TODIM method for probabilistic linguistic multiple attribute group decision making based on the similarity measures and entropy

Abstract

Recently, the TODIM (an acronym in Portuguese for Interactive Multi-criteria Decision Making) method, which can characterize the decision makers’ psychological behaviors under risk, has been introduced to handle multiple attribute group decision making (MAGDM) problems. Moreover, the probabilistic linguistic term sets (PLTSs) are effective tool for depicting uncertainty of the MAGDM problems. In this paper, we extend the TODIM method to the MAGDM with PLTSs. Firstly, the definition, comparative method and distance of PLTSs are simply introduced, and the steps of the classical TODIM method for MAGDM problems are presented. Then, on the basis of the conventional TODIM method, the extended TODIM method is proposed to deal with MAGDM problems in which the attribute values are depicted in the PLTSs, and its significant characteristic is that it can fully consider the decision makers’ bounded rationality which is a real action in decision making. Finally, a numerical example for green supplier selection is proposed to verify the developed approach and its practicality and effectiveness.

Keywords

Multiple attribute group decision making (MAGDM)probabilistic linguistic term sets (PLTSs)TODIM method Prospect theory green supplier selection

1 Introduction

With the increasing complexity of the MADM problems in real world, it may be necessary to take the decision makers (DMs)’ risk attitudes into consideration in the process of MADM [1]. A.Tversky and Kahneman [2] proposed the prospect theory which is a descriptive theory for decision making under risk. This theory incorporates three significant aspects [2]: (1) Reference dependence. The outcomes are manifested by gains and losses according to a reference alternative. (2) Diminishing sensitivity. For gains, the DMs are risk-averse. But for losses, they are risk-preference. (3) Loss aversion. The DMs are much more sensitive to losses than gains. On the basis of the prospect theory, Gomes and Lima [3] first established the TODIM (an acronym in Portuguese for Interactive Multi-Criteria Decision Making), which is effective to solve the MADM problems where the DMs’ psychological behaviors are considered. TODIM is a valuable approach to consider the DMs’ psychological behavior under risk. Gomes and Rangel [4] applied TODIM method for multicriteria rental evaluation of residential properties. Gomes, Rangel and Maranhao [5] used TODIM method for multicriteria analysis of natural gas destination in Brazil. Krohling, Pacheco and Siviero [6] defined an intuitionistic fuzzy TODIM to MADM problems. Lourenzutti and Krohling [7] gave a study of TODIM under intuitionistic fuzzy and random environment. Zhang and Xu [8] developed the hesitant fuzzy TODIM analysis approach based on novel measured functions. Wei, Ren and Rodriguez [9] designed a hesitant fuzzy linguistic TODIM method with a score function. Mishra and Rani [10] studied the biparametric information measures-based TODIM method for interval-valued intuitionistic fuzzy environment. Wei [11] analyzed the TODIM method for Picture Fuzzy MADM. Huang and Wei [12] gave the TODIM method for Pythagorean 2-tuple linguistic MADM. Wang, Wei and Lu [13] defined the TODIM for MAGDM under 2-tuple linguistic neutrosophic environment.

With the increasing complexity of the socioeconomic environment, it looks less possible for a single decision maker (DM) to consider all relevant aspects in MADM problem. As a result, many decision-making processes in the practical world take place in group settings. In many cases, due to inherent complexity, experts can’t express their opinions or preferences by exact numbers, thus resorting to other qualitative preference relations. Herrera and Martinez [14] defined a 2-tuple fuzzy linguistic representation model for computing with words. Rodriguez, Martinez and Herrera [15] proposed the hesitant fuzzy linguistic term sets s (HFLTSs) for decision making. One of the useful theories in dealing with the MADM or MAGDM problems is the theory of probabilistic linguistic term sets (PLTSs) which was proposed by Pang, Wang and Xu [16]. Recently, PLTSs have become a very hot topic in the area of HFLTSs [17 –21] and hesitant fuzzy sets (HFSs) [22 –25]. For example, Pang, Wang and Xu [16] formed a framework for ranking PLTSs and conducted a comparison method with help of the score or deviation degree of each PLTS. Bai, Zhang, Qian and Wu [26] defined more appropriate comparison method and proposed a more efficient way to handle PLTSs. Gou and Xu [27] redefined some novel operational laws for linguistic terms, HFLEs and PLTSs based on two equivalent transformation functions. Zhai, Xu and Liao [28] defined the probabilistic linguistic vector-term sets (PLVTSs) to extend the application of multi-granular linguistic information. Kobina, Liang and He [29] developed some Probabilistic linguistic power aggregation operators for MAGDM on the basis of classical power aggregation operators [30 –32]. Liao, Jiang, Xu, Xu and Herrera [33] designed a linear programming method for MADM with probabilistic linguistic information. Zhang, Xu and Liao [34] gave a consensus algorithm for GDM with probabilistic linguistic preference relations. Cheng, Gu and Xu [35] researched venture capital group decision-making with interaction under probabilistic linguistic environment. Liang, Kobina and Quan [36] proposed the grey relational analysis for probabilistic linguistic MAGDM based on geometric Bonferroni mean [37 –43]. Chen, Wang and Wang [44] developed cloud-based ERP system selection based on probabilistic linguistic MULTIMOORA method and Choquet integral operator. Feng, Liu and Wei [45] established probabilistic linguistic QUALIFLEX method with possibility degree comparison. Lin, Chen, Liao and Xu [46] designed the ELECTRE II method to deal with PLTSs for edge computing. Liao, Jiang, Lev and Fujitac [47] studied the novel operations of PLTSs based on the disparity degrees of linguistic terms in designing the probabilistic linguistic ELECTRE III method. Xie, Xu and Ren [48] researched the Influence of Chinese “New Four Inventions” under the incomplete hybrid probabilistic linguistic environment. Wang, Xu and Gou [49] proposed the nested probabilistic-numerical linguistic term sets in two-stage MAGDM.

Liu and Teng [50] developed Probabilistic linguistic TODIM method for selecting products based on online product reviews. Zhang, Xu and Liao [51] studied the water security evaluation based on the TODIM method with PLTSs. However, the employed distance measures for TODIM method in these two papers [50, 51] are more or less irrational. For example, let us consider two PLTSs L₁ (p) ={ s₀ (0.2) , s₁ (0.8) } and L₂ (p) ={ s₀ (0.6) , s₂ (0.4) }. By Equation (7) in Ref. [50] and distance measures in Ref. [51], it holds that d (L₁ (p) , L₂ (p)) = 0. Obviously, the two PLTSs are not same completely since L₁ (p) contains linguistic term s₁ but L₂ (p) contains linguistic term s₂. Furthermore, the probabilities of linguistic terms in L₁ (p) and L₂ (p) are also not identical. Therefore, the distance measure between L₁ (p) and L₂ (p) should not be equal to zero. However, based on the distance measures of Definition 10 in Ref. [52], it has d (L₁ (p) , L₂ (p)) = 0.2000 > 0. This result reflects the difference between L₁ (p) and L₂ (p), which means that the proposed distance measures of Definition 10 in Ref. [52] overcomes the deficiency in distance measures in these two papers [50, 51].

The aim of this paper is to extend the TODIM method to the MAGDM with the PLTSs based on the distance measures of Definition 10 in Ref. [52] and entropy weight. The main contribution of the paper can be summarized as follows: (1) the TODIM method is extended by PLTSs based on the distance measures of Definition 10 in Ref. [52]; (2) the probabilistic linguistic TODIM (PL-TODIM) method is proposed to solve the MAGDM problems based on the similarity measures and entropy; (3) a case study for green supplier selection is proposed to verify the developed approach; (4) some comparative studies are given with the PL-TOPSIS method and PL-GRA method to verify the rationality of PL-TODIM method.

The remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to PLTSs and conventional TODIM method for MADM problems. In Section 3 we propose the TODIM method for MAGDM problems under PLTSs with help of Shannon entropy. In Section 4, an illustrative example for green supplier selection is pointed out and some comparative analysis is conducted. In Section 5 we conclude the paper and give some remarks.

2 Preliminaries

The mathematical form of PLTS [16] is defined as follows.

Definition 1 [53]. Let L = {l_α|α = - θ, ⋯ , -2, -1, 0, 1, 2, ⋯ θ} be an LTS, the linguistic terms l_α can express the equivalent information to β is derived by the transformation function g: $g : [l_{- θ}, l_{- θ}] \to [0, 1], g (l_{α}) = \frac{α + θ}{2 θ} = β$ (1)

At the same time, β can express the equivalent information to the linguistic terms l_α β is derived by the transformation function g^-1: $g^{- 1} : [0, 1] \to [l_{- θ}, l_{- θ}], g^{- 1} (β) = l_{(2 β - 1) θ} = l_{α}$ (2)

Definition 2 [16]. Given an LTS L = {l_j|j = - θ, ⋯ , -2, -1, 0, 1, 2, ⋯ θ}, a PLTS based on L is defined as:

$\begin{matrix} L (p) & = & {l^{(ϕ)} (p^{(ϕ)}) | l^{(ϕ)} \in L, p^{(ϕ)} \geq 0, \\ ϕ & = & 1, 2, \dots, # L (p), \sum_{ϕ = 1}^{# L (p)} p^{(ϕ)} \leq 1} \end{matrix}$ (3) where l^(ϕ) (p^(ϕ)) is the ϕth linguistic term l^(ϕ) associated with the probability p^(ϕ), and # L (p) is the number of all different linguistic terms in L (p). The linguistic term l^(ϕ) in L (p) are arranged in ascending order.

In order to convenient computation, Pang, Wang and Xu [16] normalized the PLTS L (p) as $\tilde{L} (p) = {l^{(ϕ)} ({\tilde{p}}^{(ϕ)}) | l^{(ϕ)} \in L, {\tilde{p}}^{(ϕ)} \geq 0$ , $ϕ = 1, 2, \dots, # L (\tilde{p}),$ $\sum_{ϕ = 1}^{# L (p)} {\tilde{p}}^{(ϕ)} = 1}$ , where ${\tilde{p}}^{(ϕ)} = p^{(ϕ)} / - \sum_{ϕ = 1}^{# L (p)} p^{(ϕ)}$ for all $ϕ = 1, 2, \dots, # L (\tilde{p})$ .

Definition 3 [16]. Let L = {l_α|α = - θ, ⋯ , -1, 0, 1, ⋯ θ} be an LTS, ${\tilde{L}}_{1} (\tilde{p}) = {l_{1}^{(ϕ)} ({\tilde{p}}_{1}^{(ϕ)}) | ϕ = 1,$ $2, \dots, # {\tilde{L}}_{1} (\tilde{p})}$ and ${\tilde{L}}_{2} (\tilde{p}) = {l_{2}^{(ϕ)} ({\tilde{p}}_{2}^{(ϕ)}) | ϕ = 1, 2,$ $\dots, # {\tilde{L}}_{2} (\tilde{p})}$ be two PLTSs, where $# {\tilde{L}}_{1} (\tilde{p})$ and $# {\tilde{L}}_{2} (\tilde{p})$ are the numbers of LTS in ${\tilde{L}}_{1} (\tilde{p})$ and ${\tilde{L}}_{2} (\tilde{p})$ , respectively. If $# {\tilde{L}}_{1} (\tilde{p}) > # {\tilde{L}}_{2} (\tilde{p})$ , then add $# {\tilde{L}}_{1} (\tilde{p}) - # {\tilde{L}}_{2} (\tilde{p})$ linguistic terms to ${\tilde{L}}_{2} (\tilde{p})$ . Moreover, the added linguistic terms should be the smallest linguistic term in ${\tilde{L}}_{2} (\tilde{p})$ and the probabilities of added linguistic terms should be zero.

Definition 4 [16]. For a PLTS $\tilde{L} (\tilde{p}) = {l^{(ϕ)} ({\tilde{p}}^{(ϕ)}) |$ $ϕ = 1, 2, \dots, # \tilde{L} (\tilde{p})}$ , the expectation $E (\tilde{L} (\tilde{p}))$ and deviation degree $σ (\tilde{L} (\tilde{p}))$ of $\tilde{L} (\tilde{p})$ is given as follows: $E (\tilde{L} (\tilde{p})) = \sum_{ϕ = 1}^{# \tilde{L} (\tilde{p})} g (\tilde{L} (\tilde{p})) {\tilde{p}}^{(ϕ)} / \sum_{ϕ = 1}^{# \tilde{L} (\tilde{p})} {\tilde{p}}^{(ϕ)}$ (4)

$\begin{matrix} σ (\tilde{L} (\tilde{p})) = \sqrt{\sum_{ϕ = 1}^{# \tilde{L} (\tilde{p})} {(g (\tilde{L} (\tilde{p})) {\tilde{p}}^{(ϕ)} - s (\tilde{L} (\tilde{p})))}^{2}} \\ / \sum_{ϕ = 1}^{# \tilde{L} (\tilde{p})} {\tilde{p}}^{(ϕ)} \end{matrix}$ (5)

On the basis of Equations (4) and (5), the order between two PLTSs is defined as: (1) if $E ({\tilde{L}}_{1} (\tilde{p})) > E ({\tilde{L}}_{2} (\tilde{p}))$ , then ${\tilde{L}}_{1} (\tilde{p}) > {\tilde{L}}_{2} (\tilde{p})$ ; (2) if $E ({\tilde{L}}_{1} (\tilde{p})) = E ({\tilde{L}}_{2} (\tilde{p}))$ , then if $σ ({\tilde{L}}_{1} (\tilde{p})) = σ ({\tilde{L}}_{2} (\tilde{p}))$ , then ${\tilde{L}}_{1} (\tilde{p}) = {\tilde{L}}_{2} (\tilde{p})$ ; if $σ ({\tilde{L}}_{1} (\tilde{p})) < σ ({\tilde{L}}_{2} (\tilde{p}))$ , then, ${\tilde{L}}_{1} (\tilde{p}) > {\tilde{L}}_{2} (\tilde{p})$ .

Definition 5 [52]. Let L = {l_α|α = - θ, ⋯ , -1, 0, 1, ⋯ θ} be an LTS. And let ${\tilde{L}}_{1} (\tilde{p}) = {l_{1}^{(ϕ)} ({\tilde{p}}_{1}^{(ϕ)}) | ϕ = 1, 2, \dots, # {\tilde{L}}_{1} (\tilde{p})}$ and ${\tilde{L}}_{2} (\tilde{p}) = {l_{2}^{(ϕ)} ({\tilde{p}}_{2}^{(ϕ)}) | ϕ = 1, 2, \dots, # {\tilde{L}}_{2} (\tilde{p})}$ be two PLTSs with $# {\tilde{L}}_{1} (\tilde{p}) = # {\tilde{L}}_{2} (\tilde{p}) = # \tilde{L} (\tilde{p})$ , then the Euclidean distance $d ({\tilde{L}}_{1} (\tilde{p}), {\tilde{L}}_{2} (\tilde{p}))$ between ${\tilde{L}}_{1} (\tilde{p})$ and ${\tilde{L}}_{2} (\tilde{p})$ is defined as follows: $\begin{matrix} d ({\tilde{L}}_{1} (\tilde{p}), {\tilde{L}}_{2} (\tilde{p})) = \\ \sqrt{\sum_{ϕ = 1}^{# {\tilde{L}}_{1} (\tilde{p})} {({\tilde{p}}_{1}^{(ϕ)} g (l_{1}^{(ϕ)}) - {\tilde{p}}_{2}^{(ϕ)} g (l_{2}^{(ϕ)}))}^{2} / # \tilde{L} (\tilde{p})} \end{matrix}$ (6)

In a MAGDM problem, suppose that there are a set of experts E ={ E₁, E₂, ⋯ , E_q } with weight vector η = (η₁, η₂, ⋯ , η_q), where η_k ∈ [0, 1], k = 1, 2, ⋯ , q, $\sum_{k = 1}^{q} η_{k} = 1$ . If the weights of these experts are the same or not given, we can let η_k = 1/ - q, k = 1, 2, ⋯ , q. If ${\tilde{L}}_{k} (\tilde{p}) = {l_{k}^{(ϕ)} ({\tilde{p}}_{k}^{(ϕ)}) | l_{k}^{(ϕ)} \in L, ϕ = 1, 2, \dots, # {\tilde{L}}_{1} (\tilde{p})} (k =$ 1, 2 ⋯ , q′) are PLTSs on the LTS L given by q′ experts, there are q - q′ experts who don’t give any evaluation information. In order to fuse these experts’ judgments to one average opinions, Wu and Liao [54] gave an aggregation formula which was expressed as:

$\begin{matrix} \tilde{L} (\tilde{p}) = {l^{(ϕ)} ({\tilde{p}}^{(ϕ)}) | l_{k}^{(ϕ)} \in L, {\tilde{p}}^{(ϕ)} \\ = \sum_{k = 1}^{q^{'}} η_{k} \tilde{p} {^{'}}_{k}^{(ϕ)}, ϕ = 1, 2, \dots, # {\tilde{L}}_{1} (\tilde{p})} \end{matrix}$ (7) where η_k is the weight of ${\tilde{p}}_{k}^{(ϕ)}$ in ${\tilde{L}}_{k} (\tilde{p})$ and $\tilde{p} {^{'}}_{k}^{(ϕ)} = {\begin{matrix} {\tilde{p}}_{k}^{(ϕ)}, if l^{(ϕ)} \in {\tilde{L}}_{k} (\tilde{p}) \\ 0, if l^{(ϕ)} \notin {\tilde{L}}_{k} (\tilde{p}) \end{matrix}$ (8)

2.1 The TODIM method

The TODIM method [3] which was developed to consider the DM’s psychological behavior can effectively solve the MADM problems. Based on the prospect theory, this approach depicts the dominance of each alternative over others by constructing a function of multi-attribute values.

Let A ={ A₁, A₂, ⋯ , A_m } be a discrete set of alternatives. Let G ={ G₁, G₂, ⋯ , G_n } be the set of attributes, w = (w₁, w₂, ⋯ , w_n) be the weight vector of attributes G_j, where w_j ∈ [0, 1], j = 1, 2, ⋯ , n, $\sum_{j = 1}^{n} w_{j} = 1$ . Suppose that A = (a_ij) _m×n be a decision matrix, where a_ij is the attribute value, given by an expert, for the alternative A_i ∈ A with respect to the attribute G_j ∈ G, i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n. We define w_jr = w_j/ - w_r (j, r = 1, 2, ⋯ , n) are the relative weight of the attribute G_j to G_r, and w_r = max{ w_j|j = 1, 2, ⋯ , n }, and 0 ≤ w_jr ≤ 1. Then the classical TODIM approach involves the following steps:

Step 1. Normalize the decision matrix A = (a_ij) _m×n into the matrix B = (b_ij) _m×n.

Step 2. Compute the dominance degree of A_i over each alternative A_t with respect to the attribute G_j: $δ (A_{i}, A_{t}) = \sum_{j = 1}^{n} ϕ_{j} (A_{i}, A_{t}), (i, t = 1, 2, \dots, m)$ (9) where

$ϕ_{j} (A_{i}, A_{t}) = {\begin{matrix} \sqrt{w_{jr} (b_{ij} - b_{tj}) / \sum_{j = 1}^{n} w_{jr}}, if b_{ij} - b_{tj} > 0 \\ 0, if b_{ij} - b_{tj} = 0 \\ - \frac{1}{θ} \sqrt{(\sum_{j = 1}^{n} w_{jr}) (b_{tj} - b_{ij}) / w_{jr}}, if b_{ij} - b_{tj} < 0 \end{matrix}$ (10) and the parameter θ shows the attenuation factor of the losses. If b_ij - b_tj > 0, then ϕ_j (A_i, A_t) represents a gain; if b_ij - b_tj < 0, then ϕ_j (A_i, A_t) signifies a loss.

Step 3. Compute the overall value of the alternative A_i by the following formula: $\begin{matrix} ϕ (A_{i}) = \frac{\sum_{t = 1}^{m} δ (A_{i}, A_{t}) - min_{i} {\sum_{t = 1}^{m} δ (A_{i}, A_{t})}}{max_{i} {\sum_{t = 1}^{m} δ (A_{i}, A_{t})} - min_{i} {\sum_{t = 1}^{m} δ (A_{i}, A_{t})}}, \\ i = 1, 2, \dots, m . \end{matrix}$ (11)

Step 4. Rank all alternatives and select the most desirable alternative by the overall values ϕ (A_i) (i = 1, 2, ⋯ , m). The alternative with the alternative with the minimum value is the worst. Inversely, the maximum value is the most desirable one.

3 TODIM method for probabilistic linguistic MAGDM problems

In this section, we shall combine the distance measures and entropy of PLTSs to construct a novel probabilistic linguistic TODIM method for MAGDM problems. We also develop the corresponding methods to derive the weight values for decision makers and attribute.

3.1 MAGDM problems description

The following assumptions or notations are used to represent the probabilistic linguistic MAGDM problems. Let A = { A₁, A₂, ⋯ , A_m } be a discrete set of alternatives, and G ={ G₁, G₂, ⋯ , G_n } with weight vector w = (w₁, w₂, ⋯ , w_n), where ω_j ∈ [0, 1], j = 1, 2, ⋯ , n, $\sum_{j = 1}^{n} w_{j} = 1$ , and a set of experts E ={ E₁, E₂, ⋯ , E_q } with weight vector η = (η₁, η₂, ⋯ , η_q), where η_k ∈ [0, 1], k = 1, 2, ⋯ , q, $\sum_{k = 1}^{q} η_{k} = 1$ . Suppose that there are n qualitative attribute G ={ G₁, G₂, ⋯ , G_n } and their values are assessed by each expert and depicted as linguistic expressions $l_{ij}^{k} (i = 1, 2, \dots, m, j = 1, 2, \dots, n, k = 1, 2, \dots, q)$ . By using the transformation function and probability information, we may convert the linguistic information $l_{ij}^{k}$ into a PTLS ${\tilde{L}}_{ij}^{k} (\tilde{p}) = {l_{ij}^{k (ϕ)} ({\tilde{p}}_{ij}^{k (ϕ))} | ϕ = 1, 2, \dots, # {\tilde{L}}_{ij}^{k} (\tilde{p})}$ for i = 1, 2, ⋯ , m, j = 1, 2, ⋯, n, k = 1, 2, ⋯ , q.

3.2 Computing the weight values for decision makers

In this section, the weight values for decision makers are determined on the basis of similarity degree. Then the weight values for decision makers can be derived as follows:

(1) Compute the average decision matrix $A {\tilde{L}}_{ij}$ $(p) = {l_{ij}^{(ϕ)} ({\bar{p}}_{ij}^{(ϕ)}) | ϕ = 1, 2, \dots, # {\tilde{L}}_{ij} (p) .}_{m \times n}$ , where: $l_{ij}^{(ϕ)} ({\bar{p}}_{ij}^{(ϕ)}) = l_{ij}^{(ϕ)} (\sum_{k = 1}^{q^{'}} \frac{{\tilde{p}}_{ij}^{k (ϕ)}}{q^{'}}),$ (12) where q′ is the average weight of ${\tilde{p}}_{ij}^{k (ϕ)}$ in ${\tilde{L}}_{ij}^{k} (\tilde{p})$ and $\tilde{p} {^{'}}_{ij}^{k} (ϕ) = {\begin{matrix} {\tilde{p}}_{ij}^{k (ϕ)}, if l_{ij}^{k (ϕ)} \in {\tilde{L}}_{ij}^{k} (\tilde{p}) \\ 0, if l_{ij}^{k (ϕ)} \notin {\tilde{L}}_{ij}^{k} (\tilde{p}) \end{matrix}$ (13)

(2) The degree of similarity between $l_{ij}^{k (ϕ)}$ $({\tilde{p}}_{ij}^{k (ϕ)})$ and $l_{ij}^{(ϕ)} ({\bar{p}}_{ij}^{(ϕ)})$ is defined as $s (l_{ij}^{k (ϕ)}$ $({\tilde{p}}_{ij}^{k (ϕ)}), l_{ij}^{(ϕ)} ({\bar{p}}_{ij}^{(ϕ)}))$ :

$\begin{matrix} s (l_{ij}^{k (ϕ)} ({\tilde{p}}_{ij}^{k (ϕ)}), l_{ij}^{(ϕ)} ({\bar{p}}_{ij}^{(ϕ)})) \\ = 1 - d (l_{ij}^{k (ϕ)} ({\tilde{p}}_{ij}^{k (ϕ)}), l_{ij}^{(ϕ)} ({\bar{p}}_{ij}^{(ϕ)})) \end{matrix}$ (14)

Furthermore, we can calculate the $s ({\tilde{L}}_{ij}^{k} (\tilde{p}),$ $A {\tilde{L}}_{ij} (p))$ as follows:

$\begin{matrix} s ({\tilde{L}}_{ij}^{k} (\tilde{p}), A {\tilde{L}}_{ij} (p)) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} s (l_{ij}^{k (ϕ)} ({\tilde{p}}_{ij}^{k (ϕ)}), \\ l_{ij}^{(ϕ)} ({\bar{p}}_{ij}^{(ϕ)})), k = 1, 2, \dots, q . \end{matrix}$ (15)

(3) Calculate the decision makers’ weight η_k as follows: $η_{k} = \frac{s ({\tilde{L}}_{ij}^{k} (\tilde{p}), A {\tilde{L}}_{ij} (p))}{\sum_{k = 1}^{q} s ({\tilde{L}}_{ij}^{k} (\tilde{p}), A {\tilde{L}}_{ij} (p))}, k = 1, 2, \dots, q .$ (16)

The decision makers’ weight determined by this kind of method has the desirable characteristic: the closer an evaluation value is to the mean value, the larger the weight is. This can avoid the unduly high or low evaluation values induced by decision makers’ limited knowledge or expertise.

(4) Then, the experts’ individual linguistic evaluations can be fused into the collective PTLS ${\tilde{L}}_{ij} (\tilde{p}) = {l_{ij}^{(ϕ)} ({\tilde{p}}_{ij}^{(ϕ)}) | ϕ = 1, 2, \dots, # {\tilde{L}}_{ij} (\tilde{p})}$ (i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n) by the following equation:

$\begin{matrix} {\tilde{L}}_{ij} (\tilde{p}) = {l_{ij}^{(ϕ)} ({\tilde{p}}_{ij}^{(ϕ)}) | l_{ij}^{(ϕ)} \in L, {\tilde{p}}_{ij}^{(ϕ)} \\ = \sum_{k = 1}^{q^{'}} η_{k} \tilde{p} {^{'}}_{ij}^{(ϕ)}, ϕ = 1, 2, \dots, # {\tilde{L}}_{1} (\tilde{p})} \end{matrix}$ (17) where η_k is the weight of ${\tilde{p}}_{ij}^{k (ϕ)}$ in ${\tilde{L}}_{ij}^{k} (\tilde{p})$ and $\tilde{p} {^{'}}_{ij}^{k (ϕ)} = {\begin{matrix} {\tilde{p}}_{ij}^{k (ϕ)}, if l_{ij}^{k (ϕ)} \in {\tilde{L}}_{ij}^{k} (\tilde{p}) \\ 0, if l_{ij}^{k (ϕ)} \notin {\tilde{L}}_{ij}^{k} (\tilde{p}) \end{matrix}$ (18)

3.3 Computing the weight values for attributes

Most of the MAGDM in the real-life decision-making problems utilize only subjective weights determined by the decision-maker. However, when it is hard to derive reliable subjective weights, objective weights looks useful [55]. One of the methods of deriving objective weights is the application of the basic concept of Shannon entropy.

Entropy is a conventional term from information theory which is also famous as the average (expected) amount of information [56] contained in each attribute. The larger the value of entropy in a specific attribute is, the smaller the differences in the ratings of alternatives with respect to this attribute. In turn, this means that this kind of attribute supplies less information and has a smaller weight. It follows that this kind of attribute becomes less important in the MAGDM process.

Let us consider the MAGDM decision matrix ${\tilde{L}}_{ij} (\tilde{p}) = {l_{ij}^{(ϕ)} ({\tilde{p}}_{ij}^{(ϕ)}) | ϕ = 1, 2, \dots, # {\tilde{L}}_{ij} (\tilde{p})}$ (i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n). Then attribute weights can be derived as follows:

(1) Compute the normalized decision matrix $N {\tilde{L}}_{ij} (\tilde{p})$ , where: $N {\tilde{L}}_{ij} (\tilde{p}) = \frac{\sum_{ϕ = 1}^{# {\tilde{L}}_{1} (\tilde{p})} ({\tilde{p}}_{ij}^{(ϕ)} g (l_{ij}^{(ϕ)}))}{\sum_{i = 1}^{m} \sum_{ϕ = 1}^{# {\tilde{L}}_{1} (\tilde{p})} ({\tilde{p}}_{ij}^{(ϕ)} g (l_{ij}^{(ϕ)}))}, j = 1, 2, \dots, n,$ (19)

(2) Compute the vector of Shannon entropy E = (E₁, E₂, ⋯ , E_n), where: $E_{j} = - \frac{1}{ln m} \sum_{i = 1}^{m} N {\tilde{L}}_{ij} (\tilde{p}) ln N {\tilde{L}}_{ij} (\tilde{p})$ (20) and $N {\tilde{L}}_{ij} (\tilde{p}) ln N {\tilde{L}}_{ij} (\tilde{p})$ is defined as 0, if $N {\tilde{L}}_{ij} (\tilde{p}) = 0$ .

(3) Compute the vector of diversification degrees H = (H₁, H₂, ⋯ , H_n), where: $H_{j} = 1 - E_{j}, j = 1, 2, \dots, n .$ (21)

The larger the degree H_j, the more important the corresponding attribute G_j.

(4) Compute the vector of attribute weights w = (w₁, w₂, ⋯ , w_n), where $w_{j} = \frac{H_{j}}{\sum_{j = 1}^{n} H_{j}}, j = 1, 2, \dots, n .$ (22)

3.4 TODIM method for probabilistic linguistic MAGDM

To solve the MAGDM problem with probabilistic linguistic information, we try to present a probabilistic linguistic TODIM approach, which is developed in terms of the prospect theory and can depict the DMs’ behaviors under risk.

(1) Calculate the relative weight of each attribute G_j as: $w_{jr} = w_{j} / w_{r}, j, r = 1, 2, \dots, n .$ (23) where w_j is the weight of the attribute of G_j, w_r = max{ w_j|j = 1, 2, ⋯ , n }, and 0 ≤ w_jr ≤ 1.

(2) Based on the Equation (24), we can derive the dominance degree of A_i over each alternative A_t with respect to the attribute G_j:

$ϕ_{j} (A_{i}, A_{t}) = {\begin{matrix} \sqrt{w_{jr} d ({\tilde{L}}_{ij} (\tilde{p}), {\tilde{L}}_{tj} (\tilde{p})) / \sum_{j = 1}^{n} w_{jr}}, if {\tilde{L}}_{ij} (\tilde{p}) > {\tilde{L}}_{tj} (\tilde{p}) \\ 0, if {\tilde{L}}_{ij} (\tilde{p}) = {\tilde{L}}_{tj} (\tilde{p}) \\ - \frac{1}{θ} \sqrt{(\sum_{j = 1}^{n} w_{jr}) d ({\tilde{L}}_{ij} (\tilde{p}), {\tilde{L}}_{tj} (\tilde{p})) / w_{jr}}, if {\tilde{L}}_{ij} (\tilde{p}) < {\tilde{L}}_{tj} (\tilde{p}) \end{matrix}$ (24)

$\begin{matrix} d ({\tilde{L}}_{ij} (\tilde{p}), {\tilde{L}}_{tj} (\tilde{p})) \\ = \sqrt{\sum_{ϕ = 1}^{# {\tilde{L}}_{ij} (\tilde{p})} ({\tilde{p}}_{ij}^{(ϕ)} g (l_{ij}^{(ϕ)}) - {\tilde{p}}_{tj}^{(ϕ)} g (l_{tj}^{(ϕ)}))^{2} / # {\tilde{L}}_{ij} (\tilde{p})} \end{matrix}$ (25) where the parameter θ shows the attenuation factor of the losses, and $d ({\tilde{L}}_{ij} (\tilde{p}), {\tilde{L}}_{tj} (\tilde{p}))$ is to measure the distances between the PLTSs r_ij and r_tj by Definition 5. If ${\tilde{L}}_{ij} (\tilde{p}) > {\tilde{L}}_{tj} (\tilde{p})$ , then ϕ_j (A_i, A_t) represents a gain; if ${\tilde{L}}_{ij} (\tilde{p}) < {\tilde{L}}_{tj} (\tilde{p})$ , then ϕ_j (A_i, A_t) signifies a loss.

In order to indicate the functions ϕ_j (A_i, A_t) clearly, we express it in a dominance degree matrix with respect to the attribute of G_j as: $ϕ_{j} = {[ϕ_{j} (A_{i}, A_{t})]}_{m \times m} =$ $\begin{matrix} \begin{matrix} \begin{matrix} A_{1} & A_{2} & \dots & A_{m} \end{matrix} \\ \begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{m} \end{matrix} [\begin{matrix} o & ϕ_{j} (A_{1}, A_{2}) & \dots & ϕ_{j} (A_{1}, A_{m}) \\ ϕ_{j} (A_{2}, A_{1}) & 0 & \dots & ϕ_{j} (A_{2}, A_{m}) \\ ⋮ & ⋮ & \dots & ⋮ \\ ϕ_{j} (A_{m}, A_{1}) & ϕ_{j} (A_{m}, A_{2}) & \dots & 0 \end{matrix}] \end{matrix} \end{matrix}$ $j = 1, 2, \dots, n .$ (26)

(3) Calculate the overall dominance degree of the alternative A_i over each alternative A_t by the following equation: $δ (A_{i}, A_{t}) = \sum_{j = 1}^{n} ϕ_{j} (A_{i}, A_{t}), (i, t = 1, 2, \dots, m)$ (27)

Therefore, by Equation (27), the overall dominance degree matrix can be got as: $δ = {[δ (A_{i}, A_{t})]}_{m \times m} =$

$\begin{matrix} \begin{matrix} A_{1} & A_{2} & \dots & A_{m} \end{matrix} \\ \begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{m} \end{matrix} [\begin{matrix} o & δ (A_{1}, A_{2}) & \dots & δ (A_{1}, A_{m}) \\ δ (A_{2}, A_{1}) & 0 & \dots & δ (A_{2}, A_{m}) \\ ⋮ & ⋮ & \dots & ⋮ \\ δ (A_{m}, A_{1}) & δ (A_{m}, A_{2}) & \dots & 0 \end{matrix}] \end{matrix}$ (28)

(4) Finally, the overall value of each alternative A_i can be derived by the following formula: $\begin{matrix} δ (A_{i}) = \frac{\sum_{t = 1}^{m} δ (A_{i}, A_{t}) - min_{i} {\sum_{t = 1}^{m} δ (A_{i}, A_{t})}}{max_{i} {\sum_{t = 1}^{m} δ (A_{i}, A_{t})} - min_{i} {\sum_{t = 1}^{m} δ (A_{i}, A_{t})}}, \\ i = 1, 2, \dots, m . \end{matrix}$ (29)

And the order of each alternative can be ranked by the principle, that is, the greater the overall value δ (A_i) (i = 1, 2, ⋯ , m), the better the alternative A_i.

In general, probabilistic linguistic TODIM approach for MAGDM includes the following computing steps:

Step 1. By using the transformation function and probability information, we may convert the linguistic information $l_{ij}^{k}$ into a PTLS ${\tilde{L}}_{ij}^{k} (\tilde{p}) = {l_{ij}^{k (ϕ)} ({\tilde{p}}_{ij}^{k (ϕ)}) | ϕ = 1, 2, \dots, # {\tilde{L}}_{ij}^{k} (\tilde{p})}$ for i = 1, 2, ⋯, m, j = 1, 2, ⋯ , n, k = 1, 2, ⋯ , q.

Step 2. Computing the weight values for decision makers η = (η₁, η₂, ⋯ , η_q).

Step 3. The experts’ individual linguistic evaluations can be fused into the collective PTLS ${\tilde{L}}_{ij} (\tilde{p}) = {l_{ij}^{(ϕ)} ({\tilde{p}}_{ij}^{(ϕ)}) | ϕ = 1, 2, \dots, # {\tilde{L}}_{ij} (\tilde{p})}$ (i = 1, 2, ⋯, m, j = 1, 2, ⋯ , n) by the Equation (17).

Step 4. Computing the weight values for attributes w = (w₁, w₂, ⋯ , w_n).

Step 5. Calculate the relative weight of each attribute G_j using Equation (23).

Step 6. Calculate the dominance degree ϕ_j (A_i, A_t) of the alternative A_i over each alternative A_t with respect to the attribute G_j by Equation (24).

Step 7. Calculate the overall dominance degree δ (A_i, A_t) of A_i over each alternative A_t using Equation (27).

Step 8. Derive the overall value δ (A_i) (i = 1, 2, ⋯, m) of each alternative A_i using Equation (29).

Step 9. Determine the ranking of the alternatives according to the overall values δ (A_i) (i = 1, 2, ⋯, m).

4 Numerical example and comparative analysis

4.1 Numerical example

In today’s world, environmental issues have received increasingly attention in various countries and regions. Low-carbon economy and circular economy have become the mainstream of Chinese society for the sustainable development. The consciousness of environmental protection is improving among the public, so the business operators should not only create the economic benefits, but also need to consider the impact of the green image for the enterprise in the market competition. Thus, in this section we present a numerical example for green supplier selection to illustrate the method proposed in this paper. There is a panel with five possible green suppliers A_i (i = 1, 2, 3, 4, 5) to select. The experts selects four beneficial attribute to evaluate the five possible green suppliers: ①G₁ is the environmental improvement quality; ②G₂ is the price capability of suppliers; ③G₃ is the financial conditions; ④G₄ is the environmental competencies. The five possible green suppliers A_i (i = 1, 2, 3, 4, 5) are to be evaluated by using the linguistic term set $\begin{matrix} L = {l_{- 3} = extremely poor (EP), l_{- 2} = very poor (VP), \\ l_{- 1} = poor (P), l_{0} = medium (M), l_{1} = good (G), \\ l_{2} = very good (VG), l_{3} = extremely good (EG)} \end{matrix}$ by three expert group under the above four attributes (whose weighting vector ω = (0.2, 0.1, 0.3, 0.4) ^T), then the corresponding PLTSs are listed in the Tables 1 –3.

Table 1
Probabilistic linguistic decision matrix by the first expert group

Alternatives G₁ G₂ G₃ G₄

A₁ {l₁ (0.3) , l₃ (0.7)} {l₂ (0.4) , l₃ (0.6)} {l₃ (1)} {l_-3 (0.5) , l_-2 (0.5)}

A₂ {l_-2 (0.4) , l_-1 (0.6)} {l₂ (1)} {l₁ (0.6) , l₃ (0.4)} {l₁ (0.3) , l₂ (0.7)}

A₃ {l₀ (0.2) , l₁ (0.8)} {l₁ (0.8) , l₂ (0.2)} {l_-1 (0.7) , l₀ (0.3)} {l₂ (1)}

A₄ {l₂ (1)} {l_-2 (0.5) , l_-1 (0.5)} {l₂ (0.4) , l₃ (0.6)} {l₁ (0.2) , l₂ (0.6) , l₃ (0.2)}

A₅ {l₁ (0.5) , l₃ (0.5)} {l₃ (1)} {l₂ (1)} {l₂ (0.6) , l₃ (0.4)}

Alternatives	G₁	G₂	G₃	G₄
A₁	{l₁ (0.3) , l₃ (0.7)}	{l₂ (0.4) , l₃ (0.6)}	{l₃ (1)}	{l_-3 (0.5) , l_-2 (0.5)}
A₂	{l_-2 (0.4) , l_-1 (0.6)}	{l₂ (1)}	{l₁ (0.6) , l₃ (0.4)}	{l₁ (0.3) , l₂ (0.7)}
A₃	{l₀ (0.2) , l₁ (0.8)}	{l₁ (0.8) , l₂ (0.2)}	{l_-1 (0.7) , l₀ (0.3)}	{l₂ (1)}
A₄	{l₂ (1)}	{l_-2 (0.5) , l_-1 (0.5)}	{l₂ (0.4) , l₃ (0.6)}	{l₁ (0.2) , l₂ (0.6) , l₃ (0.2)}
A₅	{l₁ (0.5) , l₃ (0.5)}	{l₃ (1)}	{l₂ (1)}	{l₂ (0.6) , l₃ (0.4)}

Table 2

Probabilistic linguistic decision matrix by the second expert group

Alternatives	G₁	G₂	G₃	G₄
A₁	{l₃ (1)}	{l₂ (0.5) , l₃ (0.5)}	{l₂ (0.6) , l₃ (0.4)}	{l_-2 (1)}
A₂	{l_-2 (0.7) , l_-1 (0.3)}	{l₁ (0.2) , l₂ (0.8)}	{l₁ (0.5) , l₃ (0.5)}	{l₁ (0.5) , l₂ (0.5)}
A₃	{l₁ (1)}	{l₁ (0.6) , l₂ (0.4)}	{l_-1 (0.5) , l₀ (0.5)}	{l₁ (1)}
A₄	{l₀ (0.2) , l₂ (0.8)}	{l_-2 (0.3) , l_-1 (0.7)}	{l₂ (1)}	{l₁ (0.7) , l₂ (0.3)}
A₅	{l₁ (0.6) , l₃ (0.4)}	{l₂ (0.2) , l₃ (0.8)}	{l₂ (0.8) , l₃ (0.2)}	{l₂ (0.8) , l₃ (0.2)}

Table 3

Probabilistic linguistic decision matrix by the third expert group

Alternatives	G₁	G₂	G₃	G₄
A₁	{l₁ (1)}	{l₂ (1)}	{l₂ (1)}	{l_-3 (1)}
A₂	{l_-2 (0.6) , l_-1 (0.4)}	{l₁ (0.5) , l₂ (0.5)}	{l₃ (1)}	{l₁ (0.3) , l₂ (0.7)}
A₃	{l₀ (0.6) , l₁ (0.4)}	${l_{1} (1)}$	{l_-1 (0.6) , l₀ (0.4)}	{l₁ (0.8) , l₂ (0.2)}
A₄	{l₀ (1)}	{l_-1 (1)}	{l₂ (0.3) , l₃ (0.7)}	{l₂ (0.6) , l₃ (0.4)}
A₅	{l₁ (1)}	{l₂ (0.6) , l₃ (0.4)}	{l₂ (0.6) , l₃ (0.4)}	{l₂ (1)}

In the following, we utilize the TODIM approach developed to select the optimal green supplier.

(1) Compute the average decision matrix $A {\tilde{L}}_{5 \times 4} (p)$ , the calculating result is listed in Table 4.

Table 4

Probabilistic linguistic average decision matrix

Alternatives	G₁	G₂
A₁	{l₁ (0.4333) , l₃ (0.5667)}	{l₂ (0.6333) , l₃ (0.3667)}
A₂	{l_-2 (0.5667) , l_-1 (0.4333)}	{l₁ (0.2333) , l₂ (0.7667)}
A₃	{l₀ (0.2667) , l₁ (0.7333)}	{l₁ (0.8000) , l₂ (0.2000)}
A₄	{l₀ (0.4000) , l₂ (0.6000)}	{l_-2 (0.2667) , l_-1 (0.7333)}
A₅	{l₁ (0.7000) , l₃ (0.3000)}	{l₂ (0.2667) , l₃ (0.7333)}
Alternatives	G₃	G₄
A₁	{l₂ (0.5333) , l₃ (0.4667)}	{l_-3 (0.5000) , l_-2 (0.5000)}
A₂	{l₁ (0.3667) , l₃ (0.6333)}	{l₁ (0.3667) , l₂ (0.6333)}
A₃	{l_-1 (0.6000) , l₀ (0.4000)}	{l₁ (0.6000) , l₂ (0.4000)}
A₄	{l₂ (0.5667) , l₃ (0.4333)}	{l₁ (0.3000) , l₂ (0.5000) , l₃ (0.2000)}
A₅	{l₂ (0.8000) , l₃ (0.2000)}	{l₂ (0.8000) , l₃ (0.2000)}

(2) Compute the weight values for experts:

By Equation (15), we can have: $s ({\tilde{L}}_{5 \times 4}^{1} (\tilde{p}), A {\tilde{L}}_{5 \times 4} (p)) = 0.8705$ , $s ({\tilde{L}}_{5 \times 4}^{2} (\tilde{p}), A {\tilde{L}}_{5 \times 4} (p)) = 0.8999$ , $s ({\tilde{L}}_{5 \times 4}^{3} (\tilde{p}), A {\tilde{L}}_{5 \times 4} (p)) = 0.8290$ .

Furthermore, we can derive the weight values for experts by Equation (16):η₁ = 0.3349, η₂ = 0.3462, η₃ = 0.3189.

(3) The experts’ individual linguistic evaluations can be fused into the collective PTLS ${\tilde{L}}_{ij} (\tilde{p})$ . The collective PTLS ${\tilde{L}}_{ij} (\tilde{p})$ is listed in Table 5.

Table 5

Probabilistic linguistic collective decision matrix

Alternatives	G₁	G₂
A₁	{l₁ (0.4194) , l₃ (0.5806)}	{l₂ (0.6260) , l₃ (0.3740)}
A₂	{l_-2 (0.5676) , l_-1 (0.4324)}	{l₁ (0.2287) , l₂ (0.7713)}
A₃	{l₀ (0.2583) , l₁ (0.7417)}	{l₁ (0.7945) , l₂ (0.2055)}
A₄	{l₀ (0.3881) , l₂ (0.6119)}	{l_-2 (0.2713) , l_-1 (0.7287)}
A₅	{l₁ (0.6941) , l₃ (0.3059)}	{l₂ (0.2606) , l₃ (0.7394)}
Alternatives	G₃	G₄
A₁	{l₂ (0.5266) , l₃ (0.4734)}	{l_-3 (0.4864) , l_-2 (0.5137)}
A₂	{l₁ (0.3740) , l₃ (0.6260)}	{l₁ (0.3692) , l₂ (0.6308)}
A₃	{l_-1 (0.5989) , l₀ (0.4011)}	{l₁ (0.6013) , l₂ (0.3987)}
A₄	{l₂ (0.5758) , l₃ (0.4242)}	{l₁ (0.3093) , l₂ (0.4961) , l₃ (0.1945)}
A₅	{l₂ (0.8032) , l₃ (0.1968)}	{l₂ (0.7968) , l₃ (0.2032)}

(4) Computing the weight values for attributes:

From Equations (19)–(22), we can derive the weight values for attributes:w₁ = 0.2515, w₂ = 0.2523, w₃ = 0.2566, w₄ = 0.2396.

(5) Computing the relative weights of all the attributes:

Firstly, since w₄ = max{ w₁, w₂, w₃, w₄ }, then G₃ is the reference attribute and the reference weight is w_r = 0.2566. Therefore, the relative weights of all the attributes G_j (j = 1, 2, 3, 4) are w_1r = 0.9800, w_2r = 0.9832, w_3r = 1.0000 and w_4r = 0.9338, respectively.

(6) Then, we need to calculate the dominance degree of the candidate A_i over each candidate A_t with respect to the attributes G_j (j = 1, 2, 3, 4).

Here we take an example from the information given by the decision matrix. Let θ = 2.5, then the dominance degree matrices with respect to the attributes G_j (j = 1, 2, 3, 4) are: $\begin{matrix} \begin{matrix} A_{1} & A_{2} & A_{3} & A_{4} & A_{5} \end{matrix} \\ ϕ_{1} = \begin{matrix} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \\ A_{5} \end{matrix} [\begin{matrix} \begin{matrix} 0 . 0000 & 0 . 3615 & 0 . 2186 & 0 . 1749 & 0 . 3016 \\ - 0 . 3710 & 0 . 0000 & - 0 . 3197 & - 0 . 3317 & - 0 . 3416 \\ - 0 . 2243 & 0 . 3115 & 0 . 0000 & - 0 . 1392 & - 0 . 3335 \\ - 0 . 1795 & 0 . 3232 & 0 . 1356 & 0 . 0000 & - 0 . 3129 \\ - 0 . 3096 & 0 . 3329 & 0 . 3250 & 0 . 3049 & 0 . 0000 \end{matrix} \end{matrix}] \end{matrix}$ $\begin{matrix} \begin{matrix} A_{1} & A_{2} & A_{3} & A_{4} & A_{5} \end{matrix} \\ ϕ_{2} = \begin{matrix} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \\ A_{5} \end{matrix} [\begin{matrix} \begin{matrix} 0 . 0000 & 0 . 3547 & 0 . 2365 & 0 . 3690 & - 0 . 3716 \\ - 0 . 3641 & 0 . 0000 & 0 . 4080 & 0 . 3378 & - 0 . 1837 \\ - 0 . 2427 & - 0 . 4187 & 0 . 0000 & 0 . 3674 & - 0 . 4339 \\ - 0 . 3787 & - 0 . 3467 & - 0 . 3770 & 0 . 0000 & - 0 . 3905 \\ 0 . 3621 & 0 . 1790 & 0 . 4228 & 0 . 3805 & 0 . 0000 \end{matrix} \end{matrix}] \end{matrix}$ $\begin{matrix} \begin{matrix} A_{1} & A_{2} & A_{3} & A_{4} & A_{5} \end{matrix} \\ ϕ_{3} = \begin{matrix} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \\ A_{5} \end{matrix} [\begin{matrix} \begin{matrix} 0 . 0000 & 0 . 2589 & 0 . 3162 & 0 . 1329 & 0 . 3150 \\ - 0 . 2657 & 0 . 0000 & 0 . 3436 & - 0 . 2982 & 0 . 4068 \\ - 0 . 3245 & - 0 . 3526 & 0 . 0000 & - 0 . 3226 & - 0 . 3693 \\ - 0 . 1364 & 0 . 2906 & 0 . 3143 & 0 . 0000 & 0 . 2856 \\ - 0 . 3233 & - 0 . 4175 & 0 . 3598 & - 0 . 2931 & 0 . 0000 \end{matrix} \end{matrix}] \end{matrix}$ $\begin{matrix} \begin{matrix} A_{1} & A_{2} & A_{3} & A_{4} & A_{5} \end{matrix} \\ ϕ_{4} = \begin{matrix} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \\ A_{5} \end{matrix} [\begin{matrix} \begin{matrix} 0 . 0000 & - 0 . 3457 & - 0 . 3340 & - 0 . 3205 & - 0 . 3998 \\ 0 . 3368 & 0 . 0000 & 0 . 2361 & - 0 . 2325 & - 0 . 3537 \\ 0 . 3254 & - 0 . 2423 & 0 . 0000 & - 0 . 2608 & - 0 . 2635 \\ 0 . 3123 & 0 . 2266 & 0 . 2541 & 0 . 0000 & - 0 . 3578 \\ 0 . 3895 & 0 . 3446 & 0 . 2568 & 0 . 3486 & 0 . 0000 \end{matrix} \end{matrix}] \end{matrix}$

Secondly, the overall dominance degree of the candidate A_i over each candidate A_t can be obtained by Equation (27), and the overall dominance degree matrix is: $\begin{matrix} \begin{matrix} A_{1} & A_{2} & A_{3} & A_{4} & A_{5} \end{matrix} \\ δ = \begin{matrix} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \\ A_{5} \end{matrix} [\begin{matrix} \begin{matrix} 0 . 0000 & 0 . 6295 & 0 . 4373 & 0 . 3563 & - 0 . 1547 \\ - 0 . 6639 & 0 . 0000 & 0 . 6679 & - 0 . 5246 & - 0 . 4723 \\ - 0 . 4662 & - 0 . 7021 & 0 . 0000 & - 0 . 3552 & - 1 . 4002 \\ - 0 . 3823 & 0 . 4937 & 0 . 3270 & 0 . 0000 & - 0 . 7756 \\ 0 . 1188 & 0 . 4391 & 1 . 3643 & 0 . 7409 & 0 . 0000 \end{matrix} \end{matrix}] \end{matrix}$

Then, we derive the overall value δ (A_i) (i = 1, 2, ⋯ , m) of each alternative A_i using Equation (29): $\begin{matrix} δ (A_{1}) = 0.7504, δ (A_{2}) = 0.3456, δ (A_{3}) = 0.0000 \\ δ (A_{4}) = 0.4630, δ (A_{5}) = 1.0000 \end{matrix}$

(7) Finally, we determine the ranking of the alternatives according to the overall values δ (A_i) (i = 1, 2, 3, 4, 5): A₅ ≻ A₁ ≻ A₄ ≻ A₂ ≻ A₃, and thus the most desirable green supplier is A₅.

4.2 Comparative analysis

Furthermore, in what follows, we compare our proposed method with other existing methods including the probabilistic linguistic TOPSIS method [16], probabilistic linguistic VIKOR method [57] (let η = 0.4) and probabilistic linguistic GRA method [36] (let ρ = 0.5). The comparative results are listed in Table 6.

Table 6
Order of the green suppliers by using different methods

Methods Computing results Ordering

Probabilistic linguistic TOPSIS method [16] $\begin{matrix} CI (A_{1}) = - 0.9402, CI (A_{2}) = \begin{matrix} - 1.1693 \end{matrix}, \\ CI (A_{3}) = - 1.4585, CI (A_{4}) = \begin{matrix} - 1.0401 \end{matrix}, \\ CI (A_{5}) = \begin{matrix} 0.0000 \end{matrix} . \end{matrix}$ A₅ ≻ A₁ ≻ A₄ ≻ A₂ ≻ A₃

Probabilistic linguistic GRA method [36] $\begin{matrix} ɛ_{1}^{+} = 0.6923, ɛ_{2}^{+} = 0.5344, ɛ_{3}^{+} = 0.5040, \\ ɛ_{4}^{+} = 0.6253, ɛ_{5}^{+} = 0.7429 . \end{matrix}$ A₅ ≻ A₁ ≻ A₄ ≻ A₂ ≻ A₃

Probabilistic linguistic TODIM method $\begin{matrix} δ (A_{1}) = 0.7504, δ (A_{2}) = 0.3456 \\ δ (A_{3}) = 0.0000, δ (A_{4}) = 0.4630 \\ δ (A_{5}) = 1.0000 \end{matrix}$ A₅ ≻ A₁ ≻ A₄ ≻ A₂ ≻ A₃

Methods	Computing results	Ordering
Probabilistic linguistic TOPSIS method [16]	$\begin{matrix} CI (A_{1}) = - 0.9402, CI (A_{2}) = \begin{matrix} - 1.1693 \end{matrix}, \\ CI (A_{3}) = - 1.4585, CI (A_{4}) = \begin{matrix} - 1.0401 \end{matrix}, \\ CI (A_{5}) = \begin{matrix} 0.0000 \end{matrix} . \end{matrix}$	A₅ ≻ A₁ ≻ A₄ ≻ A₂ ≻ A₃
Probabilistic linguistic GRA method [36]	$\begin{matrix} ɛ_{1}^{+} = 0.6923, ɛ_{2}^{+} = 0.5344, ɛ_{3}^{+} = 0.5040, \\ ɛ_{4}^{+} = 0.6253, ɛ_{5}^{+} = 0.7429 . \end{matrix}$	A₅ ≻ A₁ ≻ A₄ ≻ A₂ ≻ A₃
Probabilistic linguistic TODIM method	$\begin{matrix} δ (A_{1}) = 0.7504, δ (A_{2}) = 0.3456 \\ δ (A_{3}) = 0.0000, δ (A_{4}) = 0.4630 \\ δ (A_{5}) = 1.0000 \end{matrix}$	A₅ ≻ A₁ ≻ A₄ ≻ A₂ ≻ A₃

From the above analysis, it can be seen that these above mentioned methods have the same best green supplier A₅ and five methods’ ranking results are same. This verifies the method we proposed is reasonable and effective in this paper.

5 Conclusion

In this paper, we extend the TODIM method to the MAGDM with PLTSs. Firstly, the definition, comparative method and distance of PLTSs are briefly introduced, and the steps of the conventional TODIM method for MADM problems are presented. Then, on the basis of the classical TODIM method, the extended TODIM method is proposed to deal with MAGDM problems with PLTSs and its significant characteristic is that it can fully consider the decision makers’ bounded rationality which is a real action in MAGDM problems. Finally, a practical example for green supplier selection is given to verify the developed approach and to demonstrate its practicality and effectiveness and some comparative analysis are also given to verify the developed approach. In the future, the application of the proposed models and methods with PLTSs needs to be explored in the other decision making [58 –66] and many other uncertain and fuzzy environments [67 –77].

Footnotes

Acknowledgment

The work was supported by the National Natural Science Foundation of China under Grant No. 71571128 and the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China (16XJA630005).

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