HYBRIDIZATIONS OF ARCHIMEDEAN COPULA AND GENERALIZED MSM OPERATORS AND THEIR APPLICATIONS IN INTERACTIVE DECISION-MAKING WITH Q-RUNG PROBABILISTIC DUAL HESITANT FUZZY ENVIRONMENT

: The q-rung probabilistic dual hesitant fuzzy sets (qRPDHFSs), which outperform dual hesitant fuzzy sets, probabilistic dual hesitant fuzzy sets, and probabilistic dual hesitant Pythagorean fuzzy sets, are used in this research to develop an interactive group decision-making approach. We first suggest the Archimedean Copula-based operations on q-rung probabilistic dual hesitant fuzzy (qRPDHF) components and investigate their key features before constructing the approach. We then create some new aggregation operators (AOs) in light of these operations, including the qRPDHF generalized Maclaurin symmetric mean (MSM) operator, qRPDHF geometric generalized MSM operator, qRPDHF weighted generalized MSM operator


Introduction
Finding the best option(s) from a pool of readily available possibilities based on several features, both quantitative and qualitative, is the main goal of the multiple criteria decision-making (MCDM) technique.Uncovering the numerical values of qualities can occasionally be a difficult process for an expert.Considering recent scientific and technological developments, uncertainty now dominates decisionmaking (DM) analyses.Zadeh (1965) proposed the idea of fuzzy sets (FSs) to deal with the data's ambiguity.The hesitant fuzzy set (HFS) (Torra, 2010) is an extension of FSs that permits membership degrees (MDs) to assume a limited number of likely entries as opposed to only one.By including hesitant non-membership degrees (NMDs) together with MDs in the inquiry, Zhu et al. (2012) created the concept of dual HFSs (DHFSs) and named its fundamental component dual hesitant fuzzy elements (DHFEs).The case about the presence of the probabilities of the components in the DHFSs and HFSs is not resolved even if DHFSs and HFSs are successfully used to many DM situations.Let's look at an illustration to better comprehend this: Consider a professional who indicates his propensity for anything negative as a hesitant fuzzy element (HFE) of 0.4, 0.6, or 0.7.He mentioned throughout the tests that the comfort level associated to 0.6 is the greatest when compared to others, while the comfort level related to 0.7 is the worst.Thus, under such a situation, the HFE {0.4,0.6, 0.7} isn't reasonable to depict the data.Additionally, think about a rating of an expert to assess the quality of an item as the DHFE <{0.4,0.6, 0.7}, {0.2, 0.3, 0.4}>.During the appraisals, the expert accepts that his solace level toward the item evaluating 0.4, 0.7 is twofold than 0.6 in MDs, while triple toward the 0.4 in the NMDs concerning the others.Hence, the DHFE <{0.4,0.6, 0.7}, {0.2, 0.3, 0.4}> isn't reasonable to portray the information.To handle such issues, the idea of the probabilistic HFS (PHFS) and probabilistic HFEs (PHFEs) were presented by Xu and Zhou (2017) and were extended to probabilistic DHFS (PDHFS) and probabilistic DHFEs (PDHFEs) respectively by Hao et al. (2017).The PDHFE gives a more exact depiction compared to PHFE, HFE, and DHFE and can effectively portray the data in the above-expressed example.The TODIM approach with PDHFSs was utilized by Ren et al. (2017) for enterprise strategy evaluation.New correlation coefficients were put up by Garg and Kaur (2018) and used to solve problems with PDHFSs and multi-criteria decision-making (MCDM).A strategy selection problem was handled by Ren et al. (2019) utilizing an integrated VIKOR and AHP technique using PDHF information.PDHFSs is constrained in that the total grade for both membership and non-membership should not be more than 1.Ji et al. (2021) developed the idea of probabilistic dual-hesitant.Pythagorean fuzzy sets (PDHPFSs), which adhere to the requirement that the sum of the squares of MD and NMD should not be more than 1.The q-rung probabilistic dual hesitant fuzzy sets (qRPDHFSs) sets, introduced by Li et al. (2020) hold the constraint that the addition of q th power of the MD and the NMD must accomplish the value in [0,1].The qRPDHFS reduces to PDHFS when q =1 and PDHPFS when q =2, which means that the qRPDHFSs are extended versions of PDHFSs and PDHPFSs.Thus qRPDHFSs are more powerful than PDHFSs and PDHPFSs.
In the past few decades, interactive technology has made a series of developments.Sakawa took the lead in considering the interaction between group decision makers (experts) to resolve inconsistencies (Sakawa & Yano, 1985).Some studies have shown that interactive DM gradually and dynamically learns about the personal preference structure under the continuous communication and interaction between experts, and finally obtains the most satisfactory results (Bashiri & Badri, 2011;Reverberi & Talamo, 1999;Shi & Xia, 1997).Watson et al. (1991) believed that the interactive mode within the group is a key variable that affects the rationality of the DM results.Xu and Chen (2007) believe that experts modify their preference information through interaction during the DM process, which can make the decision result more reasonable, and they use a hybrid weighted average operator to aggregate decision information in a fuzzy environment.Cheng et al. (2018) considered the consistency of evaluation results and attribute weights through the interaction between venture capital providers and between venture capital providers and entrepreneurs.Gou el al. (2019) introduced a consistency index to judge the linguistic preference relation of acceptable consistency.Thus, in the literature, there is a significant gap regarding the consideration of interactive DM problems with q-rung probabilistic dual hesitant fuzzy (qRPDHF) information.
In any MCDM method, the primary concern is how to fuse the assessment data of various criteria for alternatives, and afterward to get the fittest one.Two different ways are there to pick the most suitable alternative.One is the conventional assessment tools, and the other is the information aggregation operators (AOs).The conventional assessment tools can only generate the preference order of alternatives, while information AOs not only generate the preference order of alternatives effectively yet additionally provides comprehensive assessment value of each alternative.As a result, the information AOs can tackle MCDM issues in a more feasible way compared to the conventional assessment tools.Recently, the study of PDHF aggregation operators and their extensions has drawn significant attention to researchers.The PDHF weighted averaging (PDHFWA) operator was created by Hao et al. (2017) and used for risk assessment.To address the problem of decisionmaking, Garg and Kaur (2018) designed various PDHFS-based Einstein AOs with certain information metrics.The PDHF fuzzy power weighted Hamy mean (PDHPFPWHM) operator was created by Ji et. al. (2021) and utilised to address the MCDM issues.For resolving MCDM issues, Li et al. (2020) suggested the q-rung PDHF power weighted Muirhead mean (q-RPDHFPWMM) operator.

a) Objectives of research
Real-world multi-criteria group decision-making (MCGDM) situations allow for the observation of the relationships between a variety of factors.In this situation, it is crucial to consider how the various criteria interact to arrive at a more logical conclusion.To date, the PDHF Einstein weighted averaging operator (Garg & Kaur, 2018) and the PDHF weighted averaging AO (Hao et al., 2017) have been used to average data incorporating PDHF information.Additionally, the PDHPFPWHM operator can record the relationship between characteristics.However, they miss out on the linkages between several input criteria.The q-RPDHFPWMM operator (Li et al., 2020) can manage multi-input dependence between criteria, but it is unable to handle probability distributions, leading to information loss during the aggregation phase.The q-RPDHFPWMM operator (Li et al., 2020) was also solely used to address MCDM issues with variable weightings of criteria.To prevent information loss during aggregation, an AO that addresses the link between multiple input attributes and probability distributions in the context of an MCGDM setup with qRPDHF information is required.

b) Research gaps and motivations
A PDHFS reduces to a PHFS if the NMDs alongside their associated probabilities are ignored.Also, a PDHFS turns into a DHFS if the hesitant MDs and NMDs are equally probable.Hence, PDHFSs are generalized versions of the PHFSs and DHFSs.But, PDHFSs cannot fully express the real decision information because sum of MD and NMD must not exceed 1. Extending this restriction to their square sums, we get PDHPFSs.But PDHFSs and PDHPFSs are special cases of qRPDHFSs, since they require that the sum of the q th power of MD and q th power of NMD should not surpass 1.Thus, qRPDHFSs can express the criteria values with higher flexibility.Since Hao et al.'s method (2017) and Garg and Kaur's method (2018) are based on PDHFSs and Ji et's method (2021) is based on PDHPFSs, so Li et al's (Li et al., 2020)  Example 1: Suppose an Institute is interested in choosing an OSS-LMS package among three OSS-LMS packages, namely-Sakai (A1), eFront (A2), and Moodle (A3).These OSS-LMSs are to be assessed by three experts (E1, E2 and E3) depending on three attributes.The details of these options and attributes are presented in Table 1 and Table 2, respectively.To choose the best alternative among these five OSS-LMS, a team is formed involving three experts.Their initial evaluation results are presented in terms of PDHFEs.
Table 1.Description of the OSS-LMS alternatives

Sakai (A1)
Sakai is an OSS-LMS scheme that offers a flexible and versatile context for teaching, training, analysis, and other associations.Sakai constantly grows based on the requirements of the faculty, learners, and corporations (https://sakaiproject.org/).Functionality is the strength of the software to accommodate functions that match the user's specifications when utilizing the software under conditions.Functionality is utilized to estimate the level in which an LMS meets the functional specifications of an establishment.

Reliability (C2)
Reliability is the capacity of the software package to work consistently without falling under specific situations.Reliability is practiced evaluating the level of fault tolerance of the software packages.

Security and privacy (C3)
Security and privacy standards are required to authenticate the efficacy of a structure to safeguard private data and safeguard information from attacks and exposure on a user's computer.
The initial assessment matrices are: are given in the form of Table 3. Suppose that the consensus coefficient among experts should be above 0.97 (i.e; * 0.97
Example 2: (Li et al., 2020) "After preliminary analysis, four possible investment alternatives are considered; they are denoted by A1, A2, A3, A4.In this paper, we consider three commonly used attributes in investment evaluation decision: (1) G1 the quality of product and service; (2) G2 social and environmental impacts; (3) G3 economic benefits.The weight vector of the attributes is w = (0.3, 0.2, 0.5) T .The experts need to assess the four alternatives' performance from three aspects respectively".The initial assessment matrix is presented in Table 4".(Li et al., 2020) can produce the ranking order for any MCDM problem, but it has certain drawbacks mentioned below: 1.In Li et al's method (Li et al., 2020), experts' assessments were carried out separately, making it challenging to draw a consistent conclusion.Specifically, it fails to depict the ambiguity of articulating information with the collaboration among experts.So, the assessment outcomes get distorted.2. Sometimes, the information related to criteria weights is not known or partially unknown due to lack of data, and the expert's limited proficiency.These criteria weights can be determined by experts' personnel inclinations.In the DM technique (Li et al., 2020), due to arbitrary assignment of weights of criteria' for the final aggregation procedure, the preference ranking obtained gets affected.Moreover, the method (Li et al., 2020) leads to information loss as it doesn't not consider any information measure.3. The q-RPDHFPWMM operator (Li et al., 2020) can capture the dependency among multiple criteria, but it cannot deal with probability distributions during the aggregation process.As a result, the ranking order obtained is not reasonable.

c) Contributions
The following contributions are included in this paper: 1. We have put out a paradigm for aggregation based on interactions and qRPDHF data.The consistency harmonious weight index (CHWI) and expert assessment similarity ideas have been used in this framework to examine the expert's subjective and objective weights.The final expert weights are then calculated by considering a combination of these factors.Finally, until the required consensus value is obtained, the coefficient of consensus is computed again with expert participation.2. The cross-entropy measure considers the weight of each criterion to address the amount of unclear information.Taking use of this, an optimization model is developed in this work to determine the weights of the criterion.3. Copulas are functions that link several marginal distributions, which can indicate the correlation among variables and prevent information loss during aggregation, according to several scholars (Bacigal et al., 2015;Beliakov et al., 2007;Grabisch et al., 2011;Han et al., 2020;Nather, 2010;Nelsen, 2013;Tao et al., 2018;Poswal et al., 2022).The generalised Maclaurin Symmetric mean (GMSM) (Wang et al., 2018) operator generalises the Bonferroni mean, Hamy mean, and Maclaurin symmetric mean operators by changing the parameter values.The GMSM operator considers the connections between several criteria.Therefore, the GMSM operator has been expanded to include qRPDHF-GMSM operators with their weighted forms employing Archimedean Copula operations on qRPDHF elements (qRPDHFEs).
In section 2, we concisely discuss some essential concepts namely qRPDHFs, GMSM operator and Archimedean Copula.Section 3 investigates the shortcomings of Li et. al's Method (2017).In section 4, we present the Archimedean Copula based operations between qRPDHFEs and the associated GMSM operators.This section also puts forward the qRPDHFGMSM, qRPDHFGGMSM, qRPDHFWGMSM and qRPDHFWGGMSM operators along their characteristics.In section 5, we provide a group DM methodology using the developed AOs.A case study of open-source software LMS assessment is considered in section 6 to express the applicability of the developed approach.Section 7 deals with impact of parameters, and comparison study.The last section is the conclusions.

Preliminaries
Some basic notions are presented here that are relevant to our research.

q-Rung probabilistic dual hesitant fuzzy set (qRPDHFS)
Definition 1 (Li et al., 2020) Motivated by the score value, deviation degree and ranking rules of PDHFEs (Hao et al., 2017), we define the followings: Definition 2: The score value of Sometimes qRPDHFs cannot be compared if their score values become identical.To address this issue, their deviation degrees can be used.Definition 3: The deviation degree of


Thus, deviation degree of a qRPDHFE reflects describes the distance from the average value.

Archimedean Copula
Definition 6 (Sklar, 1959): g →is termed as a copula if: Definition 7 (Sklar, 1959): An Archimedean copula is a mapping where  is a strictly decreasing function and g is strictly increasing and  coincides with  , then g is called strict Archimedean Copula and we write:

Archimedean Copula based GMSM operators:
Based on the AC operational laws for qRPDHFEs, we first propose qRPDHFGMSM operator and study it's properties.
) ))( ) In the following, some vital properties of the qRPDHFGMSM operator are presented.
Theorem 3: (Idempotency) If be another collection of adjusted qRPDHFEs such that j, ( )

 
Proof: Similar to Theorem 2.
In the following, some vital properties of the qRPDHFGGMSM operator are presented.

Archimedean copula based weighted GMSM operator:
Although the qRPDHFGMSM operator can tackle the interrelationship among multiple input criteria, it does not consider the self-importance of the qRPDHFEs.To overcome this problem, we propose qRPDHF weighted GMSM operator (qRPDHFWGMSM operator) based on Archimedean Copula.

Group decision-making methodology with interaction of experts
as initial assessments of experts.Then the developed method has the steps mentioned below: Step 1: Derive the adjusted qRPDHF matrices .  (Han & Li, 1994).The subjective weight If the closeness between where () )  5) to obtain the objective weight of each criterion.
Utilizing a mix of , experts' final weights can be calculated by: (1 ) In Eq. ( 6),  sorts out which weight is dominating.The experts lean towards the subjective weights if  is high; and in case if  is low, experts favor the objective weights.The parameter  [0, 1] is termed as the risk attitude of experts.The greater the value of  , the more inclination of the expert towards risk.Also, he/she turns into a risk adverse person who regards the trustworthiness of the evaluation group.
Step 3: Obtain the consensus coefficient.
In an innovative MCGDM procedure, the weights of experts will change if the information provided changes.To get a more reasonable decision result, experts should interrelate with each other and lastly take a decision on assessing information.We utilize the symbol *  to express the consensus coefficient.The judgment matrix of the d th expert obtained from the  round interaction is defined by The consensus coefficient ()   which is computed by the  th adjustment can be defined by In that case the opinions of the experts are fully unified, then 01     .Thus, throughout the DM procedure, one expert should provide the weights to others based on the assessment information and estimate the consensus coefficient in the  round after he/she has obtained a consensus value *  in advance, and then check whether meets Step 4: Aggregate the qRPDHF matrices using the proposed AOs.
The weights of criteria play an important role on the final outcome.Consider an expert d E , and qRPDHF data under j C .Then the following divergence measure can be used to describe how the alternative i A differs from other alternatives.
The total divergence for the criterion j C is: Considering the importance of all experts, we can determine the overall divergences for each option over the given criterion j C .
( ) ( ) From the above discussions, it is clear that the following optimization model can be used to calculate weights of criteria.
where  is the set of partial information's about criteria weights.
Step 7: Applying the idea of adjusted probabilities, we create the adjusted aggregated qRPDHF matrix: To develop this final adjusted aggregated qRPDHF matrix Step 8: Generate the preference the options ( 1(1) ) i A i m = using the scores of and select the optimal option.

Case Study and solution
The proposed method is employed in Example-1 to assess the OSS-LMS alternatives with qRPDHF information.
Step 1: The initial assessments of experts are shown in Table-5.
(2).The connections between the experts' opinions are gained using Eq. ( 3) and Eq. ( 4) and are shown in Table-6.The objective weights of are calculated using Eq. ( 5).
Their overall weight of experts is calculated by Eq. ( 6) (choosing  =0.5) and are given in Table-7.
Step 4: Utilizing Eq. ( 8) and taking r=2, q=3, 12 2 tt == , and Solving the above optimization model, we get,  (Li et al., 2020) falls short in its attempt to reduce the impact of highly inflated attribute values from a few unreliable experts who have different biases.This unspoken problem has a negative impact on how decisions are made in any MCGDM process.By permitting the expert engagement that is lacking in the present study, this issue has been remedied utilising the suggested methodology (Li et al., 2020).4. The preference ranking produced in the DM approach (Li et al., 2020) is impacted by the random distribution of weights of criteria during the final aggregation step.Additionally, the current approach (Li et al., 2020) loses information because it doesn't take any information measures into account.Our technique computes criteria weights using an optimization model based on the cross-entropy measure.By highlighting the importance of each criterion, this optimization approach quantifies the amount of ambiguous data.

Conclusion
The qRPDHFSs can effectively portray the dubiousness and uncertainty in reality due to the inclusion of the MDs and NMDs with their corresponding probabilities.The joint occurrence of the stochastic and the non-stochastic ambiguity make the qRPDHFSs more realistic and superior.A comprehensive study on the usefulness of Archimedean Copulas under qRPDHF setting is demonstrated in our study.New operations for qRPDHF elements are formed via Archimedean Copulas.The existing AOs (Hao et al., 2017;Garg & Kaur, 2018) for aggregation PDHF data are limited to algebraic, and Einstein operators.So, they are not capable of considering dependency between multiple attributes.On the other hand, although the AOs (Ji et al., 2021;Li et al., 2020) based on Hamy mean operator and Muirhead mean operator respectively can consider dependency among criteria, cannot connect more than one marginal distribution.These facts motivated us to develop the Archimedean Copula based GMSM operators with their weighted forms under qRPDHF setting.Some pivotal qualities like idempotency, boundedness, and monotonicity, and proposed AOs are introduced.Subsequently, a MCGDM procedure is exhibited to track down the best option in qRPDHF setting.Here, the weights of criteria are determined using an optimization model and experts' weights are figured utilizing the linear combination of objective and subjective weights and interaction among experts.To give a superior comprehension of our technique, we have incorporated a case study including OSS-LMS selection.The robustness of our method has been demonstrated through sensitivity analysis of weights of criteria.The comparative study suggests that the proposed methodology can be adequately utilized in MCGDM issues containing correlated criteria in the PDHF setting.
The only limitation of the proposed method is that in absence of partial weights information of criteria the proposed method fails.In such a scenario, other objective methods like CRITIC, MEREC, entropy measure, etc can be utilized for determination of criteria weights.In further research, other aggregation operators (Saha et al., 2022a;Saha et al., 2021a;Senapati, 2021;Senapati et al., 2022;Saha et al., 2022b;Saha et al., 2021b) can also be extended to tackle the dependency among attributes with qRPDHF information and the proposed weight determination technique.Our model can be used to provide a realistic solution to well-known problems, such as sustainable supplier selection (Mishra et al., 2022a), warehouse site selection (Saha et al., 2023), bio-energy production technology selection (Hezam et al., 2023), solid waste disposal method selection (Mishra et al., 2022b), renewable energy source selection (Mishra et al., 2022c), low carbon tourism assessment (Mishra et al., 2022d), biomass feedstock selection (Saha et al., 2021c), cloud vendor selection (Krishankumar et al., 2022), and food waste treatment technology selection (Rani et al., 2021) as it can effectively avoid distorting evaluation information and handle the relationships between multiple criteria.

Conflicts of Interest:
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

List of abbreviations
a and b where 1 q  ) express the membership and non-membership degrees, respectively of i uU  and the related probabilities are qRPDHFE.Step 2: Calculate the weights of the experts.Suppose d  = weight of the expert , d E d   = subjective weight of the expert , d E d = objective weight of the expert d E .As pointed out by Cheng et al. (2018), d   gives the CHWI (consistency harmonious weight index defined by method based on qRPDHFSs is more effective compared to Hao et al.'s method (2017), and Garg and Kaur's method (2018) and Ji et's method (2021) for solving real decision-making problems.But Li et al's method has certain drawbacks too.To analyze the shortcomings of Li et al.'s method (2020), we consider the following two counter examples:

Table 2 .
Criteria details

Table 4 .
Initial assessment matrix ) Definition 11: The AC based qRPDHFWGMSM operator on qRPDHFEs is defined by: 10)

Table 7 .
Weights of experts , low level of connections is found among the 2 nd and other experts.This signifies a biased decision of 2 nd expert.Therefore, the evaluation data of 2 nd expert should get modified.The revised assessment information of the expert E2 is presented in

Table 8 .
Revised adjusted evaluation matrix for expert E2

Table 9 .
Table-9.The revised subjective weights, objective weights and final weights of experts are given in Table-10.Similarity

Table 10 .
Weights of experts