An emergency decision-making method based on D-S evidence theory for probabilistic linguistic term sets

International Journal of Disaster Risk Reduction 37 (2019) 101178

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International Journal of Disaster Risk Reduction journal homepage: www.elsevier.com/locate/ijdrr

An emergency decision-making method based on D-S evidence theory for probabilistic linguistic term sets

T

Peng Lia,∗, Cuiping Weib a b

College of Economics and Management, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu 212003, PR China College of Mathematical Sciences, Yangzhou University, Jiangsu 225002, PR China

ARTICLE INFO

ABSTRACT

Keywords: Emergency decision-making D-S evidence theory Probabilistic linguistic term set Operational law

Emergency decision-making processes, always require decision makers (DMs) to express the majority of their opinions in linguistic terms (LTs). Since there are, in most cases, many more people than pure DMs participating in the decision-making process, probabilistic linguistic term sets (PLTSs) are suitable to express the decision information. A typical feature of emergency decision-making is that even minor errors in the decision process may have dire consequences. Accurate decision information and the consequent handling of that information are absolutely essential. We show that there are significant drawbacks in the existing operational laws for PLTSs, which may result in faulty decisions. In order to address the issue, in this paper, we propose some new operational laws for PLTSs based on D-S evidence theory and discuss the properties of these new operational laws and advance a probabilistic linguistic weighted averaging operator based on D-S evidence theory (DS-PLWA). Our aggregating method can keep the form of PLTS and avoid information loss. In addition, we propose a novel comparison method for PLTSs and a method to obtain the criteria weights based on maximising deviation for the evidences. On this basis, we propose an emergency decision-making style based on D-S evidence theory and apply it to a known details of a serious mine accident that happened in Pingyi, Shandong province.

1. Introduction With the development of society and with advancing economies, unconventional emergency events will occur more frequently and have become one of the most serious problems that the world is facing. Emergency decision-making is an effective tool to reduce the huge economic losses and casualties when one unconventional emergency event occurs [1–7]. A key feature of emergency decision-making lies in the speed of response and the need for timely decisions to be made [2]. In the early stages of an unpredicted emergency event, there is axiomatically a lack of reliable information, making it difficult for DMs to express opinions in crisp numbers. They may express their opinions with uncertain information, such as fuzzy information [8], intuitionistic fuzzy information [9], hesitant fuzzy information [10] and linguistic information [11]. Linguistic information, as one type of the uncertain information, can be used to express DMs' opinions in quasi-hierarchical terms such as good, better, best; or bad, worse, worst. Therefore, linguistic information has been applied to a large amount of decision making problems. Symbolic models [12], linguistic two-tuple models [13,14] and semantic models [15] are three common ways of dealing with linguistic decision-making problems. More research on linguistic

∗

decision-making can be found in Refs. [16–20]. Due to the intrinsic complexity and uncertainty of emergency decision-making problems, it is likely that DMs will need to express their opinions with multiple linguistic terms. To address this issue, Rodríguez et al. [21] put forward the concept of hesitant fuzzy linguistic term sets (HFLTS) by combining existing hesitant fuzzy sets (HFS) [22] and linguistic term sets (LTS) [8] giving HFLTS the ability to express uncertain information. Extensive research on hesitant fuzzy linguistic decisionmaking has followed. Liao et al. [23] defined distance and similarity measures for HFLTS and on this basis put forward a decision-making method similar to extended TOPSIS [14]. Wang et al. [24] constructed a new outranking relational system for HFLTS and proposed an outranking method for hesitant fuzzy linguistic multi-criteria decisionmaking (MCDM) problems. Wang et al. [25] proposed a Definition for interval-valued hesitant fuzzy linguistic term sets and presented a novel MCDM method. Wei et al. [26] proposed the possibility of a degree formula to compare HFLTSs and offered aggregation methods for hesitant fuzzy linguistic information. Liao et al. [27] put forward some correlation coefficients of HFLTS and established a decision-making method for medical diagnosis problems. Wu and Xu [28] discussed the hesitant fuzzy linguistic preference relation (HFLPR) and researched

Corresponding author. E-mail address: [email protected] (P. Li).

https://doi.org/10.1016/j.ijdrr.2019.101178 Received 10 January 2019; Received in revised form 2 May 2019; Accepted 6 May 2019 Available online 13 May 2019 2212-4209/ © 2019 Elsevier Ltd. All rights reserved.

International Journal of Disaster Risk Reduction 37 (2019) 101178

P. Li and C. Wei

consensus and consistency for group decision-making problems. Wei et al. [29] proposed a new score function and a hesitant fuzzy linguistic TODIM method. Yavuz et al. [30] studied an evaluation method for alternative fuel vehicles by using a hesitant fuzzy linguistic decisionmaking method. Liao and Xu [31] proposed a number of cosine distance and similarity measures for HFLTS and on this basis constructed a decision-making method. Da and Xu [32] studied evaluation problems for urban waterfront redevelopment by using HFLTS. Montes et al. [33] utilised HFLTS theory to solve the decision-making problems of a housing market. Xu et al. [34] proposed a linear programming model using hesitant fuzzy linguistic information to deal with group decisionmaking problems. Song and Hu [35] put forward fuzzy linear programming ideas and proposed a new group decision-making method for HFLTS. Chen et al. [36] studied possibility distribution for HFLTS, based on the consideration of DMs' disparate attitudes to assessment. Zhang et al. [37] proposed an MCDM method for interval-valued HFLTS on the basis of Shapley fuzzy measures. Sellak et al. [38] proposed an MCDM method for HFLTS on the idea of outranking, based on knowledge. Liao et al. [39] proposed an ORESTE method for hesitant fuzzy linguistic information. Wu et al. [40] proposed two group decision making methods for HFLTSs. Later, HFLTS was generalised to incorporate extended hesitant fuzzy linguistic term set (EHFLTS) and the mathematical properties, consistency measures and total orders for EHFTS were discussed [41–43]. Traditional HFLTS expresses uncertain information by continuous linguistic terms of a certain LTS; while EFHLTS comprises a subset of discontinuous linguistic terms in a certain LTS and is able to express more abundant uncertain information. In real emergency decision-making problems, there will probably be more than one decision maker in most cases. In other words, emergency decision-making is a group decision-making problem, which has attracted many scholars [44–47]. By using EHFLTSs, the linguistic evaluations of DMs can only be simple collected, and can not carefully express the degree of subordination to each linguistic term. Therefore, Zhang et al. [48] proposed the concept of linguistic distribution and discussed some operational laws for linguistic distribution. Zhang et al. [49] applied the linguistic distribution to solve large-scale group decision making problems. Later, Pang et al. [50] introduced the Definition of probabilistic linguistic term set (PLTS) and proposed a number of operational laws and comparison methods. There is only a minor difference between the linguistic distribution and the PLTS: the sum of symbolic proportions in the former is one, while that in the latter can be less one. Gou and Xu [51] proposed further operational laws for PLTS. Bai et al. [52] put forward a novel comparison method for PLTS by constructing a possibility degree formula. Chen et al. [53] proposed two new operational laws for PLTS based on t-norm and put forward a group decision-making method. Zhou and Xu [54] discussed group consistency for probabilistic linguistic fuzzy preference and proposed a group decision-making method. Liu et al. [55] proposed a probabilistic linguistic TODIM method. There are some drawbacks in the existing operational laws for PLTS.

Dempster–Shafer theory (DST)), as an extension of Bayesian theory, is a powerful tool to handle uncertain information and can be used to integrate information from different sources of evidence by Dempster's combination rule [56]. This theory has been applied to MCDM methods, such as the ELECTRE TRI method [57] and so on [58–65]. Emergency decision-making is a particular kind of group decisionmaking style, in which DMs usually express their opinions with LTs when they take part in the decision-making process. Because of the lack of information and time pressures, DMs may be hesitant or give up commenting on some alternatives under certain criteria. These incomplete linguistic evaluation information sets, provided by DMs, can be indicated in the form of PLTSs. It is clear that in emergency decisionmaking process, even small errors in processing decision information may lead to severe consequences. Traditional methods of integrating decision information expressed with PLTSs are unlikely to meet the needs of emergency decision-making. D-S evidence theory is a powerful tool to deal with incomplete information, and we find that there exists a strong correlation between D-S evidence theory and PLTS, which leads us to believe that it is possible to apply Dempster's combination rule to aggregate criteria presented by PLTS in the MCDM problems. Inspired by this, we propose an emergency decision-making method in the evidence theory framework. We list here our principal contributions in this paper: (1) We propose a novel way to aggregate PLTSs based on D-S evidence theory, and prove that PLTS and mass function can transform mutually, which is the basis for novel operational laws for PLTS; (2) We construct some novel operational laws for PLTS based on evidence theory, which can keep the form of PLTS and need not add elements in PLTS, so can avoid information loss. (3) We propose a novel emergency decision style based on D-S evidence theory. We organise this paper as follows. Section 2 reviews the basic definitions and operations for LTS, HFLTS, EHFLTS, PLTS and D-S evidence theory. Section 3 studies the operational law for PLTS. Section 4 discusses the comparison methods for PLTS. Section 5 illustrates the idea and the steps of our novel decision-making method for PLTS. Section 6 relates to a mine accident rescue in Pingyi which illustrates the validity and effectiveness of our proposed method. Section 7 summarises the paper. 2. Preliminaries In this section, we review some basic concepts and operations for our research. 2.1. Linguistic term set In our real emergency decision-making problems, DMs may comfortably express their opinions with linguistic terms, which keep with people's cognitive habits. A subscript-symmetric linguistic term set , , 1,0,1 , } , where is a (LTS) [66] is written as S = {s | = positive integer, s0 means “indifference” and the following rules are true:

(1) A standardisation process must be carried out before the operations for PLTS: because when the probability sum of different linguistic terms is less than one, this will mean that some DMs may be hesitant or may even give up expressing their evaluations. The standardisation process is required to assign this part, or value, to other linguistic terms. Consequently, this standardisation process leads to the loss of some decision-making information. (2) The traditional operational laws tend to change the original framework for PLTS. The subscripts of linguistic terms in PLTS usually appear as decimals. An ideal operational law should keep to the original framework with PLTS.

(1) If < , then s < s . (2) The negation operator is expressed as neg (s ) = s . Example 1. Suppose that written as

S = {s

2

= very low, s

1

= 2 , then a linguistic term set S can be

= low, s0 = fair , s1 = high, s2 = very high}.

Xu [66] extended the LTS to a continuous form expressed as S = {s [ , ]} , where is a sufficient positive real number. This ensures that the operational results for any LTS are not beyond the

To solve these issues, we need to reconstruct a new series of operational laws. D-S evidence theory [51] (also called the 2

International Journal of Disaster Risk Reduction 37 (2019) 101178

P. Li and C. Wei

upper bound and the lower bound . Based on the continuous form S , four basic operational laws by subscript for LTSs are introduced by Xu [66] as follows.

promoted by Shafer [56] in 1976. It has become an effective tool to deal with uncertain systems. We now review some basic concepts about D-S evidence theory as follows.

Definition 1. [66] Let s , s [0,1], then

be a finite set (also called the frame of Definition 5. [56] Let [0,1] satisfies m ( ) = 0 , discernment). If a mapping m : 2 m (A) = 1, then the mapping m is defined as a mass function A (also called a basic probability assignment (BPA) function).

(1) (2) (3) (4)

s s

s =s s =s s =s (s ) = s

S be two arbitrary linguistic terms, and

+

For certain evidence, m (A) means a belief measure that is satisfied by a subset A under some certain conditions. If m (A) > 0 , we call A the focal element. If there is only one element in A , then A is called a singleton. In a real emergency decision-making environment, DMs may have more than one evidence sources and thus have to deal with multiple sources of information and evidence. D-S rule for evidence fusion is an effective method to cope with this problem [56].

×

2.2. Hesitant fuzzy linguistic term set Owing to the complexity of emergency decision-making environments, DMs sometimes express their opinions with hesitance within several probable values. Traditional LTS cannot accurately describe the information in these situations. To address this issue, Rodríguez et al. [21] proposed the concept of hesitant fuzzy linguistic term set (HFLTS) as follows.

Definition 6. [56] Assume two arbitrary pieces of evidence, E1 and E2 . The mass functions for the two items are m1 and m2 , respectively. The DS rule for evidence fusion is defined as follows:

m (A) = (m1

Definition 2. [21] Let S be an LTS, a HFLTS Hs is an ordered finite subset of consecutive linguistic terms in S .

, 1,0,1,

, ; pk

0,

1

1}

m1 (B ) m2 (C )

(2)

K = m1 (A) m2 (A) + m1 (B ) m2 (B ) + m1 ( ) m2 ( ) + m1 (A) m2 ( ) + m1 (B ) m2 ( ) + m2 (A) m1 ( ) + m2 (B ) m1 ( ) m = 0.3 + 0.12 + 0.01 + 0.06 + 0.03 + 0.05 + 0.04 = 0.61, 0.6 × 0.5 + 0.6 × 0.1 + 0.5 × 0.1 (A) = (m1 m2)(A) = = 0.67, 0.61

0.3 × 0.4 + 0.3 × 0.1 + 0.4 × 0.1 = 0.31, 0.61

m (B ) = (m1

m2)(B ) =

m ( ) = (m1

m2)( ) = 1

0.67

0.31 = 0.02.

Definition 6 provides a way to deal with the problem of information fusion for two elements of evidence. When facing more than two elements of evidence, the following theorem can help DMs fuse information conveniently.

, }} be an LTS. A

pk k=

,B C =A

According to the D-S rule, we combine the two elements of evidence in the following:

t

,

B, C

m2 (A) = 0.5, m2 (B ) = 0.4, m2 ( ) = 0.1.

Example 3 Let S = {s 2 s 1, s0, s1, s2} be an LTS, then the sets b1 = {s1, s2} , b2 = {s 1, s0, s1} and b3 = {s 1, s0, s2} are all EHFLTSs. In group decision-making problems, both HFLTS and EHFLTS cannot reflect the importance or weighting of DMs' opinions. To address the issue of weighting, the concept of probabilistic linguistic term sets (PLTS) was proposed [50,53].

sk, pk |k =

K

m1 (A) = 0.6, m1 (B ) = 0.3, m1 ( ) = 0.1;

Definition 3. [41] Let S be an LTS, an EHFLTS is an ordered subset of S} . the linguistic terms of S represented by HSE = {s s

L (p ) =

1 1

where K = B C = m1 (B ) m2 (C ) is called the conflicting factor. Example 5 For a frame of discernment = {A, B} , there are two mass functions m1 and m2 as follows:

S = {s 2 = very low, s 1 = low, s0 = Example 2 Let fair , s1 = high, s2 = very high} be an LTS, then the sets b1 = {s1 = high,s2 = very high} and b2 = {s 1 =low,s0 = fair ,s1 = low} are two HFLTSs. For convenience, we usually use abbreviations: b1 = {s1, s2} and b2 = {s 1, s0, s1} . HFLTS is a consecutive subset of a certain LTS and cannot deal with some group assessment problems in uncertain decision-making environments. Wang [41] extended the HFLTS to EHFLTS that can effectively express this uncertainty with discrete subset in an LTS.

Definition 4. [50,53] Let S = {sk |k { , , 1,0,1, probabilistic linguistic term set is defined as:

m2)(A) =

Theorem 1. [56] Let m1, m2 , following equations are true

(1)

pk where is the probability for the linguistic term sk (k { , , 1,0,1, , } ). Example 4 Let S = {s 2 s 1, s0, s1, s2} be an LTS, then L 2 (p ) = L1 (p) = {(s 2, 0), (s 1, 0.3), (s0, 0), (s1, 0.6), (s2, 0)} and {(s 2, 0), (s 1, 0), (s0, 0.7), (s1, 0.3), (s2, 0)} are both PLTSs. From Example 4, we observe that the sum of probabilities may be less than one in some cases, such as L1 (p) . The reason for this lies in the reality that some DMs may be hesitant to make their decisions or may fail to provide evaluations due to time pressure or lack of relevant knowledge when they evaluate some of the decision-making objectives. For simplicity, we describe the PLTSs only with probabilities greater than 0. In Example 4, we can rewrite the PLTSs as L1 (p) = {(s 1, 0.3), (s1, 0.6)} and L2 (p) = {(s0, 0.7), (s1, 0.3)} .

1) ((m1

m2 )

, mn be n arbitrary mass functions, then the

m3)(A) = (m1

2) (m1

m2)(A) = (m2

3) (m1

m2

(m2

m3))(A)

(4)

m1)(A)

mn)(A) =

1 1

(3)

K

m1 (A1 ) m2 (A2 ) mn (An ), n A =A i=1 i

(5)

mn (An ) < 1. where K = n A = m1 (A1 ) m2 (A2 ) i=1 i In some cases, evidence is not always fully credible. The credibility degrees of evidences E1, E2 , ,En are defined as Crd (E1) , Crd (E2 ), , Crd (En ) . We have the following Definition. Definition 7. [60] Let E be an arbitrary evidence, be a frame of discernment, and m be the mass function. If the credibility degree of the evidence is Crd (E ) = k (0 k 1), then the discount mass functions mCrd is defined as follows

2.3. Evidence theory In 1967, D-S evidence theory was initiated by Dempster [67] to enable reasoning with uncertain information and the theory was

mCrd (A) = k m (A) , A 3

2 ,A

(6)

International Journal of Disaster Risk Reduction 37 (2019) 101178

P. Li and C. Wei

mCrd ( ) = 1

mCrd (A) = k m ( ) + 1

k

(2) g 1: [0,1]

(7)

A

g 1 (h ) = {g

Example 6 Assume an evidence E , the frame of discernment = {A, B} and the mass function m for E : m (A) = 0.5, m (B ) = 0.3, m ( ) = 0.2 , if Crd (E ) = 0.5, we then obtain the discount mass function mCrd as follows

0.25

(1) L1 (p)

0.15 = 0.6

(2) L (p) = g 1 (

In this section, we will review some traditional operations for PLTSs and analyse their drawbacks, then propose some new operations based on D-S evidence theory.

3.1.1. Operations for PLTSs proposed by Pang et al. [50] Pang et al. [50] proposed that some operations for PLTSs rely on the computing subscripts of linguistic terms, which we saw in Definition 1. All PLTSs are normalized by the following two steps: (1) if the sum of the probabilities associated all linguistic terms is less than one, calculate the probabilities to make the sum equal to one; (2) if the number of linguistic terms of different PLTSs are not equal, add elements until the length of PLTSs are equal. In the following Definition 8, we assume that all the PLTSs have been normalized. Definition 8. [50] Let L (p) = {(sk , pk )|k = , , 1,0,1, , ; pk 0, tk = pk 1} , t L1 (p) = {(sk, pk1 )|k = , , 1,0,1, , ; pk1 0, k = pk1 1} and L2 t 2 2 (p) = {(sk , pk )|k = , , 1,0,1, , ; pk 0, k = pk2 1} be three PLTSs, then 1 {si | si L1 (p) | pi1 > 0};{sj | s j L2 (p) | pi2 > 0} {pi si

(2) L (p) = { pk sk },

{

pj2 sj ;

[0,1], g (sk ) =

{

g (Hs ) = g (sk ) =

k 2

1

+ 2 |k

+ ,

1 2 i g (L1), j g (L2 )

1 i

+

2 j

1 2 i j

pi1 pj2

,

{1 i g (L )

(1

i)

(pi )}) , {i|pi(1) > 0}.

,

, 1,0,1,

, ; pk =

pk k=

pk

}. In fact, in some

D-S evidence theory, as a useful tool to solve uncertain-system problems, is used to aggregate probabilistic linguistic information in this section. 3.2.1. Relation between PLTS and mass function According to the D-S evidence theory, an arbitrary mass function m can represent the degree to which a certain objective to be assessed belongs, using evaluation grades. For example, there is a car to be assessed associated with the grade set S = {s 1 = poor ,s0 = medium,s1 = good} . A mass function m can be written as m (s 1) = 0.6, m (s0) = 0.2, m (S ) = 0.2 . It means that the probabilities of the car belonging to “poor” and “medium” evaluation grades are 0.6 and 0.2, respectively and that there is an uncertainty degree of 0.2 in the assessment. This mass function m can be written as a PLTS {(s 1, 0.6), (s0, 0.2)} . In turn, a PLTS can be seen as a mass function. For example, let S = {s 1 = poor ,s0 = medium,s1 = good} be an LTS, L (p) = {(s 1, 0.3), (s1, 0.6)} be a PLTS on S , then the PLTS L (p) can be seen as a mass function m : m (s 1) = 0.3, m (s1) = 0.6, m (S ) = 0.1. We now illustrate the relation between PLTS and mass function.

Definition 9. [51] Let S = {sk |k { , , 1,0,1, , }} be an LTS and [0,1]} be a hesitant Hs = {sk |k [ , ]} be an HFLTS and h = { | fuzzy set. Then the membership degree and the linguistic term sk can transform into each other by the equivalent transformation functions g and g 1 as follows:

,

1

3.2. Novel operations for PLTSs based on evidence theory

3.1.2. Operations for PLTSs proposed by Gou and Xu [51] Gou and Xu [51] put forward operations for PLTSs according to two equivalent transformation functions of hesitant fuzzy sets, basing these operations on the two equivalent transformation functions and normalization of PLTSs.

(1) g :

= sk ,

cases, because of DMs' hesitant opinions, they may express their opit t pk innions by PLTSs with k = pk < 1 where the value of 1 k= dicates the DMs' hesitant degree. The normalization of PLTS usually leads to loss of information.

This type of operation for PLTSs is based on the idea of fusing the linguistic fuzzy set and the probability. We will illustrate some drawbacks in Definition 8 by the following example. Example 7 Let S = {s 3, s 2, s 1, s0, s1, s2, s3} be an LTS and let L1 (p) = {(s1, 0.2), (s3, 0.8)} and L2 (p) = {(s2, 0.1), (s3, 0.9)} be two PLTSs. Then, according to the Pang’ method [48], we have L1 (p) L2 (p) = {0.2s1 0.1s2, 0.8s3 0.9s3} = {s0.4, s4.3} . The aggregating result s4.3 exceeds the upper bound s3 in the LTS S . Furthermore, by using Pang's method, a PLTS is transformed into an EHFLTS, which is unreasonable. Effective operational laws should not change the form of expression.

1 2

1)

[0,1]} = Hs .

j|pj(2) > 0

L (p) = (sk, p k )|k =

0.

k 2

1)

|

= sk , , g 1 ( ) = s (2

We illustrate some drawbacks in this type of operation for PLTSs in the following example. Example 8. Here we use Gou and Xu's example [51]. Let S = {s 3, s 2, s 1, s0, s1, s2, s3} be an LTS, and let L3 (p) = {(s1, 0.3), (s2, 0.2), (s3, 0.5)} and L4 (p) = {(s 1, 0.2), (s0, 0.3)} be two PLTSs. Their method mainly uses t-norm to aggregate information based on operational laws for LTSs. By this method, we have s2 s3 = s3 , which is incomprehensible. Furthermore, by utilising L3 (p) L4 (p) = their method, we can obtain {(s1.67, 0.12), (s2, 0.18), (s2.33, 0.08), (s2.5, 0.12), (s3, 0.5)} . The results contain s1.67 , s2.33 and s2.5 , the meanings of which are unreasonable. Moreover, both the methods in Refs. [50,51] need normalized PLTS. That is to say, for arbitrary PLT t t L (p) = {(sk , pk ) k = , , 1,0,1, , ; pk 0, k = pk 1} , if k = pk < 1, L (p ) then must be transformed into

3.1. Traditional operations for PLTSs

L 2 (p ) =

) = s (2

L 2 (p ) = g

i|pi(1) > 0 ,

3. Some new operations based on evidence theory for PLTS

(1) L1 (p)

1(

1)

Definition 10. [51] Let S = {sk |k { , , 1,0,1, , }} be an LTS, L (p) , L1 (p) and L2 (p) be three PLTSs on S . Then the following operations are true:

mCrd (A) = 0.5 × 0.5 = 0.25, mCrd (B ) = 0.5 × 0.3 = 0.15, mCrd ( ) = 1

, ], g 1 ( ) = s(2

[

S = {s , , s 1, s0, s1, , s } be Lemma 1. Let an LTS, t , , 1,0,1, , ; pk 0, k = pk 1} be an andL (p) = {(sk , pk ) k = arbitrary PLTS on S , then L (p) can be transformed into a mass function m . Proof. See S = {s , , s 1, s0, s1, , s } as a frame of discernment. For arbitrary PLTS L (p) = {(sk , pk ) k = , , 1,0,1, , ; pk 0, tk = pk 1} , let a mass function m satisfy m (s ) = p , m (s 1) = p 1, m (s0) = p0 , pk . m (s1) = p1, m (s ) = p , m (S ) = 1 k= , , 1,0,1, } constructs a frame Obviously, the set S = {sk, k =

= ,

}=h ; 4

International Journal of Disaster Risk Reduction 37 (2019) 101178

P. Li and C. Wei

of discernment and satisfies m ( ) = 0 , sk S m (sk ) = 1. According to the Definition of mass function, we can see that m is a mass function.

1

=

=

sk , 1

pl2

l=

1

l=

pl1 +

pl1

pk1 pk2 + pk1 1

l=

pl2 + pk2 1

2) L (p) = {(sk , m (sk )} = {(sk, pk ) k = 3) L1 (p) 4) (L1 (p)

L 2 (p ) = L 2 ( p )

,

, 1,0,1 , }

l=

t = 1,2,

L 2 (p)) + L (p) = L2 (p)

(L1 (p)

m1 (S ) = 1 k=

According

(m1

m2)(sk ) =

,

to

k=

Theorem

1,

we

l=

pl2 ) + pk2 (1

l=

,

, 1,0,1,

|k = , n.

,

, 1,0,1,

(12)

, }

, n ), i.e.

, }

k=

pkt , k =

,

1,0,1,

,

, n ) are

mt , Crd (sk ) = wt mt (sk ) = wt pkt , mt , Crd (S ) = 1

wt mt (sk ) = 1 k=

k=

wt pkt |k =

,

, 1,0,1,

,

We next use mathematical induction on n . n = 2, DS PLWA (L1 (p), L 2 (p)) = w1 When we have L1 (p) w2 L 2 (p). From Theorem 3 and Definition 11, we can easily obtain

have

w1 L1 (p) = {(sk, w1 m1 (sk ) k =

,

, 1,0,1 , },

w2 L2 (p) = {(sk , w2 m2 (sk ) k =

,

, 1,0,1 , }.

The mass functions for the above two sets are discount mass functions m1, Crd and m2, Crd . Then, w1 L1 (p) w2 L 2 (p) = {(sk, m1, Crd m2, Crd (sk ))|k = , , 1,0,1, }. Suppose that n = m , Eq. (12) holds. That is

w1 L1 (p)

m1 (B ) m2 (C ) = m1 (sk ) m2 (sk ) + m1 (sk ) m2 (S ) + m1 (S ) m2 (sk )

= pk1 pk2 + pk1 (1

, Ln (p)) mn, Crd (sk )) k =

The discount mass functions for mt (t = 1,2,

Because the mass functions m1 and m2 contain only singletons, for any focal elements B for m1 and C for m2 , we have B C = for B C and B, C S . Then, we can easily obtain that B , C S, B C = sk

pl1 ).

l=

mt (sk ) = pkt , mt (S ) = 1

pk2

B, C S , B C = sk m1 (B ) m2 (C ) . 1 B C = m1 (B ) m2 (C )

pl1 ))

(9)

, 1,0,1 , ),

pk1 and m2 (S ) = 1

pl2 )(1

l=

(8)

pl1

Proof. Assume that the mass functions m1 and m2 associated with L1 (p) and L2 (p) are

m1 (sk ) = pk1 , m2 (sk ) = pk2 , (k =

pl2 ) + pk2 (1

Proof. Suppose the PLTSs are Lt (p) = {(sk , pkt )|k = , 1,0,1, , ; k = pkt 1} (t = 1,2, , n ). According to Theorem 2, we obtain the mass function associated with Lt (p) are mt

(11)

L (p))

l=

Theorem 4. The aggregated value by utilising the PLT-DSWA operator is still a PLTS and satisfies Eq. (12).

(10)

L1 ( p )

m1 (B ) m2 (C )

where mt , Crd are the discount mass functions for mt (t = 1,2, mt , Crd (sk ) = wt mt (sk ) = wt pkt , mt , Crd (S ) = 1 wt mt (sk ) , k=

, 1,0,1 , } l=

l= k=

,

m1 (B ) m2 (C )+

B C=S

pk1 (1

DS PLWA (L1 (p), L2 (p), = {(sk, m1, Crd m2, Crd

, , 1,0,1 , } is an LTS, Theorem 3. Suppose that S = {s | = L (p) = {(sk , pk )|k = , 1,0,1, , } , L1 (p) = {(sk, pk1 )|k = , 1,0,1, , ; k = pk1 1} and L2 (p) = {(sk , pk2 )|k = , 1,0,1, , ; k = pk2 1} are three arbitrary 1. The mass functions associated with L (p) , L1 (p) PLTSs on S , and 0 and L2 (p) are m , m1 and m2 , respectively. Then the following operational laws are true

pl2 + pk2 1

l=

+

B C = sk

, , 1,0,1 , } be an LTS, Definition 11. Let S = {s | = L1 (p), L 2 (p), , Ln (p) be n arbitrary PLTSs, m1, m2 , , mn be the mass L1 (p), L 2 (p), , Ln (p) , functions associated with and w = (w1, w2, , wn) be the weighting vector of Lj (p) satisfying n w = 1 and wj 0 . The DS-PLWA operator is defined as j=1 j

3.2.2. Some novel operations for PLTSs based on evidence theory Theorem 2 provides for the possibility of constructing operations based on evidence theory for PLTS. We next propose some novel operations for PLTSs based on D-S evidence theory.

m2)(sk ) k =

(pk1 pk2

m1 (B ) m2 (C ) =

B C

Therefore, Eq. (8) holds. According to Eq. (6) and Eq. (7), we have that the mass function associated with L (p) is pk . mcrd (sk ) = pk (k = , , 1,0,1 , ) and mcrd (S ) = 1 k= We can see that Eq. (9) holds. On the basis of Eq. (4) and Eq. (5), we can obtain that Eq. (10) and Eq. (11) hold. □ From the above discussion, we next propose a probabilistic linguistic weighted averaging operator based on D-S evidence theory (DSPLWA).

Theorem 2. An arbitrary PLTS is equivalent to a mass function containing only singletons.

pk1 pk2 + pk1 1

k=

+ (1

Proof. Due to the mass function m containing only singletons, any focal elements in mass function m include only one element. For any , , 1,0,1, , ) the mass function m can be focal elements sk (k = written as m (sk ) = pk . Furthermore, according to D-S evidence theory, {(sk, pk ) we have k = pk 1. Then, the set t |k = , , 1,0,1, , ; pk 0, k = pk 1} is a PLTS. Based on Lemma 1 and 2, we can obtain the relation between PLTS and mass function.

L 2 (p ) = {(sk , (m1

m1 (B ) m2 (C ) = k

Lemma 2. Let S = {s , , s 1, s0, s1, , s } be a frame of discernment, and m be a mass function containing only singletons on S , then the mass function m can be transformed into a PLTS.

1) L1 (p)

B C=

w 2 L 2 (p)

= {(sk , (m1, Crd

pl1 ).

wm Lm (p) m2, Crd

mm, Crd)(sk )) k =

,

, 1,0,1,

Then, when n = m + 1, according to Eq. (4), we have 5

, }.

International Journal of Disaster Risk Reduction 37 (2019) 101178

P. Li and C. Wei

w 1 L1 (p )

w 2 L 2 (p )

wm Lm (p)

Lm + 1 (p) = {(sk , ((m1, Crd = {(sk, (m1, Crd mm, Crd

L (p) are defined as

wm + 1

mm, Crd) mm + 1, Crd)(sk ))} mm + 1, Crd)(sk )|k = , , 1,0,1,

S (L (p)) =

, }.

1

t

kpk

(13)

k=

Then, by the principle of mathematical induction, Eq. (12) holds. mm, Crd mm + 1, Crd is still a According evidence theory, m1, Crd mass function. Based on Theorem 2, the aggregated value by utilising DS-PLWA operator is still a PLTS. □ = 3 and the LTS is taken as Example 9 Suppose that S = {s 3, s 2, s 1, s0, s1, s2, s3} . Let L1 (p) ={(s 2, 0.5), (s1, 0.3), (s2, 0.2)} and L2 (p) = {(s 2, 0.6), (s 1, 0.2), (s2, 0.1)} be two PLTSs, and the weighting vector be w = (0.6, 0.4) . We can obtain the mass functions for the two PLTSs as follows:

U (L (p)) = m (S ) = 1

pk

(14)

k=

Obviously, we can obtain S (L (p)) [ 1,1] and U (L (p)) [0,1]. Example 10 Let = 3, the LTS be S = {s 3, s 2, s 1, s0, s1, s2, s3} , and the PLTS be L (p) = {(s 2, 0.3), (s1, 0.2), (s2, 0.1), (s3, 0.3)} . Then the values of score function S (L (p)) and uncertain function U (L (p)) for L (p) are

S (L (p)) =

m1: m1 (s 3) = 0, m1 (s 2) = 0.5, m1 (s 1) = 0, m1 (s0) = 0,

1 × ( 2 × 0.3 + 1 × 0.2 + 2 × 0.1 + 3 × 0.3) = 0.23, 3

U (L (p)) = 1

m1 (s1) = 0.3, m1 (s2) = 0.2, m1 (s3) = 0

0.3

0.2

0.1

0.3 = 0.1.

m2 : m2 (s 3) = 0, m2 (s 2) = 0.6, m2 (s 1) = 0.2, m2 (s0) = 0,

We then propose a comparison method for PLTSs by the following Definition.

m2 (s1) = 0, m2 (s2 ) = 0.1, m2 (s3) = 0, m2 (S ) = 0.1,

Definition 14. Let L1 (p) and L2 (p) be two arbitrary PLTSs, then

m1, Crd : m1, Crd (s 3) = 0, m1, Crd (s 2) = 0.3, m1 (s 1) = 0, m1, Crd (s0) = 0,

(1) (2) (3) (i) (ii) (iii)

m1, Crd (s1) = 0.18, m1, Crd (s2) = 0.12, m1, Crd (s3) = 0, m2, Crd : m2, Crd (s 3) = 0, m2, Crd (s 2) = 0.24, m2, Crd (s 1) = 0.08, m2, Crd (s0) = 0, m2, Crd (s1) = 0, m2, Crd (s2) = 0.04, m2, Crd (s3) = 0, m2, Crd (S ) = 0.04. PLT DSWA (L1 (p), L 2 (p)) = {(s 2, 0.42), (s 1, 0.04), (s1, 0.13), (s2, 0.11)}

Example 11 Let = 3, the LTS be S = {s 3, s 2, s 1, s0, s1, s2, s3} . Given two PLTSs on S are L1 (p) = {(s 1, 0.2), (s2, 0.4), (s3, 0.2)} and L2 (p) = {(s 2, 0.1), (s1, 0.5), (s3, 0.3)} . Then, we have S (L1 (p)) = S (L2 (p)) = 0.4 , U (L1 (p)) = 0.2 , and U (L2 (p)) = 0.1. Therefore, we can draw the conclusion L1 (p) < L2 (p) .

4. The comparison between PLTSs The methods for comparing the PLTSs are crucial in decisionmaking problems. Pang et al. [50] proposed expectation and variance functions of PLTSs. Chen et al. [53] put forward a comparison method based on probability theory. These methods play an essential role in solving decision-making problems with PLTSs. However, these methods have a high computational complexity. To resolve the issue, we propose a novel method for comparing PLTSs. , , 1,0,1 , } , we can easily see that Given an LTS S = {s | = the higher the subscript, the better the evaluation result. The middle value 0 indicates a meaning of “indifference”. When the subscript < 0 , the evaluation result is worse than “indifference”. Instead, when the subscript > 0 , the evaluation result is better than “indifference”. Therefore, we can derive an intuitive method to compare PLTSs.

5. A novel decision-making method based on D-S evidence theory for PLTSs In this section, we will illustrate the main idea of our decisionmaking method based on D-S evidence theory for PLTSs. 5.1. Decision-making problem description For a particular emergency decision-making problem, let C = {C1, C2, , Cn} be a criteria set, whose weighting vector is n w = (w1, w2, , wn) satisfying wj 0 ( j = 1,2, , n ) and j = 1 wj = 1, and X = {X1 , X2 , , Xm } be an alternative set. DMs propose their opinions by R = [Lij (p)]m × n , a PLTS decision matrix where

, , 1,0,1 , } be an LTS, Definition 12. Let S = {s | = Li (p) = {(sk , pi k ) sk S, k = , , 1,0,1, , , k = pi k 1} and t Lj (p) = {(sk , p j k ) sk S, k = , , 1,0,1, , , k = p j k 1} be two arbitrary PLTSs on S . If 1) For any k > 0 , pki 2) For any k < 0 , pki

If S (L1 (p)) < S (L 2 (p)) , then L1 (p) < L2 (p) If S (L1 (p)) > S (L 2 (p)) , then L1 (p) > L2 (p) If S (L1 (p)) = S (L2 (p)),then If U (L1 (p)) < U (L2 (p)), then L1 (p) > L2 (p) If U (L1 (p)) = U (L2 (p)) , then L1 (p) L 2 (p) If U (L1 (p)) > U (L2 (p)), then L1 (p) < L2 (p)

t

Lij (p) = {(sk , pij k ) k { , , 1,0,1, , t ;pij k 0; k = pij k 1} means the value of alternative Xi in terms of criterion Cj . Obviously, each value Lij (p) can be transform into a mass function

pkj ; pkj

mij : mij (s ) = pij , = pij1 ,

then Li (p) Lj (p) . In fact, the conditions in the above Definition 12 are too strict and inapplicable in most cases. Meanwhile, the comparison between PLTSs is vital important in decision-making. We now introduce the score function and uncertain function for PLTSs in the D-S evidence theory framework. Definition 13. Let S = {s | = L (p) = {(sk , pk ) sk S ,

,

, 1,0,1 , }

be

an

, mij (s 1) = pij1 , mij (s0) = pij0 , mij (s1) , mij (s ) = pij

(15)

Thus, Thus, the decision-making matrix R = [Lij (p)]m × n can be transformed into mass function matrix M = [mij ]m × n , where mij can be obtained by Eq. (15). 5.2. Obtaining the criteria weights by maximising deviation for evidences

LTS,

The maximising deviation method is a useful tool to calculate the criteria weights according to decision-making information. Let mi and mj be two arbitrary mass functions in the frame of discernment S then the distance measure d (mi , mj ) between mi and mj is defined as [68]

t

k= , , 1,0,1, , , k = pk 1} be a PLTS, and m : {m (sk ) = pk k = , , 1,0,1, , } be the mass function m for L (p) . Then the score function S (L (p)) and uncertain function U (L (p)) for 6

International Journal of Disaster Risk Reduction 37 (2019) 101178

P. Li and C. Wei

d (mi , mj ) =

1 ( mi 2

mj ) T D ( mi

mj )

(16)

where D is a 2S × 2S matrix, and the elements D (A, B ) of D are defined A B as D (A, B ) = A B . By Eq. (16), in criterion Cj , the deviation value between alternative Xi and other alternatives together is m

di j ( w ) =

wj d (mij , mlj )

(17)

l=1

We then construct the following mathematical programming model to obtain the criteria weights w = (w1, w2, , wn). n

m

max

d i j (w ) =

j=1 i=1

n

m

m

wj d (mij , mlj )

(P1)

j=1 i =1 l=1

=1 s.t. wj 0 , By using the Lagrange function, we can easily obtain n w2 j=1 j

wj =

m m d (mij , mlj ) i=1 l=1 n m m d (mij , mlj ) j=1 i=1 l=1

(18) Fig. 1. The main decision process based on D-S evidence theory for PLTSs.

5.3. Aggregation for probabilistic linguistic information by DS-PLWA operator

because of continuous landslides. Drilling was required to make a suitable shaft as quickly as possible. Because the gypsum layer of rock was very loose, it was challenging to drill the shaft: any carelessness in the operation might influence the lives and safety of the mine employees and the rescue staffs. The department of emergency management invited ten experts (Ei , i = 1,2, , 10 ) to propose and evaluate four possible emergency plans ( X1, X2 , X3 and X 4 ) as follows. X1: A rescue method proposing roadway excavation before supporting the roof; X2 : A rescue method using rescue capsules sent via drilling small holes; X3: A traditional underground mine-rescue roadway excavation pattern; X 4 : A rescue method by boring big holes to the trapped staff. Three key factors (criteria) required consideration:

We will introduce an aggregation method for probabilistic linguistic information by DS-PLWA operator. Based on Eq. (12), the overall attribute values Zi (w ) for alternatives Xi (i = 1,2, , m ) can be obtained by

Zi (w ) = DS

PLWA (Li1 (p), Li2 (p),

= {(sk , (mi1

mi2

, Lin (p))

min)(sk )|k =

,

, 1,0,1,

, }

(19)

According to Eqs. (13) and (14), compute the score functions S (Zi (w )) and uncertain function U (Zi (w )) for Zi (w ) . Then rank all the alternatives according to comparison method in Definition 14. 5.4. The decision-making procedure Based on the above analysis, we summarise the decision procedure for a decision-making method based on D-S evidence theory for PLTSs by the following steps: Step 1. According to the decision-making problem, based on Eq. (15), each value Lij (p) can be transform into a mass function mij . Then the PLTS decision matrix R = [Lij (p)]m × n can be transformed into mass function matrix M = [mij ]m × n . Step 2. According to the mass function matrix M = [mij ]m × n , compute the distance measure d (mi , mj ) between mi and mj based on Eq. (16). Based on Eq. (18), calculate the criteria weights w = (w1, w2, , wn) . Step 3. According to the criteria weights w = (w1, w2, , wn) obtained by Step 2, compute the overall values Zi (w ) for alternatives Xi (i = 1,2, , m ) on the basis of Eq. (19). Step 4. Based on the overall values Zi (w ) for alternatives Xi (i = 1,2, , m ), compute the values of score function S (Zi (w )) and uncertain function U (Zi (w )) for Zi (w ) according to Eq. (13) and Eq. (14) respectively. Step 5. Rank the alternatives Xi (i = 1,2, , m ) according to the comparison rule in Definition 14 and choose the optimal alternatives. And the decision process based on D-S evidence theory for PLTSs can be described as the following Fig. 1.

• Speed of rescue (C ): embodying the speed of the total rescue process; • Efficiency of rescue (C ): embodying the ratio of rescue benefit out of cost in labour and material; • Possibility of the rescue (C ): embodying the likelihood of any pro1

2

cess succeeding.

3

In term of the three key factors, the ten experts are invited to use an LTSS = {s 3 = none (N ), s 2 = very low (VL ), s 1 = low (L), s0 = fair (F ),

s1 = high(H ), s2 = very high(VH ), s3 = perfect (P )} to express their opinions which are tabulated in Table 1. By processing the evaluated information, a decision matrix R was obtained with probabilistic linguistic information showed as Table 2. 6.1. Decision procedure Step 1. Based on the decision matrix R , the mass function matrix M = (mij ) 4× 3 can be computed as follows:

m11: m11 (s0) = 0.4, m11 (s1) = 0.6,

m12: m12 (s2) = 1,

6. A case application

m13: m13 (s 1) = 0.8, m13 (s0) = 0.2,

About 8:00 on December 25, 2015, a serious mine accident happened in Pingyi, Shandong province. There were about 29 staff trapped down a mine. The roadway in the mine was almost completely blocked

m21: m21 (s2) = 0.3, m21 (s3) = 0.7,

m22: m22 (s0) = 0.8, m22 (S ) = 0.2, 7

International Journal of Disaster Risk Reduction 37 (2019) 101178

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Table 1 Opinions presented by experts. Experts

X1 X2 X3 X4 X1 X2 X3 X4 X1 X2 X3 X4 X1 X2 X3 X4 X1 X2 X3 X4

E1

E3

E5

E7

E9

S (Z1 (w )) =

C1

C2

C3

Experts

F

VH

F

E2

VH

*

H

H

VH

VH

VL

H

F

VH

L

VH

F

VH

H

H

VH

VH

VL

H

H

VH

L

P

F

VH

H

H

VH

L

H

H

VH

L

P

F

P

H

VH

P

P

F

H

H

VH

F

P

F

P

H

VH

P

H

H

*

*

H

E4

E6

E8

E10

C1

C2

C3

F

VH

L

VH

F

H

H

H

VH

VH

VL

H

F

VH

L

P

F

VH

H

H

VH

VH

VL

H

H

VH

L

P

F

VH

X1 X2 X3 X4 X1 X2 X3 X4 X1 X2 X3 X4 X1 X2 X3 X4 X1 X2 X3 X4

H

VH F

H

H

VH

P

*

L

H

VH

P

P

H

H

H

VH

L

P

F

P

H

VH

P

P

H

P

= 0.17;

X1

X2 X3 X4

C3

{(s0 ,0.4), (s1 ,0.6)} {(s2 ,0.3), (s3 ,0.7)} {(s1 ,1)} (s2 ,0.4), (s3 ,0.4)}

{(s2 ,1)}

{(s 1,0.8), (s0 ,0.2)}

{(s0 ,0.8)}

{(s1 ,0.2), (s2 ,0.4), (s3 ,0.4)} {(s2 ,0.6), (s3 ,0.4)} {(s1 ,0.9)}

{(s1 ,0.5), (s2 ,0.5)} {(s 2 ,0.4), (s 1,0.1), (s0 ,0.2), (s1 ,0.3}

1 (0 × 0.154 + 1 × 0.042 + 2 × 0.153 + 3 × 0.244) = 0.36; 3

S (Z3 (w )) =

1 (1 × 0.315 + 2 × 0.24 + 3 × 0.076) = 0.341; 3

S (Z4 (w )) =

1 ( 2 × 0.088 + ( 1) 3 × 0.022 + 0 × 0.044 + 1 × 0.28 + 2 × 0.072 + 3 × 0.072)

= 0.147. Step 5. We can easily obtain that S (Z2 (w )) > S (Z3 (w )) > S (Z1 (w )) > S (Z4 (w )). The ranking result is X2 X3 X1 X 4 . Thus, X2 should be the best emergency plan. In the real practice of rescue, the effectiveness of rescue using the emergency plan X2 was very good. And this rescue saved people's lives and property losses to a great extent.

P

*

6.2. Comparison with existing methods In this section, we will compare the proposed method with some existing methods, including PL-TODIM [69], PL-TOPSIS [50], PLVIKOR [70] and aggregation-based method [51]. We provide the detailed decision process of TODIM [69], and only give the ranking results of PL-TOPSIS [50], PL-VIKOR [70] and aggregation-based method [51]. The PL-TODIM method [69] is a useful tool to deal with MCDM considering the DMs' preferences. The main decision process is showed as follows. Step 1. Calculate the criteria weights wj ( j = 1,2, , n ) based on the decision matrix R . Step 2. Compute the relative weights wjr = wj/ w , where w = max{wj |j = 1,2, , n} . Step 3. Compute the dominance of alternative Xi over Xk as n (Xi , Xk ) = j = 1 j (Xi , Xk ) , where

Table 2 Decision matrix R with probabilistic linguistic information. C2

S (Z2 (w )) =

VH

Note: Because the pressure of time and lack of information, experts may not be able to present their opinions for some alternatives under certain criteria and use the symbol “*” to show.

C1

1 (( 1) × 0.043 + 0 × 0.282 + 1 × 0.117 + 2 × 0.218) 3

m23 : m23 (s1) = 0.2, m23 (s2) = 0.4, m23 (s3) = 0.4, m31: m31 (s1) = 1, m32 : m32 (s1) = 0.5, m32 (s2) = 0.5,

wjr d Lij (p), Lkj (p)/

m33: m33 (s2 ) = 0.6, m33 (s3) = 0.4,

j (Xi ,

0, if Lij (p)

Xk ) =

m41: m41 (s2) = 0.4, m41 (s3) = 0.4, m41 (S ) = 0.2,

n w d j = 1 jr

1

m42 : m42 (s 2 ) = 0.4, m42 (s 1) = 0.1, m42 (s0 ) = 0.2, m42 (s1) = 0.3

n w j = 1 jr

, if Lij (p) Lkj (p)

Lkj (p)

Lij (p), Lkj (p)/ wjr , if Lij (p) Lkj (p)

Step 4. Compute the overall prospect value of alternative Xi (i = 1,2, , m )

m43: m43 (s1) = 0.9, m43 (S ) = 0.1. Step 2. Based on Eq. (16) and Eq. (18), the criteria weights are w1 = 0.319, w2 = 0.343, w3 = 0.339 . Step 3. On the basis of Eq. (19), the overall values Zi (w ) (i = 1,2,3,4 ) are computed as

(Xi ) =

m k=1

max{ i

(Xi , Xk )

m k=1

min{

(Xi , Xk )}

i

m k=1

min{

(Xi , Xk )} m k=1

(Xi , Xk )}

,

i

Step 5. Rank all the alternatives according to the values (Xi ) (i = 1,2, , m ). We use the PL-TODIM method to deal with our case in Section 6 as follows. Step 1. To make the comparison effective, we use the criteria weights computed by our proposed method w1 = 0.319, w2 = 0.343, w3 = 0.339 . Step 2. The relative weights are computed as w1r = 0.93, w2r = 1, w3r = 0.988. Step 3. The dominance of alternative Xi over Xk is influenced by the parameter . Similar to the method in Ref. [69], we choose = 1, = 10 and = 30 . We here present the dominance values for = 1 as follows.

Z1 (w ) = {(s 1, 0.043), (s0, 0.282), (s1, 0.117), (s2, 0.218)}, Z2 (w ) = {(s0, 0.154), (s1, 0.042), (s2, 0.153), (s3, 0.244)},

Z3 (w ) = {(s1, 0.315), (s2, 0.24), (s3, 0.076)}, Z4 (w ) = {(s 2, 0.088), (s 1, 0.022), (s0, 0.044), (s1, 0.28), (s2, 0.072), (s3, 0.072)} . Step 4. Compute the values of score function for Zi (w ) (i = 1,2,3,4 ) as follows 8

International Journal of Disaster Risk Reduction 37 (2019) 101178

P. Li and C. Wei

Table 3 Ranking results of different methods.

Table 4 Decision matrix in Ref. [50].

Methods

Ranking results

Methods

Ranking results

Our proposed method PL-TODIM method ( = 1) [69] PL-TODIM method ( = 10 ) [69] PL-TODIM method ( = 30 ) [69] PL-TOPSIS method [50] PL-VIKOR method [70] Aggregation-based method [51]

X2 X2 X2 X2 X2 X3 X2

Our proposed method PL-TODIM method ( = 1) [69] PL-TODIM method ( = 10 ) [69] PL-TODIM method ( = 30 ) [69] PL-TOPSIS method [50] PL-VIKOR method [70] Aggregation-based method [51]

X2 X2 X2 X2 X2 X3 X2

0 0.6 1 (Xi , Xk ) = 0.3 0.55 2 (Xi ,

Xk ) =

0 1.71 1.27 1.66

0 0.55 3 (Xi , Xk ) = 0.61 0.5

(Xi , Xk ) =

0 0.55 0.37 0.61

1.89 0.94 0 0.53 1.67 0 1.26 0.5 0.59 0 0.44 1.12 1.63 0 0.32 1.63

2.94 0 0.91 4.02

X3 X3 X3 X3 X3 X2 X3

X1 X4 X4 X1 X4 X4 X4

X4 X1 X1 X4 X1 X1 X1

1.79 0.96 0 1.44

2.29 1.7 0 2.25

X1 X4 X4 X1 X4 X4 X4

X4 X1 X1 X4 X1 X1 X1

By using our proposed method, the overall values for three alternatives can be obtained as

1.73 0.4 , 1.58 0

0.44 1.28 0 1.31

X3 X3 X3 X3 X3 X2 X3

Z1 (w ) = {(s 1, 0.03), (s0, 0.11), (s1, 0.47), (s2, 0.04)},

0.57 0.38 , 1.31 0

Z2 (w ) = {(s 2, 0.05), (s 1, 0.15), (s0, 0.29), (s1, 0.05), (s2, 0.01)}, Z3 (w ) = {(s0, 0.24), (s1, 0.21), (s2, 0.05), (s3, 0.02)}. The score function values are showed as follows.

1.48 0.55 0.49 0

S (Z1 (w )) = 0.51, S (Z2 (w )) =

0.18, S (Z3 (w )) = 0.37.

So the ranking result is X1 X3 X2 , which is same as that in Ref. [50]. 7. Conclusions

2.63 1.34 2.4 0

In this paper, we have proposed a novel way to solve emergency decision making problem with D-S evidence theory. We first discussed the drawbacks of existing operational laws for PLTSs. Then, we proposed some new operational laws for PLTSs based on D-S evidence theory which can keep the form of PLTSs, and proved that PLTSs and mass functions can transform mutually. Furthermore, we proposed a novel method to compare PLTSs and put forward a new way to obtain criteria weights by maximising deviation for evidences. Finally, we applied our method to an actual mine accident that happened in Pingyi and compared our results against the existing decision methods. Based on the case study, we can observe that our new decision making style for probabilistic linguistic information can effectively solve the emergency decision problems without the process of normalizing the PLTSs. Our future research work will mainly focuses on the information aggregation for the dynamic chain of evidence, since the emergency decision making is a dynamic process.

Step 4. The overall prospect values of alternatives are computed as

(X1) = 0, (X2) = 1, (X3) = 0.6, (X 4 ) = 0.14 Step 5. The ranking result is X2 X3 X 4 X1 for = 1. When = 10 , the overall prospect values of alternatives are

(X1) = 0, (X2) = 1, (X3) = 0.47, (X 4 ) = 0.03 The ranking result is X2 X3 X 4 X1. When = 30 , the overall prospect values of alternatives are

(X1) = 0.01, (X2) = 1, (X3) = 0.47, (X 4 ) = 0 The ranking result is X2 X3 X1 X 4 . We can see that the decision making result of the PL-TODIM method will vary slightly with different parameter . In fact, we can obtain the conclusion that X1 X 4 when > 29.3. In this situation the decision making result of our proposed method is same as the PL-TODIM method. The PL-TOPSIS method in Ref. [50], PL-VIKOR method in Ref. [70] and aggregation-based method in Ref. [51] are also applied to solve the same case. The ranking results of these methods are shown in the following Table 3. As can be seen from Table 3, the ranking result of our proposed method is slightly different with that of the other methods except the PL-TODIM method when = 30 .The main reason leading to the difference lies in the fact that our proposed method is based on D-S evidence theory that need not normalize a PLTS to a standard PLTS. On the other hand, all the other methods are based on the process of normalizing a PLTS to a standard PLTS, which usually leads to the loss of information. As we know, a minor mistake in emergency decision making may lead to massive loss. Therefore, one of the most important cores of emergency decision making is avoiding information loss. Although our proposed method may get the different ranking result with the traditional methods in some cases, it also can obtain the same ranking result with them. We analyse another example in Ref. [50]. To compare these methods conveniently we transform the linguistic scale to LTS S = {s 3, s 2, s 1, s0, s1, s2, s3} and use the criteria weights as w = (0.138, 0.304, 0.416, 0.142) in Ref. [50]. The decision matrix in Ref. [50] can be seen in Table 4.

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