Multi-Attribute Decision-Making Method Based on Prospect Theory in Heterogeneous Information Environment and Its Application in Typhoon Disaster Assessment

International Journal of Computational Intelligence Systems Vol. 12(2), 2019, pp. 881–896 DOI: https://doi.org/10.2991/ijcis.d.190722.002; ISSN: 1875-6891; eISSN: 1875-6883 https://www.atlantis-press.com/journals/ijcis/

Ruipu Tan1,2, Wende Zhang3,*, Lehua Yang2, Shengqun Chen2

1School of Economics and Management, Fuzhou University, Room 206, Library, Qi Shan Campus of Fuzhou University, 2 Xue Yuan Road, University Town, Fuzhou, 350116, China 2College of Electronics and Information Science, Fujian Jiangxia University, Fuzhou, 350108, China 3Institute of Information Management, Fuzhou University, Fuzhou, 350116, China

ARTICLEINFO A BSTRACT Article History Aiming at the decision-making problem in heterogeneous information environment and considering the influence of decision Received 26 Apr 2019 makers’ psychological behavior on decision-making results, this paper proposes a multi-attribute decision-making method based Accepted 10 Jul 2019 on prospect theory in heterogeneous information environment. The heterogeneous information in this paper indicates that the decision attribute value is represented by various types of data forms, including exact number, interval number, linguistic term, Keywords intuitionistic fuzzy number, interval intuitionistic fuzzy number, neutrosophic numbers, and trapezoidal fuzzy neutrosophic Multi-attribute decision-making numbers, and so on. Firstly, the distance and similarity measure of various heterogeneous data are introduced, and the hetero- (MADM) geneous information attribute weights are obtained using the deviation maximization method. Then, the psychological expecta- Heterogeneous information tion value of each attribute given by the decision maker is used as a reference point, thereby calculating the gain and loss of each Prospect theory attribute value relative to the reference point, and establishing a gain matrix and a loss matrix. On this basis, the prospect theory Neutrosophic set is used to obtain the comprehensive prospect value of each alternative, so as to obtain the alternative ordering result and optimal Typhoon disaster assessment alternative. Finally, an illustrative example about typhoon disaster assessment is presented to show the feasibility and effectiveness of the proposed method, and the advantages of the proposed method are illustrated by comparison with other methods.

1. INTRODUCTION methods are based on a single type of data, and there are few studies on multiple data types. However, in some complicated situations, Multi-attribute decision-making is a problem in which a decision- the decision is faced with heterogeneous data. For example, in our maker evaluates a finite set of alternatives associated with mul- typhoon disaster assessment study, the representation of the assess- tiple attributes [1]. It has been widely applied in many practical ment information is diverse. The number of population death is problems, such as supplier selection [2,3], pattern recognition [4], usually accurately obtained, which is expressed as an exact num- medical diagnosis [5,6], emergency decision [7,8], water allocation ber (EN). The number of population affected can usually obtain its [9], disaster assessment [10–12], and so on. Therefore, the multi- approximate range, which can be expressed as an interval number. attribute decision-making method has become the focus of schol- Again, the severity of economic loss tends to be expressed in lin- ars and has been extensively studied. However, the decision-making guistic terms (LTS). If the assessment of traffic flow after disaster problem has become more and more complicated because of the can get the maximum possible range and possible fluctuation range, increasing amount of decision information and alternatives, and TrFNNs can be generally used. the inherent uncertainty and complexity of decision problems, and the fuzzy nature of human thinking [12]. Especially in some sud- Heterogeneity indicates that the type and nature of information or den, complicated situations, some decision information is often not data is different [19]. In view of the high risk, complexity, and ambi- accurately represented, often expressed as fuzzy, vague, hesitant, guity of typhoon disasters, evaluation information cannot always incomplete, indeterminate, and inconsistent. So the fuzzy set (FS) be expressed as ENs, and often has the characteristics of interval, [13], hesitant fuzzy set (HFS) [14], rough set [15], grey theory [16], ambiguity, and hesitation, and is more appropriately expressed as intuitionistic fuzzy set (IFS) [17], and neutrosophic set (NS) [18] interval numbers (IVNs), fuzzy numbers (FNs), and intuitionistic are used to model the decision information. Throughout the exist- fuzzy numbers (IFNs). In addition, due to the ambiguity of human ing research literature, most of the multi-attribute decision-making thinking, it is difficult to use the quantitative numerical representation of decision information in the decision-making process, and

Pdf_Folio:881 it is more inclined to use qualitative LTs to evaluate attributes. At *Corresponding author. E-mail: [email protected] the same time, typhoon disasters also have great uncertainty. Such 882 R. Tan et al. / International Journal of Computational Intelligence Systems 12(2) 881–896 evaluation information can be characterized as neutrosophic num- Sorting method is a research hotspot in decision-making bers (NNs), trapezoidal fuzzy neutrosophic numbers (TrFNNs), including method based on scoring function and exact function, and so on. So, the processing heterogeneous information is a key similarity measure method, VIKOR method, TOPSIS method, point in the decision-making process [1,19–21]. Sun et al.[1] pro- grey theory, TODIM method, and so on, and most of the methods posed a MADM with grey multi-source heterogeneous data includ- are based on the assumption that the decision maker is completely ing ENs, interval grey numbers, and extend grey numbers. Yu et al. rational. But some studies about behavioral experiments have [19] proposed a heterogeneous MADM considering four types of shown that the decision maker is bounded rational in decision data: ENs, IVNs, triangular fuzzy numbers (TFNs), and IFNs. Pan processes and his behavior plays an important role in decision et al.[20] proposed a hybrid MADM based on VIKOR with multi- analysis [41]. In the face of emergencies, especially after the disas- granularity LNs, ENs, IVNs. Wan et al.[21] proposed a MADM for ter, the psychological factors of decision makers play a key role in heterogeneous infromation including ENs, IVNs, TFNs, and trape- decision-making, and the research on this aspect is still rare. PT zoidal fuzzy numbers (TrFNs). Lourenzutti et al.[22] proposed a introduced by Kahneman et al.[42,43] provides a simple and clear generalized TOPSIS method for MADM with heterogeneous infor- computation process to describe the psychological behavior using mation including crisp numbers, IVNs, FNs, IFNs, and random reference points, losses, gains, and overall prospect values [44]. So vectors. Zhang et al.[23] proposed a risk-based MADM based on it has been widely applied in the various fields to solve the practical three heterogeneous data: ENs, IVNs, and TFNs. Fanet et al.[24] problems considering human being’s psychological behavior. Gao proposed a hybrid MADM based on cumulative prospect theory et al.[45] and Best et al.[46] applied PT to solve asset allocation. (PT) with heterogeneous data including ENs, IVNs, and LTs. Qi Attema et al.[47] studied the application of PT in the field of et al.[25] proposed a MADM based on heterogeneous data such as health domains. Tian et al.[48] studied the park-and-ride behavior multi-granularity language numbers, IFNs, and interval intuition- in a cumulative PT-based model. Van Tol et al.[49] studied the istic fuzzy numbers (IVIFNs) into IVIFNs to calculate the compre- application of cumulative PT in option valuation and portfolio hensive evaluation value of the alternative. However, looking at the management. Zou et al.[50] studied the optimal investment with existing research literature, we have not seen the study of heteroge- transaction costs under cumulative PT. Liu et al.[51] and Zhang neous data including the NN. On the other hand, in terms of prac- et al.[52] proposed emergency decision-making methods based tical applications, the assessment of post-typhoon disasters is a very on PT. Tao et al.[53] proposed the brain mechanism of economic complicated issue, and it is sometimes difficult for experts to make management risk decision based on PT. Ning et al.[54] studied the clear decisions. For example, we invited the expert group to assess disruption management strategy based on PT. Sullivan et al.[55] the severity of social impact, 30% vote “Yes,” 20% vote “No,” 10% examined the forest certification problem based on PT. Yu et al. give up, and 40% are undecided. Such a vote is beyond the scope of [56] studied the typhoon disaster emergency scheme generation IFS to distinguish the information between “giving up” and “unde- and dynamic adjustment based on CBR and PT. However, given cided.” Such a scenario indicates that the NN needs to be studied in the existing research, there is still little literature on the application depth. Therefore, we studies the MADMs in heterogeneous infor- of PT in typhoon disaster assessment. Taking into account the mation environment including the ENs, IVNs, LTs, IFNs, IVIFNs, uncertainty of typhoon disasters and the psychological feelings of NNs, and TrFNNs, and so on. decision makers, the PT can well consider the psychological factors of decision makers to the assessment of typhoon disasters, which NSs proposed by Smarandache [26] are a powerful tool to deal with can make disaster assessment more reasonable and effective. So, incomplete, indeterminate, and inconsistent information in the real this paper studies typhoon disaster assessment based on PT. world. And it is the generalization of the theory of FSs [13], interval- valued fuzzy sets, and IFSs [17], and so on. NSs are characterized Natural disasters often lead to death and enormous property dam- by a truth-membership degree (T), an indeterminacy-membership age. Various types of natural disasters occur in China every year. degree (I), and a falsity-membership degree (F), which can repre- And the typhoon is one of the biggest disasters facing humanity. Its sent more information than other FSs. In recent years, scholars have destructive power exceeds that of the earthquake, but it has never proposed various forms of NNs. For example, Wang et al.[27] intro- been avoided. Meteorological disasters such as typhoon accounted duced the concept of single-valued neutrosophic sets (SVNSs). Ye for more than 70% of the natural disasters [57]. In China, typhoons [28] introduced the simplified neutrosophic sets (SNSs). SVNS is primarily impact the eastern coastal regions of the country, where an extension of FN and IFN. Wang and Li [29] also defined multi- the population is extremely dense, the economy is highly devel- valued neutrosophic sets (MVNSs). Wang et al.[30] proposed oped, and social wealth is notably concentrated. Once the typhoon interval neutrosophic sets (INSs). Yang and Pang [31] defined comes, it will cause huge property and economic losses, casualties, multi-valued interval neutrosophic sets (MVINSs). Deli et al.[32] and environmental damage to this area. Therefore, typhoon dis- proposed the concept of the bipolar NSs, and applied it to MADM. aster assessment is a very important issue. However, the influenc- Deli et al. proposed neutrosophic refined sets [33] and bipolar ing factors of the typhoon disasters are completely hard to describe neutrosophic refined sets [34] and applied them to medical diag- accurately. Taking economic loss, for example, it includes many nosis. Tian et al.[35] proposed the concept of the simplified neu- aspects such as the building’s collapse, the number and extent of trosophic linguistic and applied it to MADM. Broumi et al.[36–38] damage to housing, and the affected local economic conditions and Tan et al.[39] combined the NSs and graph theory to pro- [10], the degree of environmental damage, and the negative impact pose neutrosophic graphs, and used for the shortest path solving on society. The evaluation information is often hesitant, ambigu- problem. Ye [40] proposed trapezoidal fuzzy NSs and applied them ous, incomplete, inconsistent, indeterminate, and so on. Therefore, to MADM. There are many forms of NN, which are suitable for FSs, HFSs, and IFSs have been used for typhoon disaster assess- different decision-making environments. The heterogeneous data ment in recent years. For example, Shi et al.[57] proposed a fuzzy of typhoon disaster assessment we studied include single-valued MADM hybrid approach to evaluate the damage level of typhoon neutrosophicPdf_Folio:882 numbers (SVNNs) and TrFNN. based on FNs. He et al.[11] proposed a typhoon disaster assessment R. Tan et al. / International Journal of Computational Intelligence Systems 12(2) 881–896 883 method based on Dombi HFNs. Li et al.[58] and Yu [10] proposed For simplicity of calculation, a TrFNN ñ can be expressed as nTr = evaluating typhoon disasters method based on IFNs. Throughout ⟨(a1, a2, a3, a4) , (b1, b2, b3, b4) , (c1, c2, c3, c4)⟩, the parameters can the existing literature, most of the typhoon disaster assessment satisfy the following relationship: a1 ⩽ a2 ⩽ a3 ⩽ a4, b1 ⩽ b2 ⩽ studies are based on a single data form, and have not yet been con- b3 ⩽ b4, and c1 ⩽ c2 ⩽ c3 ⩽ c4. When a2 = a3, b2 = b3, c2 = ducted under heterogeneous data environment containing the NNs. c3, the TrFNN becomes triangle fuzzy neutrosophic number (TFN), Webelieve that NS will be a powerful theory and method to enhance and TFN number is a special case of TrFNN. typhoon disaster assessment. So we study the MADM based on PT in heterogeneous information environment and apply it to the typhoon disaster assessment. First we use the deviation maximiza- 2.2. IFNs, IVIFNs, and Interval Numbers tion method based on heterogeneous information to get the evalua- Definition 3. [17] Let a set X be fixed. An IFS A in X is an object tion index weights, then use the extended PT to calculate the overall having the form: prospect value of each alternative to sort the alternatives in a heterogeneous information environment. A = {⟨x, 휇A (x) , 휈A (x)⟩ |x ∈ X} . (3) This paper is organized as the following: Section 2 briefly presents some basic definitions of heterogeneous information. Section 3 where the functions 휇A ∶ X → [0, 1] and 휈A ∶ X → [0, 1] define gives the distance formula and similarity measure of various the degree of membership and the degree of nonmembership of the heterogeneous data, a MADM method based on the PT, and the element x ∈ X to A, respectively, and for every x ∈ X ∶ 0 ⩽ 휇A (x)+ compatibility and consistency measures for heterogeneous infor- 휈A (x) ⩽ 1. For each IFS A in X, if 휋A (x) = 1 – 휇A (x) – 휈A (x), for mation. Section 4 uses a typhoon disaster evaluation example to all x ∈ X, then 휋A (x) is called the degree of indeterminacy of x to illustrate the applicability of the proposed method and verifies A. For convenience, we can use 훼I = ⟨휇, 휈⟩ to represent an IFN. the advantages of the proposed method by comparative analysis. Definition 4. [59] An interval-valued intuitionistic fuzzy set (IVIFS) Finally, Section 5 gives the conclusions. Ã in the set X is an object having the following form:

Ã = {⟨x, ̃휇 ̃ ̃휈 ̃ ⟩ |x ∈ X} (4) 2. PRELIMINARIES A A = {⟨x, [inf ̃휇Ã (x) , sup ̃휇Ã (x)] , ⟩ ⟨[inf ̃휈 ̃ (x) , sup ̃휈 ̃ (x)] , |x ∈ X⟩} . In this section, we briefly introduce the basic concepts of various A A heterogeneous data used in this paper, including the NNs, IFNs, FNs, ENs, and so on. At the same time, PT will be briefly reviewed where ̃휇Ã (x) ⊂ [0, 1] and Ã휈 ̃ (x) ⊂ [0, 1] are called the degree of so that unfamiliar readers can understand our proposed method membership and the degree of nonmembership of the element x ∈ ̃ easily. X to A, respectively, and with the condition: sup ̃휇Ã (x)+sup Ã휈 ̃ (x) ⩽ 1. And ̃휋Ã (x) = [inf휋Ã (x) , sup휋Ã (x)] is called the degree of indeterminacy of x to Ã, and inf휋Ã (x) = 1 – sup휇Ã (x) – sup휈Ã (x), u ( ) ( ) ( ) 2.1. SVNN, Trapezoidal Fuzzy NN sup휋Ã x = 1 – inf휇Ã x – inf휈Ã x . Iv l u l u Definition 1. [27] Let X be a universal set. A SVNS A in X is char- For convenience, we can use 훼 = ⟨[휇 , 휇 ] , [휈 , 휈 ]⟩ to represent an interval-valued intuitionistic fuzzy number (IvIFN). If 휈l = 휈u = acterized by a truth-membership function TA (x), an indeterminacy- ( ) ( ) 0, then the IVIFN becomes a general interval number (IN): IN = membership function IA x and a falsity-membership function FA x . l u Then, a SVNS A can be denoted by [휇 , 휇 ].

A = {⟨x, TA (x) , IA (x) , FA (x)⟩ |x ∈ X} . (1) 2.3. Language Term

휏–1 휏–1 Definition 5. [60,61]: Let S = {s휃|휃 = – 2 ,⋯,–1,0,1,⋯ 2 } be where TA(x), IA(x), FA(x) ∈ [0, 1] for each x in X. Then, the sum of a finite and totally ordered discrete term set, where 휏 is odd value and ( ) ( ) ( ) ( ) ( ) TA x , IA x and FA x satisfies the condition 0 ⩽ TA x + IA x + s휃 represents a possible value for a linguistic variable. For example, s FA(x) ⩽ 3. For convenience, we can use n = ⟨T, I, F⟩ to represent a when l = 9, a set S could be given as follows: SVNN. S = {s , s , s , s , s , s , s , s , s } Definition 2. [40] Let X be a universe of discourse, then a trape- –4 –3 –2 –1 0 1 2 3 4 = {extremely poor, very poor, poor, slightly zoidal fuzzy NS Ñ in X has the following form: poor, fair, slightly good, good, very good, extremely good} . Ñ = {⟨x, TÑ (x) , IÑ (x) , FÑ (x)⟩ |x ∈ X} , (2) In these cases, it is usually required that there exist the following: where TÑ (x) ⊂ [0, 1](, IÑ (x) ⊂ [0, 1] and FÑ (x)) ⊂ [0, 1] are 1 2 3 4 three TrFNs, T ̃ (x) = t (x) , t (x) , t (x) , t (x) ∶ X → [0, 1], 1. A negation operator: Neg (si) = s–i. ( N Ñ Ñ ) Ñ Ñ 1 2 3 4 ⩽ ⩽ I ̃ (x) = i (x) , i (x) , i (x) , i (x) ∶ X → [0, 1], and F ̃ (x) = 2. The set is ordered: si sj if and only if i j. (N Ñ Ñ Ñ ) Ñ N ( ) f 1 (x) , f 2 (x) , f 3 (x) , f 4 (x) ∶ X → [0, 1] with the condition 3. Maximum operator: max si, sj = si, if i ⩾ j. Ñ Ñ Ñ Ñ ( ) 4 4 4

0Pdf_Folio:883 ⩽ t (x) + i (x) + f (x) ⩽ 3, x ∈ X. Ñ Ñ Ñ 4. Minimum operator: min si, sj = si, if i ⩽ j. 884 R. Tan et al. / International Journal of Computational Intelligence Systems 12(2) 881–896

In order to preserve all the given information, extending the discrete 3. DECISION-MAKING METHOD BASED ON term set S to a continuous term set S = {s휃|휃 ∈ [0, q]}, where, if HETEROGENEOUS INFORMATION AND s휃 ∈ S, then we call s휃 the original term, otherwise, we call s휃 the PT virtual term. In general, the decision maker uses the original LNs to evaluate alternatives, and the virtual LNs can only appear in the 3.1. Distance Measure of Heterogeneous actual calculation. Information The so-called different granularity language information refers to the preference information given by the decision makers in the The basic idea of PT is to first use the expectation of the decision group decision-making according to the language evaluation set maker as a reference point, and establish the risk return matrix and represented by the granularity of the number of different language the risk loss matrix, respectively, by calculating the gain and loss of q = q 휃 = q–1 ,⋯, 1,0,1,⋯ q–1 the program attribute value relative to the reference point, and then phrases. Let S {s휃| – 2 – 2 } be an arbitrary language evaluation set, and its granularity is q. If the value of q is to establish the prospect decision matrix. On this basis, the alter- different in the same decision process, it becomes a multi-granular natives are ranked by calculating the comprehensive prospect val- language decision. ues of each alternative. In the disaster assessment, after the typhoon disaster occurs, the decision makers will form a certain psychological expectation according to the report of the relevant personnel 2.4. Prospect Theory and their own situation, also known as the psychological reference point. At the same time, the degree of disaster of each assessment PT is a theory that describes and predicts behaviors that are object is compared with the psychological reference point of the inconsistent with traditional expectations theory and expected util- decision maker to judge the extent to which the decision maker’s ity theory in the face of risk decision-making. The theory finds that psychological perception is “gain” or “loss.” Among them, the cal- people’s risk preference behaviors are inconsistent in the face of culation of the gain and loss mainly involves the distance calcula- gains and losses, and they become risk-seeking in the face of “miss- tion of various heterogeneous information. ing,” but they are risk-averse in the face of “getting.” The establish- Let x represents the expert’s evaluation of the alternative i under ment and change of reference points affect people’s feelings of gain ij the attribute j, r represents the reference point of attribute j, and and loss, and thus influence people’s decision-making. ( ) j Dij xij, rj denote the distance between the attribute value and the In MADM problems, the attributes can be classified into two types: reference point. The attribute values of different attributes are not benefits and costs. The higher a benefit attribute is, the better the the same type of data, but heterogeneous information, including situation is, while the higher a cost attribute is, the worse the situ- ENs, IVNs, LTs, IFNs, IVIFNs, NNs, and TrFNNs, and so on, and ation is. According to different types of attributes, reference points their distance calculation formulas are as follows: changes with people’s expectations with respect to the predefined ( ) amounts to gain or lose [62]. For example, if there is a possibility 1. When the attribute value is an EN, its distance Dij xij, rj from to lose some money and predefined amounts are USD 5, 10 and 20, the reference point is calculated as follows [65]: then assuming that 10 is an acceptable loss amount to an individual (reference point of possible losses), if the final outcome is 5, he/she ( ) ( ) D x , r = 1 – Sim x , r (6) feels gains because the final losses are lower than his/her expecta- ij ij j ij ij j tion. Benefit attributes can be assessed in a similar way [52]. ( ) –|xij – rj| , = [ ] , Where Simij xij rj exp max min Gains and losses are determined by the reference point and the final dj – dj outcome with respect to different types of attributes. For measuring max i ∈ m, j ∈ n, and dj = {rj, max {xij|j ∈ n}} , the magnitude of gains and losses, an S-shaped value function is min provided in PT [42]. The value function is expressed in the form of dj = {rj, min {xij|j ∈ n}} , i ∈ m. a power law [42]:

2. When the attribute value is an IVN, that is x = [xl , xu], r = x훼, x ⩾ 0; ( ) ij ij ij j v (x) = { 훽 (5) l u –휆 (–x) , x < 0, [rj, rj ], then its distance Dij xij, rj from the reference point is calculated as follows: where x denotes the gains when x ⩾ 0, and x denotes the losses ( ) ( ) , = 1 , when x < 0, respectively; 훼 and 훽 are power parameters related Dij xij rj ( – Sim) ij xij rj (7) to gains and losses, respectively; and 0 ⩽ 훼 ⩽ 1, 0 ⩽ 훽 ⩽ 1; 휆 where Simij xij, rj ( ) ( ) is the risk aversion parameter, which has a characteristic of being ⎡ 2 2 ⎤ l l + u u steeper for losses than for gains, and 휆 > 1. Here, the values of 훼, 훽, ⎢ √ xij – rj xij – rj ⎥ and 휆 are determined through experiments [43,63,64]. For example, = exp ⎢– ⎥ , ⎢ ( )2 ( )2 ⎥ Abdellaoui et al.[63] suggest 훼 = 0.725, 훽 = 0.717, and 휆 = 2.04, ⎢ l l u u ⎥ maxi { x – r + x – r } Liu et al.[64] suggest 훼 = 0.850, 훽 = 0.850, and 휆 = 4.1, and ⎣ √ ij j ij j ⎦ Tversky et al.[43] suggest 훼 = 0.89, 훽 = 0.92, 휆 = 2.25. i ∈ m, j ∈ n.

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3. When the attribute value is an IFN, that is, xij = ⟨휇ij, 휈ij⟩ , rj = 5. When the attribute value is an IVIFN, that is, xij = l u l u l u l u ⟨휇j, 휈j⟩, the distance formula based on cosine similarity [66] is ⟨[휇(ij, 휇ij]), [휈ij, 휈ij ]⟩, rj = ⟨[휇j, 휇j ] , [휈j , 휈j ]⟩, then its distance as follows: Dij xij, rj from the reference point is calculated as follows: ( ) ( ) ( ) d z – t (9) ij i ( ) ( ) ( ) Dij xij, rj = 1 – Simij xij, rj , (8) √ 2 2 2 ( ) √ 1/6 [ 휇l – 휇l + 휇u – 휇u + 휈l – 휈l 휇i휇ij + 휈i휈ij √ ( i )ij ( i )ij ( i ij) Sim x , r = , = 2 2 2 ij ij j + 휈u – 휈u + 휋l – 휋l + 휋u – 휋u ] . 2 2 2 2 √ i ij i ij i ij √휇i + 휈i √휇ij + 휈ij i ∈ m, j ∈ n.

6. When the attribute value is a NN, that is, xij = ⟨Tij, Iij, Fij⟩, ij r = ⟨T , I , F ⟩, we use the distance measure formula proposed = j j j j 4. When the attribute value is a language term, that is, xij s휃, j ij q by Ye [67] as follows: r = s , and s ,si ∈ sq = {s |휃 = – q–1 ,⋯,–1,0,1,⋯ q–1 }, it j 휃 휃 휃 휃 2 2 ( ) will first be converted into IVIFN [25], then use the distance Dij xij, rj (10) formula of IVIFNs. When the value of q is different, it means ( ) 2 2 2 evaluation information is expressed as multi-granularity lin- = √1/3 |Tij – Ti| + |Iij – Ii| + |Fij – Fi| . guistic. However, our paper only considers one granularity, because different granularity linguistic can be converted to 7. When the attribute value is a TrFNN, that is, IVIFN by a formula [25], the conversion formula is as follows: ij ij ij ij ij ij ij ij ij ij ij ij xij = ⟨(a1, a2, a3, a4), (b1, b2, b3, b4), (c1, c2, c3, c4)⟩, ( ) j j j j j j j j j j j j tj = ⟨(a1, a2, a3, a4), (b1, b2, b3, b4), (c1, c2, c3, c4)⟩, we use the q q q q q q 휇 = 휇 q–1 , 휇 q–1 , ⋯ , 휇0, ⋯ , 휇 q–1 , 휇 q–1 distance formula based on the cosine similarity measure [36] – – +1 –1 ( 2 2 2 2 proposed by Tan et al.[68] as follows: ( ) q+1 ( ) q+1 –1 q 2 q 2 q ( ) ( ) = 휇0 , 휇0 , ⋯ , 휇0, ⋯ , ( ) ) D xij, rj = 1 – STrFNN xij, rj ( ) q+1 ( ) q+1 4 ij 4 ij 4 ij q 1/ –1 q 1/ i i i 2 2 k – ∑ =1 ahah + ∑ =1 bhbh + ∑ =1 chch 휇0 , 휇0 , = h h h . ( ( k ) ( )2 ( )2 ( )2 = k = ∑4 aij + ∑4 bij + ∑4 cij √ h=1 h h=1 h h=1 h ( ) ( )) q q q q q q ( )2 ( )2 ( )2 휈 = 휈 q–1 , 휈 q–1 , ⋯ , 휈0, ⋯ , 휈q–1 , 휈q–1 4 i 4 i 4 i – – +1 –1 × ∑ a + ∑ b + ∑ c ( 2 2 2 2 √ h=1 h h=1 h h=1 h ( ) q+1 ( ) q+1 –1 i ∈ N, j ∈ M, h = 1, 2, 3, 4. = 1 – 1 – 휈q 2 , 1 – 1 – 휈q 2 , ⋯ , 휈q, 0 0 0 (11) ( ) ) ( ) q+1 ( ) q+1 1/ –1 1/ ⋯ , 1 – 1 – 휈q 2 , 1 – 1 – 휈q 2 , 0 0 3.2. Determination of Attribute Weights in Heterogeneous Information where, Environment 1 휇q = 휈q = [0.5 – , 0.5] , For the objective of decision-making, this paper uses the method 0 0 2q of maximizing deviations to calculate the attribute weights. Due ( ) q+1 ( ) q+1 q+1 to the complexity of objective things and the ambiguity of human q 1 2 2 2 휇0 = [ 0.5 – , 0.5 ] , thinking, it is often difficult to give explicit attribute weights. Some- 2q times there is an extreme situation where the weight is completely ( ) q+1 unknown. In the multi-attribute decision-making (MADM) of the ( ) q+1 1/ q+1 q 1/ 1 2 1/ 휇 2 = [ 0.5 – , 0.5 2 ] , finite alternatives, the smaller the difference of the alternative under 0 2q the attribute cj, the smaller the effect of the attribute on the deci- ( ) q+1 sion of the alternative, otherwise it plays an important role in ( ) q+1 q+1 q 1 2 2 2 the ranking and selection of the alternatives. Therefore, the devi- 1 – 1 – 휈0 = [1 – 0.5 + , 1 – 0.5 ] , 2q ation of the attribute value of the alternative is larger. The greater ( ) q+1 the attribute weight should be given. So, the study of determin- ( ) q+1 1/ 1/ 2 q 2 1 ing attribute weight by maximizing deviation draws the attention 1 – 1 – 휈0 = [1 – 0.5 + , 2q of scholars [69,70]. In addition, Xu and Da [71] and Xu [72] put q+1 forward some similar methods, namely maximizing standard devi- 1/ 1 – 0.5 2 ] . ation method and maximizing mean deviation method, to determine attribute weights for different data forms. Based on the idea of

Pdf_Folio:885 maximizing deviations in literature [69], this paper determines the 886 R. Tan et al. / International Journal of Computational Intelligence Systems 12(2) 881–896 attribute weights in the heterogeneous information environment. 3.3. MADM Method Based on The specific formula is as follows: Heterogeneous Information and PT The weight vector 휔 can be obtained by solving the following opti- 3.3.1. Problem description and method steps mization models: This section introduces a novel MADM method based on heterogeneous information and PT that considers decision makers’ psy- n m m ( ) chological behavior and deals with different types of assessment ⎧max d (휔) = ∑ ∑ ∑ 휔 D x – x ⎪ j ij kj j=1 i=1 k=1 information. To achieve our objectives, the proposed method n (12) consists of the following several phases, depicted graphically in ⎨ s.t. ∑ 휔2 = 1, 휔2 ⩾ 0, i = 1, 2, ⋯ , m, ⎪ j=1 j j Figure 1. ⎩ j = 1, 2, ⋯ , n, j ≠ k, Consider a MADM problem, let X = {x1, x2, ⋯ , xm} be a discrete set of m alternatives, and C = {c , c , ⋯ , c } be the set of ( ) 1 2 (n ) n attributes. 휔j is the weight of the attribute cj j = 1, 2, ⋯ , n , where D xij – xkj represents the attribute value difference of any ( ) where 휔 ∈ [0, 1] j = 1, 2, ⋯ , n , and ∑n 휔 = 1. Suppose that two alternatives under the same attribute, and the distance calcu- ( j) j=1 j lation of the heterogeneous attribute information uses the distance R = x is the decision matrix, where x represents a differ- ij m×n ij measure of Section 3.1. Then, solve and normalize the above opti- ent form of data. Due to the high risk, complexity, and uncertainty mization models to get the attribute weights as follows: of typhoon disasters, decision makers have vagueness, uncertainty, and heterogeneity in the description of each attribute. In this paper, attribute information is represented by heterogeneous information m m ( ) such as ENs, IVNs, LTs, IFNs, IVIFNs, NNs, and TrFNNs, and the ∑ ∑ D x – x ij kj same attribute value is the same information form. For convenience, i=1 k=1 휔 = , j = 1, 2, ⋯ , n. (13) the attribute set is denoted as C = CEN ∪CIVN ∪CIFN ∪CLT ∪CIVIFN ∪ j n m m ( ) ( 1 2 3 )4 5 TrFNN ∪ NN ∩ = ∅, , = 1, 2, ⋯ , 7; ≠ EN ∑ ∑ ∑ D xij – xkj C6 C7 , and Ci Cj i j i j . Where C1 j=1 i=1 k=1 IVN denotes that the attribute value is expressed as a set of ENs, C2 denotes that the attribute value is expressed as a set of interval num- IFN bers, and C3 denotes that the attribute value is expressed as a set

Problem Description Calculation of gains Calculation of and losses prospect values

αβ Typhoon Disaster vG=+−()⎡⎤λ () L Gains ij ij⎣⎦⎢⎥ ij

Losses

... Calculation of comprehensive prospect values

n IFN C LT Evaluation object 1 C 4 C IVIFN Heterogeneous data U ων 3 5 ijij=∑ conversion and j=1 C IVN C TrFNN ... 2 6 normalization C NN 7 Evaluation object m Acquisition of the prospect values of Information Collection alternatives

Sorting of the Provide decision support to evaluation objects relevant departments

Figure 1 Framework of the proposed multi-attribute decision making (MADM) based on heterogeneous information and prospect theory.

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LT of IFNs, C4 denotes that the attribute value is expressed as a set of gain( and loss) in the value function. The positive ideal distance LNs, CIVIFN denotes that the attribute value is expressed as a set of + 5 Dij xij, rj represents the distance of the attribute value xij from TrFNN IVIFNs, C denotes that the attribute value is expressed as a set + 6 the positive ideal reference point r , representing the loss area. of TrFNNs, and CNN denotes that the attribute value is expressed as j 7 And the less the value, the more( serious) the assessment object is a set of the NNs. The decision-making method steps based on PT , – in a heterogeneous information environment are as follows: affected. The ideal distance Dij xij rj represents the distance of the attribute value x from the negative ideal reference point r–, Step 1: Collect heterogeneous data and get the evaluation matrix ij j which represents the income area. And the larger the value, the R = (x ) , then convert LNs into data information to quantify ij m×n larger the assessment object is affected. calculations. ( ) ′ ′ Step 5: Calculation of prospect values. According to PT, the mag- Step 2: Get the normalized evaluation matrix R = xij . m×n (nitude) of gains and losses is measured by value function. Let V = To eliminate the differences between dimensions, the attributes v be the prospect value matrix, it can be obtained by using ij m×n are normalized to ensure their consistency. In this paper, we only Equation (15) based on GM and LM, that is, need to normalize the ENs and interval numbers, the formula is as follows: ( )훼 ( )훽 vij = Gij + [–휆 Lij ] , i = 1, 2, ⋯ , m; (15) [xl /xu , xu/xu ] ifc ∈ CIVN x′ = { ij max j ij max j i 2 (14) j = 1, 2, ⋯ , n. ij EN xij/xmax j ifci ∈ C1 ,

u The prospect values indicate the earned value of each evaluation where xmax j = max {xij|i = 1, 2, ⋯ , n} and xmax j = max{xij|i = 1, 2, ⋯ , n}. object for each evaluation attribute based on the decision-maker’s psychological reference point. In the assessment of the typhoon dis- Step 3: Determine the reference point. Reference point represents aster, they indicate the severity of the evaluation objects in a certain the psychological expectations of decision makers for certain eval- aspect. uation attributes after typhoon disasters. In order to eliminate the subjective influence of the decision maker as much as possible, Step 6: Calculate the weights and calculate the attribute weights of this paper uses the positive and negative points of each attribute the heterogeneous information using Equation (13). as the reference point. Positive and negative ideal solutions rep- Step 7: The comprehensive prospect value Ui of each alternative is resent the decision makers’ maximum and minimum prediction calculated, and all alternatives are ranked according to the magni- of the disaster extent of the assessed objects. So, for c ∈ CEN, ( ) ( ) j 1 tude of the Ui value. The larger the Ui value, the higher the order of EN+ EN EN– EN the alternative, where U = ∑n 휔 v . rj = max xij , and rj = min xij , i = 1, 2, ⋯ , m. For i j=1 j ij IVN IVN+ IVN(l) IVN(u) IVN– cj ∈ C2 , rj = [ max xij , max xij ], and rj = 3.3.2. The compatibility and consistency of [ min xIVN(l), min xIVN(u)], i = 1, 2, ⋯ , m. For c ∈ CIFN, rIFN+ = ij ij j 3 j heterogeneous information IFN IFN IFN– IFN IFN ⟨max 휇ij , min 휈ij ⟩ , and r = ⟨min 휇ij , max 휈ij ⟩ , i = j Measuring the level of consistency of different opinions of experts 1, 2, ⋯ , m. For c ∈ CLT, because it is converted to an IVIFN j 4 in group decision-making is a key issue. The consistency prob- to participate in the calculation, so c ∈ CIVIFN, rIVIFN+ = j 5 j lem mainly includes two types: the consistency of the judgment IVIFN(l) IVIFN(u) IVIFN(l) IVIFN(u) ⟨[ max 휇j , max 휇j ], [min 휈ij , min 휈ij ] , matrix given by each expert and the consistency of the preference IVIFN– IVIFN(l) IVIFN(u) relationship between experts. Related to our research is the lat- and rj = ⟨[min 휇j , min 휇j ] , ter, which is the compatibility and consistency of information in [max 휈IVIFN(l), max 휈IVIFN(u)] , i = 1, 2, ⋯ , m. For c ∈ ij ij ( j ) group decision-making. When a group of decision-makers’ pref- CTrFNN, rTrFNN+ = ⟨ max aij, max aij, max aij, max aij , erence relationships are similar, there is consistency among deci- (6 j ) 1 2 3 4 sion makers. In the absence of basic compatibility, unsatisfactory min bij, min bij, min bij, min bij , 1 2 3 4 or incorrect results may result in group decision problems, and the ij ij ij ij evaluation information needs to be adjusted. The research on the (min c1, min c2, min c3, min c4), and rTrFNN– = ⟨(min aij, min aij, min aij, min aij ), compatibility and consistency of information including uncertain (j ) 1 2 3 4 information has yielded rich results [73–75], but there are not many max bij, max bij, max bij, max bij , ( 1 2 3 4) related researches on heterogeneous information, especially includ- ij ij ij ij ing NNs. We have been inspired by the literature [73] to propose max c , max c , max c , max c , i = 1, 2, ⋯ , m. For 1 2 3 4 the judgment method and adjustment strategy of compatibility and NN NN+ NN NN NN NN– C7 , rj = ⟨max Tij , min Iij , min Fij ⟩ , and rj = consistency for heterogeneous information are as follows: NN NN NN ⟨min Tij , max Iij , max Fij ⟩ , i = 1, 2, ⋯ , m. Step 1:( We) first convert the multiple evaluation matrices k k R = xij (k = 1, 2, ⋯ , e) given by the e experts into Step 4: Calculate the gains and losses. Based on the gain value and m×n the loss value, a gain matrix GM and a loss matrix( )LM are estab- unified representation, that is, the heterogeneous data eval- – uation matrix is uniformly converted into TrFNNs matrices lished. Where GM = [Gij]m×n, Gij = Dij xij, r , and LM = ( ) j ′ ′ + R k = x k (k = 1, 2, ⋯ , e) based on the relationship between [Lij]m×n, Lij = Dij(xij, r ). The distance between each attribute value ij Pdf_Folio:887 j m×n and the positive and negative ideal points represents the area of heterogeneous data and the conversion method proposed in [21]. 888 R. Tan et al. / International Journal of Computational Intelligence Systems 12(2) 881–896

Step 2: We use the TNNWAA operator to assemble the group impact (c7). Several experts and crawler software are responsible for this assessment, and the evaluation information is expressed by (evaluation) matrices into a comprehensive evaluation matrix R = heterogeneous information. Here, the population death can be xij . m×n accurately obtained through the lower-level government departments, which can be expressed by the ENs, that is, c ∈ CEN; and ′ 1 1 Step 3: We calculate the compatibility index SI(R k, R)(k = population affected can estimate the upper and lower limits, which ′ IVN k can be expressed by the IVNs, that is, c ∈ C ; and agricultural 1, 2, ⋯ , e) for each evaluation( matrix) R and comprehensive eval- 2 2 ( ) damage has fuzziness and hesitation, which can be expressed by ′ 1 ′ k m n k IFN uation matrix R. Here SI R , R = ∑ ∑ c xij , xij IFNs, that is, c ∈ C ; and for economic loss, the evaluation expert ( ) m×n i=1 j=1 3 4 ′ tends to use qualitative language information to evaluate its sever- and c xij, xij is the compatibility measure of two TrFNNs, and LT ( ) ( ) ity, which can be expressed in LTs, that is, c4 ∈ C3 ; and environ- ′ , = ′ , mental impact has ambiguity and interval, which can be expressed c xij xij D xij xij . IVIFN by the number of IVIFNs, that is, c5 ∈ C5 ; and social impact Step 4: We predetermine the critical value SI, if SIk < SI, then a can be assessed by experts in the range of maximum possible values satisfactory consistency is obtained, otherwise the corresponding and possible fluctuations, which can be expressed by TrFNNs, that ∈ TrFNN evaluation matrix is adjusted. is, c6 C6 ; and other impact factors are wide and unclear, and there is great uncertainty in expert evaluation. Some experts may Step 5: If there is an expert evaluation( matrix) Rk0 that satisfies vote for support and vote against it at the same time, the sum of the k ′ k SI 0 > SI, then the compatibility c xij 0 , xij of each pair of ele- ratios is greater than 1, exceeding IFSs represent ranges, which can NN be expressed as NNs, that is, c7 ∈ C . So the assessment matrix ′ k k ( ) 7 ments( xij 0)in R 0 and xij in R needs to be calculated separately. If R = x are given in the form of heterogeneous information ij m×n ′ k ′ k c xij 0 , xij does not meet the critical value SI, the elements xij 0 are including ENs, IVNs, LTs, IFNs, IVIFNs, NNs, and TrFNNs (see ′ k Table 1). Here LN sets are expressed as adjusted accordingly, and the adjustment strategy is to replace xij 0 S = {s , s , s , s , s , s , s } = {Absolutely low, Low, Fairly low, with xij in the corresponding comprehensive evaluation matrix, or –3 –2 –1 0 1 2 3 Medium, Fairly high, High, Absolutely high}. So the evaluation data to reevaluate the k0 expert matrix. is shown in Table 1: In particular, we only give compatibility and consistency judgment methods and adjustment strategies for heterogeneous information, Step 1: For the convenience of calculation, the language term is con- and our case study section does not make specific calculations. verted into IVIFN according to the literature [25], and the trans- formed data is shown in Table 2. Step 2: We use Equation (14) to get the normalized evaluation ′ 4. CASE STUDY AND COMPARISON WITH matrix R as shown in Table 3. EN+ OTHER APPROACHES Step 3: Determine the reference point. For c1, r1 = 1.000 and EN– IVN+ r1 = 0.200. For c2, r2 = [0.500, 1.000] and IVN– IFN+ IFN– 4.1. Case Study r2 = [0.001, 0.095]. For c3, r3 = ⟨0.90, 0.05⟩ and r3 = ⟨0.05, 0.90⟩ . For c , r LT+ = ⟨[0.8091, 0.8409] , [0.1344, 0.1591]⟩ With the rapid development of China’s economy and the growing 4 4 and r LT– = ⟨[0.0337, 0.0625], [0.9007, 0.9375]⟩. For c , development of coastal cities, the losses caused by typhoon dis- 4 5 IVIFN+ = 0 8500, 0 9000 , 0 0500, 0 1000 IVIFN– = asters have become increasingly serious. Take Fujian Province as r5 ⟨[ . . ] [ . . ]⟩ and r5 TrFNN TrFNN+ an example. It is located in the southeast coastal area of China. ⟨[0.1000, 0.3000] , [0.5500, 0.6000]⟩. For C6 , r6 = It is close to the typhoon source and is on the path of typhoon ⟨(0.7, 0.8, 0.9, 1.0) , (0.0, 0.1, 0.2, 0.3) , (0.0, 0.1, 0.2, 0.3)⟩ and TrFNN– movement. Typhoon is often landed. When the typhoon landed, r6 = ⟨(0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0)⟩. NN NN+ NN– it brought disasters such as storms and storm surges. Therefore, For C7 ,r7 = ⟨1.00, 0.00, 0.00⟩ and r7 = ⟨0.10, 0.66, 0.85⟩. this paper evaluates the typhoon disasters in Fujian Province. Tak- ing a strong typhoon “Maria” in Fujian Province in July 2018 as an example, “Maria” landed on the coast of Lianjiang River at 9:10 on November 11. When landing, the maximum wind force Table 1 Evaluation Matrix R. near the center was 14 levels (42/sec, strong typhoon level), and Index the lowest central pressure was 960 hPa. It was the strongest Cities c1 c2 c3 c4 typhoon landing in Fujian since July with meteorological records. Nanping (NP) 3 [2,100–200,000] ⟨0.05, 0.90⟩ s It was strong, fast moving, and destructive. After the disaster, we –3 Ningde (ND) 10 [65,000–1,900,000] ⟨0.80, 0.15⟩ s quickly obtains heterogeneous data from multiple sources for disas- 3 Sanming (SM) 2 [2,000–190,000] ⟨0.05, 0.90⟩ s–3 ter assessment, which provides decision support for disaster relief in Fuzhou (FZ) 7 [180,000–600,000] ⟨0.50, 0.45⟩ s2 relevant departments. This paper synthesizes the literature [56] and Putian (PT) 5 [1,000,000–2,000,000] ⟨0.80, 0.15⟩ s1 [10] to construct an assessment indicator system. The assessment Longyan (LY) 3 [5,000–300,000] ⟨0.15, 0.80⟩ s0 indicators C = {c1, c2, c3, c4, c5, c6, c7} include population death Quanzhou (QZ) 4 [18,000–500,000] ⟨0.05, 0.90⟩ s–1 0 30, 0 65 (c1), population affected (c2), agricultural damage (c3), economic Xiamen (XM) 3 [2,300–190,000] ⟨ . . ⟩ s–2 Zhangzhou (ZZ) 6 [500,000–2,000,000] ⟨0.90, 0.05⟩ s loss (c4), environmental impact (c5), social impact (c6), and other 2

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Index Cities c5 c6 c7 Nanping (NP) ⟨[0.10, 0.30] , [0.55, 0.60]⟩⟨(0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0)⟩⟨0.20, 0.60, 0.75⟩ Ningde (ND) ⟨[0.85, 0.90] , [0.05, 0.10]⟩⟨(0.7, 0.8, 0.9, 1.0) , (0.0, 0.1, 0.2, 0.3) , (0.0, 0.1, 0.2, 0.3)⟩⟨0.50, 0.50, 0.45⟩ Sanming (SM) ⟨[0.10, 0.30] , [0.55, 0.60]⟩⟨(0.0, 0.1, 0.2, 0.3) , (0.7, 0.8, 0.9, 1.0) , (0.7, 0.8, 0.9, 1.0)⟩⟨0.10, 0.66, 0.85⟩ Fuzhou (FZ) ⟨[0.55, 0.65] , [0.30, 0.35]⟩⟨(0.3, 0.4, 0.5, 0.6) , (0.2, 0.4, 0.5, 0.7) , (0.2, 0.4, 0.5, 0.7)⟩⟨0.80, 0.20, 0.15⟩ Putian (PT) ⟨[0.70, 0.80] , [0.15, 0.20]⟩⟨(0.5, 0.6, 0.7, 0.8) , (0.0, 0.2, 0.3, 0.4) , (0.0, 0.2, 0.3, 0.4)⟩⟨0.70, 0.25, 0.20⟩ Longyan (LY) ⟨[0.45, 0.50] , [0.35, 0.45]⟩⟨(0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9)⟩⟨0.30, 0.55, 0.60⟩ Quanzhou (QZ) ⟨[0.10, 0.30] , [0.55, 0.60]⟩⟨(0.3, 0.4, 0.5, 0.6) , (0.2, 0.4, 0.5, 0.7) , (0.2, 0.4, 0.5, 0.7)⟩⟨0.90, 0.10, 0.05⟩ Xiamen (XM) ⟨[0.20, 0.40] , [0.45, 0.50]⟩⟨(0.1, 0.2, 0.3, 0.4) , (0.4, 0.6, 0.7, 0.9) , (0.4, 0.6, 0.7, 0.9)⟩⟨1.00, 0.00, 0.00⟩ Zhangzhou (ZZ) ⟨[0.85, 0.90] , [0.05, 0.10]⟩⟨(0.3, 0.4, 0.5, 0.6) , (0.2, 0.4, 0.5, 0.7) , (0.2, 0.4, 0.5, 0.7)⟩⟨0.80, 0.20, 0.15⟩

Table 2 Converted evaluation matrix R.

Index Cities c1 c2 c3 c4 Nanping (NP) 3 [2,100–200,000] ⟨0.05, 0.90⟩ ⟨[0.0337, 0.0625] , [0.9007, 0.9375]⟩ Ningde (ND) 10 [65,000–1,900,000] ⟨0.80, 0.15⟩ ⟨[0.8091, 0.8409] , [0.1344, 0.1591]⟩ Sanming (SM) 2 [2,000–190,000] ⟨0.05, 0.90⟩ ⟨[0.0337, 0.0625] , [0.9007, 0.9375]⟩ Fuzhou (FZ) 7 [180,000–600,000] ⟨0.50, 0.45⟩ ⟨[0.7540, 0.7937] , [0.1751, 0.2063]⟩ Putian (PT) 5 [1,000,000–2,000,000] ⟨0.80, 0.15⟩ ⟨[0.6547, 0.7071] , [0.2507, 0.2929]⟩ Longyan (LY) 3 [5,000–300,000] ⟨0.15, 0.80⟩⟨[0.4286, 0.5000] , [0.4286, 0.5000]⟩ Quanzhou (QZ) 4 [18,000–500,000] ⟨0.05, 0.90⟩ ⟨[0.1837, 0.2500] , [0.6848, 0.7500]⟩ Xiamen (XM) 3 [2,300–190,000] ⟨0.30, 0.65⟩ ⟨[0.0787, 0.1250] , [0.8231, 0.8750]⟩ Zhangzhou (ZZ) 6 [500,000–2,000,000] ⟨0.90, 0.05⟩ ⟨[0.7540, 0.7937] , [0.1751, 0.2063]⟩

Table 3 Normalized evaluation matrix R′.

Index Cities c1 c2 c3 c4 Nanping (NP) 0.300 [0.001, 0.100] ⟨0.05, 0.90⟩⟨[0.0337, 0.0625] , [0.9007, 0.9375]⟩ Ningde (ND) 1.000 [0.033, 0.950] ⟨0.80, 0.15⟩⟨[0.8091, 0.8409] , [0.1344, 0.1591]⟩ Sanming (SM) 0.200 [0.001, 0.095] ⟨0.05, 0.90⟩⟨[0.0337, 0.0625] , [0.9007, 0.9375]⟩ Fuzhou (FZ) 0.700 [0.090, 0.300] ⟨0.50, 0.45⟩⟨[0.7540, 0.7937] , [0.1751, 0.2063]⟩ Putian (PT) 0.500 [0.500, 1.000] ⟨0.80, 0.15⟩⟨[0.6547, 0.7071] , [0.2507, 0.2929]⟩ Longyan (LY) 0.300 [0.003, 0.150] ⟨0.15, 0.80⟩⟨[0.4286, 0.5000] , [0.4286, 0.5000]⟩ Quanzhou (QZ) 0.400 [0.009, 0.250] ⟨0.05, 0.90⟩⟨[0.1837, 0.2500] , [0.6848, 0.7500]⟩ Xiamen (XM) 0.300 [0.012, 0.095] ⟨0.30, 0.65⟩⟨[0.0787, 0.1250] , [0.8231, 0.8750]⟩ Zhangzhou (ZZ) 0.600 [0.250, 1.000] ⟨0.90, 0.05⟩⟨[0.7540, 0.7937] , [0.1751, 0.2063]⟩

Step 4: Calculate the gains and losses to get the gain matrix GM and Step 5: Using Equation (15) to calculation of prospect values when a loss matrix LM as shown in Tables 4 and 5. LM represents the loss 훼 = 0.89, 훽 = 0.92, 휆 = 2.25 [43] as shown in Table 6. matrix, and each data in the matrix represents the loss value of each Step 6: The weight of the attribute calculated by java programming attribute value of each evaluation object that does not reach the pos- based on Equation (13) is as follows: itive ideal reference point (ie, the most severe value). GM denotes a gain matrix, and each data in the matrix represents a earned value 휔1 = 0.106, 휔2 = 0.156, 휔3 = 0.158, for each attribute value of each evaluation object exceeding the neg- 휔4 = 0.226, 휔5 = 0.114, 휔6 = 0.078, ative ideal reference point (ie, the lightest value). 휔7 = 0.162.

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Table 4 Loss matrix LM. 4.2. Comparative Analysis Index Cities c1 c2 c3 c4 c5 c6 c7 4.2.1. Sensitivity analysis Nanping (NP) 0.583 0.631 0.889 0.633 0.499 0.618 0.722 To illustrate the robustness of the algorithm, sensitivity analysis is Ningde (ND) 0.000 0.365 0.008 0.000 0.000 0.000 0.484 performed on three parameters in the PT under different values of Sanming (SM) 0.632 0.632 0.889 0.633 0.499 0.618 0.810 훼, 훽, and 휆. The results are as shown in Tables 7–9. Fuzhou (FZ) 0.313 0.544 0.221 0.039 0.216 0.232 0.185 Putian (PT) 0.465 0.000 0.008 0.112 0.096 0.027 0.253 Table 7 Prospect value matrix V when 훼 = 0.725, 훽 = 0.717, 휆 = 2.04. Longyan (LY) 0.583 0.614 0.761 0.280 0.301 0.483 0.620 Quanzhou (QZ) 0.528 0.580 0.889 0.483 0.499 0.232 0.087 Index Cities 훼 = 0.725, 훽 = 0.717, 휆 = 2.04. [63] Xiamen (XM) 0.583 0.630 0.531 0.582 0.421 0.483 0.000 Zhangzhou (ZZ) 0.393 0.215 0.000 0.039 0.000 0.232 0.185 c1 c2 c3 c4 c5 c6 c7 Nanping −1.173 −1.445 −1.875 −1.387 −1.239 −1.445 −1.442 Table 5 Gain matrix GM. (NP) Ningde (ND) 0.717 −0.331 0.756 0.718 0.604 0.705 −0.756 Index Sanming −1.468 −1.468 −1.875 −1.387 −1.239 −1.445 −1.754 Cities c1 c2 c3 c4 c5 c6 c7 (SM) Fuzhou (FZ) −0.313 −1.014 −0.283 0.486 −0.273 −0.490 0.107 Nanping (NP) 0.118 0.005 0.000 0.032 0.000 0.000 0.089 Putian (PT) −0.747 0.717 0.756 0.201 0.138 0.397 −0.102 Ningde (ND) 0.632 0.563 0.761 0.633 0.499 0.618 0.339 Longyan −1.173 −1.321 −1.647 −0.346 −0.547 −1.154 −1.142 Sanming (SM) 0.000 0.000 0.000 0.032 0.000 0.000 0.000 (LY) Fuzhou (FZ) 0.465 0.194 0.291 0.594 0.289 0.128 0.630 Quanzhou −0.956 −1.141 −1.875 −0.954 −1.239 −0.490 0.441 Putian (PT) 0.313 0.632 0.761 0.523 0.404 0.438 0.563 (QZ) Xiamen −1.173 −1.427 −1.150 −1.262 −0.934 −1.154 0.858 Longyan (LY) 0.118 0.052 0.008 0.356 0.204 0.019 0.195 (XM) Quanzhou (QZ) 0.221 0.139 0.000 0.153 0.000 0.128 0.729 Zhangzhou −0.536 0.010 0.918 0.486 0.604 −0.490 0.107 Xiamen (XM) 0.118 0.011 0.070 0.055 0.082 0.019 0.810 (ZZ) Zhangzhou (ZZ) 0.393 0.597 0.889 0.594 0.499 0.128 0.630

Table 6 Prospect value matrix V. Table 8 Prospect value matrix V when 훼 = 0.850, 훽 = 0.850, 휆 = 4.1. Index c c c c c c c Cities 1 2 3 4 5 6 7 Index 훼 = 0.850, 훽 = 0.850, 휆 = 4.1. [64] Nanping −1.220 −1.464 −2.019 −1.431 −1.187 −1.445 −1.551 Cities (NP) c1 c2 c4 c5 c6 c7 Ningde (ND) 0.665 −0.290 0.758 0.666 0.539 0.652 −0.772 Nanping (NP) −2.429 −2.761 −2.726 −2.271 −2.723 −2.980 Sanming −1.475 −1.475 −2.019 −1.431 −1.187 −1.445 −1.853 (SM) Ningde (ND) 0.677 −1.127 0.678 0.554 0.664 −1.814 Fuzhou (FZ) −0.267 −1.053 −0.228 0.515 −0.218 −0.426 0.186 Sanming (SM) −2.776 −2.776 −2.726 −2.271 −2.723 −3.428 Putian (PT) −0.757 0.665 0.758 0.261 0.186 0.399 −0.036 Fuzhou (FZ) −1.006 −2.196 0.382 −0.766 −1.010 −0.302 Longyan −1.220 −1.364 −1.736 −0.299 −0.503 −1.123 −1.216 Putian (PT) −1.766 0.677 −0.061 −0.097 0.305 −0.661 (LY) Longyan (LY) −2.429 −2.627 −0.974 −1.219 −2.174 −2.482 Quanzhou −0.989 −1.190 −2.019 −0.964 −1.187 −0.426 0.517 Quanzhou (QZ) −2.105 −2.394 −2.006 −2.271 −1.010 0.250 (QZ) Xiamen −1.220 −1.453 −1.163 −1.292 −0.907 −1.123 0.829 Xiamen (XM) −2.429 −2.747 −2.503 −1.846 −2.174 0.836 (XM) Zhangzhou (ZZ) −1.402 −0.465 0.382 0.554 −1.010 −0.302 Zhangzhou −0.517 0.085 0.901 0.515 0.539 −0.426 0.186 (ZZ)

훼 훽 휆 Step 7: The comprehensive prospect value Ui of each alternative is Table 9 Prospect value matrix V when = 0.89, = 0.92, = 2.25. calculated are as follows: Index Cities 훼 = 0.89, 훽 = 0.92, 휆 = 2.25 [43] c1 c2 c3 c4 c5 c6 c7 UNP = –1.499, UND = 0.282, USM = –1.577, Nanping −1.220 −1.464 −2.019 −1.431 −1.187 −1.445 −1.551 UFZ = –0.140, UPT = 0.249, ULY = –1.026, (NP) UQZ = –0.912, UXM = –0.889, UZZ = 0.275. Ningde (ND) 0.665 −0.290 0.758 0.666 0.539 0.652 −0.772 Sanming −1.475 −1.475 −2.019 −1.431 −1.187 −1.445 −1.853 (SM) From the above calculation results, we can see that UND ≻ UZZ ≻ Fuzhou (FZ) −0.267 −1.053 −0.228 0.515 −0.218 −0.426 0.186 UPT ≻ UFZ ≻ UXM ≻ UQZ ≻ ULY ≻ UNP ≻ USM. So the ranking Putian (PT) −0.757 0.665 0.758 0.261 0.186 0.399 −0.036 of disaster severity in nine cities is as follows: ND ≻ ZZ ≻ PT ≻ Longyan −1.220 −1.364 −1.736 −0.299 −0.503 −1.123 −1.216 FZ ≻ XM ≻ QZ ≻ LY ≻ NP ≻ SM. The results of this decision (LY) can be provided to relevant departments for effective disaster relief Quanzhou −0.989 −1.190 −2.019 −0.964 −1.187 −0.426 0.517 (QZ) and material distribution. Thereby providing security for the peo- Xiamen (XM) −1.220 −1.453 −1.163 −1.292 −0.907 −1.123 0.829 ple, and establishing the government’s action prestige, maintaining Zhangzhou −0.517 0.085 0.901 0.515 0.539 −0.426 0.186 social stability, and people’s lives are happy. (ZZ)

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Table 10 Comprehensive prospect value Ui and ranking result Ri with different parameter values. Index Cities 훼 = 0.725, 훼 = 0.850, 훼 = 0.89, 훽 = 0.717, 훽 = 0.850, 훽 = 0.92, 휆 = 2.04 [63] 휆 = 4.1 [64] 휆 = 2.25 [43] Ui Ri Ui Ri Ui Ri Nanping (NP) −1.447 8 −2.845 8 −1.499 8 Ningde (ND) 0.307 1 −0.015 1 0.282 1 Sanming (SM) −1.532 9 −2.956 9 −1.577 9 Fuzhou (FZ) −0.178 4 −0.702 4 −0.140 4 Putian (PT) 0.228 3 −0.074 3 0.249 3 Longyan (LY) −1.006 7 −2.109 7 −1.026 7 Quanzhou (QZ) −0.899 6 −1.933 6 −0.912 6 Xiamen (XM) −0.872 5 −1.859 5 −0.889 5 Zhangzhou (ZZ) 0.247 2 −0.057 2 0.275 2

α= 0.725 α=0.850 α= 0.725 α= 0.725 α= 0.850 α= 0.725 β= 0.717 β=0.850 β=0.92 β= 0.717 β= 0.850 β= 0.92 λ=2.04 λ= λ= 2.25 λ=2.04 λ= 4.1 λ= 2.25 4.1 NP 0.5 0.5 ZZ 0 ND 0 -0.5 e NP ND SM FZ PT LY QZ XM ZZ u

l -1

a -0.5 v

-1.5 t c

e -1 -2 p s XM -2.5 SM o r -1.5 p

-3 e v i

s -2 n e h

e -2.5 r

p QZ FZ m

o -3 C -3.5 Ranking result with different parameter values LY PT (a) (b)

Figure 2 Ranking result with different parameter values 훼, 훽, and 휆.

The comprehensive prospect value Ui of each alternative is calcu- TNNWAA (ñ1,n ̃2, ⋯ ,n ̃n) (16) lated under different values of 훼, 훽, and 휆 are as shown in Table 10. = 휔1ñ1 ⊕ 휔2ñ2 ⊕ ⋯ ⊕ 휔nñn ( As can be seen from Table 10 and Figure 2, UND ≻ UZZ ≻ UPT ≻ ( )휔 ( )휔 1 – ∏n 1 – a j , 1 – ∏n 1 – a j , UFZ ≻ UXM ≻ UQZ ≻ ULY ≻ UNP ≻ USM, so the ranking of disaster j=1 1j j=1 2j ≻ ≻ ≻ ≻ ≻ ≻ ≻ ) severity in nine cities is ND ZZ PT FZ XM QZ LY ( )휔 ( )휔 1 ∏n 1 j , 1 ∏n 1 j , NP ≻ SM. Compare the three ranking results when the parameters – j=1 – a3j – j=1 – a4j take different values, the ranking results are exactly the same. = ⟨ ( ) ⟩ . 휔 휔 휔 휔 ∏n j , ∏n j , ∏n j , ∏n j , j=1 b1j j=1 b2j j=1 b3j j=1 b4j ( ) 4.2.2. Comparative analysis with different methods 휔 휔 휔 휔 ∏n j , ∏n j , ∏n j , ∏n j j=1 c1j j=1 c2j j=1 c3j j=1 c4j In order to illustrate the validity and rationality of the algorithm, the proposed method is compared with the most widely used operator- based method. Firstly, using the heterogeneous information conversion method [21], and the relationship between the TrFNN The decision-making method steps based on the TNNWAA oper- and other data, all kinds of heterogeneous data are converted into ator are as follows: TrFNNs, and then the weighted comprehensive evaluation value of each city is calculated by using the trapezoidal fuzzy trapezoidal Step 1: By using the data conversion formula [21], the heteroge- weighted arithmetic average operator (TNNWAA) [40]. Finally, neous data of Table 3 is uniformly converted into the TrFNNs. The the exact function and scoring function of the neutrosophic in specific data are shown in Table 11: trapezoidal fuzzy are used to rank the degree of city disaster. The Step 2: Using the TNNWAA operator, the attribute values of TNNWAA operator is as follows: each scheme are assembled to obtain a comprehensive evaluation

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′ Table 11 Unified trapezoidal fuzzy neutrosophic numbers decision matrix R .

Index Cities c1 c2 c3 c4 (0.300, 0.300, 0.300, 0.300) , (0.051, 0.051, 0.051, 0.051) , (0.050, 0.050, 0.050, 0.050) , (0.0337, 0.0337, 0.0625, 0.0625) , Nanping ⟨ (0.600, 0.600, 0.600, 0.600) , ⟩⟨ (0.101, 0.101, 0.101, 0.101) , ⟩⟨ (0.050, 0.050, 0.050, 0.050) , ⟩⟨ (0.0000, 0.0000, 0.0656, 0.0656) , ⟩ (NP) (0.700, 0.700, 0.700, 0.700) (0.949, 0.949, 0.949, 0.949) (0.900, 0.900, 0.900, 0.900) (0.9007, 0.9007, 0.9375, 0.9375) (1.000, 1.000, 1.000, 1.000) , (0.491, 0.491, 0.491, 0.491) , (0.800, 0.800, 0.800, 0.800) , (0.8091, 0.8091, 0.8409, 0.8409) , Ningde ⟨ (0.000, 0.000, 0.000, 0.000) , ⟩⟨ (0.517, 0.517, 0.517, 0.517) , ⟩⟨ (0.050, 0.050, 0.050, 0.050) , ⟩⟨ (0.0000, 0.0000, 0.0565, 0.0565) , ⟩ (ND) (0.000, 0.000, 0.000, 0.000) (0.509, 0.509, 0.509, 0.509) (0.150, 0.150, 0.150, 0.150) (0.1344, 0.1344, 0.1591, 0.1591) (0.200, 0.200, 0.200, 0.200) , (0.048, 0.048, 0.048, 0.048) , (0.050, 0.050, 0.050, 0.050) , (0.0337, 0.0337, 0.0625, 0.0625) , Sanming ⟨ (0.400, 0.400, 0.400, 0.400) , ⟩⟨ (0.096, 0.096, 0.096, 0.096) , ⟩⟨ (0.050, 0.050, 0.050, 0.050) , ⟩⟨ (0.0000, 0.0000, 0.0656, 0.0656) , ⟩ (SM) (0.800, 0.800, 0.800, 0.800) (0.952, 0.952, 0.952, 0.952) (0.900, 0.900, 0.900, 0.900) (0.9007, 0.9007, 0.9375, 0.9375) (0.700, 0.700, 0.700, 0.700) , (0.195, 0.195, 0.195, 0.195) , (0.500, 0.500, 0.500, 0.500) , (0.7540, 0.7540, 0.7937, 0.7937) , Fuzhou ⟨ (0.600, 0.600, 0.600, 0.600) , ⟩⟨ (0.390, 0.390, 0.390, 0.390) , ⟩⟨ (0.050, 0.050, 0.050, 0.050) , ⟩⟨ (0.0000, 0.0000, 0.0709, 0.0709) , ⟩ (FZ) (0.300, 0.300, 0.300, 0.300) (0.805, 0.805, 0.805, 0.805) (0.450, 0.450, 0.450, 0.450) (0.1751, 0.1751, 0.2063, 0.2063) (0.500, 0.500, 0.500, 0.500) , (0.750, 0.750, 0.750, 0.750) , (0.800, 0.800, 0.800, 0.800) , (0.6547, 0.6547, 0.7071, 0.7071) , Putian ⟨ (1.000, 1.000, 1.000, 1.000) , ⟩⟨ (0.500, 0.500, 0.500, 0.500) , ⟩⟨ (0.050, 0.050, 0.050, 0.050) , ⟩⟨ (0.0000, 0.0000, 0.0946, 0.0946) , ⟩ (PT) (0.500, 0.500, 0.500, 0.500) (0.250, 0.250, 0.250, 0.250) (0.150, 0.150, 0.150, 0.150) (0.2507, 0.2507, 0.2929, 0.2929) (0.300, 0.300, 0.300, 0.300) , (0.076, 0.076, 0.076, 0.076) , (0.150, 0.150, 0.150, 0.150) , (0.4286, 0.4286, 0.5000, 0.5000) , Longyan ⟨ (0.600, 0.600, 0.600, 0.600) , ⟩⟨ (0.153, 0.153, 0.153, 0.153) , ⟩⟨ (0.050, 0.050, 0.050, 0.050) , ⟩⟨ (0.0000, 0.0000, 0.1428, 0.1428) , ⟩ (LY) (0.700, 0.700, 0.700, 0.700) (0.924, 0.924, 0.924, 0.924) (0.800, 0.800, 0.800, 0.800) (0.4286, 0.4286, 0.5000, 0.5000) (0.400, 0.400, 0.400, 0.400) , (0.130, 0.130, 0.130, 0.130) , (0.050, 0.050, 0.050, 0.050) , (0.1837, 0.1837, 0.2500, 0.2500) , Quan ⟨ (0.800, 0.800, 0.800, 0.800) , ⟩⟨ (0.259, 0.259, 0.259, 0.259) , ⟩⟨ (0.050, 0.050, 0.050, 0.050) , ⟩⟨ (0.0000, 0.0000, 0.1315, 0.1315) , ⟩ zhou (0.600, 0.600, 0.600, 0.600) (0.871, 0.871, 0.871, 0.871) (0.900, 0.900, 0.900, 0.900) (0.6848, 0.6848, 0.7500, 0.7500) (QZ) (0.300, 0.300, 0.300, 0.300) , (0.053, 0.053, 0.053, 0.053) , (0.300, 0.300, 0.300, 0.300) , (0.0787, 0.0787, 0.1250, 0.1250) , Xiamen ⟨ (0.600, 0.600, 0.600, 0.600) , ⟩⟨ (0.107, 0.107, 0.107, 0.107) , ⟩⟨ (0.050, 0.050, 0.050, 0.050) , ⟩⟨ (0.0000, 0.0000, 0.0982, 0.0982) , ⟩ (XM) (0.700, 0.700, 0.700, 0.700) (0.947, 0.947, 0.947, 0.947) (0.650, 0.650, 0.650, 0.650) (0.8231, 0.8231, 0.8750, 0.8750) (0.600, 0.600, 0.600, 0.600) , (0.625, 0.625, 0.625, 0.625) , (0.900, 0.900, 0.900, 0.900) , (0.7540, 0.7540, 0.7937, 0.7937) , Zhang ⟨ (0.800, 0.800, 0.800, 0.800) , ⟩⟨ (0.500, 0.500, 0.500, 0.500) , ⟩⟨ (0.050, 0.050, 0.050, 0.050) , ⟩⟨ (0.0000, 0.0000, 0.0709, 0.0709) , ⟩ zhou (0.400, 0.400, 0.400, 0.400) (0.375, 0.375, 0.375, 0.375) (0.050, 0.050, 0.050, 0.050) (0.1751, 0.1751, 0.2063, 0.2063) (ZZ)

Index Cities c5 c6 c7 (0.100, 0.100, 0.300, 0.300) , (0.000, 0.100, 0.200, 0.300) , (0.200, 0.200, 0.200, 0.200) , Nanping (NP) ⟨ (0.100, 0.100, 0.350, 0.350) , ⟩⟨ (0.700, 0.800, 0.900, 1.000) , ⟩⟨ (0.600, 0.600, 0.600, 0.600) , ⟩ (0.550, 0.550, 0.600, 0.600) (0.700, 0.800, 0.900, 1.000) (0.750, 0.750, 0.750, 0.750) (0.850, 0.850, 0.900, 0.900) , (0.700, 0.800, 0.900, 1.000) , (0.500, 0.500, 0.500, 0.500) , Ningde (ND) ⟨ (0.000, 0.000, 0.100, 0.100) , ⟩⟨ (0.000, 0.100, 0.200, 0.300) , ⟩⟨ (0.500, 0.500, 0.500, 0.500) , ⟩ (0.050, 0.050, 0.100, 0.100) (0.000, 0.100, 0.200, 0.300) (0.450, 0.450, 0.450, 0.450) (0.100, 0.100, 0.300, 0.300) , (0.000, 0.100, 0.200, 0.300) , (0.100, 0.100, 0.100, 0.100) , Sanming ⟨ (0.100, 0.100, 0.350, 0.350) , ⟩⟨ (0.700, 0.800, 0.900, 1.000) , ⟩⟨ (0.660, 0.660, 0.660, 0.660) , ⟩ (SM) (0.550, 0.550, 0.600, 0.600) (0.700, 0.800, 0.900, 1.000) (0.850, 0.850, 0.850, 0.850) (0.550, 0.550, 0.650, 0.650) , (0.300, 0.400, 0.500, 0.600) , (0.800, 0.800, 0.800, 0.800) , Fuzhou (FZ) ⟨ (0.000, 0.000, 0.150, 0.150) , ⟩⟨ (0.200, 0.400, 0.500, 0.700) , ⟩⟨ (0.200, 0.200, 0.200, 0.200) , ⟩ (0.300, 0.300, 0.350, 0.350) (0.200, 0.400, 0.500, 0.700) (0.150, 0.150, 0.150, 0.150) (0.700, 0.700, 0.800, 0.800) , (0.500, 0.600, 0.700, 0.800) , (0.700, 0.700, 0.700, 0.700) , Putian (PT) ⟨ (0.000, 0.000, 0.150, 0.150) , ⟩⟨ (0.000, 0.200, 0.300, 0.400) , ⟩⟨ (0.250, 0.250, 0.250, 0.250) , ⟩ (0.150, 0.150, 0.200, 0.200) (0.000, 0.200, 0.300, 0.400) (0.200, 0.200, 0.200, 0.200) (0.450, 0.450, 0.500, 0.500) , (0.100, 0.200, 0.300, 0.400) , (0.300, 0.300, 0.300, 0.300) , Longyan (LY) ⟨ (0.050, 0.050, 0.200, 0.200) , ⟩⟨ (0.400, 0.600, 0.700, 0.900) , ⟩⟨ (0.550, 0.550, 0.550, 0.550) , ⟩ (0.350, 0.350, 0.450, 0.450) (0.400, 0.600, 0.700, 0.900) (0.600, 0.600, 0.600, 0.600) (0.100, 0.100, 0.300, 0.300) , (0.300, 0.400, 0.500, 0.600) , (0.900, 0.900, 0.900, 0.900) , Quanzhou ⟨ (0.100, 0.100, 0.350, 0.350) , ⟩⟨ (0.200, 0.400, 0.500, 0.700) , ⟩⟨ (0.100, 0.100, 0.100, 0.100) , ⟩ (QZ) (0.550, 0.550, 0.600, 0.600) (0.200, 0.400, 0.500, 0.700) (0.050, 0.050, 0.050, 0.050) (0.200, 0.200, 0.400, 0.400) , (0.100, 0.200, 0.300, 0.400) , (1.000, 1.000, 1.000, 1.000) , Xiamen (XM) ⟨ (0.100, 0.100, 0.350, 0.350) , ⟩⟨ (0.400, 0.600, 0.700, 0.900) , ⟩⟨ (0.000, 0.000, 0.000, 0.000) , ⟩ (0.450, 0.450, 0.500, 0.500) (0.400, 0.600, 0.700, 0.900) (0.000, 0.000, 0.000, 0.000) (0.850, 0.850, 0.900, 0.900) , (0.300, 0.400, 0.500, 0.600) , (0.800, 0.800, 0.800, 0.800) , Zhangzhou ⟨ (0.000, 0.000, 0.100, 0.100) , ⟩⟨ (0.200, 0.400, 0.500, 0.700) , ⟩⟨ (0.200, 0.200, 0.200, 0.200) , ⟩ (ZZ) (0.050, 0.050, 0.100, 0.100) (0.200, 0.400, 0.500, 0.700) (0.150, 0.150, 0.150, 0.150)

(1.000, 1.000, 1.000, 1.000) , value TNNWAA (xi), wherein the weights use the above calculated weights, and the result data is as follows: TNNWAA (x2) = ⟨ (0.000, 0.000, 0.000, 0.000) , ⟩ , (0.000, 0.000, 0.000, 0.000)

(0.104, 0.111, 0.150, 0.159) , (0.073, 0.081, 0.121, 0.130) , TNNWAA (x1) = ⟨ (0.000, 0.000, 0.180, 0.182) , ⟩ , TNNWAA (x3) = ⟨ (0.000, 0.000, 0.174, 0.183) , ⟩ , (0.796, 0.804, 0.827, 0.834) (0.824, 0.833, 0.856, 0.863)

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(0.620, 0.624, 0.654, 0.660) , Step 3: The comprehensive evaluation values are sorted according to different methods including Prospect Theory, TOPSIS and Tra- TNNWAA (x4) = ⟨ (0.000, 0.000, 0.165, 0.169) , ⟩ , (0.286, 0.302, 0.325, 0.333) ditional Function Method (also known as Score Function, Accu- racy Function). The calculation results are shown in Table 12 and (0.690, 0.695, 0.726, 0.734) , Figure 3. TNNWAA (x5) = ⟨ (0.000, 0.000, 0.192, 0.197) , ⟩ , Step 4: It is known from the previous step the three different rank- (0.000, 0.222, 0.245, 0.251) ing results are the same, and the ranking of disaster severity in nine cities is ND ∼ XM ≻ ZZ ≻ PT ≻ FZ ≻ QZ ≻ LY ≻ NP ≻ SM. (0.286, 0.247, 0.328, 0.336) , From the comparison of the two typhoon disaster assessment meth- TNNWAA (x6) = ⟨ (0.000, 0.000, 0.208, 0.212) , ⟩ , (0.576, 0.595, 0.642, 0.654) ods, it can be seen that although the ranking results are not exactly the same, they are basically the same, and the special optimal (0.419, 0.425, 0.460, 0.469) , scheme is the same. In addition, because the TNNWAA operator is sensitive to data 0 when assembling information, the second TNNWAA (x7) = ⟨ (0.000, 0.000, 0.180, 0.185) , ⟩ , (0.425, 0.448, 0.470, 0.483) method, which is based on the TNNWAA algorithm, sometimes cannot fully sort the evaluation objects. For example, the com- (1.000, 1.000, 1.000, 1.000) , prehensive evaluation values of ND and SM are the same, that is, ∼ TNNWAA (x8) = ⟨ (0.000, 0.000, 0.000, 0.000) , ⟩ , ND XM, so they cannot be distinguished. However, this phe- (0.000, 0.000, 0.000, 0.000) nomenon does not occur in the proposed method based on PT. Compared with other related researches, the advantages of this (0.762, 0.765, 0.787, 0.791) , algorithm mainly include several aspects: TNNWAA (x9) = ⟨ (0.000, 0.000, 0.169, 0.173) , ⟩ . (0.151, 0.159, 0.182, 0.187)

Table 12 Ranking results of different methods.

Index Cities Prospect Theory TOPSIS Traditional Function Method Prospect Ranking Closeness Ranking Score Accuracy Ranking Value Results Coefficients Results Function Function Results Nanping (NP) −1.921 8 0.0011 8 0.408 −0.684 8 Ningde (ND) 0.895 1 1.0000 1 1.000 1.000 1 Sanming (SM) −2.004 9 0.0004 9 0.389 −0.743 9 Fuzhou (FZ) 0.182 5 0.7965 5 0.748 0.328 5 Putian (PT) 0.496 4 0.9168 4 0.811 0.532 4 Longyan (LY) −1.277 7 0.0858 7 0.526 −0.318 7 Quanzhou (QZ) −0.548 6 0.3834 6 0.632 −0.013 6 Xiamen (XM) 0.895 1 1.0000 1 1.000 1.000 1 Zhangzhou (ZZ) 0.583 3 0.9498 3 0.840 0.606 3

Prospect Closeness Score Prospect value(PV) Closeness Score coefficients(CC) function(SF) value coefficients function Ranking Ranking Ranking NP results based on PV results based on CC results based on SF 1 1.5 ZZ 0.5 ND 0 1 -0.5 0.5 -1 -1.5 0 XM -2 SM NP ND SM FZ PT LY QZ XM ZZ -2.5 -0.5

-1

-1.5 QZ FZ -2

-2.5 LY PT (a) (b)

Figure 3 City ranking results for different ranking methods.

Pdf_Folio:893 894 R. Tan et al. / International Journal of Computational Intelligence Systems 12(2) 881–896

1. The types of heterogeneous information studied in this paper of the research problem. Lehua Yang collected relevant data and are more comprehensive and more diverse, including ENs, implemented partial calculation through programming. The revi- interval numbers, language numbers, IFNs, IVIFNs, SVNNs, sion and submission of this paper was completed by Ruipu Tan and and TrFNNs. Compared with the literature [19], the heteroge- Shengqun Chen. neous data includes four types of data: real numbers, interval numbers, triangular fuzzy numbers, and IFNs. However, the Funding Statement information representation forms in our paper are more. Com- pared with the literature [20], the data type of our paper con- The research was funded by [National Philosophy and Social Sci- tains the NN and the TrFNN, while the literature [20] does not. ence Planning Office of China] grant number [17CGL058]. 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