**1. Introduction**

The traditional EDAS (evaluation based on distance from average solution) method [1], which can consider the conflicting attributes, has been studied in many multi-attribute decision making (MADM) problems. By computing the average solution (AV), this model can describe the difference between all the alternatives and the AV based on two distance measures which are namely PDA (Positive Distance from Average) and NDA (Negative Distance from Average), the alternative with higher values of PDA and lower values of NDA is the best choice. Until now, lots of MADM methods such as the VIKOR (VIseKriterijumska Optimizacija I KOmpromisno Resenje) method [2,3], the ELECTRE (ELimination and Choice Expressing the Reality) method [4], the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method [5], the PROMETHEE (Preference Ranking Organisation Method for Enrichment Evaluations) method [6,7], the GRA (Grey relational Analysis) method [8], the MULTIMOORA method [9] and the TODIM (an acronym in Portuguese for Interactive Multi-Criteria Decision Making) method [10–12] were broadly investigated by a large amount of scholars. Compared to the existing work, the EDAS model owns the merit of only taking AVs into account with respect to the intangibility of decision maker (DM) and the uncertainty of the decision making environment to obtain more accurate and effective aggregation results.

Atanassov [13] introduced the concept of intuitionistic fuzzy sets (IFSs), which is a generalization of the concept of fuzzy sets [14]. Atanassov and Gargov [15], and Atanassov [16] proposed the concept of interval-valued intuitionistic fuzzy sets (IVIFSs), which are characterized by a membership function, a non-membership function, and a hesitancy function whose values are intervals. Recently, Cuong and Kreinovich [17] proposed picture fuzzy sets (PFSs) and investigated some basic operations and properties of PFSs. The PFS is characterized by three functions expressing the degree of membership, the degree of neutral membership, and the degree of non-membership. The only constraint is that the sum of the three degrees must not exceed 1. Singh [18] presented the geometrical interpretation of PFSs and proposed correlation coefficients for PFSs. Son [19] presented a novel distributed picture fuzzy clustering (FC) method on PFSs. Thong and Son [20] proposed the model between picture FC and intuitionistic fuzzy recommender systems for medical diagnosis. Thong and Son [21] proposed automatic picture fuzzy clustering (AFC-PFS) for determining the most suitable number of clusters for AFC-PFS. Wei [22] proposed the MADM method based on the proposed picture fuzzy cross entropy. Son [23] defined the generalized picture distance measures and picture association measures. Son and Thong [24] developed some novel hybrid forecast models with picture FC for weather nowcasting from satellite image sequences. Wei [25] gave some cosine similarity measures of PFSs for strategic decision making on the basis of traditional similarity measures [26,27]. Wei [28] proposed some aggregation operators for MADM based on the PFSs based on traditional aggregation operators [29–35]. Wei [36] defined some similarity measures for PFSs. Wei [12] proposed the TODIM method for picture fuzzy MADM. Wei and Gao [37] developed the generalized dice similarity measures for PFSs. Wei [38] proposed some picture fuzzy Hamacher aggregation operators in MADM with traditional Hamacher operations [39–42]. Wei et al. [43] designed the projection models for MADM with picture fuzzy information. Wei et al. [44] proposed some picture 2-tuple linguistic operators in MADM. Wei [45] proposed some Bonferroni mean (BM) operators with P2TLNs in MADM. Wei [46] defined some picture uncertain linguistic BM operators for MADM.

Turskis et al. [1] originally defined the EDAS method for multi-criteria inventory classification. Keshavarz Ghorabaee et al. [47] proposed the extended EDAS method for supplier selection. Kahraman et al. [48] established the EDAS model under intuitionistic fuzzy information for solid waste disposal site selection. Keshavarz Ghorabaee et al. [49] extended the EDAS method with interval type-2 fuzzy sets. Keshavarz Ghorabaee et al. [50] defined the multi-criteria EDAS model with interval type-2 fuzzy sets. Keshavarz Ghorabaee et al. [51] proposed the stochastic EDAS method for MADM with normally distributed data. Peng and Liu [52] resolved the neutrosophic soft decision making method based on EDAS. Ecer [53] gave third-party logistics provider selection with the fuzzy AHP and the fuzzy EDAS integrated method. Feng et al. [54] developed the EDAS method for hesitant fuzzy linguistic MADM. Ilieva [55] assigned the group decision models with EDAS for interval fuzzy sets. Karasan and Kahraman [56] defined the interval-valued neutrosophic EDAS method. Keshavarz-Ghorabaee et al. [57] developed the dynamic fuzzy EDAS method for multi-criteria subcontractor evaluation. Stevic et al. [58] gave the selection of carpenter manufacturer using the fuzzy EDAS method. Keshavarz-Ghorabaee et al. [59] gave a comparative analysis of the rank reversal phenomenon with the EDAS and TOPSIS methods.

Wei et al. [44] introduced the concept of P2TLSs based on PFSs [17] and the 2-tuple linguistic information processing model [60], and developed some BM and geometric BM operators with P2TLNs. However, no studies using the EDAS model with P2TLNs were found in the literature. Hence, it was necessary to take the picture 2-tuple linguistic EDAS model into account. The purpose of our work is to establish an extended EDAS model according to the traditional EDAS method and P2TLNs to study multiple criteria group decision making (MCGDM) problems more effectively. Thus, the main contributions of this paper are (1) to extend EDAS models to picture 2-tuple linguistic sets; (2) to combine the traditional EDAS model for MCGDM with P2TLNs; (3) to provide a numerical case for green supplier selection to illustrate this new model and conduct some comparisons between the EDAS model with P2TLNs, and P2TLWA and P2TLWG aggregation operators to further illustrate advantages of the new method.

The structure of our paper is organized as follows: definition, score function, accuracy function, and operational formulas of P2TLNs are briefly introduced in Section 2. We introduce some aggregation operators of P2TLNs in Section 3. We combine the traditional EDAS model for MCGDM with P2TLNs, and the computing steps are simply depicted in Section 4. In Section 5, a numerical example for green supplier selection has been given to illustrate this new model, and some comparisons between the use of P2TLWA and P2TLWG operators in the EDAS model with P2TLNs were also conducted to further illustrate the advantages of the new method. Section 6 describes some conclusions of our work.
