# MULTIMOORA Method for Addressing Security Algorithms Evaluation Problem underq-Rung Orthopair Fuzzy Environment

Authors: Rongguo Wang; Xinmei Li; Mingwei Lin; Zhanpeng Lin
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

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. The MULTIMOORA method is an excellent decision method, which owns strong robustness. However, it has not been used to process the complex information structure ofq-rung orthopair fuzzy sets. Moreover, it cannot solve the problem that the extreme values negatively influence 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 five security algorithms is used to illustrate the decision process of the proposed q-rung orthopair fuzzy MULTIMOORA method.


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## Body

## 1. Introduction

With the quick development of multiple information technologies including cloud computing, Internet of Things, 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]. The network shows massive security loophole [4]. The companies and personals also own the special software to improve their business. The 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 efficient security algorithms to ensure the security requirements of network [7–9]. However, these security algorithms usually own different 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]. The 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. The 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.(2)
q-ROFSs have attracted many researchers since its birth. For example, linguistic q-ROFSs [18–20] and interval-valued q-ROFSs [21] are the qualitative and uncertain versions of q-ROFSs. To fuse q-ROFSs’ information, various aggregation operators have been proposed [22–28], such as Archimedean Bonferroni mean operators [22], partitioned Bonferroni mean operators [23], Heronian mean operators [24], Maclaurin symmetric mean operators [25, 26], Hamy mean operators [27], and Choquet integral operators [28]. They are the value measurement MCDM methods [29–31], which do not consider the distance between each criterion value and maximum criterion value.(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 MULTIMOORA 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.Table 1
Performance comparison among decision-making methods [12].
MethodsComputational timeSimplicityMathematical calculationsStabilityMULTIMOORAVery lessVery simpleMinimumGoodTOPSISModerateModerately criticalModeratePoorVIKORLessSimpleModerateMediumELECTREHighModerately criticalModerateMediumHence, the motivations of this study are summarized as(1)
A more flexible way ofq-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 scenarioTo overcome the challenges, a novelq-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)
The Dombi operational laws, special forms oft-norms and t-conorms, show strong flexibility when computing input values. The power average (PA) operator has the ability of alleviating negative influences of extreme input values on the decision results. The 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 q-rung 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)
The weighted forms of theq-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 decision-making 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 proposedq-rung orthopair fuzzy MULTIMOORA method. Afterwards, the influences of the parameters on the ranking results are analyzed. Then, the q-rung orthopair fuzzy MULTIMOORA method is compared with the existing decision methods that handle the q-ROFS information.The rest content of this paper is organized as follows. The basic knowledge ofq-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 MULTIMOORA 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.


## 2. Preliminaries

In this paper, the basic information of q-ROFSs, PA, Dombi T-conorm and T-norm, HM operator, and MULTIMOORA method is provided.


### 2.1.q-Rung Orthopair Fuzzy Sets

The concept ofq-ROFSs was proposed based on IFSs and PFSs. The q-ROFSs show higher flexibility and larger value range than IFSs and PFSs [37–39].Definition 1 (see [17]).
LetX=x1,x2,…,xn be a finite universe of discourse (UoD); then, a q-ROFS A on X is mathematically expressed as(1)A=x,μAx,νAx|x∈X,where μA:X⟶0,1 and νA:X⟶0,1 are the membership degree (MD) and nonmembership degree (NMD) of the element x belonging to the q-ROFS A, respectively. The constraint conditions for q-ROFS are 0≤μAx≤1, 0≤νAx≤1, and 0≤μAqx+νAqx≤1, for all q≥1. The parameter q is a positive integer. The value of πAx=1−μAxq−νAxqq is defined to be the hesitant degree (HD) of the element x belonging to the q-ROFS A. For convenience, the two-tuple μAx,νAx is simplified as μA,νA, which is also called q-rung orthopair fuzzy number (q-ROFN) by Liu and Wang [40].
For comparingq-ROFNs, the definitions of score function and accuracy function were given by Liu and Wang [40] for q-ROFNs as follows.Definition 2 (see [40]).
Given aq-ROFN o=μ,ν, then its score function and accuracy function are defined as so=μq−νq and ho=μq+νq, in which so∈−1,1 denotes the score function and ho∈0,1 is the accuracy function.
Based on the above score function and accuracy function presented in Definition2, Liu and Wang [40] gave a method for comparing two q-ROFNs as follows.Definition 3 (see [40]).
Given twoq-ROFNs o1=μ1,ν1 and o2=μ2,ν2, so1 and so2 are their score function values, and ho1 and ho2 are their accuracy function values,(1)
Ifso1>so2, then it can be considered that o1>o2.(2)
Ifso1=so2, then their accuracy function values should be further compared as follows:(1)
Ifho1>ho2, then o1>o2.(2)
Ifho1=ho2, then o1=o2.
To measure the deviation degree between any twoq-ROFNs, Liu et al. [41] provided the definition of distance between them as follows.Definition 4 (see [41]).
Forq-ROFNs, o1=μ1,ν1 and o2=μ2,ν2, the distance between them is computed as(2)do1,o2=12μ1q−μ2q+ν1q−ν2q+π1q−π2q,where π1 and π2 are the HD values of q-ROFNs o1 and o2, respectively.


### 2.2. Power Average Operator

The power average (PA) is a useful aggregation operator that was put forward by Yager [42]. The PA operator has the ability of alleviating negative influences of extreme input values on the calculation results. The original PA operator was devised to process crisp values. Its mathematical definition is given as follows:Definition 5 (see [42]).
Letoii=1,2,…,n be a series of nonnegative crisp values; then, the PA operator really acts as a function that(3)PAo1,o2,…,on=∑i=1n1+Soi∑k=1n1+Sokoi,where Soi=∑j=1,j≠inSupoi,oj is the support degree for oi from oj and Supoi,oj=1−doi,oj.
The support degree satisfies the following features:(1)
Supoi,oj∈0,1(2)
Supoi,oj=Supoj,oi(3)
Ifdoi,oj<dor,ok, then Supoi,oj>Supor,ok, where doi,oj denotes the distance between oi and oj


### 2.3. Dombi T-Norm and T-Conorm

The 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. Their 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(4)Dm,n=11+1−m/mℵ+1−n/nℵ1/ℵ,D∗m,n=1−11+m/1−mℵ+n/1−nℵ1/ℵ,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.Definition 7 (see [44]).
Given twoq-ROFNs o1=μ1,ν1 and o2=μ2,ν2, then the Dombi operational laws of q-ROFNs are defined as(1)
o1⊕o2=1−1/1+μ1q/1−μ1qℵ+μ2q/1−μ2qℵ1/ℵ1/q,1/1+1−ν1q/ν1qℵ+1−ν2q/ν2qℵ1/ℵ1/q(2)
o1⊗o2=1/1+1−μ1q/μ1qℵ+1−μ2q/μ2qℵ1/ℵ1/q,1−1/1+ν1q/1−ν1qℵ+ν2q/1−ν2qℵ1/ℵ1/q(3)
lo1=1−1/1+lμ1q/1−μ1qℵ1/ℵ1/q,1/1+l1−ν1q/ν1qℵ1/ℵ1/q(4)
o1l=1/1+l1−μ1q/μ1qℵ1/ℵ1/q,1−1/1+lν1q/1−ν1qℵ1/ℵ1/q, where ℵ>0


### 2.4. Heronian Mean and Geometric Heronian Mean Operators

The aggregation operators (AOs) [45–47] are value measurement MCDM methods. It is very simple and easy to perform AOs. The 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. The 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. The HM operators can be divided into two categories: arithmetic HM (AHM) and geometric HM (GHM) operators, which are mathematically defined as follows.Definition 8 (see [49]).
Letoii=1,2,…,n be a series of nonnegative real values; the parameters γ, η≥0; then, the AHM operator can aggregate the nonnegative real values as(5)HMγ,ηo1,o2,…,on=2nn+1∑i=1n∑j=inoiγojη1/γ+η.Definition 9 (see [49]).
Letoii=1,2,…,n be a series of nonnegative real values; the parameters γ, η≥0; then, the GHM operator can aggregate the nonnegative real values as(6)GHMγ,ηo1,o2,…,on=1γ+η∏i=1n∏j=inγoi+ηoj2/nn+1.
For the AHM operator, its aggregated values are greatly influenced by extreme values [50]. The GHM operator is capable of balancing the big differences among input values [51]. Therefore, the GHM operator performs better than the AHM operator in some cases.


### 2.5. 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. Thus, the MULTIMOORA method has three components: ratio system (RS) component, reference point (RP) component, and FMF component, respectively [52]. These 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 ofm alternatives x1,x2,…,xm and n criteria a1,a2,…,an. The weight vector of criteria is denoted as ω1,ω2,…,ωn, where ∑j=1nωj=1 and 0≤ωj≤1. The decision matrix (DM) R=oijm×n corresponding to the MCDM problem contains the evaluation information from experts. The element oij represents the evaluation information of alternative xi with respect to criterion aj.The evaluation information of alternatives across multiple criteria usually shows different dimensions, so the evaluation information in the DMR=oijm×n is suggested to be normalized as(7)o˜ij=oij∑i=1moij2.After that, the normalized DMR˜=o˜ijm×n can be derived.


#### 2.5.1. 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. The weighted arithmetic aggregation operator (AAO) is used to calculate the ranking valueYi of alternative xi as(8)Yi=∑j=1kωjo˜ij−∑j=k+1nωjo˜ij,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. Therefore, the alternatives can be ranked based on the descending order of their ranking values.


#### 2.5.2. 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(9)oj=maxio˜ij,j≤k;minio˜ij,j>k,where oj denotes the reference point of alternatives with respect to criterion aj.Then, the weighted distance between the normalized evaluation information of the alternativexi with respect to each criterion and the reference point of the same criterion is computed as dij=ωjoj−o˜ij.Finally, the ranking valueDi of alternative xi is computed as Di=maxjdij.According to the RP component, the optimal alternative should have the smallest ranking value. Thus, the alternatives can be ranked based on the ascending order of their ranking values.


#### 2.5.3. FMF Component

The 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 benefit-type criteria and lower values for cost-type criteria. The weighted geometric aggregation operator (GGO) is used to determine the ranking valueUi of alternative xi as Ui=∏j=1ko˜ijωj/∏j=k+1no˜ijωj.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.


## 2.1.q-Rung Orthopair Fuzzy Sets

The concept ofq-ROFSs was proposed based on IFSs and PFSs. The q-ROFSs show higher flexibility and larger value range than IFSs and PFSs [37–39].Definition 1 (see [17]).
LetX=x1,x2,…,xn be a finite universe of discourse (UoD); then, a q-ROFS A on X is mathematically expressed as(1)A=x,μAx,νAx|x∈X,where μA:X⟶0,1 and νA:X⟶0,1 are the membership degree (MD) and nonmembership degree (NMD) of the element x belonging to the q-ROFS A, respectively. The constraint conditions for q-ROFS are 0≤μAx≤1, 0≤νAx≤1, and 0≤μAqx+νAqx≤1, for all q≥1. The parameter q is a positive integer. The value of πAx=1−μAxq−νAxqq is defined to be the hesitant degree (HD) of the element x belonging to the q-ROFS A. For convenience, the two-tuple μAx,νAx is simplified as μA,νA, which is also called q-rung orthopair fuzzy number (q-ROFN) by Liu and Wang [40].
For comparingq-ROFNs, the definitions of score function and accuracy function were given by Liu and Wang [40] for q-ROFNs as follows.Definition 2 (see [40]).
Given aq-ROFN o=μ,ν, then its score function and accuracy function are defined as so=μq−νq and ho=μq+νq, in which so∈−1,1 denotes the score function and ho∈0,1 is the accuracy function.
Based on the above score function and accuracy function presented in Definition2, Liu and Wang [40] gave a method for comparing two q-ROFNs as follows.Definition 3 (see [40]).
Given twoq-ROFNs o1=μ1,ν1 and o2=μ2,ν2, so1 and so2 are their score function values, and ho1 and ho2 are their accuracy function values,(1)
Ifso1>so2, then it can be considered that o1>o2.(2)
Ifso1=so2, then their accuracy function values should be further compared as follows:(1)
Ifho1>ho2, then o1>o2.(2)
Ifho1=ho2, then o1=o2.
To measure the deviation degree between any twoq-ROFNs, Liu et al. [41] provided the definition of distance between them as follows.Definition 4 (see [41]).
Forq-ROFNs, o1=μ1,ν1 and o2=μ2,ν2, the distance between them is computed as(2)do1,o2=12μ1q−μ2q+ν1q−ν2q+π1q−π2q,where π1 and π2 are the HD values of q-ROFNs o1 and o2, respectively.


## 2.2. Power Average Operator

The power average (PA) is a useful aggregation operator that was put forward by Yager [42]. The PA operator has the ability of alleviating negative influences of extreme input values on the calculation results. The original PA operator was devised to process crisp values. Its mathematical definition is given as follows:Definition 5 (see [42]).
Letoii=1,2,…,n be a series of nonnegative crisp values; then, the PA operator really acts as a function that(3)PAo1,o2,…,on=∑i=1n1+Soi∑k=1n1+Sokoi,where Soi=∑j=1,j≠inSupoi,oj is the support degree for oi from oj and Supoi,oj=1−doi,oj.
The support degree satisfies the following features:(1)
Supoi,oj∈0,1(2)
Supoi,oj=Supoj,oi(3)
Ifdoi,oj<dor,ok, then Supoi,oj>Supor,ok, where doi,oj denotes the distance between oi and oj


## 2.3. Dombi T-Norm and T-Conorm

The 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. Their 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(4)Dm,n=11+1−m/mℵ+1−n/nℵ1/ℵ,D∗m,n=1−11+m/1−mℵ+n/1−nℵ1/ℵ,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.Definition 7 (see [44]).
Given twoq-ROFNs o1=μ1,ν1 and o2=μ2,ν2, then the Dombi operational laws of q-ROFNs are defined as(1)
o1⊕o2=1−1/1+μ1q/1−μ1qℵ+μ2q/1−μ2qℵ1/ℵ1/q,1/1+1−ν1q/ν1qℵ+1−ν2q/ν2qℵ1/ℵ1/q(2)
o1⊗o2=1/1+1−μ1q/μ1qℵ+1−μ2q/μ2qℵ1/ℵ1/q,1−1/1+ν1q/1−ν1qℵ+ν2q/1−ν2qℵ1/ℵ1/q(3)
lo1=1−1/1+lμ1q/1−μ1qℵ1/ℵ1/q,1/1+l1−ν1q/ν1qℵ1/ℵ1/q(4)
o1l=1/1+l1−μ1q/μ1qℵ1/ℵ1/q,1−1/1+lν1q/1−ν1qℵ1/ℵ1/q, where ℵ>0


## 2.4. Heronian Mean and Geometric Heronian Mean Operators

The aggregation operators (AOs) [45–47] are value measurement MCDM methods. It is very simple and easy to perform AOs. The 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. The 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. The HM operators can be divided into two categories: arithmetic HM (AHM) and geometric HM (GHM) operators, which are mathematically defined as follows.Definition 8 (see [49]).
Letoii=1,2,…,n be a series of nonnegative real values; the parameters γ, η≥0; then, the AHM operator can aggregate the nonnegative real values as(5)HMγ,ηo1,o2,…,on=2nn+1∑i=1n∑j=inoiγojη1/γ+η.Definition 9 (see [49]).
Letoii=1,2,…,n be a series of nonnegative real values; the parameters γ, η≥0; then, the GHM operator can aggregate the nonnegative real values as(6)GHMγ,ηo1,o2,…,on=1γ+η∏i=1n∏j=inγoi+ηoj2/nn+1.
For the AHM operator, its aggregated values are greatly influenced by extreme values [50]. The GHM operator is capable of balancing the big differences among input values [51]. Therefore, the GHM operator performs better than the AHM operator in some cases.


## 2.5. 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. Thus, the MULTIMOORA method has three components: ratio system (RS) component, reference point (RP) component, and FMF component, respectively [52]. These 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 ofm alternatives x1,x2,…,xm and n criteria a1,a2,…,an. The weight vector of criteria is denoted as ω1,ω2,…,ωn, where ∑j=1nωj=1 and 0≤ωj≤1. The decision matrix (DM) R=oijm×n corresponding to the MCDM problem contains the evaluation information from experts. The element oij represents the evaluation information of alternative xi with respect to criterion aj.The evaluation information of alternatives across multiple criteria usually shows different dimensions, so the evaluation information in the DMR=oijm×n is suggested to be normalized as(7)o˜ij=oij∑i=1moij2.After that, the normalized DMR˜=o˜ijm×n can be derived.


### 2.5.1. 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. The weighted arithmetic aggregation operator (AAO) is used to calculate the ranking valueYi of alternative xi as(8)Yi=∑j=1kωjo˜ij−∑j=k+1nωjo˜ij,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. Therefore, the alternatives can be ranked based on the descending order of their ranking values.


### 2.5.2. 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(9)oj=maxio˜ij,j≤k;minio˜ij,j>k,where oj denotes the reference point of alternatives with respect to criterion aj.Then, the weighted distance between the normalized evaluation information of the alternativexi with respect to each criterion and the reference point of the same criterion is computed as dij=ωjoj−o˜ij.Finally, the ranking valueDi of alternative xi is computed as Di=maxjdij.According to the RP component, the optimal alternative should have the smallest ranking value. Thus, the alternatives can be ranked based on the ascending order of their ranking values.


### 2.5.3. FMF Component

The 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 benefit-type criteria and lower values for cost-type criteria. The weighted geometric aggregation operator (GGO) is used to determine the ranking valueUi of alternative xi as Ui=∏j=1ko˜ijωj/∏j=k+1no˜ijωj.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.


## 2.5.1. 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. The weighted arithmetic aggregation operator (AAO) is used to calculate the ranking valueYi of alternative xi as(8)Yi=∑j=1kωjo˜ij−∑j=k+1nωjo˜ij,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. Therefore, the alternatives can be ranked based on the descending order of their ranking values.


## 2.5.2. 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(9)oj=maxio˜ij,j≤k;minio˜ij,j>k,where oj denotes the reference point of alternatives with respect to criterion aj.Then, the weighted distance between the normalized evaluation information of the alternativexi with respect to each criterion and the reference point of the same criterion is computed as dij=ωjoj−o˜ij.Finally, the ranking valueDi of alternative xi is computed as Di=maxjdij.According to the RP component, the optimal alternative should have the smallest ranking value. Thus, the alternatives can be ranked based on the ascending order of their ranking values.


## 2.5.3. FMF Component

The 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 benefit-type criteria and lower values for cost-type criteria. The weighted geometric aggregation operator (GGO) is used to determine the ranking valueUi of alternative xi as Ui=∏j=1ko˜ijωj/∏j=k+1no˜ijωj.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.


## 3.q-Rung Orthopair Fuzzy Dombi Power Heronian Mean Operators

In this section, we use the PA operator, Dombi operational laws forq-ROFNs, and arithmetic HM operator to propose q-rung orthopair fuzzy Dombi power HM (q-ROFDPHM) operator and its weighted form. Then, the features are discussed.Definition 10.
Given a set ofq-ROFNs oi=μi,νii=1,2,…,n and three parameters γ,η≥0 and ℵ>0, then the q-ROFDPHM operator is defined as(10)q−ROFDPHMo1,o2,…,on=2nn+1⊕i=1n⊕j=inn1+Soi∑k=1n1+Sokoiγ⊗n1+Soj∑k=1n1+Sokojη1/γ+η.
Based on the Dombi operational laws ofq-ROFNs [36] and HM operator, the following theorems can be derived.Theorem 1.
Givennq-ROFNs oi=μi,νii=1,2,…,n and the parameters γ,η≥0 and ℵ>0, then the aggregated result derived from equation (10) is still an q-ROFN, which is(11)q−ROFDPHMo1,o2,…,on=11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,1−11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q,where ξi=1+Soi/∑k=1n1+Sok, ψj=1+Soj/∑k=1n1+Sok, μiq/1−μiq=1/ai, μjq/1−μjq=1/aj, 1−νiq/νiq=1/bi, 1−νjq/νjq=1/bj, 1/nξi=ti, and 1/nψj=ej.Proof.
According to Definition10, we have(12)q−ROFDPHMo1,o2,…,on=2nn+1⊕i=1n⊕j=inn1+Soi∑k=1n1+Sokoiγ⊗n1+Soj∑k=1n1+Sokojη1/γ+η.
Letξi=1+Soi/∑k=1n1+Sok and ψj=1+Soj/∑k=1n1+Sok; then, we can derive(13)q−ROFDPHMo1,o2,…,on=2nn+1⊕i=1n⊕j=innξioiγ⊗nψjojη1/γ+η.
According to Definition7, it can be derived that(14)nξioi=1−11+nξiμiq/1−μiqℵ1/ℵ1/q,11+nξi1−νiq/νiqℵ1/ℵ1/q,nψjoj=1−11+nψjμjq/1−μjqℵ1/ℵ1/q,11+nψj1−νjq/νjqℵ1/ℵ1/q.
Letμiq/1−μiq=1/ai, μjq/1−μjq=1/aj, 1−νiq/νiq=1/bi, and 1−νjq/νjq=1/bj; then, we have(15)nξioi=1−11+nξi1/aiℵ1/ℵ1/q,11+nξi1/biℵ1/ℵ1/q,nψjoj=1−11+nψj1/ajℵ1/ℵ1/q,11+nψj1/bjℵ1/ℵ1/q,nξioiγ=11+γ1/nξiaiℵ1/ℵ1/q,1−11+γ1/nξibiℵ1/ℵ1/q,nψjojη=11+η1/nψjajℵ1/ℵ1/q,1−11+η1/nψjbjℵ1/ℵ1/q.
Let1/nξi=ti and 1/nψj=ej; then, the above equations can be transformed into(16)nξioiγ=11+γtiaiℵ1/ℵ1/q,1−11+γtibiℵ1/ℵ1/q,nψjojη=11+ηejajℵ1/ℵ1/q,1−11+ηejbjℵ1/ℵ1/q.
Thus,nξioiγ⊗nψjojη=1/1+γtiaiℵ+ηejajℵ1/ℵ1/q,1−1/1+γtibiℵ+ηejbjℵ1/ℵ1/q,(17)⊕i=1n⊕j=innξioiγ⊗nψjojη=1−11+∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,11+∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q,2nn+1⊕i=1n⊕j=innξioiγ⊗nψjojη=1−11+2/nn+1∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,11+2/nn+1∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q,2nn+1⊕i=1n⊕j=innξioiγ⊗nψjojη1/γ+η=11+1/γ+η1/2/nn+1∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,1−11+1/γ+η1/2/nn+1∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q.
Then, the final result can be determined as(18)2nn+1⊕i=1n⊕j=innξioiγ⊗nψjojη1/γ+η=11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,1−11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q.
Then, we need to prove that the aggregated result from theq-ROFDPHM operator is still a q-ROFN.
Letμ∧=1/1+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q and ν∧=1−1/1+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q; then, we need to prove that (1) 0≤μ∧q≤1 and 0≤ν∧q≤1 and (2) 0≤μ∧q+ν∧q≤1.
Sincenn+1/2γ+η1/ℵ≥0 and ∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ>0, then nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ≥0.
Then, we can have(19)1+nn+12γ+η1/ℵ×1∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ≥1⟹0≤11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ≤1.
Thus,0≤μ∧q≤1. Similarly, it can be proven that 0≤ν∧q≤1. It can be derived that 0≤μ∧q+ν∧q.
Becauseμiq+νiq≤1 and μjq+νjq≤1, then μiq≤1−νiq and μjq≤1−νjq. Thus, 1−μiq/μiq≥νiq/1−νiq and 1−μjq/μjq≥νjq/1−νjq. It can be derived that ai≥bi and aj≥bj. Then, we have(20)∑i=1,j=in1γtiaiℵ+ηejajℵ1/ℵ≤∑i=1,j=in1γtibiℵ+ηejbjℵ1/ℵ⟹1∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ≥1∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ⟹1+nn+12γ+η1/ℵ×1∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ≥1+nn+12γ+η1/ℵ×1∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ.
Thus,(21)11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ≤11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ.
Then, we can have(22)μ∧q+ν∧q=11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q×q+1−11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q×q=1+11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ−11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ≤1,which completes the proof of Theorem 1.Theorem 2 (idempotency).
Suppose that there are a group ofq-ROFNs oi=μi,νii=1,2,…,n and the parameters γ, η≥0 and ℵ>0. If oi=oj=o=μ,ν, for i,j=1,2,…,n, then we have(23)q−ROFDPHMo1,o2,…,on=11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,1−11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q=μ,ν=o,where ξi=1+Soi/∑k=1n1+Sok, ψj=1+Soj/∑k=1n1+Sok, μiq/1−μiq=1/ai, μjq/1−μjq=1/aj, 1−νiq/νiq=1/bi, 1−νjq/νjq=1/bj, 1/nξi=ti, and 1/nψj=ej.Proof.
Sinceoi=oj=o=μ,ν, for i,j=1,2,…,n, then Supoi,oj=1.
Thus, we have(24)ξi=1+Soi∑k=1n1+Sok=ψj=1+Soj∑k=1n1+Sok=1n,ti=ej=1,aiℵ=ajℵ=1−μiqμiqℵ=1−μjqμjqℵ=1−μqμqℵ.
Then, it can be derived that(25)∑i=1,j=in1γtiaiℵ+ηejajℵ1/ℵ=∑i=1,j=in1γ1−μq/μqℵ+η1−μq/μqℵ1/ℵ=∑i=1,j=in1γ+η×1−μq/μqℵ1/ℵ=1γ+η×nn+12×μq1−μqℵ1/ℵ⇒1∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ=11/γ+η×nn+1/2×μq/1−μqℵ1/ℵ=1nn+1/2γ+η×μq/1−μqℵ1/ℵ=2γ+ηnn+1×1−μqμqℵ1/ℵ⇒11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q=11+nn+1/2γ+η1/ℵ×2γ+η/nn+1×1−μq/μqℵ1/ℵ1/q=11+1−μq/μq1/q=μ.
Similarly,1/bi=1/bj=1−νiq/νiq=1−νjq/νjq=1−νq/νq. Then, the NMD value of the q-ROFDPHM operator can be derived as(26)1−11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q=1−11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γ+ηνq/1−νqℵ1/ℵ1/q=1−11+nn+1/2γ+η1/ℵ×1/nn+1/2γ+η1−νq/νqℵ1/ℵ1/q=1−11+νq/1−νq1/q=ν.
Thus, we haveq−ROFDPHMo1,o2,…,on=μ,ν=o, which completes the proof of Theorem 2.Theorem 3 (boundedness).
Suppose that there are a group ofq-ROFNs oi=μi,νii=1,2,…,n and the parameters γ, η≥0 and ℵ>0. If ol=mino1,o2,…,on=μl,νl and oh=maxo1,o2,…,on=μh,νh, then we have(27)ol≤q−ROFDPHMo1,o2,…,on≤oh,where(28)ol=11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,1−11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q,oh=11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,1−11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q,1nξi=ti,1nψj=ej,ξi=1+Soi∑k=1n1+Sok,ψj=1+Soj∑k=1n1+Sok,μiq1−μiq=1ai,1−νiqνiq=1bi.Proof.
According to Definition7, we have(29)nξioi=1−11+nξiμiq/1−μiqℵ1/ℵ1/q,11+nξi1−νiq/νiqℵ1/ℵ1/q,nψjoj=1−11+nψjμjq/1−μjqℵ1/ℵ1/q,11+nψj1−νiq/νiqℵ1/ℵ1/q.
Sinceol≤oi, then we have μi≥μl and νi≤νl.
Thus, it can be derived that(30)μiq≥μlq⟹11−μiq≥11−μlq⟹μiq1−μiq≥μlq1−μlq⟹1ai≥1al⟹ai≤al,νi≤νl⟹1−νiq≥1−νlq,1νiq≥1νlq⟹1bi≥1bl⟹bi≤bl.
Then, we can have(31)nξioi=1−11+nξi1/aiℵ1/ℵ1/q,11+nξi1/biℵ1/ℵ1/q≥1−11+nξi1/alℵ1/ℵ1/q,11+nξi1/blℵ1/ℵ1/q⟹nξioiγ=11+γ1/nξiaiℵ1/ℵ1/q,1−11+γ1/nξibiℵ1/ℵ1/q≥11+γ1/nξialℵ1/ℵ1/q,1−11+γ1/nξiblℵ1/ℵ1/q.
Similarly, we have(32)nψjojη=11+η1/nψjajℵ1/ℵ1/q,1−11+η1/nψjbjℵ1/ℵ1/q≥11+η1/nψjalℵ1/ℵ1/q,1−11+η1/nψjblℵ1/ℵ1/q.
Since1/nξi=ti and 1/nψj=ej, then we have(33)nξioiγ=11+γtiaiℵ1/ℵ1/q,1−11+γtibiℵ1/ℵ1/q≥11+γtialℵ1/ℵ1/q,1−11+γtiblℵ1/ℵ1/q⟹nψjojη=11+ηejajℵ1/ℵ1/q,1−11+ηejbjℵ1/ℵ1/q≥11+ηejalℵ1/ℵ1/q,1−11+ηejblℵ1/ℵ1/q,nξioiγ⊗nψjojη=11+γtiaiℵ+ηejajℵ1/ℵ1/q,1−11+γtibiℵ+ηejbjℵ1/ℵ1/q≥11+γtialℵ+ηejalℵ1/ℵ1/q,1−11+γtiblℵ+ηejblℵ1/ℵ1/q,⊕i=1n⊕j=innξioiγ⊗nψjojη=1−11+∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,11+∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q≥1−11+∑i=1,j=in1/γtiaiℵ+ηejalℵ1/ℵ1/q,11+∑i=1,j=in1/γtibiℵ+ηejblℵ1/ℵ1/q,2nn+1⊕i=1n⊕j=innξioiγ⊗nψjojη=1−11+2/nn+1∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,11+2/nn+1∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q≥1−11+2/nn+1∑i=1,j=in1/γtiaiℵ+ηejalℵ1/ℵ1/q,11+2/nn+1∑i=1,j=in1/γtibiℵ+ηejblℵ1/ℵ1/q,2nn+1⊕i=1n⊕j=innξioiγ⊗nψjojη1/γ+η=11+1/γ+η1/2/nn+1∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,1−11+1/γ+η1/2/nn+1∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q≥11+1/γ+η1/2/nn+1∑i=1,j=in1/γtialℵ+ηejalℵ1/ℵ1/q,1−11+1/γ+η1/2/nn+1∑i=1,j=in1/γtiblℵ+ηejblℵ1/ℵ1/q=ol.
Similarly, it can be proven thatq−ROFDPHMo1,o2,…,on≤oh in the same way, which completes the proof of Theorem 3.
It can be noted that the proposedq-ROFDPHM operator uses the PA operator and Dombi operational laws to optimize the HM operator. Its significance can be listed as follows. (1) It can alleviate negative influences of extreme input values on the calculation results. (2) It shows strong flexibility for computing input values. (3) It is capable of capturing the complex interrelationships among criteria values. (4) It can process the complex information structure of q-ROFSs. Nevertheless, during the aggregation processes, it does not consider the weight values of criteria, which is very important in the MCDM contexts. To tackle this deficiency, a novel q-rung orthopair fuzzy weighted Dombi power Heronian mean (q-ROFWDPHM) operator is put forward in the following part.Definition 11.
Given a set ofq-ROFNs oi=μi,νii=1,2,…,n, three parameters γ,η≥0 and ℵ>0, and the weight values ω1,ω2,…,ωn of q-ROFNs, then the q-ROFWDPHM operator is defined as(34)q−ROFWDPHMo1,o2,…,on=2nn+1⊕i=1n⊕j=innωi1+Soi∑k=1nωk1+Sokoiγ⊗nωj1+Soj∑k=1nωk1+Sokojη1/γ+η.Theorem 4.
Given a set ofq-ROFNs oi=μi,νii=1,2,…,n, three parameters γ,η≥0 and ℵ>0, and the weight values ω1,ω2,…,ωn of q-ROFNs, then the aggregated result obtained from the q-ROFWDPHM operator is still a q-ROFN, which is(35)q−ROFWDPHMo1,o2,…,on=11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,1−11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q,where ξi=1+Soi/∑k=1nωk1+Sok, ψj=1+Soj/∑k=1nωk1+Sok, μiq/1−μiq=1/ai, μjq/1−μjq=1/aj, 1−νiq/νiq=1/bi, 1−νjq/νjq=1/bj, ti=1/nωiξi, and ej=1/nωjψj.Proof.
Letξi=1+Soi/∑k=1nωk1+Sok and ψj=1+Soj/∑k=1nωk1+Sok; then, equation (34) can be transformed into(36)q−ROFWDPHMo1,o2,…,on=2nn+1⊕i=1n⊕j=innωiξioiγ⊗nωjψjojη1/γ+η.
According to Definition7, we have(37)nωiξioi=1−11+nωiξiμiq/1−μiqℵ1/ℵ1/q,11+nωiξi1−νiq/νiqℵ1/ℵ1/q,nωjψjoj=1−11+nωjψjμjq/1−μjqℵ1/ℵ1/q,11+nωjψj1−νjq/νjqℵ1/ℵ1/q.
Letμiq/1−μiq=1/ai, μjq/1−μjq=1/aj, 1−νiq/νiq=1/bi, and 1−νjq/νjq=1/bj; then, these equations can be transformed into(38)nωiξioi=1−11+nωiξi1/aiℵ1/ℵ1/q,11+nωiξi1/biℵ1/ℵ1/q,nωjψjoj=1−11+nωjψj1/ajℵ1/ℵ1/q,11+nωjψj1/bjℵ1/ℵ1/q.
According to Definition7, then we have(39)nωiξioiγ=11+γ1/nωiξiaiℵ1/ℵ1/q,1−11+γ1/nωiξibiℵ1/ℵ1/q,nωjψjojη=11+η1/nωjψjajℵ1/ℵ1/q,1−11+η1/nωjψjbjℵ1/ℵ1/q.
Letti=1/nωiξi and ej=1/nωjψj; then, the above two equations can be transformed into(40)nωiξioiγ=11+γtiaiℵ1/ℵ1/q,1−11+γtibiℵ1/ℵ1/q,nωjψjojη=11+ηejajℵ1/ℵ1/q,1−11+ηejbjℵ1/ℵ1/q,nωiξioiγ⊗nωjψjojη=11+γtiaiℵ+ηejajℵ1/ℵ1/q,1−11+γtibiℵ+ηejbjℵ1/ℵ1/q,⊕i=1n⊕j=innωiξioiγ⊗nωjψjojη=1−11+∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,11+∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q,2nn+1⊕i=1n⊕j=innωiξioiγ⊗nωjψjojη=1−11+2/nn+1∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,11+2/nn+1∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q,2nn+1⊕i=1n⊕j=innωiξioiγ⊗nωjψjojη1/γ+η=11+1/γ+η1/2/nn+1∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,1−11+1/γ+η1/2/nn+1∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q,q−ROFWDPHMo1,o2,…,on=2nn+1⊕i=1n⊕j=innωiξioiγ⊗nωjψjojη1/γ+η=11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtiaiℵ+ηejajℵ1/ℵ1/q,1−11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γtibiℵ+ηejbjℵ1/ℵ1/q.
The process for proving that the aggregation result of theq-ROFWDPHM operator is a q-ROFN is the same as that of Theorem 1. Thus, it is omitted here.
The proposedq-ROFWDPHM operator also owns the features of idempotency and boundedness as the proposed q-ROFDPHM operator. Their proof processes are similar to those of Theorems 2 and 3. Due to the limited space, the proof processes are omitted here.


## 4.q-Rung Orthopair Fuzzy Dombi Power Geometric Heronian Mean Operators

In this section, we use the PA operator, Dombi operational laws forq-ROFNs, and geometric HM operator to develop a novel q-ROFDPGHM operator and its weighted form. Then, the features are discussed.Definition 12.
Given a set ofq-ROFNs oi=μi,νii=1,2,…,n and three parameters γ,η≥0 and ℵ>0, then the q-ROFDPGHM operator is defined as(41)q−ROFDPGHMo1,o2,…,on=1γ+η⊗i=1n⊗j=inγoin1+Soi/∑k=1n1+Sok⊕ηojn1+Soj/∑k=1n1+Sok2/nn+1.
Based on the Dombi operational laws ofq-ROFNs and GHM operator, a theorem is derived.Theorem 5.
Givennq-ROFNs oi=μi,νii=1,2,…,n and the parameters γ,η≥0 and ℵ>0, then the aggregated result derived from equation (41) is still a q-ROFN, which is(42)q−ROFDPGHMo1,o2,…,on=1−11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γti1/aiℵ+ηej1/ajℵ1/ℵ1/q,11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γti1/biℵ+ηej1/bjℵ1/ℵ1/q,where ξi=1+Soi/∑k=1n1+Sok, ψj=1+Soj/∑k=1n1+Sok, ai=1−μiq/μiq, aj=1−μjq/μjq, bi=νiq/1−νiq, bj=νjq/1−νjq, ti=1/nξi, and ej=1/nψj.
The proof process of this theorem is similar to that of Theorem1. Thus, it is omitted here.
The proposedq-ROFDPGHM operator also owns the features of idempotency and boundedness as the proposed q-ROFDPHM operator. Their proof processes are similar to those of Theorems 2 and 3. Due to the limited space, the proof processes are omitted here.
Similar to the proposedq-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-ROFNsoi=μi,νii=1,2,…,n, three parameters γ,η≥0 and ℵ>0, and the weight values ω1,ω2,…,ωn of q-ROFNs, then the q-ROFWDPGHM operator is defined as(43)q−ROFWDPGHMo1,o2,…,on=1γ+η⊗i=1n⊗j=inγoinωi1+Soi/∑k=1nωk1+Sok⊕ηojnωj1+Soj/∑k=1nωk1+Sok2/nn+1.
Based on the Dombi operational laws ofq-ROFNs and GHM operator, a theorem is derived.Theorem 6.
Given a set ofq-ROFNs oi=μi,νii=1,2,…,n, the parameters γ, η≥0 and ℵ>0, and the weight values ω1,ω2,…,ωn of q-ROFNs, then the aggregated result that is obtained from equation (43) is still a q-ROFN, which is(44)q−ROFWDPGHMo1,o2,…,on=1−11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γti1/aiℵ+ηej1/ajℵ1/ℵ1/q,11+nn+1/2γ+η1/ℵ×1/∑i=1,j=in1/γti1/biℵ+ηej1/bjℵ1/ℵ1/q,where ξi=1+Soi/∑k=1nωk1+Sok, ψj=1+Soj/∑k=1nωk1+Sok, ti=1/nωiξi, ej=1/nωjψj, ai=1−μiq/μiq, aj=1−μjq/μjq, bi=νiq/1−νiq, and bj=νjq/1−νjq.Proof.
The proof process is similar to that of Theorem1. Thus, it is omitted here.
The proposedq-ROFWDPGHM operator also has the features of idempotency and boundedness as the proposed q-ROFDPHM operator. Their proof processes are similar to those of Theorems 2 and 3. Due to the limited space, the proof processes are omitted here.


## 5. MULTIMOORA Method forq-Rung Orthopair Fuzzy Sets

In this section, the MULTIMOORA method is improved for processing the MCDM problems with theq-ROFS information. There 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.


### 5.1. Problem Description

Let us suppose that there exists an MCDM problem consisting ofm alternatives x1,x2,…,xm and n criteria a1,a2,…,an. The weight values of criteria are denoted as ω1,ω2,…,ωn, where ∑j=1nωj=1 and 0≤ωj≤1. The decision matrix (DM) R=oijm×n corresponding to this MCDM problem consists of the evaluation information from experts. The element oij denotes the evaluation information of alternative xi with respect to criterion aj. In this MCDM problem, experts use the flexible q-ROFNs for expressing the evaluation information of alternative xi with respect to criterion aj, namely, oij=μij,νij. Here, the criteria are divided into two different categories: benefit-type criteria and cost-type criteria.Before processing DMR=oijm×n, equation (45) is used to transform the values of cost-type criteria for deriving the transformed DM R˜=o˜ijm×n:(45)o˜ij=μij,νij,for benefit−type criteriona,νij,μij,for cost−type criterionaj.


### 5.2.q-Rung Orthopair Fuzzy MULTIMOORA Method

According to the above problem description, we introduce theq-ROFWDPHM and q-ROFWDPGHM operators to improve the original MULTIMOORA method so as to propose a novel q-rung orthopair fuzzy MULTIMOORA (q-ROF-MULTIMOORA) 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˜ijm×n, these three components compute the ranking values of alternatives as follows.


#### 5.2.1.q-ROF-RS Component

In this component, theq-ROFWDPHM operator is applied to aggregate the evaluation information of each alternative xi with respect to its n criteria. Therefore, using (34), the aggregated criteria value of alternative xi can be computed as(46)fi1=2nn+1⊕h=1n⊕g=hnnωh1+So˜ih∑k=1nωk1+So˜iko˜ihγ⊗nωg1+So˜ig∑k=1nωk1+So˜iko˜igη1/γ+η,where So˜ik=∑l=1,l≠knSupo˜ik,o˜il and Supo˜ik,o˜il=1−do˜ik,o˜il.Since the aggregated value is aq-ROFN, then the score function in Definition 2 is used to derive the crisp ranking value of alternative xi as(47)fi1=sfi1=s2nn+1⊕h=1n⊕g=hnnωh1+So˜ih∑k=1nωk1+So˜iko˜ihγ⊗nωg1+So˜ig∑k=1nωk1+So˜iko˜igη1/γ+η=s11+nn+1/2γ+η1/ℵ×1/∑h=1,g=hn1/γtihaihℵ+ηeigaigℵ1/ℵ1/q,1−11+nn+1/2γ+η1/ℵ×1/∑h=1,g=hn1/γtihbihℵ+ηeigbigℵ1/ℵ1/q,where ξih=1+So˜ih/∑k=1nωk1+So˜ik, ψig=1+So˜ig/∑k=1nωk1+So˜ik, μihq/1−μihq=1/aih, μigq/1−μigq=1/aig, 1−νihq/νihq=1/bih, 1−νigq/νigq=1/big, tih=1/nωhξih, and eig=1/nωgψig.The alternative with larger ranking value is better. Hence, all the alternatives can be ranked according to the descending order of their ranking values.


#### 5.2.2.q-ROF-RP Component

In this component, the reference point of each criterion is first derived as(48)ρj=argmaxi=1mso˜ij,j=1,2,…,n.In the second step, Definition4 is applied to compute the distance between the evaluation information of alternative xi with respect to each criterion and the reference point of the same criterion as(49)εij=do˜ij,ρj.It can be known thatεij is a real value and εij≥0. Considering the interrelationships among criteria, the ranking value of alternative xi is computed by aggregating the criteria distances of alternative xi as(50)fi2=2nn+1∑h=1n∑g=hnnωh1+Sεih∑k=1nωk1+Sεikεihγ×nωg1+Sεig∑k=1nωk1+Sεikεigη1/γ+η,where Sεik=∑l=1,l≠knSupεik,εil and Supεik,εil=1−εik−εil.In this component, the alternative with smaller ranking value is better. Thus, all the alternatives should be ranked according to the ascending order of their ranking values.


#### 5.2.3.q-ROF-FMF Component

In this component, the proposedq-ROFWDPGHM operator is applied to aggregate the evaluation information of each alternative xi with respect to its n criteria. Thus, using equation (43), the aggregated criteria value of alternative xi can be computed as(51)fi3=1γ+η⊗h=1n⊗g=hnγo˜ihnωh1+So˜ih/∑k=1nωk1+So˜ik⊕ηo˜ignωg1+So˜ig/∑k=1nωk1+So˜ik2/nn+1,where So˜ik=∑l=1,l≠knSupo˜ik,o˜il and Supo˜ik,o˜il=1−do˜ik,o˜il.Since the aggregated value is aq-ROFN, then the score function in Definition 2 is used to derive the crisp ranking value of alternative xi as(52)fi3=sfi3=s1−11+nn+1/2γ+η1/ℵ×1/∑h=1,g=hn1/γtih1/aihℵ+ηeig1/aigℵ1/ℵ1/q,11+nn+1/2γ+η1/ℵ×1/∑h=1,g=hn1/γtih1/bihℵ+ηeig1/bigℵ1/ℵ1/q,where ξih=1+So˜ih/∑k=1nωk1+So˜ik, ψig=1+So˜ig/∑k=1nωk1+So˜ik, aih=1−μihq/μihq, aig=1−μigq/μigq, bih=νihq/1−νihq, big=νigq/1−νigq, tih=1/nωhξih, and eig=1/nωgψig.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 MULTIMOORA 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. The HM operator owns the advantage of capturing the interrelationships hiding behind input values. Afterwards, by using the ranking values obtained from equations (47)–(52), here a new DM is constructed, where the three components of the q-ROF-MULTIMOORA method are regarded as criteria of alternatives: q-ROF-RS component (c1), q-ROF-RP component (c2), and q-ROF-FMF component (c3). Hence, the new DM M is constructed as(53)M=c1c2c3x1x2⋮xmf11f12f13f21f22f23⋮⋮⋮fm1fm2fm3,where x1,x2,…,xm denotes the set of alternatives and c1,c2,c3 denotes the set of criteria. The element fiyy=1,2,3 in the DM M denotes the ranking value of the alternative xi with respect to the criterion cy. Let χ=χ1,χ2,χ3 be the weight values of criteria c1,c2,c3, satisfying 0≤χy≤1 and ∑y=13χy=1. In general, the weight values of criteria are set to χ1=χ2=χ3=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 fiyy=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 fiyy=1,2,3 are normalized as(54)f˜iy=fiy−min1≤i≤mfiymax1≤i≤mfiy−min1≤i≤mfiy,for benefit−type criterioncy,max1≤i≤mfiy−fiymax1≤i≤mfiy−min1≤i≤mfiy,for cost−type criterioncy,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 xi with respect to three criteria for deriving the final ranking value of this alternative as(55)Fi=233+1∑h=13∑g=h3χhf˜ihγχgf˜igη1/γ+η.The 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.


### 5.3. Decision-Making Procedure

Based on the discussion and results in Section5.2, 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 DMR=oijm×n=μij,νijm×n. At the same time, the values of the parameters q, γ, 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 transform DM R=oijm×n into DM R˜=o˜ijm×n.Step 3: for the transformed DMR˜=o˜ijm×n, (47) is applied to compute the ranking value fi1 of each alternative xi with respect to the q-ROF-RS component. The alternatives can be ranked according to the descending order of their ranking values.Step 4: for the transformed DMR˜=o˜ijm×n, (50) is applied to compute the ranking value fi2 of each alternative xi with respect to the q-ROF-RP component. The alternatives can be ranked according to the ascending order of their ranking values.Step 5: for the transformed DMR˜=o˜ijm×n, (52) is applied to compute the ranking value fi3 of each alternative xi with respect to the q-ROF-FMF component. The 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 DMM=fiym×3 is constructed. Afterwards, (54) is applied to transform DM M=fiym×3 into M˜=f˜iym×3.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. Then, all the alternatives are ranked according to the descending order of their final ranking values.The above steps are also shown in Figure1.Figure 1
The frame diagram of the proposed method.Theq-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.(2)
It shows strong flexibility when computing the criteria values due to the Dombi operational laws ofq-ROFNs.(3)
The HM and GHM operators are capable of capturing the complex interrelationships hiding behind the criteria values. Moreover, the MULTIMOORA method integrates the ranking values obtained from three components for deriving the final ranking values. Thus, the decision results of theq-ROF-MULTIMOORA method are more reasonable and effective.


## 5.1. Problem Description

Let us suppose that there exists an MCDM problem consisting ofm alternatives x1,x2,…,xm and n criteria a1,a2,…,an. The weight values of criteria are denoted as ω1,ω2,…,ωn, where ∑j=1nωj=1 and 0≤ωj≤1. The decision matrix (DM) R=oijm×n corresponding to this MCDM problem consists of the evaluation information from experts. The element oij denotes the evaluation information of alternative xi with respect to criterion aj. In this MCDM problem, experts use the flexible q-ROFNs for expressing the evaluation information of alternative xi with respect to criterion aj, namely, oij=μij,νij. Here, the criteria are divided into two different categories: benefit-type criteria and cost-type criteria.Before processing DMR=oijm×n, equation (45) is used to transform the values of cost-type criteria for deriving the transformed DM R˜=o˜ijm×n:(45)o˜ij=μij,νij,for benefit−type criteriona,νij,μij,for cost−type criterionaj.


## 5.2.q-Rung Orthopair Fuzzy MULTIMOORA Method

According to the above problem description, we introduce theq-ROFWDPHM and q-ROFWDPGHM operators to improve the original MULTIMOORA method so as to propose a novel q-rung orthopair fuzzy MULTIMOORA (q-ROF-MULTIMOORA) 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˜ijm×n, these three components compute the ranking values of alternatives as follows.


### 5.2.1.q-ROF-RS Component

In this component, theq-ROFWDPHM operator is applied to aggregate the evaluation information of each alternative xi with respect to its n criteria. Therefore, using (34), the aggregated criteria value of alternative xi can be computed as(46)fi1=2nn+1⊕h=1n⊕g=hnnωh1+So˜ih∑k=1nωk1+So˜iko˜ihγ⊗nωg1+So˜ig∑k=1nωk1+So˜iko˜igη1/γ+η,where So˜ik=∑l=1,l≠knSupo˜ik,o˜il and Supo˜ik,o˜il=1−do˜ik,o˜il.Since the aggregated value is aq-ROFN, then the score function in Definition 2 is used to derive the crisp ranking value of alternative xi as(47)fi1=sfi1=s2nn+1⊕h=1n⊕g=hnnωh1+So˜ih∑k=1nωk1+So˜iko˜ihγ⊗nωg1+So˜ig∑k=1nωk1+So˜iko˜igη1/γ+η=s11+nn+1/2γ+η1/ℵ×1/∑h=1,g=hn1/γtihaihℵ+ηeigaigℵ1/ℵ1/q,1−11+nn+1/2γ+η1/ℵ×1/∑h=1,g=hn1/γtihbihℵ+ηeigbigℵ1/ℵ1/q,where ξih=1+So˜ih/∑k=1nωk1+So˜ik, ψig=1+So˜ig/∑k=1nωk1+So˜ik, μihq/1−μihq=1/aih, μigq/1−μigq=1/aig, 1−νihq/νihq=1/bih, 1−νigq/νigq=1/big, tih=1/nωhξih, and eig=1/nωgψig.The alternative with larger ranking value is better. Hence, all the alternatives can be ranked according to the descending order of their ranking values.


### 5.2.2.q-ROF-RP Component

In this component, the reference point of each criterion is first derived as(48)ρj=argmaxi=1mso˜ij,j=1,2,…,n.In the second step, Definition4 is applied to compute the distance between the evaluation information of alternative xi with respect to each criterion and the reference point of the same criterion as(49)εij=do˜ij,ρj.It can be known thatεij is a real value and εij≥0. Considering the interrelationships among criteria, the ranking value of alternative xi is computed by aggregating the criteria distances of alternative xi as(50)fi2=2nn+1∑h=1n∑g=hnnωh1+Sεih∑k=1nωk1+Sεikεihγ×nωg1+Sεig∑k=1nωk1+Sεikεigη1/γ+η,where Sεik=∑l=1,l≠knSupεik,εil and Supεik,εil=1−εik−εil.In this component, the alternative with smaller ranking value is better. Thus, all the alternatives should be ranked according to the ascending order of their ranking values.


### 5.2.3.q-ROF-FMF Component

In this component, the proposedq-ROFWDPGHM operator is applied to aggregate the evaluation information of each alternative xi with respect to its n criteria. Thus, using equation (43), the aggregated criteria value of alternative xi can be computed as(51)fi3=1γ+η⊗h=1n⊗g=hnγo˜ihnωh1+So˜ih/∑k=1nωk1+So˜ik⊕ηo˜ignωg1+So˜ig/∑k=1nωk1+So˜ik2/nn+1,where So˜ik=∑l=1,l≠knSupo˜ik,o˜il and Supo˜ik,o˜il=1−do˜ik,o˜il.Since the aggregated value is aq-ROFN, then the score function in Definition 2 is used to derive the crisp ranking value of alternative xi as(52)fi3=sfi3=s1−11+nn+1/2γ+η1/ℵ×1/∑h=1,g=hn1/γtih1/aihℵ+ηeig1/aigℵ1/ℵ1/q,11+nn+1/2γ+η1/ℵ×1/∑h=1,g=hn1/γtih1/bihℵ+ηeig1/bigℵ1/ℵ1/q,where ξih=1+So˜ih/∑k=1nωk1+So˜ik, ψig=1+So˜ig/∑k=1nωk1+So˜ik, aih=1−μihq/μihq, aig=1−μigq/μigq, bih=νihq/1−νihq, big=νigq/1−νigq, tih=1/nωhξih, and eig=1/nωgψig.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 MULTIMOORA 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. The HM operator owns the advantage of capturing the interrelationships hiding behind input values. Afterwards, by using the ranking values obtained from equations (47)–(52), here a new DM is constructed, where the three components of the q-ROF-MULTIMOORA method are regarded as criteria of alternatives: q-ROF-RS component (c1), q-ROF-RP component (c2), and q-ROF-FMF component (c3). Hence, the new DM M is constructed as(53)M=c1c2c3x1x2⋮xmf11f12f13f21f22f23⋮⋮⋮fm1fm2fm3,where x1,x2,…,xm denotes the set of alternatives and c1,c2,c3 denotes the set of criteria. The element fiyy=1,2,3 in the DM M denotes the ranking value of the alternative xi with respect to the criterion cy. Let χ=χ1,χ2,χ3 be the weight values of criteria c1,c2,c3, satisfying 0≤χy≤1 and ∑y=13χy=1. In general, the weight values of criteria are set to χ1=χ2=χ3=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 fiyy=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 fiyy=1,2,3 are normalized as(54)f˜iy=fiy−min1≤i≤mfiymax1≤i≤mfiy−min1≤i≤mfiy,for benefit−type criterioncy,max1≤i≤mfiy−fiymax1≤i≤mfiy−min1≤i≤mfiy,for cost−type criterioncy,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 xi with respect to three criteria for deriving the final ranking value of this alternative as(55)Fi=233+1∑h=13∑g=h3χhf˜ihγχgf˜igη1/γ+η.The 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.


## 5.2.1.q-ROF-RS Component

In this component, theq-ROFWDPHM operator is applied to aggregate the evaluation information of each alternative xi with respect to its n criteria. Therefore, using (34), the aggregated criteria value of alternative xi can be computed as(46)fi1=2nn+1⊕h=1n⊕g=hnnωh1+So˜ih∑k=1nωk1+So˜iko˜ihγ⊗nωg1+So˜ig∑k=1nωk1+So˜iko˜igη1/γ+η,where So˜ik=∑l=1,l≠knSupo˜ik,o˜il and Supo˜ik,o˜il=1−do˜ik,o˜il.Since the aggregated value is aq-ROFN, then the score function in Definition 2 is used to derive the crisp ranking value of alternative xi as(47)fi1=sfi1=s2nn+1⊕h=1n⊕g=hnnωh1+So˜ih∑k=1nωk1+So˜iko˜ihγ⊗nωg1+So˜ig∑k=1nωk1+So˜iko˜igη1/γ+η=s11+nn+1/2γ+η1/ℵ×1/∑h=1,g=hn1/γtihaihℵ+ηeigaigℵ1/ℵ1/q,1−11+nn+1/2γ+η1/ℵ×1/∑h=1,g=hn1/γtihbihℵ+ηeigbigℵ1/ℵ1/q,where ξih=1+So˜ih/∑k=1nωk1+So˜ik, ψig=1+So˜ig/∑k=1nωk1+So˜ik, μihq/1−μihq=1/aih, μigq/1−μigq=1/aig, 1−νihq/νihq=1/bih, 1−νigq/νigq=1/big, tih=1/nωhξih, and eig=1/nωgψig.The alternative with larger ranking value is better. Hence, all the alternatives can be ranked according to the descending order of their ranking values.


## 5.2.2.q-ROF-RP Component

In this component, the reference point of each criterion is first derived as(48)ρj=argmaxi=1mso˜ij,j=1,2,…,n.In the second step, Definition4 is applied to compute the distance between the evaluation information of alternative xi with respect to each criterion and the reference point of the same criterion as(49)εij=do˜ij,ρj.It can be known thatεij is a real value and εij≥0. Considering the interrelationships among criteria, the ranking value of alternative xi is computed by aggregating the criteria distances of alternative xi as(50)fi2=2nn+1∑h=1n∑g=hnnωh1+Sεih∑k=1nωk1+Sεikεihγ×nωg1+Sεig∑k=1nωk1+Sεikεigη1/γ+η,where Sεik=∑l=1,l≠knSupεik,εil and Supεik,εil=1−εik−εil.In this component, the alternative with smaller ranking value is better. Thus, all the alternatives should be ranked according to the ascending order of their ranking values.


## 5.2.3.q-ROF-FMF Component

In this component, the proposedq-ROFWDPGHM operator is applied to aggregate the evaluation information of each alternative xi with respect to its n criteria. Thus, using equation (43), the aggregated criteria value of alternative xi can be computed as(51)fi3=1γ+η⊗h=1n⊗g=hnγo˜ihnωh1+So˜ih/∑k=1nωk1+So˜ik⊕ηo˜ignωg1+So˜ig/∑k=1nωk1+So˜ik2/nn+1,where So˜ik=∑l=1,l≠knSupo˜ik,o˜il and Supo˜ik,o˜il=1−do˜ik,o˜il.Since the aggregated value is aq-ROFN, then the score function in Definition 2 is used to derive the crisp ranking value of alternative xi as(52)fi3=sfi3=s1−11+nn+1/2γ+η1/ℵ×1/∑h=1,g=hn1/γtih1/aihℵ+ηeig1/aigℵ1/ℵ1/q,11+nn+1/2γ+η1/ℵ×1/∑h=1,g=hn1/γtih1/bihℵ+ηeig1/bigℵ1/ℵ1/q,where ξih=1+So˜ih/∑k=1nωk1+So˜ik, ψig=1+So˜ig/∑k=1nωk1+So˜ik, aih=1−μihq/μihq, aig=1−μigq/μigq, bih=νihq/1−νihq, big=νigq/1−νigq, tih=1/nωhξih, and eig=1/nωgψig.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 MULTIMOORA 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. The HM operator owns the advantage of capturing the interrelationships hiding behind input values. Afterwards, by using the ranking values obtained from equations (47)–(52), here a new DM is constructed, where the three components of the q-ROF-MULTIMOORA method are regarded as criteria of alternatives: q-ROF-RS component (c1), q-ROF-RP component (c2), and q-ROF-FMF component (c3). Hence, the new DM M is constructed as(53)M=c1c2c3x1x2⋮xmf11f12f13f21f22f23⋮⋮⋮fm1fm2fm3,where x1,x2,…,xm denotes the set of alternatives and c1,c2,c3 denotes the set of criteria. The element fiyy=1,2,3 in the DM M denotes the ranking value of the alternative xi with respect to the criterion cy. Let χ=χ1,χ2,χ3 be the weight values of criteria c1,c2,c3, satisfying 0≤χy≤1 and ∑y=13χy=1. In general, the weight values of criteria are set to χ1=χ2=χ3=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 fiyy=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 fiyy=1,2,3 are normalized as(54)f˜iy=fiy−min1≤i≤mfiymax1≤i≤mfiy−min1≤i≤mfiy,for benefit−type criterioncy,max1≤i≤mfiy−fiymax1≤i≤mfiy−min1≤i≤mfiy,for cost−type criterioncy,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 xi with respect to three criteria for deriving the final ranking value of this alternative as(55)Fi=233+1∑h=13∑g=h3χhf˜ihγχgf˜igη1/γ+η.The 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.


## 5.3. Decision-Making Procedure

Based on the discussion and results in Section5.2, 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 DMR=oijm×n=μij,νijm×n. At the same time, the values of the parameters q, γ, 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 transform DM R=oijm×n into DM R˜=o˜ijm×n.Step 3: for the transformed DMR˜=o˜ijm×n, (47) is applied to compute the ranking value fi1 of each alternative xi with respect to the q-ROF-RS component. The alternatives can be ranked according to the descending order of their ranking values.Step 4: for the transformed DMR˜=o˜ijm×n, (50) is applied to compute the ranking value fi2 of each alternative xi with respect to the q-ROF-RP component. The alternatives can be ranked according to the ascending order of their ranking values.Step 5: for the transformed DMR˜=o˜ijm×n, (52) is applied to compute the ranking value fi3 of each alternative xi with respect to the q-ROF-FMF component. The 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 DMM=fiym×3 is constructed. Afterwards, (54) is applied to transform DM M=fiym×3 into M˜=f˜iym×3.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. Then, all the alternatives are ranked according to the descending order of their final ranking values.The above steps are also shown in Figure1.Figure 1
The frame diagram of the proposed method.Theq-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.(2)
It shows strong flexibility when computing the criteria values due to the Dombi operational laws ofq-ROFNs.(3)
The HM and GHM operators are capable of capturing the complex interrelationships hiding behind the criteria values. Moreover, the MULTIMOORA method integrates the ranking values obtained from three components for deriving the final ranking values. Thus, the decision results of theq-ROF-MULTIMOORA method are more reasonable and effective.


## 6. Illustrative Example and Comparison Analysis

In this section, a practical case concerning the evaluation of security algorithm is shown to illustrate the decision-making procedure of the proposedq-ROF-MULTIMOORA 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.


### 6.1. Decision Process Using theq-ROF-MULTIMOORA Method

In this section, a real case concerning the evaluation of security algorithms is provided to illustrate the decision procedure of the proposedq-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 (c1), reliability (c2), usability (c3), performance (c4), portability (c5), and complexity (c6). Hence, an MCDM problem composed of 5 security algorithms x1,x2,x3,x4,x5 and 6 criteria c1,c2,c3,c4,c5,c6 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. The 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=oij5×6=μij,νij5×6, as shown in Table 2.Step 1: the values of the parametersγ, 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=oij5×6 in Table 2 into DM R˜=o˜ij5×6 as depicted in Table 3.Step 3: for the transformed DMR˜=o˜ij5×6, (47) is applied to compute the ranking value fi1 of each security algorithm xi with respect to the q-ROF-RS component as(56)f11=0.036,f21=0.027,f31=0.033,f41=0.111,f51=0.047.Hence, these security algorithms can be ranked asx4≻x5≻x1≻x3≻x2.Step 4: for the transformed DMR˜=o˜ij5×6, (50) is applied to compute the ranking value fi2 of each security algorithm xi with respect to the q-ROF-RP component as(57)f12=0.179,f22=0.225,f32=0.213,f42=0.031,f52=0.229.Hence, these security algorithms can be ranked asx4≻x1≻x3≻x2≻x5.Step 5: for the transformed DMR˜=o˜ij5×6, (52) is applied to compute the ranking value fi3 of each security algorithm xi with respect to the q-ROF-FMF component as(58)f13=−0.004,f23=−0.018,f33=0.007,f43=−0.001,f53=−0.035.Hence, these security algorithms can be ranked asx3≻x4≻x1≻x2≻x5.Step 6: based on the ranking values of security algorithms obtained from three components in Steps 3–5, a new DMM=fiym×3 is formed. Afterwards, (54) is applied to transform the DM M=fiym×3 into M˜=f˜iym×3 as(59)M˜=0.1070.2530.7380.0000.0200.4050.0710.0811.0001.0001.0000.8100.2380.0000.000.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(60)F1=0.130,F2=0.057,F3=0.147,F4=0.313,F5=0.032.
Then, the final ranking order of these security algorithms isx4≻x3≻x1≻x2≻x5. Thus, the security algorithm x4 is the suitable one for the organization when building the security system.Table 2
Theq-rung orthopair fuzzy DM.
c1c2c3c4c5c6x1(0.1,0.2)(0.3,0.3)(0.5,0.3)(0.6,0.3)(0.2,0.1)(0.1,0.1)x2(0.3,0.2)(0.1,0.1)(0.5,0.1)(0.4,0.5)(0.3,0.2)(0.6,0.1)x3(0.6,0.1)(0.2,0.2)(0.4,0.3)(0.2,0.4)(0.3,0.2)(0.1,0.2)x4(0.2,0.5)(0.1,0.3)(0.7,0.1)(0.8,0.1)(0.4,0.3)(0.3,0.3)x5(0.4,0.4)(0.4,0.5)(0.5,0.2)(0.1,0.4)(0.3,0.4)(0.4,0.4)Table 3
The transformedq-rung orthopair fuzzy DM.
c1c2c3c4c5c6x1(0.1,.02)(0.3,0.3)(0.5,0.3)(0.6,0.3)(0.2,0.1)(0.1,0.1)x2(0.3,0.2)(0.1,0.1)(0.5,0.1)(0.4,0.5)(0.3,0.2)(0.1,0.6)x3(0.6,0.1)(0.2,0.2)(0.4,0.3)(0.2,0.4)(0.3,0.2)(0.2,0.1)x4(0.2,0.5)(0.1,0.3)(0.7,0.1)(0.8,0.1)(0.4,0.3)(0.3,0.3)x5(0.4,0.4)(0.4,0.5)(0.5,0.2)(0.1,0.4)(0.3,0.4)(0.4,0.4)


### 6.2. Influences of the Parameters on the Ranking Results

In this section, the influences of the parameters on the ranking results are discussed.


#### 6.2.1. Influence of the Parameterq on the Final Ranking Results

The influence of the parameterq on the final ranking results of the q-ROF-MULTIMOORA method is first discussed. In this case, the parameters γ=η=ℵ=1. For the transformed DM R=oij5×6 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.Table 4
Ranking results of theq-ROF-MULTIMOORA method when the value of q varies.
Final ranking values of security algorithmsRanking resultsq=1F1=0.210, F2=0.096, F3=0.192, F4=0.333, and F5=0.000x4≻x1≻x3≻x2≻x5q=2F1=0.093, F2=0.191, F3=0.069, F4=0.312, and F5=0.043x4≻x2≻x1≻x3≻x5q=3F1=0.130, F2=0.057, F3=0.147, F4=0.313, and F5=0.032x4≻x3≻x1≻x2≻x5q=5F1=0.136, F2=0.019, F3=0.142, F4=0.302, and F5=0.014x4≻x3≻x1≻x2≻x5Figure 2
Ranking results of theq-ROF-MULTIMOORA method when the value of q varies.From Table4, 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 x4≻x1≻x3≻x2≻x5. When q=2, the ranking result of security algorithms is x4≻x2≻x1≻x3≻x5. When q=3 or q=5, the ranking results of security algorithms are x4≻x3≻x1≻x2≻x5. Although the ranking result of security algorithms changes when the value of the parameter q varies, the most suitable security algorithm keeps unchanged, namely, x4. 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. The smallest value of the parameter q should satisfy μq+νq≤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 0.97+0.97<1.


#### 6.2.2. Influences of the Parametersγ and η on the Ranking Results

The influences of the parametersγ 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=oij5×6 in Table 2, the ranking results of security algorithms are shown in Table 5 and Figure 3 when the values of the parameters γ and η vary.Table 5
Ranking results of theq-ROF-MULTIMOORA method when the values of γ and η vary.
Final ranking values of security algorithmsRanking resultsγ=1 and η=0F1=0.094, F2=0.038, F3=0.118, F4=0.307, and F5=0.021x4≻x3≻x1≻x2≻x5γ=1 and η=0.5F1=0.112, F2=0.041, F3=0.131, F4=0.312, and F5=0.024x4≻x3≻x1≻x2≻x5γ=1 and η=1F1=0.130, F2=0.057, F3=0.147, F4=0.313, and F5=0.032x4≻x3≻x1≻x2≻x5γ=1 and η=2F1=0.151, F2=0.070, F3=0.187, F4=0.310, and F5=0.044x4≻x3≻x1≻x2≻x5γ=1 and η=3F1=0.174, F2=0.082, F3=0.215, F4=0.312, and F5=0.051x4≻x3≻x1≻x2≻x5γ=1 and η=4F1=0.185, F2=0.086, F3=0.235, F4=0.315, and F5=0.055x4≻x3≻x1≻x2≻x5γ=0 and η=1F1=0.197, F2=0.003, F3=0.170, F4=0.333, and F5=0.070x4≻x1≻x3≻x5≻x2γ=0.5 and η=1F1=0.139, F2=0.043, F3=0.155, F4=0.308, and F5=0.024x4≻x3≻x1≻x2≻x5γ=2 and η=1F1=0.144, F2=0.073, F3=0.184, F4=0.313, F5=0.047x4≻x3≻x1≻x2≻x5γ=3 and η=1F1=0.162, F2=0.093, F3=0.213, F4=0.317, and F5=0.051x4≻x3≻x1≻x2≻x5γ=4 and η=1F1=0.181, F2=0.106, F3=0.233, F4=0.318, and F5=0.054x4≻x3≻x1≻x2≻x5Figure 3
Ranking results of theq-ROF-MULTIMOORA method when the values of γ and η vary.From Table5, it can be seen that the ranking result obtained from the q-ROF-MULTIMOORA method is always x4≻x3≻x1≻x2≻x5 except when γ=0 and η=1. Nevertheless, the most suitable security algorithm is always x4. When γ=0 and η=1, the ranking result of security algorithms changes into x4≻x1≻x3≻x5≻x2. Thus, 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.


#### 6.2.3. Influence of the Parameterℵ on the Ranking Results

The influence of the parameterℵ on the ranking results of the q-ROF-MULTIMOORA method is analyzed in this part. In this case, the parameters γ=η=1 and q=3. For DM R=oij5×6 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.Table 6
Ranking results of theq-ROF-MULTIMOORA method when the value of ℵ varies.
Final ranking values of security algorithmsRanking resultsℵ=1F1=0.130, F2=0.057, F3=0.147, F4=0.313, and F5=0.032x4≻x3≻x1≻x2≻x5ℵ=2F1=0.153, F2=0.003, F3=0.150, F4=0.297, and F5=0.027x4≻x1≻x3≻x5≻x2ℵ=3F1=0.170, F2=0.003, F3=0.154, F4=0.296, and F5=0.054x4≻x1≻x3≻x5≻x2ℵ=4F1=0.172, F2=0.003, F3=0.151, F4=0.291, and F5=0.063x4≻x1≻x3≻x5≻x2ℵ=5F1=0.173, F2=0.003, F3=0.149, F4=0.286, and F5=0.065x4≻x1≻x3≻x5≻x2ℵ=6F1=0.173, F2=0.003, F3=0.147, F4=0.285, and F5=0.067x4≻x1≻x3≻x5≻x2ℵ=7F1=0.173, F2=0.003, F3=0.145, F4=0.283, and F5=0.066x4≻x1≻x3≻x5≻x2ℵ=8F1=0.174, F2=0.003, F3=0.145, F4=0.283, and F5=0.066x4≻x1≻x3≻x5≻x2ℵ=9F1=0.174, F2=0.003, F3=0.144, F4=0.282, and F5=0.067x4≻x1≻x3≻x5≻x2ℵ=10F1=0.174, F2=0.003, F3=0.142, F4=0.282, and F5=0.067x4≻x1≻x3≻x5≻x2Figure 4
Ranking results of theq-ROF-MULTIMOORA method when the value of ℵ varies.From Table6, 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 x4≻x3≻x1≻x2≻x5. 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 x4≻x1≻x3≻x5≻x2. However, the most suitable security algorithm always keeps unchanged, namely, x4 no matter how the value of the parameter ℵ varies. Hence, the ranking result that is obtained from the q-ROF-MULTIMOORA method is relatively stable. Because of the Dombi operational laws for q-ROFNs, the q-ROF-MULTIMOORA 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.


### 6.3. Comparative Analysis

For the proposedq-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.
The 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 cloud storage and health care, as well as payment and transfers. To benefit from the blockchain technology, organizations across the globe try to introduce the blockchain technology for building their own decentralized systems. The blockchain platforms are a good solution, which help organizations to facilitate the development and deployment of decentralized systems. Suppose that an organization wants to evaluate five blockchain platforms with respect to five criteria: usability, performance, scalability, security, and cost. Therefore, evaluating these five blockchain platforms can be formulated as an MCDM composed of blockchain platformsx1,x2,x3,x4,x5 with respect to criteria c1,c2,c3,c4,c5. The weights of criteria are set to ω=0.10,0.25,0.35,0.2,0.1. The transformed DM R=oij5×5=μij,νij5×5 is given in Table 7.
The original MULTIMOORA method andq-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 Table8, 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 ranking results obtained from the q-ROF-MULTIMOORA method and original MULTIMOORA method are different. Moreover, the ranking result obtained from the q-ROF-RS component of the q-ROF-MULTIMOORA method is different from that obtained from the RS component of the original MULTIMOORA method, so do the ranking results of other two components in the q-ROF-MULTIMOORA method and original MULTIMOORA method. The reasons are analyzed as follows:(1)
In the process of evaluating blockchain platforms, there are the interrelationships hiding behind the criteria values. Theq-ROF-MULTIMOORA method has been equipped with the AHM and GHM operators to process the interrelationships, while the original MULTIMOORA method is unable to process the hiding interrelationships.(2)
For theq-rung orthopair fuzzy DM R, there exists relatively great difference among criteria values. The q-ROF-MULTIMOORA method is integrated with the PA operator to alleviate the negative impact of extreme criteria values on the ranking results, while the original MULTIMOORA method ignores this case.
From the above analysis, it can be noted that theq-ROF-MULTIMOORA method performs better than the original MUTLIMOORA method because the q-ROF-MULTIMOORA method derives more robust and reasonable ranking results.Table 7
The transformedq-rung orthopair fuzzy DM for evaluating blockchain platforms.
c1c2c3c4c5x1(0.5,0.1)(0.5,0.1)(0.6,0.1)(0.8,0.1)(0.5,0.3)x2(0.1,0.2)(0.2,0.2)(0.3,0.4)(0.4,0.1)(0.2,0.2)x3(0.3,0.1)(0.2,0.4)(0.5,0.3)(0.5,0.4)(0.3,0.1)x4(0.4,0.2)(0.7,0.2)(0.6,0.1)(0.5,0.1)(0.7,0.2)x5(0.5,0.4)(0.2,0.1)(0.2,0.4)(0.3,0.5)(0.2,0.4)Table 8
Ranking results of five blockchain platforms when using different methods.
Decision methodRanking results1RS component [32]x1≻x4≻x3≻x2≻x52RP component [32]x1≻x4≻x3≻x2≻x53FMF component [32]x4≻x1≻x3≻x2≻x54The original MULTIMOORA [32]x1≻x4≻x3≻x2≻x55q-ROF-RS componentx4≻x1≻x3≻x5≻x26q-ROF-RP componentx4≻x1≻x3≻x2≻x57q-ROF-FMF componentx1≻x4≻x3≻x2≻x58q-ROF-MULTIMOORAx4≻x1≻x3≻x2≻x5


## 6.1. Decision Process Using theq-ROF-MULTIMOORA Method

In this section, a real case concerning the evaluation of security algorithms is provided to illustrate the decision procedure of the proposedq-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 (c1), reliability (c2), usability (c3), performance (c4), portability (c5), and complexity (c6). Hence, an MCDM problem composed of 5 security algorithms x1,x2,x3,x4,x5 and 6 criteria c1,c2,c3,c4,c5,c6 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. The 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=oij5×6=μij,νij5×6, as shown in Table 2.Step 1: the values of the parametersγ, 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=oij5×6 in Table 2 into DM R˜=o˜ij5×6 as depicted in Table 3.Step 3: for the transformed DMR˜=o˜ij5×6, (47) is applied to compute the ranking value fi1 of each security algorithm xi with respect to the q-ROF-RS component as(56)f11=0.036,f21=0.027,f31=0.033,f41=0.111,f51=0.047.Hence, these security algorithms can be ranked asx4≻x5≻x1≻x3≻x2.Step 4: for the transformed DMR˜=o˜ij5×6, (50) is applied to compute the ranking value fi2 of each security algorithm xi with respect to the q-ROF-RP component as(57)f12=0.179,f22=0.225,f32=0.213,f42=0.031,f52=0.229.Hence, these security algorithms can be ranked asx4≻x1≻x3≻x2≻x5.Step 5: for the transformed DMR˜=o˜ij5×6, (52) is applied to compute the ranking value fi3 of each security algorithm xi with respect to the q-ROF-FMF component as(58)f13=−0.004,f23=−0.018,f33=0.007,f43=−0.001,f53=−0.035.Hence, these security algorithms can be ranked asx3≻x4≻x1≻x2≻x5.Step 6: based on the ranking values of security algorithms obtained from three components in Steps 3–5, a new DMM=fiym×3 is formed. Afterwards, (54) is applied to transform the DM M=fiym×3 into M˜=f˜iym×3 as(59)M˜=0.1070.2530.7380.0000.0200.4050.0710.0811.0001.0001.0000.8100.2380.0000.000.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(60)F1=0.130,F2=0.057,F3=0.147,F4=0.313,F5=0.032.
Then, the final ranking order of these security algorithms isx4≻x3≻x1≻x2≻x5. Thus, the security algorithm x4 is the suitable one for the organization when building the security system.Table 2
Theq-rung orthopair fuzzy DM.
c1c2c3c4c5c6x1(0.1,0.2)(0.3,0.3)(0.5,0.3)(0.6,0.3)(0.2,0.1)(0.1,0.1)x2(0.3,0.2)(0.1,0.1)(0.5,0.1)(0.4,0.5)(0.3,0.2)(0.6,0.1)x3(0.6,0.1)(0.2,0.2)(0.4,0.3)(0.2,0.4)(0.3,0.2)(0.1,0.2)x4(0.2,0.5)(0.1,0.3)(0.7,0.1)(0.8,0.1)(0.4,0.3)(0.3,0.3)x5(0.4,0.4)(0.4,0.5)(0.5,0.2)(0.1,0.4)(0.3,0.4)(0.4,0.4)Table 3
The transformedq-rung orthopair fuzzy DM.
c1c2c3c4c5c6x1(0.1,.02)(0.3,0.3)(0.5,0.3)(0.6,0.3)(0.2,0.1)(0.1,0.1)x2(0.3,0.2)(0.1,0.1)(0.5,0.1)(0.4,0.5)(0.3,0.2)(0.1,0.6)x3(0.6,0.1)(0.2,0.2)(0.4,0.3)(0.2,0.4)(0.3,0.2)(0.2,0.1)x4(0.2,0.5)(0.1,0.3)(0.7,0.1)(0.8,0.1)(0.4,0.3)(0.3,0.3)x5(0.4,0.4)(0.4,0.5)(0.5,0.2)(0.1,0.4)(0.3,0.4)(0.4,0.4)


## 6.2. Influences of the Parameters on the Ranking Results

In this section, the influences of the parameters on the ranking results are discussed.


### 6.2.1. Influence of the Parameterq on the Final Ranking Results

The influence of the parameterq on the final ranking results of the q-ROF-MULTIMOORA method is first discussed. In this case, the parameters γ=η=ℵ=1. For the transformed DM R=oij5×6 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.Table 4
Ranking results of theq-ROF-MULTIMOORA method when the value of q varies.
Final ranking values of security algorithmsRanking resultsq=1F1=0.210, F2=0.096, F3=0.192, F4=0.333, and F5=0.000x4≻x1≻x3≻x2≻x5q=2F1=0.093, F2=0.191, F3=0.069, F4=0.312, and F5=0.043x4≻x2≻x1≻x3≻x5q=3F1=0.130, F2=0.057, F3=0.147, F4=0.313, and F5=0.032x4≻x3≻x1≻x2≻x5q=5F1=0.136, F2=0.019, F3=0.142, F4=0.302, and F5=0.014x4≻x3≻x1≻x2≻x5Figure 2
Ranking results of theq-ROF-MULTIMOORA method when the value of q varies.From Table4, 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 x4≻x1≻x3≻x2≻x5. When q=2, the ranking result of security algorithms is x4≻x2≻x1≻x3≻x5. When q=3 or q=5, the ranking results of security algorithms are x4≻x3≻x1≻x2≻x5. Although the ranking result of security algorithms changes when the value of the parameter q varies, the most suitable security algorithm keeps unchanged, namely, x4. 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. The smallest value of the parameter q should satisfy μq+νq≤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 0.97+0.97<1.


### 6.2.2. Influences of the Parametersγ and η on the Ranking Results

The influences of the parametersγ 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=oij5×6 in Table 2, the ranking results of security algorithms are shown in Table 5 and Figure 3 when the values of the parameters γ and η vary.Table 5
Ranking results of theq-ROF-MULTIMOORA method when the values of γ and η vary.
Final ranking values of security algorithmsRanking resultsγ=1 and η=0F1=0.094, F2=0.038, F3=0.118, F4=0.307, and F5=0.021x4≻x3≻x1≻x2≻x5γ=1 and η=0.5F1=0.112, F2=0.041, F3=0.131, F4=0.312, and F5=0.024x4≻x3≻x1≻x2≻x5γ=1 and η=1F1=0.130, F2=0.057, F3=0.147, F4=0.313, and F5=0.032x4≻x3≻x1≻x2≻x5γ=1 and η=2F1=0.151, F2=0.070, F3=0.187, F4=0.310, and F5=0.044x4≻x3≻x1≻x2≻x5γ=1 and η=3F1=0.174, F2=0.082, F3=0.215, F4=0.312, and F5=0.051x4≻x3≻x1≻x2≻x5γ=1 and η=4F1=0.185, F2=0.086, F3=0.235, F4=0.315, and F5=0.055x4≻x3≻x1≻x2≻x5γ=0 and η=1F1=0.197, F2=0.003, F3=0.170, F4=0.333, and F5=0.070x4≻x1≻x3≻x5≻x2γ=0.5 and η=1F1=0.139, F2=0.043, F3=0.155, F4=0.308, and F5=0.024x4≻x3≻x1≻x2≻x5γ=2 and η=1F1=0.144, F2=0.073, F3=0.184, F4=0.313, F5=0.047x4≻x3≻x1≻x2≻x5γ=3 and η=1F1=0.162, F2=0.093, F3=0.213, F4=0.317, and F5=0.051x4≻x3≻x1≻x2≻x5γ=4 and η=1F1=0.181, F2=0.106, F3=0.233, F4=0.318, and F5=0.054x4≻x3≻x1≻x2≻x5Figure 3
Ranking results of theq-ROF-MULTIMOORA method when the values of γ and η vary.From Table5, it can be seen that the ranking result obtained from the q-ROF-MULTIMOORA method is always x4≻x3≻x1≻x2≻x5 except when γ=0 and η=1. Nevertheless, the most suitable security algorithm is always x4. When γ=0 and η=1, the ranking result of security algorithms changes into x4≻x1≻x3≻x5≻x2. Thus, 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.


### 6.2.3. Influence of the Parameterℵ on the Ranking Results

The influence of the parameterℵ on the ranking results of the q-ROF-MULTIMOORA method is analyzed in this part. In this case, the parameters γ=η=1 and q=3. For DM R=oij5×6 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.Table 6
Ranking results of theq-ROF-MULTIMOORA method when the value of ℵ varies.
Final ranking values of security algorithmsRanking resultsℵ=1F1=0.130, F2=0.057, F3=0.147, F4=0.313, and F5=0.032x4≻x3≻x1≻x2≻x5ℵ=2F1=0.153, F2=0.003, F3=0.150, F4=0.297, and F5=0.027x4≻x1≻x3≻x5≻x2ℵ=3F1=0.170, F2=0.003, F3=0.154, F4=0.296, and F5=0.054x4≻x1≻x3≻x5≻x2ℵ=4F1=0.172, F2=0.003, F3=0.151, F4=0.291, and F5=0.063x4≻x1≻x3≻x5≻x2ℵ=5F1=0.173, F2=0.003, F3=0.149, F4=0.286, and F5=0.065x4≻x1≻x3≻x5≻x2ℵ=6F1=0.173, F2=0.003, F3=0.147, F4=0.285, and F5=0.067x4≻x1≻x3≻x5≻x2ℵ=7F1=0.173, F2=0.003, F3=0.145, F4=0.283, and F5=0.066x4≻x1≻x3≻x5≻x2ℵ=8F1=0.174, F2=0.003, F3=0.145, F4=0.283, and F5=0.066x4≻x1≻x3≻x5≻x2ℵ=9F1=0.174, F2=0.003, F3=0.144, F4=0.282, and F5=0.067x4≻x1≻x3≻x5≻x2ℵ=10F1=0.174, F2=0.003, F3=0.142, F4=0.282, and F5=0.067x4≻x1≻x3≻x5≻x2Figure 4
Ranking results of theq-ROF-MULTIMOORA method when the value of ℵ varies.From Table6, 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 x4≻x3≻x1≻x2≻x5. 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 x4≻x1≻x3≻x5≻x2. However, the most suitable security algorithm always keeps unchanged, namely, x4 no matter how the value of the parameter ℵ varies. Hence, the ranking result that is obtained from the q-ROF-MULTIMOORA method is relatively stable. Because of the Dombi operational laws for q-ROFNs, the q-ROF-MULTIMOORA 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.


## 6.2.1. Influence of the Parameterq on the Final Ranking Results

The influence of the parameterq on the final ranking results of the q-ROF-MULTIMOORA method is first discussed. In this case, the parameters γ=η=ℵ=1. For the transformed DM R=oij5×6 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.Table 4
Ranking results of theq-ROF-MULTIMOORA method when the value of q varies.
Final ranking values of security algorithmsRanking resultsq=1F1=0.210, F2=0.096, F3=0.192, F4=0.333, and F5=0.000x4≻x1≻x3≻x2≻x5q=2F1=0.093, F2=0.191, F3=0.069, F4=0.312, and F5=0.043x4≻x2≻x1≻x3≻x5q=3F1=0.130, F2=0.057, F3=0.147, F4=0.313, and F5=0.032x4≻x3≻x1≻x2≻x5q=5F1=0.136, F2=0.019, F3=0.142, F4=0.302, and F5=0.014x4≻x3≻x1≻x2≻x5Figure 2
Ranking results of theq-ROF-MULTIMOORA method when the value of q varies.From Table4, 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 x4≻x1≻x3≻x2≻x5. When q=2, the ranking result of security algorithms is x4≻x2≻x1≻x3≻x5. When q=3 or q=5, the ranking results of security algorithms are x4≻x3≻x1≻x2≻x5. Although the ranking result of security algorithms changes when the value of the parameter q varies, the most suitable security algorithm keeps unchanged, namely, x4. 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. The smallest value of the parameter q should satisfy μq+νq≤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 0.97+0.97<1.


## 6.2.2. Influences of the Parametersγ and η on the Ranking Results

The influences of the parametersγ 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=oij5×6 in Table 2, the ranking results of security algorithms are shown in Table 5 and Figure 3 when the values of the parameters γ and η vary.Table 5
Ranking results of theq-ROF-MULTIMOORA method when the values of γ and η vary.
Final ranking values of security algorithmsRanking resultsγ=1 and η=0F1=0.094, F2=0.038, F3=0.118, F4=0.307, and F5=0.021x4≻x3≻x1≻x2≻x5γ=1 and η=0.5F1=0.112, F2=0.041, F3=0.131, F4=0.312, and F5=0.024x4≻x3≻x1≻x2≻x5γ=1 and η=1F1=0.130, F2=0.057, F3=0.147, F4=0.313, and F5=0.032x4≻x3≻x1≻x2≻x5γ=1 and η=2F1=0.151, F2=0.070, F3=0.187, F4=0.310, and F5=0.044x4≻x3≻x1≻x2≻x5γ=1 and η=3F1=0.174, F2=0.082, F3=0.215, F4=0.312, and F5=0.051x4≻x3≻x1≻x2≻x5γ=1 and η=4F1=0.185, F2=0.086, F3=0.235, F4=0.315, and F5=0.055x4≻x3≻x1≻x2≻x5γ=0 and η=1F1=0.197, F2=0.003, F3=0.170, F4=0.333, and F5=0.070x4≻x1≻x3≻x5≻x2γ=0.5 and η=1F1=0.139, F2=0.043, F3=0.155, F4=0.308, and F5=0.024x4≻x3≻x1≻x2≻x5γ=2 and η=1F1=0.144, F2=0.073, F3=0.184, F4=0.313, F5=0.047x4≻x3≻x1≻x2≻x5γ=3 and η=1F1=0.162, F2=0.093, F3=0.213, F4=0.317, and F5=0.051x4≻x3≻x1≻x2≻x5γ=4 and η=1F1=0.181, F2=0.106, F3=0.233, F4=0.318, and F5=0.054x4≻x3≻x1≻x2≻x5Figure 3
Ranking results of theq-ROF-MULTIMOORA method when the values of γ and η vary.From Table5, it can be seen that the ranking result obtained from the q-ROF-MULTIMOORA method is always x4≻x3≻x1≻x2≻x5 except when γ=0 and η=1. Nevertheless, the most suitable security algorithm is always x4. When γ=0 and η=1, the ranking result of security algorithms changes into x4≻x1≻x3≻x5≻x2. Thus, 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.


## 6.2.3. Influence of the Parameterℵ on the Ranking Results

The influence of the parameterℵ on the ranking results of the q-ROF-MULTIMOORA method is analyzed in this part. In this case, the parameters γ=η=1 and q=3. For DM R=oij5×6 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.Table 6
Ranking results of theq-ROF-MULTIMOORA method when the value of ℵ varies.
Final ranking values of security algorithmsRanking resultsℵ=1F1=0.130, F2=0.057, F3=0.147, F4=0.313, and F5=0.032x4≻x3≻x1≻x2≻x5ℵ=2F1=0.153, F2=0.003, F3=0.150, F4=0.297, and F5=0.027x4≻x1≻x3≻x5≻x2ℵ=3F1=0.170, F2=0.003, F3=0.154, F4=0.296, and F5=0.054x4≻x1≻x3≻x5≻x2ℵ=4F1=0.172, F2=0.003, F3=0.151, F4=0.291, and F5=0.063x4≻x1≻x3≻x5≻x2ℵ=5F1=0.173, F2=0.003, F3=0.149, F4=0.286, and F5=0.065x4≻x1≻x3≻x5≻x2ℵ=6F1=0.173, F2=0.003, F3=0.147, F4=0.285, and F5=0.067x4≻x1≻x3≻x5≻x2ℵ=7F1=0.173, F2=0.003, F3=0.145, F4=0.283, and F5=0.066x4≻x1≻x3≻x5≻x2ℵ=8F1=0.174, F2=0.003, F3=0.145, F4=0.283, and F5=0.066x4≻x1≻x3≻x5≻x2ℵ=9F1=0.174, F2=0.003, F3=0.144, F4=0.282, and F5=0.067x4≻x1≻x3≻x5≻x2ℵ=10F1=0.174, F2=0.003, F3=0.142, F4=0.282, and F5=0.067x4≻x1≻x3≻x5≻x2Figure 4
Ranking results of theq-ROF-MULTIMOORA method when the value of ℵ varies.From Table6, 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 x4≻x3≻x1≻x2≻x5. 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 x4≻x1≻x3≻x5≻x2. However, the most suitable security algorithm always keeps unchanged, namely, x4 no matter how the value of the parameter ℵ varies. Hence, the ranking result that is obtained from the q-ROF-MULTIMOORA method is relatively stable. Because of the Dombi operational laws for q-ROFNs, the q-ROF-MULTIMOORA 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.


## 6.3. Comparative Analysis

For the proposedq-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.
The 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 cloud storage and health care, as well as payment and transfers. To benefit from the blockchain technology, organizations across the globe try to introduce the blockchain technology for building their own decentralized systems. The blockchain platforms are a good solution, which help organizations to facilitate the development and deployment of decentralized systems. Suppose that an organization wants to evaluate five blockchain platforms with respect to five criteria: usability, performance, scalability, security, and cost. Therefore, evaluating these five blockchain platforms can be formulated as an MCDM composed of blockchain platformsx1,x2,x3,x4,x5 with respect to criteria c1,c2,c3,c4,c5. The weights of criteria are set to ω=0.10,0.25,0.35,0.2,0.1. The transformed DM R=oij5×5=μij,νij5×5 is given in Table 7.
The original MULTIMOORA method andq-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 Table8, 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 ranking results obtained from the q-ROF-MULTIMOORA method and original MULTIMOORA method are different. Moreover, the ranking result obtained from the q-ROF-RS component of the q-ROF-MULTIMOORA method is different from that obtained from the RS component of the original MULTIMOORA method, so do the ranking results of other two components in the q-ROF-MULTIMOORA method and original MULTIMOORA method. The reasons are analyzed as follows:(1)
In the process of evaluating blockchain platforms, there are the interrelationships hiding behind the criteria values. Theq-ROF-MULTIMOORA method has been equipped with the AHM and GHM operators to process the interrelationships, while the original MULTIMOORA method is unable to process the hiding interrelationships.(2)
For theq-rung orthopair fuzzy DM R, there exists relatively great difference among criteria values. The q-ROF-MULTIMOORA method is integrated with the PA operator to alleviate the negative impact of extreme criteria values on the ranking results, while the original MULTIMOORA method ignores this case.
From the above analysis, it can be noted that theq-ROF-MULTIMOORA method performs better than the original MUTLIMOORA method because the q-ROF-MULTIMOORA method derives more robust and reasonable ranking results.Table 7
The transformedq-rung orthopair fuzzy DM for evaluating blockchain platforms.
c1c2c3c4c5x1(0.5,0.1)(0.5,0.1)(0.6,0.1)(0.8,0.1)(0.5,0.3)x2(0.1,0.2)(0.2,0.2)(0.3,0.4)(0.4,0.1)(0.2,0.2)x3(0.3,0.1)(0.2,0.4)(0.5,0.3)(0.5,0.4)(0.3,0.1)x4(0.4,0.2)(0.7,0.2)(0.6,0.1)(0.5,0.1)(0.7,0.2)x5(0.5,0.4)(0.2,0.1)(0.2,0.4)(0.3,0.5)(0.2,0.4)Table 8
Ranking results of five blockchain platforms when using different methods.
Decision methodRanking results1RS component [32]x1≻x4≻x3≻x2≻x52RP component [32]x1≻x4≻x3≻x2≻x53FMF component [32]x4≻x1≻x3≻x2≻x54The original MULTIMOORA [32]x1≻x4≻x3≻x2≻x55q-ROF-RS componentx4≻x1≻x3≻x5≻x26q-ROF-RP componentx4≻x1≻x3≻x2≻x57q-ROF-FMF componentx1≻x4≻x3≻x2≻x58q-ROF-MULTIMOORAx4≻x1≻x3≻x2≻x5


## 7. Conclusions

To solve the security algorithms’ evaluation problem, we propose an efficientq-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 theq-ROFDPHM, q-ROFWDPHM, q-ROFDPGHM, and q-ROFWDPGHM operators to aggregate q-ROFNs.(2)
The proposedq-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 theq-ROF-MULTIMOORA method. The influences of the parameters on the ranking results are analyzed.(4)
To validate the effectiveness of the proposedq-ROF-MULTIMOORA method, a new example of evaluating blockchain platforms is given.The proposed methods also have some limitations:(1)
theq-ROFSs model, the uncertain information uses only three characteristic functions and does not have the characteristic function that denotes the degree of abstinence. This 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)
The weights of attributes are directly given in this study. It ignores the objective significance. The method combining the objective weights and subjective weights should be considered in the future.(3)
In the proposedq-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.The proposedq-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.



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*Source: 3179688-2022-08-09.xml*