℘ϵ/τqφϵqΔτ2q∏ϵ=1nΔ−1srϵ,ℜϵ/τφϵ∏ϵ=1n2−Δ−1srϵ,ℜϵ/τqφϵ+∏ϵ=1nΔ−1srϵ,ℜϵ/τqφϵqΔτ2q∏ϵ=1nωΔ−1srϵ,ℜϵ/τφϵ∏ϵ=1n2−ωΔ−1srϵ,ℜϵ/τqφϵ+∏ϵ=1nωΔ−1srϵ,ℜϵ/τqφϵq.Theorem 2.
Letς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵandς˜ϵ′=spϵ′,℘ϵ′,ωspϵ′,℘ϵ′,srϵ′,ℜϵ′,ωsrϵ′,ℜϵ′ϵ=1,2,…,nbe two sets of 2TLCq-ROFNs; then, the 2TLCq-ROFHWA operator has the following properties:(1)
Idempotency. If all 2TLCq-ROFNsς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵϵ=1,2,…,nare equal, that is,ς˜ϵ=ς˜=sp,℘,ωsp,℘,sr,ℜ,ωsr,ℜ, then(26)2TLCq−ROFHWAφς˜1,ς˜2,…,ς˜n=ς˜.(2)
Boundedness. Letς˜ϵϵ=1,2,…,nbe a collection of 2TLCq-ROFNs, and letς˜−=minς˜ϵandς˜+=maxς˜ϵ; then,(27)ς˜−≤2TLCq−ROFHWAφς˜1,ς˜2,…,ς˜n≤ς˜+.(3)
Monotonicity. Letς˜ϵandς˜ϵ′ϵ=1,2,…,nbe two collections of 2TLCq-ROFNs, ifς˜ϵ≤ς˜ϵ′, for allϵ; then,(28)2TLCq−ROFHWAφς˜1,ς˜2,…,ς˜n≤2TLCq−ROFHWAφς˜1′,ς˜2′,…,ς˜n′.Definition 12.
Letς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵϵ=1,2,…,n be a collection of 2TLCq-ROFNs, then the 2TLCq-ROFHOWA operator is defined as(29)2TLCq−ROFHOWAψς˜1,ς˜2,…,ς˜n=⊕ϵ=1nψϵς˜♭ϵ,where ♭1,♭2,…,♭n is a permutation of 1,2,…,n, such that ς˜♭ϵ−1≥ς˜♭ϵ for all ϵ=2,…,n, and ψ=ψ1,ψ2,…,ψnT is the aggregation-associated weight vector such that ψϵ∈0,1 and ∑ϵ=1nψϵ=1, ϱ>0.
Based on Hamacher operations of the 2TLCq-ROF values described, we can derive Theorem 3.Theorem 3.
Letς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵϵ=1,2,…,nbe a collection of 2TLCq-ROFNs, whereϱ>0. Then, for anyq≥0, the aggregated value by utilizing 2TLCq-ROFHOWA operator is also a 2TLCq-ROFN, and(30)2TLCq−ROFHOWAψς˜1,ς˜2,…,ς˜n=⊕ϵ=1nψϵς˜♭ϵ=Δτ∏ϵ=1n1+ϱ−1Δ−1sp♭ϵ,℘♭ϵ/τqψϵ−∏ϵ=1n1−Δ−1sp♭ϵ,℘♭ϵ/τqψϵ∏ϵ=1n1+ϱ−1Δ−1sp♭ϵ,℘♭ϵ/τqψϵ+ϱ−1∏ϵ=1n1−Δ−1sp♭ϵ,℘♭ϵ/τqψϵqΔτ∏ϵ=1n1+ϱ−1ωΔ−1sp♭ϵ,℘♭ϵ/τqψϵ−∏ϵ=1n1−ωΔ−1sp♭ϵ,℘♭ϵ/τqψϵ∏ϵ=1n1+ϱ−1ωΔ−1sp♭ϵ,℘♭ϵ/τqψϵ+ϱ−1∏ϵ=1n1−ωΔ−1sp♭ϵ,℘♭ϵ/τqψϵqΔτϱq∏ϵ=1nΔ−1sr♭ϵ,ℜ♭ϵ/τψϵ∏ϵ=1n1+ϱ−11−Δ−1sr♭ϵ,ℜ♭ϵ/τqψϵ+ϱ−1∏ϵ=1nΔ−1sr♭ϵ,ℜ♭ϵ/τqψϵqΔτϱq∏ϵ=1nωΔ−1sr♭ϵ,ℜ♭ϵ/τψϵ∏ϵ=1n1+ϱ−11−ωΔ−1sr♭ϵ,ℜ♭ϵ/τqψϵ+ϱ−1∏ϵ=1nωΔ−1sr♭ϵ,ℜ♭ϵ/τqψϵq,where ♭1,♭2,…,♭nis a permutation of1,2,…,n, such thatς˜♭ϵ−1≥ς˜♭ϵfor allϵ=2,…,n, andψ=ψ1,ψ2,…,ψnTis the aggregation-associated weighted vector such thatψϵ∈0,1and∑ϵ=1nψϵ=1,ϱ>0.
The 2TLCq-ROFHOWA operator has the same properties as Theorem 2.Definition 13.
Letς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵϵ=1,2,…,n be a collection of 2TLCq-ROFNs, then the 2TLCq-ROFHHA operator is defined as(31)2TLCq−ROFHHAψ,φς˜1,ς˜2,…,ς˜n=⊕ϵ=1nψϵς˜˙♭ϵ,where ψ=ψ1,ψ2,…,ψn is the associated weighting vector such that ψϵ∈0,1 and ∑ϵ=1nψϵ=1, and ς˜˙♭ϵ is the ϵ-th largest element of 2TLCq-ROF arguments ς˜˙ϵς˜˙ϵ=nφϵς˜ϵ,ϵ=1,2,…,n, φ=φ1,φ2,…,φn is the weighted vector of 2TLCq-ROF arguments ς˜ϵ, with φϵ∈0,1 and ∑ϵ=1nφϵ=1, and n is the balancing coefficient. Especially, if ψ=1/n,1/n,…,1/nT, then 2TLCq-ROFHHA is reduced to the 2TLCq-ROFHWA operator; if φ=1/n,1/n,…,1/n, then 2TLCq-ROFHHA is reduced to the 2TLCq-ROFHOWA operator.
Based on Hamacher sum operations of the 2TLCq-ROF values described, we can derive Theorem 4.Theorem 4.
Letς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵϵ=1,2,…,nbe a collection of 2TLCq-ROFNs, whereϱ>0. Then, its aggregated value by utilizing the 2TLCq-ROFHHA operator is also a 2TLCq-ROFN, and(32)2TLCq−ROFHHAψ,φς˜1,ς˜2,…,ς˜n=⊕ϵ=1nψϵς˜˙♭ϵ=Δτ∏ϵ=1n1+ϱ−1Δ−1s˙p♭ϵ,℘˙♭ϵ/τqψϵ−∏ϵ=1n1−Δ−1s˙p♭ϵ,℘˙♭ϵ/τqψϵ∏ϵ=1n1+ϱ−1Δ−1s˙p♭ϵ,℘˙♭ϵ/τqψϵ+ϱ−1∏ϵ=1n1−Δ−1s˙p♭ϵ,℘˙♭ϵ/τqψϵqΔτ∏ϵ=1n1+ϱ−1ωΔ−1s˙p♭ϵ,℘˙♭ϵ/τqψϵ−∏ϵ=1n1−ωΔ−1s˙p♭ϵ,℘˙♭ϵ/τqψϵ∏ϵ=1n1+ϱ−1ωΔ−1s˙p♭ϵ,℘˙♭ϵ/τqψϵ+ϱ−1∏ϵ=1n1−ωΔ−1s˙p♭ϵ,℘˙♭ϵ/τqψϵqΔτϱq∏ϵ=1nΔ−1s˙r♭ϵ,ℜ˙♭ϵ/τψϵ∏ϵ=1n1+ϱ−11−Δ−1s˙r♭ϵ,ℜ˙♭ϵ/τqψϵ+ϱ−1∏ϵ=1nΔ−1s˙r♭ϵ,ℜ˙♭ϵ/τqψϵqΔτϱq∏ϵ=1nωΔ−1s˙r♭ϵ,ℜ˙♭ϵ/τψϵ∏ϵ=1n1+ϱ−11−ωΔ−1s˙r♭ϵ,ℜ˙♭ϵ/τqψϵ+ϱ−1∏ϵ=1nωΔ−1s˙r♭ϵ,ℜ˙♭ϵ/τqψϵq,where ψ=ψ1,ψ2,…,ψnis the aggregation-associated weighting vector such thatψϵ∈0,1and∑ϵ=1nψϵ=1, andς˜˙♭ϵis theϵ-th largest element of 2TLCq-ROF argumentsς˜˙ϵς˜˙ϵ=nφϵς˜ϵ,ϵ=1,2,…,n,φ=φ1,φ2,…,φnis the weighting vector of 2TLCq-ROF argumentsς˜ϵ, withφϵ∈0,1and∑ϵ=1nφϵ=1, andnis the balancing coefficient.The 2TLCq-ROFHHA operator has the same properties as Theorem 2.Definition 14.
Letς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵϵ=1,2,…,n be a collection of 2TLCq-ROFNs; then, the 2TLCq-ROFHWG operator is defined as(33)2TLCq−ROFHWGφς˜1,ς˜2,…,ς˜n=⊗ϵ=1nς˜ϵφϵ,where φ=φ1,φ2,…,φnT be the weight vector of ς˜ϵϵ=1,2,…,n, and φϵ>0, ∑ϵ=1nφϵ=1.Using the 2TLCq-ROFH operations, we deduce the following theorem from Def. 14.Theorem 5.
Letς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵϵ=1,2,…,nbe a collection of 2TLCq-ROFNs, whereϱ>0. Then, its aggregated value by utilizing 2TLCq-ROFHWG operator is also a 2TLCq-ROFN, and(34)2TLCq−ROFHWGφς˜1,ς˜2,…,ς˜n=⊗ϵ=1nς˜ϵφϵ=Δτϱq∏ϵ=1nΔ−1spϵ,℘ϵ/τφϵ∏ϵ=1n1+ϱ−11−Δ−1spϵ,℘ϵ/τqφϵ+ϱ−1∏ϵ=1nΔ−1spϵ,℘ϵ/τqφϵqΔτϱq∏ϵ=1nωΔ−1spϵ,℘ϵ/τφϵ∏ϵ=1n1+ϱ−11−ωΔ−1spϵ,℘ϵ/τqφϵ+ϱ−1∏ϵ=1nωΔ−1spϵ,℘ϵ/τqφϵqΔτ∏ϵ=1n1+ϱ−1Δ−1srϵ,ℜϵ/τqφϵ−∏ϵ=1n1−Δ−1srϵ,ℜϵ/τqφϵ∏ϵ=1n1+ϱ−1Δ−1srϵ,ℜϵ/τqφϵ∏ϵ=1n+ϱ−1∏ϵ=1n1−Δ−1srϵ,ℜϵ/τqφϵqΔτ∏ϵ=1n1+ϱ−1ωΔ−1srϵ,ℜϵ/τqφϵ−∏ϵ=1n1−ωΔ−1srϵ,ℜϵ/τqφϵ∏ϵ=1n1+ϱ−1ωΔ−1srϵ,ℜϵ/τqφϵ+ϱ−1∏ϵ=1n1−ωΔ−1srϵ,ℜϵ/τqφϵq,whereφ=φ1,φ2,…,φnTbe the weight vector ofς˜ϵϵ=1,2,…,n, andφϵ>0, ∑ϵ=1nφϵ=1,ϱ>0.Based on the parameterϱ, we can derive the following special cases of Theorem 5.Remark 3.
Whenϱ=1, the 2TLCq-ROFHWG operator transforms to the 2TLCq-ROF weighted geometric (2TLCq-ROFWG) operator as follows:(35)2TLCq−ROFWGφς˜1,ς˜2,…,ς˜n=Δτ∏ϵ=1nΔ−1spϵ,℘ϵ/τφϵ,Δτ∏ϵ=1nωΔ−1spϵ,℘ϵ/τφϵ,Δτq1−∏ϵ=1n1−Δ−1srϵ,ℜϵ/τqφϵ,Δτq1−∏ϵ=1n1−ωΔ−1srϵ,ℜϵ/τqφϵ.Remark 4.
Whenϱ=2, the 2TLCq-ROFHWG operator reduces to the 2TLCq-ROF Einstein weighted geometric (2TLCq-ROFEWG) operator as follows:(36)2TLCq−ROFEWGφς˜1,ς˜2,…,ς˜n=Δτ2q∏ϵ=1nΔ−1spϵ,℘ϵ/τφϵ∏ϵ=1n2−Δ−1spϵ,℘ϵ/τqφϵ+∏ϵ=1nΔ−1spϵ,℘ϵ/τqφϵqΔτ2q∏ϵ=1nωΔ−1spϵ,℘ϵ/τφϵ∏ϵ=1n2−ωΔ−1spϵ,℘ϵ/τqφϵ+∏ϵ=1nωΔ−1spϵ,℘ϵ/τqφϵqΔτ∏ϵ=1n1+Δ−1srϵ,ℜϵ/τqφϵ−∏ϵ=1n1−Δ−1srϵ,ℜϵ/τqφϵ∏ϵ=1n1+Δ−1srϵ,ℜϵ/τqφϵ+∏ϵ=1n1−Δ−1srϵ,ℜϵ/τqφϵqΔτ∏ϵ=1n1+ωΔ−1srϵ,ℜϵ/τqφϵ−∏ϵ=1n1−ωΔ−1srϵ,ℜϵ/τqφϵ∏ϵ=1n1+ωΔ−1srϵ,ℜϵ/τqφϵ+∏ϵ=1n1−ωΔ−1srϵ,ℜϵ/τqφϵq.Theorem 6.
Letς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵ and ς˜ϵ′=spϵ′,℘ϵ′,ωspϵ′,℘ϵ′,srϵ′,ℜϵ′,ωsrϵ′,ℜϵ′ϵ=1,2,…,nbe two sets of 2TLCq-ROFNs; then, the 2TLCq-ROFHWG operator has the following properties:(1)
Idempotency. If all 2TLCq-ROFNs ς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵϵ=1,2,…,nare equal, that is,,ς˜ϵ=ς˜=sp,℘,ωsp,℘,sr,ℜ,ωsr,ℜ, then(37)2TLCq−ROFHWGφς˜1,ς˜2,…,ς˜n=ς˜.(2)
Boundedness. Letς˜ϵϵ=1,2,…,nbe a collection of 2TLCq-ROFNs, and letς˜−=minς˜ϵandς˜+=maxς˜ϵ; then,(38)ς˜−≤2TLCq−ROFHWGφς˜1,ς˜2,…,ς˜n≤ς˜+.(3)
Monotonicity. Letς˜ϵandς˜ϵ′ϵ=1,2,…,nbe the two collections of 2TLCq-ROFNs, ifς˜ϵ≤ς˜ϵ′, for allϵ; then,(39)2TLCq−ROFHWGφς˜1,ς˜2,…,ς˜n≤2TLCq−ROFHWGφς˜1′,ς˜2′,…,ς˜n′.Definition 15.
Letς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵϵ=1,2,…,n be a collection of 2TLCq-ROFNs, then the 2TLCq-ROFHOWG operator is defined as(40)2TLCq−ROFHOWGψς˜1,ς˜2,…,ς˜n=⊗ϵ=1nς˜♭ϵψϵ,where ♭1,♭2,…,♭n is a permutation of 1,2,…,n, such that ς˜♭ϵ−1≥ς˜♭ϵ for all ϵ=2,…,n, and ψ=ψ1,ψ2,…,ψnT is the aggregation-associated weighted vector such that ψϵ∈0,1 and ∑ϵ=1nψϵ=1.Theorem 7.
Letς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵϵ=1,2,…,nbe a collection of 2TLCq-ROFNs, whereϱ>0. Then, its aggregated value by utilizing the 2TLCq-ROFHOWG operator is also a 2TLCq-ROFN, and(41)2TLCq−ROFHOWGψς˜1,ς˜2,…,ς˜n=⊗ϵ=1nς˜♭ϵψϵ=Δτϱq∏ϵ=1nΔ−1sp♭ϵ,℘♭ϵ/τψϵ∏ϵ=1n1+ϱ−11−Δ−1sp♭ϵ,℘♭ϵ/τqψϵ+ϱ−1∏ϵ=1nΔ−1sp♭ϵ,℘♭ϵ/τqψϵqΔτϱq∏ϵ=1nωΔ−1sp♭ϵ,℘♭ϵ/τψϵ∏ϵ=1n1+ϱ−11−ωΔ−1sp♭ϵ,℘♭ϵ/τqψϵ+ϱ−1∏ϵ=1nωΔ−1sp♭ϵ,℘♭ϵ/τqψϵqΔτ∏ϵ=1n1+ϱ−1Δ−1sr♭ϵ,ℜ♭ϵ/τqψϵ−∏ϵ=1n1−Δ−1sr♭ϵ,ℜ♭ϵ/τqψϵ∏ϵ=1n1+ϱ−1Δ−1sr♭ϵ,ℜ♭ϵ/τqψϵ+ϱ−1∏ϵ=1n1−Δ−1sr♭ϵ,ℜ♭ϵ/τqψϵqΔτ∏ϵ=1n1+ϱ−1ωΔ−1sr♭ϵ,ℜ♭ϵ/τqψϵ−∏ϵ=1n1−ωΔ−1sr♭ϵ,ℜ♭ϵ/τqψϵ∏ϵ=1n1+ϱ−1ωΔ−1sr♭ϵ,ℜ♭ϵ/τqψϵ+ϱ−1∏ϵ=1n1−ωΔ−1sr♭ϵ,ℜ♭ϵ/τqψϵq,where♭1,♭2,…,♭nis a permutation of1,2,…,n, such thatς˜♭ϵ−1≥ς˜♭ϵfor allϵ=2,…,n, andψ=ψ1,ψ2,…,ψnTis the aggregation-associated weighted vector such thatψϵ∈0,1and∑ϵ=1nψϵ=1,ϱ>0.The 2TLCq-ROFHOWG operator has the same properties as Theorem 2.Definition 16.
Letς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵϵ=1,2,…,n be a collection of 2TLCq-ROFNs; then, the 2TLCq-ROFHHG operator is defined as(42)2TLCq−ROFHHGψ,φς˜1,ς˜2,…,ς˜n=⊗ϵ=1nς˜˙♭ϵψϵ,where ψ=ψ1,ψ2,…,ψn be the associated weighting vector with ψϵ∈0,1, ∑ϵ=1nψϵ=1, and ς˜˙♭ϵ is the ϵ-th largest element of 2TLCq-ROF arguments ς˜˙ϵς˜˙ϵ=nφϵς˜ϵ,ϵ=1,2,…,n, φ=φ1,φ2,…,φn is the weighting vector of 2TLCq-ROF arguments ς˜ϵ, with φϵ∈0,1 and ∑ϵ=1nφϵ=1, and n is the balancing coefficient, ϱ>0. Especially, if ψ=1/n,1/n,…,1/nT, then 2TLCq-ROFHHG is reduced to the 2TLCq-ROFHWG operator; if φ=1/n,1/n,…,1/n, then 2TLCq-ROFHHG is reduced to the 2TLCq-ROFHOWG operator.
Based on Hamacher sum operations of the 2TLCq-ROF values described, we can derive Theorem 8.Theorem 8.
Letς˜ϵ=spϵ,℘ϵ,ωspϵ,℘ϵ,srϵ,ℜϵ,ωsrϵ,ℜϵϵ=1,2,…,n be a collection of 2TLCq-ROFNs, whereϱ>0. Then, its aggregated value by utilizing the 2TLCq-ROFHHG operator is also a 2TLCq-ROFN, and(43)2TLCq−ROFHHGψ,φς˜1,ς˜2,…,ς˜n=⊗ϵ=1nς˜♭ϵψϵ=Δτϱq∏ϵ=1nΔ−1s˙p♭ϵ,℘˙♭ϵ/τψϵ∏ϵ=1n1+ϱ−11−Δ−1s˙p♭ϵ,℘˙♭ϵ/τqψϵ+ϱ−1∏ϵ=1nΔ−1s˙p♭ϵ,℘˙♭ϵ/τqψϵqΔτϱq∏ϵ=1nωΔ−1s˙p♭ϵ,℘˙♭ϵ/τψϵ∏ϵ=1n1+ϱ−11−ωΔ−1s˙p♭ϵ,℘˙♭ϵ/τqψϵ+ϱ−1∏ϵ=1nωΔ−1s˙p♭ϵ,℘˙♭ϵ/τqψϵqΔτ∏ϵ=1n1+ϱ−1Δ−1s˙r♭ϵ,ℜ˙♭ϵ/τqψϵ−∏ϵ=1n1−Δ−1s˙r♭ϵ,ℜ˙♭ϵ/τqψϵ∏ϵ=1n1+ϱ−1Δ−1s˙r♭ϵ,ℜ˙♭ϵ/τqψϵ+ϱ−1∏ϵ=1n1−Δ−1s˙r♭ϵ,ℜ˙♭ϵ/τqψϵqΔτ∏ϵ=1n1+ϱ−1ωΔ−1s˙r♭ϵ,ℜ˙♭ϵ/τqψϵ−∏ϵ=1n1−ωΔ−1s˙r♭ϵ,ℜ˙♭ϵ/τqψϵ∏ϵ=1n1+ϱ−1ωΔ−1s˙r♭ϵ,ℜ˙♭ϵ/τqψϵ+ϱ−1∏ϵ=1n1−ωΔ−1s˙r♭ϵ,ℜ˙♭ϵ/τqψϵq,whereψ=ψ1,ψ2,…,ψnbe the associated weighting vector withψϵ∈0,1,∑ϵ=1nψϵ=1, andς˜˙♭ϵis theϵ-th largest element of 2TLCq-ROF argumentsς˜ϵ.ς˜ϵ.=ς˜ϵnφϵ,ϵ=1,2,…,n,φ=φ1,φ2,…,φnis the weighting vector of 2TLCq-ROF argumentsς˜ϵ, withφϵ∈0,1and∑ϵ=1nφϵ=1, andnis the balancing coefficient,ϱ>0.The 2TLCq-ROFHHG operator has the same properties as Theorem 2.


## 5. 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.


### 5.1. 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. Therefore, 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 ℷκ based on attributes ℏϵ is calculated, the result is computed as follows.Phase 1.
Establish the attributes as well as the alternatives.
The goal of the MADM process is to choose the best alternative from a set ofm alternatives ℷ=ℷ1,ℷ2,…,ℷm under the attributes set ℏ=ℏ1,ℏ2,…,ℏn. Assume a group of DEs appointed to serve on a panel E=e1,e2,…,eΛ, which was formulated in order to find the optimal alternative(s). Let ℷ=ℷκϵλ,κ=11m,ϵ=11n be the linguistic decision matrix provided by the DEs, where ℷκϵλ shows the assessed values of an alternative ℷκ over attributes ℏϵ in the form of linguistic values for λth the expert.Phase 2.
Construct the aggregated 2TLCq-ROF decision matrix.
We stimulate the 2TLCq-ROFHWA and 2TLCq-ROFHWG operators to obtain the decision matrix subsequently and achieve R=ηλϵΛ×n from (20) and (34).Phase 3.
Calculate the assessment values of the favorable-type and nonfavorable-type attributes.
Each alternative is defined throughout the designed model in terms of its total of maximaα⌣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.
LetΔ=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:(44)α⌣κ=⊕ϵ=1lφϵηκϵ,κ=11m.
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:(45)β⌣κ=⊕ϵ=l+1nφϵηκϵ,κ=11m,where l represents favorable types and n represents the attributes.Phase 4.
Furthermore, we calculate the relative degreeΓκ of each alternative ℷκκ=11m. Obviously, the bigger the value of Γκ, the higher the importance of the alternative. Γκ can be obtained as follows:(46)Γκ=Ϝα⌣κ+minκϜβ⌣κ∑κ=1pϜβ⌣κϜβ⌣κ∑κ=1pminκϜβ⌣κ/Ϝβ⌣κ,κ=11m,where Ϝα⌣κ is the score value of α⌣κ and Ϝβ⌣κ is the score value of β⌣κ.
Equation (46) can be simplified as(47)Γκ=Ϝα⌣κ+∑κ=1pϜβ⌣κϜβ⌣κ∑κ=1p1/Ϝβ⌣κ,κ=11m.
Γκ from (47) reflects the satisfaction measure of each alternative. Based on the Γκ, maximal value Ϝ can be determined.Phase 5.
Calculate the summary of priority.(48)Ϝ=maxκΓκ,κ=11m.
Thus, the alternative(s) with the associated maximal relative degree is selected among the possible alternatives. Moreover, we can ascertain the utility degreeUκ of each alternative with the aid of the Γκ. Uκ can be determined by using the following formula:(49)Uκ=ΓκΓmax×100%,κ=11m.
Hence, the bigger the valueUκ, the higher the rank of the alternative ℷκ.


## 5.1. 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. Therefore, 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 ℷκ based on attributes ℏϵ is calculated, the result is computed as follows.Phase 1.
Establish the attributes as well as the alternatives.
The goal of the MADM process is to choose the best alternative from a set ofm alternatives ℷ=ℷ1,ℷ2,…,ℷm under the attributes set ℏ=ℏ1,ℏ2,…,ℏn. Assume a group of DEs appointed to serve on a panel E=e1,e2,…,eΛ, which was formulated in order to find the optimal alternative(s). Let ℷ=ℷκϵλ,κ=11m,ϵ=11n be the linguistic decision matrix provided by the DEs, where ℷκϵλ shows the assessed values of an alternative ℷκ over attributes ℏϵ in the form of linguistic values for λth the expert.Phase 2.
Construct the aggregated 2TLCq-ROF decision matrix.
We stimulate the 2TLCq-ROFHWA and 2TLCq-ROFHWG operators to obtain the decision matrix subsequently and achieve R=ηλϵΛ×n from (20) and (34).Phase 3.
Calculate the assessment values of the favorable-type and nonfavorable-type attributes.
Each alternative is defined throughout the designed model in terms of its total of maximaα⌣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.
LetΔ=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:(44)α⌣κ=⊕ϵ=1lφϵηκϵ,κ=11m.
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:(45)β⌣κ=⊕ϵ=l+1nφϵηκϵ,κ=11m,where l represents favorable types and n represents the attributes.Phase 4.
Furthermore, we calculate the relative degreeΓκ of each alternative ℷκκ=11m. Obviously, the bigger the value of Γκ, the higher the importance of the alternative. Γκ can be obtained as follows:(46)Γκ=Ϝα⌣κ+minκϜβ⌣κ∑κ=1pϜβ⌣κϜβ⌣κ∑κ=1pminκϜβ⌣κ/Ϝβ⌣κ,κ=11m,where Ϝα⌣κ is the score value of α⌣κ and Ϝβ⌣κ is the score value of β⌣κ.
Equation (46) can be simplified as(47)Γκ=Ϝα⌣κ+∑κ=1pϜβ⌣κϜβ⌣κ∑κ=1p1/Ϝβ⌣κ,κ=11m.
Γκ from (47) reflects the satisfaction measure of each alternative. Based on the Γκ, maximal value Ϝ can be determined.Phase 5.
Calculate the summary of priority.(48)Ϝ=maxκΓκ,κ=11m.
Thus, the alternative(s) with the associated maximal relative degree is selected among the possible alternatives. Moreover, we can ascertain the utility degreeUκ of each alternative with the aid of the Γκ. Uκ can be determined by using the following formula:(49)Uκ=ΓκΓmax×100%,κ=11m.
Hence, the bigger the valueUκ, the higher the rank of the alternative ℷκ.


## 6. 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, letL=ℷ1,ℷ2,ℷ3,ℷ4,ℷ5,ℷ6,ℷ7 be a set of seven network security service providers (see Table 1) and let A=ℏ1,ℏ2,ℏ3,ℏ4 be a set of four attributes with weighting vector ξ=0.17,0.31,0.27,0.25T. Suppose, seven network security service providers are evaluated by three experts E=e1,e2,e3, with weighting vector φ′=0.2,0.5,0.3T 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)
ℏ1: web security(2)
ℏ2: data loss prevention(3)
ℏ3: antivirus and anti-malware software(4)
ℏ4: mobile device securityTable 1
Network security service providers (alternatives).
ℷ1Delta techℷ2Catalytic securityℷ3PakCERTℷ4Tranchulasℷ5Tier 3 cyber security servicesℷ6Cyber security consultancy companyℷ7Institute of cyber securityExperts should evaluate the effectiveness of network security service provider concerning all attributes in 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.


### 6.1. The Outcomes of a Case Study

#### 6.1.1. Decision-Making Procedure Based on the 2TLCq-ROFHWA Operator

On the basis of 2TLCq-ROFNs matrix (see Tables 2–4) and Table 3 by utilizing (15), the collective 2TLCq-ROF assessing matrix is computed. The aggregated outcomes are listed in Table 5.Table 2
2TLCq-ROF decision matrix by e1.
ℏ1ℏ2ℏ3ℏ4ℷ1s1,0,s4,0,s7,0,s3,0s2,0,s3,0,s3,0,s5,0s5,0,s2,0,s6,0,s63,0s4,0,s3,0,s2,0,s3,0ℷ2s1,0,s4,0,s5,0,s3,0s3,0,s4,0,s2,0,s3,0s2,0,s4,0,s2,0,s4,0s2,0,s4,0,s5,0,s3,0ℷ3s3,0,s2,0,s2,0,s3,0s3,0,s3,0,s4,0,s2,0s2,0,s4,0,s3,0,s1,0s3,0,s3,0,s2,0,s5,0ℷ4s2,0,s2,0,s1,0,s3,0s2,0,s5,0,s5,0,s4,0s4,0,s1,0,s4,0,s5,0s4,0,s3,0,s2,0,s3,0ℷ5s2,0,s3,0,s2,0,s4,0s3,0,s2,0,s4,0,s2,0s3,0,s2,0,s1,0,s5,0s2,0,s3,0,s5,0,s1,0ℷ6s1,0,s5,0,s6,0,s3,0s5,0,s1,0,s3,0,s6,0s5,0,s6,0,s3,0,s2,0s6,0,s5,0,s2,0,s3,0ℷ7s3,0,s4,0,s5,0,s2,0s5,0,s4,0,s2,0,s3,0s3,0,s4,0,s5,0,s2,0s6,0,s3,0,s1,0,s3,0Table 3
2TLCq-ROF decision matrix by e2.
ℏ1ℏ2ℏ3ℏ4ℷ1s4,0,s2,0,s2,0,s4,0s1,0,s5,0,s6,0,s5,0s2,0,s3,0,s2,0,s3,0s3,0,s5,0,s3,0,s2,0ℷ2s1,0,s3,0,s4,0,s3,0s7,0,s2,0,s2,0,s5,0s3,0,s4,0,s2,0,s4,0s2,0,s4,0,s3,0,s2,0ℷ3.s4,0,s2,0,s4,0,s3,0s2,0,s2,0,s5,0,s3,0s3,0,s4,0,s3,0,s3,0s2,0,s3,0,s5,0,s2,0ℷ4s3,0,s4,0,s2,0,s3,0s3,0,s4,0,s3,0,s5,0s5,0,s3,0,s2,0,s6,0s4,0,s2,0,s4,0,s2,0ℷ5s1,0,s5,0,s4,0,s2,0s5,0,s1,0,s2,0,s4,0s5,0,s2,0,s4,0,s6,0s2,0,s5,0,s6,0,s4,0ℷ6s6,0,s5,0,s3,0,s4,0s5,0,s6,0,s4,0,s3,0s1,0,s2,0,s5,0,s4,0s2,0,s1,0,s4,0,s5,0ℷ7s3,0,s2,0,s5,0,s3,0s3,0,s4,0,s4,0,s5,0s4,0,s2,0,s3,0,s2,0s4,0,s2,0,s5,0,s3,0Table 4
2TLCq-ROF decision matrix by e3.
ℏ1ℏ2ℏ3ℏ4ℷ1s5,0,s3,0,s3,0,s2,0s2,0,s3,0,s3,0,s2,0s6,0,s2,0,s4,0,s5,0s71,0,s2,0,s1,0,s2,0ℷ2s6,0,s1,0,s3,0,s2,0s2,0,s3,0,s5,0,s3,0s5,0,s2,0,s2,0,s6,0s3,0,s2,0,s4,0,s7,0ℷ3s3,0,s2,0,s3,0,s5,0s2,0,s3,0,s2,0,s7,0s4,0,s2,0,s3,0,s5,0s7,0,s3,0,s1,0,s3,0ℷ4s2,0,s4,0,s3,0,s4,0s4,0,s2,0,s3,0,s7,0s5,0,s2,0,s1,0,s2,0s4,0,s3,0,s2,0,s3,0ℷ5s6,0,s4,0,s1,0,s5,0s4,0,s6,0,s5,0,s1,0s2,0,s2,0,s1,0,s1,0s1,0,s1,0,s2,0,s2,0ℷ6s2,0,s1,0,s2,0,s1,0s1,0,s2,0,s1,0,s2,0s3,0,s2,0,s6,0,s5,0s2,0,s3,0,s5,0,s6,0ℷ7s2,0,s3,0,s4,0,s5,0s1,0,s3,0,s4,0,s3,0s2,0,s7,0,s3,0,s3,0s5,0,s1,0,s4,0,s2,0Table 5
Collective the 2TLCq-ROF assessing matrix by the 2TLCq-ROFHWA operator.
Collective assessment matrixAlternativesℏ1ℏ2ℷ1s4,0.1104,s3,0.2121,s3,0.3992,s3,−0.0131s2,−0.2220,s4,0.1377,s4,0.0177,s4,−0.1636ℷ2s4,0.3900,s3,0.2269,s4,−0.0636,s3,−0.3423s6,−0.3618,s3,0.2121,s3,−0.3501,s4,−0.3015ℷ3s4,−0.4988,s2,−0,s3,−0.0131,s4,−0.4888s2,0.4392,s3,−0.2782,s4,−0.4214,s4,−0.4626ℷ4s3,−0.4560,s4,−0.3231,s2,−0.1639,s3,0.2730s3,0.2607,s4,0.1274,s4,−0.4888,s5,0.2652ℷ5s4,0.4042,s4,0.3162,s2,0.1513,s3,0.2636s4,0.3162,s4,0.4042,s3,0.2636,s2,0.1513ℷ6s5,−0.2594,s5,−0.4421,s3,0.3176,s2,0.4301s5,−0.4421,s5,−0.2594,s2,0.4301,s3,0.3176ℷ7s3,−0.2008,s3,0.2121,s5,−0.3164,s3,0.1141s4,−0.1736,s4,−0.2260,s3,0.2588,s4,−0.3015Alternativesℏ3ℏ4ℷ1s5,−0.1136,s3,−0.4560,s3,0.4760,s4,0.3572s5,0.3895,s4,0.0637,s2,−0.0876,s2,0.2596ℷ2s4,−0.1528,s4,−0.3231,s2,−0,s5,−0.4502s2,0.4392,s4,−0.3231,s4,−0.1739,s3,0.3992ℷ3s3,0.2607,s4,−0.3231,s3,−0,s3,−0.4655s5,0.2476,s3,0,s2,0.3664,s3,−0.0087ℷ4s5,−0.2419,s2,0.4721,s2,0.0056,s4,0.1656s4,−0,s3,−0.2782,s3,−0.3540,s3,−0.4478ℷ5s4,0.0637s2,−0,s2,−0.2521,s3,0.4067s2,−0.1587,s4,0.0459,s4,0.1656,s2,0.1513ℷ6s4,−0.2112,s4,0.4229,s5,−0.4196,s3,0.4961s4,0.4229,s4,−0.2112,s3,0.4961,s5,−0.4196ℷ7s3,0.3791,s5,0.3429,s4,−0.4888,s2,0.2596s5,0.0936,s2,0.3572,s3,−0.0828,s3,−0.3423Construct the assessing matrix (see Table6) of favorable- and nonfavorable-type attributes by utilizing the (15) and (44), and (45).Table 6
Assessing values of favorable- and nonfavorable-type attributes by the 2TLCq-ROFHWA operator.
Assessed values of favorable attributesAssessed values of nonfavorable attributesAlternativesℷ1s4,−0.0223,s3,0.0357,s5,0.4604,s6,−0.4488s4,−0.4918,s3,0.1659,s6,−0.4243,s6,−0.1948ℷ2s3,−0.1058,s3,−0.1778,s6,0.4078,s6,−0.0561s4,0.3583,s3,0.0065,s4,0.3236,s6,−0.1917ℷ3s4,−0.2054,s2,0.1860,s6,−0.3941,s6,0.0175s3,−0.4656,s3,−0.1525,s5,0.2148,s5,−0.0104ℷ4s3,−0.1086,s3,−0.4202,s5,0.3485,s6,−0.2301s4,−0.3919,s3,0.1491,s5,−0.3007,s6,0.2438ℷ5s3,−0.1620,s3,0.3348,s6,0.0219,s6,−0.4349s4,−0.3433,s3,0.2963,s4,0.4429,s5,−0.3404ℷ6s4,−0.3514,s3,0.3253,s6,0.1586,s6,0.2394s4,−0.3057,s4,−0.0024,s5,0.2002,s5,0.2980ℷ7s4,−0.3787,s2,0.2501,s6,0.2304,s6,−0.2224s3,0.1682,s4,0.0718,s5,0.2776,s5,−0.0865Calculate the scoring outcomes of alternatives for favorableϜα⌣κ- and nonfavorable Ϝβ⌣κ-type attributes by utilizing (12), and establish the ranking order by using the 2TLCq-ROF-COPRAS method. The evaluation outcomes are listed in Table 7.Table 7
Assessing outcomes of alternatives and ranking order by utilizing the 2TLCq-ROF-COPRAS.
AlternativesBenefit attributes scoresCost attributes scoresRelative degreeUtility degreeRankingϜα⌣κϜβ⌣κΓκUκℷ10.40830.38710.87491.00001ℷ20.32910.43620.74320.84944ℷ30.37370.42360.80020.91463ℷ40.38940.39380.84810.96937ℷ50.37270.46560.76070.86955ℷ60.33800.43430.75400.86176ℷ70.35210.44000.76260.87162


#### 6.1.2. Decision-Making Procedure Based on the 2TLCq-ROFHWG Operator

On the basis of 2TLCq-ROFN matrix (see Tables 2–4) and by utilizing (34), the collective 2TLCq-ROF assessing matrix is computed. The aggregated outcomes are listed in Table 8.Table 8
Collective 2TLCq-ROF assessing matrix by the 2TLCq-ROFHWG operator.
Collective assessment matrixAlternativesℏ1ℏ2ℷ1s3,−0.1510,s3,−0.2137,s5,0.2476,s3,0.3791s2,−0.4840,s4,−0.3015,s5,−0.1524,s5,−0.4294ℷ2s2,−0.2581,s2,0.3599,s4,0.1984,s3,−0.2008s4,−0.1147,s3,−0.2137,s4,−0.2766,s4,0.1377ℷ3s3,0.3691,s2,−0,s3,0.3791,s4,−0.0669s2,0.2596,s3,−0.4478,s4,0.2507,s5,0.2600ℷ4s2,0.3533,s3,0.2588,s2,0.3572,s3,0.3951s3,−0.0989,s3,0.4961,s4,−0.0669,s6,−0.2770ℷ5s2,0.1438,s4,0.0278,s3,0.2084,s4,0.0368s4,0.0278,s2,0.1438,s4,0.0368,s3,0.2084ℷ6s3,−0.4227,s3,0.1313,s5,−0.4991,s3,0.3488s3,0.1313,s3,−0.4227,s3,0.3488,s5,−0.4991ℷ7s3,−0.3423,s3,−0.2137,s5,−0.2419,s4,−0.1528s3,−0.4655,s4,−0.3273,s4,−0.3231,s4,0.1377Alternativesℏ3ℏ4ℷ1s4,−0.2726,s2,0.3533,s5,−0.3732,s5,−0.0540s4,0.3320,s3,0.2820,s2,0.4721,s2,0.4392ℷ2s3,0.1141,s3,0.2588,s2,−0,s5,−0.1351s2,0.2596,s3,0.2588,s4,0.1365,s5,0.2476ℷ3s3,−0.0989,s3,0.2588,s3,0,s4,−0.1736s3,0.3992,s3,−0,s4,−0.0332,s4,−0.1827ℷ4s5,−0.3164,s2,−0.0876,s3,0.0107,s5,0.1435s4,0,s3,−0.4478,s3,0.2427,s3,−0.2782ℷ5s3,0.2820,s2,−0,s3,0.1730,s5,0.1343s2,−0.3752,s3,−0.3275,s5,0.1435,s3,0.2084ℷ6s2,0.2714,s3,−0.1754,s5,0.0585,s4,0.1274s3,−0.1754,s2,0.2714,s4,0.1274,s5,0.0585ℷ7s3,−0.0131,s4,−0.2947,s4,−0.0669,s2,0.4392s5,−0.1395,s2,−0.1639,s4,0.2350,s3,−0.2008Construct the assessing matrix (see Table9) of favorable- and nonfavorable-type attributes by utilizing the (34) and (44), and (45).Table 9
Assessing values of favorable- and nonfavorable-type attributes by the 2TLCq-ROFHWG operator.
Assessed values of favorable attributesAssessed values of nonfavorable attributesAlternativesℷ1s6,0.2971,s6,−0.0599,s3,0.3893,s2,0.3551s4,0.3090,s5,−0.0370,s4,0.1242,s4,0.1305ℷ2s5,0.1244,s6,−0.2031,s3,0.3356,s4,−0.2780s5,0.3839,s5,−0.0397,s3,−0.1818,s4,−0.0743ℷ3s6,0.1372,s6,−0.4356,s3,0.0175,s3,0.1010s5,−0.4634,s5,−0.1616,s3,0.3172,s4,0.1169ℷ4s6,0.0442,s6,−0.2335,s2,0.3901,s2,0.4500s5,0.4866,s5,−0.3619,s3,0.1240,s5,−0.2501ℷ5s5,−0.0879,s6,−0.0028,s4,−0.3074,s3,−0.1011s6,−0.4945,s4,0.0610,s3,0.2265,s4,−0.1764ℷ6s6,−0.3032,s6,−0.4044,s3,0.4362,s4,−0.3434s5,−0.3116,s5,−0.3189,s4,−0.1944,s4,−0.2280ℷ7s6,0.3865,s5,0.2536,s4,−0.4234,s3,−0.3159s5,−0.2756,s6,−0.4764,s3,0.3102,s3,0.1518Calculate the scoring outcomes of alternatives for favorableϜα⌣κ- and nonfavorable Ϝβ⌣κ-type attributes by utilizing (12) and establish the ranking order by using the 2TLCq-ROF-COPRAS method. The evaluation outcomes are listed in Table 10.Table 10
Assessing outcomes of alternatives and ranking order by utilizing the 2TLCq-ROF-COPRAS.
AlternativesBenefit attributes scoresCost attributes scoresRelative degreeUtility degreeRankingϜα⌣κϜβ⌣κΓκUκℷ10.66200.52271.23541.00001ℷ20.59170.56991.11760.90473ℷ30.63440.53441.19530.96754ℷ40.64470.54671.19300.96566ℷ50.59880.55301.14070.92347ℷ60.60470.53361.16630.94415ℷ70.63490.57391.15710.93662


### 6.2. Parameter Analysis

Surely, the parameterϱ has a great influence on the ranking results. The influence of parameters on score values and ranking results based on 2TLCq-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. Then, these scores are used to rank the alternatives. Ranking results are used to select the best alternative, and ℷ1 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. The 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. The decision preference can be represented in the actual DM process by varying the values of q and ϱ to obtain the best decision result.Table 11
Score values and ranking outcomes according to the parameterq by the 2TLCq-ROFHWA operator.
ParametersϜℷ1Ϝℷ2Ϝℷ3Ϝℷ4Ϝℷ5Ϝℷ6Ϝℷ7Rankingq=21.00000.78770.88650.92860.80960.82460.8228ℷ1>ℷ4>ℷ3>ℷ6>ℷ7>ℷ5>ℷ2q=31.00000.81770.89750.95130.83710.83940.8457ℷ1>ℷ4>ℷ3>ℷ7>ℷ6>ℷ5>ℷ2q=51.00000.87830.93230.98190.80960.88550.8962ℷ1>ℷ4>ℷ3>ℷ7>ℷ6>ℷ2>ℷ5q=71.00000.92360.96150.99500.94450.92780.9359ℷ1>ℷ4>ℷ3>ℷ5>ℷ7>ℷ6>ℷ2q=110.99990.97120.99011.00000.98450.82460.9776ℷ4>ℷ1>ℷ3>ℷ5>ℷ7>ℷ2>ℷ6q=131.00000.98210.99530.99990.99160.98660.9865ℷ1>ℷ4>ℷ3>ℷ5>ℷ6>ℷ7>ℷ2q=171.00000.99240.99900.99930.99700.99550.9945ℷ1>ℷ4>ℷ3>ℷ5>ℷ6>ℷ7>ℷ2q=191.00000.99480.99960.99930.99810.99730.9962ℷ1>ℷ3>ℷ4>ℷ5>ℷ6>ℷ7>ℷ2q=231.00000.99730.99990.99940.99910.99880.9981ℷ1>ℷ3>ℷ4>ℷ5>ℷ6>ℷ7>ℷ2q=291.00000.78770.88650.92860.80960.82460.8228ℷ1>ℷ4>ℷ3>ℷ6>ℷ7>ℷ5>ℷ2Table 12
Score values and ranking outcomes according to the parameterϱ by the 2TLCq-ROFHWA operator.
ParametersϜℷ1Ϝℷ2Ϝℷ3Ϝℷ4Ϝℷ5Ϝℷ6Ϝℷ7Rankingϱ=21.00000.86070.92540.97090.88220.87390.8820ℷ1>ℷ4>ℷ3>ℷ5>ℷ7>ℷ6>ℷ2ϱ=31.00000.84940.91460.96930.86950.86180.8717ℷ1>ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ2ϱ=51.00000.83330.89950.96620.85110.84520.8568ℷ1>ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ2ϱ=71.00000.82170.88870.96330.83750.83380.8460ℷ1>ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ2ϱ=111.00000.80480.87330.95820.81740.81830.8303ℷ1>ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ2ϱ=131.00000.79840.86730.95600.80950.81250.8243ℷ1>ℷ4>ℷ3>ℷ7>ℷ6>ℷ5>ℷ2ϱ=171.00000.78770.85760.95190.79620.80340.8143ℷ1>ℷ4>ℷ3>ℷ7>ℷ6>ℷ5>ℷ2ϱ=191.00000.78320.85350.95010.79050.79970.8102ℷ1>ℷ4>ℷ3>ℷ7>ℷ6>ℷ5>ℷ2ϱ=231.00000.77540.84640.94670.78050.79350.8030ℷ1>ℷ4>ℷ3>ℷ7>ℷ6>ℷ5>ℷ2ϱ=291.00000.76590.83770.94220.76800.78610.7942ℷ1>ℷ4>ℷ3>ℷ7>ℷ6>ℷ5>ℷ2We fix the several values ofq 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. Then, these scores are used to rank the alternatives. The ranking results are used to select the best alternative, and ℷ1 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. The change in values of q and ϱ has a significant influence on the results of the alternative 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. The decision preference can be represented in the actual DM process by varying the values of q and ϱ to obtain the best decision result.Table 13
Score values and ranking outcomes according to the parameterq by the 2TLCq-ROFHWG operator.
ParametersϜℷ1Ϝℷ2Ϝℷ3Ϝℷ4Ϝℷ5Ϝℷ6Ϝℷ7Rankingq=21.00000.90170.96160.96610.92310.94130.9254ℷ1>ℷ4>ℷ3>ℷ6>ℷ7>ℷ5>ℷ2q=31.00000.89840.96320.96350.91980.94100.9274ℷ1>ℷ4>ℷ3>ℷ6>ℷ7>ℷ5>ℷ2q=51.00000.91700.97300.97060.93240.94970.9492ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2q=71.00000.94610.98430.98280.95620.96500.9731ℷ1>ℷ3>ℷ4>ℷ7>ℷ6>ℷ5>ℷ2q=111.00000.98280.99800.99790.98770.98880.9960ℷ1>ℷ3>ℷ4>ℷ7>ℷ6>ℷ5>ℷ2q=130.99930.99010.99991.00000.99350.99380.9986ℷ4>ℷ3>ℷ1>ℷ7>ℷ6>ℷ5>ℷ2q=170.99790.99510.99991.00000.99710.99700.9988ℷ4>ℷ3>ℷ7>ℷ1>ℷ5>ℷ6>ℷ2q=190.99800.99650.99991.00000.99800.99790.9989ℷ4>ℷ3>ℷ7>ℷ1=ℷ5>ℷ6>ℷ2q=230.99860.99800.99991.00000.99900.99890.9992ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ1>ℷ2q=290.99930.99910.99991.00000.99950.99940.9996ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ1>ℷ2Table 14
Score values and ranking outcomes according to the parameterϱ by the 2TLCq-ROFHWG operator.
ParametersϜℷ1Ϝℷ2Ϝℷ3Ϝℷ4Ϝℷ5Ϝℷ6Ϝℷ7Rankingϱ=21.00000.90750.96930.96980.92600.94510.9392ℷ1>ℷ4>ℷ3>ℷ6>ℷ7>ℷ5>ℷ2ϱ=31.00000.90460.96740.96560.92330.94400.9366ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=51.00000.90230.96630.96140.92140.94400.9342ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=71.00000.90170.96630.95940.92100.94490.9333ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=111.00000.90200.96720.95770.92180.94710.9329ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=131.00000.90250.96770.95730.92250.94820.9330ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=171.00000.90370.96880.95690.92400.95020.9334ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=191.00000.90440.96940.95690.92480.95110.9337ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=231.00000.90560.97030.95690.92630.95280.9342ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=291.00000.90750.97160.95700.92850.95500.9351ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2


### 6.3. 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. The COPRAS 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-ROFHWG operators. Comparative analysis with different FSs (2TLCIFH, 2TLCPFH, 2TLCFFH), aggregation operators (2TLCq-ROFWA, 2TLCq-ROFEWA, 2TLCq-ROFWG, 2TLCq-ROFEWG), and methods (CODAS, EDAS, TOPSIS) is illustrated as follows.


#### 6.3.1. Comparative Analysis with Different Aggregation Operators

Our proposed operator is more realistic and superior because it can evaluate the interrelationship of fused arguments and scientifically consider the complexities of humans in practical MAGDM problems, whereas the 2TLCq-ROFWA (2TLCq-ROFWG) and 2TLCq-ROFEWA (2TLCq-ROFEWG) operators are unable to evaluate the interrelationships between fused arguments and cannot consider the complexity of DMs. As a result, proposed operators are more general in expressing fuzzy information.The 2TLCq-ROFHWA and 2TLCq-ROFHWG operators can provide more effective and flexible information fusion, making it more suitable for dealing with MAGDM problems with dependent attributes, as demonstrated in Tables 15 and 16. As a result, our technique overcomes the constraints of 2TLCq-ROFWA (2TLCq-ROFWG) and 2TLCq-ROFEWA (2TLCq-ROFEWG) aggregation operators. As a result, 2TLCq-ROFHWA and 2TLCq-ROFHWG operators are highly suitable for dealing with MAGDM in a 2TLCq-ROF environment.Table 15
Evaluation outcomes by utilizing different aggregation operators based on the 2TLCq-ROFHWA operator.
Alternatives2TLCq-ROFHRanking2TLCq-ROFWARanking2TLCq-ROFEWARankingℷ11.0000I1.0000I1.0000Iℷ20.8494IV0.8771IV0.8607IVℷ30.9146III0.9413III0.9254IIIℷ40.9693VII0.9724V0.9709Vℷ50.8695V0.9006VII0.8822VIIℷ60.8617VI0.8925VI0.8739VIℷ70.8716II0.8970II0.8820IITable 16
Evaluation outcomes by utilizing different aggregation operators based on the 2TLCq-ROFHWG operator.
Alternatives2TLCq-ROFHRanking2TLCq-ROFWGRanking2TLCq-ROFEWGRankingℷ11.0000I1.0000I1.0000Iℷ20.9047III0.9138IV0.9075IVℷ30.9675IV0.9741III0.9693IIIℷ40.9656IV0.9784VI0.9698VIℷ50.9234VII0.9321VII0.9260VIIℷ60.9441V0.9486V0.9451Vℷ70.9366II0.9447II0.9392II


#### 6.3.2. Comparative Analysis with Different Fuzzy Sets

We compared our suggested operators to the 2TLCIFH (forq=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.The characteristic of the proposed set is that the sum ofq 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. The 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 ℷ1 based on the 2TLCq-ROFHWA (2TLCq-ROFHWG) operator.Table 17
Evaluation outcomes by utilizing different FSs based on the 2TLCq-ROFHWA operator.
Alternatives2TLCq-ROFHRanking2TLCFFHRanking2TLCPFHRanking2TLCIFHRankingℷ11.0000I1.0000I1.0000I1.0000Iℷ20.8494IV0.8177IV0.7877IV0.7833IVℷ30.9146III0.8975III0.8865III0.8990IIIℷ40.9693VII0.9513VII0.9286VI0.9121VIℷ50.8695V0.8371VI0.8096VII0.8134VIIℷ60.8617VI0.8394V0.8246V0.8387Vℷ70.8716II0.8457II0.8228II0.8252IITable 18
Evaluation outcomes by utilizing different FSs based on the 2TLCq-ROFHWG operator.
Alternatives2TLCq-ROFHRanking2TLCFFHRanking2TLCPFHRanking2TLCIFHRankingℷ11.0000I1.0000I1.0000I1.0000Iℷ20.9047III0.8984IV0.9017IV0.9171IVℷ30.9675IV0.9632III0.9616III0.9639IIIℷ40.9656VI0.9635VI0.9661VI0.9745VIℷ50.9234VII0.9198VII0.9231VII0.9330VIIℷ60.9441V0.9410V0.9413V0.9464Vℷ70.9366II0.9274II0.9254II0.9341II


#### 6.3.3. Comparative Analysis with Different MAGDM Methods

This subsection compares the proposed method to some existing methodologies in order to highlight the superiority of our method. To begin, our developed method is compared to CODAS, EDAS, and TOPSIS methods. We carefully calculate the results of these methods and compare them with our proposed method. For the CODAS, EDAS, and TOPSIS methods, the ranking order isℷ6>ℷ7>ℷ2>ℷ1>ℷ3>ℷ4>ℷ5, ℷ6>ℷ7>ℷ5>ℷ2>ℷ1>ℷ3>ℷ4, and ℷ5>ℷ3>ℷ4>ℷ2>ℷ1>ℷ7>ℷ6, respectively, based on the 2TLCq-ROFHWA operator in Table 19. Therefore, the best alternatives are ℷ6, ℷ6, and ℷ5 according to CODAS, EDAS, and TOPSIS methods, respectively. However, in our purposed method (COPRAS), the ranking order is ℷ1>ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ2, and the best (worst) alternative is ℷ1ℷ2, based on the 2TLCq-ROFHWA operator as shown in Table 19.Table 19
Evaluation outcomes by utilizing different methodologies based on the 2TLCq-ROFHWA operator.
AlternativesCOPRASRankingCODASRankingEDASRankingTOPSISRankingℷ11.0000I−0.0439IV0.5277VI−1.3229Vℷ20.8494IV0.0539VII0.5331VII−1.9971IIIℷ30.9146III−0.2050II0.2846V−1.1370IVℷ40.9693VII−0.5126I0.0000II−1.1445Iℷ50.8695V−3.3930III0.5648I0.0000IIℷ60.8617VI2.7821IV0.7304III−2.7610VIIℷ70.8716II1.3185V0.5820IV−2.2180VIIn Table20 the ranking order is ℷ6>ℷ2>ℷ3>ℷ5>ℷ7>ℷ1>ℷ4, ℷ7>ℷ5>ℷ4>ℷ2>ℷ3>ℷ1>ℷ6, and ℷ7>ℷ4>ℷ1>ℷ2>ℷ6>ℷ5>ℷ3 for the CODAS, EDAS, and TOPSIS methods, respectively, based on the 2TLCq-ROFHWG operator. Therefore, the best alternatives are ℷ6, ℷ7, and ℷ7 according to CODAS, EDAS, and TOPSIS methods, respectively. In our purposed method (COPRAS), the ranking order is ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2, based on the 2TLCq-ROFHWG operator as shown in Table 20. However, the best and the worst alternative remains the same that are ℷ1 and ℷ2 that show the reliability and effectiveness of our purposed method.Table 20
Evaluation outcomes by utilizing different methodologies based on the 2TLCq-ROFHWG operator.
AlternativesCOPRASRankingCODASRankingEDASRankingTOPSISRankingℷ11.0000I−1.4587VI0.2452VII−0.2422VIIℷ20.9047III1.4557II0.4219V−0.5476IVℷ30.9675IV0.8752III0.2846IV−0.6408Iℷ40.9656VI−1.6952VII0.4511II−0.1472IIℷ50.9234VII0.4201V0.4725III−0.6044VIℷ60.9441V1.6987I0.0913I−0.5873Vℷ70.9366II−1.2959IV0.9024VI−0.0653III


### 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. The merits of our proposed method are summarized as follows:(i)
The 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 parameterϱ and can impact the risk perspective of the DMs. Thus, we can conclude that the proposed methods are significantly preferable and have more comprehensive applications. The 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. Therefore, the proposed approach is clear and has less loss of data.(ii)
The 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. Thus, 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.


## 6.1. The Outcomes of a Case Study

### 6.1.1. Decision-Making Procedure Based on the 2TLCq-ROFHWA Operator

On the basis of 2TLCq-ROFNs matrix (see Tables 2–4) and Table 3 by utilizing (15), the collective 2TLCq-ROF assessing matrix is computed. The aggregated outcomes are listed in Table 5.Table 2
2TLCq-ROF decision matrix by e1.
ℏ1ℏ2ℏ3ℏ4ℷ1s1,0,s4,0,s7,0,s3,0s2,0,s3,0,s3,0,s5,0s5,0,s2,0,s6,0,s63,0s4,0,s3,0,s2,0,s3,0ℷ2s1,0,s4,0,s5,0,s3,0s3,0,s4,0,s2,0,s3,0s2,0,s4,0,s2,0,s4,0s2,0,s4,0,s5,0,s3,0ℷ3s3,0,s2,0,s2,0,s3,0s3,0,s3,0,s4,0,s2,0s2,0,s4,0,s3,0,s1,0s3,0,s3,0,s2,0,s5,0ℷ4s2,0,s2,0,s1,0,s3,0s2,0,s5,0,s5,0,s4,0s4,0,s1,0,s4,0,s5,0s4,0,s3,0,s2,0,s3,0ℷ5s2,0,s3,0,s2,0,s4,0s3,0,s2,0,s4,0,s2,0s3,0,s2,0,s1,0,s5,0s2,0,s3,0,s5,0,s1,0ℷ6s1,0,s5,0,s6,0,s3,0s5,0,s1,0,s3,0,s6,0s5,0,s6,0,s3,0,s2,0s6,0,s5,0,s2,0,s3,0ℷ7s3,0,s4,0,s5,0,s2,0s5,0,s4,0,s2,0,s3,0s3,0,s4,0,s5,0,s2,0s6,0,s3,0,s1,0,s3,0Table 3
2TLCq-ROF decision matrix by e2.
ℏ1ℏ2ℏ3ℏ4ℷ1s4,0,s2,0,s2,0,s4,0s1,0,s5,0,s6,0,s5,0s2,0,s3,0,s2,0,s3,0s3,0,s5,0,s3,0,s2,0ℷ2s1,0,s3,0,s4,0,s3,0s7,0,s2,0,s2,0,s5,0s3,0,s4,0,s2,0,s4,0s2,0,s4,0,s3,0,s2,0ℷ3.s4,0,s2,0,s4,0,s3,0s2,0,s2,0,s5,0,s3,0s3,0,s4,0,s3,0,s3,0s2,0,s3,0,s5,0,s2,0ℷ4s3,0,s4,0,s2,0,s3,0s3,0,s4,0,s3,0,s5,0s5,0,s3,0,s2,0,s6,0s4,0,s2,0,s4,0,s2,0ℷ5s1,0,s5,0,s4,0,s2,0s5,0,s1,0,s2,0,s4,0s5,0,s2,0,s4,0,s6,0s2,0,s5,0,s6,0,s4,0ℷ6s6,0,s5,0,s3,0,s4,0s5,0,s6,0,s4,0,s3,0s1,0,s2,0,s5,0,s4,0s2,0,s1,0,s4,0,s5,0ℷ7s3,0,s2,0,s5,0,s3,0s3,0,s4,0,s4,0,s5,0s4,0,s2,0,s3,0,s2,0s4,0,s2,0,s5,0,s3,0Table 4
2TLCq-ROF decision matrix by e3.
ℏ1ℏ2ℏ3ℏ4ℷ1s5,0,s3,0,s3,0,s2,0s2,0,s3,0,s3,0,s2,0s6,0,s2,0,s4,0,s5,0s71,0,s2,0,s1,0,s2,0ℷ2s6,0,s1,0,s3,0,s2,0s2,0,s3,0,s5,0,s3,0s5,0,s2,0,s2,0,s6,0s3,0,s2,0,s4,0,s7,0ℷ3s3,0,s2,0,s3,0,s5,0s2,0,s3,0,s2,0,s7,0s4,0,s2,0,s3,0,s5,0s7,0,s3,0,s1,0,s3,0ℷ4s2,0,s4,0,s3,0,s4,0s4,0,s2,0,s3,0,s7,0s5,0,s2,0,s1,0,s2,0s4,0,s3,0,s2,0,s3,0ℷ5s6,0,s4,0,s1,0,s5,0s4,0,s6,0,s5,0,s1,0s2,0,s2,0,s1,0,s1,0s1,0,s1,0,s2,0,s2,0ℷ6s2,0,s1,0,s2,0,s1,0s1,0,s2,0,s1,0,s2,0s3,0,s2,0,s6,0,s5,0s2,0,s3,0,s5,0,s6,0ℷ7s2,0,s3,0,s4,0,s5,0s1,0,s3,0,s4,0,s3,0s2,0,s7,0,s3,0,s3,0s5,0,s1,0,s4,0,s2,0Table 5
Collective the 2TLCq-ROF assessing matrix by the 2TLCq-ROFHWA operator.
Collective assessment matrixAlternativesℏ1ℏ2ℷ1s4,0.1104,s3,0.2121,s3,0.3992,s3,−0.0131s2,−0.2220,s4,0.1377,s4,0.0177,s4,−0.1636ℷ2s4,0.3900,s3,0.2269,s4,−0.0636,s3,−0.3423s6,−0.3618,s3,0.2121,s3,−0.3501,s4,−0.3015ℷ3s4,−0.4988,s2,−0,s3,−0.0131,s4,−0.4888s2,0.4392,s3,−0.2782,s4,−0.4214,s4,−0.4626ℷ4s3,−0.4560,s4,−0.3231,s2,−0.1639,s3,0.2730s3,0.2607,s4,0.1274,s4,−0.4888,s5,0.2652ℷ5s4,0.4042,s4,0.3162,s2,0.1513,s3,0.2636s4,0.3162,s4,0.4042,s3,0.2636,s2,0.1513ℷ6s5,−0.2594,s5,−0.4421,s3,0.3176,s2,0.4301s5,−0.4421,s5,−0.2594,s2,0.4301,s3,0.3176ℷ7s3,−0.2008,s3,0.2121,s5,−0.3164,s3,0.1141s4,−0.1736,s4,−0.2260,s3,0.2588,s4,−0.3015Alternativesℏ3ℏ4ℷ1s5,−0.1136,s3,−0.4560,s3,0.4760,s4,0.3572s5,0.3895,s4,0.0637,s2,−0.0876,s2,0.2596ℷ2s4,−0.1528,s4,−0.3231,s2,−0,s5,−0.4502s2,0.4392,s4,−0.3231,s4,−0.1739,s3,0.3992ℷ3s3,0.2607,s4,−0.3231,s3,−0,s3,−0.4655s5,0.2476,s3,0,s2,0.3664,s3,−0.0087ℷ4s5,−0.2419,s2,0.4721,s2,0.0056,s4,0.1656s4,−0,s3,−0.2782,s3,−0.3540,s3,−0.4478ℷ5s4,0.0637s2,−0,s2,−0.2521,s3,0.4067s2,−0.1587,s4,0.0459,s4,0.1656,s2,0.1513ℷ6s4,−0.2112,s4,0.4229,s5,−0.4196,s3,0.4961s4,0.4229,s4,−0.2112,s3,0.4961,s5,−0.4196ℷ7s3,0.3791,s5,0.3429,s4,−0.4888,s2,0.2596s5,0.0936,s2,0.3572,s3,−0.0828,s3,−0.3423Construct the assessing matrix (see Table6) of favorable- and nonfavorable-type attributes by utilizing the (15) and (44), and (45).Table 6
Assessing values of favorable- and nonfavorable-type attributes by the 2TLCq-ROFHWA operator.
Assessed values of favorable attributesAssessed values of nonfavorable attributesAlternativesℷ1s4,−0.0223,s3,0.0357,s5,0.4604,s6,−0.4488s4,−0.4918,s3,0.1659,s6,−0.4243,s6,−0.1948ℷ2s3,−0.1058,s3,−0.1778,s6,0.4078,s6,−0.0561s4,0.3583,s3,0.0065,s4,0.3236,s6,−0.1917ℷ3s4,−0.2054,s2,0.1860,s6,−0.3941,s6,0.0175s3,−0.4656,s3,−0.1525,s5,0.2148,s5,−0.0104ℷ4s3,−0.1086,s3,−0.4202,s5,0.3485,s6,−0.2301s4,−0.3919,s3,0.1491,s5,−0.3007,s6,0.2438ℷ5s3,−0.1620,s3,0.3348,s6,0.0219,s6,−0.4349s4,−0.3433,s3,0.2963,s4,0.4429,s5,−0.3404ℷ6s4,−0.3514,s3,0.3253,s6,0.1586,s6,0.2394s4,−0.3057,s4,−0.0024,s5,0.2002,s5,0.2980ℷ7s4,−0.3787,s2,0.2501,s6,0.2304,s6,−0.2224s3,0.1682,s4,0.0718,s5,0.2776,s5,−0.0865Calculate the scoring outcomes of alternatives for favorableϜα⌣κ- and nonfavorable Ϝβ⌣κ-type attributes by utilizing (12), and establish the ranking order by using the 2TLCq-ROF-COPRAS method. The evaluation outcomes are listed in Table 7.Table 7
Assessing outcomes of alternatives and ranking order by utilizing the 2TLCq-ROF-COPRAS.
AlternativesBenefit attributes scoresCost attributes scoresRelative degreeUtility degreeRankingϜα⌣κϜβ⌣κΓκUκℷ10.40830.38710.87491.00001ℷ20.32910.43620.74320.84944ℷ30.37370.42360.80020.91463ℷ40.38940.39380.84810.96937ℷ50.37270.46560.76070.86955ℷ60.33800.43430.75400.86176ℷ70.35210.44000.76260.87162


### 6.1.2. Decision-Making Procedure Based on the 2TLCq-ROFHWG Operator

On the basis of 2TLCq-ROFN matrix (see Tables 2–4) and by utilizing (34), the collective 2TLCq-ROF assessing matrix is computed. The aggregated outcomes are listed in Table 8.Table 8
Collective 2TLCq-ROF assessing matrix by the 2TLCq-ROFHWG operator.
Collective assessment matrixAlternativesℏ1ℏ2ℷ1s3,−0.1510,s3,−0.2137,s5,0.2476,s3,0.3791s2,−0.4840,s4,−0.3015,s5,−0.1524,s5,−0.4294ℷ2s2,−0.2581,s2,0.3599,s4,0.1984,s3,−0.2008s4,−0.1147,s3,−0.2137,s4,−0.2766,s4,0.1377ℷ3s3,0.3691,s2,−0,s3,0.3791,s4,−0.0669s2,0.2596,s3,−0.4478,s4,0.2507,s5,0.2600ℷ4s2,0.3533,s3,0.2588,s2,0.3572,s3,0.3951s3,−0.0989,s3,0.4961,s4,−0.0669,s6,−0.2770ℷ5s2,0.1438,s4,0.0278,s3,0.2084,s4,0.0368s4,0.0278,s2,0.1438,s4,0.0368,s3,0.2084ℷ6s3,−0.4227,s3,0.1313,s5,−0.4991,s3,0.3488s3,0.1313,s3,−0.4227,s3,0.3488,s5,−0.4991ℷ7s3,−0.3423,s3,−0.2137,s5,−0.2419,s4,−0.1528s3,−0.4655,s4,−0.3273,s4,−0.3231,s4,0.1377Alternativesℏ3ℏ4ℷ1s4,−0.2726,s2,0.3533,s5,−0.3732,s5,−0.0540s4,0.3320,s3,0.2820,s2,0.4721,s2,0.4392ℷ2s3,0.1141,s3,0.2588,s2,−0,s5,−0.1351s2,0.2596,s3,0.2588,s4,0.1365,s5,0.2476ℷ3s3,−0.0989,s3,0.2588,s3,0,s4,−0.1736s3,0.3992,s3,−0,s4,−0.0332,s4,−0.1827ℷ4s5,−0.3164,s2,−0.0876,s3,0.0107,s5,0.1435s4,0,s3,−0.4478,s3,0.2427,s3,−0.2782ℷ5s3,0.2820,s2,−0,s3,0.1730,s5,0.1343s2,−0.3752,s3,−0.3275,s5,0.1435,s3,0.2084ℷ6s2,0.2714,s3,−0.1754,s5,0.0585,s4,0.1274s3,−0.1754,s2,0.2714,s4,0.1274,s5,0.0585ℷ7s3,−0.0131,s4,−0.2947,s4,−0.0669,s2,0.4392s5,−0.1395,s2,−0.1639,s4,0.2350,s3,−0.2008Construct the assessing matrix (see Table9) of favorable- and nonfavorable-type attributes by utilizing the (34) and (44), and (45).Table 9
Assessing values of favorable- and nonfavorable-type attributes by the 2TLCq-ROFHWG operator.
Assessed values of favorable attributesAssessed values of nonfavorable attributesAlternativesℷ1s6,0.2971,s6,−0.0599,s3,0.3893,s2,0.3551s4,0.3090,s5,−0.0370,s4,0.1242,s4,0.1305ℷ2s5,0.1244,s6,−0.2031,s3,0.3356,s4,−0.2780s5,0.3839,s5,−0.0397,s3,−0.1818,s4,−0.0743ℷ3s6,0.1372,s6,−0.4356,s3,0.0175,s3,0.1010s5,−0.4634,s5,−0.1616,s3,0.3172,s4,0.1169ℷ4s6,0.0442,s6,−0.2335,s2,0.3901,s2,0.4500s5,0.4866,s5,−0.3619,s3,0.1240,s5,−0.2501ℷ5s5,−0.0879,s6,−0.0028,s4,−0.3074,s3,−0.1011s6,−0.4945,s4,0.0610,s3,0.2265,s4,−0.1764ℷ6s6,−0.3032,s6,−0.4044,s3,0.4362,s4,−0.3434s5,−0.3116,s5,−0.3189,s4,−0.1944,s4,−0.2280ℷ7s6,0.3865,s5,0.2536,s4,−0.4234,s3,−0.3159s5,−0.2756,s6,−0.4764,s3,0.3102,s3,0.1518Calculate the scoring outcomes of alternatives for favorableϜα⌣κ- and nonfavorable Ϝβ⌣κ-type attributes by utilizing (12) and establish the ranking order by using the 2TLCq-ROF-COPRAS method. The evaluation outcomes are listed in Table 10.Table 10
Assessing outcomes of alternatives and ranking order by utilizing the 2TLCq-ROF-COPRAS.
AlternativesBenefit attributes scoresCost attributes scoresRelative degreeUtility degreeRankingϜα⌣κϜβ⌣κΓκUκℷ10.66200.52271.23541.00001ℷ20.59170.56991.11760.90473ℷ30.63440.53441.19530.96754ℷ40.64470.54671.19300.96566ℷ50.59880.55301.14070.92347ℷ60.60470.53361.16630.94415ℷ70.63490.57391.15710.93662


## 6.1.1. Decision-Making Procedure Based on the 2TLCq-ROFHWA Operator

On the basis of 2TLCq-ROFNs matrix (see Tables 2–4) and Table 3 by utilizing (15), the collective 2TLCq-ROF assessing matrix is computed. The aggregated outcomes are listed in Table 5.Table 2
2TLCq-ROF decision matrix by e1.
ℏ1ℏ2ℏ3ℏ4ℷ1s1,0,s4,0,s7,0,s3,0s2,0,s3,0,s3,0,s5,0s5,0,s2,0,s6,0,s63,0s4,0,s3,0,s2,0,s3,0ℷ2s1,0,s4,0,s5,0,s3,0s3,0,s4,0,s2,0,s3,0s2,0,s4,0,s2,0,s4,0s2,0,s4,0,s5,0,s3,0ℷ3s3,0,s2,0,s2,0,s3,0s3,0,s3,0,s4,0,s2,0s2,0,s4,0,s3,0,s1,0s3,0,s3,0,s2,0,s5,0ℷ4s2,0,s2,0,s1,0,s3,0s2,0,s5,0,s5,0,s4,0s4,0,s1,0,s4,0,s5,0s4,0,s3,0,s2,0,s3,0ℷ5s2,0,s3,0,s2,0,s4,0s3,0,s2,0,s4,0,s2,0s3,0,s2,0,s1,0,s5,0s2,0,s3,0,s5,0,s1,0ℷ6s1,0,s5,0,s6,0,s3,0s5,0,s1,0,s3,0,s6,0s5,0,s6,0,s3,0,s2,0s6,0,s5,0,s2,0,s3,0ℷ7s3,0,s4,0,s5,0,s2,0s5,0,s4,0,s2,0,s3,0s3,0,s4,0,s5,0,s2,0s6,0,s3,0,s1,0,s3,0Table 3
2TLCq-ROF decision matrix by e2.
ℏ1ℏ2ℏ3ℏ4ℷ1s4,0,s2,0,s2,0,s4,0s1,0,s5,0,s6,0,s5,0s2,0,s3,0,s2,0,s3,0s3,0,s5,0,s3,0,s2,0ℷ2s1,0,s3,0,s4,0,s3,0s7,0,s2,0,s2,0,s5,0s3,0,s4,0,s2,0,s4,0s2,0,s4,0,s3,0,s2,0ℷ3.s4,0,s2,0,s4,0,s3,0s2,0,s2,0,s5,0,s3,0s3,0,s4,0,s3,0,s3,0s2,0,s3,0,s5,0,s2,0ℷ4s3,0,s4,0,s2,0,s3,0s3,0,s4,0,s3,0,s5,0s5,0,s3,0,s2,0,s6,0s4,0,s2,0,s4,0,s2,0ℷ5s1,0,s5,0,s4,0,s2,0s5,0,s1,0,s2,0,s4,0s5,0,s2,0,s4,0,s6,0s2,0,s5,0,s6,0,s4,0ℷ6s6,0,s5,0,s3,0,s4,0s5,0,s6,0,s4,0,s3,0s1,0,s2,0,s5,0,s4,0s2,0,s1,0,s4,0,s5,0ℷ7s3,0,s2,0,s5,0,s3,0s3,0,s4,0,s4,0,s5,0s4,0,s2,0,s3,0,s2,0s4,0,s2,0,s5,0,s3,0Table 4
2TLCq-ROF decision matrix by e3.
ℏ1ℏ2ℏ3ℏ4ℷ1s5,0,s3,0,s3,0,s2,0s2,0,s3,0,s3,0,s2,0s6,0,s2,0,s4,0,s5,0s71,0,s2,0,s1,0,s2,0ℷ2s6,0,s1,0,s3,0,s2,0s2,0,s3,0,s5,0,s3,0s5,0,s2,0,s2,0,s6,0s3,0,s2,0,s4,0,s7,0ℷ3s3,0,s2,0,s3,0,s5,0s2,0,s3,0,s2,0,s7,0s4,0,s2,0,s3,0,s5,0s7,0,s3,0,s1,0,s3,0ℷ4s2,0,s4,0,s3,0,s4,0s4,0,s2,0,s3,0,s7,0s5,0,s2,0,s1,0,s2,0s4,0,s3,0,s2,0,s3,0ℷ5s6,0,s4,0,s1,0,s5,0s4,0,s6,0,s5,0,s1,0s2,0,s2,0,s1,0,s1,0s1,0,s1,0,s2,0,s2,0ℷ6s2,0,s1,0,s2,0,s1,0s1,0,s2,0,s1,0,s2,0s3,0,s2,0,s6,0,s5,0s2,0,s3,0,s5,0,s6,0ℷ7s2,0,s3,0,s4,0,s5,0s1,0,s3,0,s4,0,s3,0s2,0,s7,0,s3,0,s3,0s5,0,s1,0,s4,0,s2,0Table 5
Collective the 2TLCq-ROF assessing matrix by the 2TLCq-ROFHWA operator.
Collective assessment matrixAlternativesℏ1ℏ2ℷ1s4,0.1104,s3,0.2121,s3,0.3992,s3,−0.0131s2,−0.2220,s4,0.1377,s4,0.0177,s4,−0.1636ℷ2s4,0.3900,s3,0.2269,s4,−0.0636,s3,−0.3423s6,−0.3618,s3,0.2121,s3,−0.3501,s4,−0.3015ℷ3s4,−0.4988,s2,−0,s3,−0.0131,s4,−0.4888s2,0.4392,s3,−0.2782,s4,−0.4214,s4,−0.4626ℷ4s3,−0.4560,s4,−0.3231,s2,−0.1639,s3,0.2730s3,0.2607,s4,0.1274,s4,−0.4888,s5,0.2652ℷ5s4,0.4042,s4,0.3162,s2,0.1513,s3,0.2636s4,0.3162,s4,0.4042,s3,0.2636,s2,0.1513ℷ6s5,−0.2594,s5,−0.4421,s3,0.3176,s2,0.4301s5,−0.4421,s5,−0.2594,s2,0.4301,s3,0.3176ℷ7s3,−0.2008,s3,0.2121,s5,−0.3164,s3,0.1141s4,−0.1736,s4,−0.2260,s3,0.2588,s4,−0.3015Alternativesℏ3ℏ4ℷ1s5,−0.1136,s3,−0.4560,s3,0.4760,s4,0.3572s5,0.3895,s4,0.0637,s2,−0.0876,s2,0.2596ℷ2s4,−0.1528,s4,−0.3231,s2,−0,s5,−0.4502s2,0.4392,s4,−0.3231,s4,−0.1739,s3,0.3992ℷ3s3,0.2607,s4,−0.3231,s3,−0,s3,−0.4655s5,0.2476,s3,0,s2,0.3664,s3,−0.0087ℷ4s5,−0.2419,s2,0.4721,s2,0.0056,s4,0.1656s4,−0,s3,−0.2782,s3,−0.3540,s3,−0.4478ℷ5s4,0.0637s2,−0,s2,−0.2521,s3,0.4067s2,−0.1587,s4,0.0459,s4,0.1656,s2,0.1513ℷ6s4,−0.2112,s4,0.4229,s5,−0.4196,s3,0.4961s4,0.4229,s4,−0.2112,s3,0.4961,s5,−0.4196ℷ7s3,0.3791,s5,0.3429,s4,−0.4888,s2,0.2596s5,0.0936,s2,0.3572,s3,−0.0828,s3,−0.3423Construct the assessing matrix (see Table6) of favorable- and nonfavorable-type attributes by utilizing the (15) and (44), and (45).Table 6
Assessing values of favorable- and nonfavorable-type attributes by the 2TLCq-ROFHWA operator.
Assessed values of favorable attributesAssessed values of nonfavorable attributesAlternativesℷ1s4,−0.0223,s3,0.0357,s5,0.4604,s6,−0.4488s4,−0.4918,s3,0.1659,s6,−0.4243,s6,−0.1948ℷ2s3,−0.1058,s3,−0.1778,s6,0.4078,s6,−0.0561s4,0.3583,s3,0.0065,s4,0.3236,s6,−0.1917ℷ3s4,−0.2054,s2,0.1860,s6,−0.3941,s6,0.0175s3,−0.4656,s3,−0.1525,s5,0.2148,s5,−0.0104ℷ4s3,−0.1086,s3,−0.4202,s5,0.3485,s6,−0.2301s4,−0.3919,s3,0.1491,s5,−0.3007,s6,0.2438ℷ5s3,−0.1620,s3,0.3348,s6,0.0219,s6,−0.4349s4,−0.3433,s3,0.2963,s4,0.4429,s5,−0.3404ℷ6s4,−0.3514,s3,0.3253,s6,0.1586,s6,0.2394s4,−0.3057,s4,−0.0024,s5,0.2002,s5,0.2980ℷ7s4,−0.3787,s2,0.2501,s6,0.2304,s6,−0.2224s3,0.1682,s4,0.0718,s5,0.2776,s5,−0.0865Calculate the scoring outcomes of alternatives for favorableϜα⌣κ- and nonfavorable Ϝβ⌣κ-type attributes by utilizing (12), and establish the ranking order by using the 2TLCq-ROF-COPRAS method. The evaluation outcomes are listed in Table 7.Table 7
Assessing outcomes of alternatives and ranking order by utilizing the 2TLCq-ROF-COPRAS.
AlternativesBenefit attributes scoresCost attributes scoresRelative degreeUtility degreeRankingϜα⌣κϜβ⌣κΓκUκℷ10.40830.38710.87491.00001ℷ20.32910.43620.74320.84944ℷ30.37370.42360.80020.91463ℷ40.38940.39380.84810.96937ℷ50.37270.46560.76070.86955ℷ60.33800.43430.75400.86176ℷ70.35210.44000.76260.87162


## 6.1.2. Decision-Making Procedure Based on the 2TLCq-ROFHWG Operator

On the basis of 2TLCq-ROFN matrix (see Tables 2–4) and by utilizing (34), the collective 2TLCq-ROF assessing matrix is computed. The aggregated outcomes are listed in Table 8.Table 8
Collective 2TLCq-ROF assessing matrix by the 2TLCq-ROFHWG operator.
Collective assessment matrixAlternativesℏ1ℏ2ℷ1s3,−0.1510,s3,−0.2137,s5,0.2476,s3,0.3791s2,−0.4840,s4,−0.3015,s5,−0.1524,s5,−0.4294ℷ2s2,−0.2581,s2,0.3599,s4,0.1984,s3,−0.2008s4,−0.1147,s3,−0.2137,s4,−0.2766,s4,0.1377ℷ3s3,0.3691,s2,−0,s3,0.3791,s4,−0.0669s2,0.2596,s3,−0.4478,s4,0.2507,s5,0.2600ℷ4s2,0.3533,s3,0.2588,s2,0.3572,s3,0.3951s3,−0.0989,s3,0.4961,s4,−0.0669,s6,−0.2770ℷ5s2,0.1438,s4,0.0278,s3,0.2084,s4,0.0368s4,0.0278,s2,0.1438,s4,0.0368,s3,0.2084ℷ6s3,−0.4227,s3,0.1313,s5,−0.4991,s3,0.3488s3,0.1313,s3,−0.4227,s3,0.3488,s5,−0.4991ℷ7s3,−0.3423,s3,−0.2137,s5,−0.2419,s4,−0.1528s3,−0.4655,s4,−0.3273,s4,−0.3231,s4,0.1377Alternativesℏ3ℏ4ℷ1s4,−0.2726,s2,0.3533,s5,−0.3732,s5,−0.0540s4,0.3320,s3,0.2820,s2,0.4721,s2,0.4392ℷ2s3,0.1141,s3,0.2588,s2,−0,s5,−0.1351s2,0.2596,s3,0.2588,s4,0.1365,s5,0.2476ℷ3s3,−0.0989,s3,0.2588,s3,0,s4,−0.1736s3,0.3992,s3,−0,s4,−0.0332,s4,−0.1827ℷ4s5,−0.3164,s2,−0.0876,s3,0.0107,s5,0.1435s4,0,s3,−0.4478,s3,0.2427,s3,−0.2782ℷ5s3,0.2820,s2,−0,s3,0.1730,s5,0.1343s2,−0.3752,s3,−0.3275,s5,0.1435,s3,0.2084ℷ6s2,0.2714,s3,−0.1754,s5,0.0585,s4,0.1274s3,−0.1754,s2,0.2714,s4,0.1274,s5,0.0585ℷ7s3,−0.0131,s4,−0.2947,s4,−0.0669,s2,0.4392s5,−0.1395,s2,−0.1639,s4,0.2350,s3,−0.2008Construct the assessing matrix (see Table9) of favorable- and nonfavorable-type attributes by utilizing the (34) and (44), and (45).Table 9
Assessing values of favorable- and nonfavorable-type attributes by the 2TLCq-ROFHWG operator.
Assessed values of favorable attributesAssessed values of nonfavorable attributesAlternativesℷ1s6,0.2971,s6,−0.0599,s3,0.3893,s2,0.3551s4,0.3090,s5,−0.0370,s4,0.1242,s4,0.1305ℷ2s5,0.1244,s6,−0.2031,s3,0.3356,s4,−0.2780s5,0.3839,s5,−0.0397,s3,−0.1818,s4,−0.0743ℷ3s6,0.1372,s6,−0.4356,s3,0.0175,s3,0.1010s5,−0.4634,s5,−0.1616,s3,0.3172,s4,0.1169ℷ4s6,0.0442,s6,−0.2335,s2,0.3901,s2,0.4500s5,0.4866,s5,−0.3619,s3,0.1240,s5,−0.2501ℷ5s5,−0.0879,s6,−0.0028,s4,−0.3074,s3,−0.1011s6,−0.4945,s4,0.0610,s3,0.2265,s4,−0.1764ℷ6s6,−0.3032,s6,−0.4044,s3,0.4362,s4,−0.3434s5,−0.3116,s5,−0.3189,s4,−0.1944,s4,−0.2280ℷ7s6,0.3865,s5,0.2536,s4,−0.4234,s3,−0.3159s5,−0.2756,s6,−0.4764,s3,0.3102,s3,0.1518Calculate the scoring outcomes of alternatives for favorableϜα⌣κ- and nonfavorable Ϝβ⌣κ-type attributes by utilizing (12) and establish the ranking order by using the 2TLCq-ROF-COPRAS method. The evaluation outcomes are listed in Table 10.Table 10
Assessing outcomes of alternatives and ranking order by utilizing the 2TLCq-ROF-COPRAS.
AlternativesBenefit attributes scoresCost attributes scoresRelative degreeUtility degreeRankingϜα⌣κϜβ⌣κΓκUκℷ10.66200.52271.23541.00001ℷ20.59170.56991.11760.90473ℷ30.63440.53441.19530.96754ℷ40.64470.54671.19300.96566ℷ50.59880.55301.14070.92347ℷ60.60470.53361.16630.94415ℷ70.63490.57391.15710.93662


## 6.2. Parameter Analysis

Surely, the parameterϱ has a great influence on the ranking results. The influence of parameters on score values and ranking results based on 2TLCq-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. Then, these scores are used to rank the alternatives. Ranking results are used to select the best alternative, and ℷ1 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. The 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. The decision preference can be represented in the actual DM process by varying the values of q and ϱ to obtain the best decision result.Table 11
Score values and ranking outcomes according to the parameterq by the 2TLCq-ROFHWA operator.
ParametersϜℷ1Ϝℷ2Ϝℷ3Ϝℷ4Ϝℷ5Ϝℷ6Ϝℷ7Rankingq=21.00000.78770.88650.92860.80960.82460.8228ℷ1>ℷ4>ℷ3>ℷ6>ℷ7>ℷ5>ℷ2q=31.00000.81770.89750.95130.83710.83940.8457ℷ1>ℷ4>ℷ3>ℷ7>ℷ6>ℷ5>ℷ2q=51.00000.87830.93230.98190.80960.88550.8962ℷ1>ℷ4>ℷ3>ℷ7>ℷ6>ℷ2>ℷ5q=71.00000.92360.96150.99500.94450.92780.9359ℷ1>ℷ4>ℷ3>ℷ5>ℷ7>ℷ6>ℷ2q=110.99990.97120.99011.00000.98450.82460.9776ℷ4>ℷ1>ℷ3>ℷ5>ℷ7>ℷ2>ℷ6q=131.00000.98210.99530.99990.99160.98660.9865ℷ1>ℷ4>ℷ3>ℷ5>ℷ6>ℷ7>ℷ2q=171.00000.99240.99900.99930.99700.99550.9945ℷ1>ℷ4>ℷ3>ℷ5>ℷ6>ℷ7>ℷ2q=191.00000.99480.99960.99930.99810.99730.9962ℷ1>ℷ3>ℷ4>ℷ5>ℷ6>ℷ7>ℷ2q=231.00000.99730.99990.99940.99910.99880.9981ℷ1>ℷ3>ℷ4>ℷ5>ℷ6>ℷ7>ℷ2q=291.00000.78770.88650.92860.80960.82460.8228ℷ1>ℷ4>ℷ3>ℷ6>ℷ7>ℷ5>ℷ2Table 12
Score values and ranking outcomes according to the parameterϱ by the 2TLCq-ROFHWA operator.
ParametersϜℷ1Ϝℷ2Ϝℷ3Ϝℷ4Ϝℷ5Ϝℷ6Ϝℷ7Rankingϱ=21.00000.86070.92540.97090.88220.87390.8820ℷ1>ℷ4>ℷ3>ℷ5>ℷ7>ℷ6>ℷ2ϱ=31.00000.84940.91460.96930.86950.86180.8717ℷ1>ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ2ϱ=51.00000.83330.89950.96620.85110.84520.8568ℷ1>ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ2ϱ=71.00000.82170.88870.96330.83750.83380.8460ℷ1>ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ2ϱ=111.00000.80480.87330.95820.81740.81830.8303ℷ1>ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ2ϱ=131.00000.79840.86730.95600.80950.81250.8243ℷ1>ℷ4>ℷ3>ℷ7>ℷ6>ℷ5>ℷ2ϱ=171.00000.78770.85760.95190.79620.80340.8143ℷ1>ℷ4>ℷ3>ℷ7>ℷ6>ℷ5>ℷ2ϱ=191.00000.78320.85350.95010.79050.79970.8102ℷ1>ℷ4>ℷ3>ℷ7>ℷ6>ℷ5>ℷ2ϱ=231.00000.77540.84640.94670.78050.79350.8030ℷ1>ℷ4>ℷ3>ℷ7>ℷ6>ℷ5>ℷ2ϱ=291.00000.76590.83770.94220.76800.78610.7942ℷ1>ℷ4>ℷ3>ℷ7>ℷ6>ℷ5>ℷ2We fix the several values ofq 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. Then, these scores are used to rank the alternatives. The ranking results are used to select the best alternative, and ℷ1 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. The change in values of q and ϱ has a significant influence on the results of the alternative 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. The decision preference can be represented in the actual DM process by varying the values of q and ϱ to obtain the best decision result.Table 13
Score values and ranking outcomes according to the parameterq by the 2TLCq-ROFHWG operator.
ParametersϜℷ1Ϝℷ2Ϝℷ3Ϝℷ4Ϝℷ5Ϝℷ6Ϝℷ7Rankingq=21.00000.90170.96160.96610.92310.94130.9254ℷ1>ℷ4>ℷ3>ℷ6>ℷ7>ℷ5>ℷ2q=31.00000.89840.96320.96350.91980.94100.9274ℷ1>ℷ4>ℷ3>ℷ6>ℷ7>ℷ5>ℷ2q=51.00000.91700.97300.97060.93240.94970.9492ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2q=71.00000.94610.98430.98280.95620.96500.9731ℷ1>ℷ3>ℷ4>ℷ7>ℷ6>ℷ5>ℷ2q=111.00000.98280.99800.99790.98770.98880.9960ℷ1>ℷ3>ℷ4>ℷ7>ℷ6>ℷ5>ℷ2q=130.99930.99010.99991.00000.99350.99380.9986ℷ4>ℷ3>ℷ1>ℷ7>ℷ6>ℷ5>ℷ2q=170.99790.99510.99991.00000.99710.99700.9988ℷ4>ℷ3>ℷ7>ℷ1>ℷ5>ℷ6>ℷ2q=190.99800.99650.99991.00000.99800.99790.9989ℷ4>ℷ3>ℷ7>ℷ1=ℷ5>ℷ6>ℷ2q=230.99860.99800.99991.00000.99900.99890.9992ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ1>ℷ2q=290.99930.99910.99991.00000.99950.99940.9996ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ1>ℷ2Table 14
Score values and ranking outcomes according to the parameterϱ by the 2TLCq-ROFHWG operator.
ParametersϜℷ1Ϝℷ2Ϝℷ3Ϝℷ4Ϝℷ5Ϝℷ6Ϝℷ7Rankingϱ=21.00000.90750.96930.96980.92600.94510.9392ℷ1>ℷ4>ℷ3>ℷ6>ℷ7>ℷ5>ℷ2ϱ=31.00000.90460.96740.96560.92330.94400.9366ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=51.00000.90230.96630.96140.92140.94400.9342ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=71.00000.90170.96630.95940.92100.94490.9333ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=111.00000.90200.96720.95770.92180.94710.9329ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=131.00000.90250.96770.95730.92250.94820.9330ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=171.00000.90370.96880.95690.92400.95020.9334ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=191.00000.90440.96940.95690.92480.95110.9337ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=231.00000.90560.97030.95690.92630.95280.9342ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2ϱ=291.00000.90750.97160.95700.92850.95500.9351ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2


## 6.3. 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. The COPRAS 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-ROFHWG operators. Comparative analysis with different FSs (2TLCIFH, 2TLCPFH, 2TLCFFH), aggregation operators (2TLCq-ROFWA, 2TLCq-ROFEWA, 2TLCq-ROFWG, 2TLCq-ROFEWG), and methods (CODAS, EDAS, TOPSIS) is illustrated as follows.


### 6.3.1. Comparative Analysis with Different Aggregation Operators

Our proposed operator is more realistic and superior because it can evaluate the interrelationship of fused arguments and scientifically consider the complexities of humans in practical MAGDM problems, whereas the 2TLCq-ROFWA (2TLCq-ROFWG) and 2TLCq-ROFEWA (2TLCq-ROFEWG) operators are unable to evaluate the interrelationships between fused arguments and cannot consider the complexity of DMs. As a result, proposed operators are more general in expressing fuzzy information.The 2TLCq-ROFHWA and 2TLCq-ROFHWG operators can provide more effective and flexible information fusion, making it more suitable for dealing with MAGDM problems with dependent attributes, as demonstrated in Tables 15 and 16. As a result, our technique overcomes the constraints of 2TLCq-ROFWA (2TLCq-ROFWG) and 2TLCq-ROFEWA (2TLCq-ROFEWG) aggregation operators. As a result, 2TLCq-ROFHWA and 2TLCq-ROFHWG operators are highly suitable for dealing with MAGDM in a 2TLCq-ROF environment.Table 15
Evaluation outcomes by utilizing different aggregation operators based on the 2TLCq-ROFHWA operator.
Alternatives2TLCq-ROFHRanking2TLCq-ROFWARanking2TLCq-ROFEWARankingℷ11.0000I1.0000I1.0000Iℷ20.8494IV0.8771IV0.8607IVℷ30.9146III0.9413III0.9254IIIℷ40.9693VII0.9724V0.9709Vℷ50.8695V0.9006VII0.8822VIIℷ60.8617VI0.8925VI0.8739VIℷ70.8716II0.8970II0.8820IITable 16
Evaluation outcomes by utilizing different aggregation operators based on the 2TLCq-ROFHWG operator.
Alternatives2TLCq-ROFHRanking2TLCq-ROFWGRanking2TLCq-ROFEWGRankingℷ11.0000I1.0000I1.0000Iℷ20.9047III0.9138IV0.9075IVℷ30.9675IV0.9741III0.9693IIIℷ40.9656IV0.9784VI0.9698VIℷ50.9234VII0.9321VII0.9260VIIℷ60.9441V0.9486V0.9451Vℷ70.9366II0.9447II0.9392II


### 6.3.2. Comparative Analysis with Different Fuzzy Sets

We compared our suggested operators to the 2TLCIFH (forq=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.The characteristic of the proposed set is that the sum ofq 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. The 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 ℷ1 based on the 2TLCq-ROFHWA (2TLCq-ROFHWG) operator.Table 17
Evaluation outcomes by utilizing different FSs based on the 2TLCq-ROFHWA operator.
Alternatives2TLCq-ROFHRanking2TLCFFHRanking2TLCPFHRanking2TLCIFHRankingℷ11.0000I1.0000I1.0000I1.0000Iℷ20.8494IV0.8177IV0.7877IV0.7833IVℷ30.9146III0.8975III0.8865III0.8990IIIℷ40.9693VII0.9513VII0.9286VI0.9121VIℷ50.8695V0.8371VI0.8096VII0.8134VIIℷ60.8617VI0.8394V0.8246V0.8387Vℷ70.8716II0.8457II0.8228II0.8252IITable 18
Evaluation outcomes by utilizing different FSs based on the 2TLCq-ROFHWG operator.
Alternatives2TLCq-ROFHRanking2TLCFFHRanking2TLCPFHRanking2TLCIFHRankingℷ11.0000I1.0000I1.0000I1.0000Iℷ20.9047III0.8984IV0.9017IV0.9171IVℷ30.9675IV0.9632III0.9616III0.9639IIIℷ40.9656VI0.9635VI0.9661VI0.9745VIℷ50.9234VII0.9198VII0.9231VII0.9330VIIℷ60.9441V0.9410V0.9413V0.9464Vℷ70.9366II0.9274II0.9254II0.9341II


### 6.3.3. Comparative Analysis with Different MAGDM Methods

This subsection compares the proposed method to some existing methodologies in order to highlight the superiority of our method. To begin, our developed method is compared to CODAS, EDAS, and TOPSIS methods. We carefully calculate the results of these methods and compare them with our proposed method. For the CODAS, EDAS, and TOPSIS methods, the ranking order isℷ6>ℷ7>ℷ2>ℷ1>ℷ3>ℷ4>ℷ5, ℷ6>ℷ7>ℷ5>ℷ2>ℷ1>ℷ3>ℷ4, and ℷ5>ℷ3>ℷ4>ℷ2>ℷ1>ℷ7>ℷ6, respectively, based on the 2TLCq-ROFHWA operator in Table 19. Therefore, the best alternatives are ℷ6, ℷ6, and ℷ5 according to CODAS, EDAS, and TOPSIS methods, respectively. However, in our purposed method (COPRAS), the ranking order is ℷ1>ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ2, and the best (worst) alternative is ℷ1ℷ2, based on the 2TLCq-ROFHWA operator as shown in Table 19.Table 19
Evaluation outcomes by utilizing different methodologies based on the 2TLCq-ROFHWA operator.
AlternativesCOPRASRankingCODASRankingEDASRankingTOPSISRankingℷ11.0000I−0.0439IV0.5277VI−1.3229Vℷ20.8494IV0.0539VII0.5331VII−1.9971IIIℷ30.9146III−0.2050II0.2846V−1.1370IVℷ40.9693VII−0.5126I0.0000II−1.1445Iℷ50.8695V−3.3930III0.5648I0.0000IIℷ60.8617VI2.7821IV0.7304III−2.7610VIIℷ70.8716II1.3185V0.5820IV−2.2180VIIn Table20 the ranking order is ℷ6>ℷ2>ℷ3>ℷ5>ℷ7>ℷ1>ℷ4, ℷ7>ℷ5>ℷ4>ℷ2>ℷ3>ℷ1>ℷ6, and ℷ7>ℷ4>ℷ1>ℷ2>ℷ6>ℷ5>ℷ3 for the CODAS, EDAS, and TOPSIS methods, respectively, based on the 2TLCq-ROFHWG operator. Therefore, the best alternatives are ℷ6, ℷ7, and ℷ7 according to CODAS, EDAS, and TOPSIS methods, respectively. In our purposed method (COPRAS), the ranking order is ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2, based on the 2TLCq-ROFHWG operator as shown in Table 20. However, the best and the worst alternative remains the same that are ℷ1 and ℷ2 that show the reliability and effectiveness of our purposed method.Table 20
Evaluation outcomes by utilizing different methodologies based on the 2TLCq-ROFHWG operator.
AlternativesCOPRASRankingCODASRankingEDASRankingTOPSISRankingℷ11.0000I−1.4587VI0.2452VII−0.2422VIIℷ20.9047III1.4557II0.4219V−0.5476IVℷ30.9675IV0.8752III0.2846IV−0.6408Iℷ40.9656VI−1.6952VII0.4511II−0.1472IIℷ50.9234VII0.4201V0.4725III−0.6044VIℷ60.9441V1.6987I0.0913I−0.5873Vℷ70.9366II−1.2959IV0.9024VI−0.0653III


## 6.3.1. Comparative Analysis with Different Aggregation Operators

Our proposed operator is more realistic and superior because it can evaluate the interrelationship of fused arguments and scientifically consider the complexities of humans in practical MAGDM problems, whereas the 2TLCq-ROFWA (2TLCq-ROFWG) and 2TLCq-ROFEWA (2TLCq-ROFEWG) operators are unable to evaluate the interrelationships between fused arguments and cannot consider the complexity of DMs. As a result, proposed operators are more general in expressing fuzzy information.The 2TLCq-ROFHWA and 2TLCq-ROFHWG operators can provide more effective and flexible information fusion, making it more suitable for dealing with MAGDM problems with dependent attributes, as demonstrated in Tables 15 and 16. As a result, our technique overcomes the constraints of 2TLCq-ROFWA (2TLCq-ROFWG) and 2TLCq-ROFEWA (2TLCq-ROFEWG) aggregation operators. As a result, 2TLCq-ROFHWA and 2TLCq-ROFHWG operators are highly suitable for dealing with MAGDM in a 2TLCq-ROF environment.Table 15
Evaluation outcomes by utilizing different aggregation operators based on the 2TLCq-ROFHWA operator.
Alternatives2TLCq-ROFHRanking2TLCq-ROFWARanking2TLCq-ROFEWARankingℷ11.0000I1.0000I1.0000Iℷ20.8494IV0.8771IV0.8607IVℷ30.9146III0.9413III0.9254IIIℷ40.9693VII0.9724V0.9709Vℷ50.8695V0.9006VII0.8822VIIℷ60.8617VI0.8925VI0.8739VIℷ70.8716II0.8970II0.8820IITable 16
Evaluation outcomes by utilizing different aggregation operators based on the 2TLCq-ROFHWG operator.
Alternatives2TLCq-ROFHRanking2TLCq-ROFWGRanking2TLCq-ROFEWGRankingℷ11.0000I1.0000I1.0000Iℷ20.9047III0.9138IV0.9075IVℷ30.9675IV0.9741III0.9693IIIℷ40.9656IV0.9784VI0.9698VIℷ50.9234VII0.9321VII0.9260VIIℷ60.9441V0.9486V0.9451Vℷ70.9366II0.9447II0.9392II


## 6.3.2. Comparative Analysis with Different Fuzzy Sets

We compared our suggested operators to the 2TLCIFH (forq=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.The characteristic of the proposed set is that the sum ofq 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. The 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 ℷ1 based on the 2TLCq-ROFHWA (2TLCq-ROFHWG) operator.Table 17
Evaluation outcomes by utilizing different FSs based on the 2TLCq-ROFHWA operator.
Alternatives2TLCq-ROFHRanking2TLCFFHRanking2TLCPFHRanking2TLCIFHRankingℷ11.0000I1.0000I1.0000I1.0000Iℷ20.8494IV0.8177IV0.7877IV0.7833IVℷ30.9146III0.8975III0.8865III0.8990IIIℷ40.9693VII0.9513VII0.9286VI0.9121VIℷ50.8695V0.8371VI0.8096VII0.8134VIIℷ60.8617VI0.8394V0.8246V0.8387Vℷ70.8716II0.8457II0.8228II0.8252IITable 18
Evaluation outcomes by utilizing different FSs based on the 2TLCq-ROFHWG operator.
Alternatives2TLCq-ROFHRanking2TLCFFHRanking2TLCPFHRanking2TLCIFHRankingℷ11.0000I1.0000I1.0000I1.0000Iℷ20.9047III0.8984IV0.9017IV0.9171IVℷ30.9675IV0.9632III0.9616III0.9639IIIℷ40.9656VI0.9635VI0.9661VI0.9745VIℷ50.9234VII0.9198VII0.9231VII0.9330VIIℷ60.9441V0.9410V0.9413V0.9464Vℷ70.9366II0.9274II0.9254II0.9341II


## 6.3.3. Comparative Analysis with Different MAGDM Methods

This subsection compares the proposed method to some existing methodologies in order to highlight the superiority of our method. To begin, our developed method is compared to CODAS, EDAS, and TOPSIS methods. We carefully calculate the results of these methods and compare them with our proposed method. For the CODAS, EDAS, and TOPSIS methods, the ranking order isℷ6>ℷ7>ℷ2>ℷ1>ℷ3>ℷ4>ℷ5, ℷ6>ℷ7>ℷ5>ℷ2>ℷ1>ℷ3>ℷ4, and ℷ5>ℷ3>ℷ4>ℷ2>ℷ1>ℷ7>ℷ6, respectively, based on the 2TLCq-ROFHWA operator in Table 19. Therefore, the best alternatives are ℷ6, ℷ6, and ℷ5 according to CODAS, EDAS, and TOPSIS methods, respectively. However, in our purposed method (COPRAS), the ranking order is ℷ1>ℷ4>ℷ3>ℷ7>ℷ5>ℷ6>ℷ2, and the best (worst) alternative is ℷ1ℷ2, based on the 2TLCq-ROFHWA operator as shown in Table 19.Table 19
Evaluation outcomes by utilizing different methodologies based on the 2TLCq-ROFHWA operator.
AlternativesCOPRASRankingCODASRankingEDASRankingTOPSISRankingℷ11.0000I−0.0439IV0.5277VI−1.3229Vℷ20.8494IV0.0539VII0.5331VII−1.9971IIIℷ30.9146III−0.2050II0.2846V−1.1370IVℷ40.9693VII−0.5126I0.0000II−1.1445Iℷ50.8695V−3.3930III0.5648I0.0000IIℷ60.8617VI2.7821IV0.7304III−2.7610VIIℷ70.8716II1.3185V0.5820IV−2.2180VIIn Table20 the ranking order is ℷ6>ℷ2>ℷ3>ℷ5>ℷ7>ℷ1>ℷ4, ℷ7>ℷ5>ℷ4>ℷ2>ℷ3>ℷ1>ℷ6, and ℷ7>ℷ4>ℷ1>ℷ2>ℷ6>ℷ5>ℷ3 for the CODAS, EDAS, and TOPSIS methods, respectively, based on the 2TLCq-ROFHWG operator. Therefore, the best alternatives are ℷ6, ℷ7, and ℷ7 according to CODAS, EDAS, and TOPSIS methods, respectively. In our purposed method (COPRAS), the ranking order is ℷ1>ℷ3>ℷ4>ℷ6>ℷ7>ℷ5>ℷ2, based on the 2TLCq-ROFHWG operator as shown in Table 20. However, the best and the worst alternative remains the same that are ℷ1 and ℷ2 that show the reliability and effectiveness of our purposed method.Table 20
Evaluation outcomes by utilizing different methodologies based on the 2TLCq-ROFHWG operator.
AlternativesCOPRASRankingCODASRankingEDASRankingTOPSISRankingℷ11.0000I−1.4587VI0.2452VII−0.2422VIIℷ20.9047III1.4557II0.4219V−0.5476IVℷ30.9675IV0.8752III0.2846IV−0.6408Iℷ40.9656VI−1.6952VII0.4511II−0.1472IIℷ50.9234VII0.4201V0.4725III−0.6044VIℷ60.9441V1.6987I0.0913I−0.5873Vℷ70.9366II−1.2959IV0.9024VI−0.0653III


## 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. The merits of our proposed method are summarized as follows:(i)
The 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 parameterϱ and can impact the risk perspective of the DMs. Thus, we can conclude that the proposed methods are significantly preferable and have more comprehensive applications. The 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. Therefore, the proposed approach is clear and has less loss of data.(ii)
The 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. Thus, 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.


## 7. Conclusions

In this work, we contributed to the development of MAGDM by analyzing difficulties in a 2TLCq-ROF context. The theoretical basis of aggregation operators must be carefully addressed in preparation for their use in decision-making. The 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. The following conclusions can be drawn in summary:(1)
The 2TLCq-ROFS idea has been used to describe uncertain data. The Cq-ROFS is an extended version of the CIFS and CPyFS. The Cq-ROFS is defined by two functions that express the degree of complex-valued membership and nonmembership. A flexible parameter q in C q-ROFS will influence its values to reflect information in a broader domain.(2)
The 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. This 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. Therefore, 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. The 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. Therefore, 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 [47–51], 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.



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*Source: 4523287-2022-06-26.xml*