Interval Grey Hesitant Fuzzy Set and Its Applications in Decision-Making

. Te use of interval-valued hesitant fuzzy sets (IVHFS) can aid decision-makers in evaluating a variable using multiple interval numbers, making it a valuable tool for addressing decision-making problems. However, it fails to obtain information with greyness. Te grey fuzzy set (GFS) can improve this problem but studies on it have lost the advantages of IVHFS. In order to improve the accuracy of decision-making and obtain more reasonable results, it is important to enhance the description of real-life information. We combined IVHFS and GFS and defned a novel fuzzy set named interval grey hesitant fuzzy set (IGHFS), in which possible degrees of grey numbers are designed to indicate the upper and lower limits of the interval number. Meanwhile, its basic operational laws, score function, entropy method, and distance measures are proposed. And then, a multicriteria decision-making (MCDM) model IGHFS-TOPSIS is developed based on them. Finally, an example of MOOC platform selection issues for teaching courses illustrates the efectiveness and feasibility of the decision model under the IGHFS.


Introduction
Te limitations of the human cognitive level and subjective judgment, as well as the widespread ambiguity and uncertainty in real life, make it impossible for the exact number to completely cover the decision information [1]. Terefore, Zadeh [2] proposed fuzzy sets with a number between 0 and 1 to describe the afliation of the information to better describe these uncertainties. However, there are various reallife cases such as a membership degree of an element in the considered set cannot be accurately expressed by a single value [3,4]. Afterward, some other fuzzy sets were extended to overcome the drawback of fuzzy sets. A review of some of the most signifcant research advances in fuzzy sets is presented in Table 1. Tese theories have been widely used in the feld of multicriteria decision-making (MCDM) [20][21][22].
Among them, the hesitant fuzzy set (HFS), which was proposed by Torra [8], provides an expression of the membership degree of an element of a specifed set with multiple values. Tis can address the problem of inaccurate representation of fuzzy sets and is applied to indecisive environments where decision-makers feel inconsistent, discrepant situations [23,24]. Wan et al. have dedicated to study the application of HFSs in MCDM problems. For example, they proposed a multicriteria group decision-making (MCGDM) method where the information is represented by the hybrid of HFSs and other fuzzy sets [25]; they developed a hesitant fuzzy version of the preference ranking organization method for enrichment evaluations (PROMETHEE) and applied it to select green suppliers [26]; they also proposed a consistency index for hesitant fuzzy preference relations (HFPRs) and developed a new group decision-making method based on it [27]. HFSs have spawned many variants and gained extensive studies. For instance, Batool et al. [28] improved the Pythagorean probabilistic hesitant fuzzy set (PyPHFS) and proposed a decision-making model based on a new entropy measure, which performed excellently in identifying the most critical factors infuencing haze pollution. Zadeh [5] Type-2 fuzzy sets Type-2 FS A Type-2 FS can be represented by a fuzzy membership function, whose grade (or fuzzy grade) is a fuzzy set within the unit interval [0, 1], rather than a point within Bonissone [6] Linguistic fuzzy sets LFS Te LFSs are designed to handle qualitative settings where experts struggle to determine the appropriate linguistic term to assess the membership of an element Atanassov [7] Interval-valued intuitionistic fuzzy sets IVIFS In the IVIFSs, both the membership degree and nonmembership degree are included, and their sum cannot exceed 1 Torra [8] Hesitant fuzzy sets HFS Te HFSs address situations where experts struggle to determine a numerical value to defne the membership of an element in a quantitative setting Yager [9] Pythagorean fuzzy sets PFS Te PFSs handle situations where the sum of membership and nonmembership degrees for a given attribute exceeds 1 Chen et al. [10] Interval-valued hesitant fuzzy sets IVHFS Te IVHFSs allow for membership within a certain interval, which is an extension of hesitant fuzzy sets Cuong et al. [11] Picture fuzzy set PTFS Te PTFSs are characterized by three degrees of membership: positive, neutral, and negative. Te total of these memberships should not exceed 1 Pang et al. [12] Probabilistic linguistic term sets PLTS Te PLTSs enable experts to express their preference for one linguistic term over another Ashraf et al. [13] Spherical linguistic fuzzy sets SLFS Te SLFS consist of linguistic terms, positive membership degree, neutral membership degree, and negative membership degree Lin et al. [14] Linguistic Pythagorean fuzzy sets LPFS Te LPFSs allow experts to express their language preferences using Pythagorean fuzzy numbers Ashraf et al. [15,16] Spherical fuzzy sets SFS Te SFSs are characterized by a membership degree, a nonmembership degree, and a hesitancy degree. Te squared sum of these degrees must be equal to or less than 1 Ali et al. [17] Complex T-spherical fuzzy sets CTSFS Te CTSFSs incorporate truth, abstinence, and falsity grades within a unit disc on a complex plane. Te sum of the q-power of their real and imaginary parts must not exceed a unit interval Akram et al. [18] Complex spherical fuzzy sets CSFS Te CSFSs include three polar coordinates (truth, abstinence, and falsity) within a unit disc on a complex plane. Te sum of squares of their real parts should not exceed a unit interval Mahmood and Ali [19] Fuzzy superior Mandelbrot sets FSMS Te FSMSs are designed to handle awkward and inconsistent information in real-life problems 2 Journal of Mathematics Te notion of HFS is limited in dealing with complex MCDM problems as it can be difcult to express the membership degrees of an object with one or more exact numbers. To address this issue, Chen et al. [10] introduced the concept of an interval-valued hesitant fuzzy set (IVHFS), which represents membership as an interval value instead of an exact number. Tis approach has gained attention in recent years. Zeng et al. [29] defned some basic operations and score functions for a weighted IVHFS. Farhadinia et al. [30] proposed an approach for deriving modifed correlation coefcients of HFSs and extended it to IVHFSs. Ma [31] studied an IVHFS Maclaurin symmetric mean information aggregation method. Bharati [32] developed a new IVHFS-based optimization method for production planning. Finally, Rashid and Sindhu [33], Zhang and Gao [34], and Luo [35] have utilized this approach to solve problems related to MCDM.
Tere are certain real-life problems where the properties of objects are not black or white but may lie between them [36], that is, the information has greyness in solving the MCDM problem, and hence, the IVHFS fails to achieve reasonable results. Chen [37] proposes the grey fuzzy set (GFS), which has provided successful results in dealing with those problems. It integrates the advantages of grey numbers and fuzzy sets and can be regarded as an extension of both grey numbers and fuzzy sets. Te research of GFS has attracted great attention, such as Jiang et al. [38] extend it and propose the intuitionistic grey number set (IGNS), which combines intuitionistic fuzzy sets with grey numbers, and then constructs a grey absolute correlation closeness decision model based on the score function value of the IGNS [39]. On the basis of this, Liu et al. [40] and Li and Liu [41] introduce the concept of interval intuitionistic grey number (IIGN), which integrates the intuitionistic fuzzy method and the kernel and greyness degree method of grey numbers to solve the problem of nonideal decision precision. Li and Liu [36] propose grey hesitant fuzzy sets (GHFS) as a solution to MCDM problems involving partially certain and partially uncertain information. Te approach combines hesitant fuzzy sets and grey numbers. Liu et al. [42] puts forward a grey correlation decision-making method based on GHFS. Te method adopts the concepts of "kernel" and "grey scale" in grey systems, replaces the traditional data supplementation method of predetermined risk preferences of decision-makers with the grey system "kernel and grey scale" method, and then extends the "kernel and grey scale" algorithm to the grey correlation degree decision-making. It improves credibility without changing the information domain.
However, these methods cannot handle some complex decision problems in which uncertain, ambiguous, fexible, and grey information may appear. For example, certain criteria evaluation scores may be arbitrary real numbers and cannot be represented by membership degrees between [0,1]. Te limitations of existing decision-making methods have led to the proposal of interval grey hesitant fuzzy sets (IGHFSs) as a generalization of IVHFS and GFS. Unlike IVHFS, IGHFS refers to multiple uncertain numbers that take values within a certain interval or a general set of numbers, while IVHFS refers to membership degrees within multiple [0, 1] intervals. But in reality, hesitant fuzzy decision information often only knows the approximate range but does not know its exact value, rather than any number within a given range. Terefore, IGHFS has practical signifcance for dealing with many realistic problems. Te IGHFS-TOPSIS model is then developed to address those complex MCDM problems.
Te main contributions of the presented IGHFS-TOPSIS model are provided as follows: (1) Presenting IGHFS and its basic operations, possible degrees formula, and score function (2) Proposing an interval grey hesitant fuzzy entropy method (IGHFEM) based on the score function to get criteria weights (3) Providing distance measure for two interval grey hesitant fuzzy numbers (IGHFN) and justifying it (4) Introducing IGHFS-TOPSIS based on the Euclidean distance measure of IGHFNs and the novel IGHFEM to calculate the relative closeness coefcient of each candidate for prioritizing them Te remaining sections of this paper are structured as follows: in Section 2, we provide a review of fundamental concepts related to grey numbers and interval hesitant fuzzy sets and then introduce interval grey hesitant fuzzy sets and their operation properties, possible degrees formula, and score function. Section 3 proposes the interval grey hesitant fuzzy entropy. Distance measure for IGHFNs is studied in Section 4. Section 5 shows the detailed steps of the IGHFS-TOPSIS model. A case about a selection of MOOC platforms illustrates the feasibility and applicability of the proposed IGHFS-TOPSIS in Section 6. In particular, we conducted sensitivity analysis and comparative analysis to verify the validity of the method. And concluding remarks are made in Section 7. Table 2 shows the meaning of the main symbols.

Preliminaries
Defnition 1 (see [8,43]). Set X as where where c � [c L , c U ] is an interval number, and c L � infc and c U � supc indicate the upper bound and lower bound of c, respectively.

Journal of Mathematics 3
is called a general grey number, where ⊗ i is an interval grey number and where a i and a i are the lower and upper limits of the information separately in formula (4) and meet two conditions: a i , a i ∈ R and a i− 1 ≤ a i ≤ a i ≤ a i+1 . We give a new expression of the lower and upper bounds of the range of values of the grey number g ± for convenience, that is, g − � inf a i ∈ g ± and g + � sup a i ∈ g ± . It is worth noting that, g ± degenerates to an interval grey number when i � 1, and further degenerates to a real number if a i � a i under this condition.
Defnition 3 (see [45]). Let the "possibility" or probability that the grey number g ± is a certain true value be denoted as P( ⊗ i ). Ten, a continuous probability function about the grey number can be formed within the range of grey values, which is a continuous function and needs to satisfy the condition of being monotonically increasing before becoming monotonically decreasing. Tese monotonically increasing functions or monotonically decreasing functions are usually simplifed as linear equations in practice, so their typical grey number probability functions are given by the following equation: and are shown in Figure 1.
Defnition 4. Set X as a fxed set.
is called the interval grey hesitant fussy set (IGHFS), where where P − and P

Defnition 5.
Let gc a � [P − a , P + a ] and gc b � [P − b , P + b ] be two interval numbers, and λ > 0, the basic operations among them are expressed as follows: , and when λ � 0, λgc a � 0 are three IGHFNs, the basic operations among them are expressed as follows: Defnition 7 (see [45]). Set gc a and gc b as two interval numbers, and suppose that l gc a � P + a − P − a and l gc b the possible degrees formula of P(gc a > gc b ) should be as follows: Defnition 8. Suppose that the desirability function of [P − , P + ] was E(I) � 1/2(P − + P + ), the score function of an IGHFN igh should be as follows: where #igh is the number of interval numbers, and S(igh) is an interval number in [0, 1]. Te value of the score function is the main impact on numeric comparisons, where there is a higher value of S(igh), there is a larger IGHFN.

Interval Grey Hesitant Fuzzy Entropy Method
It is common for the weight information to be completely unknown in decision-making problems. In this case, the entropy measure can be used to determine the criterion weights. However, it is crucial to establish an appropriate entropy method. In this section, we will introduce a new entropy method in the IGHS environment to address this issue. Te entropy refects the average uncertainty of the sources [46][47][48]. In the decision system, smaller information entropy represents that the options are less fuzzy on an attribute value, refecting that the attribute provides a larger amount of information, that is, the attribute plays a greater role in the ranking of the options, and thus the weight of the attribute should be larger [49][50][51]. Terefore, the information entropy on each criterion can be used to determine the size of the corresponding criteria weights.
Defnition 9 (see [52]). If a discrete source can be expressed as follows: F: where p i is the prior probability of the random variable F, which takes values in the range [0, 1], and the sum of all p i is 1. Te average uncertainty of this discrete source is defned as follows: where the constant k is determined by the units and normally, k � 1. Te base of the logarithmic function in the abovementioned equation is usually 2, 10, or e. Te method used to derive the criteria weights in this study is based on the average information entropy of each criterion calculated from the score function of interval grey Figure 1: Te probability function of a grey number.
Journal of Mathematics 5 hesitant fuzzy numbers. Tis approach is derived from previous studies [53] on interval intuitionistic fuzzy numbers and is simple and easy to implement. Te score function refects the degree of fuzziness, so the average information entropy of each criterion can be calculated from the score function, and then the relevant criteria weights can be obtained [54,55].

Defnition 10.
Suppose H could be called interval grey hesitant fuzzy entropy. Inspired by Wu's entropy measure for interval hesitant fuzzy elements, the entropy measure of an IGHES is defned as follows: where S(igh) is the normalized value of the scoring function S(igh): and m is the number of attributes of the alternative.

Distance Measure for IGHFNs
Defnition 11 (see [56,57] Considering the parameter features in IGHFs, a set of distance measures of IGHFs is put forward.
as two IGHFNs, the normalized Euclidean distance between them can be calculated as follows: where P − 1 (k)) and P + 1 (k)) are the kth largest number of p 1 or p 2 , and l is the number of intervals in p 1 or p 2 .

TOPSIS Method with IGHFSs
We propose a TOPSIS model for IGHFS to solve MCDM problems in interval grey hesitant fuzzy environments. Tis model is called IGHFS-TOPSIS, and Figure 2 shows its fowchart. Te specifc steps of the model are as follows: Let A � A 1 , A 2 , . . . , A m be the set of alternatives, and x � x 1 , x 2 , . . . , x n � X 1 ∪ X 2 be the set of all attributes, where X 1 is the set of beneft attributes, X 2 is the set of cost attributes with the condition X 1 ∩ X 2 � ∅, and l be the number of interval numbers in each IGHFN.
Step 1. Te grey number possibility function of each criterion is constructed based on the specifc problem context.
Step 2. Te fuzzy decision matrix M is constructed as follows: where a ij is a grey number and a ij � ⋃ l k�1 ⊗ k .
Step 3. Using the probability function, the decision information expressed by general grey numbers is converted into an IGHFN representation set, and the IGHFN matrix M is shown as follows: where a ij is an IGHFN, and a ij � ⋃ l k�1 [P(⊗ k ) min , P(⊗ k ) max ].
Step 4. Compute the entropy H S (f j ) of various criteria in IGHFs using equation (13); Step 5. Calculate the weight w j of criterion x j using the following equation: Step 6. Generating a new matrix M by normalizing the IGHFN matrix M. Tis involves converting the cost-based criteria data into beneft-based criteria data and reordering the decision matrix elements and adding new ones. Te resulting normalized decision matrix M is composed of normalized IGHFNs a.
Step 7. According to the score function, the optimal ideal scheme A + and the worst ideal scheme A − of each criterion are obtained.
Step 8. Too calculate the distances between each scheme and both the optimal and worst ideal solutions by using equations (21) and (22). It should be noted that the distance here is a weighted distance because the importance of each criterion is considered to be diferent. Specifcally, the distance from each solution to the optimal ideal solution is denoted as d + and the distance from each solution to the worst ideal solution is denoted as d − .

Pre-work
Expert opinion

Expert opinion
Step 1 Step 2 Step 3 Calculate normalized value of the scoring function

Caculate entropy of each critecia
Determine weights of critecia Step 4 Step 6 Step 7 Step 8 Step 9 Step 5 Define criteria for assessment Step 9. To determine the relative closeness of each scheme to the rational solution by employing equation (23). Higher relative proximity values indicate better solutions.

Case Analysis.
Te COVID-19 pandemic has created a huge challenge for basic and higher education in China [58,59]. Chinese schools have had to stop teaching in the classroom and switch to online teaching [60]. Te boom of online education has promoted the emergence of various online teaching platforms and tools, which provide an online learning environment for students to learn "everywhere and all the time" [61][62][63]. At the same time, a large number of online education courses are being created [64,65]. An education teacher at a university in Wuhan recorded a series of education courses but did not know which platform to choose to upload them. In this study, to address this issue, we researched MOOC platforms in China and decided to choose one between A1: "XuetangX," A2: "China University MOOC," A3: "Tencent Classroom," and A4: "Rain Classroom." Based on the relevant evaluation standards, and taking into account the actual situation of universities and the epidemic situation, the expert group determined four key attributes that afect the selection of online teaching platforms. All mathematical calculations in this paper were performed on MATLAB R2020b. Te attributes used in this study are: "ease of use," "maturity," "specialty," and "popularity." All of these attributes are in the beneft criteria set and take values between 0 and 10. Te specifc decision-making process is as follows: Step 1: Tree experts made possibility functions of the criteria under three round table meetings. Te fnal result is all possibility functions of these criteria are given in f 1 (x).
Step 2: Tree experts evaluated the alternatives under the criteria. Te opinion of each expert is given in Table 3.
Step 3: Te IGHF matrix given in Table 4 is transformed from the grey matrix shown in Table 3 according to f 1 (x).
Step 4: Te entropy is calculated by using equation (23), and the results are shown in Table 5.
Step 5: Te weight of every attribute is calculated by using equation (24), and the results are also shown in Table 5.
Step 6: According to the risk-averse principle, the minimum number of intervals is added so that the number of intervals corresponding to each IGFHN is equal. Te results of data processing are shown in Table 6.
Step 7: Te IGHFN with the largest score function in each attribute is the ideal solution and the IGHFN with the smallest score function is the worst ideal solution.
Te ideal solution set of the matrix of Table 6 is as follows: Te worst ideal solution is as follows: Step 8: Apply equation (15) to calculate the distance between each MOOC platform's t IGHFS and both the optimal and worst ideal solution sets. Te results are presented in Table 7.
Step 9: Compute the relative closeness of each MOOC platform to the positive ideal solution using equation (20). Te results are presented in Table 7.
Sorting the fnal scores c(A i ) of alternatives, our optimum selection to the MOOC platform is in the order of XuetangX ≫ China University MOOC ≫ Rain Classroom ≫ Tencent Classroom.

Sensitivity Analysis.
Te sensitivity analysis aims to investigate the impact of changes in criteria weights on the fnal ranking of MOOC platforms. By exchanging the weights of each criterion, the relative closeness values of each alternative are recalculated, which may lead to changes in the priority order of the alternatives. Te results of the sensitivity analysis can provide insights into the relative importance of each criterion and how changes in their weights afect the decisionmaking process. Ultimately, the goal is to determine the robustness or sensitivity of the IGHFS-TOPSIS method and validate its efectiveness for MOOC platform selection.
Tis study involves six experiments that investigate the efect of criteria weights on the ranking order. One of the criteria weights obtained from the IGHFEM is exchanged with another one to conduct the six experiments. Ten, recalculate the relative closeness c of each alternative in these six experimental environments. Te relative closeness c of each alternative is then recalculated using the exchanged weight set of the criteria. Te exchanged criteria weights change the distances of each alternative from the optimal ideal solution (d + ) and the worst ideal solution (d − ), which means that the relative closeness c of each alternative is expected to change in each experiment. By substituting each criterion weight with a diferent weight, the sensitivity analysis can be conducted to examine the impact of each criterion on the results. Te relative closeness obtained from each experiment has a diferent name, such as C12 or C23,   which indicates which criteria weights were exchanged. Te results of the sensitivity analysis are shown in Figures 3 and  4, as well as Table 8. Figure 3 presents the relative closeness values obtained from six experiments in a radar chart, while their corresponding value ranges are presented in Figure 4. Te clearer results and fnal ranking for each experiment are shown in Table 8. Based on these fgures and the table, it is apparent that some variances occur in some experiments.
Changes in the criteria weights can result in changes in the fnal ranking. For instance, exchanging the weights of A1 and A2, as well as those of A3 and A2, leads to slight changes in the fnal ranking, whereas severe changes occur when the weights of A1 and A2, or those of A3 and A2, are exchanged. Tis means that A1 stands as the frst alternative among the three experiments only when their weights are not exchanged, while A2 has the highest value among the remaining experiments. Based on this, it can be concluded that the decision process is sensitive to changes in the criteria weights.

Comparative Analysis.
To demonstrate the validity and robustness of the IGHFEM, a comparative analysis is performed by comparing the criteria weights obtained using the IGHFEM with those obtained using another method called the interval hesitant fuzzy entropy method (IHFEM), as described in [66]. Tis analysis is necessary based on the results of the sensitivity analysis. Te initial matrix of the IHFEM consists of interval hesitant fuzzy elements (IHFE), which are several scoring intervals between the range of [0.0, 1.0]. It can weaken the uncertainty of experts' scoring when they score the indexes. However, interval hesitant fuzzy elements may not be able to capture the greyness of information in real-world problems. To address this limitation, IGHFNs have been proposed as a more general form that can handle greyness, vagueness, and uncertainty in real-life issues. On the other hand, unlike IHFEM, IGHFEM calculates the criteria weights from the information entropy perspective. Terefore, a comparison was made between the criterion weight calculation of IHFEM and IGHFEM. It is noted that since the initial matrix of IHFEM consists of IHFEs, the data in Table 6 are considered as the initial matrix. Figure 5 provides a visual comparison between the IHFEM and IGHFEM. Each criterion weight derived from the two methods is shown in this histogram. Te weight ranking result of IGHFEM is popularity ≫ specialty ≫ ease of use ≫ maturity while IHFEM is popularity ≫ ease of use ≫ specialty ≫ maturity. Te ranking results of both methods show that popularity has the largest weight and maturity has the smallest weight. Furthermore, it is not difcult to see that even with diferent methods of computing entropy, the weight values of each criterion are quite similar.
Te correlation analysis further validates the rationality of IGHFEM. In this analysis, the Pearson correlation coefcient is used to measure the correlation between IHFEM and IGHFEM, and the results are shown in Table 9, which indicates a strong positive correlation between the proposed method and IHFEM. Tis further demonstrates the efectiveness of using the weight calculated by IGHFEM.
Another comparative analysis is performed to demonstrate the scientifc efectiveness of the IGHFS-TOPSIS method. Tis involves comparing the results obtained using IGHFS-TOPSIS with those obtained using other methods, such as IVHF-TOPSIS and GHFS-TOPSIS. Te former was proposed by [67], while the latter was proposed by [41]. Te initial data matrix of IVHF-TOPSIS consists of IHFE. Te initial dataset of GHFS-TOPSIS is a grey hesitant fuzzy element (GHFE) matrix, which is composed of several grey  numbers. IGHFN can be seen as an extension of IHFE or GHFS. Te initial data IGHFNs of IGHFS-TOPSIS can be considered as an extension of IHFE or GHFE. Terefore, this analysis compares the rankings of MOOC platforms using three diferent approaches: IVHF-TOPSIS, GHFS-TOPSIS, and IGHFS-TOPSIS. In order to eliminate the sensitivity of indicator weights to the ranking results, the weights of the criteria in these three methods are the data in Table 5. Table 10 shows the results obtained by the three methods, including the relative closeness degree c(A i ) of each MOOC platform to the rational solution and the fnal ranking of the platforms. Te fnal new score of each alternative obtained by these three methods and the fnal rankings of those alternatives are shown in Table 10. For more intuitive expressions of the similarities and diferences in these methods, the fnal rankings are displayed in Figure 6. According to these results, in all methods, XuetangX is

12
Journal of Mathematics always top-ranked, followed by China University MOOC. It is worth noting that the GHFS-TOPSIS method is consistent with the ranking of this paper's method. Te results of the IVHF-TOPSIS method only difer in the third and fourth places with this paper's ranking. One reason why IVHF-TOPSIS does not agree well with the ranking results of our method is that the two methods difer in defning their respective positive and negative ideal solutions for fuzzy sets, resulting in not exactly the same ordering. In conclusion, the results of the comparison show that the method IGHFS-TOPSIS proposed in this paper is reasonable.

Conclusion
In this paper, we have presented a new concept of fuzzy sets called interval grey hesitant fuzzy sets. One of the key advantages of this type of fuzzy set is its ability to expand the scope of real-world information, thereby overcoming limitations associated with traditional studies that struggle with handling grey information or situations where people are hesitant to provide interval-valued assessments. Another is to ofer a convenient frame for one to evaluate an attribute that consists of some interval grey numbers instead of interval numbers between 0 and 1. Obviously, two defciencies can be solved. One is those criteria weights can afect the fnal sorting results. Another is that the method relies on more behaviors of experts, which makes the method more subjective. Our contributions to the manuscript are as follows: frstly, some basic operational laws, possible degree formulas, and score functions for IGHFS are defned for subsequent calculations. Ten, an interval grey hesitant fuzzy entropy has been developed for calculating the weights of the criteria. In addition, we instructed the distance measure for IGHFNs and proved its validity. Ultimately, an approach called IGHFS-TOPSIS has been constructed, which is designed to address MCDM problems involving weighted interval grey hesitant fuzzy information. An example related to the selection of MOOC platforms validates the rationality and practicality of our decision-making method.
In future works, we intend to further optimize the interval grey hesitant fuzzy entropy method by using machine learning, particle swarm optimization, and other algorithms. Extending grey numbers to other types of fuzzy sets will be an interesting work. Moreover, we plan to use grey fuzzy set theory to explore other types of fuzzy sets, such as picture fuzzy sets and spherical fuzzy sets. Tis will allow us to derive novel information measures.

Data Availability
All data produced or analyzed during this research are available within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.