key: cord-0741574-5yet2kjr
authors: Mishra, Arunodaya Raj; Rani, Pratibha; Pandey, Kiran
title: Fermatean fuzzy CRITIC-EDAS approach for the selection of sustainable third-party reverse logistics providers using improved generalized score function
date: 2021-02-09
journal: J Ambient Intell Humaniz Comput
DOI: 10.1007/s12652-021-02902-w
sha: 0f403c44ac41f712fa93884cb2e50e0b401a44eb
doc_id: 741574
cord_uid: 5yet2kjr

In today’s world, the demand for sustainable third-party reverse logistics providers (S3PRLPs) becomes an increasingly considerable issue for industries seeking improved customer service, cost reduction and sustainability perspectives. However, the assessment and selection of right S3PRLP is a complex uncertain decision-making problem due to involvement of numerous conflicting attributes, imprecise human mind and lack of information. Recently, Fermatean fuzzy set (FFS) has been recognized as one of the suitable tools to tackle the uncertain and inaccurate information. In this paper, we introduce a hybrid methodology based on CRITIC and EDAS methods with Fermatean fuzzy sets (FFSs) to solve the S3PRLP selection problem in which the attributes and decision makers’ weights are completely unknown. In this framework, CRITIC approach is applied to calculate the attribute weight and EDAS method is used to evaluate the priority order of S3PRLP options. To do this, a new improved generalized score function (IGSF) is developed with its elegant properties. Also, a formula is discussed to calculate the decision makers’ weights based on the developed IGSF. Next, developed framework is applied to assess a case study of S3PRLP selection problem with Fermatean fuzzy information, which elucidates the usefulness and practicality of the proposed method. Finally, comparative study is implemented to show the strength of introduced framework with extant approaches. The outcomes of the work confirm that the introduced approach is more feasible and well-consistent with the other extant approaches.

Step-wise weight assessment ratio analysis S3PRLP

Sustainable third-party reverse logistics provider TOPSIS Technique for order of preference by similarity to ideal solution VIKOR VlseKriterijumska Optimizcija I Kaompromisno Resenje WASPAS Weighted aggregated sum product assessment WPM Weighted product model

During these times, good quality product, satisfaction of customers' requirements and existence in competitive marketplaces are elementary needs for any business. Indeed, these requirements have become business principles, prominent corporations to pursuit for different facets that influence the purchasing options of users. Subsequently, reverse logistics (RL) has become main feature for contributing to the desirable outcomes of several enterprises. RL comprises the actions related with the collection and succeeding retrieval of used products (Fattahi and Govindan 2017) . The emerging implication of RL has supervised numerous enterprises to design and reconstruct procedures as a part of their sustainable development initiatives (Govindan et al. 2015; Banihashemi et al. 2019) . Through RL, products are displaced from their final terminus to a new position, where their worth is considered and they are managed to the manufacturing line again or appropriately disposed (Tavana et al. 2016; Kannan et al. 2017) . Furthermore, eco-conscious customers incline to give extra on eco-friendly products, increasing the revenue of those businesses that utilize RL to achieve with the needs of such consumers (Mavi et al. 2017; Zarbakhshnia et al. 2018) . RL management is one of the important issues in the SCM, which mostly emphases on backward flow of products and raw materials from users to suppliers (Mavi et al. 2017) . Growing environmental responsiveness and prospective economic growth have determined ever more corporations to outsource their logistics operations to S3PRLPs (Mavi et al. 2017; Zarbakhshnia et al. 2018; Li et al. 2018) . In order to achieve the objectives of cost savings and environmental sustainability, it is significant for the firms to select the best S3PRLPs option. In recent times, the assessment of 3PRLPs selection process has received great attentions from the researchers. Numerous scholarly articles have been presented for selecting the best 3PRLP alternative in the literature, however, more studies are required to manage the prioritizations of different expertise, different environments and knowledge levels on reverse logistics with the consideration of social, environmental and economic dimensions simultaneously. Because of increasing complexity and several constraints, it is not always possible to explore the priorities more proficiently and accurately in the best 3PRLPs selection.

The FSs doctrine (Zadeh 1965) has successfully been employed in diverse 3PRLPs selection problem and proved its powerful ability to tackle with imprecise and uncertain information. As an extension of FSs, the theory of FFSs (Senapati and Yager 2019a) has been proven as one of the powerful platforms to deal with the imprecise and uncertain information. The key characteristic of FFS is the cube addition of BD and NBD is less than or equal to 1. Thus, the FFSs theory is more superior tool than FSs, IFSs and PFSs. Based on its unique advantage, the paper focuses under the environment of FFSs for the assessment of S3PRLP selection. Inspired by the above studies and literature, we introduce Fermatean fuzzy-CRITIC-EDAS framework for assessing the S3PRLPs selection. Thus, this is the first study which proposes a hybrid framework under FFSs. The main contributions of the work are discussed as follows:

1. A New improved generalized score function (IGSF) is proposed for FFNs with their elegant properties. 2. An FF-CRITIC-EDAS framework is introduced to handle the MCDM problemon FFSs.

3. To determine the practicality and effectiveness of the introduced framework, a case study of S3PRLP selection is taken with FFNs. 4. A Comparative discussion is made with the extant models to validate of the developed framework.

The rest of paper is arranged as follows: Sect. 2 depicts a comprehensive review related to present study. Section 3 shows the basic notions on FFSs and proposes a new IGSF with its elegant properties. Section 4 introduces novel Fermatean fuzzy-CRITIC-EDAS approach to elucidate the MCDM problems. Section 5 deliberates a case study of S3PRLP selection and also discusses a comparative discussion with the extant models. Section 6 spectacles the conclusions and scope for future study.

In the current section, we present literature survey related to the present study.

Various criteria are involved in the evaluation of 3PRLPs selection process, consequently, this selection process can be observed as a MCDM problem. Existing studies on 3PRLP selection problem confirm the emergent interest of scholars and manufacturers. Over the last few years, copious MCDM models have been established in the setting of 3PRLP assessment problem. Realistic reverse logistics outsourcing assessments are commonly prepared under imprecise and vague environment due to multiple indicators, like as partial ignorance, imprecise estimation, partial or inaccessible decision information (Saen 2010; Azadi and Saen 2011) . Consequently, crisp values are usually inappropriate to model such type of practical decision conditions. FS theory and their extensions have widely been employed to cope with uncertain and vague information occurred in realistic MCDM applications. Senthil et al. (2014) suggested a combined model with AHP and TOP-SIS approaches for evaluating an ideal reverse logistics contractor. In a further study by Tajik et al. (2014) , a decision-making framework was introduced for choosing most suitable 3PRLP alternative by taking all three aspects of sustainability on FSs. Later, Uygun et al. (2015) planned and selected an outsourcing provider for a telecommunications business by employing DEMATEL and ANP approaches. Tavana et al. (2016) suggested a conceptual analytic network model to thoroughly tackle the complex behavior of interactions among the 3PRLPs assessment factors. Mavi et al. (2017) presented SWARA method for weighting the assessment criteria of 3PRLP in the plastics industry and further, ranked the sustainable 3PRLP alternatives through MOORA model within FSs context. Tavana et al. (2018) suggested a combined method with the integration of ANP and grey superiority and inferiority methods on IFSs for the assessment of 3PRLPs selection process. Li et al. (2018) used a combined cumulative prospect doctrine with hybrid-information MCDM methodology for the evaluation of 3PRLPs from sustainability perspectives. Zarbakhshnia et al. (2018) weighted the assessment criteria through fuzzy SWARA method and ranked the sustainable 3PRLPs by employing COPRAS method under fuzzy environment. Liu et al. (2019) suggested the BWM to research the selection of 3PRLPs on IVPHFSs. Bai and Sarkis (2019) pioneered multi-stage, multi-method, and multi-metric decision-making tool based on TOPSIS, VIKOR and neighborhood rough set for the evaluation of 3PRLP selection decision. Recently, Zarbakhshnia et al. (2020) established a framework with the AHP and MOORA-G models for the assessment of 3PRLPs under FSs. Zhang and Su (2020) introduced the heterogeneous linguistic model with dominance degree to assess the best S3PRLP for a car manufacture industry.

The theory of FSs has broadly been received great attention from the researchers for dealing with uncertain and imprecise information. In 1986, Atanassov (1986) presented the idea of IFS, which is termed as belongingness degree (BD), non-belongingness degree (NBD), and holds the condition that the addition of BD and NBD is less than or equal to 1. As an extension of IFSs, Yager (2014) initiated the notion of PFSs. The PFSs are more powerful tool than IFSs for handling the uncertain, vague and imprecise information arisen in the realistic problems. Recently, numerous researchers have explored the different concepts by considering the theoretical and practical aspects of PFSs. For instance, Rani et al. (2019) proposed the VIKOR approach-based on entropy and discrimination measures to handle the renewable energy resources assessment in India. Rani et al. (2020a) developed COPRAS method to select the pharmacological therapies for type-2 diabetes disease. Rani et al. (2020b) assessed and selected the healthcare waste treatment options using SWARA and ARAS methods.

However, in numerous practical concerns, a group of DMs may give the BD to which an alternative satisfies the attribute is 0.8 and the NBD to which an alternative dissatisfies the attribute is 0.7. Here, we observe that 0.8 + 0.7 > 1 and 0.8 2 + 0.7 2 > 1 , as a result, the IFS and PFS are incapable to tackle this concern. To handle such information, Senapati and Yager (2019a) gave the doctrine of Fermatean fuzzy set (FFS), which is termed as the BD and NBD and the constraint that the cube sum of BD and NBD is less than or equal to 1. Consequently, the FFS can efficiently solve the aforementioned concern. Also, FFSs can solve the MCDM problems with more effective way and handle the complex uncertain information. For example, Senapati and Yager (2019a) pioneered the doctrine of FFSs and proposed various basic concepts for solving decision-making problem. Senapati and Yager (2019b) discussed numerous aggregation operators with their elegant properties for FFSs and used WPM to solve the MCDM problems (Senapati and Yager 2019c) . Garg et al. (2020) studied several Fermatean fuzzy aggregation operators and presented their application in COVID-19 facility selection. Aydemir and Gunduz (2020) proposed Dombi operators and their properties for FFSs. Recently, there is no study related to the S3PRLP assessment on FFSs context.

In the MCDM process, various criteria weighting methods have been developed in the literature (Suh et al. 2019; Mishra et al. 2020a; Rani et al. 2020c ). The criteria weight computation procedures are classified as objective weight and subjective weight (Peng 2019) . The CRITIC approach (Diakoulaki et al. 1995) is one of the weighting models to estimate the objective weights of the attributes using the standard deviation and the correlation coefficient to quantify the value of each attribute and computes the attribute weights of MCDM procedure. Recently, several fusion models have been introduced by applying CRITIC with various other MCDM methods (Yalcin and 

The objective of MCDM procedure is to select the best option from a set of options under a set of various attributes. Currently, numerous MCDM approaches namely COPRAS (Kumari and Mishra 2020; Mishra and Rani 2018) and others have been discussed to solve the MCDM problems on different uncertain environments. Each MCDM procedure has been developed with different advantages and disadvantages, though, the scholars usually select an approach which is based on the nature and intricacy of the problem.

The EDAS method (Ghorabaee et al. 2015) is an original and efficient tool to solve the MCDM problem with conflicting attributes. It utilizes the AVS for prioritizing the options and describes the discrimination between the options and the AVS according to the measures PDA and NDA (Ghorabaee et al. 2016) . Kahraman et al. (2017) used EDAS method for assessing solid waste disposal location on IFSs. Gundogdu et al. (2018) extended the EDAS model for assessing and choosing the suitable hospital under HFSs environment. Mi and Liao (2019) discussed a combined framework with BWM and EDAS method with HFSs to assess insurance projects. Zhang et al. (2019) extended the EDAS approach to select the best green supplier. Han and Wei (2020) gave multivalued neutrosophic EDAS model for dealing with the MCDM problems. Mishra et al. (2020d) discussed parametric discrimination measure-based EDAS framework for assessing the HCWD method on IFSs. In this study, we develop a combined framework with CRITIC and EDAS approaches for FFSs.

This section introduces a new score function for Fermatean Fuzzy Numbers (FFNs) which avoids the shortcoming of existing score function (Senapati & Yager, 2019a,b) . Here, the first subsection presents the concept, score and accuracy functions, and operational laws of FFSs. Based on this concept, an improved score function for FFNs is developed in the next subsection.

Definition 3.1. Let Δ be a limited universe of discourse. In 2019, Senapati and Yager (2019a) firstly presented the concept of FFS, which is mathematically expressed as.

wherein b T ∶ Δ → [0, 1] represent the belongingness degree (BD) of an element t i ∈ Δ in FFS and n T ∶ Δ → [0, 1] signify the non-belongingness degree (NBD) of an element t i ∈ Δ in FFS. For every t i ∈ Δ, it satisfies the condition

For simplicity, Senapati and Yager (2019a) 

Definition 3.2. Let = b , n be a FFN. Then, the score and accuracy functions of are defined as follows (Senapati and Yager 2019a, b) :

Thus, to rank two FFNs 1 = b 1 , n 1 and 2 = b 2 , n 2 , we have the following comparative scheme (Senapati and Yager 2019a, b) :

are three FFNs. Now, the operational laws are summarized as follows:

This section develops an IGSF for any FFN j = b j , n j , and is presented as

Here, 1 , 2 > 0 signifies the performance of the proposed IGSF and 1 + 2 = 1 . In addition, 1 and 2 provide the weighted average of the indeterminacy degree between the BD and NBD.

Theorem 3.1: For a FFN j = b j , n j , the IGSF ℘ * s (.) satisfies the following:

Proof p1. When j = (0, 1) or j = (1, 0), then according to Eq. (1), the IGSF ℘ * s j attains the least value '0′ or utmost value '1′, respectively. It follows that 0 ≤ ℘ * s j ≤ 1. p2. To prove this part, differentiate Eq. (1) partially w.r.t. b j , then we have

In the similar way, the first partial derivative of Eq. (1) w.r.t. n j is presented as follows:

Thus, the theorem is proved.

Remark 3.1 When we comparing any two FFNs 1 = (0.5, 0.5) and 2 = (0.4, 0.4) , we can observe that the score function given by Definition 3.2 (Senapati and Yager 2019a, b) is unable to distinguish the difference between the given FFNs because ℘ s 1 = ℘ s 2 = 0 , whilst the developed IGSF (1) can successfully deal with this example and therefore, we have ℘ * s 1 = 0.2188 and ℘ * s 2 = 0.1198. Hence, 1 ≻ 2 . This verifies the validity of the proposed score function over the existing one.

Here, Fig. 1 presents the value of score function ℘ * s j w. r. t. 1 , 2 > 0, 1 + 2 = 1 and at b j , n j = (0.8, 0.7). The color of each pair 1 , 2 on the simplex demonstrates the score value of the fixed Fermatean fuzzy numbers. As the value of 1 and 2 become larger, the value of ℘ * s j becomes better. Similarly, Fig. 2 presents the value of score function ℘ * s j w.r.t. b j and n j at 1 = 2 = 0.5. The color of each pair b j , n j on the simplex presents the deviation on score function of the fixed Fermatean fuzzy numbers. As the value of b j become larger, the value of ℘ * s j becomes superior.

(1)

Alternatively, the effects of diverse values of 1 and 2 on the preference order of the FFNs has been examined and their equivalent score values are summarized in Fig. 1 and Fig. 2 . From the Figs. 1 and 2, it has been concluded that with the increase of 1 from 0 to 1 (or simultaneously decreasing the value of 2 from 1 to 0) the relative score of the FFNs increases. Also, from Remark 3.1, relative score of the FFNs increases but the overall ranking of the FFNs remain unaltered. Thus, based on these different parameter values, system analysis or decision maker may possess the power to alter the ranks of the FFNs. Ghorabaee et al. (2015) originated the notion of EDAS method as an effective tool for solving MCDM problems with conflicting criteria. In this method, the optimal solution is estimated based on the distance from average solution. Therefore, there is no need to estimate the ideal and anti-ideal solutions. This is the key benefit of the EDAS approach. As FFS is one of the new ways to deal with the uncertainty of real-life applications. So, in this section, we combine the CRITIC and EDAS approaches with FFSs for solving complicated decision-making problems and named as Fermatean fuzzy CRITIC-EDAS (FF-CRITIC-EDAS) (see Fig. 3 ).

Step 1. Problem description Consider that K 1 , K 2 , ..., K p be a set of options and N 1 , N 2 , ..., N q be a criterion set. Let us suppose that a group of decision makers ( ij signifies the evaluation information of an option K i w.r.t. criterion N j in the form of FFNs.

Step 2. Determination of DMs' weights In order to compute the significance degrees of DMs, first of all, consider the significance degrees of the DMs in term of FFNs. Now, suppose W = b k , n k be the importance rating of k th DM expressed by an authority in terms of FFN, then the formula for the computation of k th DM's weight is presented as follows:

Here, k ≥ 0 and r ∑ k=1 k = 1.

Step 3. Aggregate the individual opinions of DMs In the MCDM problem, it is important to merge all the individuals' opinions of DMs into a collective opinion to form the aggregated Fermatean fuzzy decision matrix To facilitate this, FF-weighted averaging operator (FFWAO) is employed on FF-DM and let Z = z ij p × q , (i = 1, 2, ..., p,j = 1, 2, ..., q) be the A-FFDM, where

Step 4. Evaluate the criteria weight by employing CRITIC technique Firstly, consider that X = 1 , 2 , ..., q T be the weight of the criterion set with j ∈ [0, 1] and q ∑ j=1 j = 1 . Now, the computation procedure of CRITIC method is presented in the following steps:

Step 4.1. Evaluate the score matrix Ξ = ij p × q , wherein

Step 4.2. Switch the score matrix Ξ into a standard Fer-

Step 4.3. Estimate the criteria standard deviations with the use of following formula:

Step 4.4. Calculate the correlation between criteria with the use of following formula:

Step 4.5. Evaluate the quantity of information of each criterion by using

The better the c j is, the more information an attribute contains, consequently the weight of evaluation attribute is superior than that of other attributes.

Step 4.6. Calculate the weight of each criterion as follows:

Step 5. Assess the average solution (AVS) associated to the criteria

Step 6. Estimate the positive distance and the negative distance from average solution matrix according as the benefit and cost criteria If N b and N n are the set of benefit and cost criteria, respectively, then the positive distance from average (PDA) and the negative distancefrom average (NDA) are computed as below:

such that wherein h ij and ij describe the positive and the negative distances of assessment degrees of i th option from the AVS on jth criterion, respectively, and ℘ * s i shows the score value of AVS.

Step 7 Step 8. For all options, determine the normalize values of ℤ (+) i and ℤ (−) i , given as

z ij , i = 1, 2, ..., p, j = 1, 2, ..., q.

Here, the terms ' ℘ * s ℤ (+) i ' and ' ℘ * s ℤ (−) i ' denote the score values of weighted sum of PDA and NDA, respectively.

Step 9. Evaluate the appraisal score for all options, given as

Step 10. Based on the score values ℘ * s ℂ i of appraisal scores ℂ i (i = 1, 2, ..., p), determine the ranking of the alternatives. As a result, the alternative with the highest appraisal score is the most appropriate choice among the others.

Step 11. End.

In this section, we present an illustrative example of Indian electronics manufacturing firm to reveal the effectiveness and potentiality of the proposed FF-CRITIC-EDAS methodology. The considered manufacturing firm is situated in Gurugram, India and at present, this firm has five S3PRLP options (K 1 , K 2 , K 3 , K 4 , K 5 ), correspondingly. In order to process the present decision-making problem, we form a group of three skilled decision makers ( O 1 , O 2 , O 3 ). The given S3PRLP options are evaluated on the basis of following 13 attributes/criteria: Cost of pollution control (N 1 ), Cost of green product and eco-design (N 2 ), Green warehousing (N 3 ), Green R & D and innovation (N 4 ), Environmental management system (N 5 ), Costs (N 6 ), Flexibility (N 7 ), Quality (N 8 ), Technology capability (N 9 ), Health and safety practices (N 10 ), Social responsibility (N 11 ), Education Infrastructure (N 12 ) and Employment Practices (N 13 ). The preferred criteria in this study are taken from different resources. The descriptions of these attributes are shown in Table 1 . In this study, the criteria N 1 , N 2 and N 6 are of cost type and remaining others are of benefit type. The implementation procedures of FF-CRITIC-EDAS method for the evaluation of S3PRLPs are as given below:

Step 1. Let us assume that the importance ratings of DMs' judgements are given in the form of FFNs which as {(0.80, 0.50, 0.7133), (0.85, 0.45, 0.6655), (0.75, 0.55, 0.7440)}. Now, the FF-DM is constructed in Table 2 .

Steps 2-3: As the weights ofthe DMs' as expressed by the experts are conveyed in the form of FFNs. Next, with the use of Eq. (2), the DMs' weights are obtained in the following crisp form: { Ψ 1 = 0.3341, Ψ 2 = 0.3807, Ψ 3 = 0.2852}. Further, the evaluation opinions provided by three DMs are aggregated with the use of Eq. (3) and then required A-FFDM are presented in Table 3 .

Step 4. In the step, the criteria weights are determined by CRITIC approach, which as

Step 4.1. First, with the use of Eq. (4) and Table 3 , the score values of A-FFDM are evaluated.

Step 4.2. Convert the score matrix Ξ = ij p × q into the standard FF-matrix Ξ = ̃i j p × q by employing the Eq. (5).

Steps 4.3-4.6. With the use of Eqs. (6)-(8), the standard deviation, correlation coefficient and quantity of information of each factor are computed and presented in Table 4 . By employing Eq. (9), the criteria weights ( j ) are estimated and then shown in Table 4 .

Steps 5-9. By employing Eq. (10) and Table 3 , the AVS matrix is computed and shown in Table 3 . With the use of formulae (11)-(17), the PDA and NDA based on types of criteria, weighted sum of PDA and NDA for all alternatives, the normalize values of weighted sum of PDA and NDA and the appraisal score ℂ i for all options are estimated and presented in Tables 5, 6, respectively.

Step 10. On the basis of score function ℘ * s ℂ i of appraisal score ℂ i , the option K 3 is the most optimal choice and the ranking order of the S3PRLP alternatives is

In this section, a comparison with Fermatean Fuzzy-TOP-SIS (Senapati & Yager, 2019a) and Fermatean Fuzzy-WPM (Senapati & Yager, 2019b) is presented to verify the robustness of the proposed methodology.

The Fermatean Fuzzy-TOPSIS method involves the following calculation procedures:

Steps 1-4. Same as FF-CRITIC-EDAS framework.

Step 5. In the present case study, N 1 , N 2 and N 6 are cost criteria and rest all are benefit criteria, therefore, there is a need to normalize the A-FFDM.

Step 6 Now, find out the distances between the alternatives K i and the Fermatean fuzzy positive and negative ideal solutions over the attribute N j .

Step 7. The relative closeness to the Fermatean fuzzy positive ideal solution is determined by utilizing

and Therefore, the obtained values are R K 1 = 0.589, R K 2 = 0.343, R K 3 = 0.499, R K 4 = 0.472, R K 5 = 0.480.

Step 8: The preference order of S3PRLP options are K 1 ≻ K 3 ≻ K 5 ≻ K 4 ≻ K 2 , thus, the option K 1 is the best S3PRLP.

Steps 1-5. Same as FF-CRITIC-EDAS approach.

Step 6. Based on Weighted Product Model (WPM), the total relative significance of option K i is calculated using (2002) preference order of options are as K 3 ≻ K 4 ≻ K 5 ≻ K 1 ≻ K 2 . Hence, the alternative K 3 is the best choice among the given S3PRLPoptions.

Based on Fermatean fuzzy-TOPSIS method, the preference ordering of the S3PRLP alternatives is K 1 ≻ K 3 ≻ K 5 ≻ K 4 ≻ K 2 , and thus, the option K 1 is the best choice. Similarly, with the use of Fermatean fuzzy-WPM, the final ranking of the S3PRLP alternatives is K 3 ≻ K 4 ≻ K 5 ≻ K 1 ≻ K 2 and thus, K 3 is the most desirable option. Figure 4 presents the graphical representation of ranking of the options by different approaches. Consequently, we can see that the desirable S3PRLP alternative, i.e., (K 3 ) is same by using Fermatean fuzzy-WPM and introduced approach, whereas the ranking result somewhat differ by Fermatean fuzzy-TOPSIS approach and the desirable choice is K 1 . Also, by comparing with Tavana et al. (2018) and Li et al. (2018) approaches, the final ranking order of the S3PRLP alternatives is K 3 ≻ K 5 ≻ K 4 ≻ K 2 ≻ K 1 and K 3 ≻ K 4 ≻ K 5 ≻ K 2 ≻ K 1 , respectively, therefore, the most appropriate choice is K 3 among all other S3PRLPs. The introduced methodology discussed in this study is found proficient for solving the MCDM problems with conflicting criteria. The main advantages of the proposed method are listed as below:

-To deal with the ambiguity in the MCDM problems, all input variables are taken into account as uncertain issues described by Fermatean fuzzy numbers. The indeterminacy degree is considered necessary independently in the whole method and the options are put in rank utilizing trade-off values of all three parameters, unlike Tavana (Senapati and Yager 2019a ) and fuzzy-TOPSIS (Tavana et al. 2018) approaches, in which the "positive ideal solution" and "negative ideal solution" are calculated by the experts as per their own facts, whilst in EDAS method, Fermatean fuzzy weighted averaging operator is employed for the determination of AVS, which is simple and free from the impact of human concerns. Furthermore, the subtraction procedure and IGSF of Fermatean fuzzy numbers is used . 4 The significance degrees of alternatives over different methods to estimate the "PDA" and the "NDA" of each alternative. It can fruitfully evade the selecting distance measure which needs to add or reduce the number of objects of FFSs and lead to the distortion of information. -The developed FF-CRITIC-EDAS framework is not only appropriate for evaluating the MCDM problems under FFSs context, but can also successfully tackle with the MCDM problems under fuzzy sets, intuitionistic fuzzy sets and Pythagorean fuzzy sets contexts. The introduce methodology has the benefits of easy computation process and fast information processing.

Some limitations of the proposed method are as follows:

-In the proposed FF-CRITIC-EDAS method, all criteria are assumed to be independent. In reality, there are interrelationships among the criteria. -This method has limitation in order to deal with a large number of criteria.

The past period has witnessed ever-increasing issue about the disposal of customer goods, since numerous of these materials comprise both large quantities of waste and considerable amounts of toxic heavy metals. Manufacturers have faced intensifying pressure from both governments and environmentally concentrated committee to 'reduce', 'recycle', and 'reuse' their industrial waste. Presently, RL has been considered as a main concern. The operative management of RL is useful to environmental safety, and it can carry the evident economic benefits to organizations. Many corporations do not hold sufficient resource or capability to accomplish their RL activities, thus they have to select the S3PRLP to those activities. In this study, a novel FF-CRITIC-EDAS method was used to choose the suitable S3PRLPs for Indian manufacturing company located in Gurugram, India. A case study is taken to display some important insights regarding assessment criteria and prominent S3PRLP options. To do this, proposed IGSF is applied to compute the DMs' weights, CRITIC model is used to assess the importance value of criteria and proposed EDAS method is implemented to rank the S3PRLP options. The obtained results by proposed method show that the option K 3 is the most appropriate provider for this case. Moreover, the comparative discussion with extant models is also presented to elucidate the rationality of the introduced method. Thus, we found that K 3 is the most suitable choice among a set of S3PRLPs. As a consequence, the proposed model has significant information that can be utilized by administrators in taking strategic or operational decisions in S3PRLPs evaluation.

Without loss of generality, the proposed framework would be correspondingly appropriate to other concerns and different organizations. It can also be applied as a standard procedure for service providers in guiding their modifications to the processes and strategic instructions, so that they can well assistance with client and societal potentials. Simultaneously, administrations and governing bodies can employ the introduced method to study the relationships among economic, environmental, and social concerns, and utilize the outcomes to influence and reassure stronger law and strategy execution on the sustainability.

The assessment of S3PRLP has become one of important decisions for the enterprises in the modern competitive market. The objective of this study is to introduce a MCDM methodology for assessing and selecting the optimal S3PRLP option on FFSs. To do this, firstly novel IGSF has been proposed to compare the options. Secondly, a hybrid framework based on CRITIC and EDAS methods with FFSs has been developed to solve the MCDM problems, wherein the DMs and attribute weights are completely unknown. In this framework, IGSF-based procedure has been proposed to compute the DMs' weights and the attribute weights have been calculated by applying CRITIC approach. Further, a case study of S3PRLP selection has been taken to elucidate the practicality and effectiveness of the introduced framework. For this, an evaluation index process for S3PRLPs has been organized, which comprises three prime aspects. These aspects are characterized into five, four and four criteria, respectively, which are broadly deliberated according to the existing literatures. The CRITIC approach determines the weights of the considered criteria, which as Education infrastructure (0.1183), Flexibility (0.1097), Cost of green product and eco-design (0.1013), Green R & D and innovation (0.0975), Green warehousing (0.0745), Quality (0.0731), Environmental management system (0.0680), Technology capability (0.0644), Health and safety practices (0.0605), Costs (0.0603), Social responsibility (0.0587), Cost of pollution control (0.0569) and Employment practices (0.0567). Next, by employing the EDAS method, the priority order of S3PRLPs is obtained as K 3 ≻ K 4 ≻ K 5 ≻ K 1 ≻ K 2 . Next, comparison with extant models has been made to validate the introduced framework. The outcomes verify that the introduced model has good proficiency and strength than the extant models. In addition, the introduced approach not only offers the priority order of the S3PRLP options but also illustrates the attributes performances in the S3PRLP selection.

In future, we will work on diverse MCDM approaches (namely, CoCoSo, WASPAS, MULTIMOORA or DNMA) to select the optimal S3PRLP under FFSs context. Also, we will implement the introduced framework to the different problems, namely, EVCS site evaluation, HCWD method assessment, green supplier assessment, and other decisionmaking problems.

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A framework for sustainable forest resource allocation: a Canadian case study

A novel fuzzy multi-criteria decision framework for sustainable supplier selection with incomplete information

A new ranking method for TOPSIS and VIKOR under interval valued intuitionistic fuzzy sets and possibility measures

Analysis of enablers for implementation of sustainable supply chain management-a textile case

Determining objective weights in multiple criteria problems: The CRITIC method

Integrated forward/reverse logistics network design under uncertainty with pricing for collection of used products

An ELECTRE-based multiple criteria decision making method for supplier selection using Dempster-shafer theory

Decision-making analysis based on fermatean fuzzy yager aggregation operators with application in COVID-19 testing facility

Multi-criteria inventory classification using a new method of evaluation based on distance from average solution (EDAS)

Extended EDAS method for fuzzy multi-criteria decision-making: an application to supplier selection

Assessment of third-party logistics providers using a CRITIC-WASPAS approach with interval type-2 fuzzy sets

A new hybrid fuzzy MCDM approach for evaluation of construction equipment with sustainability considerations

Analysis of third party reverse logistics provider using interpretive structural modeling

Reverse logistics and closed-loop supply chain: A comprehensive review to explore the future

Multicriteria group decision making for the third party reverse logistics service provider in the supply chain model using fuzzy TOPSIS for transportation services

A novel hesitant fuzzy EDAS method and its application to hospital selection

An extended EDAS method for multicriteria decision-making based on multivalued neutrosophic sets

CODAS method for Pythagorean 2-tuple linguistic multiple attribute group decision making

Evaluation of third party logistics providers considering social sustainability

Intuitionistic fuzzy EDAS method: an application to solid waste disposal site selection

Fuzzy axiomatic design approach based green supplier selection: a case study from Singapore

Integrating disassembly line balancing in the planning of a reverse logistics network from the perspective of a third party provider

A novel extension to VIKOR method under intuitionistic fuzzy context for solving personnel selection problem

Multi-criteria COPRAS method based on parametric measures for intuitionistic fuzzy sets: application of green supplier selection

Integration of artificial neural network and MADA methods for green supplier selection

A 3PL supplier selection model based on fuzzy sets

Third party reverse logistics provider selection approach based on hybrid-information MCDM and cumulative prospect theory

An EDAS method for multiple attribute group decision-making under intuitionistic fuzzy environment and its application for evaluating green building energy-saving design projects

A hybrid model for selection of outsourcing provider

An integrated fuzzy approach for provider evaluation and selection in third-party logistics

The selection of 3PRLs on self-service mobile recycling machine: Interval-valued Pythagorean hesitant fuzzy best-worst multi-criteria group decision-making

Sustainable third party reverse logistic provider selection with fuzzy SWARA and fuzzy MOORA in plastic industry

A conceptual model for selecting and evaluating third-party reverse logistics providers

An integrated approach to multiple criteria decision making based on the average solution and normalized weights of criteria deduced by the hesitant fuzzy best worst method

Interval-Valued Intuitionistic Fuzzy WAS-PAS method: application in reservoir flood control management policy

A novel EDAS approach on intuitionistic fuzzy set for assessment of health-care waste disposal technology using new parametric divergence measures

Healthcare evaluation in hazardous waste recycling using novel interval-valued intuitionistic fuzzy information based on complex proportional assessment method

A novel entropy and divergence measures with multicriteria service quality assessment using interval-valued intuitionistic fuzzy TODIM method

Intuitionistic fuzzy divergence measure-based ELECTRE method for performance of cellular mobile telephone service providers

New multiparametric similarity measure and distance measure for interval neutrosophic set with IoT industry evaluation

Fuzzy decision making method based on CoCoSo with CRITIC for financial risk evaluation

Pythagorean fuzzy MCDM method based on CoCoSo and CRITIC with score function for 5G industry evaluation

A novel VIKOR approach based on entropy and divergence measures of Pythagorean fuzzy sets to evaluate renewable energy technologies in India

An extended Pythagorean fuzzy complex proportional assessment approach with new entropy and score function: application in pharmacological therapy selection for type 2 diabetes

A novel WASPAS approach for multi-criteria physician selection problem with intuitionistic fuzzy type-2 sets

A new Pythagorean fuzzy based decision framework for assessing healthcare waste treatment

Evaluation and selection of third party logistics provider under sustainability perspectives: an interval valued fuzzy-rough approach

A mathematical model for selecting third-party reverse logistics providers

A new model for selecting third-party reverse logistics providers in the presence of multiple dual-role factors

Decision support framework for selection of 3PL providers: Dominance-based approach in combination with grey set theory

Fermatean fuzzy sets

Some new operations over Fermatean fuzzy numbers and application of Fermatean fuzzy WPM in multiple criteria decision making

Fermatean fuzzy weighted averaging/geometric operators and its application in multi-criteria decision-making methods

A robust hybrid multi-criteria decision making methodology for contractor evaluation and selection in third-party reverse logistics

Evaluating mobile services using integrated weighting approach and fuzzy VIKOR

Hybrid Fuzzy MCDM Approach for Sustainable Third-Party Reverse Logistics Provider Selection

A conceptual analytic network model for evaluating and selecting third-party reverse logistics providers

An intuitionistic fuzzy-grey superiority and inferiority ranking method for third-party reverse logistics provider selection

An improved MUL-TIMOORA approach for multi-criteria decision-making based on interdependent inputs of simplified neutrosophic linguistic information

An integrated DEMA-TEL and Fuzzy ANP techniques for evaluation and selection of outsourcing provider for a telecommunication company

Algorithms for probabilistic uncertain linguistic multiple attribute group decision making based on the GRA and CRITIC method: application to location planning of electric vehicle charging stations

Improving quality function deployment analysis with the cloud MULTIMOORA method

Pythagorean membership grades in multicriteria decision making

A multi-criteria performance analysis of initial public offering (IPO) firms using CRITIC and VIKOR methods

A hybrid data analytic methodology for 3PL transportation provider evaluation using fuzzy multi-criteria decision making

Fuzzy sets

Sustainable thirdparty reverse logistics provider evaluation and selection using fuzzy SWARA and developed fuzzy COPRAS in the presence of risk criteria

A novel hybrid multiple attribute decision-making approach for outsourcing sustainable reverse logistics

The dominance degree-based heterogeneous linguistic decision-making technique for sustainable S3PRLP selection

Evaluation Based on Distance from Average Solution Method for Multiple Criteria Group Decision Making under Picture 2-Tuple Linguistic Environment

Pythagorean fuzzy combinative distance-based assessment with pure linguistic information and its application to financial strategies of multi-national companies

Interval-valued intuitionistic fuzzy TODIM method based on Schweizer-Sklar power aggregation operators and their applications to group decision making