A Decision-Making Approach for the Evaluation of Information Security Management under Complex Intuitionistic Fuzzy Set Environment

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Introduction
e multicriteria group decision-making (MCGDM) approach is one of the methods for selecting the best option from a collection of alternatives.We come across different forms of decision-making (DM) problems in our everyday lives.erefore, in order to solve such problems, we need to learn how decisions are made.Typically, the DM method requires crisp information, but complexity plays an important role in any DM problem in many everyday practical problems, and as a result, the information may not be in the form of a crisp collection.e notion of the fuzzy set (FS) was therefore developed by Zadeh [1] to deal with these situations and is characterized by a membership degree (MD) belonging to the closed interval [0, 1].However, the fuzzy set only addresses the level of satisfaction or dissatisfaction with the DM problem.erefore, to overcome this situation, Atanassov [2] proposed the idea of intuitionistic fuzzy set (IFS), which combines both qualities by adding the non-membership degree to the FS and satisfying the condition that the number of the membership degrees is less than or equal to 1. Several scholars and authors have presented their theories and methodologies [3][4][5][6][7] to explore the problems of DM and extend them to different disciplines under this theory.Song et al. [8] developed a similarity measure for IFSs and applied the definition to a medical diagnosis problem.Furthermore, Liu et al. [9] proposed some new operators based on Dombi operational laws [10] and Bonferroni mean (BM) operators [11], such as intuitionistic fuzzy Dombi Bonferroni mean (IFDBM) operators, to deal with the multicriteria group decision-making (MCGDM) problem.Khan et al. [12] presented prioritized aggregation operators for Pythagorean fuzzy information, and also Khan et al. [13] developed Einstein T-norm and Tconorm-based operators to deal with MCDM problems under the PF environment.Khan et al. [14,15] developed interval-valued Pythagorean fuzzy Choquet integral operators.However, the PFN has also some restrictions on the scope of information.Khan et al. [16] introduced the notion of Pythagorean hesitant fuzzy sets and proposed decisionmaking approaches based on the Pythagorean hesitant fuzzy TOPSIS method [17] and Pythagorean hesitant fuzzy Choquet integral [18] to solve MCGDM problems.Moreover, Garg and Rani [19] proposed a new decision-making technique based on a hesitant fuzzy set.Batool et al. [20] proposed an MCDM technique for the fog-haze factor assessment problem under a Pythagorean probabilistic hesitant fuzzy environment.Based on maximizing deviation and the TOPSIS method, Garg and Rani [21] developed a MADM method under a simplified neutrosophic hesitant fuzzy environment.Khan et al. [22] developed a new ranking method for q-Rung Orthopair fuzzy values and proposed a new graphical ranking method based on hesitancy index and entropy.Khan et al. [23] introduced a linguistic q-Rung Orthopair fuzzy set and applied the notion to decisionmaking problems.As discussed, that these theories have their own advantages, however, the theory of soft sets introduced by Molodtsov [24] carried out a shovel work as it generalizes all the theories.In 2000, Lee [25] introduced another generalization of these theories in the form of bipolar-value fuzzy sets.Recently, Mahmood [26] introduced a new type of bipolar soft set called "T-bipolar soft set" and showed that the new approach is closer to the concept of bipolarity as compared to the previous ones.
Researchers have considered the MCGDM problems with FS and IFS, which are only able to deal with uncertainty and vagueness that occurs in results, according to the above discussion.ese models are unable to account for the data's partial effect when dealing with data and its uncertainty at a particular point in time.However, in complex datasets, uncertainty and vagueness in the data occur simultaneously with changes in the data's process.To address these issues, Romot et al. [27] proposed the concept of a complex fuzzy set (CFS), which is defined as a complexvalued membership degree with a codomain unit disc in a complex plane.Many researchers [28][29][30] have conducted research in the area of CFS.However, CFS only models the agreement of elements in any set and does not discuss the disagreement.us, to handle such situations, Alkouri and Salleh [31] proposed the concept of a complex intuitionistic fuzzy set (CIFS), an extension of CFS, and is characterized by a complex-valued membership degree.In [32], the authors discussed the relation, projection, composition between CIFSs.ey also developed distance measure for CIFS and proposed some operational laws for CIFS.To solve an MCDM problem, Rani and Garg [33,34] developed some T-norm and T-conorm-based generalized averaging operators, power operators, and distance measures, respectively.
Garg and Rani [21] developed a variety of similarity measures and entropy measures for CIFS; they discussed some properties and applied the concept to MCDM problems and extended the concept of Bonferroni mean [11] operators.Garg and Rani [19] proposed some complex intuitionistic fuzzy Bonferroni mean operators with their different properties and applied the concept to the MCGDM problem.However, Bonferroni mean cannot be processed by Dombi operations.In addition, because CIFNs can describe fuzzy information easier, it is an increasing demand to combine the BM operator and the Dombi operations to deal with CIFNs for dealing with MCGDM problems.e goal of this paper is to propose some new complex intuitionistic fuzzy Dombi Bonferroni mean (CIFDBM) operators for dealing with MCGDM problems with complex intuitionistic fuzzy information.CIFDBM operators not only discuss interrelationships among criteria but are also flexible to deal with MCGDM problem.Motivated by these, the main objectives of this article are as follows: (a) Some new complex intuitionistic fuzzy Dombi Bonferroni mean (CIFDBM) operators with complex intuitionistic fuzzy setting are introduced in order to project time-periodic problems and twodimensional information simultaneously in one set.(b) Some properties of the proposed operators are discussed in detail.(c) A MCGDM method is proposed under the CIF environment.e applicability of the proposed method is explored on a real-life decision-making problem related to the evaluation of information security management.Finally, the proposed method is compared with existing methods in order to demonstrate the application and validity of the proposed approach.e remaining of the paper is arranged as follows.Some basis definitions, operational laws, and aggregation operators are presented in Section 2. Section 3 deals with the complex intuitionistic fuzzy Dombi Bonferroni mean (CIFDBM) operator, complex intuitionistic fuzzy Dombi weighted Bonferroni mean (CIFDWBM) operator, complex intuitionistic fuzzy Dombi geometric Bonferroni mean (CIFDGBM) operator, and complex intuitionistic fuzzy Dombi weighted geometric Bonferroni mean (CIFDWGBM) operator.Some properties of the proposed operators are investigated, and various cases are discussed in detail.A MCGDM approach is proposed in Section 4. In Section 5, a numerical example for firewall selection problem is presented, and comparative study is also conducted.Concluding remark is presented in Section 6.

Preliminaries
is section comprises some basic definitions and operational laws for complex intuitionistic fuzzy numbers.

Complex Intuitionistic Fuzzy Set and eir Operations.
Alkouri and Salleh [30] first suggested the theory of complex intuitionistic fuzzy set as a generalization of the complex 2 Journal of Mathematics fuzzy set and intuitionistic fuzzy set.It is characterized by a complex-valued MD and complex-valued NMD.
en, the score and accuracy function for CIFNs are defined as and , and S(Al 2 ) ≤ S(A 2 ).Some basic operational laws for CIFNs were proposed by Garg and Rani [33].
Definition 3 (see [33]).Let L j � 〈〈Υ j , η j 〉, 〈Λ j , λ j 〉〉 (j � 1, 2) be CIFNs, and E ≥ 0 be any real scalar.en, the algebraic operational laws for CIFNs are define as In literature [21], Garg and Rani introduced different entropy measures and applied the concept to the MCDM problem.e CIF entropy measure can be defined as follows: Definition 4 (see [21]).For any set H ∈ ψ( Ã), entropy measure Ê: ψ( Ã) ⟶ [0, 1] is a real-valued function satisfying the following properties: en, the entropy measure can be defined by As we have discussed in the previous section, the BM operator is one of the better tools to deal with a DM problem.
e BM operators not only deal interrelationships among criteria but are also more flexible.erefore, Garg and Rani [19] proposed CIFBM operators as Definition 6 (see [19]).Let In 1982, Dombi [10] introduced a generator and produced Dombi T-norm and Dombi T-conorm shown as where developed operators are discussed, and various cases are investigated in detail.
en, a mapping CIFDBM p,q : L n ⟶ L is called a complex intuitionistic fuzzy Dombi Bonferroni mean operator if en, the resultant aggregated values by using the CIFDBM p,q operator is a CIFN, and Proof.First, we prove that the above equation holds.Since and en, and It implies that Journal of Mathematics Furthermore, 6 Journal of Mathematics en, Put, Next, we prove that equation ( 9) is CIFN.Let Journal of Mathematics Next, we prove that CVMD and CVNMD fulfill the condition as follows: (a) First, we prove part (a).Since μ j , μ i , τ j , and τ i are monotonically decreasing functions, ] j , ] i , ρ j , and ρ i are monotonically increasing functions, and k > 0. As and en, we have It implies that Journal of Mathematics at is 0 ≤ α ≤ 1, and 0 ≤ c ≤ 1.In the similar way, we can show that 0 ≤ β ≤ 1, and 0 ≤ δ ≤ 1.
Step 4. Aggregate the total assessment information by utilizing the CIFDWBM/CIFDWGBM operator.
Step 5. Rank the alternatives by calculating the score values of the CIFNs and then select the best alternative.

Illustrative Example
Due to the rapid advancement of computer and information technology, information has become a new commodity for businesses and has become increasingly relevant.In recent years, it is the need of all companies together that how to secure information security, which is a big problem.erefore, in this section, we take a numerical example that how to evaluate the enterprise's information security management.We look at an issue of information security management assessment in a particular organization.Let DM e three experts presented their preference values in the form of CIFNs as presented in Tables 1-3.
Expert weight is calculated using the method presented by Xu [35].
Step 1.Using the CIFWDBM/CIFDGBM operator to aggregate the assessment information presented by the DMs.We obtain the collective CIF decision matrices presented in Table 4.
Step 2. Compute the attribute weight by using the CIF entropy measure presented in equation ( 4 Based on score values, the ranking of alternatives is Hence, the most desirable alternative is Al 5 .
For CIFWDGBM operator Step 1'.Using the CIFWDGBM operator to aggregate the assessment information presented by the DMs.We obtain the collective CIF decision matrices presented in Table 5 Step 2'.Compute the attribute weight by using the CIF entropy measure presented in equation ( 4), and we get the attribute weight as Based on score values the ranking of alternatives is Hence, the most desirable alternative is Al 5 .
5.1.Impact of Different Parameters p, q, and Delta on the Ranking of Alternatives.Since the above result is based on the parameters p, q, and delta having fixed values.Tables 6  and 7 show a more detailed overview of the effect of the various parameters by adjusting the parameters p, q, and k.
It is seen that corresponding to p � 1, q � 1andp � 1 q � 2, p � 2, q � 1, p � 2, q � 2, the order of the alternatives is Al 5 ≻Al 3 ≻Al 2 ≻Al 1 ≻Al 4 .However, to increase the value of δ � 2 to δ � 3, the score values of the alternative tend to increase.Also, we get the same score values for, p � 1, q � 1 and p � 1, q � 2, p � 2, q � 1, p � 2, q � 2 when k � 2 to k � 3. us, for different values of parameters, the DMs can pick the nature of the problem.

Comparison Analysis.
In this section, we looked at how the proposed findings are compared to all of the current comparisons in the CIFS environment.To compare and describe the proposed method's success with a number of current CIFS studies, a comparative study is conducted, as presented in Table 8.
From Table 8, we see that we get the same best alternatives by applying the existing methods and proposed operators.e ranking of alternatives by using the CIFWA/ CIFWG operator [33], CIFEWA/CIFFWG operator [34], , which is the same as obtained by using the proposed operators.However, the existing CIFWBM operator only has an interrelationship among the criteria but there is no flexibility, while the proposed operators not only have an interrelationship among the criteria but also have more flexibility as compared to existing operators.Hence, the present method is more reliable and valid as compared with the existing methods.

Conclusion and Further Studies
Since Bonferroni mean operators are very effective tools to discuss interrelation among the criteria.Also, CIFS is more suitable in order to deal with the phase term as well as the amplitude term during the decision-making process.erefore, keeping the advantages of the CIFS and Bonferroni mean operators, in this paper, we developed some new aggregation operators based on Dombi T-norm and Tconorm with Bonferroni mean operators under the CIFS environment.We proposed a CIFDBM operator and a CIFDGBM operator and their weighted forms for CIFNs.Moreover, some desirable properties of the proposed operators are discussed and investigated in different cases.Further, we proposed an MCGDM approach based on the developed operators.Furthermore, to show the application and effectiveness of the developed approach, we presented a real-world decision-making problem of the evaluation of information security management.Finally, we compared the proposed approach with existing methods and showed that our proposed approach is more effective as compared to existing ones.In the future, we will develop some more generalized interactive aggregation operators by using various fuzzy environments [37,38].
1 , DM 2 , and DM 3 be three experts that have been invited from some famous universities and play a role of DMs. e DMs select five large state-owned enterprises and are denoted by Al 1 , Al 2 , Al 3 , Al 4 , and Al 5 .Meanwhile, they describe four characteristics based on the previous study denoted as Cr 1 , Cr 2 , Cr 3 , and Cr 4 , which are presented as follows: