MULTIMOORA Method for Addressing Security Algorithms Evaluation Problem under q-Rung Orthopair Fuzzy Environment

How to determine a suitable security algorithm for a special application scenario is a complex problem. In this paper, this complex problem is formulated as a multicriteria decision-making (MCDM) problem, and we propose a novel MULTIMOORA (multiobjective optimization on the basis of a ratio analysis plus the full MULTIpevaluation information in the security algorithms evaluation problem.eMULTIMOORAmethod is an excellent decisionmethod, which owns strong robustness. However, it has not been used to process the complex information structure of q-rung orthopair fuzzy sets. Moreover, it cannot solve the problem that the extreme values negatively inuence the ranking results, and it also cannot capture the interrelationship hiding behind the criteria. To overcome the above challenges, we propose novel q-rung orthopair fuzzy Dombi power Heronian mean (DPHM) operator and q-rung orthopair fuzzy Dombi power geometric Heronian mean (DPGHM) operator. Based on these two operators, the MULTIMOORA method is improved for solving the security algorithms’ evaluation problem. Finally, a practical example for evaluating ve security algorithms is used to illustrate the decision process of the proposed q-rung orthopair fuzzy MULTIMOORA method.


Introduction
With the quick development of multiple information technologies including cloud computing, Internet of ings, and edge computing [1], more and more companies and personals choose to upload their private data to the network [2]. However, as the scale of network becomes larger, the whole network becomes more complicated [3]. e network shows massive security loophole [4]. e companies and personals also own the special software to improve their business. e software also has massive security loophole and risks. To ensure the reliability of software and network, researchers and scholars have provided some solutions. For example, Abdel-Basset et al. [5] have put forward a neutrosophic decision-making model for evaluating the e-government website according to the quality, security, and accessibility. Wang et al. [6] have combined the TOPSIS (technique for order of preference by similarity to ideal solution) approach with the 0-1 integer programming method to choose an intelligent web service for improving the reliability of network.
Researchers and scholars also designed a number of e cient security algorithms to ensure the security requirements of network [7][8][9]. However, these security algorithms usually own di erent characteristics and advantages [10]. For a special application scenario, a suitable security algorithm should be selected for satisfying the requirements of this application scenario. How to choose the most suitable security algorithm for a special application scenario is a big challenge. To address this problem, Ning et al. [11] formulated this problem as a multicriteria decision-making (MCDM) problem and proposed a hybrid model for selecting the best encryption algorithm according to several requirements such as the performance, physical, and security. However, the study [11] still has some shortcomings.
(1) In the study [11], crisp values are used to evaluate security algorithms. Since the security algorithm evaluation problem becomes more and more complex, it is not easy for decision makers to use accurate crisp values for evaluating security algorithms [13]. e birth of fuzzy sets (FSs) [14] provides decision makers with a new way to express uncertain evaluation information. However, FSs only describe the membership degree (MD) information. To enhance the uncertain information modeling capability, intuitionistic fuzzy sets (IFSs) [15] were proposed to express the MD and nonmembership degree (NMD) information. In IFSs, the sum of MD and NMD values is not larger than 1. To provide the decision makers with more freedom for expressing the evaluation information, the concept of Pythagorean fuzzy sets (PFSs) was proposed by Yager and Abbasov [16], where the square sum of MD and NMD is not larger than 1. To generalize the concepts of IFSs and PFSs, a generic version, called q-rung orthopair fuzzy set (q-ROFS), was proposed by Yager [17]. In this study, we intend to use q-ROFSs to express the uncertain information. e significance of q-ROFSs is that this information representation way is flexible, and it provides the decision makers with more freedom than PFSs and IFSs.
(3) As one of the efficient decision methods, the MULTIMOORA (multiobjective optimization on the basis of a ratio analysis plus the full MULTIplicative form) method [32] consists of three submodels for comprehensively determining the decision results. As shown in Table 1, the decision results that are obtained from the MULTIMOORA method are robust and the MULTIMOORA method outperforms than some other decision methods [12].
Because of its excellent characteristics, the MUL-TIMOORA method has been used to process various evaluation information, such as interval numbers [33], IFSs [34], picture fuzzy sets [35], and probabilistic linguistic term sets [36]. To the best of our knowledge, there have been no research results on the combination of q-ROFSs and MULTIMOORA method to date. In this paper, we intend to extend the MULTIMOORA method for processing the q-ROFS information in the MCDM problems. Nevertheless, the MULTIMOORA method cannot handle the case that extreme values influence the reliability of the decision results. Moreover, it is incapable of processing the complex interrelationships hiding behind criteria values.
Hence, the motivations of this study are summarized as (1) A more flexible way of q-ROFSs is used to express the uncertain and vague evaluation information for the security algorithms evaluation problems (2) A novel decision-making method is developed to solve the security algorithms evaluation problems and select an appropriate algorithm for a special application scenario To overcome the challenges, a novel q-rung orthopair fuzzy MULTIMOORA method based on Dombi power Heronian mean aggregation operators is proposed in this paper, and it is applied to solve the security algorithms' evaluation problem.
(1) e Dombi operational laws, special forms of t-norms and t-conorms, show strong flexibility when computing input values. e power average (PA) operator has the ability of alleviating negative influences of extreme input values on the decision results. e Heronian mean (HM) acts as a mapping function that can capture the complex interrelationships among input values. Considering the excellent characteristics, in this paper, some Dombi power Heronian mean aggregation operators are proposed to fuse q-rung orthopair fuzzy numbers (q-ROFNs), which are qrung orthopair fuzzy Dombi power Heronian mean (q-ROFDPHM) operator and q-rung orthopair fuzzy Dombi power geometric Heronian mean (q-ROFDPGHM) operator, as well as their weighted forms. Afterwards, their features are discussed.
(2) e weighted forms of the q-ROFDPHM and q-ROFDPGHM operators are applied to improve the MULTIMOORA method so that a novel q-rung orthopair fuzzy MULTIMOORA method is put forward for handling the security algorithms' evaluation problem. After that, the detailed decisionmaking procedure of the proposed q-rung orthopair fuzzy MULTIMOORA method is provided. (3) A case concerning the evaluation of five security algorithms is provided to show the implementation processes of the proposed q-rung orthopair fuzzy MULTIMOORA method. Afterwards, the influences of the parameters on the ranking results are analyzed. en, the q-rung orthopair fuzzy MULTI-MOORA method is compared with the existing decision methods that handle the q-ROFS information.
e rest content of this paper is organized as follows. e basic knowledge of q-rung orthopair fuzzy sets, PA, Dombi T-conorm and T-norm, HM operator, and MULTIMOORA method is provided in Section 2. In Section 3, the q-ROFDPHM operator and its weighted form are put forward. Section 4 puts forward the q-ROFDPGHM operator and its weighted form. In Section 5, we apply the proposed operators to propose a novel q-rung orthopair fuzzy MULTI-MOORA method and also present the decision procedure. In Section 6, an illustrative example of evaluating of security algorithms is provided to show the implementation process of the proposed q-rung orthopair fuzzy MULTIMOORA method. In Section 7, some valuable conclusions are listed.

Preliminaries
In this paper, the basic information of q-ROFSs, PA, Dombi T-conorm and T-norm, HM operator, and MULTIMOORA method is provided.
Definition 1 (see [17]). Let X � x 1 , x 2 , . . . , x n be a finite universe of discourse (UoD); then, a q-ROFS A on X is mathematically expressed as where μ A : X ⟶ [0, 1] and ] A : X ⟶ [0, 1] are the membership degree (MD) and nonmembership degree (NMD) of the element x belonging to the q-ROFS A, respectively. e constraint conditions for q-ROFS are q is defined to be the hesitant degree (HD) of the element x belonging to the q-ROFS A. For convenience, the two-tuple (μ A (x), ] A (x)) is simplified as (μ A , ] A ), which is also called q-rung orthopair fuzzy number (q-ROFN) by Liu and Wang [40].
For comparing q-ROFNs, the definitions of score function and accuracy function were given by Liu and Wang [40] for q-ROFNs as follows.
Based on the above score function and accuracy function presented in Definition 2, Liu and Wang [40] gave a method for comparing two q-ROFNs as follows.
Definition 3 (see [40]). Given two q-ROFNs o 1 � (μ 1 , ] 1 ) and o 2 � (μ 2 , ] 2 ), s(o 1 ) and s(o 2 ) are their score function values, and h(o 1 ) and h(o 2 ) are their accuracy function values, , then their accuracy function values should be further compared as follows: To measure the deviation degree between any two q-ROFNs, Liu et al. [41] provided the definition of distance between them as follows.

Power Average
Operator. e power average (PA) is a useful aggregation operator that was put forward by Yager [42]. e PA operator has the ability of alleviating negative influences of extreme input values on the calculation results. e original PA operator was devised to process crisp values. Its mathematical definition is given as follows: Definition 5 (see [42]). Let o i (i � 1, 2, . . . , n) be a series of nonnegative crisp values; then, the PA operator really acts as a function that where e support degree satisfies the following features:

Dombi T-Norm and T-Conorm.
e Dombi T-norm (TNM) and T-conorm (TCNM), which were proposed by Dombi [43], are referred to as special forms of t-norms and t-conorms. eir mathematical expressions are provided as follows.
Definition 6 (see [43]). Given any two real values, m and n, then the Dombi TNM and Dombi TCNM act as two functions, which are mathematically defined as where ℵ > 0, m, n ∈ [0, 1]. Based on the above Dombi TNM and Dombi TCNM, Jana et al. [44] gave the Dombi operational laws for computing q-ROFNs as follows.

Heronian Mean and Geometric Heronian Mean
Operators.
e aggregation operators (AOs) [45][46][47] are value measurement MCDM methods. It is very simple and easy to perform AOs. e AOs are the processes, which fuse given input values into a single value [48]. For aggregating the complicated information structures of various fuzzy sets, researchers have put forward various AOs. e Heronian mean (HM) operator [49], an excellent and useful AO, is capable of processing the complicated interrelationships among input values, which are common in the MCDM contexts. e HM operators can be divided into two categories: arithmetic HM (AHM) and geometric HM (GHM) operators, which are mathematically defined as follows.

(5)
Definition 9 (see [49]). Let o i (i � 1, 2, . . . , n) be a series of nonnegative real values; the parameters c, η ≥ 0; then, the GHM operator can aggregate the nonnegative real values as For the AHM operator, its aggregated values are greatly influenced by extreme values [50]. e GHM operator is capable of balancing the big differences among input values [51]. erefore, the GHM operator performs better than the AHM operator in some cases.

MULTIMOORA.
To obtain more robust decision results, the full multiplicative form (FMF) was applied by Brauers and Zavadskas [32] to extend the initial MOORA (multiobjective optimization on the basis of ratio analysis) method.
us, the MULTIMOORA method has three components: ratio system (RS) component, reference point (RP) component, and FMF component, respectively [52]. ese three components derive the decision results independently. For the purpose of determining the final decision result, the decision results obtained from these three components are processed by the dominance theory [32]. In the following part, the process for implementing the MULTIMOORA method is listed as follows.
Let us suppose that there exists an MCDM problem consisting of m alternatives x 1 , x 2 , . . . , x m and n criteria a 1 , a 2 , . . . , a n . e weight vector of criteria is denoted as [ω 1 , ω 2 , . . . , ω n ], where n j�1 ω j � 1 and 0 ≤ ω j ≤ 1. e decision matrix (DM) R � (o ij ) m×n corresponding to the MCDM problem contains the evaluation information from experts. e element o ij represents the evaluation information of alternative x i with respect to criterion a j . e evaluation information of alternatives across multiple criteria usually shows different dimensions, so the evaluation information in the DM R � (o ij ) m×n is suggested to be normalized as After that, the normalized DM R � (o ij ) m×n can be derived.

RS Component.
In this component, the criteria should be divided into two categories: benefit-type (BT) criteria and cost-type (CT) criteria. For BT criteria, the larger the evaluation information of alternative, the better the alternative. For CT criteria, the larger the evaluation information of alternative, the worse the alternative. e weighted 4 Mathematical Problems in Engineering arithmetic aggregation operator (AAO) is used to calculate the ranking value Y i of alternative x i as where k represents the number of benefit-type criteria and n − k means the number of cost-type criteria. From the above equation, it is noted that the alternative in the RS component having the maximum ranking value is considered as the best one. erefore, the alternatives can be ranked based on the descending order of their ranking values.

RP Component.
For the RP component, the worst criterion value of each alternative that is farthest from the reference point of the corresponding criterion should be first derived, and then, the alternative with the smallest worst criterion value is considered as the optimal one.
In this component, the reference point of each criterion is first determined as where o j denotes the reference point of alternatives with respect to criterion a j . en, the weighted distance between the normalized evaluation information of the alternative x i with respect to each criterion and the reference point of the same criterion is computed as Finally, the ranking value D i of alternative x i is computed as D i � max j d ij .
According to the RP component, the optimal alternative should have the smallest ranking value. us, the alternatives can be ranked based on the ascending order of their ranking values.

FMF Component.
e design idea of FMF component is the same as that of RS component. In the FMF component, the better alternative should have higher values for benefittype criteria and lower values for cost-type criteria. e weighted geometric aggregation operator (GGO) is used to determine the ranking value U i of alternative x i as According to the design idea, the alternative having the largest ranking value should be considered as the best one in the FMF component. Hence, the alternatives can be ranked based on the descending order of their ranking values.
To aggregate the ranking orders of alternatives obtained from these three components, the dominance theory was suggested by Brauers and Zavadskas [32] to be used for deriving the final decision results.

q-Rung Orthopair Fuzzy Dombi Power Heronian Mean Operators
In this section, we use the PA operator, Dombi operational laws for q-ROFNs, and arithmetic HM operator to propose q-rung orthopair fuzzy Dombi power HM (q-ROFDPHM) operator and its weighted form. en, the features are discussed.

Definition
10. Given a set of q-ROFNs o i � (μ i , ] i )(i � 1, 2, . . . , n) and three parameters c, η ≥ 0 and ℵ > 0, then the q-ROFDPHM operator is defined as Based on the Dombi operational laws of q-ROFNs [36] and HM operator, the following theorems can be derived.
. . , n) and the parameters c, η ≥ 0 and ℵ > 0, then the aggregated result derived from equation (10) is still an q-ROFN, which is Mathematical Problems in Engineering Proof. According to Definition 10, we have Let According to Definition 7, it can be derived that

Mathematical Problems in Engineering
Let (1/nξ i ) � t i and (1/nψ j ) � e j ; then, the above equations can be transformed into us, en, the final result can be determined as Mathematical Problems in Engineering en, we need to prove that the aggregated result from the q-ROFDPHM operator is still a q-ROFN.

Mathematical Problems in Engineering
Proof. According to Definition 7, we have Since us, it can be derived that en, we can have Similarly, we have Since (1/nξ i ) � t i and (1/nψ j ) � e j , then we have Similarly, it can be proven that q − ROFDPHM , and e j � (1/nω j ψ j ).

Mathematical Problems in Engineering
According to Definition 7, then we have Let t i � (1/nω i ξ i ) and e j � (1/nω j ψ j ); then, the above two equations can be transformed into Mathematical Problems in Engineering e process for proving that the aggregation result of the q-ROFWDPHM operator is a q-ROFN is the same as that of eorem 1. us, it is omitted here. e proposed q-ROFWDPHM operator also owns the features of idempotency and boundedness as the proposed q-ROFDPHM operator. eir proof processes are similar to those of eorems 2 and 3. Due to the limited space, the proof processes are omitted here.

q-Rung Orthopair Fuzzy Dombi Power Geometric Heronian Mean Operators
In this section, we use the PA operator, Dombi operational laws for q-ROFNs, and geometric HM operator to develop a novel q-ROFDPGHM operator and its weighted form. en, the features are discussed.

Theorem 5. Given n q-ROFNs
where e proof process of this theorem is similar to that of eorem 1. us, it is omitted here. e proposed q-ROFDPGHM operator also owns the features of idempotency and boundedness as the proposed q-ROFDPHM operator. eir proof processes are similar to those of eorems 2 and 3. Due to the limited space, the proof processes are omitted here.

Similar to the proposed q-ROFDPHM operator, the q-ROFDPGHM also does not consider the weight values of criteria. To tackle this deficiency, a new q-rung orthopair fuzzy weighted Dombi power geometric Heronian mean (q-ROFWDPGHM) operator is put forward in the following part.
Definition 13. Given a set of q-ROFNs 1, 2, . . . , n), three parameters c, η ≥ 0 and ℵ > 0, and the weight values [ω 1 , ω 2 , . . . , ω n ] of q-ROFNs, then the q-ROFWDPGHM operator is defined as Based on the Dombi operational laws of q-ROFNs and GHM operator, a theorem is derived. , n), the parameters c, η ≥ 0 and ℵ > 0, and the weight values [ω 1 , ω 2 , . . . , ω n ] of q-ROFNs, then the aggregated result that is obtained from equation (43) is still a q-ROFN, which is

Proof.
e proof process is similar to that of eorem 1. us, it is omitted here. e proposed q-ROFWDPGHM operator also has the features of idempotency and boundedness as the proposed q-ROFDPHM operator. eir proof processes are similar to those of eorems 2 and 3. Due to the limited space, the proof processes are omitted here.

MULTIMOORA Method for q-Rung Orthopair Fuzzy Sets
In this section, the MULTIMOORA method is improved for processing the MCDM problems with the q-ROFS information. ere usually exist the interrelationships among the criteria in the MCDM problems. Moreover, there may be extreme criteria values in the MCDM problems. To tackle these two problems, we use the proposed q-ROFWDPHM and q-ROFWDPGHM operators to modify the MULTIMOORA method.

Problem Description.
Let us suppose that there exists an MCDM problem consisting of m alternatives x 1 , x 2 , . . . , x m and n criteria a 1 , a 2 , . . . , a n . e weight values of criteria are denoted as [ω 1 , ω 2 , . . . , ω n ], where n j�1 ω j � 1 and 0 ≤ ω j ≤ 1. e decision matrix (DM) R � (o ij ) m×n corresponding to this MCDM problem consists of the evaluation information from experts. e element o ij denotes the evaluation information of alternative x i with respect to criterion a j . In this MCDM problem, experts use the flexible q-ROFNs for expressing the evaluation information of alternative x i with respect to criterion a j , namely, o ij � (μ ij , ] ij ). Here, the criteria are divided into two different categories: benefit-type criteria and cost-type criteria.
Before processing DM R � (o ij ) m×n , equation (45) is used to transform the values of cost-type criteria for deriving the transformed DM R � (o ij ) m×n : ] ij , μ ij , for cost − type criterion a j .

q-Rung Orthopair Fuzzy MULTIMOORA Method.
According to the above problem description, we introduce the q-ROFWDPHM and q-ROFWDPGHM operators to improve the original MULTIMOORA method so as to propose a novel q-rung orthopair fuzzy MULTIMOORA (q-ROF-MULTI-MOORA) method. Similar to the original MULTIMOORA method [53], the q-ROF-MULTMOORA method is also composed of three components, which are the q-rung orthopair fuzzy RS (q-ROF-RS) component, q-rung orthopair fuzzy RP (q-ROF-RP) component, and q-rung orthopair fuzzy FMF (q-ROF-FMF) component, respectively. Based on the transformed DM R � (o ij ) m×n , these three components compute the ranking values of alternatives as follows.

q-ROF-RS Component.
In this component, the q-ROFWDPHM operator is applied to aggregate the evaluation information of each alternative x i with respect to its n criteria. erefore, using (34), the aggregated criteria value of alternative x i can be computed as where Since the aggregated value is a q-ROFN, then the score function in Definition 2 is used to derive the crisp ranking value of alternative x i as Mathematical Problems in Engineering , and e ig � (1/nω g ψ ig ).
e alternative with larger ranking value is better. Hence, all the alternatives can be ranked according to the descending order of their ranking values.

q-ROF-RP Component. In this component, the reference point of each criterion is first derived as
In the second step, Definition 4 is applied to compute the distance between the evaluation information of alternative x i with respect to each criterion and the reference point of the same criterion as It can be known that ε ij is a real value and ε ij ≥ 0. Considering the interrelationships among criteria, the ranking value of alternative x i is computed by aggregating the criteria distances of alternative x i as where In this component, the alternative with smaller ranking value is better. us, all the alternatives should be ranked according to the ascending order of their ranking values.

q-ROF-FMF Component.
In this component, the proposed q-ROFWDPGHM operator is applied to aggregate the evaluation information of each alternative x i with respect to its n criteria. us, using equation (43), the aggregated criteria value of alternative x i can be computed as where Since the aggregated value is a q-ROFN, then the score function in Definition 2 is used to derive the crisp ranking value of alternative x i as where ξ ih � ((1 + S(o ih ))/ n k�1 ω k (1 + S(o ik ))), ψ ig � ((1+ S(o ig ))/ n k�1 ω k (1 + S(o ik ))), a ih � ((1 − μ , and e ig � (1/nω g ψ ig ).
In this component, the alternative with larger ranking value is better. Hence, all the alternatives can be ranked according to the descending order of their ranking values.
After obtaining the ranking values of all the alternatives from these three components, we need to fuse them for deriving the final ranking values. In the original MULTI-MOORA method, the dominance theory is usually used to aggregate three ranking orders for deriving the final ranking order. However, it is incapable of handling massive operations resulting from its cumbersome pairwise comparison processes [54]. For the purpose of overcoming the deficiency of dominancy theory, the HM operator is put forward for integrating the ranking values of alternatives obtained from three components of the proposed q-ROF-MULTIMOORA method. e HM operator owns the advantage of capturing the interrelationships hiding behind input values. Afterwards, by using the ranking values obtained from equations (47) where x 1 , x 2 , . . . , x m denotes the set of alternatives and c 1 , c 2 , c 3 denotes the set of criteria. e element f iy (y � 1, 2, 3) in the DM M denotes the ranking value of the alternative x i with respect to the criterion c y . Let χ � χ 1 , χ 2 , χ 3 be the weight values of criteria c 1 , c 2 , c 3 , satisfying 0 ≤ χ y ≤ 1 and 3 y�1 χ y � 1. In general, the weight values of criteria are set to χ 1 � χ 2 � χ 3 � (1/3). For the DM M, the ranking values of each alternative with respect to three criteria should be aggregated for determining the final ranking values. However, the ranking values f iy (y � 1, 2, 3) show different dimensions because they are obtained from the different components. For the purpose of making them dimensionless, all the ranking values f iy (y � 1, 2, 3) are normalized as where 1 ≤ i ≤ m and 1 ≤ y ≤ 3. Afterwards, the weighted HM operator [55] is used to aggregate the normalized ranking values f iy of each alternative x i with respect to three criteria for deriving the final ranking value of this alternative as . (55) e alternative with larger final ranking value is better. Hence, all the alternatives can be ranked based on the descending order of their final ranking values.

Decision-Making
Procedure. Based on the discussion and results in Section 5.2, the decision-making procedure of the proposed q-ROF-MULTIMOORA method is summarized using the following 7 steps.
Step 1: all the evaluation information is collected for constructing DM R � (o ij ) m×n � (μ ij , ] ij ) m×n . At the same time, the values of the parameters q, c, η, and ℵ should be provided.
Step 2: to transform the criteria values of each alternative with respect to cost-type criteria, (45) is used to Step 3: for the transformed DM R � (o ij ) m×n , (47) is applied to compute the ranking value f i1 of each alternative x i with respect to the q-ROF-RS component. e alternatives can be ranked according to the descending order of their ranking values.
Step 4: for the transformed DM R � (o ij ) m×n , (50) is applied to compute the ranking value f i2 of each alternative x i with respect to the q-ROF-RP component. e alternatives can be ranked according to the ascending order of their ranking values.
Step 5: for the transformed DM R � (o ij ) m×n , (52) is applied to compute the ranking value f i3 of each alternative x i with respect to the q-ROF-FMF component. e alternatives can be ranked according to the descending order of their ranking values.
Step 6: based on the ranking values of alternatives obtained from three components in Steps 3-5, a new DM M � (f iy ) m×3 is constructed. Afterwards, (54) is Step 7: in the final step, (55) is used to aggregate the ranking values of alternatives with respect to three components of the q-ROF-MULTIMOORA method for deriving the final ranking values. en, all the alternatives are ranked according to the descending order of their final ranking values. e above steps are also shown in Figure 1. e q-ROF-MULTIMOORA method is a combination of PA operator, Dombi operational laws, AHM, GHM, and MULTIMOORA. It shows the following advantages: (1) It has the ability of alleviating negative influences of extreme criteria values on the decision results, which makes the decision results more stable and robust.

Illustrative Example and Comparison Analysis
In this section, a practical case concerning the evaluation of security algorithm is shown to illustrate the decisionmaking procedure of the proposed q-ROF-MULTI-MOORA method. Afterwards, the influences of the parameters on the decision results are analyzed. Finally, the proposed q-ROF-MULTIMOORA method is compared with the original MULTIMOORA method for processing q-ROFNs.

Decision Process Using the q-ROF-MULTIMOORA
Method. In this section, a real case concerning the evaluation of security algorithms is provided to illustrate the decision procedure of the proposed q-ROF-MULTIMOORA method.

Example 1.
With the quick development of Internet applications, more and more user data are stored online. Hackers frequently attack the Internet applications for obtaining the privacy data. To protect users' privacy data, various security algorithms have been designed and implemented. However, these security algorithms show different features. How to choose the suitable security algorithm is a big challenge for organizations since multiple criteria should be considered. Here, we try to formulate the process of evaluating the security algorithms and selecting a suitable one as a classical MCDM problem. Suppose organization plans to evaluate 5 candidates of security algorithms and select the suitable one by considering 6 criteria: function (c 1 ), reliability (c 2 ), usability (c 3 ), performance (c 4 ), portability (c 5 ), and complexity (c 6 ). Hence, an MCDM problem composed of 5 security algorithms x 1 , x 2 , x 3 , x 4 , x 5 and 6 criteria c 1 , c 2 , c 3 , c 4 , c 5 , c 6 can be constructed. According to the real requirements for building the security system, the organization sets the weights of criteria as ω � (0.10, 0.15, 0.35, 0.20, 0.10, 0.10). e technical panel of this organization uses the q-ROFNs to evaluate these five security algorithms with respect to their criteria. All the q-ROFNs are collected to form the DM R � (o ij ) 5×6 � (μ ij , ] ij ) 5×6 , as shown in Table 2.
Step 1: the values of the parameters c, η, and ℵ are set to 1 and the value of the parameter q is set to 3.
Step 2: the first five criteria are benefit-type criteria, while the maintenance cost is cost-type criteria. Hence, (45) is used to transform DM R � (o ij ) 5×6 in Table 2 into DM R � (o ij ) 5×6 as depicted in Table 3.
Step 3: for the transformed DM R � (o ij ) 5×6 , (47) is applied to compute the ranking value f i1 of each security algorithm x i with respect to the q-ROF-RS component as Hence, these security algorithms can be ranked as x 4 ≻x 5 ≻x 1 ≻x 3 ≻x 2 .
Step 4: for the transformed DM R � (o ij ) 5×6 , (50) is applied to compute the ranking value f i2 of each security algorithm x i with respect to the q-ROF-RP component as Hence, these security algorithms can be ranked as x 4 ≻x 1 ≻x 3 ≻x 2 ≻x 5 .
Step 5: for the transformed DM R � (o ij ) 5×6 , (52) is applied to compute the ranking value f i3 of each security algorithm x i with respect to the q-ROF-FMF component as

22
Mathematical Problems in Engineering Hence, these security algorithms can be ranked as x 3 ≻x 4 ≻x 1 ≻x 2 ≻x 5 . Step Step 7: in the final step, equation (55) is used to aggregate the ranking values of five security algorithms with respect to three components for deriving the final ranking values as en, the final ranking order of these security algorithms is x 4 ≻x 3 ≻x 1 ≻x 2 ≻x 5 . us, the security algorithm x 4 is the suitable one for the organization when building the security system.

Influences of the Parameters on the Ranking Results.
In this section, the influences of the parameters on the ranking results are discussed.

Influence of the Parameter q on the Final Ranking
Results. e influence of the parameter q on the final ranking results of the q-ROF-MULTIMOORA method is first discussed. In this case, the parameters c � η � ℵ � 1. For the transformed DM R � (o ij ) 5×6 in Table 2, the ranking results of security algorithms are shown in Table 4 and Figure 2 when the value of the parameter q varies.
From Table 4, it can be known that the ranking results of security algorithms are different when the value of q varies. When q � 1, the ranking result of security algorithms is x 4 ≻x 1 ≻x 3 ≻x 2 ≻x 5 . When q � 2, the ranking result of security algorithms is x 4 ≻x 2 ≻x 1 ≻x 3 ≻x 5 . When q � 3 or q � 5, the ranking results of security algorithms are x 4 ≻x 3 ≻x 1 ≻x 2 ≻x 5 . Although the ranking result of security algorithms changes when the value of the parameter q varies, the most suitable security algorithm keeps unchanged, namely, x 4 . When q � 1, then q-ROFNs reduce to IFNs. When q � 2, then q-ROFNs reduce to PFNs. How to determine the reasonable value of q depends on the evaluation information provided by the expert. e smallest value of the parameter q should satisfy μ q + ] q ≤ 1. For instance, if the evaluation information given by the expert is (0.9, 0.9), then the smallest value of the parameter q should be 7 so that 0.9 7 + 0.9 7 < 1.

Influences of the Parameters c and η on the Ranking
Results.
e influences of the parameters c and η on the ranking results of the q-ROF-MULTIMOORA method are analyzed in this part. In this case, the parameters ℵ � 1 and q � 3. For DM R � (o ij ) 5×6 in Table 2, the ranking results of security algorithms are shown in Table 5 and Figure 3 when the values of the parameters c and η vary.
From Table 5, it can be seen that the ranking result obtained from the q-ROF-MULTIMOORA method is always x 4 ≻x 3 ≻x 1 ≻x 2 ≻x 5 except when c � 0 and η � 1. Nevertheless, the most suitable security algorithm is always x 4 . When c � 0 and η � 1, the ranking result of security algorithms changes into x 4 ≻x 1 ≻x 3 ≻x 5 ≻x 2 .
us, the ranking result obtained from the q-ROF-MULTIMOORA method is not sensitive to the values of these two parameters. In other words, the q-ROF-MULTIMOORA method is robust and effective.

Influence of the Parameter ℵ on the Ranking Results.
e influence of the parameter ℵ on the ranking results of the q-ROF-MULTIMOORA method is analyzed in this part. In this case, the parameters c � η � 1 and q � 3. For DM R � (o ij ) 5×6 in Table 2, the ranking results of security algorithms are listed in Table 6 and Figure 4 when the value of the parameter ℵ varies.
From Table 6, it can be seen that the ranking result obtained from the q-ROF-MULTIMOORA method slightly changes when the value of the parameter ℵ varies. When the value of the parameter ℵ is set to ℵ � 1, then the ranking result of security algorithms is x 4 ≻x 3 ≻x 1 ≻x 2 ≻x 5 . When the value of the parameter ℵ is set to a value in the integer set 2, 3, . . . , 10 { }, then the ranking result of security algorithms is changed into x 4 ≻x 1 ≻x 3 ≻x 5 ≻x 2 . However, the most suitable security algorithm always keeps unchanged, namely, x 4 no matter how the value of the parameter ℵ varies. Hence, the ranking result that is obtained from the q-ROF-MUL-TIMOORA method is relatively stable. Because of the Dombi operational laws for q-ROFNs, the q-ROF-MUL-TIMOORA method has high flexibility by providing the parameter ℵ. Experts can adjust the value of the parameter ℵ according to the actual situation of MCDM problems.

Comparative Analysis.
For the proposed q-ROF-MULTIMOORA method, it applies the PA operator to alleviate the negative influence of extreme values on the ranking results and integrates the AHM and GHM operators to handle the interrelationships hiding behind criteria values. For the purpose of verifying the effectiveness of the q-ROF-MULTIMOORA method, it is compared with the original MULTIMOORA method [32,56] for handling the q-ROFNs. Different from the q-ROF-MULTIMOORA method, the original MULTIMOORA method does not contain the PA operator to solve the problem of extreme values and also does not integrate the AHM and GHM operators to handle the interrelationships among criteria values. Hence, it is a suitable way for comparing the q-ROF-MULTIMOORA method with the original MULTIMOORA method. For the purpose of conducting this comparative analysis, an example of evaluating blockchain platforms is given.

Example 2.
e blockchain technology has the ability to solve the problems resulting from our increasingly connected society and tackle real-world business concerns. It has been broadly applied to many fields such as distributed   Table 7. e original MULTIMOORA method and q-ROF-MULTIMOORA method are applied to process the transformed q-rung orthopair fuzzy DM in Table 7. Because of the limited space, the computation processes are omitted here and the ranking results of different methods are provided in Table 8.
In Table 8, the ranking results obtained from the q-ROF-MULTIMOORA method and original method are provided. Moreover, the ranking results obtained from the three components of q-ROF-MULTIMOORA and original method are also given. From Table 8, it can be noted that the   From the above analysis, it can be noted that the q-ROF-MULTIMOORA method performs better than the original MUTLIMOORA method because the q-ROF-MULTI-MOORA method derives more robust and reasonable ranking results.

Conclusions
To solve the security algorithms' evaluation problem, we propose an efficient q-ROF-MULTIMOORA method in this paper. Our contributions are listed as follows: (1) We combine the PA operator, Dombi operational laws, and AHM and GHM operators to design the q-ROFDPHM, q-ROFWDPHM, q-ROFDPGHM, and q-ROFWDPGHM operators to aggregate q-ROFNs. (2) e proposed q-ROFWDPHM and q-ROFWDPGHM operators are applied to modify the original MULTIMOORA method for proposing a novel q-ROF-MULTIMOORA method.
(3) A practical case of evaluating five security algorithms is given to show the decision procedure of the q-ROF-MULTIMOORA method. e influences of the parameters on the ranking results are analyzed. (4) To validate the effectiveness of the proposed q-ROF-MULTIMOORA method, a new example of evaluating blockchain platforms is given.
e proposed methods also have some limitations: (1) the q-ROFSs model, the uncertain information uses only three characteristic functions and does not have the characteristic function that denotes the degree of abstinence. is limitation can be removed by introducing the concept of T-spherical fuzzy sets, which was proposed by Mahmood et al. [57]. It has been studied by many scholars [58,59]. (2) e weights of attributes are directly given in this study. It ignores the objective significance. e method combining the objective weights and subjective weights should be considered in the future.
(3) In the proposed q-ROF-MULTIMOORA method, the q-ROFDPHM and q-ROFDPGHM operators do not consider the interaction between the membership degree and the nonmembership degree of q-ROFSs, which will produce unreasonable aggregated results.
e proposed q-ROF-MULTIMOORA method has some potential applications. In the future research plan, we intend to apply the proposed method into the sustainable supplier selection [60]. According to the third limitation mentioned in the above paragraph, the idea of interaction operational rules [61] will be used to improve the proposed method.

Data Availability
e data used to support the findings of this study are included within the article.