Online teaching quality evaluation based on multi-granularity probabilistic linguistic term sets

Abstract

In today’s education industry, online teaching is increasingly becoming an important teaching way, and it is necessary to evaluate the quality of online teaching so as to improve the overall level of the education industry. The online teaching quality evaluation is a typical multi-attribute group decision-making (MAGDM) problem, and its evaluation index can be expressed by linguistic term sets (LTSs) by decision makers (DMs). Especially, multi-granularity probabilistic linguistic term sets (MGPLTSs) produced from many DMs are more suitable to express complex fuzzy evaluation information, and they can not only provide different linguistic term set for different DMs the give their preferences, but also reflect the importance of each linguistic term. Based on the advantages of MGPLTSs, in this paper, we propose a transformation function of MGPLTSs based on proportional 2-tuple fuzzy linguistic representation model. On this basis, the operational laws and comparison rules of MGPLTSs are given. Then, we develop a new Choquet integral operator for MGPLTSs, which considers the relationship among attributes and does not need to consider the process of normalizing the probabilistic linguistic term sets (PLTSs), and can effectively avoid the loss of evaluation information. At the same time, the properties of the proposed operator are also proved. Furthermore, we propose a new MAGDM method based on the new operator, and analyze the effectiveness of the proposed method by online teaching quality evaluation. Finally, by comparing with some existing methods, the advantages of the proposed method are shown.

Keywords

Multiple-attribute group decision-making online teaching quality evaluation multi-granularity probabilistic linguistic term sets Choquet integral

1 Introduction

With the development of the Internet, the traditional teaching way is combined with the Internet, online teaching appeared. Online teaching is a long-distance education model based on the Internet. Due to the flexibility and diversity of online teaching, it is developing rapidly. He et al. [1] mentioned that online teaching is becoming an increasingly important teaching way in higher educational institutions. Now, online teaching is a research hotspot. Bennett et al. [2] analyzed the learning differences between online teaching and traditional classes. Martin et al. [3] studied how to determine curriculum design and evaluation from the perspective of award-winning online faculty. Jones and Meyer [4] pointed out that the effective teachers are the key to the success of students in online teaching. In recent years, there have been many studies on the impact of online teaching quality. Schmidt et al. [5] believed that curriculum development and online teaching methods are two key factors for effective online teaching. Rhode et al. [6] gave the research results that online teaching quality should be improved from teachers’ attitude, belief and technical level. Diaz et al. [7] gave the research results that teacher development and information technology have an important impact on the quality of online teaching. Online teaching quality evaluation is a comprehensive evaluation process in which multiple decision makers (DMs) make some value judgments on online teaching based on evaluation methods, analyzing teaching information and teaching resources. Therefore, online teaching quality evaluation is a typical multi-attribute group decision-making (MAGDM) problem. Some scholars studied the MAGDM method for online teaching quality evaluation. Liu and Rong [50] selected four evaluation attributes to evaluate the online courses quality of China National Open University, Liu et al. [51] evaluated the quality of the live platform online education from four evaluation attributes. In order to evaluate online teaching from more appropriate attributes, based on some standards for online course evaluation proposed by the Ministry of Education of China, the online education evaluation includes the following evaluation attributes: teaching philosophy and curriculum design, teaching contents and learning resources, faculty and teaching activities, user interface design and technical support.

MAGDM can be recognized as a procedure of evaluating several alternatives or selecting the optimal alternative by some DMs according to multiple attributes [8 , 54]. Owing to the complexity of decision-making and the fuzziness of human thinking, DMs cannot accurately express their evaluation information with real numbers. In order to overcome this obstacle, Zadeh [11] introduced linguistic variable and Herrera et al. [12] presented the concept of linguistic term sets (LTSs). Due to the different knowledge backgrounds and educational experiences of DMs, “the granularity of uncertainty” (cardinality of LTSs) used by each DM is different. Herrera et al. [13] proposed a fusion approach of multi-granularity linguistic term sets (MGLTSs), Herrera and Martínez [14] proposed a 2-tuple fuzzy linguistic model to compute with words, which can avoid the problem of information loss. Based on the 2-tuple fuzzy linguistic model, Wang and Hao [15] proposed proportional 2-tuple fuzzy linguistic representation model.

However, due to complexity of the problem, DMs may be hesitant in several linguistic terms. Rodriguez et al. [16] proposed concept of hesitant fuzzy LTSs (HFLTSs) based on hesitant fuzzy sets (HFSs) and LTSs to deal with this problem. After that, scholars focus on the researches from three aspects of HFLTSs.

The basic theories of HFLTSs, such as some operations [17, 18], distance measures [19, 20], and preference relations [21, 22].

The decision-making methods based on HFLTSs. Wu et al. [23] developed compromise solutions for MAGDM problem using HFLTSs. Lin et al. [24] extended the traditional TODIM method to deal with the HFLTSs based on the novel comparison function and distance measure.

The aggregation operators (AOs) based on HFLTSs. Wei [25] studied arithmetic and geometric aggregation operators with interval valued hesitant fuzzy uncertain linguistic information. Liu et al. [26] proposed the hesitant fuzzy linguistic (HFL) Muirhead mean operator and the weighted HFL Muirhead mean (WHFLMM) operator. Zhu and Li [27] presented some HFL aggregation operators based on the Hamacher t-norm and t-conorm.

However, the HFLTSs also have some shortcomings. It cannot reflect the different importance and different frequencies of each linguistic term in HFLTS given by DMs. In order to solve this shortcoming, Pang et al. [28] proposed probabilistic LTSs (PLTSs) with the different importance of each linguistic term. Then, Bai et al. [29] proposed possibility degree of PLTSs to compare two PLTSs. Liu and You [30] extended TODIM to PLTSs to solve the multi-attribute decision-making (MADM) problem. Lin et al. [52] combined PLTSs and Best-Worst method to calculate the criteria weights, and proposed a new probabilistic linguistic TODIM method to rank Internet of Things platforms. Liu and Teng [31] established an extended probabilistic linguistic TODIM method to assess online product. In the real decision environment, because the preferences of DMs are different, they can use PLTSs with different granularity levels to give decision information. Therefore, in order to improve the applicability of PLTSs in complex environment, the multi-granularity PLTSs (MGPLTSs) were developed. Wang et al. [32] determined the distance measure between two MGPLTSs, and proposed a novel MAGDM method. Wang [33] proposed some novel distance measures between two MGPLTSs. Song and Li [34] presented a large-scale group decision-making with incomplete MGPLTSs. Obviously, MGPLTSs have widely been used in MAGDM problems.

In MAGDM process, aggregation operators (AOs) are a powerful tool for information fusion. Many researches on AOs can be divided into two aspects: operations and the functions.

About operations. Pang et al. [28] put forward some basic operational laws and the comparison method of PLTSs. Gou and Xu [35] proposed some logical operational laws for PLTSs to overcome some unreasonable problems. Lin et al. [56] proposed new distance measure and comparison method for PLTSs. Wang et al. [36] introduced a rational comparison method and proposed extended Hausdorff distance of PLTSs. In order to measure the relationship between two PLTSs, Lin et al. [57] proposed correlation coefficient. Wang et al. [32] defined generalized probabilistic linguistic Hamming distance, generalized probabilistic linguistic Euclidean distance and generalized probabilistic linguistic Hausdorff distance of MGPLTSs, which improved the accuracy of MGPLTSs in MAGDM problems.

About functions. Pang et al. [28] proposed probabilistic linguistic averaging (PLA) operator and probabilistic linguistic weighted averaging (PLWA) operator, Liu and Li [37] proposed a probabilistic linguistic-dependent weighted average (PLDWA) operator, and then combined the PLDWA operator with the MULTIMOORA method. Kobina et al. [38] proposed the probabilistic linguistic power average (PLPA), the weighted probabilistic linguistic power average (WPLPA) operators, the probabilistic linguistic power geometric (PLPG) and the weighted probabilistic linguistic power geometric (WPLPG) operators. Lin et al. [55] defined probabilistic uncertain linguistic term set and proposed four operators. Wang [33] proposed an MAGDM algorithm with MGPLTSs based on the distance measures and prospect theory (PT), and verified the validity of the proposed MAGMD method.

In practical decision-making, interrelationship among attributes is very common. Sugeno [39] proposed the fuzzy measure and defined λ-fuzzy measure for the MAGDM problem to handle the interrelationship among attributes. Then Murofushi and Sugeno [40] proposed Choquet integrals of fuzzy measures, and discussed the rationality. Chen et al. [41] proposed probabilistic linguistic Choquet integral operator based on PLTSs to aggregate the enterprise resource planning package evaluation matrices. Choquet integral is widely used in MAGDM method, which has strong practicability.

At present, there are few researches on MGPLTSs. Most of the existing decision-making methods for dealing with MGPLTSs are to transform MGPTLSs into PTLSs with same granularity level according to the transformation function. This means that they cannot directly handle MGPLTSs. No matter in MGPLTSs or PLTSs, the existing operations and AOs cannot avoid the process of normalizing the PLTSs in the calculation process. Based on the above discussion, the motivations and contributions of this paper are as follows:

A new transformation function for MGPLTSs based on the proportional 2-tuple fuzzy linguistic representation model is proposed, which avoids the process of normalizing the PLTSs and reduces the complexity of existing operations of PLTSs [33 , 42–44], then the operational laws and comparison rules of MGPLTSs are given.

Based on the new transformation function, a new Choquet integral operator for MGPLTSs is proposed, which simplifies the steps of transforming MGPLTSs into the PLTSs with same granularity level before aggregation [33].

A new MAGDM method based on the created operator is proposed to solve decision-making problems with relationship between attributes.

The validity of the created MAGDM method is verified, and the new MAGDM method is applied to solve the problem of online teaching quality evaluation.

The rest of this paper is organized as follows. Section 2 briefly introduces the basic concepts. Section 3 proposes some new operations for MGPLTSs. In Section 4, based on the Choquet integral, a new operator for MGPLTSs is proposed and its properties are proved. Section 5 proves the effectiveness of the proposed method by a case study. Then by comparative analysis, the advantages of the created MAGDM method are reflected. In Section 6, concluding remarks are drawn.

2 Preliminaries

2.1 PLTS

A LTS can be denoted as S ={ ς_i|i = 0, . . . , 2g, g ∈ N⁺ }, and 2g + 1 is the granularity of LTS. In general, suppose that S^MG ={ S^t|t = 1, 2 . . . , T } is a MGLTS, where S^t (t = 1, 2, . . . , T) is a LTS. It is assumed that $S^{t} = {ς_{0}^{t}, ς_{1}^{t}, . . ., ς_{2 g^{(t)}}^{t}}$ , the granularity of S^t is 2g^(t) + 1, and $ς_{i}^{t} \in S^{t}$ means $ς_{i}^{t}$ is the ith tern of S^t.

Definition 1 [28]. Assumed that S ={ ς₀, ς₁, . . . , ς_2g } is a LTS, then a PLTS can be described as

$L (p) = {ς_{i} (p_{i}) | ς_{i} \in S, p_{i} ⩾ 0, i = 0, 1, . . ., # L (p), \sum_{i = 0}^{# L (p)} p_{i} ⩽ 1} (1)$ where ς_i (p_i) represents the linguistic term ς_i with its probability p_i, and # L (p) represents the number of linguistic terms in PLTS.

If $\sum_{i = 0}^{# L (p)} p_{i} = 1$ , it means that the probabilistic information of each LTS in PLTS is completely known;

If $0 < \sum_{i = 0}^{# L (p)} p_{i} < 1$ , it means that probabilistic information is partly unknown;

If $\sum_{i = 0}^{# L (p)} p_{i} = 0$ , it means that the probabilistic information is completely unknown.

2.2 The relative theories of 2-tuple linguistic model

Definition 2 [45]. Assumed that S ={ ς_i|i = 0, . . . , g } is a LTS, and β ∈ [0, g] means the result of a symbolic aggregation operation. Then, the function Δ used to obtain the 2-tuple linguistic information equivalent to β can be described as: $Δ : [0, g] \to S \times [- 0.5, 0.5)$

$Δ (β) = {\begin{matrix} ς_{i} i = round (β) \\ α = β - i α = [- 0.5, 0.5) \end{matrix}$ where round (·) means the rounding operation, ς_i has the closest index label to β, and α is the value of the symbolic translation.

Definition 3 [45]. Assumed that S ={ ς_i|i = 0, . . . , g } is a LTS, and (ς_i, α) is a 2-tuple linguistic representation model. Then a 2-tuple linguistic representation model can be transformed into equivalent numerical value β ∈ [0, g] by the function Δ^- 1.

$\begin{matrix} Δ^{- 1} : S \times [- 0.5, 0.5) \\ Δ^{- 1} (ς_{i}, α) = i + α . \end{matrix}$

A representational model of MGLTS is linguistic hierarchy [9], which can be denoted as

$LH = ⋃_{k}^{K} l (k, n (k))$

where l (k, n (k)) is the LTS of the level k, and the granularity is n (k), and can be denoted as $S^{n (k)} = {ς_{0}^{n (k)}, ς_{1}^{n (k)}, . . ., ς_{n (k) - 1}^{n (k)}}$ , k = 1, 2, . . . , K.

Definition 4 [45]. Assumed that $LH = ⋃_{k}^{K} l (k, n (k))$ is a linguistic hierarchy, and LTSs are $S^{n (k)} = {ς_{0}^{n (k)}, ς_{1}^{n (k)}, . . ., ς_{n (k) - 1}^{n (k)}}$ , k = 1, 2, . . . , K. Therefore, the transformation function ${TF}_{k^{'}}^{k}$ to map the set of linguistic variables between different levels can be obtained as:

$\begin{matrix} {TF}_{k^{'}}^{k} (ς_{i}^{n (k)}, α^{n (k)}) \\ = Δ (\frac{Δ^{- 1} (ς_{i}^{n (k)}, α^{n (k)}) \cdot (n (k^{'} - 1))}{n (k) - 1}) \end{matrix}$

To extend the application scope of 2-tuple linguistic representation model, Wang and Hao [15] proposed proportional 2-tuple fuzzy linguistic representation model.

2.3 The relative theories of proportional 2-tuple linguistic model

Definition 5 [15]. Assumed that S ={ ς_i|i = 0, . . . , g } is a LTS. Then the proportional 2-tuple fuzzy linguistic model is (ας_i, (1 - α) ς_i+1), α ∈ [0, 1], ς_i, ς_i+1 ∈ S. The set of all ordinal proportional 2-tuple on LTS S is shown as following.

$\bar{S} = {(ς_{i}, (1 - α) ς_{i + 1}) | α \in [0, 1], i = 0, . . ., g - 1} .$

Assumed that (ας_i, (1 - α) ς_i+1) and (βς_i, (1 - β) ς_i+1) are two proportional 2-tuple linguistic terms, then we have

$\begin{matrix} (α ς_{i}, (1 - α) ς_{i + 1}) < (β ς_{i}, (1 - β) ς_{i + 1}) \\ \Leftrightarrow α i + (1 - α) (i + 1) < β j + (1 - β) (j + 1) \\ \Leftrightarrow i + (1 - α) < j + (1 - β) . \end{matrix}$

Thus, for any two proportional 2-tuples (ας_i, (1 - α) ς_i+1) and (βς_i, (1 - β) ς_i+1), the comparison rules between them are as follows:

if i < j, then

if i = j - 1 and α = 0, β = 1, then (ας_i, (1 - α) ς_i+1) and (βς_i, (1 - β) ς_i+1) represent the same linguistic information,

otherwise: (βς_i, (1 - β) ς_i+1) < (βς_i, (1 - β) ς_i+1);

if i = j, then

if α = β, then (ας_i, (1 - α) ς_i+1) and (βς_i, (1 - β) ς_i+1) represent the same linguistic information,

if α < β, then (ας_i, (1 - α) ς_i+1) < (βς_i, (1 - β) ς_i+1),

if α > β, then (ας_i, (1 - α) ς_i+1) > (βς_i, (1 - β) ς_i+1).

2.4 Choquet integral

Definition 6 [40]. Assumed that P (X) is the power set of X = {x₁, . . . , x_h}. A fuzzy measure on the set X is a function μ : P (X) → [0, 1], and it satisfies the following two conditions:

μ (ϕ) = 0, μ (X) = 1;

μ (B) ⩽ μ (C) , ∀ B, C ∈ P (X) and B ⊂ C.

If X is infinite, it need to add a continuity condition [46]. However, in practical decision problems, the set X is generally limited. According to Definition 6, in order to determine the fuzzy measure, 2^h - 2 parameters need to be calculated. To reduce the calculation complexity of the fuzzy measure, the general fuzzy measure can be replaced by the λ-fuzzy measure.

Definition 7 [40]. Assumed that P (X) is the power set of X = {x₁, . . . , x_h}, for ∀A, B ∈ P (X) , A ∩ B = ϕ. If the fuzzy measure μ satisfies the following conditions:

$μ (A \cup B) = μ (A) + μ (B) + λ μ (A) μ (B)$ where λ ∈ (- 1, ∞), μ is called λ-fuzzy measure (μ_λ).

According to Definition 7, if λ = 0, it means that λ-fuzzy measure μ_λ has additivity, in other words, set A and set B are independent of each other. If λ ≠ 0, it means that λ-fuzzy measure μ_λ has no additivity, in other words, there is an interrelationship between set A and set B; if λ > 0, then μ_λ (A ∪ B) > μ_λ (A) + μ_λ (B), it means that set A and set B have super-additivity; if λ < 0, then μ_λ (A ∪ B) < μ_λ (A) + μ_λ (B), it means that set A and set B have sub-additivity.

According to the definition of λ-fuzzy measure μ_λ, μ_λ (X) =1, then μ_λ can be expressed as follows: $μ_{λ} (X) = {\begin{matrix} \frac{1}{λ} (\prod_{i = 1}^{h} (1 + λ μ_{λ} (x_{i})) - 1), λ \neq 0 \\ \sum_{i = 1}^{h} μ_{λ} (x_{i}), λ = 0 \end{matrix}$ (1)

Due to μ_λ (X) = 1, when λ ≠ 0, the parameter λ can be determined by the following formula: $λ + 1 = \prod_{i = 1}^{h} (1 + λ μ (x_{i}))$ (2)

Definition 8 [47]. Assumed that f is a nonnegative function defined on X = {x₀, . . . , x_2h} and μ is a fuzzy measure defined on X, then the discrete Choquet integral of f with respect to the fuzzy measure μ is: $\int fd μ = \sum_{i = 1}^{h} f (x_{(i)}) [μ (A_{(i)}) - μ (A_{(i + 1)})]$ (3) where subscript (i) is permutation of f (x_(i)), making 0 ⩽ f (x₍₁₎) ⩽ f (x₍₂₎) ⩽ . . . ⩽ f (x_(i)); A_(i) = (x_(i), x_(i+1), . . . , x_(h)), and A_(h+1) = ϕ.

3 Some new operations for MGPLTSs

3.1 The transformation function for MGPLTSs

The transformation function of MGPLTSs based on the proportional 2-tuple fuzzy linguistic representation model is shown as follows.

Definition 9. Assumed that L₁ (p) and L₂ (p) are two arbitrary PLTSs with different granularity levels, $L_{1} (p) = {ς_{i}^{1, n (1)}$ $(p_{i}^{1, n (1)}) | ς_{i}^{1, n (1)} \in S^{n (1)}, p_{i}^{1, n (1)} ⩾ 0, i = 0, 1, . . ., # L_{1} (p),$ $. \sum_{i = 0}^{# L_{1} (p)} p_{i}^{1, n (1)} ⩽ 1}$ and $L_{2} (p) = {ς_{i}^{2, n (2)} (p_{i}^{2, n (2)}) |$ $ς_{i}^{2, n (2)} \in S^{n (2)}$ , $p_{i}^{2, n (2)} ⩾ 0, i = 0, 1, . . ., # L_{2} (p)$ , $\sum_{i = 0}^{# L_{2} (p)} p_{i}^{2, n (2)} ⩽ 1}$ . Then, the transformation function TF from L₁ (p) to L₂ (p) is

$\begin{matrix} {TF}_{2}^{1} (ς_{i}^{1, n (1)} (p_{i}^{1, n (1)})) \\ = {ς_{β}^{1, n (2)} (γ_{β}^{i} p_{i}^{1, n (1)}), ς_{β + 1}^{1, n (2)} (α_{β + 1}^{i} p_{i}^{1, n (1)})} \end{matrix}$ (4) where (i = 0, 1, . . . , n (1) -1), β = round $(\frac{i \times (n (2) - 1)}{n (1) - 1})$ , $α_{β + 1}^{i} = \frac{i \times (n (2) - 1)}{n (1) - 1} - β$ , $γ_{β}^{i} = 1 - α_{β + 1}^{i}$ , $ς_{β}^{1, n (2)}$ is a linguistic term transformed from L₁ (p) to granularity n (2).

After the transformation of each probabilistic linguistic term in L₁ (p), we can calculate the probabilities of the same linguistic terms, and the result is as follows:

$\begin{matrix} {TF}_{2}^{1} (L_{1} (p) = {ς_{0}^{1, n (1)} (p_{0}^{1, n (1)}), ς_{1}^{1, n (1)} (p_{1}^{1, n (1)}), . . ., ς_{n (1) - 1}^{1, n (1)} (p_{n (1) - 1}^{1, n (1)})}) \\ = (ς_{0}^{1, n (2)} (\sum_{i = 0}^{n (1) - 1} (γ_{0}^{i} p_{i}^{1, n (1)} + α_{0}^{i} p_{i}^{1, n (1)})), ς_{1}^{1, n (2)} (\sum_{i = 0}^{n (1) - 1} (γ_{1}^{i} p_{i}^{1, n (1)} + α_{1}^{i} p_{i}^{1, n (1)})), . . ., \\ ς_{j}^{1, n (2)} (\sum_{i = 0}^{n (1) - 1} (γ_{j}^{i} p_{i}^{1, n (1)} + α_{j}^{i} p_{i}^{1, n (1)})), . . ., ς_{n (2) - 1}^{1, n (2)} (\sum_{i = 0}^{n (1) - 1} (γ_{n (2) - 1}^{i} p_{i}^{1, n (1)} + α_{n (2) - 1}^{i} p_{i}^{1, n (1)}))) \\ = {ς_{0}^{1, n (2)} (p_{0}^{1, n (2)}), ς_{1}^{1, n (2)} (p_{1}^{1, n (2)}), . . ., ς_{j}^{1, n (2)} (p_{j}^{1, n (2)}), . . ., ς_{n (2) - 1}^{1, n (2)} (p_{n (2) - 1}^{1, n (2)})} \end{matrix}$ (6)

Example 1. Assumed that $L_{1} (p) = (ς_{0}^{1, n (1)} (0.1), ς_{1}^{1, n (1)} (0.2), ς_{2}^{1, n (1)} (0.7))$ and L₂ (p) = ${ς_{0}^{2, n (2)} (0.2),$ $ς_{1}^{2, n (2)} (0.2)$ , $ς_{2}^{2, n (2)} (0.2)$ , $ς_{3}^{2, n (2)} (0.2)$ , $ς_{4}^{2, n (2)} (0.2)}$ are two PLTSs,

With respect to the linguistic level (2, n (2)), L₁ (p) can be transformed into $L_{1}^{'} (p)$ as follows.

${TF}_{2}^{1} (ς_{0}^{1, n (1)} (0.1)) = {ς_{0}^{1, n (2)} (1 \times 0.1)} = {ς_{0}^{1, n (2)} (0.1)}$ ,

${TF}_{2}^{1} (ς_{1}^{1, n (1)} (0.2)) = {ς_{2}^{1, n (2)} (1 \times 0.2)} = {ς_{2}^{1, n (2)} (0.2)}$ ,

${TF}_{2}^{1} (ς_{2}^{1, n (1)} (0.7)) = {ς_{4}^{1, n (2)} (1 \times 0.7)} = {ς_{4}^{1, n (2)} (0.7)}$ .

The result is $L_{1}^{'} (p) = {ς_{0}^{1, n (2)} (0.1)$ $ς_{2}^{1, n (2)} (0.2)$ , $ς_{4}^{1, n (2)} (0.7)}$

With respect to the linguistic level (1, n (1)), L₂ (p) can be transformed into $L_{2}^{'} (p)$ as follows.

${TF}_{1}^{2} (ς_{0}^{2, n (2)} (0.2)) = {ς_{0}^{2, n (1)} (1 \times 0.2)} = {ς_{0}^{2, n (1)} (0.2)}$ ,

${TF}_{1}^{2} (ς_{1}^{2, n (2)} (0.2)) = {ς_{0}^{2, n (1)} (0.5 \times 0.2), ς_{1}^{2, n (1)} (0.5 \times 0.2)} = {ς_{0}^{2, n (1)} (0.1), ς_{1}^{2, n (1)} (0.1)}$ ,

${TF}_{1}^{2} (ς_{2}^{2, n (2)} (0.2)) = {ς_{1}^{2, n (1)} (1 \times 0.2)} = {ς_{1}^{2, n (1)} (0.2)}$ ,

${TF}_{1}^{2} (ς_{3}^{2, n (2)} (0.2)) = {ς_{1}^{2, n (1)} (0.5 \times 0.2), ς_{2}^{2, n (1)} (0.5 \times 0.2)} = {ς_{1}^{2, n (1)} (0.1), ς_{2}^{2, n (1)} (0.1)}$ ,

${TF}_{1}^{2} (ς_{4}^{2, n (2)} (0.2)) = {ς_{2}^{2, n (1)} (1 \times 0.2)} = {ς_{2}^{2, n (1)} (0.2)}$ .

The result is $L_{2}^{'} (p) =$ ${ς_{0}^{2, n (1)} (0.3)$ , $ς_{1}^{2, n (1)} (0.4)$ , $ς_{2}^{2, n (1)} (0.3)}$ .

Therefore, the PLTSs with different granularity levels can be transformed into the same granularity level.

3.2 The operational laws for MGPLTSs

Theorem 1 [48]. Assumed that S ={ ς₀, ς₁, . . . , ς_2g } is a LTS. L₁ (p), L₂ (p) and L₃ (p) are three PLTSs on S, and 0 ⩽ a ⩽ 1, where $L_{1} (p) = {. ς_{i} (p_{i}^{1}) | ς_{i} \in S, p_{i}^{1} ⩾ 0, i = 0, 1, . . ., 2 g, \sum_{i = 0}^{2 g} p_{i}^{1} ⩽ 1}$ , $L_{2} (p) = {. ς_{i} (p_{i}^{2}) | .$ $ς_{i} \in S, p_{i}^{2} ⩾ 0$ , i = 0, 1,..., 2g,

$. \sum_{i = 0}^{2 g} p_{i}^{2} ⩽ 1}$ , $L_{3} (p) = {. ς_{i} (p_{i}^{3}) | .$ ς_i ∈ S, $p_{i}^{3} ⩾ 0$ , i = 0, 1, . . . , 2g, $. \sum_{i = 0}^{2 g} p_{i}^{3} ⩽ 1}$ .

Then we have the following operational laws [48]: $\begin{matrix} L_{1} (p) \oplus L_{2} (p) = \\ {ς_{i} (\frac{p_{i}^{1} p_{i}^{2} + p_{i}^{1} (1 - \sum_{i = 0}^{2 g} p_{i}^{2}) + p_{i}^{2} (1 - \sum_{i = 0}^{2 g} p_{i}^{1})}{(1 - \sum_{i = 0}^{2 g} p_{i}^{2}) (1 - \sum_{i = 0}^{2 g} p_{i}^{1}) + \sum_{i = 0}^{2 g} (p_{i}^{1} p_{i}^{2} + p_{i}^{1} (1 - \sum_{i = 0}^{2 g} p_{i}^{2}) + p_{i}^{2} (1 - \sum_{i = 0}^{2 g} p_{i}^{1}))})} \end{matrix}$ (7)

${aL}_{3} (p) = {ς_{i} ({ap}_{i}^{3}) | i = 0, . . ., 2 g}$ (8) $L_{1} (p) \oplus L_{2} (p) = L_{2} (p) \oplus L_{1} (p)$ (9) $\begin{matrix} (L_{1} (p) \oplus L_{2} (p)) \oplus L_{3} (p) = L_{2} (p) \\ \oplus (L_{1} (p) \oplus L_{3} (p)) \end{matrix}$ (10)

In order to calculate MGPLTSs directly, we propose the operational laws of MGPLTSs based on Definition 9 and Theorem 1.

Theorem 2. Assumed that L₁ (p), L₂ (p), L₃ (p) are three PLTSs with different granularity levels, $L_{1} (p) = {ς_{i}^{1, n (1)} (p_{i}^{1, n (1)}) | .$ $ς_{i}^{1, n (1)} \in S$ , $p_{i}^{1, n (1)} ⩾ 0$ , i = 0, 1, . . . , # L₁ (p), $. \sum_{i = 0}^{# L_{1} (p)} p_{i}^{1, n (1)} ⩽ 1}$ , (n (1) = 2g⁽¹⁾ + 1, g⁽¹⁾ ∈ N⁺), $L_{2} (p) = {ς_{i}^{2, n (2)} (p_{i}^{2, n (2)}) |$ $ς_{i}^{2, n (2)} \in S$ , $p_{i}^{2, n (2)} ⩾ 0$ , i = 0, 1, . . . , # L₂ (p), $\sum_{i = 0}^{# L_{2} (p)} p_{i}^{2, n (2)} ⩽ 1}$ , (n (2) = 2g⁽²⁾ + 1, g⁽²⁾ ∈ N⁺), $L_{3} (p) = {ς_{i}^{3, n (3)} (p_{i}^{3, n (3)}) |$ $ς_{i}^{3, n (3)} \in S$ , $p_{i}^{3, n (3)} ⩾ 0$ , i = 0, 1, . . . , # L₃ (p), $\sum_{i = 0}^{# L_{3} (p)} p_{i}^{3, n (3)} ⩽ 1}$ , (n (3) = 2g⁽³⁾ + 1, g⁽³⁾ ∈ N⁺), and L₁ (p) is the basic LTS (BLTS). The operational laws are as follows:

$\begin{matrix} L_{1} (p) \oplus L_{2} (p) = L_{4} (p) = \\ {ς_{i}^{4, n (1)} (\frac{p_{i}^{1, n (1)} p_{i}^{2, n (1)} + p_{i}^{1, n (1)} ϕ_{2} + p_{i}^{2, n (1)} ϕ_{1}}{ϕ_{2} ϕ_{1} + \sum_{i = 0}^{n (1) - 1} (p_{i}^{1, n (1)} p_{i}^{2, n (1)} + p_{i}^{1, n (1)} ϕ_{2} + p_{i}^{2, n (1)} ϕ_{1})})} \end{matrix}$ (11)

where $p_{i}^{2, n (1)} = \sum_{j = 0}^{n (2) - 1} (γ_{i}^{j} p_{j}^{2, n (2)} + α_{i}^{j} p_{j}^{2, n (2)})$ , ϕ₁ = $(1 - \sum_{i = 0}^{n (1) - 1} p_{i}^{1, n (1)})$ , $ϕ_{2} = (1 - \sum_{i = 0}^{n (1) - 1} p_{i}^{2, n (1)})$ , (i = 0, 1, …, n (1)).

Some other operational laws:

$\begin{matrix} {aL}_{3} (p) = \\ {ς_{i}^{3, n (3)} ({ap}_{i}^{3, n (3)}) | i = 0, . . ., n (3) - 1}, \\ (0 ⩽ a ⩽ 1) \end{matrix}$ (12)

$L_{1} (p) \oplus L_{2} (p) = L_{2} (p) \oplus L_{1} (p)$ (13) $\begin{matrix} (L_{1} (p) \oplus L_{2} (p)) \oplus L_{3} (p) = L_{2} (p) \\ \oplus (L_{1} (p) \oplus L_{3} (p)) \end{matrix}$ (14)

Proof. According to Equations (5)–(7), we can easily derive Equation (11). According to Equations (5) and Equation (8), we can derive Equation (13). The expressions of Equation (13) and Equation (14) are consistent with Equation (9) and Equation (10).

3.3 The comparison between MGPLTSs

According to Equation (5) and Equation (6), the comparison rules MGPLTSs can be given.

Definition 10 [48]. Assumed that L₁ (p), L₂ (p) are two PLTSs with different granularity levels, $L_{1} (p) = {ς_{i}^{1, n (1)} (p_{i}^{1, n (1)}) |$ $ς_{i}^{1, n (1)} \in S$ , $p_{i}^{1, n (1)} ⩾ 0$ , i = 0, 1, . . . , # L₁ (p), $\sum_{i = 0}^{# L_{1} (p)} p_{i}^{1, n (1)} ⩽ 1}$ , (n (1) = 2g⁽¹⁾ + 1, g⁽¹⁾ ∈ N⁺), $L_{2} (p) = {ς_{j}^{2, n (2)} (p_{j}^{2, n (2)}) |$ $ς_{j}^{2, n (2)} \in S$ , $p_{j}^{2, n (2)} ⩾ 0$ , j = 0, 1, . . . , # L₂ (p), $\sum_{j = 0}^{# L_{2} (p)} p_{j}^{2, n (2)} ⩽ 1}$ , (n (2) =2g⁽²⁾ + 1, g⁽²⁾ ∈ N⁺), and L₁ (p) is the BLTS. Then the score function D (L₁ (p)) and uncertain function F (L₁ (p)) of L₁ (p) can be expressed as: $D (L_{1} (p)) = \frac{2}{n (1) - 1} \sum_{i = 0}^{n (1) - 1} (i \times p_{i}^{1, n (1)})$ (15) $F (L_{1} (p)) = 1 - \sum_{i = 0}^{n (1) - 1} p_{i}^{1, n (1)}$ (16) where D (L₁ (p)) ∈ [0, 2] and F (L₁ (p)) ∈ [0, 1]. The score function D (L₂ (p)), and uncertain function F (L₂ (p)) of L₂ (p) can be expressed as:

$\begin{matrix} D (L_{2} (p)) = \frac{2}{n (1) - 1} \\ \sum_{i = 0}^{n (1) - 1} (i \times (\sum_{j = 0}^{n (2) - 1} (γ_{i}^{j} p_{j}^{2, n (2)} + α_{i}^{j} p_{j}^{2, n (2)}))) \end{matrix}$ (17)

$\begin{matrix} F (L_{2} (p)) = \\ 1 - \sum_{i = 0}^{n (1) - 1} (\sum_{j = 0}^{n (2) - 1} (γ_{i}^{j} p_{j}^{2, n (2)} + α_{i}^{j} p_{j}^{2, n (2)})) \end{matrix}$ (18) where D (L₁ (p)) ∈ [0, 2] and F (L₁ (p)) ∈ [0, 1].

According to Definition 10, Equation (15) and Equation (16) are used to compare two PLTSs with the same granularity level. Based on score function D (L₂ (p)) and uncertain function F (L₂ (p)), the comparison rules of MGPLTSs are as follows:

Definition 11. Assumed that L₁ (p) and L₂ (p) are two PLTSs with the different granularity levels, then:

If D (L₁ (p)) < D (L₂ (p)), then L₁ (p) ≺ L₂ (p);

If D (L₁ (p)) > D (L₂ (p)), then L₁ (p) ≻ L₂ (p);

If D (L₁ (p)) = D (L₂ (p)), then

If F (L₁ (p)) < F (L₂ (p)), then L₁ (p) ≻ L₂ (p),

If F (L₁ (p)) = F (L₂ (p)), then L₁ (p) ∼ L₂ (p),

If F (L₁ (p)) > F (L₂ (p)), then L₁ (p) ≺ L₂ (p).

Definition 11 is also applicable to compare two PLTs with the same granularity level.

4 The decision making method based on multi-granularity probabilistic linguistic Choquet integral operator

4.1 The multi-granularity probabilistic linguistic Choquet integral operator

Based on Equations (2)–(7), we propose a new operator for MGPLTSs.

Definition 12. Assumed that L₁ (p) , L₂ (p) , . . . , L_m (p) are m arbitrary PLTSs with the different granularity levels, and [μ (A_(i)) - u (A_(i+1))] _a is the weight of the L_a (p) , (a = 1, 2, . . , m). Suppose L₁ (p) is the BLTS. The multi-granularity probabilistic linguistic Choquet integral (MGPLCA) operator is defined as $\begin{matrix} MGPLCA (L_{1} (p), L_{2} (p), . . ., L_{m} (p)) \\ = {{[μ (A_{(i)}) - u (A_{(i + 1)})]}_{1} {ς_{j}^{1, n (1)} (p_{j}^{1, n (1)})} \\ \oplus {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{2} {ς_{l}^{2, n (2)} (p_{l}^{2, n (2)})} \\ \oplus . . . \oplus {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{m} {ς_{s}^{m, n (m)} (p_{s}^{m, n (m)})}} \\ = {{[μ (A_{(i)}) - u (A_{(i + 1)})]}_{1} {ς_{j}^{1, n (1)} (p_{j}^{1, n (1)})} \\ \oplus {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{2} {ς_{j}^{2, n (1)} (p_{j}^{2, n (1)})} \\ \oplus . . . \oplus {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{m} {ς_{s}^{m, n (1)} (p_{s}^{m, n (1)})}} \\ = {ς_{j}^{m + 1, n (1)} ({[μ (A_{(i)}) - u (A_{(i + 1)})]}_{1} p_{j}^{1, n (1)} \\ \oplus {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{2} p_{j}^{2, n 1} \\ \oplus . . . \oplus {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{m} p_{j}^{m, n (1)}) \\ | j = 0, . . ., n (1) - 1} \end{matrix}$ (19)

Theorem 3. Assumed that L₁ (p) , L₂ (p) , . . . , L_m (p) are m arbitrary PLTSs with the different granularity levels. Suppose L₁ (p) is the BLTS, the aggregation outputs by MGPLCA operator is also a PLTS, where

$\begin{matrix} MGPLCA (L_{1} (p), L_{2} (p), . . ., L_{m} (p)) \\ = {ς_{j}^{m + 1, n (1)} (p_{j}^{m + 1,, n (1)})} . \end{matrix}$ (20)

Proof. We can use mathematical induction to prove this theorem.

(1) When m = 2, we derive

$\begin{matrix} MGPLCA (L_{1} (p), L_{2} (p)) = \\ {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{1} L_{1} (p) \\ \oplus {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{2} L_{2} (p) . \end{matrix}$

According to Theorem 1 and Definition 12, we can derive

$\begin{matrix} {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{1} L_{1} (p) = {ς_{j}^{1, n (1)} \\ ({[μ (A_{(i)}) - u (A_{(i + 1)})]}_{1} p_{j}^{1, n (1)}) | j = 0, . . ., n (1) - 1} \end{matrix}$

$\begin{matrix} {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{2} L_{2} (p) \\ = {ς_{l}^{2, n (2)} ({[μ (A_{(i)}) - u (A_{(i + 1)})]}_{2} p_{l}^{2, n (1)}) |} \end{matrix}$

$\begin{matrix} | l = 0, . . ., n (2) - 1} \\ = {ς_{j}^{2, n (1)} ({[μ (A_{(i)}) - u (A_{(i + 1)})]}_{2} p_{j}^{2, n (2)}) \\ | j = 0, . . ., n (1) - 1} . \end{matrix}$

Then,

$\begin{matrix} {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{1} L_{1} (p) \\ \oplus {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{2} L_{2} (p) \\ = {ς_{j}^{1, n (1)} ({[μ (A_{(i)}) - u (A_{(i + 1)})]}_{1} p_{j}^{1, n (1)} \end{matrix}$

$\begin{matrix} \oplus {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{2} p_{j}^{2, n (1)}) \\ | j = 0, . . ., n (1) - 1} \\ = {ς_{j}^{3, n (1)} p_{j}^{3, n (1)} | j = 0, . . ., n (1) - 1} . \end{matrix}$

i.e., when m = 2, Equation (20) holds.

(2) Assume that when m = q, Equation (20) holds. That is

$\begin{matrix} {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{1} L_{1} (p) \oplus . . . \\ \oplus {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{q} L_{q} (p) \\ = {ς_{j}^{1, n (1)} ({[μ (A_{(i)}) - u (A_{(i + 1)})]}_{1} p_{j}^{1, n (1)} \oplus . . . \\ \oplus {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{q} p_{j}^{q, n (1)}) \\ | j = 0, . . ., n (1) - 1} \\ = {ς_{j}^{q + 1, n (1)} p_{j}^{q + 1, n (1)} | j = 0, . . ., n (1) - 1} . \end{matrix}$

Then, when m = q + 1, from Equation (9), we have

$\begin{matrix} {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{1} L_{1} (p) \oplus . . . \oplus {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{q} L_{q} (p) \oplus {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{q + 1} L_{q + 1} (p) \\ = {ς_{j}^{q + 2, n (1)} (\begin{matrix} {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{1} p_{j}^{1, n (1)} \oplus . . . \oplus {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{q} p_{j}^{q, n (1)} \oplus \\ {[μ (A_{(i)}) - u (A_{(i + 1)})]}_{q + 1} p_{j}^{q + 1, n (1)} \end{matrix}) | j = 0, . . ., n (1) - 1} \\ = {ς_{j}^{q + 2, n (1)} p_{j}^{q + 2, n (1)} | j = 0, . . ., n (1) - 1} . \end{matrix}$

i.e., when m = q + 1, Equation (20) holds.

According to (1) and (2), Equation (20) holds.

For Equation (19) and Equation (20), we can see that the calculation method of PLTSs with the same granularity level or MGPLTSs are the same. Therefore, in the following, we use PLTSs with the same granularity level to prove the properties of MGPLCA operator.

(1) Commutativity: Assumed that L₁ (p) = ${ς_{0}^{1} (p_{0}^{1}), . . ., ς_{2 g}^{1} (p_{2 g}^{1})}$ , $L_{2} (p) = {ς_{0}^{2} (p_{0}^{2}), . . ., ς_{2 g}^{2} (p_{2 g}^{2})}$ are two different PLTSs with the same granularity level, then MGPLCA (L₁ (p) , L₂ (p)) = MGPLCA (L₂ (p) , L₁ (p)).

Based on the Equation (9) and Equation (10), we can easily get this result, it can be omitted here.

(2) Boundedness: Assumed that $L_{1} (p) = {ς_{0}^{1} (p_{0}^{1}), . . ., ς_{2 g}^{1} (p_{2 g}^{1})}$ , $L_{2} (p) = {ς_{0}^{2} (p_{0}^{2}), . . ., ς_{2 g}^{2} (p_{2 g}^{2})}$ are two different PLTSs with the same granularity level. $MGPLCA (L_{1} (p), L_{2} (p)) = {ς_{0}^{3} (p_{0}^{3}), . . ., ς_{2 g}^{3} (p_{2 g}^{3})}$ = L₃ (p). Then $0 < \sum_{i = 0}^{2 g} p_{i}^{3} < 1$ , (i = 0, 1, . . . , 2g).

Proof. When g = 1, suppose there are two PLTSs, $L_{1} (p) = {ς_{0}^{1} (p_{0}^{1}), ς_{1}^{1} (p_{1}^{1}), ς_{2}^{1} (p_{2}^{1})} = {ς_{0}^{1} (a),$ $ς_{1}^{1} (b), ς_{2}^{1} (e)}$ , (0 ⩽ a + b + e ⩽ 1) and $L_{2} (p) = {ς_{0}^{2} (p_{0}^{2}), ς_{1}^{2} (p_{1}^{2}), ς_{2}^{2} (p_{2}^{2})} = {ς_{0}^{2} (c), ς_{1}^{2} (d), ς_{2}^{2} (f)}$ , (0 ⩽ c + d + f ⩽ 1). And the fuzzy measure of L₁ (p) is 0 < μ (L₁ (p)) < 1, the fuzzy measure of L₂ (p) is 0 < μ (L₂ (p)) < 1.

Calculate the weights of L₁ (p) and L₂ (p) by Equation (3) and Equation (4). For simplicity, suppose the weight of L₁ (p) is ω₁, the weight of L₂ (p) is ω₂. Then based on Equation (11), we can obtain

$\begin{matrix} L_{1} (p) \oplus L_{2} (p) = {ς_{i} (ω_{1} p_{i}^{1} \oplus ω_{2} p_{i}^{2}) | i = 0, 1, 2} \\ = {ς_{0}^{3} (\frac{ac + a (1 - ω_{2}) + c (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + ac + a (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})}), \\ ς_{1}^{3} (\frac{bd + b (1 - ω_{2}) + d (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + ac + a (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})}), \\ ς_{2}^{3} (\frac{ef + e (1 - ω_{2}) + f (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + ac + a (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})})} \\ = {ς_{0}^{3} (p_{0}^{3}), ς_{1}^{3} (p_{1}^{3}), ς_{2}^{3} (p_{2}^{3})} = L_{3} (p) . \end{matrix}$

Then

$\begin{matrix} p_{0}^{3} + p_{1}^{3} + p_{2}^{3} = \\ \frac{ac + a (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})}{ac + a (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1}) + (1 - ω_{2}) (1 - ω_{1})} \end{matrix}$

Therefore, $0 < p_{0}^{3} + p_{1}^{3} + p_{2}^{3} < 1$ . Further, when g = 2, 3, . . ., $0 < \sum_{i = 0}^{2 g} p_{i}^{3} < 1, (i = 0, 1, . . ., 2 g)$ holds.

(3) Monotonicity: Assumed that $L_{1} (p) = {ς_{0}^{1} (p_{0}^{1}), . . ., ς_{2 g}^{1} (p_{2 g}^{1})}$ , $L_{2} (p) = {ς_{0}^{2} (p_{0}^{2}), . . ., ς_{2 g}^{2} (p_{2 g}^{2})}$ are two different PLTSs with the same granularity level. The weights of L₁ (p) and L₂ (p) are ω₁ and ω₂. Then, adjust the probability of L₁ (p), get $L_{3} (p) = {ς_{0}^{3} (p_{0}^{3}), ς_{1}^{3} (p_{1}^{1}), . . ., ς_{2 g}^{3} (p_{2 g}^{1})}$ , $(p_{0}^{3} > p_{0}^{1})$ and to make ω₁ = ω₃. We have $MGPLCA (L_{1} (p), L_{2} (p)) = {ς_{0}^{4} (p_{0}^{4}), ς_{0}^{4} (p_{1}^{4}), . . ., ς_{2 g}^{4} (p_{2 g}^{4})} = L_{4} (p)$ , $MGPLCA (L_{3} (p), L_{2} (p)) = {ς_{0}^{5} (p_{0}^{5}), ς_{0}^{5} (p_{1}^{5}), . . ., ς_{2 g}^{5} (p_{2 g}^{5})} = L_{5} (p)$ . Then $p_{0}^{4} < p_{0}^{5}, p_{i}^{4} > p_{i}^{5}, (i = 1, 2, . . ., 2 g)$ .

Proof. When g = 1, suppose there are two different PLTSs, $L_{1} (p) = {ς_{0}^{1} (p_{0}^{1}), ς_{1}^{1} (p_{1}^{1}), ς_{2}^{1} (p_{2}^{1})} =$ ${ς_{0}^{1} (a), ς_{1}^{1} (b), ς_{2}^{1} (e)}$ , (0 ⩽ a + b + e ⩽ 1) and $L_{2} (p) = {ς_{0}^{2} (p_{0}^{2}), ς_{1}^{2} (p_{1}^{2}), ς_{2}^{2} (p_{2}^{2})} = {ς_{0}^{2} (c), ς_{1}^{2} (d)$ , $ς_{2}^{2} (f)}$ , (0 ⩽ c + d + f ⩽ 1).

For simplicity, suppose the weight of L₁ (p) is ω₁, the weight of L₂ (p) is ω₂. Then, adjust the probability of L₁ (p), get $L_{3} (p) = {ς_{0}^{3} (p_{1}^{3}), ς_{1}^{3} (p_{1}^{3}), ς_{2}^{3} (p_{2}^{3})} = {ς_{0}^{3} (A), ς_{1}^{3} (b), ς_{2}^{3} (e)}$ , (A = a + B ⩾ a, B > 0, 0 ⩽ A + b + e ⩽ 1), and to make ω₁ = ω₃.

Based on Equation (11), we have

$\begin{matrix} L_{1} (p) \oplus L_{2} (p) = {ς_{i} (ω_{1} p_{i}^{1} \oplus ω_{2} p_{i}^{2}) | i = 0, 1, 2} \\ = {ς_{0}^{4} (\frac{ac + a (1 - ω_{2}) + c (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + ac + a (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})}), \\ ς_{1}^{4} (\frac{bd + b (1 - ω_{2}) + d (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + ac + a (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})}), \\ ς_{2}^{4} (\frac{ef + e (1 - ω_{2}) + f (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + ac + a (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})})} \\ = {ς_{0}^{4} (p_{0}^{4}), ς_{1}^{4} (p_{1}^{4}), ς_{2}^{4} (p_{2}^{4})} = L_{4} (p) \end{matrix}$

$\begin{matrix} L_{3} (p) \oplus L_{2} (p) = {ς_{i} (ω_{3} p_{i}^{3} \oplus ω_{2} p_{i}^{2}) | i = 0, 1, 2} \\ = {ς_{0}^{5} (\frac{Ac + A (1 - ω_{2}) + c (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + Ac + A (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})}), \\ ς_{1}^{5} (\frac{bd + b (1 - ω_{2}) + d (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + Ac + A (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})}), \\ ς_{2}^{5} (\frac{ef + e (1 - ω_{2}) + f (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + Ac + A (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})})} \\ = {ς_{0}^{5} (p_{0}^{5}), ς_{1}^{5} (p_{1}^{5}), ς_{2}^{5} (p_{2}^{5})} = L_{5} (p) . \end{matrix}$

Because of A = a + B > a, we can easily obtain

$\begin{matrix} \frac{bd + b (1 - ω_{2}) + d (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + ac + a (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})} > \\ \frac{bd + b (1 - ω_{2}) + d (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + Ac + A (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})}; \end{matrix}$

$\begin{matrix} \frac{ef + e (1 - ω_{2}) + f (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + ac + a (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})} > \\ \frac{ef + e (1 - ω_{2}) + f (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + Ac + A (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})}; \end{matrix}$ and

$\begin{matrix} \frac{Ac + A (1 - ω_{2}) + c (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + Ac + A (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})} > \\ \frac{ac + a (1 - ω_{2}) + c (1 - ω_{1})}{(1 - ω_{2}) (1 - ω_{1}) + ac + a (1 - ω_{2}) + c (1 - ω_{1}) + bd + b (1 - ω_{2}) + d (1 - ω_{1}) + ef + e (1 - ω_{2}) + f (1 - ω_{1})} . \end{matrix}$

Obviously, $p_{0}^{4} < p_{0}^{5}, p_{1}^{4} > p_{1}^{5}, p_{2}^{4} > p_{2}^{5}$ . And so on, When g = 2, 3, . . ., it holds.

Example 2. It is assumed that there are four PLTSs:

$\begin{matrix} L_{1} (p) = {ς_{0}^{1} (0), ς_{1}^{1} (0), ς_{2}^{1} (0.0833), ς_{3}^{1} (0.8167), ς_{4}^{1} (0.1)}, L_{2} (p) = {ς_{0}^{2} (0), ς_{1}^{2} (0), ς_{2}^{2} (0.05), ς_{3}^{2} (0.8), ς_{4}^{2} (0.15)}, \\ L_{3} (p) = {ς_{0}^{3} (0), ς_{1}^{3} (0), ς_{2}^{3} (0.1), ς_{3}^{3} (0.7944), ς_{4}^{3} (0.1056)}, L_{4} (p) = {ς_{0}^{4} (0), ς_{1}^{4} (0), ς_{2}^{4} (0.05), ς_{3}^{4} (0.8944), ς_{4}^{4} (0.0556)} \end{matrix}$

and μ (ϕ) = 0, μ (L₁ (p)) = 3/9, μ (L₂ (p)) = 4/9, μ (L₃ (p)) = 2/9, μ (L₄ (p)) = 3/9.

According to Equation (2), Equation (3), we derive λ = -0.3333, then

$\begin{matrix} μ (L_{1} (p), L_{2} (p)) = 0.692, μ (L_{1} (p), L_{3} (p)) = 0.513, \\ μ (L_{1} (p), L_{4} (p)) = 0.602, \\ μ (L_{2} (p), L_{3} (p)) = 0.61, μ (L_{2} (p), L_{4} (p)) = 0.692, \\ μ (L_{3} (p), L_{4} (p)) = 0.513, \\ μ (L_{1} (p), L_{2} (p), L_{3} (p)) = 0.825, μ (L_{1} (p), L_{2} (p), L_{4} (p)) \\ = 0.892, \\ μ (L_{1} (p), L_{3} (p), L_{4} (p)) = 0.747, μ (L_{2} (p), L_{3} (p), L_{4} (p)) \\ = 0.825, \\ μ (L_{1} (p), L_{2} (p), L_{3} (p), L_{4} (p)) = 1 . \end{matrix}$

The according to Equation (19), we can get the result:

$\begin{matrix} MGPLCA (L_{1} (p), L_{2} (p), L_{3} (p), L_{4} (p)) \\ = {ς_{2} (0.0309), ς_{3} (0.5826), ς_{4} (0.0612)} . \end{matrix}$

4.2 The procedure of proposed method

Based on the above calculations and discussion, we summarize the decision-making process for a MAGDM method by using MGPLCA operator. For a MAGDM problem with MGPLTSs, let X ={ x₁, x₂, . . . , x_m } be a set of alternatives, C ={ c₁, c₂, . . . , c_n } be the set of attributes, and E ={ e₁, e₂, . . . , e_d } be a set of DMs, where, x_i (i = 1, 2, . . . , m) represents the ith alternative, c_j = (j = 1, 2, . . . , n) represents the jth attribute and let u = (u (c₁) , u (c₂) , . . . , u (c_n)) be the fuzzy measure vector of attributes c_j = (j = 1, 2, . . . , n), and u (c₁, c₂, . . . , c_n) = 1, moreover, e_v (v = 1, 2, . . . , d) represents the vth DM, and let y = (y₁, y₂, . . . , y_d) ^T be the weight vector of the DMs with e_v (v = 1, 2, . . . , d), y_v ∈ [0, 1] , v = 1, 2, . . . , d and $\sum_{v = 1}^{d} y_{v} = 1$ . Let S^MG ={ S^t|t = 1, 2 . . . , T } be a MGLTS, DM will choose a LTS S^t to give the decision matrix $D_{v} = {[L_{t}^{v, ij} (p)]}_{m \times n} (v = 1, 2, . . ., d)$ , where $L_{t}^{v, ij} (p)$ is a PLTS, which represents the evaluation information of the alternative x_i (i = 1, 2, . . . , m) about the attribute c_j (j = 1, 2, . . . , n) given by the DM e_v (v = 1, 2, . . . , d), Fig. 1 shows the flow chart of the proposed MAGDM methodology, and the specific steps of our proposed method are as follows:

Fig. 1

Procedures involved in the created MAGDM method.

Step 1. According to their own evaluation habits of DMs, DM e_v (v = 1, 2, . . . , d) can choose a LTS S^t to give the decision matrix of MGPTLSs, respectively. Then, d decision matrices are obtained. Then multiply each probability

value by the weight of the corresponding DM, and integrate d decision matrices into one matrix.

Step 2. Calculate λ by Equation (3) and calculate fuzzy measure by Equation (2). Then, calculate weights w_j of attributes by Equation (4).

Step 3. Choose a LTS as the BLTS. According to the weights between different attributes, calculate the aggregation result R_i = MGPLCA (L₁ (p) , L₂ (p) , . . . , L_n (p)), i = 1, 2, . . . , m of each alternative by MGPLCA operator.

Step 4. The score function D (R_i (w)) can be calculated by Equation (15) and the uncertainty function F (R_i (w)) can be calculated by Equation (16) based on the aggregation results R_i (i = 1, 2, . . . , m).

Step 5. According to Definition 11, first compare score function D (R_i (w)) of each alternative, if there are several alternatives with the same score function D (R_i (w)), then compare the uncertainty function F (R_i (w)) of these alternatives. Finally, we can obtain the ranking of alternatives and the optimal alternative.

5 Application examples

In this section, an example about the online teaching quality evaluation is used to illustrate the application of the created MAGDM method, and some comparisons of the created MAGDM method with other existing methods are given.

5.1 Application of the created multi-attribute group decision-making method

Example 3. We invited three DMs E ={ e₁, e₂, e₃ }, and assume that the weights of DMs are y = (y₁, y₂, y₃) ^T = (0.5, 0.3, 0.2) ^T. There are four online teaching courses X ={ x₁, x₂, x₃, x₄ } to be evaluated based on four attributes which are as follows: (1) teaching philosophy and curriculum design (c₁), (2) teaching content and learning resources (c₂), (3) faculty and teaching activities (c₃), user interface design and technical support (c₄). Assume that the fuzzy measure of each attribute is μ (ϕ) =0, μ (c₁) =0.2, μ (c₂) =0.3, μ (c₃) =0.4, μ (c₄) =0.2. Three PLTSs are with three, five and nine granularity level.

$\begin{matrix} S^{1, n (1)} = {ς_{0}^{1, n (1)} = bad, ς_{1}^{1, n (1)} = medium, ς_{2}^{1, n (1)} = good} \\ S^{2, n (2)} = {ς_{0}^{2, n (2)} = very bad, ς_{1}^{2, n (2)} = bad, ς_{2}^{2, n (2)} = medium, ς_{3}^{2, n (2)} = good, ς_{4}^{2, n (2)} = very good} \\ S^{3, n (3)} = {\begin{matrix} ς_{0}^{3, n (3)} = extremely bad, ς_{1}^{3, n (3)} = very bad, ς_{2}^{3, n (3)} = a little bad, ς_{3}^{3, n (3)} = bad, ς_{4}^{3, n (3)} \\ = medium, ς_{5}^{3, n (3)} = good, ς_{6}^{3, n (3)} = a little good, ς_{7}^{3, n (3)} = very bad, ς_{8}^{3, n (3)} = emely good \end{matrix}} . \end{matrix}$

Step 1. The evaluation information given by three DMs with MGPLTSs are shown in Tables 1 –3.

Table 1
The decision matrix D₁ provided by the DM e₁

c ₁ c ₂ c ₃ c ₄

x ₁ $S^{3, n (3)} = {ς_{7}^{3, n (3)} (1)}$ ${ς_{7}^{3, n (3)} (1)}$ ${\begin{matrix} ς_{6}^{3, n (3)} (0.4), \\ ς_{7}^{3, n (3)} (0.6) \end{matrix}}$ ${\begin{matrix} ς_{6}^{3, n (3)} (0.3), \\ ς_{7}^{3, n (3)} (0.7) \end{matrix}}$

x ₂ ${\begin{matrix} ς_{5}^{3, n (3)} (0.7), \\ ς_{6}^{3, n (3)} (0.3) \end{matrix}}$ ${\begin{matrix} ς_{4}^{3, n (3)} (0.2), \\ ς_{5}^{3, n (3)} (0.8) \end{matrix}}$ ${ς_{4}^{3, n (3)} (1)}$ ${ς_{4}^{3, n (3)} (1)}$

x ₃ ${\begin{matrix} ς_{5}^{3, n (3)} (0.3), \\ ς_{6}^{3, n (3)} (0.7) \end{matrix}}$ ${\begin{matrix} ς_{5}^{3, n (3)} (0.3), \\ ς_{6}^{3, n (3)} (0.7) \end{matrix}}$ ${\begin{matrix} ς_{6}^{3, n (3)} (0.8), \\ ς_{7}^{3, n (3)} (0.2) \end{matrix}}$ ${ς_{4}^{3, n (3)} (1)}$

x ₄ ${\begin{matrix} ς_{2}^{3, n (3)} (0.2), \\ ς_{3}^{3, n (3)} (0.8) \end{matrix}}$ ${\begin{matrix} ς_{3}^{3, n (3)} (0.8), \\ ς_{4}^{3, n (3)} (0.2) \end{matrix}}$ ${ς_{2}^{3, n (3)} (1)}$ ${\begin{matrix} ς_{2}^{3, n (3)} (0.5), \\ ς_{3}^{3, n (3)} (0.5) \end{matrix}}$

	c ₁	c ₂	c ₃	c ₄
x ₁	$S^{3, n (3)} = {ς_{7}^{3, n (3)} (1)}$	${ς_{7}^{3, n (3)} (1)}$	${\begin{matrix} ς_{6}^{3, n (3)} (0.4), \\ ς_{7}^{3, n (3)} (0.6) \end{matrix}}$	${\begin{matrix} ς_{6}^{3, n (3)} (0.3), \\ ς_{7}^{3, n (3)} (0.7) \end{matrix}}$
x ₂	${\begin{matrix} ς_{5}^{3, n (3)} (0.7), \\ ς_{6}^{3, n (3)} (0.3) \end{matrix}}$	${\begin{matrix} ς_{4}^{3, n (3)} (0.2), \\ ς_{5}^{3, n (3)} (0.8) \end{matrix}}$	${ς_{4}^{3, n (3)} (1)}$	${ς_{4}^{3, n (3)} (1)}$
x ₃	${\begin{matrix} ς_{5}^{3, n (3)} (0.3), \\ ς_{6}^{3, n (3)} (0.7) \end{matrix}}$	${\begin{matrix} ς_{5}^{3, n (3)} (0.3), \\ ς_{6}^{3, n (3)} (0.7) \end{matrix}}$	${\begin{matrix} ς_{6}^{3, n (3)} (0.8), \\ ς_{7}^{3, n (3)} (0.2) \end{matrix}}$	${ς_{4}^{3, n (3)} (1)}$
x ₄	${\begin{matrix} ς_{2}^{3, n (3)} (0.2), \\ ς_{3}^{3, n (3)} (0.8) \end{matrix}}$	${\begin{matrix} ς_{3}^{3, n (3)} (0.8), \\ ς_{4}^{3, n (3)} (0.2) \end{matrix}}$	${ς_{2}^{3, n (3)} (1)}$	${\begin{matrix} ς_{2}^{3, n (3)} (0.5), \\ ς_{3}^{3, n (3)} (0.5) \end{matrix}}$

Table 2

The decision matrix D₂ provided by the DM e₂

	c ₁	c ₂	c ₃	c ₄
x ₁	${ς_{3}^{2, n (2)} (1)}$	${\begin{matrix} ς_{3}^{2, n (2)} (0.4), \\ ς_{4}^{2, n (2)} (0.6) \end{matrix}}$	${\begin{matrix} ς_{3}^{2, n (2)} (0.9), \\ ς_{4}^{2, n (2)} (0.1) \end{matrix}}$	${ς_{3}^{2, n (2)} (1)}$
x ₂	${\begin{matrix} ς_{2}^{2, n (2)} (0.5), \\ ς_{3}^{2, n (2)} (0.5) \end{matrix}}$	${\begin{matrix} ς_{2}^{2, n (2)} (0.4), \\ ς_{3}^{2, n (2)} (0.6) \end{matrix}}$	${\begin{matrix} ς_{1}^{2, n (2)} (0.3), \\ ς_{2}^{2, n (2)} (0.7) \end{matrix}}$	${\begin{matrix} ς_{2}^{2, n (2)} (0.3), \\ ς_{3}^{2, n (2)} (0.7) \end{matrix}}$
x ₃	${\begin{matrix} ς_{2}^{2, n (2)} (0.3), \\ ς_{3}^{2, n (2)} (0.7) \end{matrix}}$	${ς_{3}^{2, n (2)} (1)}$	${\begin{matrix} ς_{2}^{2, n (2)} (0.6), \\ ς_{3}^{2, n (2)} (0.4) \end{matrix}}$	${\begin{matrix} ς_{3}^{2, n (2)} (0.8), \\ ς_{4}^{2, n (2)} (0.2) \end{matrix}}$
x ₄	${ς_{2}^{2, n (2)} (1)}$	${ς_{1}^{2, n (2)} (1)}$	${\begin{matrix} ς_{1}^{2, n (2)} (0.8), \\ ς_{2}^{2, n (2)} (0.2) \end{matrix}}$	${\begin{matrix} ς_{2}^{2, n (2)} (0.7), \\ ς_{3}^{2, n (2)} (0.3) \end{matrix}}$

Table 3

The decision matrix D₃ provided by the DM e₃

	c ₁	c ₂	c ₃	c ₄
x ₁	${\begin{matrix} ς_{1}^{1, n (1)} (0.2), \\ ς_{2}^{1, n (1)} (0.8) \end{matrix}}$	${ς_{2}^{1, n (1)} (1)}$	${\begin{matrix} ς_{1}^{1, n (1)} (0.3), \\ ς_{2}^{1, n (1)} (0.7) \end{matrix}}$	${\begin{matrix} ς_{1}^{1, n (1)} (0.6), \\ ς_{2}^{1, n (1)} (0.4) \end{matrix}}$
x ₂	${\begin{matrix} ς_{1}^{1, n (1)} (0.1), \\ ς_{2}^{1, n (1)} (0.9) \end{matrix}}$	${\begin{matrix} ς_{1}^{1, n (1)} (0.7), \\ ς_{2}^{1, n (1)} (0.3) \end{matrix}}$	${ς_{1}^{1, n (1)} (1)}$	${\begin{matrix} ς_{1}^{1, n (1)} (0.9), \\ ς_{2}^{1, n (1)} (0.1) \end{matrix}}$
x ₃	${\begin{matrix} ς_{1}^{1, n (1)} (0.3), \\ ς_{2}^{1, n (1)} (0.7) \end{matrix}}$	${\begin{matrix} ς_{1}^{1, n (1)} (0.8), \\ ς_{2}^{1, n (1)} (0.2) \end{matrix}}$	${\begin{matrix} ς_{1}^{1, n (1)} (0.4), \\ ς_{2}^{1, n (1)} (0.6) \end{matrix}}$	${\begin{matrix} ς_{1}^{1, n (1)} (0.5), \\ ς_{2}^{1, n (1)} (0.5) \end{matrix}}$
x ₄	${\begin{matrix} ς_{0}^{1, n (1)} (0.5), \\ ς_{1}^{1, n (1)} (0.5) \end{matrix}}$	${\begin{matrix} ς_{0}^{1, n (1)} (0.6), \\ ς_{1}^{1, n (1)} (0.4) \end{matrix}}$	${\begin{matrix} ς_{1}^{1, n (1)} (0.8), \\ ς_{2}^{1, n (1)} (0.2) \end{matrix}}$	${ς_{1}^{1, n (1)} (1)}$

Multiply the probability in the transformed evaluation information by the weight of corresponding DM, and we can integrate 3 decision matrices into one matrix shown in Table 4.

Table 4

The decision matrix D₄ after weighting

	c ₁	c ₂	c ₃	c ₄
x ₁	${\begin{matrix} {ς_{7}^{3, n (3)} (0.5)}, \\ ς_{3}^{2, n (2)} (0.3), \\ ς_{1}^{1, n (1)} (0.04), \\ ς_{2}^{1, n (1)} (0.16) \end{matrix}}$	${\begin{matrix} ς_{7}^{3, n (3)} (0.5), \\ ς_{3}^{2, n (2)} (0.12), \\ ς_{4}^{2, n (2)} (0.18), \\ ς_{2}^{1, n (1)} (0.2) \end{matrix}}$	${\begin{matrix} ς_{6}^{3, n (3)} (0.2), \\ ς_{7}^{3, n (3)} (0.3), \\ ς_{3}^{2, n (2)} (0.27), \\ ς_{4}^{2, n (2)} (0.03), \\ ς_{1}^{1, n (1)} (0.06), \\ ς_{2}^{1, n (1)} (0.14) \end{matrix}}$	${\begin{matrix} ς_{6}^{3, n (3)} (0.15), \\ ς_{7}^{3, n (3)} (0.35), \\ ς_{3}^{2, n (2)} (0.3), \\ ς_{1}^{1, n (1)} (0.12), \\ ς_{2}^{1, n (1)} (0.08) \end{matrix}}$
x ₂	${\begin{matrix} ς_{5}^{3, n (3)} (0.35), \\ ς_{6}^{3, n (3)} (0.15), \\ ς_{2}^{2, n (2)} (0.15), \\ ς_{3}^{2, n (2)} (0.15), \\ ς_{1}^{1, n (1)} (0.02), \\ ς_{2}^{1, n (1)} (0.18) \end{matrix}}$	${\begin{matrix} ς_{4}^{3, n (3)} (0.1), \\ ς_{5}^{3, n (3)} (0.4), \\ ς_{2}^{2, n (2)} (0.12), \\ ς_{3}^{2, n (2)} (0.18), \\ ς_{1}^{1, n (1)} (0.14), \\ ς_{2}^{1, n (1)} (0.06) \end{matrix}}$	${\begin{matrix} ς_{4}^{3, n (3)} (0.5), \\ ς_{1}^{2, n (2)} (0.09), \\ ς_{2}^{2, n (2)} (0.21), \\ ς_{1}^{1, n (1)} (0.2) \end{matrix}}$	${\begin{matrix} ς_{4}^{3, n (3)} (0.5), \\ ς_{2}^{2, n (2)} (0.09), \\ ς_{3}^{2, n (2)} (0.21), \\ ς_{1}^{1, n (1)} (0.18), \\ ς_{2}^{1, n (1)} (0.02) \end{matrix}}$
x ₃	${\begin{matrix} ς_{5}^{3, n (3)} (0.15), \\ ς_{6}^{3, n (3)} (0.35), \\ ς_{2}^{2, n (2)} (0.09), \\ ς_{3}^{2, n (2)} (0.21), \\ ς_{1}^{1, n (1)} (0.06), \\ ς_{2}^{1, n (1)} (0.14) \end{matrix}}$	${\begin{matrix} ς_{5}^{3, n (3)} (0.15), \\ ς_{6}^{3, n (3)} (0.35), \\ ς_{3}^{2, n (2)} (0.3), \\ ς_{1}^{1, n (1)} (0.16), \\ ς_{2}^{1, n (1)} (0.04) \end{matrix}}$	${\begin{matrix} ς_{6}^{3, n (3)} (0.4), \\ ς_{7}^{3, n (3)} (0.1), \\ ς_{2}^{2, n (2)} (0.18), \\ ς_{3}^{2, n (2)} (0.12), \\ ς_{1}^{1, n (1)} (0.08), \\ ς_{2}^{1, n (1)} (0.12) \end{matrix}}$	${\begin{matrix} ς_{4}^{3, n (3)} (0.5), \\ ς_{3}^{2, n (2)} (0.24), \\ ς_{4}^{2, n (2)} (0.06), \\ ς_{1}^{1, n (1)} (0.1), \\ ς_{2}^{1, n (1)} (0.1) \end{matrix}}$
x ₄	${\begin{matrix} ς_{2}^{3, n (3)} (0.1), \\ ς_{3}^{3, n (3)} (0.4), \\ ς_{2}^{2, n (2)} (0.3), \\ ς_{0}^{1, n (1)} (0.1), \\ ς_{1}^{1, n (1)} (0.1) \end{matrix}}$	${\begin{matrix} ς_{3}^{3, n (3)} (0.4), \\ ς_{4}^{3, n (3)} (0.1), \\ ς_{1}^{2, n (2)} (0.3), \\ ς_{0}^{1, n (1)} (0.12), \\ ς_{1}^{1, n (1)} (0.08) \end{matrix}}$	${\begin{matrix} ς_{2}^{3, n (3)} (0.5), \\ ς_{1}^{2, n (2)} (0.24), \\ ς_{2}^{2, n (2)} (0.06), \\ ς_{1}^{1, n (1)} (0.16), \\ ς_{2}^{1, n (1)} (0.04) \end{matrix}}$	${\begin{matrix} ς_{2}^{3, n (3)} (0.25), \\ ς_{3}^{3, n (3)} (0.25), \\ ς_{2}^{2, n (2)} (0.21), \\ ς_{3}^{2, n (2)} (0.09), \\ ς_{1}^{1, n (1)} (0.2) \end{matrix}}$

Step 2. According to Equation (3), we get λ = -0.2368.

Based on Equation (2), the fuzzy measures of the attributes are obtained as follows:

$\begin{matrix} μ (c_{1}, c_{2}) = 0.4858, μ (c_{1}, c_{3}) = 0.5811, μ (c_{1}, c_{4}) \\ = 0.3905, μ (c_{2}, c_{3}) = 0.6716, μ (c_{2}, c_{4}) = 0.4858, \\ μ (c_{3}, c_{4}) = 0.5811, μ (c_{1}, c_{2}, c_{3}) = 0.8398, \\ μ (c_{1}, c_{2}, c_{4}) = 0.6628, μ (c_{1}, c_{3}, c_{4}) = 0.7536, \\ μ (c_{2}, c_{3}, c_{4}) = 0.8398, μ (c_{1}, c_{2}, c_{3}, c_{4}) = 1 . \end{matrix}$

Step 3. Suppose S^2,n(2) is the BLTS. According to the MGPLCA operator, we can get the result R_i (w) , (i = 1, 2, 3, 4) as follows:

$\begin{matrix} R_{1} (w) = \\ {ς_{2}^{4, n (2)} (0.0242), ς_{3}^{4, n (2)} (0.3500), ς_{4}^{4, n (2)} (0.2604)}, \end{matrix}$

$\begin{matrix} R_{2} (w) = \\ {ς_{2}^{5, n (2)} (0.3665), ς_{3}^{5, n (2)} (0.2363), ς_{4}^{5, n (2)} (0.0703)}, \end{matrix}$

$\begin{matrix} R_{3} (w) = \\ {ς_{2}^{6, n (2)} (0.1972), ς_{3}^{6, n (2)} (0.3623), ς_{4}^{6, n (2)} (0.0718)}, \end{matrix}$

$\begin{matrix} R_{4} (w) = \\ {ς_{0}^{7, n (2)} (0.0237), ς_{1}^{7, n (2)} (0.3493), ς_{2}^{7, n (2)} \\ (0.2374), ς_{3}^{7, n (2)} (0.0084), ς_{4}^{7, n (2)} (0.0085)} . \end{matrix}$

Step 4. The score function $D (R_{i} (w)) = \frac{2}{n (1) - 1} \sum_{s = 0}^{n (1) - 1} (s \times p_{s}^{1, n (1)})$ for R_i (w) , (i = 1, 2, 3, 4) are shown as follows:

$D (R_{1} (w)) = \frac{1}{2} \times (2 \times 0.0242 + 3 \times 0.35 + 4 \times 0.2604) = 1.07;$

$D (R_{2} (w)) = \frac{1}{2} \times (2 \times 0.3665 + 3 \times 0.2363 + 4 \times 0.0703) = 0.8616;$

$D (R_{3} (w)) = \frac{1}{2} \times (2 \times 0.1972 + 3 \times 0.3623 + 4 \times 0.0718) = 0.8843;$

$\begin{matrix} D (R_{4} (w)) = \frac{1}{2} \times \\ (0 \times 0.0237 + 1 \times 0.3493 + 2 \times 0.2374 \\ + 3 \times 0.0084 + 4 \times 0.0085) = 0.4416 . \end{matrix}$

Step 5. According to the score function results, we know D (R₁ (w)) > D (R₃ (w)) > D (R₂ (w)) > D (R₄ (w)).

The ranking of alternatives is x₁ ≻ x₃ ≻ x₂ ≻ x₄. Therefore, the optimal alternative is x₁.

5.2 Validity and reliability test of the created MAGDM method

For the same decision-making problem, different MAGDM methods will lead to different results. Wang and Triantaphyllou [49] presented three test criteria to verify the reliability and validity of the MAGDM methods.

Test criteria 1. An effective MAGDM method, without changing the attribute weights, if a non-optimal alternative replaces another non-optimal alternative, the optimal alternative should not be changed.

According to Test criteria 1, change the evaluation information of three DMs on alternative x₄ in Example 3.

Based on the above steps, we can get $\begin{matrix} R_{4} (w) = {ς_{0}^{8, n (2)} (0.0758), ς_{1}^{8, n (2)} (0.2697), \\ ς_{2}^{8, n (2)} (0.2691), ς_{4}^{8, n (2)} (0.0086)}, \\ D (R_{4} (w)) = \frac{1}{2} \times (0 \times 0.0745 + 1 \times 0.2693 \\ + 2 \times 0.2651 + 4 \times 0.0083) = 0.4212 . \end{matrix}$

Because D (R₁ (w)) > D (R₃ (w)) > D (R₂ (w)) > D (R₄ (w)), the ranking of alternatives is x₁ ≻ x₃ ≻ x₂ ≻ x₄, and the best alternative is x₁. Therefore, the created MAGDM method passed the Test criteria 1.

Test criterion 2. An effective MAGDM method should have transitivity.

Test criterion 3. A MAGDM problem is divided into several sub-problems, which are calculated by the same MAGDM method. The comprehensive ranking of the alternatives should be the same as the ranking of the original MAGDM problem.

According to Test criteria 2 and Test criteria 3, we divide the four alternatives in Example 3 into two parts. The first part is to evaluate {x₁, x₂, x₃}, and the second part is to evaluate {x₂, x₃, x₄}, and other information in Example 3 remains unchanged.

Based on the proposed MAGDM method, we get that the ranking of the first part is x₁ ≻ x₃ ≻ x₂, and the ranking of the second part is x₃ ≻ x₂ ≻ x₄. Combine the two rankings to get the comprehensive ranking x₁ ≻ x₃ ≻ x₂ ≻ x₄. This ranking result is the same as the original MAGDM problem. Therefore, the created MAGDM method passed the Test criteria 2 and Test criteria 3.

5.3 Comparison analysis and discussion

In order to further verify the effectiveness of the proposed method, it is compared with the existing four MAGDM methods: Lei et al.’s method [42] based on the TOPSIS; Lu et al.’s method [43] based on the TOPSIS; Mao et al.’s method [44] based on the Generalized probabilistic linguistic Hamacher weighted averaging (GPLHWA) operator; Wang’s method [33] based on prospect theory (PT). It is assumed that γ = 1, λ = 1 for Mao et al.’s method [44], and it is assumed that λ = 1, θ = -2.25, α = 0.88, and β = 0.88 for Wang’s method [33]. We set the attribute weight vector in Example 3 to w = (w₁, w₂, w₃, w₄) ^T = (0.18, 0.27, 0.37, 0.18) ^T, other information remains unchanged. The ranking results of the online teaching courses in Example 3 for the different methods are listed in Table 8.

Table 5
The evaluation information after changed by the DM e₁

c ₁ c ₂ c ₃ c ₄

x ₄ ${\begin{matrix} ς_{2}^{3, n (3)} (0.3), \\ ς_{3}^{3, n (3)} (0.7) \end{matrix}}$ ${\begin{matrix} ς_{2}^{3, n (3)} (0.2), \\ ς_{3}^{3, n (3)} (0.8) \end{matrix}}$ ${\begin{matrix} ς_{1}^{3, n (3)} (0.7), \\ ς_{2}^{3, n (3)} (0.3) \end{matrix}}$ ${\begin{matrix} ς_{2}^{3, n (3)} (0.5), \\ ς_{3}^{3, n (3)} (0.5) \end{matrix}}$

	c ₁	c ₂	c ₃	c ₄
x ₄	${\begin{matrix} ς_{2}^{3, n (3)} (0.3), \\ ς_{3}^{3, n (3)} (0.7) \end{matrix}}$	${\begin{matrix} ς_{2}^{3, n (3)} (0.2), \\ ς_{3}^{3, n (3)} (0.8) \end{matrix}}$	${\begin{matrix} ς_{1}^{3, n (3)} (0.7), \\ ς_{2}^{3, n (3)} (0.3) \end{matrix}}$	${\begin{matrix} ς_{2}^{3, n (3)} (0.5), \\ ς_{3}^{3, n (3)} (0.5) \end{matrix}}$

Table 6

The evaluation information after changed by the DM e₂

	c ₁	c ₂	c ₃	c ₄
x ₄	${ς_{2}^{2, n (2)} (1)}$	${ς_{1}^{2, n (2)} (1)}$	${\begin{matrix} ς_{1}^{2, n (2)} (0.1), \\ ς_{2}^{2, n (2)} (0.9) \end{matrix}}$	${\begin{matrix} ς_{2}^{2, n (2)} (0.9), \\ ς_{3}^{2, n (2)} (0.1) \end{matrix}}$

Table 7

The evaluation information after changed by the DM e₃

	c ₁	c ₂	c ₃	c ₄
x ₄	${\begin{matrix} ς_{0}^{1, n (1)} (0.8), \\ ς_{1}^{1, n (1)} (0.2) \end{matrix}}$	${\begin{matrix} ς_{0}^{1, n (1)} (0.6), \\ ς_{1}^{1, n (1)} (0.4) \end{matrix}}$	${\begin{matrix} ς_{1}^{1, n (1)} (0.8), \\ ς_{2}^{1, n (1)} (0.2) \end{matrix}}$	${\begin{matrix} ς_{0}^{1, n (1)} (0.1), \\ ς_{1}^{1, n (1)} (0.9) \end{matrix}}$

Table 8

Ranking of alternatives from two methods for Example 3

Approaches	Score values	Ranking
Lei et al.’s method [42]	Cannot be Obtained	—
(Based on the PL- TOPSIS method)
Lu et al.’s method [43] (Based on the PL- TOPSIS method)	Cannot be Obtained	—
Mao et al.’s method [44] (Based on the GPLHWA operator)	Cannot be Obtained	—
Wang’s method [33] based on the (PLTSs operator)	$\begin{matrix} C_{1} = 0.87, C_{2} = 0.36, \\ C_{3} = 0.49, C_{4} = 0.31 . \end{matrix}$	x₁ ≻ x₃ ≻ x₂ ≻ x₄
The created method (Based on the MGPLCA operator)	$\begin{matrix} D (R_{1} (w)) = 1.06, D (R_{2} (w)) = 0.79, \\ D (R_{3} (w)) = 0.88, D (R_{4} (w)) = 0.44 . \end{matrix}$	x₁ ≻ x₃ ≻ x₂ ≻ x₄

From Table 8, we can find that only the proposed method and Wang’s method [33] can get the same ranking result, i.e., x₁ ≻ x₃ ≻ x₂ ≻ x₄, whereas Lei et al.’s method [42], Lu et al.’s method [43] and Mao et al.’s method [44] cannot get the ranking results. Because these three methods cannot handle MGPLTSs. Therefore, in order to make these three methods can get ranking results, we transform MGPLTSs into the same granularity level by Equation (5) and Equation (6) show in Tables 9, 10 and 11. Then, we use these three methods to calculate and rank four alternatives, and the results are listed in Table 11.

Table 9

The decision information provided by the DM e₁ transformed into 5 granularity level

	c ₁	c ₂	c ₃	c ₄
x ₁	${\begin{matrix} ς_{3}^{3, n (2)} (0.5), \\ ς_{4}^{3, n (2)} (0.5) \end{matrix}}$	${\begin{matrix} ς_{3}^{3, n (2)} (0.5), \\ ς_{4}^{3, n (2)} (0.5) \end{matrix}}$	${\begin{matrix} ς_{3}^{3, n (2)} (0.7), \\ ς_{4}^{3, n (2)} (0.3) \end{matrix}}$	${\begin{matrix} ς_{3}^{3, n (2)} (0.65), \\ ς_{4}^{3, n (2)} (0.35) \end{matrix}}$
x ₂	${\begin{matrix} ς_{2}^{3, n (2)} (0.35), \\ ς_{3}^{3, n (2)} (0.65) \end{matrix}}$	${\begin{matrix} ς_{2}^{3, n (2)} (0.6), \\ ς_{3}^{3, n (2)} (0.4) \end{matrix}}$	${ς_{2}^{3, n (2)} (1)}$	${ς_{2}^{3, n (2)} (1)}$
x ₃	${\begin{matrix} ς_{2}^{3, n (2)} (0.15), \\ ς_{3}^{3, n (2)} (0.85) \end{matrix}}$	${\begin{matrix} ς_{2}^{3, n (2)} (0.15), \\ ς_{3}^{3, n (2)} (0.85) \end{matrix}}$	${\begin{matrix} ς_{3}^{3, n (2)} (0.9), \\ ς_{4}^{3, n (2)} (0.1) \end{matrix}}$	${ς_{2}^{3, n (2)} (1)}$
x ₄	${\begin{matrix} ς_{1}^{3, n (2)} (0.6), \\ ς_{2}^{3, n (2)} (0.4) \end{matrix}}$	${\begin{matrix} ς_{1}^{3, n (2)} (0.4), \\ ς_{2}^{3, n (2)} (0.6) \end{matrix}}$	${ς_{1}^{3, n (2)} (1)}$	${\begin{matrix} ς_{1}^{3, n (2)} (0.75), \\ ς_{2}^{3, n (2)} (0.25) \end{matrix}}$

Table 10

The decision information provided by the DM e₂ transformed into 5 granularity level

	c ₁	c ₂	c ₃	c ₄
x ₁	${ς_{3}^{2, n (2)} (1)}$	${\begin{matrix} ς_{3}^{2, n (2)} (0.4), \\ ς_{4}^{2, n (2)} (0.6) \end{matrix}}$	${\begin{matrix} ς_{3}^{2, n (2)} (0.9), \\ ς_{4}^{2, n (2)} (0.1) \end{matrix}}$	${ς_{3}^{2, n (2)} (1)}$
x ₂	${\begin{matrix} ς_{2}^{2, n (2)} (0.5), \\ ς_{3}^{2, n (2)} (0.5) \end{matrix}}$	${\begin{matrix} ς_{2}^{2, n (2)} (0.4), \\ ς_{3}^{2, n (2)} (0.6) \end{matrix}}$	${\begin{matrix} ς_{1}^{2, n (2)} (0.3), \\ ς_{2}^{2, n (2)} (0.7) \end{matrix}}$	${\begin{matrix} ς_{2}^{2, n (2)} (0.3), \\ ς_{3}^{2, n (2)} (0.7) \end{matrix}}$
x ₃	${\begin{matrix} ς_{2}^{2, n (2)} (0.3), \\ ς_{3}^{2, n (2)} (0.7) \end{matrix}}$	${ς_{3}^{2, n (2)} (1)}$	${\begin{matrix} ς_{2}^{2, n (2)} (0.6), \\ ς_{3}^{2, n (2)} (0.4) \end{matrix}}$	${\begin{matrix} ς_{3}^{2, n (2)} (0.8), \\ ς_{4}^{2, n (2)} (0.2) \end{matrix}}$
x ₄	${ς_{2}^{2, n (2)} (1)}$	${ς_{1}^{2, n (2)} (1)}$	${\begin{matrix} ς_{1}^{2, n (2)} (0.8), \\ ς_{2}^{2, n (2)} (0.2) \end{matrix}}$	${\begin{matrix} ς_{2}^{2, n (2)} (0.7), \\ ς_{3}^{2, n (2)} (0.3) \end{matrix}}$

Table 11

The decision information provided by the DM e₃ transformed into 5 granularity level

	c ₁	c ₂	c ₃	c ₄
x ₁	${\begin{matrix} ς_{2}^{1, n (2)} (0.2), \\ ς_{4}^{1, n (2)} (0.8) \end{matrix}}$	${ς_{4}^{1, n (2)} (1)}$	${\begin{matrix} ς_{2}^{1, n (2)} (0.3), \\ ς_{4}^{1, n (2)} (0.7) \end{matrix}}$	${\begin{matrix} ς_{2}^{1, n (2)} (0.6), \\ ς_{4}^{1, n (2)} (0.4) \end{matrix}}$
x ₂	${\begin{matrix} ς_{2}^{1, n (2)} (0.1), \\ ς_{4}^{1, n (2)} (0.9) \end{matrix}}$	${\begin{matrix} ς_{2}^{1, n (2)} (0.7), \\ ς_{4}^{1, n (2)} (0.3) \end{matrix}}$	${ς_{2}^{1, n (2)} (1)}$	${\begin{matrix} ς_{2}^{1, n (2)} (0.9), \\ ς_{4}^{1, n (2)} (0.1) \end{matrix}}$
x ₃	${\begin{matrix} ς_{2}^{1, n (2)} (0.3), \\ ς_{4}^{1, n (2)} (0.7) \end{matrix}}$	${\begin{matrix} ς_{2}^{1, n (2)} (0.8), \\ ς_{4}^{1, n (2)} (0.2) \end{matrix}}$	${\begin{matrix} ς_{2}^{1, n (2)} (0.4), \\ ς_{4}^{1, n (2)} (0.6) \end{matrix}}$	${\begin{matrix} ς_{2}^{1, n (2)} (0.5), \\ ς_{4}^{1, n (2)} (0.5) \end{matrix}}$
x ₄	${\begin{matrix} ς_{0}^{1, n (2)} (0.5), \\ ς_{2}^{1, n (2)} (0.5) \end{matrix}}$	${\begin{matrix} ς_{0}^{1, n (2)} (0.6), \\ ς_{2}^{1, n (2)} (0.4) \end{matrix}}$	${\begin{matrix} ς_{2}^{1, n (2)} (0.8), \\ ς_{4}^{1, n (2)} (0.2) \end{matrix}}$	${ς_{2}^{1, n (2)} (1)}$

In Table 12, all methods derive the same Ranking result, i.e., x₁ ≻ x₃ ≻ x₂ ≻ x₄. This can prove that the method in this paper is effective. But Lei et al.’s method [42], Lu et al.’s method [43], Mao et al.’s method [44] and Wang’s method [33] have different shortcomings in processing PLTSs. Thus, in what follows, we specifically analyze these methods.

Table 12

Ranking of alternative from two methods for Example 3

Approaches	Score values	Ranking
Lei et al.’s method [42] (Based on the PL- TOPSIS method)	$\begin{matrix} PLRCD ({PLA}_{1}, PLPIS) = 0, \\ PLRCD ({PLA}_{2}, PLPIS) = 0.62, \\ PLRCD ({PLA}_{3}, PLPIS) = 0.5, \\ PLRCD ({PLA}_{4}, PLPIS) = 1 . \end{matrix}$	x₁ ≻ x₃ ≻ x₂ ≻ x₄
Lu et al.’s method [43] (Based on the PL- TOPSIS method)	$\begin{matrix} PLRCD ({PLA}_{1}, PLNIS) = 1, \\ PLRCD ({PLA}_{2}, PLNIS) = 0.35, \\ PLRCD ({PLA}_{3}, PLNIS) = 0.50, \\ PLRCD ({PLA}_{4}, PLNIS) = 0 . \end{matrix}$	x₁ ≻ x₃ ≻ x₂ ≻ x₄
Mao et al.’s method [44] (Based on the GPLHWA operator)	$\begin{matrix} {CD}_{1} = 0.88, {CD}_{2} = 0.29, \\ {CD}_{3} = 0.53, {CD}_{4} = 0 . \end{matrix}$	x₁ ≻ x₃ ≻ x₂ ≻ x₄
Wang’s method [33] (based on the PLTSs operator)	$\begin{matrix} C_{1} = 0.87, C_{2} = 0.36, \\ C_{3} = 0.49, C_{4} = 0.31 . \end{matrix}$	x₁ ≻ x₃ ≻ x₂ ≻ x₄
The created method (Based on the MGPLCA operator)	$\begin{matrix} D (R_{1} (w)) = 1.06, D (R_{2} (w)) = 0.79, \\ D (R_{3} (w)) = 0.88, D (R_{4} (w)) = 0.44 . \end{matrix}$	x₁ ≻ x₃ ≻ x₂ ≻ x₄

(1) The Shortcomings of Lei et al.’s method [42], Lu et al.’s method [43]: First, these two methods cannot directly handle MGPLTSs. Before calculation, these two methods need to transform MGPLTSs into the same granularity level. Of course, the transformation function we proposed can handle this problem well. Second, these two methods are based on TOPSIS, so they can only give the relative ranking result, and cannot give the comprehensive values of alternatives, further, they can also lead to the change of ranking results when the alternatives are increased or decreased. As mentioned in the previous section, the effective MAGDM method should be verified by the three test criteria presented by Wang and Triantaphyllou [49]. Among the three test criteria, Test criteria 2 is necessary for an effective MAGDM method, i.e., it should have transitivity, and Test criteria 3 is the consistency between the smaller MAGDM problems and the original MAGDM problem. Obviously, these two methods do not have transitivity, and cannot pass the verification of consistency. Therefore, these two methods are not completely effective. We have verified the created method has passed Test criterions 1, 2, 3 and is an effective MAGDM method. Third, in the process of processing PLTSs by these two methods, the sum of PLTS may be less than one, which requires the normalization of PLTS.

(2) The Shortcomings of Mao et al.’s method [44]: First, this method is the same as Lei et al.’s method [42], Lu et al.’s method [43], they can only be used in the PLTSs environment, and can’t directly deal with MGPLTSs. By combined with our transformation function, we can get the ranking results in the complex MGPLTSs environment. It is worth mentioning that our transformation function is not only applicable to these three methods, but also effective for many operators and methods in PLTSs environment. Second, this method cannot avoid the normalization of PLTS.

(3) The Shortcomings of Wang’s method [33]: First, this method can handle MGPLTSs, but it also needs to transform the original MGPLTSs into the same granularity level before starting the calculation. That is, this method cannot directly aggregate MGPLTSs. But the proposed method based on the MGPLCA operator can directly aggregate the original MGPLTSs. Second, the method in [33] needs to be normalized twice, which increases the complexity of the operations. Different from this method [33], the proposed MAGDM method based on the MGPLCA operator omits the normalization of PLTS, which can undoubtedly reduce the complexity of the calculation. Third, the calculation results of this method [33] are not PLTSs, which means that operator leads to loss of probability information. On the contrary, Theorem 3 shows that the calculation result of MGPLCA operator in the proposed method is still PLTS. So, the proposed MAGDM method avoids the loss of decision information, especially probability information, and the ranking results are more credible.

In summary, the advantages of the proposed method are that it can not only directly deal with MGPLTSs, but also avoids the problem of loss of decision information during the calculation process, and there is no need to normalizing the PLTSs, which reduces complexity.

6 Conclusion

In the Internet era, especially due to the impact of the COVID-19, online teaching has become particularly significant, and online teaching quality evaluation is a key for improving the quality of online teaching. Therefore, we choose MGPLTSs to reflect the preferences and habits of different DMs and the importance of different linguistic terms, and propose a new MAGDM method to evaluate the quality of online teaching. First, propose the transformation function of MGPLTSs based on the proportional 2-tuple fuzzy linguistic representation model. On this basis, the operational laws and comparison rules of MGPLTSs are presented. Then, considering the relationship among attributes, propose MGPLCA operator based on PLTSs and Choquet integral, and prove the properties of the MGPLCA operator. At the same time, develop a MAGDM method which does not need to consider the process of normalizing the PLTSs. Furthermore, provide a case of online teaching quality evaluation to illustrate reliability and validity of the created MAGDM method. Finally, compare the created MAGDM method with some existing methods, and illustrate the advantages of our MAGDM methods. Based on the case study, we can know that the created MAGDM method can be effectively applied to online teaching quality evaluation. However, there are also some limitations of the proposed method. First, we have verified the properties of the proposed MGPLCA operator, including commutativity, boundedness and monotonicity, but the MGPLCA operator has no idempotency. Second, the proposed MGPLCA operator, combined with Choquet integral, can be used to deal with the problem of the relationship between attributes, but it cannot handle the situation where the attribute weight is completely unknown.

In the future, we can extend the proposed method to the situation where the attribute weights are completely unknown, at the same time, we will also study consensus measure by considering the consensus among DMs with different cognitive and knowledge backgrounds.

Footnotes

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (No. 71771140), Project of cultural masters and “the four kinds of a batch” talents”, the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045).

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