PROBABILISTIC LINGUISTIC Q-RUNG ORTHOPAIR FUZZY ARCHIMEDEAN AGGREGATION OPERATORS FOR GROUP DECISION-MAKING

: To express uncertain and imprecise information systematically, the concept of probabilistic linguistic q-rung ortho-pair fuzzy set (PLqROFS), which is an advanced version of linguistic intuitionistic fuzzy set and linguistic Pythagorean fuzzy set, considering the instantaneous occurrence of stochastic and non-stochastic-uncertainty. There isn't yet any literature on PLqROFSs that addresses the issue of the relative importance of experts and criteria. The evaluation's


Introduction
"Group decision-making (GDM) (Saha et al., 2021;Mishra et al., 2022;Ivanovic et al., 2022;Saha et al., 2022;Krishankumar et al., 2022;Senapati et al., 2023), is a complex and attractive decision problem that gets ratings/opinions from multiple experts to choose a suitable element from the set of elements based on diverse competing criteria (Riaz et al., 2021).In recent times, researchers widely adopted qualitative preferences in the GDM process to flexibly share her/his opinions on objects/criteria.Herrera & Martínez (2000) framed the idea of a "linguistic term set" (LTS) and promoted linguistic decision-making that considers qualitative terms directly as preference values and decision methods attempt to select suitable objects based on such rating information.Rodriguez et al. (2012) showed that LTS was unable to accept more than one qualitative term as a rating argument, which is unreasonable due to the practical uncertainty that exists in the decision process.To handle the issue, a "hesitant fuzzy linguistic term set" (HFLTS) was proposed that could flexibly accept more than one term as rating information thereby allowing experts to effectively share their opinions.To achieve this flexibility, HFLTS integrated the idea of LTS and "hesitant fuzzy set" (HFS) Torra (2010) Although HFLTS is attractive, it cannot assign weights to the diverse terms, which indicates that all the terms are of equal importance and that is unreasonable in practical decision problems.To resolve the issue, Pang et al. (2016) came up with a "probabilistic linguistic term set" (PLTS) that associates occurrence probability to the qualitative rating thereby assigning unequal weights to the terms.Attracted by the PLTS, scholars adopted it for GDM by proposing operators (Kobina et al., 2017;P. Liu & Li, 2019;P. Liu & Teng, 2018), ranking methods (Krishankumar et al., 2019;Ramadass et al., 2020;Sivagami et al., 2019), entropy/distance measures (Lin & Xu, 2018;Su et al., 2019), and others (Krishankumar et al., 2019;Liao et al., 2017Liao et al., , 2020;;X. Zhang & Xing, 2017)." To cope with practical situations, equivocal human judgments were taken into consideration, which gave rise to the idea of fuzzy sets (FSs) (Zadeh, 1965).The FS theory, on the other hand, can control reality emerging from computational observation and comprehension, which includes ambiguity, partial belongingness, inaccuracy, sharpness limitations, and so forth.In the GDM model, "decision experts (DEs)" might assess the "belongingness grade (BG)" of an element to a set of diverse grades in various realistic settings due to their individual opinion, time constraints, and the lack of information.To evade the concern, an HFS was developed by (Torra & Narukawa, 2009), according to which doctrine a BG should comprise several distinct BGs.As an FS extension, HFS has attracted much researchers' interest in treating with vagueness in realistic problems.Recently, the HFS has powerfully been associated with the "intuitionistic fuzzy set (IFS)" (Atanassov, 1986) an extension of the FS.The relevance of the IFS, however, is restricted because the sum of the BG µ and the "non-belongingness grade (NBG)"  cannot exceed 1, that is µ+ ≤1.
However, it was later noticed that, depending on the preferences suggested by DEs for complex GDM issues, the given constraint was not satisfied.For example, if a DE favors BG 0.7 and NBG 0.5 while using IFSs, their sum exceeds 1 at that point.To obtain this circumstance type, Yager (2013) pioneered the notion of "Pythagorean fuzzy sets (PFSs)" with the BG µ and the NBG , complying with the condition µ 2 + 2 ≤1.The "q rung ortho-pair fuzzy sets (qROFSs)" pioneered by Yager (2017) hold the constraint that the q th powers sum of the BG and the NBG lies between 0 and 1, i.e. 0 ≤µ q + q ≤1.When q=1, the qROFSs are reduced to IFSs, and when q=2, to PFSs, which means that qROFSs are the extended versions of IFSs and PFSs.
Inspired by the flexibility of q-ROFS, Liu & Liu (2019) presented a "linguistic qrung ortho-pair fuzzy set (L-qROFS)" along with power operator to utilize the advantages of both qualitative terms and q-ROFS so that flexibility in rating improves.The "linguistic intuitionistic fuzzy sets (LIFSs)" (Zhang, 2014) and "linguistic Pythagorean fuzzy sets (LPFSs) (Garg, 2018) are particular cases of L-qROFS for q=1 and q=2 respectively.Lin et al. (2020) extended the Heronian operator along with weighted variants for GDM with L-qROFS.(Akram et al., 2021) put forward the Einstein operator with weighted variants under L-qROFS context for GDM.Liu et al. (2022) put forward the generalized point operator along with the weighted versions for aggregating L-qROFS information to perform GDM.Though the L-qROFS is attractive, the assignment of weights to the multiple terms is missing and that is considered to be crucial information in the decision process.Inspired by the claim, Liu & Huang (2020) proposed a "probabilistic linguistic q-rung ortho-pair fuzzy set" (PL-qROFS) which is a generalization of L-qROFS that includes probability for the terms that potentially adds support to the decision process.Earlier works on L-qROFS have primarily focused on preference aggregation, but other decision phases have to be still explored for rational decision-making.

Research gaps and our motivation
Our motivations are as follows: 1.The instantaneous occurrence of stochastic and non-stochastic ambiguity in genuine issues is not taken into account by the LIFSs (Garg, 2018;Zhang, 2014).
2. Works from Zhang (2014), Garg (2018), and Liu & Huang (2020) form some theoretical base by presenting operational laws, but it fails to provide a rational and flexible decision process.3. Aggregation operators (AOs) are utilized to combine all the input data into a single entity.They are effectively used for information processing, specifically decision-making, pattern recognition, data mining, and machine learning for the last two decades.Aggregation operators in Zhang's method (2014), Garg's method (2018), Liu & Huang's method (2020) cannot effectively handle extreme values provided by some experts who tend to be biased or unwilling to participate in the decision process.4. In practice, not all of the requirements are equally important.For instance, a teaching faculty member's qualifications and experience in the field are valued more highly than their age.As a result, priority must be supplied logically to determine the weights of the criteria.Weights of criteria are not methodically derived in the relevant methodologies now in use (Garg, 2018;Liu & Huang, 2020;Zhang, 2014), which could lead to subjectivity and mistakes in the process.These methods (Garg, 2018;Liu & Huang, 2020;Zhang, 2014) are also unable to solve the issue that arises when applying the priority of a link among criteria for the evaluation of criteria weights. 5. Experts' weight assignment is a matter of significant concern for the process of aggregation.Experts' weight must be assessed systematically to mitigate subjective randomness from.This is completely missing in Zhang's method (2014), Liu & Huang's method (2020).6.A completely aggregation-based method under hesitant and probabilistic information for ranking is still unexplored.

Contribution of the paper
"Motivated by these claims, a new integrated method for GDM is put forward, which consists of the following: 1.To cope with ambiguous data, we use PLqROFSs.In fact, by enabling stochastic and non-stochastic uncertainty to emerge immediately in realworld circumstances, PLqROFS restores the dependability of the GDM techniques (Garg, 2018;Zhang, 2014).Therefore, PLqROFSs outperform LIFSs and LPFSs (Garg, 2018;Zhang, 2014).2. Archimedean t-norms (t-Nms) and t-conorms (t-Cnms) are the generalizations of a large number of other t-Nms and t-Cnms.So, some new operational laws are developed by taking the advantage of Archimedean tnorm and t-conorm (Klement & Mesiar, 2005;Klir & Yuan, 1996;Nguyen et al., 2018) for the theoretical superiorities.3. To provide an aggregation operator that enables argument values to support one another during the aggregate process, a power average must be used.So combining Archimedean operators and power averaging operators, Archimedean power weighted average and geometric operators are developed with their properties for handling extreme value situations from experts. 4. Criteria weights determination tools are divided into two categories: subjective and objective.The subjective methods namely AHP, FUCOM, and BWM select weights based on the consideration or judgments of decisionmakers.FUCOM (Pamučar et al., 2018a) technique is extended to PL-qROFS for criteria weight determination.so that consistent weights are obtained with a rational understanding of the views of experts. 5. Also, the variance approach is put forward for experts' weight assessment through methodical procedure.This aids in the reduction of subjectivity and biases in the process.6.A new ranking algorithm is developed by utilizing the developed AOs. 7. Sensitivity analysis of weights reveals the robustness of the developing ranking technique with PL-qROFS information."

Arrangement of the paper
We summarize the remaining paper below.Some vital concepts related to PLqROFS, Archimedean operators, and PAO are presented in Section 2. Section 3 deals with the presented Archimedean operations for the PLqROFNs and the associated PLqROF-Archimedean AOs, such as PLqROFAWAA, PLqROFAWGA, PLqROFAPWAA, and PLqROFAPWGA.The MCGDM method is discussed in the PLqROFSs context in section 4. A case study of the choice of CO2 storage location is taken in Section 5. .Section 6 deals with the results and discussions.Finally, we wrap up the entire research in section 7.

Preliminaries
Here, we concisely review the existing concepts.For this, we first listed all the abbreviations, in Table 1, used in the entire paper for the better readability of the paper.

Linguistic term set
Definition 1 (Zadeh, 1975): A linguistic term set (LTS) is a set ( u signifies a "linguistic value (LV)" and z being a non-negative integer), holding the constraints: Definition 2 (Xu, 2004;2005) The most frequently utilized LSF is:

Probabilistic linguistic q-rung orthopair fuzzy sets
Definition 3 (Liu & Huang, 2020): For a given set U and a LST , a probabilistic linguistic q-rung orthopair fuzzy set (PLqROFS) where the LTs r and s are associated with probabilities To handle the aggregation process in a simplistic way, Wu et al. (2018) introduced the concept of "adjustment of probabilities".In this paper, we extend it under the PLqROF setting.To understand the process of adjustment, example 1 is employed.
Example 1: (Wu et al., 2018) For a LST Definition 4 (Liu & Huang, 2020): For a PLqROFN Sometimes score values become insufficient for the comparison of PLqROFNs.As an instance, take two PLqROFNs . Score values can't efficiently deal with this situation.To solve the concern, Liu and Huang (2020) defined the accuracy value.

Archimedean operations
Definition 7: For the adjusted PLqROFNs ( propose the Archimedean operations among PLqROFNs as Then, for any 12 , , 0

Archimedean Power weighted operators
Here, based on the power operator (Yager, 2013), we show the development of the operators PLqROFAPWAA and PLqROFAPWGA.
   be an assortment of adjusted PLqROFNs.Then the PLqROFAPWAA operator is given by Here, where symbolizes the distance between () i and () j .
Theorem 4: Then the aggregation of
   be an assortment of adjusted PLqROFNs.Then the PLqROFAPWGA given by Theorem 5: The aggregated value n PLqROFAPWGA is a PLqROFN and (1) (2) ( )

Proposed MCGDM Methodology
Consider a group decision-making problem where m different alternatives

Formation of the initial assessment matrices
Step 1: Prepare PLqROF-matrices representing the initial evaluations of DEs.Consider as the initial assessment of the DE Dk.
For evaluation, we take the LST Step 2: Find the DEs' original assessment rating in the updated PLqROFNs forms

Determination of Experts' weights
Here, we offer a new approach for DEs weight calculation under the PLqROFS context.Popular methods from the latter context are the "analytical hierarchy process (AHP)" (Saaty, 2002), "stepwise weight assessment ratio analysis (SWARA)" (Koksalmis & Kabak, 2019), and others.Statistical variance is a useful and straightforward tool for weighting value, which considers the doctrine of variation (Liu et al., 2016).Kao (2010) correctly mentioned the effectiveness of the variance tool and concluded the model deals the hesitancy/ambiguity efficiently.Koksalmis & Kabak (2019) discussed the significance of DEs' weight and its usage in mitigating biases from direct elicitation.Driven by these claims, we plan to lengthen the variance approach for DEs weight calculation under the PLqROFSs context.Steps for calculation are given by: Step 1: Obtain l matrices of order mn with adjusted PLqROF information.
where ijk c is the single value from the membership side and ijk d is the single value from the non-membership side for the k th DE.
Step 3: Compute the net variance exhibited by each DE using Eq. ( 19).Step 4: Obtain the confidence/non-hesitation factor for each DE by taking the complement of the normalized variance.Specifically, a DE with high hesitancy will produce a low confidence factor and ultimately, the weight is low.This concept is used to obtain the weights of DEs in Eqs. ( 20)-( 22) as

Computation of supports and power weights
Step 1: Estimate the supports ) Step 2: Calculate the values ijk  utilizing Eq. ( 12) assuming that


 are weights of the decision experts () kl Dk  .
The PLqROFAPWAA or PLqROFAPWGA operator is employed to obtain the A-PLqROF-M () ij mn    as follows: Probabilistic linguistic q-rung orthopair fuzzy Archimedean aggregation operators for group… 649 Suppose the aggregated PLqROF matrix is Step 2: Obtain the normalized A-PLqROF-M () . Here, where , BC QQ denote the beneficial and cost criteria, respectively.

Determination of criteria weights
The FUCOM is defined following the concepts of comparisons in pairs of characteristics and the validation of the outcomes by defining the "deviation from the maximum consistency (DMC)" (Pamučar et al., 2018a).In recent times, there are various disciplines in which the FUCOM has been implemented successfully such as, evaluation of the airline traffic (Badi & Abdulshahed, 2019), evaluation of the period of installation of security procedure (Pamučar et al., 2018b), road traffic route evaluation for hazardous products (Noureddine & Ristic, 2019), selection of equipment for storage schemes in the logistics (Fazlollahtabar et al., 2019), evaluation urban mobility scheme (Pamucar et al., 2020), evaluation of a suitable territory in Spain's autonomous societies (Yazdani et al., 2020) and others.Saha et al. (2022a) solved the "healthcare waste treatment method (HCWTM) assessment problem using q-ROFSs, FUCOM, and "double normalization based multi-aggregation (DNMA)" methods.Mishra et al. (2022a) developed a DEA-FUCOM-MABAC methodology on HFSs for "sustainable supplier selection (SSS) in the automotive industry.
In this paper, for estimating the criteria weights, we apply the FUC0M method (Pamučar et al., 2018a).
Step 2: Estimate the scores values of the A-PLqROF-M () () Step 3: Prioritize the alternatives i A () m i  with the use of Definition 6.

Case study: CO2 storage location selection
"The commencement of investigation on the option of CO2 neutralization by receiving and its storage in suitably chosen geological surroundings took place at the initiating in the 1990s (Bachu, 2000;Koteras et al., 2020).Sequestration of CO2 is a significant system to obtain CO2 emission reduction.CO2 storage locations can be categorized into the following types: geological, biological, and oceanic sequestration, respectively (Hsu et al., 2012).From the safe CO2 storage location, CO2 can be injected into deep geological storage (GS) in the supercritical state.The GS of CO2 is the most appropriate location selection.Based on the research, three kinds of GS can be applied in the procedure of CO2 GS as deep saline structures, oil and gas reservoirs, and unmixable coal sheets (Guo et al., 2020).Following these, the deep saline water sheet has the leading storage capacity.The structure in which CO2 is stored is known as a reservoir, and the upper portion is known as a cap rock layer.Based on the numerous parameters namely geological circumstances, engineering approaches, and force majeure of storage location, CO2 may escape from the GS reservoir and harm the environment and human beings, so the assessment of CO2 storage location is a significant portion of whole Carbon capture, utilization and storage (CCUS) project management.In India, a huge stable CO2 sink has been utilized in the Deccan volcanic region, which comprises the drainage of Kutch, Deccan, and Saurashtra.The potentials of CO2 storage in geological structures are deliberated in the characteristic of a suitable storage location of the geological structure, along with the tightness and veracity of contiguous layers, which may establish natural insulation of suitable location.The further features are associated with natural storage settings, which have an impact on sustaining the integrity of the location.To decide on GS of CO2, it is essential to assess various attributes that assess the procedure employment from sustainability perspective.The suitability of any specific location, is consequently, based on various concerns, containing the nearness to CO2 sources and other reservoirs with definite assets namely porosity, permeability, and leakage capacity.For CCUS to flourish, it is expected that each storage variety would always store massive amounts of injected CO2, keeping the gas sequestered from the environment in perpetuity (Folger, 2021).
For ecological storage, CO2 is saturated underground in a variety of topographical situations in muddy bowls.In the bowls, oil and gas basins and vacant areas, unmineable coal layers, and saline designs are possible regions.Additional likely storing terminuses for the sequestration of CO2 contain assimilated sinkholes, basalt rocks, and natural shale.These kinds of land provisions exist on land as well as seaward in several regions all over the world.Nevertheless, to appropriately release the injurious natural effects on environment from CO2 accretion, the capacity must be tenacious.A storage space with lifelong worth shows that CO2, it comprises will not stumble over the climate at a huge rate for several years.The development of CO2 storage is stirring progressively in India (Kumar et al., 2019).Based on the literature review and DEs opinions, five possible option storage locations in the Indian context, named Coal deposits (A1), Gas-field (A2), Basalt (A3), Aquifer (A4), and Oilfield (A5) are selected.Assume that a committee of three DEs D1, D2, D3 to choose the suitable storage location over 12 criteria, and depicted in Table 2.

Criteria
Description Type Cost (C1) Considers the overall cost including initial cost, transportation costs, maintenance costs, and others.

Cost
Storage capacity (C2) Considers the capacity of underground geological structures.

Benefit
Regional risks (C3) Considers the risks namely earthquake risk, natural risk, and others in the region.

Cost
Reservoir area and net thickness (C4) The

Problem solution
We have applied the PLqROFAWAA operator (taking q=3) to solve the case study defined in Section 5.1.Here, we consider the LST ={ 0 =extremely bad, 1 =bad, 2 =moderately bad, 3 =moderate, 4 =good, 5 =very good, and 6 =very very good}.Table 3 presents the initial assessment matrices.The initial evaluation ratings of the DEs in the form of adjusted PLqROFNs are depicted in Table S1 of the Supplementary material.The variance, normalized variance, confidence factor and weight of DEs are estimated by Eqs. ( 19), ( 20), ( 21), and ( 22) respectively.These are given in Table 4.We compute the supports and denote them as in Table S2 of the supplementary material.From Eq. ( 12), values of are evaluated (Table S3 of the supplementary file) by From Eq. ( 23) and taking , we obtain A-PLqROF-M.The entries are then normalized using Eq. ( 25).The final normalized A-PLqROF-M is given in Table S4 of the supplementary material.
"Next, to determine criteria weights, we assume that the ranking of criteria: C1>C2>C3>C4>C5>C6>C7>C8>C9>C10>C11>C12.The comparison is prepared with the first-ranked C1 criterion and using the scale [1,9].Hence, the preferences of criteria ( ) for each attribute ranked in Step 1 are achieved (Table 5).The final model for predicting the weight values uses the comparative preferences of the attributes, which are computed based on the attained preferences of the attributes as described below." For solving the model with Lingo 17.0 tool, the weight values of attributes and DFC are computed (Table 6).

Sensitivity investigation (SI)
"Here, we utilize "sensitivity investigation (SI)" to assess the influence of an appropriate attribute on the results of the introduced model.An attribute is chosen as the "most significant attribute" if it has the highest weight value.It was suggested by several authors (Saha et al., 2021a)] that Eq. ( 28) can be applied to assess the weights proportionality through the assessment.
The assumptions during the SA are as follows: (1) s  (Weight coefficient of elasticity of an appropriate attribute) is given; (2) The ratio of weight values remains unchanged in the process of SA.
(28), we observe that the variation amount applied to a weight set is signified by x  based on weight elasticity values.We can compute the limit values of x  as: The original weights of the characteristics are then approximated based on the preset parameters after we have expressed the limitations for x  .A set of attribute weights is computed by Eq. ( 31) and Eq.(32).Here, the highest weight coefficient degree w1 = 0.1750 and C1 is considered as the most influential attribute.Subsequently, the weight elasticity coefficients (Table 7) are assessed and, the variation of weight coefficient ( x  ) is found to lie in the range - 0.1750≤ x  ≤0.8251.According to the given limits for the variation of weight coefficient values of an attribute, various attribute weight sets (CWS1, CWS2, CWS3, … , CWS12) for SA are calculated.The interval -0.1750≤ x  ≤0.8251 is separated into twelve sets.For each set, original values of criteria weights are computed by Eq. ( 31) and Eq. ( 32), and are depicted in Table 8.Thus, the scores of options are calculated for various attribute weights and are represented in Figure 1.A1 is rated fourth in all cases, A2 is ranked first in all cases, A3 is ranked third in all cases, A4 is ranked fifth in all situations, and A5 is ranked second in all cases after evaluating the ranking positions of the alternatives for various attribute weight sets.This analysis demonstrates that, in comparison to all other options, alternative A2 is more palatable.We can see from Figure 1 that the ranking order does not change, and as a result, the average of the SRCC (rA) values is "1," demonstrating a "very high correlation" between alternative rankings.As a consequence, the findings demonstrate the validity and dependability of the priority of alternatives determined using the established methodology.Aquifer (A4) Oilfield (A5)

Comparative investigation
This section compares results from theoretical and numerical angles.To compare the presented method 5 with some extant techniques on PLqROF, LPF and, LIF settings, respectively, we assess different extant methods, such as Liu and Huang's method (2020), Garg's method (2018) and, Zhang's method (2014).To elucidate the usefulness of the introduced model, we apply these methods to the aforementioned case study.Table 9 provides a summary of the results.2018) and Zhang's method (2014) to the same case study described earlier with DEs weights (initial weights) 0.32834, 0.33417, 0.33749 and criteria weights 0.1750, 0.1463, 0.1249, 0.0974, 0.0832, 0.0732, 0.0625, 0.0585, 0.0500, 0.0461, 0.0439, 0.0391, we do not obtain any ranking order for the reason that Garg's method (2018) and Zhang's method (2014) do not deal with the probabilistic hesitant uncertainty.The major disadvantages of the existing methods are: 1. Liu and Huang's method ( 2020) is based on the behavioral TOPSIS method, which is an extension of TOPSIS methods under the PLqROF environment.However, Opricovic & Tzeng (2004) have proposed a guarantee of the nonexactness of the solution obtained using TOPSIS with the perfect solution.It means that the method developed by Liu and Huang's method ( 2020) is not that useful.2. In Garg's technique (2018) and Zhang's method (2014), DEs represent information using only one LT as a membership value and only one LT as a nonmembership value.However, in practice, DEs occasionally struggle to explain the outcomes of their assessments using a single LT format and are reluctant to use anyone in particular.3.In Garg's method (2018) and, Zhang's method (2014), all assessment values are assumed to have equal importance.However, in practice, DEs can have varying levels of liking for several potential LTs. 4. In real situations, all criteria don't have equal importance.The weight of the criteria needs to be assessed very logically.In the decision-making methodologies developed by Liu and Huang (2020), Liu and Huang's method (2020), criteria weights were given arbitrarily during the aggregation of criteria values.Consequently, the final ranking gets influenced. 5. Experts' weight assignment, which is a matter of significant concern for the process of aggregation, is completely missing in Liu and Huang (2020), and, Zhang's method (2014).Table 9 demonstrates the advantages of the introduced method.From the assessment, we can deduce the following: 1. We used PLqROFS-based information, a dependable technique, to manage ambiguous data.The dependability and adaptability of conventional DM approaches are significantly improved by PLqROFSs by accounting for the simultaneous occurrence of stochastic and non-stochastic uncertainty in real issues (Zhang, 2014;Garg, 2018).PLqROFSs are superior to LIFSs and LPFSs as a result.2. The PLqROF-Archimedean weighted average and geometric AOs can efficiently aggregate the PLqROF information with greater generality and flexibility because PLqROF weighted average and geometric AOs (2020), PLqROF Einstein weighted average and geometric AOs, and PLqROF Hamachar weighted average and geometric AOs are specific examples of introduced AOs. 3. DEs' weights are calculated by extending the variance approach in PLqROF environment.The advantages of the variance approach are (i) it is straightforward; (ii) it can effectively reflect the DEs' hesitation during preference expression; and (iii) it assumes all preferences (data points) before assessing the variability in the distribution, unlike other statistical measures such as minimum/maximum.4. Our method for determining the weights of the criterion makes use of the FUCOM technique.The FUCOM indicated fewer variances to obtain the criteria weights from the most favorable ratings as compared to the BWM, AHP, and others.Thus, the method that is presented lessens MCGDM process errors.5.The two-way comparative approach (stochastic approach like Liu and Huang's method (2020), and non-stochastic approaches like Garg's method (2018) and, Zhang's method ( 2014)) establishes our model as a superior and most effective one in tackling DEs' judgments in MCGDM problems.

Conclusions
In this paper, we have used PLqROFSs, which are generalizations of qROFSs, LIFSs and, LPFSs to handle uncertain and inaccurate information in decision-making.The existing tools, proposed so far for aggregating PLqROF data, are classified into algebraic operations, and even we observe both lacking flexibility and generality during the aggregation process.Due to this reason, we have suggested new operations amongst PLqROFNs in this study using Archimedean operations.The evolved operations' refined characteristics are examined.Additionally, we have spoken with the developed operations about several PLqROF AOs, including PLqROFWAA, PLqROFWGA, PLqROFPWAA, and PLqROFPWGA operators.In the suggested methodology, DEs weights are estimated using the variance approach, whereas criteria weights are derived using FUCOM.Here, one case study addressing the choice of a CO2 storage location is taken into consideration to better understand the created method we previously exhibited.The sensitivity assessment reveals the suggested operator's robustness.Furthermore, by drawing comparisons, we are better able to state categorically that the developed approach can be applied to resolve MCGDM issues in a PLqROF environment.It is pertinent to state here that many existing operators in connection with the PLqROF -information can be considered special cases of the developed AOs.
The managerial implications are discussed related to the study as follows: (i) The developed model utilizes the PLqROF doctrine to offer a decision structure that integrates inaccurate information inherent in the CO2 storage site selection.(ii) It improves the theoretical perception of PLqROFSs by offering a new structural base for MCGDM where DEs gain flexibility from the generic structure and allows ease of decision-making.
The main limitation of the developed model is that it can't deal with any consensus-reaching process when expert(s) opinion(s) is(are) biased.Moreover, the proposed model doesn't consider dependency among multiple numbers of criteria.To overcome all these things, in the future, one can develop consensus-based decision-making models with Bonferroni mean or Hamy Mean or Heronian mean or Maclaurin Symmetric mean operators.Besides, for the determination of criteria weights, objective methods like MEREC, CRITIC, entropy measure, maximum deviation method, optimization models can be utilized.Although, we have used a case study related to sustainable material selection, other case studies (Deb et al., 2022;Hezam et al., 2023;Krishankumar et al., 2021;Liu et al., 2022;Saha et al., 2023) can also be considered.
. Our proposed methodology is as follows: k denotes the mean and variance respectively for k th DE.
variance value, and k  is the weight of k th DE.
in attribute weights in the SA, s w = Weight of the most prominent attribute, 0 c w =Original values of the attribute weights, 0 c W = Sum of actual values of modified attribute weights, c  = Weight coefficient of elasticity.The relative significance for different values of weights is articulated by c  , when we relate the variations made in the most important weight.
Probabilistic linguistic q-rung orthopair fuzzy Archimedean aggregation operators for group… 657 the general form of proportionality of weights, which holds the expression.The priority of the options is estimated by consideration of new attributes weights."

Table 1 .
List of abbreviations : Suppose u

Table 5 .
Preferences of attributes

Table 7 .
weight coefficient of elasticity of attributes