Improved TOPSIS Method Considering Fuzziness and Randomness in Multi-Attribute Group Decision Making

Abstract

Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) is a commonly used decision model in multi-attribute group decision making (MAGDM), and a probabilistic linguistic term set (PLTS) is the linguistic variable that can effectively express the fuzziness of decision makers’ (DMs’) preference. However, in actual decision use, PLTS type decision preference needs to be processed before use, which can distort the decision results. The randomness of DM’s preference which also affects the final decision making is often ignored. Therefore, in order to better serve the MAGDM problem, this paper proposes an asymmetric probabilistic linguistic cloud TOPSIS (ASPLC-TOPSIS) method. First, the basic theories of linguistic variables and cloud model (CM) are introduced. Second, the conversation model between linguistic variables and CM is defined along with the operation formula of ASPLC. Third, considering the importance of the DMs’ subjective weights, a DM trust network is established to calculate the DMs’ weights. Finally, the decision process of ASPLC-TOPSIS is proposed and the superiority of this method is proved through experimental studies.

1. Introduction

Multi-attribute group decision making (MAGDM) is a process wherein a limited number of decision makers (DMs) evaluate limited alternatives according to limited attributes to select the best alternative in a complex decision-making environment. MAGDM has been widely used in various fields by researchers, such as emergency decisions [1], public safety [2], supplier selection [3,4], and equipment evaluation [5].

The problem of MAGDM is often divided into two stages: (i) expression of DMs’ preferences, and (ii) aggregation of DMs’ evaluation. In the first stage of decision making, the focus is on collecting DMs’ preferences. With increasing complexity of the decision-making environment of DMs, it is impossible for each DM to express preference with clear values [6]. To better describe the fuzziness of human preference, Zadeh [7] proposed the concept of a fuzzy set, Torra and Narukawa [8] introduced the concept of a hesitant fuzzy set; Rodriguez and Martinez [9] developed the hesitant fuzzy linguistic term set (HFLT) based on linguistic term sets and hesitant fuzzy sets to better reflect hesitation of the DMs among several fuzzy elements. To overcome the loss of evaluation information resulting from the HFLT during aggregation, Pang and Wang [10] proposed a probabilistic linguistic term set (PLTS) and defined the basic operations and aggregation formulas [11]. A PLTS can express the fuzziness of DMs’ decision preference well; however, it cannot express the randomness of DM’s preference. For example, when a shooter with perfect design skill shoots, in the ideal state, the result of each shot is 10 rings, but in the actual shooting, the shooting results may be scattered across 10 rings owing to uncertain factors. For a sufficient number of shots, the shooting results are normally distributed around 10 rings (see Figure 1). In the example, the randomness of DM’s preference is explored.

To characterize the randomness of DM’s preference, Li and Liu [12] proposed the cloud model (CM). The emergence of CM is a solution to the inability of linguistic models in expressing the randomness of DMs’ preference. The expectation, entropy, and hyper entropy [12] in the CM can replace the linguistic variables well to describe both fuzziness and randomness of DMs’ preference. The extended CM method can be listed as follows: the AHP-cloud model [5], the entropy weight CM method [13] and the trapezoid CM [14] method. However, the particularities of CM preclude its direct use as a tool for expressing the randomness of DMs’ preference. Therefore, several scholars introduced CM into the TOPSIS method [15], analytic hierarchy process (AHP) method [16], and CRITIC method [17].

The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method ranks the alternatives according to their closeness to the ideal solution. This method can make full use of the information of the original data, and its results can accurately reflect the gap between the evaluation schemes [18]. The method has no strict restrictions on data distribution and sample size, and can be applied to various decision-making environments [19,20,21]. Because of these advantages, TOPSIS is widely used in public infrastructure project evaluation [22,23], the selection of business partners and suppliers [24,25,26], daily family decision making [27,28,29], and other fields. However, in the second stage of decision making, TOPSIS method needs to normalize the linguistic variables, that is, probability elements need to be added before subsequent operations. This processing significantly reduces the truthfulness and receptivity of the decision-making results. On the contrary, due to the particularity of CM composition, subsequent aggregation and comparison can be carried out directly without dealing with the DMs’ preferences. In order to maintain the wide applicability of the TOPSIS method and solve the problem that the TOPSIS method artificially adds probability elements in the decision-making process, many experts try to combine CM with the TOPSIS method and apply it to many fields, for example, (1) green renovation: Liu and Chen [19] combined the CM and TOPSIS model to evaluate the green renovation project of university public buildings; (2) electric power development: Zhao et al. [20] proposed a model composed of CM and TOPSIS, and used this model to evaluate the development of national electric power development; (3) sustainable supply chain management: Li and Fang [21] introduced normal cloud into the TOPSIS model to evaluate fuzzy, subjective, and random information in sustainable supplier selection; (4) household finance: Samriya and Kumar [30] used TOPSIS to select the best task under the cloud environment; (5) enterprise operation and supplier selection: Garg and Sidhu [31] used the improved CM and TOPSIS method to evaluate enterprise productivity.

As a key issue in MAGDM, the distribution of DM’s weight sometimes is unreasonable. In existing studies, objective weight determination methods were often used [32,33], for example, calculating the DMs’ weight according to the decision matrix [10]. This kind weight determination method pays too much attention to DMs’ preference and ignores the connection between DMs. Most existing group decision-making (GDM) models assume that the DMs make independent decisions, which is unreasonable in reality [34]. Many cases have shown that people’s views are easily affected by the external environment. With the development of the social network, individuals have more chances to pick up and update information. In the process of their network communication, the opinions of DMs with more knowledge and experience will affect the attitudes of others [35]. This paper introduces a novel subjective weight determination method. Although a few scholars recognize that the DMs’ subjective weight is an important factor to improve the consistency of decision results [33], they have not yet proposed a reasonable solution. This paper establishes a DM trust network. This network does not need a complex consensus model, and only calculates the DM’s weight according to the trust relationship between DMs, thereby rendering the aggregated group decision evaluation more easily acceptable to individuals and improving decision efficiency.

Summarizing the existing research, we found that the decision-making model under the linguistic environment represented by TOPSIS has the following problems:

(1)

The research on the conversion between probabilistic linguistic variables and asymmetric probabilistic linguistic CM (ASPLC) is lacking.

(2)

The influence of random factors on decision evaluations cannot be presented in the final decision result.

(3)

Because of incomplete preference in the decision matrix, it is inevitable to complete missing information which may be a subjective process.

(4)

The weight of the criterion is determined by the decision matrix, and the weight of the DM is also determined by the decision matrix. Too much attention is paid to the decision-making information of the DM’s evaluation indicators, which weakens the acceptability of the decision results to a certain extent.

In order to solve the above problems of existing decision models, this paper proposes ASPLC-TOPSIS method. The contributions of this research can be summarized as follows:

(1)

The fuzziness and randomness of DMs’ preference are presented by the combination of probabilistic linguistic variables and CMs simultaneously.

(2)

This paper establishes a conversion model to realize the conversion between linguistic variables and ASPLC, and proposes some basic calculation formulas of the ASPLC.

(3)

To improve the acceptability and efficiency of the decision results, a trust network is established, which is the basis of calculating DMs’ weights.

The structure of the rest of this paper is as follows. In part 2, basic theories of PLTS and CM are introduced. The conversion model between PLTS and asymmetric probability CM is established in Section 3. In part 4, a new decision method is proposed by combining the TOPSIS and ASPLC under PLTS. In Section 5, a case is illustrated about supplier selection and used to verify the feasibility of this model, and the advantages of the ASPLC-TOPSIS method can be obtained by comparing with the other existing methods. In the last part, we summarize the merits and disadvantages of this proposed method.

2. Preliminary Methods

In this section, we introduce the cloud model, probability linguistic term sets, and some other theories. These theories will be used in the following chapters.

2.1. Linguistic Variables

Linguistic variables are widely used in MAGDM because they are most suitable to express the fuzziness of DMs’ preference.

Definition 1. 

[36,37]. Let $L=\left\{l_i\left|i=-\tau ,\dots ,0,\dots ,\tau \right.\right\}$ is a subscript balanced linguistic term set, where $l_0$ represent insensibility, other fuzzy elements are evenly distributed on both sides of $l_0$.

Pang and Wang [10] presented the PLTS that can better describe the fuzziness of DM’s preference.

Definition 2. 

[10]. Let $L\left(P\right)=\left\{l_i\left(p_i^r\right)\left|i=-\tau ,\dots ,0\dots ,\tau ;r=1,2,\dots ,\#r,p_i^r\in \left[0,1\right];{\displaystyle \sum }_{i=-\tau }^{\tau }{p}_i^r\le 1\right.\right\}$ is the subscript balanced PLTS, where $\#r$ represents total number of fuzzy elements in $L\left(P\right)$, $p_i^r$ is the $r$ th element in $L\left(P\right)$.

The subscript balanced PLTS satisfies the following operations [38]:

(1)

If $\zeta <\theta$, then $l_{\zeta }<l_{\theta }$

(2)

$l_{\zeta }\oplus l_{\theta }=l_{\zeta +\theta }$

(3)

$\lambda l_{\theta }=l_{\lambda \theta }$

(4)

$\lambda \left(l_{\zeta }\oplus l_{\theta }\right)=\lambda l_{\zeta }\oplus \lambda l_{\theta }$

(5)

$neg\left(l_{-\theta }\right)=l_{\theta }$

Where $l_{\zeta },l_{\theta }ϵL\left(P\right)$, $\zeta ,\theta \in \left[-\tau ,\tau \right]$, $\lambda \in \left[0,1\right]$. Considering that PLTS can fully reflect the fuzziness of DMs’ preferences, this paper uses PLTS as a tool to collect DMs’ preferences.

2.2. Cloud Model

CM proposed by Li and Liu [12] is a model that can better describe the psychological state of DMs and the randomness of DMs’ preference [39]. At present, it has been widely used and developed. The specific definitions are as follows:

Definite a universe $U=\left\{u\right\}$. $L=\left\{l_i\left|i=-\tau ,\dots ,0,\dots ,\tau \right.\right\}$ is a qualitative term in $U$, $Q\in L$. ${\varphi }_L\left(Q\right)\in \left[0,1\right]$ is a random mapping of qualitative term $L$ on $U$. ${\varphi }_L\left(Q\right)\in \left[0,1\right]$ represents the certainty degree of $Q$ belonging to $L$, where $Q$ is a random instantiation of $L$ [40]. Meanwhile, the distribution of $l_i$ in the universe $U$ can be represented as $Y_i=\left(E{x}_i,E{n}_i,H{e}_i\right)$ [12].

$E{x}_i$ is the expected value of cloud drops and the most important sign of qualitative linguistic term sets transforming into the cloud; $E{n}_i$ represents the uncertainty of qualitative evaluation; $E{n}_i$ also represents the acceptable degree of qualitative concepts in quantitative terms and reflects the fuzziness of DMs’ decision evaluation. $H{e}_i$ can measure the uncertainty of entropy.

If $l_i$ satisfying $l_i~N\left(E{x}_i,E{n}_i^{*2}\right)$, $E{n}_i^{*2}~N\left(E{n}_i,H{e}_i^{*2}\right)$, the distribution of $l_i$ in $X$ is named a normal cloud [12,41,42], and it can be expressed as:

$$Y_i=exp\left(-\frac{{\left(x_i-E{x}_i\right)}^2}{2{\left(E{n}_i\right)}^2}\right)$$

(1)

2.3. Convert Linguistic Variables to Asymmetric Cloud Model

TOPSIS and CM are presented in combination to handle the MAGDM problem in the probabilistic linguistic environment. Some researchers have transformed linguistic variables into ASCM, and the specific process can be outlined as follows [43,44,45]:

$${\vartheta }_i=\left\{\begin{array}{c}\frac{{\sigma }^{\tau }-{\sigma }^{-i}}{2{\sigma }^{\tau }-2},\left(-\tau \le i\le 0\right)\\ \frac{{\sigma }^{\tau }+{\sigma }^i-2}{2{\sigma }^{\tau }-2},\left(0\le i\le \tau \right)\end{array},\right.{\vartheta }_i\in \left[0,1\right]$$

(2)

According to many experimental results, it is found that $\sigma$ is most likely to be obtained in the interval of [1.36, 1.4] [46], which is used as 1.4 in this paper.

Give a universe $U=\left[U_{min},U_{max}\right]$ and consider the “$3\sigma$ principle” to calculate $E{x}_i$, $\underset{¯}{E{n}_i}$, $\overline{E{n}_i}$, $\underset{¯}{H{e}_i}$, and $\overline{H{e}_i}$, then the calculation process is as follows:

$$E{x}_i=U_{min}+{\vartheta }_i\left(U_{max}-U_{min}\right)$$

(3)

when $x_i=0$, $E{x}_0=\left(U_{min}+U_{max}\right)/2$.

$$\underset{¯}{E{n}_i}=\underset{¯}{E{n}_i^{\prime }}=\frac{E{x}_i-E{x}_{i-1}}{3}$$

(4)

$$\overline{E{n}_i}=\overline{E{n}_i^{\prime }}=\frac{E{x}_{i+1}-E{x}_i}{3}$$

(5)

$$\underset{¯}{H{e}_i}=\frac{max\left\{max\left(\underset{¯}{E{n}_i^{\prime }}\right)-\underset{¯}{E{n}_i},\underset{¯}{E{n}_i}-min\left(\underset{¯}{E{n}_i^{\prime }}\right)\right\}}{3}$$

(6)

$$\overline{H{e}_i}=\frac{max\left\{max\left(\overline{E{n}_i^{\prime }}\right)-\overline{E{n}_i^{\prime }},\overline{E{n}_i^{\prime }}-min\left(\overline{E{n}_i^{\prime }}\right)\right\}}{3}$$

(7)

Through the above process, we can convert linguistic term sets into ASCM and get $Y_i=\left(E{x}_i,\underset{¯}{E{n}_i},\overline{E{n}_i},\underset{¯}{H{e}_i},\overline{H{e}_i}\right)$.

Similar to TOPSIS model, in CM, scholars often use distance measure to rank alternative projects. Expectation, entropy, and super entropy of CM have different meanings and weights. In fact, DMs have different attitudes towards entropy and super entropy values (the randomness part of the decision result) and expectation (the certainty part). In order to better reflect the DMs’ different attitudes towards the three parts, Wang and Huang [42] proposed a scoring method for CM:

$$score\left(x_i\right)=E{x}_i-\xi \sqrt{E{n}_i^2+H{e}_i^2}$$

(8)

where $\xi \in \left[0,1\right]$ is the coefficient reflecting standard deviation attitude $\sqrt{E{n}_i^2+H{e}_i^2}$. The value of $\xi$ is determined by the DMs.

Remark: The closer the value of $\xi$ is to 1, the more the DMs care about the deviation of decision results. $\xi =0$ stands for DMs only considering the expectation value.

3. Establish a New Conversion Model between Probability Linguistic Variables and Asymmetric Probability Cloud Model

3.1. Conversion Algorithm

To realize the combination of probability linguistic variables and ASPLC, this paper establishes the conversion model between probabilistic linguistic variables and ASPLC in detail. First, calculate the conversion coefficient in ASCM; second, convert ASCM into ASPLC; third, the basic calculation formulas of ASLC are defined; last, the comparison rules of ASPLC are introduced. The specific steps are as follows:

Step 1: Transform linguistic variables into ASCM $\left(Y_i\right)$ through Algorithm 1.

Algorithm 1: Convert Linguistic Variables to ASCM

Input: Universe $U=\left[U_{min},U_{max}\right]$, subscript symmetry linguistic term $L\left(P\right)=\left\{\left(l_i\left(p_i^r\right)|i\in \left[-\tau ,\tau \right],r\in 1,\dots ,\#r\right)\right\}$

Output: $Y_i=\left(\underset{¯}{H{e}_i},\underset{¯}{E{n}_i},E{x}_i,\overline{E{n}_i},\overline{H{e}_i}\right)$

Calculate ${\vartheta }_i$ by Equation (2)

Calculate $E{x}_i$ by Equation (3)

Calculate $\underset{¯}{E{n}_i}$ and $\overline{E{n}_i}$ using Equations (4) and (5)

Calculate $\underset{¯}{H{e}_i}$ and $\overline{H{e}_i}$ using Equations (6) and (7)

Return $Y_i=\left(\underset{¯}{H{e}_i},\underset{¯}{E{n}_i},E{x}_i,\overline{E{n}_i},\overline{H{e}_i}\right)$

Step 2: Convert ASCM $\left(Y_i\right)$ to ASPLC $\left(P{Y}_i\right)$:

$$\begin{array}{c}P{Y}_i=\left(P\underset{¯}{H{e}_i},P\underset{¯}{E{n}_i},PE{x}_i,P\overline{E{n}_i},P\overline{H{e}_i}\right)\hfill \\ =\left(\sqrt{{\displaystyle {\displaystyle \sum }_{r=1}^{\#r}}{\left(p_i^r\right)}^2\underset{¯}{H{e}_i^2}},\sqrt{{\displaystyle {\displaystyle \sum }_{r=1}^{\#r}}{\left(p_i^r\right)}^2\underset{¯}{E{n}_i^2}},{\displaystyle {\displaystyle \sum }_{r=1}^{\#r}}{p}_i^{2}E{x}_i,\sqrt{{\displaystyle {\displaystyle \sum }_{l=1}^{\#l}}{\left(p_i^r\right)}^2\overline{E{n}_i^2}},\sqrt{{\displaystyle {\displaystyle \sum }_{l=1}^{\#l}}{\left(p_i^r\right)}^2\overline{H{e}_i^2}}\right)\hfill \end{array}$$

(9)

In particular, $P{Y}_i$ and $P{Y}_j$ are two ASPLCs, which meet the following operation rules [41,42,44]:

$$P{Y}_i+P{Y}_j=\sqrt{P\underset{¯}{H{e}_i^2}+P\underset{¯}{H{e}_j^2}},\sqrt{P\underset{¯}{E{n}_i^2}+P\underset{¯}{E{n}_j^2}},PE{x}_i+PE{x}_j,\sqrt{P\overline{E{n}_i^2}+P\overline{E{n}_j^2}},\sqrt{P\overline{H{e}_i^2}+P\overline{E{n}_j^2}}$$

(10)

$$P{Y}_i-P{Y}_j=\sqrt{P\underset{¯}{H{e}_i^2}+P\underset{¯}{H{e}_j^2}},\sqrt{P\underset{¯}{E{n}_i^2}+P\underset{¯}{E{n}_j^2}},PE{x}_i-PE{x}_j,\sqrt{P\overline{E{n}_i^2}+P\overline{E{n}_j^2}},\sqrt{P\overline{H{e}_i^2}+P\overline{E{n}_j^2}}$$

(11)

The extension above operation rules, $P{Y}_{\mathrm{i}}$, $P{Y}_{\mathrm{j}}$, and $P{Y}_{\mathrm{k}}$, are three PLCs, which meet the following operation rules:

$$P{Y}_i+P{Y}_j=P{Y}_i+P{Y}_j$$

(12)

$$\left(P{Y}_i+P{Y}_j\right)+P{Y}_k=P{Y}_i+\left(P{Y}_j+P{Y}_k\right)$$

(13)

Step 3: Aggregate the ASPLC:

$$\begin{array}{c}PLCAA\left(P{Y}_1,P{Y}_2,\dots ,P{Y}_d\right)={\displaystyle {\displaystyle \sum }_{i=1}^d}\frac{1}{n}P{Y}_i\hfill \\ =\left(\sqrt{{\displaystyle {\displaystyle \sum }_{i=1}^d}\frac{1}{d}\left(P{\underset{¯}{He}}_i^2\right)},\sqrt{{\displaystyle {\displaystyle \sum }_{i=1}^d}\frac{1}{d}\left(P{\underset{¯}{En}}_i^2\right)},{\displaystyle {\displaystyle \sum }_{i=1}^d}\frac{1}{d}\left(PE{x}_i\right),\sqrt{{\displaystyle {\displaystyle \sum }_{i=1}^d}\frac{1}{d}\left(P{\overline{En}}_i^2\right)},\sqrt{{\displaystyle {\displaystyle \sum }_{i=1}^d}\frac{1}{d}\left(P{\overline{He}}_i^2\right)}\right)\hfill \end{array}$$

(14)

Owing to the different meanings and weights expressed by the expectations of CM, entropy, and hyper entropy, this paper improves the cloud score calculation method proposed by Wang and Huang [42] and applies it to the ASPLC score calculation.

3.2. Comparison Rules

$P{Y}_i$ and $P{Y}_j$ are two ASPLCs, which meet the following comparison rules:

$$score\left(P{Y}_i\right)=PE{x}_i-\xi \sqrt{P\underset{¯}{H{e}_i^2}+P{\underset{¯}{E{n}_i}}^2+P{\overline{E{n}_i}}^2,P{\overline{H{e}_i}}^2}$$

(15)

$$sd\left(P{Y}_i\right)=\sqrt{P\underset{¯}{H{e}_i^2}+P{\underset{¯}{E{n}_i}}^2+P{\overline{E{n}_i}}^2,P{\overline{H{e}_i}}^2}$$

(16)

If $score\left(P{Y}_i\right)>score\left(P{Y}_j\right)$, then $P{Y}_i$ better than $P{Y}_j$.

If $score\left(P{Y}_i\right)<score\left(P{Y}_j\right)$, then $P{Y}_j$ better than $P{Y}_i$.

If $score\left(P{Y}_i\right)=score\left(P{Y}_j\right)$, it can be divided into the following three cases:

If $sd\left(P{Y}_i\right)<sd\left(P{Y}_j\right)$, then $P{Y}_i$ better than $P{Y}_j$;

If $sd\left(P{Y}_i\right)>sd\left(P{Y}_j\right)$, then $P{Y}_i$ better than $P{Y}_j$;

If $sd\left(P{Y}_i\right)=sd\left(P{Y}_j\right)$, it indicates that there is no difference between $P{Y}_i$ and $P{Y}_j$.

4. ASPLC-TOPSIS Method Based on the Asymmetric Probability Cloud Model and TOPSIS

4.1. Problem Description

We can suppose that there are restricted alternatives $X=\left(x_1,x_2,\dots ,x_z\right)$, limited DMs $E=\left(e_1,e_2,\dots ,e_q\right)$, limited attribute $A=\left(a_1,a_2,\dots ,a_j\right)$ in MAGDM problem. $l_{mn}^k\left(p\right)$ is the evaluation of the $m$th alternative by the $k$th DM under the $n$th attribute. The weight of each DM is $\omega =\left({\omega }_1,{\omega }_2,\dots ,{\omega }_q\right)$, $0\le {\omega }_k\le 1$, ${\displaystyle \sum }_{k=1}^q{\omega }_k=1$. The weight of each attribute is $\gamma =\left({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_j\right)$, $0\le {\gamma }_n\le 1$, ${\displaystyle \sum }_{n=1}^j{\gamma }_n=1$. In decision making, $L^c\left(P\right)={\left[l_{mn}^c\left(p\right)\right]}_{z\times j}$ represents the evaluation matrix that aggregates the evaluation of all DMs. The linguistic evaluation scale is:

$$L=\left\{l_{-2}=verydislike,l_{-1}=dislike,l_0=medium,l_1=like,l_2=verylike\right\}$$

4.2. ASLC-TOPSIS Method Procedure

In MAGDM problems, using the PLTS to collect the decision preference can express the fuzziness of DMs’ decision preference well. However, in existing TOPSIS methods, it is necessary to process the decision matrix, such as normalizing and adding fuzzy linguistic elements. These behaviors distort the decision-making results, resulting in the decision-making results being different from the initial opinions of the DMs. In order to solve the above problem, this paper proposes an ASPLC-TOPSIS method, which combine ASPLC and TOPSIS under probability environment. In order to intuitively display the decision-making steps of ASPLC-TOPSIS method, we have established a flowchart (see Figure 2). (It is worth noting that in order to make the article clearer and reduce redundant words, we have abbreviated some noun phrases. For the convenience of readers, these abbreviations and their original meanings are placed in

Table A1. The notation list.

Variables	Description
${\vartheta }_i$	Linguistic scaling function
$\xi$	Random partial weight of cloud model
$Y_i$	Asymmetric linguistic cloud
$P{Y}_i$	Asymmetric probabilistic linguistic cloud
$x_z$	z-th alternative
$a_j$	j-th attribute
${\omega }_k$	k-th DM’s weight
${\gamma }_j$	j-th attribute’s weight
$L_{mn}^k$	Evaluation of the $m$th alternative by the $k$th DM under the $n$th attribute.
$E_{ok}$	o-th DM’s trust in the k-th DM
MAGDM	Multi-attribute group decision making
DM	Decision maker
CM	Cloud model
ASCM	Asymmetric linguistic cloud model
ASPLC	Asymmetric probability linguistic cloud model

in the Appendix A).

To better describe the fuzziness of DM’s preference, this study uses the PLTS to collect each DM’s preference. The connections between DMs may affect the acceptance of the final decision result. To enhance the acceptability of the decision result by DMs, this study considers the trust between DMs, and the DMs’ weight is calculated based on trust network. The most trusted DMs have a greater influence in the final decision result, improving the acceptability of the decision-making results. Simultaneously, considering the importance of attribute preference in decision support systems, Pang’s [10] method of using a decision matrix is adopted to calculate the attributes’ weight. The specific steps of the ASLC-TOPSIS model proposed in this paper are as follows:

Step 1: Collect each DM’s decision evaluation.

$L^k\left(P\right)={\left[l_{mn}^k\left(p\right)\right]}_{z\times j}$ is the decision matrix for $k$th DM.

Step 2: Calculate each DM’s weight and aggregate the decision evaluation.

Various relationships and different degrees of trust among DMs have been reported, whereby DMs can express their trust in other members of the decision-making group through linguistic elements on the linguistic evaluation scale: $E_{ok}=\left\{h_{-2}=totaldistrust,h_{-1}=distrust,h_0=medium,h_1=trust,h_2=totaltrust\right\}$ represents $e_o$’s trust in $e_k$, the mapping of the weight evaluation scale on $\left[0,1\right]$ can be expressed as $E_{ok}=\left\{h_{-2}=0,h_{-1}=0.25,h_0=0.5,h_1=0.75,h_2=1\right\}$. In the following steps, we will calculate the weight of DMs on this basis. To facilitate understanding, we visualized the establishment of the trust network between DMs (see Figure 3).

Step 2.1: Calculate the weight of DMs through the trust degree.

$${\omega }^k=\frac{{{\displaystyle \sum }}_{\begin{array}{c}o=1\\ o\ne k\end{array}}^q{E}_{ok}}{{{\displaystyle \sum }}_{\begin{array}{c}o=1\\ o\ne k\end{array}}^q{{\displaystyle \sum }}_{k=1}^q{E}_{ok}}$$

(17)

where $0\le E_{ok}\le 1$, ${\omega }^k\in \left[0,1\right]$, ${\displaystyle \sum }_{k=1}^q{\omega }^k=1$.

Step 2.2: Aggregate decision evaluation reflected in DMs’ weights.

Considering the generality and particularity, we introduce the probabilistic linguistic weighted average (PLWA) operator [10,47] to aggregate DMs’ evaluation matrix.

$$PLWA\left(L^1\left(P\right),L^2\left(P\right),\dots ,L^q\left(P\right)\right)={\omega }_1{L}^1\left(P\right)\oplus {\omega }_2{L}^2\left(P\right)\oplus \dots \oplus {\omega }_q{L}^q\left(P\right)$$

(18)

Step 3: Obtain the significance of each attribute.

Use Equation (19) to obtain the weight of each attribute and aggregate decision evaluation by Formula (20).

$${\gamma }_n=\frac{{{\displaystyle \sum }}_{m=1}^z{{\displaystyle \sum }}_{m\ne t}d\left(L_{mn}\left(P\right),L_{tn}\left(P\right)\right)}{{{\displaystyle \sum }}_{m=1}^z{{\displaystyle \sum }}_{n=1}^j{{\displaystyle \sum }}_{m\ne t}d\left(L_{mn}\left(P\right),L_{tn}\left(P\right)\right)}=\frac{{{\displaystyle \sum }}_{m=1}^z{{\displaystyle \sum }}_{m\ne t}\sqrt{\frac{1}{\#L_{mn}}{{\displaystyle \sum }}_{l=1}^{\#L_{mn}}{\left(p_{mn}^l{r}_{mn}^l-p_{tn}^l{r}_{tn}^l\right)}^2}}{{{\displaystyle \sum }}_{m=1}^z{{\displaystyle \sum }}_{n=1}^j{{\displaystyle \sum }}_{m\ne t}\sqrt{\frac{1}{\#L_{mn}}{{\displaystyle \sum }}_{l=1}^{\#L_{mn}}{\left(p_{mn}^l{r}_{mn}^l-p_{tn}^l{r}_{tn}^l\right)}^2}}$$

(19)

where $m\ne t$, $m,t\in \left[0,z\right]$, $n\in \left[0,j\right]$, $\#L_{mn}$ are the numbers of the PLTS in $L_{mn}\left(p\right)$.

$$PLWA\left[L_{m1}\left(P\right),L_{m2}\left(P\right),\dots ,L_{mj}\left(P\right)\right]={\gamma }_1{L}_{m1}\left(P\right)⨁{\gamma }_2{L}_{m2}\left(P\right)⨁\dots ⨁{\gamma }_j{L}_{mj}\left(P\right)$$

(20)

Step 4: Convert linguistic variables into ASCM $\left(Y_i\right)$ through Algorithm 1.

Step 5: Convert ASCM $\left(Y_i\right)$ into ASLC $\left(P{Y}_i\right)$ through Equation (9).

Step 6: Aggregate each alternative ASLC through Equations (10)–(14).

Step 7: Calculate score of ASLC through Equation (15) and rank it through Equation (16).

5. Experimental Studies

In this section, we use the example in Pang and Wang [10] to verify the feasibility of the ASPLC-TOPSIS method.

5.1. Case Study

Strategic investment is a necessity of the company’s survival. A company has a board of directors consisting of five DMs $\left(e_q|q=1,2,3,4,5\right)$ to evaluate three alternative projects $\left(x_m|m=1,2,3\right)$. Five decision-making members need to evaluate these companies and rank them. To select the most worthwhile investment company, four attributes are considered (all attributes belong to the benefits type): (1) $a_1$ is economic performance; (2) $a_2$ is customer attitude; (3) $a_3$ is internal business operation of the company; (4) $a_4$ is future growth potential of the company.

Step 1: Collect decision evaluation of each DM and the results are presented in

Table 1. The DMs’ evaluation.

		${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$
${\mathit{e}}_{\mathbf{1}}$	${\mathit{x}}_{\mathbf{1}}$	$l_0$	$l_1$	$l_1$	$l_2$
	${\mathit{x}}_{\mathbf{2}}$	$l_0$	$l_0$	$l_{-1}$	$l_0$
	${\mathit{x}}_{\mathbf{3}}$	$l_1$	$l_0$	-	$l_1$
${\mathit{e}}_{\mathbf{2}}$	${\mathit{x}}_{\mathbf{1}}$	$l_1$	$l_{-1}$	$l_1$	$l_2$
	${\mathit{x}}_{\mathbf{2}}$	$l_0$	-	$l_{-2}$	$l_0$
	${\mathit{x}}_{\mathbf{3}}$	$l_1$	$l_0$	$l_2$	$l_1$
${\mathit{e}}_{\mathbf{3}}$	${\mathit{x}}_{\mathbf{1}}$	$l_1$	$l_1$	$l_1$	$l_0$
	${\mathit{x}}_{\mathbf{2}}$	$l_2$	$l_{-1}$	-	$l_1$
	${\mathit{x}}_{\mathbf{3}}$	$l_0$	$l_0$	$l_1$	$l_3$
${\mathit{e}}_{\mathbf{4}}$	${\mathit{x}}_{\mathbf{1}}$	$l_1$	$l_1$	$l_1$	$l_0$
	${\mathit{x}}_{\mathbf{2}}$	$l_0$	$l_1$	$l_0$	$l_0$
	${\mathit{x}}_{\mathbf{3}}$	$l_0$	-	$l_0$	$l_1$
${\mathit{e}}_{\mathbf{5}}$	${\mathit{x}}_{\mathbf{1}}$	$l_0$	$l_1$	$l_0$	$l_2$
	${\mathit{x}}_{\mathbf{2}}$	$l_0$	$l_0$	$l_{-1}$	$l_0$
	${\mathit{x}}_{\mathbf{3}}$	$l_0$	$l_1$	-	$l_1$

.

Step 2: Calculate each DM’s weight and aggregate the decision evaluation.

Step 2.1: Calculate each DM’s weight through the trust degree.

The collected trust degree of DMs is listed in

Table 2. Trust among DMs.

	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_5$
${\mathit{e}}_{\mathbf{1}}$	-	$E_{12}=h_0$	$E_{13}=h_1$	$E_{14}=h_{-2}$	$E_{15}=h_1$
${\mathit{e}}_{\mathbf{2}}$	$E_{21}=h_1$	-	$E_{23}=h_0$	$E_{24}=h_1$	$E_{25}=h_1$
${\mathit{e}}_{\mathbf{3}}$	$E_{31}=h_2$	$E_{32}=h_1$	-	$E_{34}=h_0$	$E_{35}=h_1$
${\mathit{e}}_{\mathbf{4}}$	$E_{41}=h_1$	$E_{42}=h_1$	$E_{43}=h_0$	-	$E_{45}=h_0$
${\mathit{e}}_{\mathbf{5}}$	$E_{51}=h_1$	$E_{52}=h_1$	$E_{53}=h_{-1}$	$E_{54}=h_0$	-

:

The significance of DMs can be calculated based on the trust between DMs using Equation (17). The results are as follows: ${\omega }_1=0.26,{\omega }_2=0.22,{\omega }_3=0.16,{\omega }_4=0.14,{\omega }_5=0.22$.

Step 2.2: Aggregate the decision evaluation reflected in DMs’ weights using Equation (18).

Aggregate the group decision-making evaluation matrix, and the results shown in

Table 3. Aggregate DMs’ decision evaluation.

	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$
${\mathit{x}}_{\mathbf{1}}$	$\left\{l_0\left(0.48\right),l_1\left(0.52\right)\right\}$	$\left\{l_{-1}\left(0.22\right),l_1\left(0.78\right)\right\}$	$\left\{l_0\left(0.22\right),l_1\left(0.78\right)\right\}$	$\left\{l_0\left(0.3\right),l_2\left(0.7\right)\right\}$
${\mathit{x}}_{\mathbf{2}}$	$\left\{l_0\left(0.84\right),l_2\left(0.16\right)\right\}$	$\left\{\begin{array}{c}{l}_{-1}\left(0.16\right),l_0\left(0.48\right),\\ l_1\left(0.14\right)\end{array}\right\}$	$\left\{\begin{array}{c}{l}_{-2}\left(0.22\right),l_{-1}\left(0.48\right),\\ l_0\left(0.14\right)\end{array}\right\}$	$\left\{l_0\left(0.84\right),l_1\left(0.16\right)\right\}$
${\mathit{x}}_{\mathbf{3}}$	$\left\{l_0\left(0.52\right),l_1\left(0.48\right)\right\}$	$\left\{l_0\left(0.64\right),l_1\left(0.22\right)\right\}$	$\left\{\begin{array}{c}{l}_0\left(0.14\right),l_1\left(0.16\right),\\ l_2\left(0.22\right)\end{array}\right\}$	$\left\{l_1\left(0.84\right),l_3\left(0.16\right)\right\}$

.

Step 3: Obtain the significance of each attribute.

The weight is obtained by Formula (19): ${\gamma }_1=0.14,{\gamma }_2=0.3,{\gamma }_3=0.42,{\gamma }_4=0.14$.

Using Equation (20) to aggregate the decision matrix according to the weight of each attribute. The results are presented in

Table 4. Aggregate group decision matrix.

Alternative	Evaluation Results
${\mathit{x}}_{\mathbf{1}}$	$\left\{l_{-1}\left(0.2\right),l_0\left(0.63\right),l_2\left(0.1\right)\right\}$
${\mathit{x}}_{\mathbf{2}}$	$\left\{l_{-2}\left(0.09\right),l_{-1}\left(0.25\right),l_0\left(0.44\right),l_1\left(0.06\right),l_2\left(0.02\right)\right\}$
${\mathit{x}}_{\mathbf{3}}$	$\left\{l_0\left(0.32\right),l_1\left(0.32\right),l_2\left(0.09\right),l_3\left(0.09\right)\right\}$

below.

Step 4: Convert linguistic variables into ASCM through Algorithm 1.

Given a universe $U=\left[0,10\right]$.

After the calculation of Step 4.1–4.4, the ASCM is obtained, and the results are listed following:

$$Y_{-3}=\left(0,0,0,0.73,0.24\right),Y_{-2}=\left(0.24,0.73,2.2,0.57,0.19\right),Y_{-1}=\left(0.19,0.57,3.9,0.37,0.12\right),$$

$$Y_0=\left(0.12,0.37,5,0.37,0.12\right),Y_1=\left(0.12,0.37,6.1,0.57,0.19\right),Y_2=\left(0.19,0.57,7.8,0.73,0.24\right),$$

$$Y_3=\left(0.24,0.73,10,0,0\right)$$

Step 5: Convert ASCM $\left(Y_i\right)$ into ASPLC $\left(P{Y}_i\right)$ through Equation (9).

The decision evaluation is expressed in ASPLC, and the results are in

Table 5. Decision evaluation use for ASPLC.

Alternative	ASPLC
${\mathit{x}}_{\mathbf{1}}$	$\left\{\begin{array}{c}\left(0.038,0.114,0.78,0.074,0.024\right),\left(0.076,0.233,3.15,0.233,0.076\right),\\ \left(0.019,0.037,0.78,0.073,0.024\right)\end{array}\right\}$
${\mathit{x}}_{\mathbf{2}}$	$\left\{\begin{array}{c}\left(0.021,0.066,0.198,0.051,0.017\right),\left(0.048,0.143,0.975,0.093,0.03\right),\\ \left(0.053,0.163,2.2,0.163,0.053\right),\left(0.007,0.022,0.366,0.034,0.011\right),\\ \left(0.004,0.011,0.156,0.015,0.005\right)\end{array}\right\}$
${\mathit{x}}_{\mathbf{3}}$	$\left\{\begin{array}{c}\left(0.038,0.118,1.6,0.118,0.038\right),\left(0.038,0.118,1.952,0.182,0.061\right),\\ \left(0.017,0.051,0.702,0.066,0.022\right),\left(0.005,0.015,0.2,0,0\right)\end{array}\right\}$

:

Step 6: Aggregate each alternative ASPLC through Equation (14).

Use Equation (14) to aggregate ASPLC, and the results are presented in

Table 6. ASPLC aggregation results.

Alternatives	ASPLC
${\mathit{x}}_{\mathbf{1}}$	$\left(0.034,0.0102,0.779,0.089,0.029\right)$
${\mathit{x}}_{\mathbf{2}}$	$\left(0.05,0.024,1.57,0.147,0.048\right)$
${\mathit{x}}_{\mathbf{3}}$	$\left(0.028,0.088,1.1135,0.113,0.038\right)$

.

Step 7: Calculate the score of each candidate using Equation (15) and rank candidate companies according to their score.

$$score\left(x_1\right)=0.7075,score\left(x_2\right)=1.488,score\left(x_3\right)=1.038$$

(21)

Rank the three companies: ${\mathrm{x}}_2\succ {\mathrm{x}}_3\succ {\mathrm{x}}_1$

5.2. Sensitivity Analysis

In this section, we keep other coefficients in the ASPLC-TOPSIS model unchanged, and observe the impact of $\xi$ on the score of each candidate company. The specific results are shown in Figure 4.

It can be found that the value of $\xi$ affects the score of each candidate, but the significant effect on the best alternative is not obvious. Through sensitivity analysis, we can conclude that the ASPLC-TOPSIS method is stable and feasible.

5.3. Comparison and Discussion

This subsection compares the ASPLC-TOPSIS method with the extended-TOPSIS model proposed by Pang and Wang [10], the improved TODIM method proposed by Liu and You [33], the different Entropy method proposed by Liu and Jiang [48], and the fuzzy linguistic multiset (FLM)-TOPSIS model proposed by Pei and Liu [49]. The merits and shortcomings of these methods are summarized in

Table 7. Comparison with other methods.

Methods	Merits	Disadvantages
Pang and Wang [10] method	(a) Describe the fuzziness of DMs’ preference; (b) Consider the weight of each attribute.	(a) Estimate decision matrix unknown decision preference; (b) Add linguistic elements artificially; (c) Ignore DMs’ weight; (d) Cannot show the randomness of DMs’ decision evaluation.
Liu and You [33] method	(a) Describe the fuzziness of DMs’ preference; (b) Consider the DMs’ psychology.	(a) Estimate decision matrix unknown decision preference; (b) Ignore DMs’ weight; (c) Cannot show the randomness of DMs’ decision evaluation.
Liu and Jiang [48] method	(a) Describe the fuzziness of DMs’ preference; (b) Different DMs can choose different forms of entropy; (c) Consider DMs’ weight.	(a) Estimate decision matrix unknown decision preference; (b) Preset DMs’ weight; (c) The randomness of DMs’ decision preference is not considered.
Pei and Liu [49] method	(a) Describe the fuzziness of DMs’ preference; (b) Consider DMs’ weight; (c) Solve the problem that the decision probability is less than 1.	(a)  Preset the weight of each DM; (b)  Solve the problem that the total probability is less than 1 by adding 0 elements; (c)  The randomness of DMs’ preference is not considered.
The proposed method	(a) Reflect the fuzziness and randomness of DMs’ preference; (b) Consider the DMs’ subjective weight; (c) Do not need to estimate the decision matrix unknown decision preference or complete decision evaluation.

it should be noted that all methods have been proposed in the context of PLTS).

Pang and Wang [10] systematically introduced PLTS in the TOPSIS method, only linguistic variables were used to reflect the fuzziness among DMs, which appears more suitable for practical MAGDM problems. This method calculates the distance between each alternative and positive ideal solution (PIS) or negative ideal solution (NIS) to obtain the closeness coefficient $CI\left(x_d\right)$ of each alternative as the basis for the final ranking. The method improves the existing decision-making model and linguistic variables and appears feasible for practical decision making. However, when aggregating the decision matrix, the method needs to estimate the unknown information of the original evaluation matrix to ensure that all fuzzy linguistic sets of each DM have the same fuzzy number; this consequently distorts the decision of the original DMs to a certain extent and leads to a loss of authenticity of the decision results.

Liu and You [33] considered the different psychological states of DMs during evaluation of the project and introduced TODIM method. The aversion coefficient ($\theta$) was used to calculate the prospect value ($\delta \left(x_d\right)$) of each decision-making scheme as the basis for scheme ranking. This method is reasonable to a certain extent but ignores the fact that PLTS includes the preference of DMs, and this method also needs to normalize the decision evaluation.

Liu and Jiang [48] proposed a variety of methods based on entropy measures in different linguistic environments, which improved existing entropy methods. Liu used entropy to calculate the distance $d^{-}\left(x_d\right)$ between each alternative and NIS (PIS), which was then used as a basis for ranking the alternative. This method entirely reflects the fuzziness and hesitation of DMs; however, it is not significantly different from the method proposed by Pang.

The FLM-TOPSIS model proposed by Pei and Liu [49] was developed using FLMs with the help of a 0–1 matrix and fuzzy envelope. The specific steps of the FLM-TOPSIS method include adopting MAGDM for different single attribute decision making and using the pseudo-distance formula to calculate the distance between each alternative PIS and NIS; finally, the closeness degree $C\left(x_1\right)$ is calculated as the decision basis. This method provided certain improvements but artificially creates an HFLT out of the initial fuzzy elements, which expanded the degree of fuzziness of the DMs’ evaluation, thereby distorting the decision-making evaluation.

The ASPLC-TOPSIS method uses the particularity of CM to depict the randomness of DM’s preference while simultaneously improving the conversion operator of the existing linguistic CM to convert the PLTS into CM without any processing. Considering the different weights of ASPLC components, this study abandons the distance measure and directly uses the score function. The DM can use this formula to express randomness or fuzziness of the decision results.

The ranking results obtained using the actual cases in Section 5.1 are displayed in

Table 8. Comparison of ranking results.

	Sort Key	Ranking
Pang and Wang’s [10] method	$CI\left(x_1\right)=0,$ $CI\left(x_2\right)=-1.80,$ $CI\left(x_3\right)=-0.6$	$x_3\succ x_1\succ x_2$
Liu and You’s [33] method	$\delta \left(x_1\right)=1,$ $\delta \left(x_2\right)=0,$ $\delta \left(x_3\right)=0.7952$	$x_1\succ x_3\succ x_2$
Liu and Jiang’s [48] method	$d^{-}\left(x_1\right)=0.4737,$ $d^{-}\left(x_2\right)=0.3379,$ $d^{-}\left(x_3\right)=0.4733,$	$x_1\succ x_3\succ x_2$
Pei and Liu’s [49] method	$C\left(x_1\right)=1,$ $C\left(x_2\right)=0.61,$ $C\left(x_3\right)=0.67$	$x_3\succ x_1\succ x_2$
This article’s method	$Score\left(x_1\right)=0.7075,$ $Score\left(x_2\right)=1.488,$ $Score\left(x_3\right)=1.038$	$x_2\succ x_3\succ x_1$

. There are obvious differences between the methods proposed in this paper and the other existing methods. The reasons for this phenomenon can be summarized as follows:

(1)

This method does not need to estimate the decision matrix of the DMs, nor does it need to deal with the decision matrix of the DMs. It can reflect the most real preference of DM.

(2)

The participation of CM can reflect the randomness of DMs’ preferences caused by uncertain factors, which makes this method different from existing methods.

(3)

The DMs’ subjective weight is considered, in which the trust between DMs is the basis of the weight calculation, which makes the trusted DMs have a higher weight in the final decision and improves the acceptability of the decision results.

6. Conclusions

When existing decision models use linguistic variables to deal with MAGDM problems, they are required to estimate the unknown probability of the initial preference of the DM, which is unwise and can distort the decision results to a certain extent. Similarly, we find that the randomness of DM decision evaluations will also affect the final decision results. At the same time, how to determine the weights of DMs is also a problem encountered in existing research. To address the above problems, this paper improves the TOPSIS method to generate ASPLC-TOPSIS, which effectively avoids the above problems. The ASPLC-TOPSIS method has the following advantages:

(1)

The ASPLC-TOPSIS method can fully reflect the influence of randomness on the decision result.

(2)

The ASPLC-TOPSIS can deal with an incomplete linguistic preference matrix.

(3)

The ASPLC-TOPSIS method reduces the possibility that the decision results will not be accepted by DMs.

(4)

The ASPLC-TOPSIS method is more suitable for the decision-making scenarios where the results may be biased due to random factors.

However, ASPLC-TOPSIS still has the following shortcomings:

(1)

When making large-scale decisions, with the increase in the number of DMs, it will become very difficult to establish a trust network between DMs.

(2)

When the number of alternative and considered standards is very large, the conversion process between PLTS and ASPLC will consume a lot of time.

Therefore, the ASPLC-TOPSIS method still has substantial room for development. In future research, first, we will try to apply the ASPLC-TOPSIS method to more practical fields; second, we will continue to explore more practical methods to determine the weight of DMs; and last, we will try to introduce more theories, such as quantum probability theory, to explain whether there is interference between DMs.