Abstract
Since intuitionistic fuzzy set only deals with uncertainty but not periodicity, therefore to overcome, this situation complex intuitionistic fuzzy set is a better tool that deals with both uncertainty and periodicity. Also, Bonferroni mean operator has the advantage of considering interrelationships between parameters, but it deals with only the crisp data. Recently, to deal with fuzzy data, many extensions of Bonferroni operators have been developed. Motivated by the CIFS and BM operators, in this paper, we proposed some Dombi Bonferroni mean operators to deal with CIF information. Dombi Bonferroni mean operators are special cases of general T-conorm and T-norm, which have the advantage of good flexibility with a general parameter. We proposed 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 developed operators are discussed in detail, and different cases are investigated. Moreover, a multicriteria group decision-making (MCGDM) method is developed based on the proposed aggregation operators. Finally, a numerical example of information security management evaluation is given in order to demonstrate the application and effectiveness of the proposed approach. A comparative study is also conducted in order to show the advantage of the developed method.
1. Introduction
The 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. Therefore, 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. The 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. Therefore, 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 [3–7] 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 T-conorm-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 decision-making 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 decision-making 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. These 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 complex-valued membership degree with a codomain unit disc in a complex plane. Many researchers [28–30] 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. Thus, 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. They 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. The 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 two-dimensional 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. The 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.
The 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.
2. Preliminaries
This section comprises some basic definitions and operational laws for complex intuitionistic fuzzy numbers.
2.1. Complex Intuitionistic Fuzzy Set and Their Operations
Alkouri and Salleh [30] first suggested the theory of complex intuitionistic fuzzy set as a generalization of the complex fuzzy set and intuitionistic fuzzy set. It is characterized by a complex-valued MD and complex-valued NMD.
Definition 1 (see [30]). Let be a universal set. Then, a complex intuitionistic fuzzy set can be defined bywhere , defined by and , where and . A CIFN is denoted by such that .
For two CIFNs , Alkouri and Salleh [23] defined a relation between them as(1) if , , , and (2) if and only if and (3)Garg and Rani [33] developed a score and accuracy function in order to compare two CIFNs. These can be defined by the following definition:
Definition 2 (see [33]). Let be CIFNs. Then, the score and accuracy function for CIFNs are defined asandrespectively. Moreover, if are CIFNs, then , if either or , and .
Some basic operational laws for CIFNs were proposed by Garg and Rani [33].
Definition 3 (see [33]). Let be CIFNs, and be any real scalar. Then, the algebraic operational laws for CIFNs are define as(1)(2)(3)(4)In literature [21], Garg and Rani introduced different entropy measures and applied the concept to the MCDM problem. The CIF entropy measure can be defined as follows:
Definition 4 (see [21]). For any set , entropy measure is a real-valued function satisfying the following properties:(1)(2), if is a crisp set(3)(4)(5) if either 𝒦 with and or 𝒦 with and
Definition 5 (see [21]). Let be a CIFS. Then, the entropy measure can be defined byAs we have discussed in the previous section, the BM operator is one of the better tools to deal with a DM problem. The BM operators not only deal interrelationships among criteria but are also more flexible. Therefore, Garg and Rani [19] proposed CIFBM operators as
Definition 6 (see [19]). Let be CIFNs, . Then, a mapping is called CIFBM operator ifIn 1982, Dombi [10] introduced a generator and produced Dombi T-norm and Dombi T-conorm shown aswhere , .
3. Complex Intuitionistic Fuzzy Dombi Bonferroni Mean Operators
In this section, we develop some new operators based on Bonferroni mean operators and Dombi operational laws to aggregate the CIFNs. Some desirable properties of the developed operators are discussed, and various cases are investigated in detail.
Definition 7. Let be CIFNs, . Then, a mapping is called a complex intuitionistic fuzzy Dombi Bonferroni mean operator if
Theorem 1. Let be CIFNs, . Then, the resultant aggregated values by using the operator is a CIFN, and
Proof. First, we prove that the above equation holds. SinceandNow, let , , , , , , , and . Then,andIt implies thatFurthermore,Then,Put, , , , , , , , and . Then,Next, we prove that equation (9) is CIFN. LetandNext, we prove that CVMD and CVNMD fulfill the condition as follows:(a) First, we prove part (a). Since , , , and are monotonically decreasing functions, , , , and are monotonically increasing functions, and .AsandThen, we haveandIt implies thatThat is , and . In the similar way, we can show that , and .
(b) We prove that and .
Since , and , . We need only to prove that and for this we have, as , , and , , then, , , and , implies that , , , and . That is, , , , and . Furthermore, we can getandandSo,andThus, and . Hence from (a) and (b), we conclude that equation (9), is a CIFN. For , some cases regarding parameters p and q are presented as follows:
Case a. When , , then we can have
Case b. When , then we can have
Case c. When , then we can have
Case d. When , then we can get
Example 1. Let , , and be three CIFNS. Then, we use the CIFDBM to aggregate the three CIFNs and generate a comprehensive value. The steps are shown as follows: (suppose , = 2, and ). SinceWe present some desirable properties for the operator as follows.
Idempotency 1. Let be CIFNs and . If , that is, , then
Proof. Suppose . First, we prove that , . Since , and , , therefore,This means that .
In the similar way, we can also prove that . This shows that
Monotonicity 1. Let and and be two collections of CIFNs, . If , , , and for all , then
Proof. Let and . We need to prove that , , , and . Since , , , and for all , then we have , and , . So, we haveandandThus,andThis shows that and . In the similar way, we can show that and . Therefore, we have
Boundedness 1. Let be CIFNs and . Then,
Proof. Based on Property 1, we haveAlso, by Property 2, we haveTherefore, we can get .
Definition 8. Let be CIFNs and . Then, a mapping is called CIF weighted Dombi Bonferroni mean operator ifwhere is a weighted vector of such that , .
Theorem 2. Let CIFNs and . Then, by using operators, the aggregated value is a CIFN and where is a weighted vector of such that , .where is a weighted vector of such that , . Idempotency 2: let be CIFNs and . If , that is, , then Monotonicity 2: let and be two collections of CIFNs and . If , , , and for all , then Boundedness 2: let be CIFNs and , and . Then,
Definition 9. Let be CIFNs and . Then, a mapping is called CIF Dombi geometric Bonferroni mean operator if
Theorem 3. Let be CIFNs and . Then, the resultant aggregated values by using the operator is a CIFN, andWe present some desirable properties for operator as follows: Idempotency 3: let be CIFNs and . If , that is, , then Monotonicity 3: let and be two collections of CIFNs and . If , , ,and for all , then Boundedness 3: let be CIFNs and , and . Then,
Definition 10. Let be CIFNs and . Then, a mapping is called a complex intuitionistic fuzzy weighted Dombi Bonferroni mean operator ifwhere is a weighted vector of such that , .
Theorem 4. Let beCIFNs and . Then by using the operators, the resultant aggregated value is a CIFN andwhere is a weighted vector of such that ,.
We present some desirable properties for CIFWDGBM operator as follows: Idempotency 4: let be CIFNs and . If , that is, , then Monotonicity 4: let and be two collections of CIFNs,. If , , , and for all, then Boundedness 4: let beCIFNs and , and . Then,
4. MCGDM Approach Based on CIFDBM Operators
In this section, a MCGDM method is proposed based on CIFDBM operators. The advantages of the proposed operators are that it not only considers interrelationships between criteria but are also more flexible in the decision-making process.
Generally, the MCGDM problem involves data and knowledge that is vague and imprecise. We consider MCGDM problems in which all criteria values expressed in CIFNs and interrelationships between decision-making criteria or decision-makers' preferences are considered. The collection of expert viewpoints is critical in MCGDM problems in order to properly conduct the assessment process. Based on CIFDBM operators in the following, a MCGDM approach is proposed under CIFNs where the assessment values are aggregated by utilizing the CIFDBM operators: Step 1. Since for each alternative , each is invited to express their individual evaluation or preference according to each criterion by CIFNs . Then, we obtain a CIF MCGDM matrix as follows: Let be an CIF decision matrix, provided by decision makers , as the following form: where is the CIFN representing the alternative's efficiency ranking with respect to the criteria provided by the DMs . Step 2. Aggregating the assessment information provided by the DMs by means of CIFDWBM/CIFDWGBM operator. or Step 3. Utilize the entropy measure defined in equation (4) to find the attribute weight of the assessment information. Step 4. Aggregate the total assessment information by utilizing the CIFDWBM/CIFDWGBM operator. or Step 5. Rank the alternatives by calculating the score values of the CIFNs and then select the best alternative.
5. 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. Therefore, 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 , , and be three experts that have been invited from some famous universities and play a role of DMs. The DMs select five large state-owned enterprises and are denoted by , , , , and . Meanwhile, they describe four characteristics based on the previous study denoted as , , , and , which are presented as follows: = control of physical protection (it includes the security management of facilities and the security management of the environment) = control of operational protection (it includes stuff-like operating system security management, network security management, and database security management) = risk management (environment risk, process risk, database risk, and control risk are all included.) = human resource and policy management (it includes human resource management, strategy management, and applicable policy management for the organization, among other things).
The 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), and we get the attribute weight as Step 4. Utilizing equations (80) and (81) aggregate to overall CIFNs, and we get the final aggregated values as Step 5. Utilizing equation (2), we get the score values as follows:
Based on score values, the ranking of alternatives is
Hence, the most desirable alternative is .
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 Step 3'. Utilizing equations (80) and (81) aggregate to overall CIFNs, and we get the final aggregated values as Step 4'. Utilizing equation (2), we get the score values as follows:
Based on score values the ranking of alternatives is
Hence, the most desirable alternative is .
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 , and , , , , , the order of the alternatives is . However, to increase the value of to , the score values of the alternative tend to increase. Also, we get the same score values for, , and , , , , , when 𝓀 = 2 to 𝓀 = 3. Thus, for different values of parameters, the DMs can pick the nature of the problem.
5.2. 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. The ranking of alternatives by using the CIFWA/CIFWG operator [33], CIFEWA/CIFFWG operator [34], and CIFLWA operator [36] is . Also, by using proposed operators, we have the same ranking of alternatives as . However, there is no interrelationship among the criteria by using existing operators. But in the proposed method, there is an interrelationship among the criteria. Moreover, by using the CIFWBM operator [19], we get the ranking of alternatives as , 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.
6. 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. Therefore, 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 T-conorm 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].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors have no conflicts of interest.