Multi-Attribute Group Decision-Making Based on Interval-Valued q -Rung Orthopair Fuzzy Power Generalized Maclaurin Symmetric Mean Operator and Its Application in Online Education Platform Performance Evaluation

: This paper aims to propose a novel multi-attribute group decision-making (MAGDM) method based on interval-valued q-rung orthopair fuzzy sets (IVq-ROFSs). The IVq-ROFSs have been proved to be effective in handling MAGDM problems, and several novel decision-making methods have been proposed. Nevertheless, it is worth pointing out that these approaches still have some limitations, and there still exist some realistic situations that cannot be solved by existing MAGDM methods. Hence, the objective of this paper is to introduce a novel MAGDM method, which can overcome some of the drawbacks of existing approaches. To effectively and appropriately aggregate interval-valued q-rung orthopair fuzzy numbers (IVq-ROFNs), we combine the power average with generalized Maclaurin symmetric mean (GMSM), propose the power GMSM operator and extend it into IVq-ROFSs. Afterwards, a collection of new aggregation operators for IVq-ROFNs are developed. In this paper, we study deﬁnitions of these operators and investigate their characteristics as well as special cases. Then, based on the new aggregation operators, we present a new MAGDM method. Finally, we apply the proposed MAGDM method in online education platform performance evaluation to illustrate its effectiveness and validity. In addition, we also provide comparative analysis to explain why decision-makers should use our method instead of the others.


Introduction
With the continuous development of information and communication technologies, the online education industry has ushered in new development opportunities, and many new online education platforms have emerged. Especially under the influence of the COVID-19 epidemic, many countries and regions have taken measures to close schools and switch to online teaching. Hence, online education platforms have obtained new development opportunities. In addition, the promotion of online education platforms can alleviate the unbalanced distribution of educational resources between regions, thereby promoting educational equity to a certain extent. Different online education platforms are quite different, and their quality and performance are also uneven. Therefore, it is necessary to evaluate the performance of online education platforms according to some methods.
The evaluation problem of online education platforms can be regarded as a multiattribute group decision-making (MAGDM) problem essentially. Decision-makers evaluate different online education platforms from multiple dimensions, and on this basis, they can Considering that HM has the similar function as BM, motivated by PBM, Liu [34] combined PA with HM and proposed power Heronian mean (PHM) operator and applied it in interval-valued intuitionistic fuzzy sets. Liu's [34] contributions demonstrated the advantages and superiorities of PHM. Based on Liu's [34] pioneering works, Ju et al. [35], Liu et al. [36], Wang et al. [37], Liu et al. [38], and Jiang et al. [39] investigated PHM under hesitant fuzzy linguistic sets, linguistic neutrosophic sets, q-rung orthopair hesitant fuzzy sets, neutrosophic cubic sets, and interval-valued dual hesitant fuzzy sets, respectively.
Recent publications reveal the advantages of PBM and PHM in aggregating fuzzy information, however, they still have drawbacks. One of the prominent drawbacks of PBM and PHM is that they only consider the interrelationship among any two attributes, which is insufficient and inadequate to handle some real MAGDM problems. More and more researches have indicated the importance and necessity of considering the interrelationship and interaction among multiple attributes [40]. Hence, it is necessary to combine PA with an AO, which considers the interrelationship among multiple arguments. The generalized Maclaurin symmetric mean (GMSM) [41] is an operator, which has the ability of reflecting the interrelationship among multiple input variables. In addition, GMSM has the ability of reflecting the individual importance of aggregated arguments, which makes it more powerful than the classical Maclaurin symmetric mean (MSM) [42]. Due to this reason, GMSM has been widely applied in Pythagorean fuzzy sets [43], q-rung orthopair fuzzy sets (q-ROFSs) [44], multi-hesitant fuzzy linguistic sets [45], intuitionistic fuzzy soft sets [46], probabilistic linguistic sets [47], etc. In this paper, considering the ability and flexibility of GMSM, we combine PA with GMSM and propose the power generalized Maclaurin symmetric mean (PGMSM) operator. Evidently, PGMSM is more powerful than PBM, PHM, and GMSM and is more suitable to deal with actual-life MAGDM problems. Additionally, we notice that the interval-valued q-ROFS (IVq-ROFS) proposed by Ju et al. [48] is a powerful information representation tool that can depict fuzziness and uncertainty efficiently. The IVq-ROFS is an extension of q-ROFS proposed by Yager [49]. MAGDM methods based on q-ROFSs have received much attention [50][51][52]. Similarly, some MAGDM methods under IVq-ROFSs have been proposed [53][54][55][56]. Nevertheless, these methods still have a drawback, i.e., they fail to handle the complicated interrelationship between attributes and simultaneously eliminate the bad influence of decision-makers' evaluation values. In order to overcome this drawback, this paper extends PGMSM into IVq-ROFSs to propose a new MAGDM method.
The main contributions of this paper contain three aspects. First, a novel AO, called PGMSM is developed. Second, some new AOs for interval-valued q-rung orthopair fuzzy numbers (IVq-ROFNs) are developed. Finally, a new MAGDM method under IVq-ROFS circumstances is put forward. The rest of this paper is organized as follows. Section 2 recalls basic notions. Section 3 presents several new AOs for IVq-ROFNs and discusses their properties and characteristics. Based on the developed AOs, Section 4 presents a novel MAGDM method. Section 5 applies the proposed MAGDM method in online education performance evaluation. Section 6 summarizes the paper.

Basic Notions
This section introduces some basic notions that will be used in the following sections.

The Interval-Valued q-Rung Orthopair Fuzzy Sets
Definition 1 [48]. Let X be an ordinary fixed set, an IVq-ROFS A defined on X is expressed as where µ A (x), v A (x) ⊆ [0, 1] are two interval-valued values, denoting the membership and nonmembership degrees of element x ∈ X to the set A, such that (sup(µ A (x))) q + (sup(v A (x))) q ≤ 1(q ≥ 1). For convenience, the ordered pair A = (µ A (x), v A (x)) is called an IVq-ROFN, which can be denoted as α = ([a, b], [c, d]) for simplicity, such that a ≤ b, c ≤ d, and b q + d q ≤ 1(q ≥ 1).
The method to compare any two IVq-ROFNs is presented as follows: Definition 3 [48].
be an IVq-ROFN, then the score function of α is expressed as and the accuracy function of α is expressed as For any two IVq-ROFNs α 1 and α 2 The distance measure between two IVq-ROFNs is defined as follows: ) be two IVq-ROFNs, then the distance between α 1 and α 2 is defined as
In addition, we can obtain some special cases of the proposed PGMSM operator.

Novel Aggregation Operators for IVq-ROFNs
In MAGDM problems, AOs are used to aggregate attribute values to determine comprehensive evaluation values of alternatives. Hence, based on the above-mentioned basic AOs, we introduce some novel AOs for IVq-ROFNs and discuss their properties.

The Interval-Valued q-Rung Orthopair Fuzzy Power Average Operator
We first extend the classical PA operator into IVq-ROFSs to propose a PA operator for IVq-ROFNs.
Based on the operational rules presented in Definition 2, the following aggregated results can be obtained. a collection of IVq-ROFNs, then the  aggregated value by the IVq-ROFPA operator is still an IVq-ROFN and IVqROFPA(α 1 , α 2 , . . . , α n ) Theorem 1 is trivial and we omit its proof. In addition, it is easy to prove that the IVq-ROFPA operator has the following properties. where

The Interval-Valued q-Rung Orthopair Fuzzy Power Weighted Average Operator
In MAGDM problem, when aggregating a collection of IVq-ROFNs, not only the IVq-ROFNs themselves, but also their corresponding weight information should be taken into account. Hence, we take into consideration of the weights of aggregated IVq-ROFNs in the IVq-ROFPA operator and propose its weighted form.
Simialrily, we can gain the following theorem.
. . , n) be a collection of IVq-ROFNs, the aggregated value by the IVq-ROFPWA operator is still an IVq-ROFN and IVqROFPWA(α 1 , α 2 , . . . , α n ) In addition, it is easy to prove that the IVq-ROFPWA operator has the properties of idempotency and boundedness.

The Interval-Valued q-Rung Orthopair Fuzzy Power Generalized Maclaurin Symmetric Mean Operator
In MAGDM problems, when fusing attribute values, we usually have to consider two significant issues, i.e., taking the interrelationship among attributes into consideration and reducing or eliminating the negative influence of decision-makers' extreme evaluation values. The above-mentioned PGMSM takes the advantages of both PA and GMSM and, hence, it can resolve the above two issues. Therefore, in this subsection, we extend the PGMSM operator into IVq-ROFSs to propose a novel AO for IVq-ROFNs.
Additionally, if the value Sup α i , α j is a non-negative number, then IVq-ROFPGMSM operator further reduces to the interval-valued q-rung orthopair fuzzy Maclaurin symmetric mean (IVq-ROFMSM) operator, i.e.,

The Interval-Valued q-Rung Orthopair Fuzzy Power Weighted Generalized Maclaurin Symmetric Mean Operator
In addition, when fusing a set of IVq-ROFNs, their weight information should also be taken into account. Hence, we take the weight vector of aggregated IVq-ROFNs in the IVq-ROFPGMSM operator and propose its weighted form.
. . , n) be a collection of IVq-ROFNs, then the aggregated valued by the IVq-ROFPWGMSM operator is also an IVq-ROFN and The proof of Theorem 6 is similar to that of Theorem 5.

A New Multi-Attribute Group Decision-Making Method Based on Interval-Valued q-Rung Orthopair Fuzzy Numbers
This section investigates a new MAGDM method under IVq-ROFSs. We assume there are m alternatives to be evaluated, which can be denoted as A = {A 1 , A 2 , . . . , A m }. There are l decision-makers that are asked to evaluate the performance of the candidates. The group of decision-makers can be denoted as {D 1 , D 2 , . . . , D l }. The weight vector of decision-makers is λ = (λ 1 , λ 2 , . . . , λ l ) T , such that ∑ l t=1 λ t = 1 and 0 ≤ λ t ≤ 1. Suppose that decision-makers are required to evaluate the alternatives under n attributes, and the attribute set can be expressed as G = (G 1 , G 2 , . . . , G n ) T . Weight vector of these attributes is w = (w 1 , w 2 , . . . , w n ) T , satisfying ∑ n j=1 w j = 1 and 0 ≤ w j ≤ 1. In the following, we present a novel method to determine the optimal alternative.
Step 1. Construct the original decision matrices. Decision-makers are asked for evaluating the properties of the candidates under the n attributes by using IVq-ROFNs. For instance, decision-maker D t uses an IVq- ) to express his/her evaluation value of the attribute G j (j = 1, 2, . . . , n) of alternative A i (i = 1, 2, . . . , m). Afterwards, a series of interval-valued q-rung orthopair fuzzy decision matrices are obtained.
Step 2. Normalize the original decision matrices. Considering the fact that there exist two kinds of attributes, i.e., benefit type (T 1 ) and cost type (T 2 ), the original decision matrices provided by decision-makers should be normalized according to the following formula, i.e., Step 3. Determine the comprehensive decision matrix. Step where s, f = 1, 2, . . . , l(s = f ) and d(α s ij , α f ij ) denotes the distance between α s ij and α f ij .
Step 3.2. Calculate the overall support T(α s ij ) by Step 3.3. Compute the power weight associated with α s ij by Step 3.4. Utilize the IVq-ROFPWA operator to compute the overall decision matrix, i.e., Step 4. Compute the overall evaluation value for each alternative.
Step 4.2. Determine the overall support Step 4.3. Compute the power weight of α ij by Step 4.4. Obtain the overall evaluation values by using the IVq-ROFPWGMSM operator, i.e., Step 5. Compute the score values of alternatives X i (i = 1, 2, . . . , m).
Step 6. Determine the rank of their scores and choose the best one.
To better demonstrate the steps of our new proposed MAGDM method, we provide the following Figure 1. Step 1 Step2 Step 3 Step 4 Step 5 Step6

An Application of the Proposed Method in Online Education Platform Performance Evaluation
With the rise of the Internet, the education industry has ushered in new development opportunities, and increasingly more online education platforms have sprung up. Mobile

An Application of the Proposed Method in Online Education Platform Performance Evaluation
With the rise of the Internet, the education industry has ushered in new development opportunities, and increasingly more online education platforms have sprung up. Mobile Internet and 5G communication technology have promoted the development of the online education industry, and the development of online education platforms is showing a blowout trend. In addition, affected by COVID-19, more and more schools and educational institutions adopt online teaching models, which objectively promotes the development of online education platforms. Online teaching platforms cannot only realize distance teaching, but are also considered to be an important measure to promote the flow and reasonable distribution of teaching resources, and can alleviate the structural contradictions caused by the unbalanced distribution of educational resources. Therefore, all countries are vigorously developing online education platforms. With the continuous increase in the number of online education platforms, their uneven quality has also received widespread attention from the public. Therefore, it is very necessary to evaluate the performance of online education platforms. Assume that three decision-makers (D 1 , D 2 , and D 3 ) are invited to evaluate the performance of four famous online education platforms, which can be denoted as A 1 , A 2 , A 3 , and A 4. Weight vector of decision-maker is λ = (0.3, 0.3, 0.4) T . Decision-makers evaluate the performance of the four education platforms under four attributes, i.e., availability (G 1 ), interactive quality (G 2 ), system quality (G 3 ), and quality of service (G 4 ). Weight vector of attributes is w = (0.40, 0.25, 0.20, 0.15) T .

Online Education Platforms Performance Evaluation Process
Step 1. Require the three decision-makers to provide their evaluation values in the form of IVq-ROFNs. Hence, we can obtain three interval-valued q-rung orthopair fuzzy decision matrices, which are presented in Tables 1-3.   Step 2. Normalize the original decision matrices. It is noted that all attributes are benefit type and hence the original decision matrices do not need to be normalized.
Step 3. Determine the integrated decision matrix.
Step 3.1. According to Equation (40) Here we use the symbol η (s) to denote the power weights associated with the IVq-ROFN α s ij , where s = 1, 2, 3.
Step 3.4. Use the IVq-ROFPWA operator to determine the integrated decision matrix, which is listed in Table 4. Step 4. Compute the overall evaluation value for each alternative. Here we employ the symbol Sup j,h to denote the supports for α ij from α ih , where j, h = 1, 2, 3, 4 and j = h.
Step 4.2. Determine the overall support T α ij by using Equation (44)  Step 6. Determine the rank of alternatives and we can obtain A 2 A 4 A 1 A 3 . Hence, A 2 is the best online education platform.

Parameter Analysis
It is noted that the parameters in our proposed MAGDM method have significant impact on the decision results. Hence, it is worth investigating the effect of the parameters on the decision results.

The Impact of q
We first investigate the influence of the parameter q on the decision results. We use different values of q in our MAGDM method and present the final decision results in Table 5.
In addition, we also provide Figure 2 to better illustrate the decision-making results. Table 5. Ranking results by the proposed method when k = 2, λ 1 = 1, λ 2 = 1 for different values of the parameter q.

q
Scores S(α i )(i=1,2,3,4) Ranking Orders It is noted that when the variables k and remain unchanged, with the increase in the value of q, all the results show a trend of decreasing firstly and increasing afterwards. When the value of q is small, the result value decreases gradually with the increase in q. When the value of q is large, the result value gradually increases with the increase in q. In addition, different q values lead to different sorting results but 3 is always the worst alternative, whereas the optimal alternative keeps in changing. More specifically, 2 is the optimal alternative when = 1, 2 and 3. If = 4 or 10, the optimal alternative is 4 . Hence, the value of q has significant impact on the decision results. In practical MAGDM problems, decision-makers can choose a proper value of q according to evaluation values provided by decision-makers. The principle of determining the value of q is that its should take the smallest integer that satisfies the constraint + ≤ 1. For instance, if the at- It is noted that when the variables k and λ remain unchanged, with the increase in the value of q, all the results show a trend of decreasing firstly and increasing afterwards. When the value of q is small, the result value decreases gradually with the increase in q. When the value of q is large, the result value gradually increases with the increase in q. In addition, different q values lead to different sorting results but A 3 is always the worst alternative, whereas the optimal alternative keeps in changing. More specifically, A 2 is the optimal alternative when q = 1, 2 and 3. If q = 4 or 10, the optimal alternative is A 4 . Hence, the value of q has significant impact on the decision results. In practical MAGDM problems, decision-makers can choose a proper value of q according to evaluation values provided by decision-makers. The principle of determining the value of q is that its should take the smallest integer that satisfies the constraint b q + d q ≤ 1. For instance, if the attribute value is [(0.6, 0.9), (0.7, 0.8)], then the value can be taken as 0.9 4 + 0.8 4 = 1.0657 > 1 and 0.9 5 + 0.8 5 = 0.91817 < 1, hence, the value of q is 5.

The Impact of λ
In order to evaluate the impact of different λ on the final evaluation results, we keep k and q unchanged and use different λ in the IVq-ROFPWGMSM operator to determine the final decision results. The corresponding results are shown in Table 6. For the sake of easy description, we divide the researches into three groups to comprehensively analyze the influence of the parameters on the decision results. The decision-making results of each group are in Figures 3-5. Table 6. Ranking results by the proposed method when q = 3, k = 2 for different values of the parameter λ 1 and λ 2 . First, we assume 1 to be a constant ( 1 = 1) and investigate how the parameter 2 affects the decision results. In Figure 3, we discover that when the value 2 increases, the score values of 2 and 3 also increase. In addition, the score values of 1 decreases. The change in score values of 4 is special. With the increase in value 2 , the score value of 4 increases at the beginning and decreases soon afterwards. Moreover, different values of 2 lead to different score values and ranking orders of alternatives. Nevertheless, the optimal alternative is always 2 .  Second, we assume 2 to be a constant ( 2 = 1) and study the influence the parameter 1 on the decision results. It can be seen from Figure 4 that the increase in value 1 leads the increase of the score values of 1 , 2 , and 3 . The change in the score value of 4 is special. With the increase in the value 1 , the score value of 4 decreases first and increases afterwards. In addition, different values of 1 in the IVq-ROFPWGMSM operator result in different score values and ranking orders of alternatives. Moreover, the worst alternative is always 4 . Third, we investigate the influence of the parameters 1 and 2 on the final decision results simultaneously (we assume 1 = 2 ). As seen from Figure 5, with the increase in values 1 and 2 , some score value also increase and some other decrease. In addition, different values of parameters 1 and 2 lead to different score values and the worst optimal alternative is always 3 . From the above analysis, we can discover that the values of parameters 1 , 2 , … , have significant impact on the final decision results.  Second, we assume 2 to be a constant ( 2 = 1) and study the influence the parameter 1 on the decision results. It can be seen from Figure 4 that the increase in value 1 leads the increase of the score values of 1 , 2 , and 3 . The change in the score value of Third, we investigate the influence of the parameters 1 and 2 on the final decision results simultaneously (we assume 1 = 2 ). As seen from Figure 5, with the increase in values 1 and 2 , some score value also increase and some other decrease. In addition, different values of parameters 1 and 2 lead to different score values and the worst optimal alternative is always 3 . From the above analysis, we can discover that the values of parameters 1 , 2 , … , have significant impact on the final decision results. First, we assume λ 1 to be a constant (λ 1 = 1) and investigate how the parameter λ 2 affects the decision results. In Figure 3, we discover that when the value λ 2 increases, the score values of A 2 and A 3 also increase. In addition, the score values of A 1 decreases. The change in score values of A 4 is special. With the increase in value λ 2 , the score value of A 4 increases at the beginning and decreases soon afterwards. Moreover, different values of λ 2 lead to different score values and ranking orders of alternatives. Nevertheless, the optimal alternative is always A 2 .

2,3,4) Ranking Orders
Second, we assume λ 2 to be a constant (λ 2 = 1) and study the influence the parameter λ 1 on the decision results. It can be seen from Figure 4 that the increase in value λ 1 leads the increase of the score values of A 1 , A 2 , and A 3 . The change in the score value of A 4 is special. With the increase in the value λ 1 , the score value of A 4 decreases first and increases afterwards. In addition, different values of λ 1 in the IVq-ROFPWGMSM operator result in different score values and ranking orders of alternatives. Moreover, the worst alternative is always A 4 .
Third, we investigate the influence of the parameters λ 1 and λ 2 on the final decision results simultaneously (we assume λ 1 = λ 2 ). As seen from Figure 5, with the increase in values λ 1 and λ 2 , some score value also increase and some other decrease. In addition, different values of parameters λ 1 and λ 2 lead to different score values and the worst optimal alternative is always A 3 . From the above analysis, we can discover that the values of parameters λ 1 , λ 2 , . . . , λ k have significant impact on the final decision results.

The Impact of k
In the following, we continue to investigate the influence of the parameter k on the decision results. We use different values of k in our proposed method and present the decision results in Table 7 and Figure 6. Table 7. Ranking results by the proposed method when q = 3, λ 1 = λ 2 = · · · = λ k = 1 for different values of the parameter k.
k Score Values S(α i )(i=1,2,3,4) Ranking Orders Information 2021, 12, x FOR PEER REVIEW 24 of 29 5.2.3. The Impact of k In the following, we continue to investigate the influence of the parameter k on the decision results. We use different values of k in our proposed method and present the decision results in Table 7 and Figure 6.  Figure 6 shows the score values and ranking orders of alternatives with different values of k in the IVq-ROFPWGMSM operator, when = 3 and 1 = 2 = ⋯ = = 1. It is easy to find out that different score values and ranking orders of alternatives are obtained when we use different values of k in the IVq-ROFPWGMSM operator. Hence, it is necessary to determine a proper value of k. Basically, when = 1, our proposed method does not consider the interrelationship between attributes. When k = 2 or k = 3, then our method has the capability of capturing the interrelationship among any two or three attributes. In addition, when = 4, then our method takes the interrelationship among all the four attributes into account. Therefore, the proposed novel MAGDM method is effective for its ability to capture the interrelationship among attributes and it is suitable for handling realistic MAGDM problems, as in most decision-making problems attributes are usually interrelated. In MAGDM problems, decision-makers can determine a proper value of k according to actual needs.

Comparison Analysis
This subsection conducts comparison analysis. Through comparison analysis, the advantages and merits of our proposed method can be better demonstrated. The score values of alternatives by different k (q = 3, λ 1 = λ 2 = · · · = λ k = 1). Figure 6 shows the score values and ranking orders of alternatives with different values of k in the IVq-ROFPWGMSM operator, when q = 3 and λ 1 = λ 2 = · · · = λ k = 1. It is easy to find out that different score values and ranking orders of alternatives are obtained when we use different values of k in the IVq-ROFPWGMSM operator. Hence, it is necessary to determine a proper value of k. Basically, when k = 1, our proposed method does not consider the interrelationship between attributes. When k = 2 or k = 3, then our method has the capability of capturing the interrelationship among any two or three attributes. In addition, when k = 4, then our method takes the interrelationship among all the four attributes into account. Therefore, the proposed novel MAGDM method is effective for its ability to capture the interrelationship among attributes and it is suitable for handling realistic MAGDM problems, as in most decision-making problems attributes are usually interrelated. In MAGDM problems, decision-makers can determine a proper value of k according to actual needs.

Comparison Analysis
This subsection conducts comparison analysis. Through comparison analysis, the advantages and merits of our proposed method can be better demonstrated.

Compared with Power Bonferroni Mean Based Decision-Making Method
First of all, we compare our method with that based on the PBM operator. More specifically, we compare our proposed method with that developed by Liu and Li [32] based on the interval-valued intuitionistic fuzzy weighted power Bonferroni mean (IVIFWPBM) operator. We use both Liu and Li's [32] method and our proposed method to solve the air quality evaluation problem presented in [32]. (The original decision matrices can be found in Tables 1-3 in [32]). Score values and the final ranking orders derived by the two methods are presented in Table 8. Table 8. Decision making results of the air quality evaluation problem presented in [32] by using Liu and Li' [32] method and our proposed method.

Score Values Ranking Orders
Liu and Li's [32] method (x = y = 1 ) As seen from Table 8, the score values derived by the two methods are different whereas the ranking orders are the same. This also indicates the validity of our proposed method. Although the final ranking orders of alternatives are the same, our proposed method has obvious advantages over Liu and Li's [32] approach. First, the fuzzy set used in Liu and Li's [32] method is interval-valued intuitionistic fuzzy set (IVIFS), which is only a special case of IVq-ROFS (when q = 1). IVq-ROFS has laxer constraint, which means that our proposed method provides larger information space and more freedom for decisionmakers to express their evaluation values. Second, Liu and Li [32] use PBM operator to aggregate decision information. It can handle interrelationship between attributes which is same as our method. However, the PBM fails to handle the interrelationship among more than two attributes. In other words, if the interrelationship exists among multiple attributes, then Liu and Li's [32] method is powerless to handle such a problem. Contrary, our method is more useful and powerful due to its ability of considering the interrelationship among multiple attributes. Hence, our method is more powerful than that developed by Liu and Li [32].

Compared with Power Heronian Mean Based Decision-Making Method
Second, we compare our proposed method with that developed by Liu [34] based on the interval-valued intuitionistic fuzzy power weighted Heronian mean (IVIFPWHM) operator. We use Liu's [34] method and our novel MAGDM method to solve a management information system selection problem presented in reference [34] (The original decision matrices are listed in Tables 3-5 in reference [34]). The score values and ranking orders derived by the two methods are presented in Table 9. Table 9. Decision making results of the management information system selection problem presented in [34] by using Liu's [34] method and our proposed method.

Score Values Ranking Orders
Liu's [34] method (p = q = 2 ) S( z 1 ) = 0.0719, S(z 2 ) = 0.1090, S( z 3 ) = 0.2846, ( z 4 ) = 0.3022 Our proposed method (q = 1 ; k = λ 1 = λ 2 = 2) S( z 1 ) = 0.1278, S(z 2 ) = 0.1021, S( z 3 ) = 0.2784, S( z 4 ) = 0.2957 It can be found from Table 9 that the score values derived by the two methods are different. In addition, we notice that although the ranking orders produced by the two methods are slightly different, the optimal alternatives are the same. This result shows the effectiveness of our proposed method. However, our proposed method still has advantages over Liu's [34] MAGDM method. This is because that Liu's [34] method is based on IVIFSs, which have limited information space. Our method is based on IVq-ROFSs, which provide decision-makers enough space to express their evaluation information. In addition, Liu's [34] method can only consider the interrelationship between any two attributes, whereas our proposed method not only considers the interrelationship between any two attributes, but also has the capability of taking the interrelationship among multiple attributes into account. Therefore, our method is more powerful and flexible than Liu's [34] method.

Compared with Power Maclaurin Symmetric Mean Based Decision-Making Methods
Third, we compare our method with those based on PMSM operator. More specific, we compare our method with that developed by Liu et al. [57] based on the weighted interval-valued intuitionistic fuzzy power Maclaurin symmetric mean (WIVIFPMSM) and that put forward by Mu et al. [58] based on the weighted interval-valued Pythagorean fuzzy power Maclaurin symmetric mean (WIVPFPMSM) operator. We use these three methods to solve the partner country selection problem presented in [57] and list the decision results in Table 10 (The original decision matrices are listed in Tables 1-3 in reference [57]).  Our proposed method (k = 2 ; q = λ 1 = λ 2 = 1) S( α 1 ) = 0.3317, S( α 2 ) = 0.2147, S( a 3 ) = 0.1075, S( a 4 ) = 0.1304 As seen from Table 10, the three methods generate different score values, and ranking orders of alternatives are slightly different. In addition, the optimal alternatives derived by the three methods are the same. The advantages of our method over Liu et al.'s [57] method and Mu et al.' [58] method are two-folds: (1) Our method is based on IVq-ROFSs. The method of Liu et al. [57] is based on IVIFSs, and the method of Mu et al. [58] is based on interval-valued Pythagorean fuzzy sets (IVPFSs). It is noted that both IVIFS and IVPFS are special cases of IVq-ROFS. Compared with IVIFSs and IVPFSs, IVq-ROFSs provide decision-makers more freedom to express their evaluation values. (2) The methods of Liu et al. [57] and Mu et al. [58] are based on PMSM operator, which is a special case of our proposed PGMSM operator. In other word, the methods of Liu et al. [57] and Mu et al. [58] do not consider the importance of aggregated values. Hence, our proposed method is more powerful and flexible than those developed by Liu et al. [57] and Mu et al. [58].

Conclusions
This paper presents a novel MAGDM method under IVq-ROFS decision-making environment. In order to do this, we firstly put forward a novel compound AO, called PGMSM operator, by integrating PA with GMSM operators. Afterwards, we generalized the PGMSM operator into IVq-ROFSs and introduced a new series of AOs for IVq-ROFNs. Considering it is important to investigate the properties of an AO, we studied the desirable properties of the newly developed AOs. Based on these AOs, we introduced a novel MAGDM method under IVq-ROFSs decision-making situations. Finally, we applied the proposed method in an online education platform performance evaluation problem, which helps decision-makers choose the optimal platform. In future works, we plan to continue our work from two aspects. First, we will investigate more MAGDM methods under IVq-ROFSs situations and study their applications in real problems. Second, we will generalize the PGMSM operator into more decision-making environments, such as probabilistic dual hesitant Pythagorean fuzzy sets [59], q-rung orthopair linguistic sets [60], q-rung orthopair uncertain linguistic sets [61], linguistic q-rung orthopair fuzzy sets [62], etc. By so doing, more novel AOs and corresponding MAGDM methods can be obtained.