The idea of intuitionistic fuzzy sets (IFSs) was initially developed by Atanassov (1986) to generalize the notion of fuzzy set (Zadeh, 1965). Zhou et al. (2019) defined the normalized weighted Bonferroni Harmonic mean-based intuitionistic fuzzy operators for sustainable selection of search and rescue robots. There were two described variables in IFSs, including the degrees of membership and non-membership. In terms of IFSs, Atanassov and Gargov (1989) and Atanassov (1994) gave the theory of interval-valued intuitionistic fuzzy sets (IVIFSs) which can describe fuzzy numbers more exactly and reasonably. Lu and Wei (2019) designed the TODIM method for performance appraisal on social-integration-based rural reconstruction under IVIFSs. Wu et al. (2019a) gave the VIKOR method for financing risk assessment of rural tourism projects under IVIFSs. Wu et al. (2020) proposed some interval-valued intuitionistic fuzzy Dombi Heronian mean operators for evaluating the ecological value of forest ecological tourism demonstration areas. Wu et al. (2019b) designed the algorithms for competitiveness evaluation of tourist destination with some interval-valued intuitionistic fuzzy Hamy mean operators. However, in reality, there exist some particular situations when the neutral membership degree is needed to be calculated independently. Thus, to conquer this defect and obtain more rigorous information, Cuong and Kreinovich (2015) initiated the theory of picture fuzzy sets (PFSs) which took another described variable (neutral membership) into consideration. There are three described variables in PFSs which are the degrees of membership, neutral membership and non-membership. The only condition that must be fulfilled is that the three described variables’ sum cannot exceed 1. As a powerful tool, PFSs deliver more comprehensive information which the application of some particular situations required more answer types of human ideas: yes, abstain, no, refusal. Cuong et al. (2015) found PFSs’ main logic operators and developed the main operations of reasoning process in PFSs by linking the triple picture fuzzy operators of De Morgan. Garg (2017) investigated several PFSs’ aggregation operators, including PFWA, PFOWA and PFMA aggregation operators. Xu et al. (2018) combined Muirhead mean (MM) operator with PFSs to develop PFMM operator and created a novel method which can be widely applied in attribute values to address MADM issues. Zhang et al. (2018a) found several novel operational rules of PFSs relying on Dombi t-conorm and t-norm (DTT) and made use of the information aggregation technology of Heronian mean (HM) to integrate PFNs. Jana et al. (2018) put forward a model which was related to picture fuzzy Dombi aggregation operators to address MADM issues in an updated way. Wei (2016) gave the notion about picture fuzzy cross entropy and established the entropy of the alternative attribute value of PFNs. Joshi and Kumar (2018) pointed out an approach for MADM issues derived from the Dice similarity and weighted Dice similarity measures for PFSs. Son (2017) extended the fundamental distance measure in PFSs to the generalized picture distance measures and picture association measures. Liu et al. (2019) explored some distance measures for PFSs and proposed Picture fuzzy ordered weighted distance measure and Picture fuzzy hybrid weighted distance measure for MAGDM method in an updated way. Singh (2015) presented the concept about the PFSs’ geometrical interpretation and made a correlation coefficient of PFSs. Son (2015) found DPFCM method which was an innovative distributed picture fuzzy clustering method. Thong and Son (2015) put forward a novel hybrid model which was an application of medical diagnosis on the basis of picture fuzzy clustering and intuitionistic fuzzy recommender systems. Wang et al. (2018a) utilized picture fuzzy information to formulate a framework which was related to hybrid fuzzy MADM to sort the EPC projects’ risk factors. Wang et al. (2018b) integrated the PFNP model with VIKOR method to create a method called picture fuzzy normalized projection-based VIKOR. Liang et al. (2018) integrated EDAS method with ELECTRE module in PFSs to infer the level of cleaner production. Ashraf et al. (2018) made a discussion about the weighted geometric aggregation operator’s generalized form in PFSs and proposed TOPSIS method to aggregate PFNs. Ju et al. (2018) extended the classical GRP approach to PFSs and calculated each EVCS site’s relative grey relational projection. Wei et al. (2019b) defined an extended bidirectional projection algorithms for picture fuzzy MAGDM issue for safety assessment of construction project. Furthermore, Wei et al. (2018) put forward the concept of P2TLSs on the basis of PFSs and 2-tuple linguistic term sets. Wei (2017) developed the P2TLWBM operator and the P2TLWGBM operator on the basis of Bonferroni mean. Zhang et al. (2018b) presented P2TLNs’ novel operational laws which can conquer the limitation of existing operations relating to PFNs and P2TLNs. Zhang et al. (2019b) designed the MABAC method for MAGDM issue under P2TLSs.

With the continuous destruction of the human environment and the shortage of earth resources, the traditional supply chain has gradually failed to adapt to the current era and the needs of society, thus introducing the concept of green supply chain. The establishment of green supply chain has become the main challenge and trend for enterprises to provide green products and move towards a sustainable development society. Among them, the important link and core content of implementing green supply chain is the evaluation and selection of green suppliers, especially those with sustainable development who meet the requirements of green environmental protection. Because supplier selection plays an important role in green supply chain management, it directly determines the optimization of the whole chain and the core competitiveness of the enterprise. Therefore, how to efficiently determine the required suppliers from a large number of suppliers is the key problem to be solved in modern green supply chain management. The green supplier selection problem is based on multiple attributes and many experts, as it is not a single-attribute problem (He et al., 2019b; Hu et al., 2016; Lei et al., 2019; Li et al., 2020; Wang et al., 2019c; Wang P. et al., 2019a, 2019b). In this respect, multiple attribute group decision making (MAGDM) techniques or tools can be used to investigate this problem in a better way (Deng and Gao, 2019; Gao et al., 2019; Li and Lu, 2019; Wang et al., 2019a; Wang, 2019). MAGDM methods are used to rank suppliers or to choose the most appropriate and favourable supplier on the basis of multiple attributes and many experts (Mohammadi et al., 2017; Paydar and Saidi-Mehrabad, 2017). Many researchers have employed different techniques to select green suppliers. Gao et al. (2020) developed the VIKOR method for MAGDM based on q-rung interval-valued orthopair fuzzy information for supplier selection of medical consumption products. Hashemi et al. (2015) defined an integrated green supplier selection approach with analytic network process and improved Grey relational analysis. Awasthi and Kannan (2016) designed the green supplier development program selection by using NGT and VIKOR under fuzzy environment. Liou et al. (2016) developed a new hybrid COPRAS-G MADM model for improving and selecting suppliers in green supply chain management. Wei et al. (2019a) proposed the supplier selection of medical consumption products with the probabilistic linguistic MABAC method. In Wang et al. (2019b), the q-rung orthopair hesitant fuzzy weighted power generalized Heronian mean (q-ROHFWPGHM) operator and the q-rung orthopair hesitant fuzzy weighted power generalized geometric Heronian mean (q-ROHFWPGGHM) operator are applied to deal with green supplier selection in supply chain management. Liu and Wang (2018) designed some interval-valued intuitionistic fuzzy Schweizer–Sklar power aggregation operators for supplier selection. Kannan et al. (2013) integrated fuzzy multi criteria decision making method and multi-objective programming approach for supplier selection and order allocation in a green supply chain.

CODAS (Combinative Distance-based Assessment) method was initially developed by Ghorabaee et al. (2017) to tackle the multi-criteria decision making issues. In recent years, there existed various related extensions to enrich this method. Bolturk (2018) integrated CODAS method with Pythagorean fuzzy environment. Ghorabaee et al. (2016) utilized linguistic variables and trapezoidal fuzzy numbers to extend the CODAS method. Badi et al. (2017) made use of a novel CODAS method to address MCDM issues for a steelmaking company in Libya. Pamucar et al. (2018) presented an original MCDM Pairwise-CODAS method which was the modification of the classical CODAS method. Roy et al. (2019) built an assessment framework for addressing MCDM issues by extending CODAS method with interval-valued intuitionistic fuzzy numbers. So far, we have failed to find the work of the CODAS method with P2TLNs in the existing literature. Thus, investigating the CODAS method with P2TLNs is essential. The fundamental objective of our research is to develop an original method which can be more effective to address some MAGDM issues within the CODAS method and P2TLSs. Hence, the highlights of this essay are illustrated subsequently. Above all, we intend to extend the CODAS method to the picture 2-tuple linguistic environment. In addition, since the DMs are restrained by their knowledge, it is tricky to assign the criteria weights directly. Hence, CRITIC method is utilized to decide each attribute’s weight. Last but not least, an empirical application is offered to demonstrate this novel approach and several comparative analysis between the CODAS method with P2TLNs and other methods are also offered to further demonstrate the merits of the novel approach.

The reminder of our essay proceeds as follows. Some fundamental knowledge of PFSs and P2TLSs is concisely reviewed in Section 2. Several aggregation operators of P2TLNs are presented in Section 3. The CODAS approach is integrated with P2TLNs and the calculating procedures are simply depicted in Section 4. An empirical application of green supplier selection is given to show the merits of this approach and some comparative analysis is also offered to further prove the merits of this method in Section 5. At last, we make an overall conclusion of our work in Section 6.

In this chapter, under the picture 2-tuple linguistic environment, some arithmetic aggregation operators are introduced, including picture 2-tuple linguistic weighted averaging (P2TLWA) operator and picture 2-tuple linguistic weighted geometric (P2TLWG) operator.

Derived from the Definition 9, the subsequent result can be easily acquired:

Derived from the Definition 10, the subsequent result can be easily acquired: Theorem 2.

The aggregated value by utilizing P2TLWG operator is also a P2TLN, where (14) P2TLWGω(pˆ1,pˆ2,…,pˆn)= ⨂j=1n(pˆj)ωj=⟨Δ( ∏j=1n(Δ−1(sj,ℓj)ωj)),( ∏j=1n(μj+ηj)ωj− ∏j=1n(ηj)ωj, ∏j=1n(ηj)ωj,1− ∏j=1n(1−νj)ωj)⟩, where ω=(ω1,ω2,…,ωn)T is the weight vector of pˆj (j=1,2,…,n) and ωj>0 0$]]>, ∑j=1nωj=1 .

In this chapter, the CODAS method will be integrated with P2TLNs, which can be utilized to conquer the limitations of the existing multi-attribute value method.

Let C={C1,C2,…,Cn} be the collection of attributes, c={c1,c2,…,cn} be the weight vector of attributes Cj , where cj∈[0,1] , j=1,2,…,n , ∑j=1ncj=1 . Assume L={L1,L2,…,Lr} is a collection of decision makers that have significant degree of l={l1,l2,…,lr} , where lk∈[0,1] , k=1,2,…,r . ∑k=1rlk=1 . Let κ={κ1,κ2,…,κm} be a discrete collection of alternatives. And G=(gij)m×n is the comprehensive picture 2-tuple linguistic decision matrix, gij means the value of alternative κi regarding the attribute Cj .

After that, the specific calculation procedures will be depicted in Fig. 1.

(I) Phase 1: Obtain the assessment information

Step 1. Build each decision maker’s picture 2-tuple linguistic decision matrix Gk=(gijk)m×n and then calculate the comprehensive picture 2-tuple linguistic decision matrix G=(gij)m×n . (15) G(k)=[gijk]m×n=g11kg12k…g1nkg21kg22k…g2nk⋮⋮⋮⋮gm1kgm2k…gmnk, (16) G=[gij]m×n=g11g12…g1ng21g22…g2n⋮⋮⋮⋮gm1gm2…gmn, (17) gij=⟨Δ( ∑k=1rlkΔ−1(sijk,ℓijk)),gij=(1− ∏k=1r(1−μijk)lk, ∏k=1r(ηijk)lk, ∏k=1r(νijk+ηijk)lk− ∏k=1r(ηijk)lk)⟩, where gijk is the assessment value of the alternative κi (i=1,2,…,m) on the basis of the attribute Cj (j=1,2,…,n) and the decision maker Lk (k=1,2,…,r) .

Step 2. Normalize the comprehensive picture 2-tuple linguistic decision matrix by utilizing the following Eq. (18). (18) GN=(gijN)m×n=⟨(sij,ℓij),(μij,ηij,νij)⟩,Cjis a benefit criterion,⟨Δ(T−Δ−1(sij,ℓij)),(νij,ηij,μij)⟩,Cjis a benefit criterion.

(II) Phase 2: Determine the comprehensive criteria weight values

Step 3. Determine the criterion’s weighting matrix by utilizing CRITIC method.

CRITIC (CRiteria Importance Through Intercriteria Correlation) method will be proposed in this part which is utilized to decide attributes’ weights. This method was initially put forward by Diakoulaki et al. (1995) which took the correlations between attributes into consideration. Subsequently, the calculation procedures of this method will be presented.

(1) Depending on the comprehensive picture 2-tuple linguistic decision matrix GN=(gijN)m×n , the correlation coefficient matrix ϑ=(ιjt)n×n is built by calculating the correlation coefficient between attributes. (19) ιjt=∑i=1m(S(gijN)−S(gjN))(S(gitN)−S(gtN))∑i=1m(S(gijN)−S(gjN))2∑i=1m(S(gitN)−S(gtN))2,j,t=1,2,…,n, where S(gjN)=1m∑i=1mS(gijN) and S(gtN)=1m∑i=1mS(gitN) .

(2) Calculate attribute’s standard deviation. (20) SDj=1m ∑i=1m(S(gijN)−S(gjN))2,j=1,2,…,n, where S(gjN)=1m∑i=1mS(gijN) .

(3) Calculate attributes’ weights. (21) cj=SDj∑t=1n(1−ιjt)∑j=1n(SDj∑t=1n(1−ιjt)),j=1,2…,n, where cj∈[0,1] and ∑j=1ncj=1 .

(III) Phase 3: Acquire the ranking results with the CODAS method

Step 4. Calculate the weighted normalized matrix. The weighted normalized performance values (g˜ij) are calculated as in Eqs. (22) and (23): (22) G˜=[g˜ij]m×n, (23) g˜ij=cj⊗gijN, where cj denotes the weights of the jth criterion.

Step 5. The negative-ideal solution is defined using Eqs. (24) and (25): (24) NS˜=[nsj˜]1×n, (25) nsj˜=minig˜ij, where minig˜ij={g˜hj|S(g˜hj)=mini(S(g˜hj)),h∈{1,2,…,n}} .

Step 6. Calculate alternatives’ weighted Euclidean ( EDi ) and weighted Hamming ( HDi ) distances from the negative-ideal solution as in Eqs. (26) and (27): (26) EDi= ∑j=1ndE(g˜ij,nsj˜), (27) HDi= ∑j=1ndH(g˜ij,nsj˜).

Step 7. Determine relative assessment matrix (RA) as in Eqs. (28) and (29): (28) RA=[pih]n×m, (29) pih=(EDi−EDh)+(t(EDi−EDh)×(HDi−HDh)), where h∈{1,2,…,n} and t is a threshold function that is defined in Eq. (30): (30) t(x)=1if|x|⩾θ,0if|x|<θ. In this function, θ is the threshold parameter of this function that can be set by decision maker. In this study, θ=0.05 is taken for the calculations.