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

: 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.


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 1.
PROMETHEE is improved by PLTS so that it can be used to evaluate meteorological disaster risk in an uncertain and imprecision environment.

2.
The relative importance of the criteria is converted to a constraint optimization problem in meteorological disasters. The problem is solved by the evolutionary algorithm. 3.
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.

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

HFLTS
Let S = {s t |t = −τ, . . . , −1, 0, 1, . . . , τ} be a linguistic term set; x i ∈ X, i = 1, 2, . . . , N be fixed. A HFLTS can be described as follows: 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 φl (x i )|s φl (x i ) ∈ S, l = 1, . . . , L with L being the number of linguistic terms in h s (x i ) [13,27].

PLTSs
Let S = {s 0 , s 1 , . . . , s τ } be the linguistic term sets; a probabilistic linguistic term set (PLTS) is given as follows: 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].

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: 1.
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.

2.
Establish the criteria. Generally, alternatives are evaluated based on the established criteria. The decision-makers give their rating according to the performance of criteria.

3.
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.

4.
Evaluate performances of alternatives. Decision-makers are asked to make an objective evaluation of each alternative based on their professional background.

5.
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 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.
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: where w j is the weight of the jth criterion; P j (x i , x k ) is the preference function.

Compute the leaving and entering flow
The leaving and entering flow can be computed by the following two equations: where H(x i , x k ) is the preference index and obtained in the previous step. The leaving flow ϕ + (x i ) is the dominance of the ith alternative over other alternatives. The entering flow ϕ − (x i ) has the opposite meaning.

Calculate the net flow
The net flow is determined by the leaving flow and entering flow as follows: The higher the net flow is, the better the alternative is.

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.

Problem Description
In this section, we use multi-criteria group decision-making with probabilistic linguistic information to solve the above problem. Let x = {x 1 , x 2 , . . . , x 8 } be the set of eight areas, Let c Sustainability 2019, 11, 1405 6 of 13 = {c1,c2,c3,c4} be the criteria, in which c 1 is the crop disaster area, c 2 is the flood-hit population, c 3 is the death population, c 4 is the direct economic loss. The weight of the four criteria w = {w 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 . . , 4 denotes the value over the province x i with respect to the criterion j by decision-makers, in which L ij . When four decision-makers have given their evaluations, we combine results from four decision-makers together to form the matrix R d .
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.

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 . where is the subscript of linguistic term L (k) ; i = 1, 2, . . . , m; j = 1, 2, . . . , n; m = 8, n = 4.
Step 2: Normalize the matrix R .
where r * i = max j r ij ; r − i = min j r ij . Step 3: Motived by AHP theory, we can convert Equation (20) to Equation (22) when the matrix Equation (20) is completely consistent.
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: 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].

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.
where σ 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.

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= {s 0 = none, s 1 = very low, s 2 = low, s 3 = medium, s 4 = high, s 5 = very high}. The evaluation results are listed in Tables 1 and 2, in which c 1 is the crop disaster area, c 2 is the flood-hit population, c 3 is the death population, and c 4 is the direct economic loss. These evaluation results are aggregated to generate results in Table 3. Take the results {s 2 (0.75), s 3 (0.25)} as an example, three experts give s 2 , and one expert gives s 3 with respect to the criterion c1 in Shandong province. Therefore, three of four suggest s 2 and one of four deems s 3 .   Table 3. The aggregation results. Step 2 Weights of the criteria. Convert the Table 3 to establish Table 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 4  For example, the preference index of Shandong province and Shanghai city can be written as H (1, 2), which can be calculated as follows: 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, All the leaving and the entering flow can be obtained by similar steps. The results are demonstrated in Table 6. 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. 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.

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 (x 1 , x 2 , x 3 ). The criteria are as follows: (1) c 1 : Financial perspective; (2) c 2 : Customer satisfaction; (3) c 3 : Internal business process perspective; (4) c 4 : Learning and growth perspective. The final decision matrix is shown in Table 7.  Then, after calculating the preference function and preference function index, the leaving, entering and net flows are listed in Table 8. Therefore, the final ranking is A 1 > A 3 > A 2 . 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 (h 1 , h 2 , h 3 and h 4 ). Three main criteria are as follows: the environmental factor of medical and health service (c 1 ); personalized diagnosis and treatment optimization (c 2 ); and social resource allocation optimization under the pattern of wisdom medical and health services (c 3 ). The weights are respectively 0.2, 0.1, and 0.7. As a result, the probabilistic linguistic decision matrix is shown in Table 9. Table 9. The probabilistic linguistic decision matrix.  Table 10. 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.

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.