Combination of Probabilistic Linguistic Term Sets and PROMETHEE to Evaluate Meteorological Disaster Risk: Case Study of Southeastern China

Abstract

The meteorological disasters have brought destructive damages all around the world in the past decades. These disasters have posed great threats to sustainable development. It is necessary to evaluate meteorological disaster risk to make corresponding emergency measures. The process is uncertain and fuzzy regarding the experts’ preferences. To deal with the problem, a novel evaluation approach based on PROMETHEE method and probabilistic linguistic term set (PLTS) is firstly proposed. First of all, PLTS is adopted to express preferences’ of experts. Then, the weights of criteria are obtained by the differential evolution (DE) algorithm, and steps of the method are proposed. Finally, the proposed method is used to evaluate the whole meteorological disaster risk in the southeast coastal areas of China and results have verified the effectiveness of the method. By comparing with some similar methods, results have demonstrated the advantages of the approach.

1. Introduction

Sustainable development should be considered in the following decades. Sustainability is to maintain change in the environment. During the process, environment, economic, and social need to be in harmony [1]. However, meteorological disasters have risen in the past decades. These disasters have posed great threats to sustainable development. To deal with these disasters, emergency management is adopted. Planning for emergencies is essential. Emergency management is the organization and management of the resources and responsibilities for dealing with all aspects of emergencies (preparedness, response, mitigation, and recovery). The aim is to reduce the harmful effects of disasters and realize sustainable development. In the process of emergency management, risk analysis is indispensable [2,3].

The number of meteorological disasters has steadily risen in the past decades. Meteorological disasters have brought destructive damages to infrastructures and people’s lives globally. For example, the typhoon is a kind of meteorological disasters. Typhoons result in catchment, landslides, and rainfall. In November 2013, Typhoon Haiyan passed through the Philippines. It caused major damages. More than 3 million families were affected, with more than 6300 persons killed, more than 28,000 injured, and more than 1000 missing. Buildings, infrastructure, and ecosystems suffered [4]. As a country with frequent typhoon disasters, China had over 260 typhoon landing events from 1949 to 2017, which caused more than 23 billion RMB in direct economic losses from 1991 to 2013 [5,6,7]. Most of them occurred in the southeast coastal areas. These areas are exposed to the threat of meteorological disasters. But these areas are highly-developed with a large population. There are many modern buildings in these areas, such as the 632 m high Shanghai Tower and 600 m high Ping-An Finance Center (PAFC) [8]. Therefore, evaluating the risk of meteorological disasters in these areas is of great significance, which can assist in taking corresponding emergency measures in advance.

Evaluation often requires decision-makers to offer certain and precise performance for each alternative with respect to each criterion. However, it is difficult to obtain certain and precise information when evaluating meteorological disasters risk as techniques and statistics are limited. The actual evaluation process is full of complexity and difficulty. In fact, decision-makers use linguistic terms to express their preferences when they are asked to make evaluations. Linguistic terms are useful approaches to solve uncertain and imprecise problems, such as bad, good. Linguistic terms are widely used in decision-making. To make operational laws of linguistic terms reasonable, some operational laws for linguistic terms are proposed [9]. Linguistic terms are used to design fuzzy rule-based classifiers [10]. The fuzzy linguistic TOPSIS approach is developed to plan the development of projects [11].

However, it is still difficult for decision-makers to express their preferences by just one linguistic term when they encounter complicated problems in real applications. To deal with the problem, hesitant fuzzy linguistic term sets (HFLTSs) were proposed by Rodriguez et al. [12]. HFLTSs improved the elicitation of linguistic terms. Decision-makers can express their preferences by more than one linguistic term with the help of linguistic terms. The performance of HFLTSs is very promising, which has attracted much attention in the past few years. According to the definition of HFLTSs, Liao et al. defined the mathematical form of HFLTSs [13]. Then, they defined cosine distance and similarity measures for HFLTSs and applied them in decision-making [14]. An outranking method based on a novel knowledge-based paradigm for comparing HFLTSs was developed [15]. Current developments, issues, and challenges of HFLTSs were fully reviewed by Wang et al. [16].

All the linguistic terms have the same status in the HFLTSs. Linguistic terms have the same weights when they are used to express opinions of decision experts. In fact, information may be lost when decision-makers have preferences for some linguistic terms. To solve the defects of HFLTSs, the probabilistic linguistic term set (PLTS) is proposed [17]. Obviously, PLTS can depict preferences more reasonably and rationally [18]. PLTS is used in customer relationship management, which is full of uncertainty and fuzziness regarding the users’ preferences [19]. An efficient decision-making framework is proposed to solve real-life problems by PLTS [20]. The extended TODIM method for PLTSs was developed to multi-attribute decision-making [18]. A new method for MAGDM with PLTSs was developed to select technology companies under the background of financial technology [21]. Meteorological disasters create uncertain and imprecise information. Decision-makers are asked to give their evaluation. It is difficult for them to give certain and precise opinions as techniques and statistics are limited. They have their preferences. They may prefer some linguistic terms to other linguistic terms according to their knowledge, experience, and background. Therefore, it can be noticed that PLTS is very suitable to depict features of meteorological disasters.

Decision-making methods are essential to evaluate meteorological disasters risk. In the past two decades, they have been many methods proposed, such as TOPSIS [22], VIKOR [22], TODIM [18], PROMETHEE [23,24,25,26], and so on. Among these methods, PROMETHEE is one of the most popular. It takes advantage of the outranking principle to rank alternatives. As the procedure of PROMETHEE is very easy and transparent, it can be easily understood by decision-makers [26]. The method can offer a reasonable ranking of all alternatives. Therefore, they are widely adopted in energy projects [23], regional tourism competitiveness [24], and airport location selection [25].

As the performance of PROMETHEE is superior, it is employed to evaluate meteorological disasters risk. However, decision-making problems have become increasingly complicated in the past two decades. Especially, precision information of meteorological disasters cannot be obtained, and the criteria of relative importance for meteorological disasters is not easily determined. Some uncertain and imprecise information has to be provided. Thus, it is necessary to improve PROMETHEE so that it can be used in an uncertain and imprecise environment. In addition, it is necessary to develop an approach to determine the criteria of relative importance for meteorological disasters. Therefore, the main contribution of the paper is as follows:

• PROMETHEE is improved by PLTS so that it can be used to evaluate meteorological disaster risk in an uncertain and imprecision environment.
• The relative importance of the criteria is converted to a constraint optimization problem in meteorological disasters. The problem is solved by the evolutionary algorithm.
• The proposed method is applied to evaluate meteorological disaster risk in the southeast coastal areas of China. Compared with other approaches, the proposed method is effective.

The rest of the paper is organized as follows: Some basic concept and operational laws are reviewed in Section 2. PROMETHEE is briefly introduced in Section 3. Section 4 proposes the extended model based on PROMETHEE and PLTS. An empirical study is implemented, and some comparisons are made in Section 5.

2. Preliminaries

Some concepts and operational laws of HFLTS and PLTSs are introduced briefly in this section.

2.1. HFLTS

Let $S=\{s_t|t=-\tau ,\dots ,-1,0,1,\dots ,\tau \}$ be a linguistic term set; $x_i\in X,i=1,2,\dots ,N$ be fixed. A HFLTS can be described as follows:

$$H_s=\{\langle x_i,h_s(x_i)\rangle |x_i\in X\}$$

(1)

where $h_s(x_i)$ is a collection of values. These values are from the linguistic term set $S$ and can be depicted as $h_s(x_i)=\{s_{\varphi l}(x_i)|s_{\varphi l}(x_i)\in S,l=1,\dots ,L\}$ with $L$ being the number of linguistic terms in $h_s(x_i)$ [13,27].

2.2. PLTSs

Let $S=\left\{s_0,s_1,\dots ,s_{\tau }\right\}$ be the linguistic term sets; a probabilistic linguistic term set (PLTS) is given as follows:

$$L(p)=\{L^{(k)}(p^{(k)})|L^{(k)}\in S,p^{(k)}\ge 0,k=1,2,\dots ,\#L(p),{\displaystyle \sum }_{k=1}^{\#L(p)}{p}^{(k)}\le 1\}$$

(2)

where $L^{(k)}$ is the linguistic term; $p^{(k)}$ is the probability; $L^{(k)}(p^{(k)})$ is the $L^{(k)}$ associated with $p^{(k)}$; $\#L(p)$ is the number of all different linguistic terms in $L(p)$ [17].

Let $L(p)=\{L^{(k)}(p^{(k)})|k=1,2,\dots ,\#L(p)\}$ be an ordered PLTS; $r^{(k)}$ is the subscript of linguistic term $L^{(k)}$. Then, the score $E(L(p))$ and the deviation degree $\sigma (L(p))$ of $L(p)$ are defined as follows:

$$E(L(p))=S_{\overline{\alpha }}$$

(3)

$$\overline{\alpha }={\displaystyle \sum }_{k=1}^{\#L(p)}{r}^{(k)}{p}^{(k)}/{\displaystyle \sum }_{k=1}^{\#L(p)}{p}^{(k)}$$

(4)

$$\sigma (L(p))=({\displaystyle \sum }_{k=1}^{\#L(p)}(p^{(k)}{r}^{(k)}-\stackrel{)}{\alpha }){)}^2/{\displaystyle \sum }_{k=1}^{\#L(p)}{p}^{(k)}$$

(5)

The main functions of score $E(L(p))$ and deviation degree $\sigma (L(p))$ of $L(p)$ are used to make comparisons. Let $L_1(p)$ and $L_2(p)$ be two PLTSs. The following rules can be used to compare the two PLTSs [17]. If $E(L_1(p))>E(L_2(p))$ then $L_1(p)>L_2(p)$. If $E(L_1(p))=E(L_2(p))$ and $\sigma (L_1(p))<(L_2(p))$ then $L_1(p)>L_2(p)$.

Let a PTLS $L(p)$ with ${\displaystyle \sum }_{k=1}^{\#L_{(p)}}{p}^{(k)}<1$, then the $L^{\prime }(p)$ can be defined as the associated PLTS by the following equation:

$$L^{\prime }(p)=\{L^{(k)}(p^{\prime (k)})|k=1,2,\dots ,\#L(p)\}$$

(6)

where $p^{\prime (k)}=p^{(k)}/{\displaystyle \sum }_{k=1}^{\#L_{(p)}}{p}^{(k)}$.

Let $L_a(p)$ and $L_b(p)$ be two PLTSs, $L_a(p)=\{L_a^{(k)}(p_a^{(k)})|k=1,2,\dots ,\#L_a(p)\}$, $L_b(p)=\{L_b^{(k)}(p_b^{(k)})|k=1,2,\dots ,\#L_b(p)\}$, $\#L_a(p)=\#L_b(p)$, then the deviation degree between the two PLTSs can be computed as follows:

$$d(L_a(p),L_b(p))=\sqrt{{\displaystyle \sum }_{k=1}^{\#L_{a(p)}}{(p_a^{(k)}{r}_a^{(k)}-p_b^{(k)}{r}_b^{(k)})}^2/\#L_{a(p)}}$$

(7)

3. PROMETHEE

PROMETHEE was developed in 1982. Because the procedure of PROMETHEE is very easy, it can be understood by decision-makers. The method attracts much attention. It is widely used in different fields, such as energy sources [28,29], sub-watersheds ranking [30], military airport location selection [25]. PROMETHEE has been extended to different versions, such as PROMETHEE II, PROMETHEE III, and PROMETHEE IV [31]. The main steps of the PROMETHEE are as follows:

• Identify decision-makers. Before making evaluation, some decision-makers should be invited. Furthermore, experts with abundant experience in this area are desirable. They can give more reasonable results.
• Establish the criteria. Generally, alternatives are evaluated based on the established criteria. The decision-makers give their rating according to the performance of criteria.
• Weights of the criteria. The weight is the relative importance among the criteria. Since the weights are not calculated, they are the decision-maker’s preferences. The weights are subjective, not objective. Nowadays, there are many methods to obtain weights of the criteria, such as AHP, Entropy, and Delphi. In addition, the requirement that the sum of weights is equal to one is needed.
• Evaluate performances of alternatives. Decision-makers are asked to make an objective evaluation of each alternative based on their professional background.
• Select the preference function. The preference function $P_j(x_i,x_k)$ is to describe the difference between alternatives $x_i$ and $x_k$. The difference can be denoted as

  $$d_j(x_i,x_k)=f_j(x_i)-f_j(x_k)$$

  (8)

  where $x_i$ and $x_k$ are two alternatives; j is the jth criterion; $f_j(x_i)$ and $f_j(x_k)$ are the evaluation results of alternatives $x_i$ and $x_k$ from decision-makers on the criterion j. Six preference types are put forward to simplify calculation [32]. These types include the usual criterion, quasi-criterion, criterion with linear preference, level criterion, criterion with linear preference and indifference area, and Gaussian criterion.

  $$P(d)=\{\begin{array}{c}0\begin{array}{cc}& d\le 0\end{array}\\ 1\begin{array}{cc}& d>0\end{array}\end{array}$$

  (9)

  $$P(d)=\{\begin{array}{c}0\begin{array}{cc}& d\le q\end{array}\\ 1\begin{array}{cc}& d>q\end{array}\end{array}$$

  (10)

  $$P(d)=\{\begin{array}{ll}0& d\le q\\ \frac{1}{2}& q<d\le p\\ 1& d>p\end{array}$$

  (11)

  $$P(d)=\{\begin{array}{ll}0& d\le q\\ \frac{d-q}{p-q}& q<d\le p\\ 1& d>p\end{array}$$

  (12)

  $$P(d)=\{\begin{array}{ll}0& d\le 0\\ 1-e^{-\frac{d^2}{2s^2}}& d>0\end{array}$$

  (13)

In the Equations (9)–(13), p and q are the thresholds. Take Equation (10) as an example, if the difference of two alternatives on criterion j is less than p, they are indifferent. Otherwise, they are strictly different.

1. Calculate the preference index

The preference index can be calculated as follows:

$$H(x_i,x_k)={\displaystyle \sum }_{j=1}^n{w}_j\times P_j(x_i,x_k)$$

(14)

where $w_j$ is the weight of the jth criterion; $P_j(x_i,x_k)$ is the preference function.

2. Compute the leaving and entering flow

The leaving and entering flow can be computed by the following two equations:

$${\phi }^{+}(x_i)={\displaystyle \sum }_{k=1}^{m}H(x_i,x_k)$$

(15)

$${\phi }^{-}(x_i)={\displaystyle \sum }_{k=1}^{m}H(x_k,x_i)$$

(16)

where $H(x_i,x_k)$ is the preference index and obtained in the previous step. The leaving flow ${\phi }^{+}(x_i)$ is the dominance of the ith alternative over other alternatives. The entering flow ${\phi }^{-}(x_i)$ has the opposite meaning.

3. Calculate the net flow

The net flow is determined by the leaving flow and entering flow as follows:

$$\phi (x_i)={\phi }^{+}(x_i)-{\phi }^{-}(x_i)$$

(17)

The higher the net flow is, the better the alternative is.

4. The Extended PROMETHEE to Evaluate Meteorological Disaster Risk

4.1. Background

In the recent years, frequent meteorological disasters have posed a serious threat to the life and property safety of people, especially for the southeast coastal areas in China. To effectively evaluate the risk of meteorological disaster and reduce the adverse effect, we choose eight regions which are often hit by meteorological disasters, respectively, Shanghai city, Shandong, Jiangsu, Zhejiang, Fujian, Guangdong, Guangxi, and Hainan provinces.

4.2. Problem Description

In this section, we use multi-criteria group decision-making with probabilistic linguistic information to solve the above problem. Let $x=\left\{x_1,x_2,\dots ,x_8\right\}$ be the set of eight areas, Let c = {c1,c2,c3,c4} be the criteria, in which c₁ is the crop disaster area, c₂ is the flood-hit population, c₃ is the death population, c₄ is the direct economic loss. The weight of the four criteria $w=\left\{w_1,w_2,w_3,w_4\right\}$, where $w_i\ge 0,{{\displaystyle \sum }}_{i=1}^4{w}_i=1$. As the complexity of the problem, we invite four decision-makers (DM_d, d = 1,2,3,4) to evaluate the meteorological risk of eight regions via PTLSs that are namely affiliated with China’s emergency management department, China’s meteorological administration, China’s ministry of civil affairs, and Nanjing University of Information Science and Technology. All of them are top research institutions for meteorological disasters in China.

These decision-makers use PTLSs to evaluate the meteorological disasters risk of above eight areas. $L_{ij}(p)=\{L_{ij}^{(k)}(p_{ij}^{(k)})|k=1,2,\dots ,\#L_{ij}^k(p)\};i=1,2,\dots ,8,j=1,2,\dots ,4$ denotes the value over the province x_i with respect to the criterion j by decision-makers, in which $L_{ij}^{(k)}$ is the kth value of $L_{ij}(p)$; $p_{ij}^{(k)}$ is the probability; $\#L_{ij}^k(p)$ is the number of $L_{ij}^{(k)}$. When four decision-makers have given their evaluations, we combine results from four decision-makers together to form the matrix $R_d$.

$$R_d={\left[L_{ij}(p)\right]}_{d,m\times n}$$

(18)

where $R_d(d=1,2,3,4)$ is the evaluation result from the dth decision-maker.

Then, we aggregate $R_d$ to construct the decision matrix $R$.

$$R={\left[L_{ij}(p)\right]}_{m\times n}=\left[\begin{array}{cccc}{L}_{11}(p)& L_{12}(p)& \dots & L_{1n}(p)\\ L_{21}(p)& L_{22}(p)& \dots & L_{2n}(p)\\ \dots & \dots & \dots & \dots \\ L_{m1}(p)& L_{m2}(p)& \dots & L_{mn}(p)\end{array}\right]$$

(19)

where $L_{ij}(p)=\{L_{ij}^{(k)}(p_{ij}^{(k)})|k=1,2,\dots ,\#L_{ij}^k(p)\},i=1,2,\dots ,8;j=1,2,\dots ,4.$

4.3. Weights of the Criteria Conversion

In the decision-making problem, the weight represents the relative significance among criterion. The Lagrange function is introduced to get weights of the criteria where PLTs are firstly proposed [17]. The calculation is a little complicated. Here, motived by AHP, we convert the weights of the criteria to a constraint optimization problem by the following steps:

Step 1: Convert the decision matrix R to R′.

$$R^{\prime }={\left[r_{ij}\right]}_{m\times n}=\left[\begin{array}{cccc}{r}_{11}& r_{12}& \dots & r_{1n}\\ r_{21}& r_{22}& \dots & r_{2n}\\ \dots & \dots & \dots & \dots \\ r_{m1}& r_{m2}& \dots & r_{mn}\end{array}\right]$$

(20)

where $r_{ij}={\displaystyle \sum }_{k=1}^{\#L_{ij}(p)}{r}^{(k)}{p}^{(k)}/{\displaystyle \sum }_{k=1}^{\#L_{ij}(p)}{p}^{(k)}$; $r^{(k)}$ is the subscript of linguistic term $L^{(k)}$; $i=1,2,\dots ,m;j=1,2,\dots ,n;m=8,n=4.$

Step 2: Normalize the matrix R′.

$$r_{ij}{}^{\prime }=\frac{r_{ij}-r_i^{-}}{r_i^{*}-r_i^{-}},i\in benefits{r}_{ij}{}^{\prime }=\frac{r_i^{*}-r_{ij}}{r_i^{-}-r_i^{*}},i\in \cos t s$$

(21)

where $r_i^{*}=\underset{j}{\max }{r}_{ij};r_i^{-}=\underset{j}{\min }{r}_{ij}$.

Step 3: Motived by AHP theory, we can convert Equation (20) to Equation (22) when the matrix Equation (20) is completely consistent.

$${\displaystyle \sum }_{k=1}^n(r_{ik}{}^{\prime }{w}_k)={\displaystyle \sum }_{k=1}^n(\raisebox{1ex}{$w_i$}\!\left/ \!\raisebox{-1ex}{$w_k$}\right.)w_k= n{w}_i,i=1,2,\dots ,4$$

(22)

$${\displaystyle \sum }_{i=1}^n|{\displaystyle \sum }_{k=1}^n(r_{ik}{}^{\prime }{w}_k)-n{w}_i|=0$$

(23)

However, it is very hard to meet the condition that the matrix Equation (20) is completely consistent in real applications. Then, the Equation (22) can be converted to the following format:

$$minf(w)=\frac{{{\displaystyle \sum }}_{i=1}^n|{{\displaystyle \sum }}_{k=1}^n(r_{ik}{}^{\prime }{w}_k)-n{w}_i|}{n}$$

(24)

$$0<w_k<1,{\displaystyle \sum }_{k=1}^n{w}_k=1$$

(25)

This is the constraint optimization problem. The objective function is Equation (24), and constraint is Equation (25). Now, there are many evolutionary algorithms, such as the differential evolution (DE) algorithm, genetic algorithm (GA), particle swarm optimization (PSO), and so on. They are available to optimize constraint optimization problems. As the performance of the DE is attractive, it is adopted to solve the constraints optimization problem [33].

4.4. Combine Probabilistic Lingustic Information and PROMETHEE

Based on the above analysis, we further combine PROMETHEE and PLTS together. The specific steps are as follows.

Step 1

Four above decision-makers are invited to give their evaluations in the form of PLTSs.

Step 2

Aggregate these evaluation results in the form of Equation (19).

Step 3

According to Section 4.3, the weights of the criteria can be determined.

Step 4

Select the preference function. GAUSSIAN function is often applied in real applications [23]. Therefore, the GAUSSIAN preference function is adopted as the preference function.

$$p_j(d_j(r_{ij},r_{kj}))=1-e^{-\frac{{(d_j(r_{ij},r_{kj}))}^2}{{\sigma }^2}}$$

(26)

where $\sigma$ is the threshold value, and $d_j(r_{ij},r_{kj})$ can be computed by Equation (7).

Step 1

Calculate the preference index $H(x_i,x_k)$ through Equation (14).

Step 2

Obtain the leaving and entering flow by Equations (15) and (16), respectively.

Step 3

Obtain the net flow according to Equation (17).

Step 4

Rank the risk of meteorological disaster from eight areas.

5. Case Study

5.1. Evaluate Meteorological Risk in the Southeast Coastal Areas

Now, we use the proposed method to evaluate the meteorological risk in the Shanghai city, Shandong, Jiangsu, Zhejiang, Fujian, Guangdong, Guangxi, and Hainan provinces. These areas are highly-developed with a large population. However, they often suffer from meteorological disasters. we invited four experts to make evaluations as follows:

Step 1

The four experts use the linguistic term sets to evaluate the above eight areas. S= {s₀ = none, s₁ = very low, s₂ = low, s₃ = medium, s₄ = high, s₅ = very high}.

The evaluation results are listed in

Table 1. The evaluation results by the 1st expert and the 2nd expert.

	the 1st Expert				the 2nd Expert
	c1	c2	c3	c4	c1	c2	c3	c4
Shandong	s2	s2	s1	s2	s2	s1	s2	s3
Shanghai	s1	s2	s1	s3	s2	s1	s1	s4
Jiangsu	s2	s3	s1	s4	s3	s2	s2	s3
Zhejiang	s2	s2	s2	s3	s3	s2	s3	s3
Fujian	s3	s4	s3	s3	s3	s4	s2	s2
Guangdong	s4	s3	s2	s3	s3	s3	s1	s4
Guangxi	s2	s2	s1	s2	s2	s3	s2	s2
Hainan	s3	s2	s2	s2	s3	s2	s1	s2

and

Table 2. The evaluation results by the 3nd expert and 4th expert.

	the 3rd Expert				the 4th Expert
	c1	c2	c3	c4	c1	c2	c3	c4
Shandong	s3	s1	s2	s3	s3	s1	s2	s3
Shanghai	s1	s2	s1	s4	s1	s2	s1	s4
Jiangsu	s3	s3	s2	s3	s3	s3	s2	s3
Zhejiang	s3	s3	s3	s3	s3	s3	s3	s3
Fujian	s2	s3	s2	s2	s2	s3	s2	s2
Guangdong	s3	s3	s1	s3	s3	s3	s1	s3
Guangxi	s2	s3	s1	s2	s2	s3	s1	s2
Hainan	s3	s2	s1	s3	s3	s2	s1	s3

, in which c₁ is the crop disaster area, c₂ is the flood-hit population, c₃ is the death population, and c₄ is the direct economic loss. These evaluation results are aggregated to generate results in

Table 3. The aggregation results.

	C1	C2	C3	C4
Shandong	{s2(0.75), s3(0.25)}	{s1(0.5), s2(0.5)}	{s1(0.5), s2(0.5)}	{s2(0.5), s3(0.5)}
Shanghai	{s1(0.75), s2(0.25)}	{s1(0.5), s2(0.5)}	{s0(0.25), s1(0.75)}	{s3(0.5), s4(0.5)}
Jiangsu	{s2(0.5), s3(0.5)}	{s2(0.5), s3(0.5)}	{s1(0.5), s2(0.5)}	{s3(0.5), s4(0.5)}
Zhejiang	{s2(0.25), s3(0.75)}	{s2(0.5), s3(0.5)}	{s2(0.5), s3(0.5)}	{s3(0.75), s4(0.25)}
Fujian	{s2(0.25), s3(0.75)}	{s3(0.25), s4(0.75)}	{s2(0.5), s3(0.5)}	{s2(0.5), s3(0.5)}
Guangdong	{s3(0.5), s4(0.5)}	{s3(0.75), s4(0.25)}	{s1(0.5), s2(0.5)}	{s3(0.5), s4(0.5)}
Guangxi	{s1(0.25), s2(0.75)}	{s2(0.5), s3(0.5)}	{s1(0.5), s2(0.5)}	{s2(0.75), s3(0.25)}
Hainan	{s2(0.25), s3(0.75)}	{s2(0.75), s3(0.25)}	{s1(0.5), s2(0.5)}	{s2(0.5), s3(0.5)}

. Take the results {s₂(0.75), s₃(0.25)} as an example, three experts give s₂, and one expert gives s₃ with respect to the criterion c1 in Shandong province. Therefore, three of four suggest s₂ and one of four deems s₃.

Step 2

Weights of the criteria. Convert the Table 3 to establish

Table 4. The decision matrix.

	c1	c2	c3	c4
Shandong	2.25	1.5	1.5	2.5
Shanghai	1.25	1.5	0.75	3.5
Jiangsu	2.5	2.5	1.5	3.5
Zhejiang	2.75	2.5	2.5	3.25
Fujian	2.75	3.75	2.5	2.5
Guangdong	3.5	3.25	1.5	3.5
Guangxi	1.75	2.5	1.5	2.25
Hainan	2.75	2.25	1.5	2.5

by Equation (20). The weights of the criteria can be converted to a constraint optimization problem, and the DE algorithm is used to solve the problem. The weights are as follows: w = [0.0429, 0.2157, 0.2093, 0.5322].

Step 3

Calculate the preference index function. As discussed above, GAUSSIAN function is chosen as the preference index function. The procedure is based on Equation (13). It determines the deviations $d_j(r_{ij},r_{kj})$ between alternatives i and k with respect to criterion j. The results are obtained.

Step 4

Calculate the preference index by Equation (14). The results of preference index are listed in

Table 5. Preference index.

H (1)	0.0193	0	0	0	0	0.0055	0
H (2)	0.0625	0	0.0041	0.0625	0	0.0944	0.0625
H (3)	0.0882	0.0472	0.0041	0.0625	0	0.0973	0.0642
H (4)	0.0874	0.1024	0.0249	0.0361	0.0246	0.0922	0.0624
H (5)	0.1271	0.1782	0.0632	0.0383	0.0312	0.0720	0.0775
H (6)	0.1388	0.1029	0.0197	0.0217	0.0654	0.1227	0.0908
H (7)	0.0253	0.0409	0	0	0	0	0.0017
H (8)	0.0160	0.0394	0.0003	0	0	0	0.0092

, in which H (1) deems H (1, 2), H (1, 3), H (1, 4), H (1, 5), H (1, 6), H (1, 7), and H (1, 8). H (1, 2) = 0.0193, H (1, 3) = 0, H (1, 4) = 0, H (1, 5) = 0, H (1, 6) = 0, H (1, 7) = 0.0055, and H (1, 8) = 0.

For example, the preference index of Shandong province and Shanghai city can be written as H (1, 2), which can be calculated as follows:

$$H(1,2)={\displaystyle \sum }_{j=1}^4{w}_j\times P_j(r_{1j},r_{2j})=0.0193$$

Step 5

According to the results of the preference index, the leaving and the entering flow can be calculated by Equations (15) and (16). For example,

$$\begin{gathered} {\phi }^{+}(1)={\displaystyle \sum }_{k=1}^{8}H(1,k)=0.0247 \\ {\phi }^{-}(1)={\displaystyle \sum }_{k=1}^{8}H(k,1)=0.5453 \end{gathered}$$

All the leaving and the entering flow can be obtained by similar steps. The results are demonstrated in

Table 6. Leaving, entering, and net flows.

	1	2	3	4	5	6	7	8
${\phi }^{+}$	0.0247	0.2862	0.3636	0.4301	0.5875	0.5620	0.0679	0.0649
${\phi }^{-}$	0.5453	0.5303	0.1081	0.0683	0.2267	0.0558	0.4933	0.3591
$\phi$	−0.5205	−0.2441	0.2555	0.3618	0.3608	0.5062	−0.4254	−0.2942

.

Step 6

The net flow can be obtained. According to Equation (17), the net flow is computed and listed in Table 6. The results are depicted in Figure 1. Therefore, it can be concluded that the order of meteorological disasters risk is Guangdong > Zhejiang > Fujian > Jiangsu > Shanghai > Hainan > Guangxi > Shandong.

5.2. Methodology Validation

To further validate the effectiveness and advantages of the proposed method; the method is used to compare with TOPSIS based on PLTSs. Now, we give a comparison between the proposed method and TOPSIS method.

The selected case is from the study of Parreiras et al. [34], which needs five decision-makers to evaluate three projects (x₁, x₂, x₃). The criteria are as follows: (1) c₁: Financial perspective; (2) c₂: Customer satisfaction; (3) c₃: Internal business process perspective; (4) c₄: Learning and growth perspective. The final decision matrix is shown in

Table 7. The probabilistic linguistic decision matrix of the group.

	a1	a2	a3	a4
x1	{s3(0.4), s4(0.6)}	{s2(0.2), s4(0.8)}	{s3(0.2), s4(0.8)}	{s3(0.4), s5(0.6)}
x2	{s3(0.8), s5(0.2)}	{s2(0.2), s3(0.4), s4(0.2)}	{s1(0.2), s2(0.4), s3(0.2)}	{s4(0.2), s3(0.8)}
x3	{s3(0.6), s4(0.4)}	{s3(0.6), s4(0.2)}	{s3(0.2), s4(0.2), s5(0.2)}	{s4(0.8), s6(0.2)}

.

From our proposed method, the weights can be obtained as follows: [0.3020, 0.2603, 0.2178, 0.2199].

Then, after calculating the preference function and preference function index, the leaving, entering and net flows are listed in

Table 8. Leaving, entering and net flows.

	A1	A2	A3
${\phi }^{+}$	0.2115	0	0.0618
${\phi }^{-}$	0.0013	0.2104	0.0615
$\phi$	0.2101	−0.2104	0.0002

. Therefore, the final ranking is A₁ > A₃ > A₂. The result is consistent with the original paper. However, the proposed method has its advantages. The proposed method is very easy to implement. Especially, the weights of criteria are very easy to acquire. In addition, the proposed method is based on PROMETHEE. The main advantages of PROMETHEE is that simple and different preference functions give us more choices. It can provide a useful approach to handle with Multiple Criteria Decision Making (MCDM) problems.

In addition, we further compare the proposed method with the possibility degree formula of PLTSs [20]. The example is to evaluate four domestic hospitals (h₁, h₂, h₃ and h₄). Three main criteria are as follows: the environmental factor of medical and health service (c₁); personalized diagnosis and treatment optimization (c₂); and social resource allocation optimization under the pattern of wisdom medical and health services (c₃). The weights are respectively 0.2, 0.1, and 0.7. As a result, the probabilistic linguistic decision matrix is shown in

Table 9. The probabilistic linguistic decision matrix.

	a1	a2	a3
h1	{s0(0.4), s1(0.6)}	{s2(1)}	{s−1(0.2), s0(0.8)}
h2	{s2(0.3), s3(0.7)}	{s0(1), s3(0.4), s4(0.2)}	{s1(0.2), s2(0.4), s3(0.4)}
h3	{s1(1)}	{s1(0.5), s2(0.5)}	{s2(0.6), s3(0.4)}
h4	{s2(0.5), s3(0.5)}	{s2(0.4), s1(0.1), s0(0.2), s1(0.3)}	{s1(1)}

.

According to the proposed method, the results of net flow are −0.9751, 0.6776, 0.4854, and −0.1879, respectively. The ranking order is h₂ > h₃ > h₄ > h₁. To further compare the result with three other MCDM methods, we put all outcomes into

Table 10. The final results based on the existing methods.

	Ranking Order	the Optimal Alternative
The TOPSIS method by conventional operational laws [27]	h2 > h1 > h4 > h3	h2
The TOPSIS method by novel operational laws [30]	h2 > h3 > h4 > h1	h2
Probabilistic linguistic term sets for multi-criteria decision making	h2 > h3 > h4 > h1	h2
The proposed method	h2 > h3 > h4 > h1	h2

. Apparently, three methods including the proposed method have the same ranking results. However, the TOPSIS method by conventional operational laws obtains a different ranking order for its defects. It cannot comprehensively make good use of the original linguistic information provided by decision-makers, and, thus, may produce distorted results. Furthermore, when our method is relative with possibility degree formula of PLTSs, the proposed method is relatively simple.

6. Conclusions

To realize sustainable development, it is necessary to learn about the risk of meteorological disasters. Governments and citizens can take useful measures to deal with these disasters. This is a typical multi-criteria group decision-making problem with imprecise information and more than one expert. To solve the problem, a novel evaluation approach is proposed to combine PROMETHE and PLTSs together. We use PLTS to deal with uncertain and fuzzy problems, and PROMETHE is applied to provide the ranking of alternatives. In addition, the proposed technique has been compared with other methods by case studies.

As the eight coastal regions Shanghai city, Shandong, Jiangsu, Zhejiang, Fujian, Guangdong, Guangxi, and Hainan provinces are often hit by meteorological disasters in China, they were selected as objectives. The study has shown that the proposed method is effective; Guangdong, Zhejiang, and Fujian provinces are the most vulnerable to meteorological disasters among the eight regions. For these provinces, more efforts should be made to deal with these disasters so that sustainable development can be implemented. Finally, the method is compared with similar approaches, and the results have denoted that the extended PROMETHE has strengths.