# Hybrid TODIM Method with Crisp Number and Probability Linguistic Term Set for Urban Epidemic Situation Evaluation

Authors: Wenshuai Wu; Zeshui Xu
Journal: Complexity
 (2020)

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## Abstract

Emergency medical services during the COVID-19 epidemic have become the focus of worldwide attention, and how to effectively respond to urban epidemic situation during a complex environment has become a global challenge. Emergency decision-making can be considered as a multicriteria decision-making (MCDM) problem, which involves multiple criteria or attributes about qualitative and quantitative aspects. So, in this paper, based on the TODIM method, a hybrid TODIM method with crisp number and probability linguistic term set is first provided to evaluate the severity of urban COVID-19 epidemic situation during a complex humanitarian crisis environment. In this hybrid method, the quantitative aspects are evaluated on the basis of precise numerical values, and the qualitative aspects are evaluated by means of probability linguistic term set, which can not only express their judgments or linguistic preference with multiple linguistic terms but also reflect different importance degrees or probability degrees of all the possible linguistic information or preference information. In addition, the concept of entropy and probability linguistic entropy is applied to induce hybrid criteria weight information. Furthermore, sensitivity analysis of the parameter about attenuation factor of the losses in the hybrid TODIM method, which considers the psychology factors and cognitive behavior of the DMs, is further conducted on a case study, to verify the effectiveness and stability of the proposed method for urban epidemic situation evaluation according to the results of this study.


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## Body

## 1. Introduction

The frequent occurrence of emergency medical incidents has caused a great deal of property losses and a lot of casualties [1]. Until April 5, 2020, the global COVID-19 epidemic has gotten further intensified as the global death toll due to this deadly virus soars to 64,701, and the total confirmed cases surge to 1,201,591 around the world according to the statistical data provided by Johns Hopkins University. The highest number of confirmed cases in the world is in the United States, with a cumulative total of more than 310,000 cases, and the highest number of deaths is in Italy, with a cumulative total of 15,362 cases. Emergency medical services during the COVID-19 epidemic have become the focus of worldwide attention, and the response to the COVID-19 outbreak has become a global challenging issue.Emergency medical management and epidemic situation evaluation have a complex evolutionary process, and emergency decision-making is very difficult to deal with the traditional decision theory. The decision-making on epidemic situation evaluation of accidental medical emergencies usually concerns about multiple criteria or attributes about qualitative and quantitative aspects, a finite number of alternatives, semantic benefits, and interests of multiple stakeholders, and it can be considered as a complex multicriteria decision-making (MCDM) problem. Kahneman and Tversky [2] pointed out that the decision-makers (DMs) play a limited rational role in the complex decision-making process; in other words, the psychology and behavior of DMs may be the important factors which can affect the decision-making results. The TODIM method, first proposed by Gomes and Lima [3], is an interactive MCDM method considering the psychology factors and cognitive behavior of the DMs [4]. Liu and Teng [5] proposed an extended TODIM method based on intuitionistic uncertain linguistic variables for multiple attribute group decision-making. Rani et al. [6] proposed an extended intuitionistic fuzzy TODIM technique for multicriteria decision-making method based on Shapley weighted divergence measure. Davoudabadi et al. [7] proposed a new last aggregation method of multiattribute group decision-making based on concepts of TODIM, WASPAS, and TOPSIS under interval-valued intuitionistic fuzzy uncertainty. Although this TODIM method has done a lot of outreach and extension, it is still not well studied and applied for epidemic situation evaluation in the emergency medical decision-making process.The DMs usually need data and information to make effective decisions, especially for complex decisions. In general, emergency decision-making on the COVID-19 outbreak is considered as a multicriteria decision-making (MCDM) problem, which involves multiple criteria or attributes about qualitative and quantitative aspects. The quantitative aspects are evaluated on the basis of precise numerical values, and the qualitative aspects are evaluated by means of uncertainty including vagueness and ambiguity [8], such as fuzzy set theory, rough set theory, and neutrosophic set theory. However, vague data are uncertain because they lack detail or precision, and ambiguous data are uncertain because they are subject to multiple interpretations [9]. Probability linguistic term set (PLTS) is provided by Pang et al. [10] as a new type of linguistic variable, which can not only express their judgments or linguistic preference with multiple linguistic terms but also reflect different importance degrees or probability of all the possible linguistic information or preference information. Liu and You [4] extended the TODIM with PLTS to solve the multiattribute decision-making problems by an example of corporate project evaluation, and the objective weights are determined based on the traditional measure of information entropy under probabilistic linguistic environment. In this paper, firstly, the extended hybrid TODIM method with crisp number and PLTS is first provided to evaluate the severity of urban COVID-19 epidemic situation. Secondly, the objective weights for PLTS are induced and determined based on the concept of probability linguistic entropy, which has the advantages of simple calculation and clear concept.In emergency medical management, decision-making process has become more and more complex because many of the attributes are difficult to quantify [11–13] and decisions are usually made with subjective preference information in human perceptual timeframes under pressure, lack of knowledge, and data cases [14, 15]. Furthermore, their effectiveness can be constrained due to the complexity of the decision-making process and the intricacy of systems [13, 16]. How to effectively respond to the accidental medical emergencies and evaluate the severity of urban COVID-19 epidemic situation during a complex environment has become a global challenge.To sum up, this paper focuses on the development of a hybrid TODIM method with crisp number and probability linguistic term set, based on the TODIM method which can consider the psychology factors and cognitive behavior of the DMs, for aiming to assist DMs to evaluate the severity of urban COVID-19 epidemic situation involved in emergency medical management. Five main works of this paper that have led to its novelty are as follows: (1) A hybrid TODIM method with crisp number and probability linguistic term set is first provided to evaluate the severity of urban COVID-19 epidemic situation during a complex humanitarian crisis environment. (2) The proposed hybrid TODIM method involves multiple criteria or attributes about qualitative and quantitative aspects, considering the complexity of emergency decision-making process. (3) The proposed hybrid TODIM method can not only express their judgments or linguistic preference with multiple linguistic terms, but also reflect different importance degrees or probability degrees of all the possible linguistic information or preference information. (4) The concept of entropy and probability linguistic entropy is applied to induce hybrid criteria weight information, which has the advantages of simple calculation and clear concept. (5) Sensitivity analysis of the parameter in the hybrid TODIM method, which considers the psychology factors and cognitive behavior of the DMs, is further carried out to demonstrate the effectiveness and stability of the proposed hybrid method for urban epidemic situation evaluation.The rest of this paper is organized as follows. Section2 reviews the related research. Section 3 introduces some brief backgrounds required for this paper. In Section 4, a hybrid TODIM method with crisp number and PLTS is proposed. In Section 5, an empirical study on urban COVID-19 epidemic situation evaluation during a complex humanitarian crisis environment is conducted to verify our provided method. Finally, Section 6 concludes the paper.


## 2. Related Works

The frequent occurrence of emergency medical incidents has caused a great deal of property losses and a lot of casualties [17]. In the meantime, it has also promoted the theoretical research and technology development to deal with emergency problems. Pullum and Ramanathan [18] indicated that data-driven machine learning tools should be developed to automate discovery for helping DMs gain quantifiable insights on emerging disease. Wang et al. [19] identified EMS emergency responses with attempted advanced airway management. Temple et al. [20] employed the GRADE method to conduct a systematic literature review and search multiple databases to examine the effects of caffeine on results of interest. Lazzara et al. [21] synthesized the literature pertinent to multiteam systems and emergency medical services to assist the prehospital community in strengthening patient care. However, epidemic situation evaluation has a complex evolutionary process. The emergency decision-making on epidemic situation evaluation usually concerns about multiple criteria or attributes about qualitative and quantitative aspects, a finite number of alternatives, semantic benefits, and interests of multiple stakeholders, which is essentially considered as a complex multicriteria decision-making (MCDM) problem. Based on the complexity of the emergency decision-making process, the TODIM method is still not well studied and applied for epidemic situation evaluation in the emergency medical decision-making, although it has done a lot of outreach and extension.MCDM has made remarkable progress and has evolved into a mature discipline, which concerns about multiple conflicting criteria, a finite number of alternatives, and multiple opinions of DMs [1, 22–24]. Kahneman and Tversky [2] pointed out that the DMs play a limited rational role in the complex decision-making process; in other words, the psychology and behavior of DMs may be the important factors which affect the decision-making results. TODIM method, an interactive MCDM method, is proposed by Gomes and Lima [3], which can consider the psychology factors and cognitive behavior of the DMs [4] to address decision-making problems [10, 25–27].Decisions usually need data information, particularly for complex decisions, which involve multiple criteria about qualitative and quantitative aspects. The quantitative aspects are usually evaluated on the base of accurate numerical values, and the qualitative aspects are usually evaluated by means of uncertainty including vagueness and ambiguity [8]. However, vague data are uncertain because they lack detail or precision, and ambiguous data are uncertain because they are subject to multiple interpretations [9]. It is impossible to deal with this imprecise situation with the traditional MCDM method. PLTS is provided by Pang et al. [10] as a new type of linguistic variable for indicating fuzzy data information, which can express and reflect different importance degrees or probability of all the possible linguistic information with multiple linguistic terms. For instance, when one DM is evaluating the severity of the urban epidemic situation, based on self-awareness of the epidemic, the DM may think he/she is 30% sure the urban epidemic is “very serious,” 50% sure it is “serious,” and 20% sure it is “general”. It is obvious that the problem of urban epidemic situation evaluation needs to be integrated with PTLS to express complex and uncertain information. So, this paper, based on the TODIM method, provides a hybrid TODIM method with crisp number and PLTS to evaluate the severity of urban COVID-19 epidemic situation.Besides, in the application of TODIM method, some scholars have done some research and development. Fan et al. [28] proposed an extended TODIM method to solve the hybrid MADM problem with crisp numbers, interval numbers, and fuzzy numbers. Liu and Teng [29] extended the TODIM method to 2-dimensional uncertain linguistic information for multiple attribute group decision-making. Liu and You [4] extended the TODIM with PLTS to solve the multiattribute decision-making problems according to an example of corporate project evaluation, and during the probabilistic linguistic environment, the objective weights are determined based on the traditional measure of information entropy. Wei [26] expanded the TODIM model to the MADM with the picture fuzzy numbers. Although some achievements have been made in the application of TODIM method, the hybrid TODIM method with crisp number and PLTS has not been fully studied. Furthermore, in this paper, the extended hybrid TODIM method is first proposed to evaluate the severity of urban COVID-19 epidemic situation. In addition, Zhang et al. [27] explored the water security evaluation based on the TODIM method with probabilistic linguistic term sets, where a programming model is developed to derive the attribute weights. The attribute weights determined by mathematical programming may be separated from the characteristics of the data itself to some extent. So, in this paper, the concept of entropy and probability linguistic entropy is applied to induce hybrid criteria weight information, which has the advantages of simple calculation and clear concept.In view of the foregoing, this paper based on the TODIM method provides a hybrid TODIM method with crisp number and PLTS to evaluate the severity of urban COVID-19 epidemic situation for aiding DMs involved in emergency medical management during a complex humanitarian crisis environment.


## 3. Preliminaries

In this section, some brief backgrounds of PLTS, the comparison between PLTSs, the normalization of PLTS, and the TODIM method are introduced separately.


### 3.1. Probabilistic Linguistic Term Set

Deﬁnition 1 (see [10]).
Based on the additive linguistic term setS=sα=−τ,…,−1,0,…,τ [30,31], the probabilistic linguistic term set (PLTS) can be defined by Pang et al. [10] as follows:(1)Lp=LkpkLk∈S,pk≥0,k=1,2,…,#Lp,∑k=1#Lppk≤1,where Lkpk represents the linguistic term Lk associated with the probability pk and #Lp is the number of all different linguistic terms in Lp.
If∑k=1#Lppk=1, then the PLTSs have complete probabilistic information of all possible linguistic terms if ∑k=1#Lppk<1, then the PLTSs have partial probabilistic information; if ∑k=1#Lppk=0, then the PLTSs have completely unknown probabilistic information.Definition 2 (see [27]).
Given a probabilistic linguistic term setLp=Lkpkk=1,2,…,#Lp,∑k=1#Lppk≤1, on the additive linguistic term set S=sα=−τ,…,−1,0,…,τ, the negation operation for Lp is given by(2)negLp=L^kpkL^k=negLk=negSα,k=1,2,…,#Lp,where negLp is the negation operation for linguistic terms [31], defined as negSα=S−α, especially, negS0=S0.


### 3.2. The Comparison between PLTSs

First, the score of PLTS is defined and introduced by Pang et al. [10] as follows.Deﬁnition 3 (see [32]).
LetLp=Lkpkk=1,2,…,#Lp be a PLTS and rk be the subscript of the Lk, then the score of Lp is given by(3)ELp=sα¯,where α¯=∑k=1#Lprkpk/∑k=1#Lppk.
For any two PLTSsL1p and L2p, if EL1p>EL2p, then L1p is superior to L2p, denoted by L1p≻L2p; if EL1p<EL2p, then L2p is superior to L1p, denoted by L1p≺L2p; if EL1p=EL2p, then these two PLTSs cannot be distinguished. In this case, the deviation degree of a PLTS is further defined by Pang et al. [10] as follows.Deﬁnition 4 (see [10]).
LetLp=Lkpkk=1,2,…,#Lp be a PLTS and rk be the subscript of the Lk, and ELp=sα¯, then the deviation degree of Lp is given by(4)σLp=∑k=1#Lppkrk−α¯21/2∑k=1#Lppk.
For two PLTSsL1p and L2p with EL1p=EL2p, if σL1p>σL2p, then L1p≺L2p; if σL1p=σL1p, then L1p is indifferent to L2p, denoted by L1p∼L2p.Definition 5 (see [10]).
LetL1p=L1kpkk=1,2,…,#L1p and L2p=L2kpkk=1,2,…,#L2p be any two PLTSs, #L1p=#L2p, the distance between L1p and L2p can be defined as follows:(5)dL1p,L1p=∑k=1#L1pp1kr1k−p2kr2k2#L1p.


### 3.3. The Normalization of PLTS

The normalization of PLTSs usually has two distinct tasks. One is to estimate the ignorance degree of probabilistic information, and the other is to normalize the cardinality for computational convenience.Definition 6 (see [10]).
Given a PLTSLp with ∑k=1#Lppk<1, then the normalized PLTS L˜p is defined as follows:(6)L˜p=Lkpkk=1,2,…,#Lp,where p˜k=pk/∑k=1#Lppk, for all k=1,2,…,#Lp.
LetL1p=L1kp1kk=1,2,…,#Lp and L2p=L2kp2kk=1,2,…,#Lp, then the normalization process can be obtained as follows:(1)
If∑k=1#Lppk<1, then according to formula (6), L˜p can be calculated.(2)
If#L1p≠#L2p, then some elements can be added to the one with the smaller number of elements. That is to say, if #L1p>#L2p, then the #L1p−#L2p linguistic terms need to be added to L2p, which make the numbers of all different linguistic terms in L1p and L2p equal. The added linguistic terms should be the smallest ones in L2p, and their probabilities should be zero.


### 3.4. The TODIM Method

The TODIM method, proposed by Gomes and Lima [3], is an interactive MCDM method, which can capture psychology factors and cognitive behavior of DMs. On the base of prospect theory [2], the alternatives are ranked according to their overall dominance degree based on the dominance degree of each alternative over others with respect to each criterion. The steps of TODIM method can be summarized as follows [3].Suppose there areAii=1,2,…,m alternatives and Cjj=1,2,…,n criteria.Step 1: identify the decision matrixX=xijm×n, where xiji=1,2,…,m,j=1,2,…,n is the jth criteria value with respect to the ith alternative, and xij are all crisp numbers.Step 2: obtain the normalized decision matrixG=gijm×n. The normalized decision matrix can be obtained by using the normalization method.Step 3: calculate the relative weightwjr of the criteria Cj to the reference criteria Cr as follows:(7)wjr=wjwrj=1,2,…,n,wherewj is the weight of the criteria Cj and wr=maxwjj∈1,2,…,n.Step 4: obtain the dominance degree of alternativeAi over the alternative At, as follows:(8)ϑAi,At=∑j=1nϕjAi,At,∀i,t.whereϕjAi,At is the dominance degree of the alternative Ai over the alternative At with respect to the criterion Cj, and it can be calculated as follows:(9)ϕjAi,At=wjrgij−gtj∑j=1nwjr,ifgij−gtj>0,0,ifgij−gtj=0,−1θ∑j=1nwjrgij−gtjwjr,ifgij−gtj<0,where the parameterθ is the attenuation factor of the losses. There are three cases: (1) if gij−gtj>0, then ϕjAi,At represents a gain; (2) if gij−gtj=0, then ϕjAi,At represents a nil; and (3) if gij−gtj<0, then ϕjAi,At represents a loss.Step 5: get the overall dominance degree of the alternativeAi as follows:(10)δAi=∑t=1mϑAi,At−mini∑t=1mϑAi,Atmaxi∑t=1mϑAi,At−mini∑t=1mϑAi,At.Step 6: sort the alternatives according to the overall dominance degreeδAi.The larger the value ofδAi is, the better the alternative is.


## 3.1. Probabilistic Linguistic Term Set

Deﬁnition 1 (see [10]).
Based on the additive linguistic term setS=sα=−τ,…,−1,0,…,τ [30,31], the probabilistic linguistic term set (PLTS) can be defined by Pang et al. [10] as follows:(1)Lp=LkpkLk∈S,pk≥0,k=1,2,…,#Lp,∑k=1#Lppk≤1,where Lkpk represents the linguistic term Lk associated with the probability pk and #Lp is the number of all different linguistic terms in Lp.
If∑k=1#Lppk=1, then the PLTSs have complete probabilistic information of all possible linguistic terms if ∑k=1#Lppk<1, then the PLTSs have partial probabilistic information; if ∑k=1#Lppk=0, then the PLTSs have completely unknown probabilistic information.Definition 2 (see [27]).
Given a probabilistic linguistic term setLp=Lkpkk=1,2,…,#Lp,∑k=1#Lppk≤1, on the additive linguistic term set S=sα=−τ,…,−1,0,…,τ, the negation operation for Lp is given by(2)negLp=L^kpkL^k=negLk=negSα,k=1,2,…,#Lp,where negLp is the negation operation for linguistic terms [31], defined as negSα=S−α, especially, negS0=S0.


## 3.2. The Comparison between PLTSs

First, the score of PLTS is defined and introduced by Pang et al. [10] as follows.Deﬁnition 3 (see [32]).
LetLp=Lkpkk=1,2,…,#Lp be a PLTS and rk be the subscript of the Lk, then the score of Lp is given by(3)ELp=sα¯,where α¯=∑k=1#Lprkpk/∑k=1#Lppk.
For any two PLTSsL1p and L2p, if EL1p>EL2p, then L1p is superior to L2p, denoted by L1p≻L2p; if EL1p<EL2p, then L2p is superior to L1p, denoted by L1p≺L2p; if EL1p=EL2p, then these two PLTSs cannot be distinguished. In this case, the deviation degree of a PLTS is further defined by Pang et al. [10] as follows.Deﬁnition 4 (see [10]).
LetLp=Lkpkk=1,2,…,#Lp be a PLTS and rk be the subscript of the Lk, and ELp=sα¯, then the deviation degree of Lp is given by(4)σLp=∑k=1#Lppkrk−α¯21/2∑k=1#Lppk.
For two PLTSsL1p and L2p with EL1p=EL2p, if σL1p>σL2p, then L1p≺L2p; if σL1p=σL1p, then L1p is indifferent to L2p, denoted by L1p∼L2p.Definition 5 (see [10]).
LetL1p=L1kpkk=1,2,…,#L1p and L2p=L2kpkk=1,2,…,#L2p be any two PLTSs, #L1p=#L2p, the distance between L1p and L2p can be defined as follows:(5)dL1p,L1p=∑k=1#L1pp1kr1k−p2kr2k2#L1p.


## 3.3. The Normalization of PLTS

The normalization of PLTSs usually has two distinct tasks. One is to estimate the ignorance degree of probabilistic information, and the other is to normalize the cardinality for computational convenience.Definition 6 (see [10]).
Given a PLTSLp with ∑k=1#Lppk<1, then the normalized PLTS L˜p is defined as follows:(6)L˜p=Lkpkk=1,2,…,#Lp,where p˜k=pk/∑k=1#Lppk, for all k=1,2,…,#Lp.
LetL1p=L1kp1kk=1,2,…,#Lp and L2p=L2kp2kk=1,2,…,#Lp, then the normalization process can be obtained as follows:(1)
If∑k=1#Lppk<1, then according to formula (6), L˜p can be calculated.(2)
If#L1p≠#L2p, then some elements can be added to the one with the smaller number of elements. That is to say, if #L1p>#L2p, then the #L1p−#L2p linguistic terms need to be added to L2p, which make the numbers of all different linguistic terms in L1p and L2p equal. The added linguistic terms should be the smallest ones in L2p, and their probabilities should be zero.


## 3.4. The TODIM Method

The TODIM method, proposed by Gomes and Lima [3], is an interactive MCDM method, which can capture psychology factors and cognitive behavior of DMs. On the base of prospect theory [2], the alternatives are ranked according to their overall dominance degree based on the dominance degree of each alternative over others with respect to each criterion. The steps of TODIM method can be summarized as follows [3].Suppose there areAii=1,2,…,m alternatives and Cjj=1,2,…,n criteria.Step 1: identify the decision matrixX=xijm×n, where xiji=1,2,…,m,j=1,2,…,n is the jth criteria value with respect to the ith alternative, and xij are all crisp numbers.Step 2: obtain the normalized decision matrixG=gijm×n. The normalized decision matrix can be obtained by using the normalization method.Step 3: calculate the relative weightwjr of the criteria Cj to the reference criteria Cr as follows:(7)wjr=wjwrj=1,2,…,n,wherewj is the weight of the criteria Cj and wr=maxwjj∈1,2,…,n.Step 4: obtain the dominance degree of alternativeAi over the alternative At, as follows:(8)ϑAi,At=∑j=1nϕjAi,At,∀i,t.whereϕjAi,At is the dominance degree of the alternative Ai over the alternative At with respect to the criterion Cj, and it can be calculated as follows:(9)ϕjAi,At=wjrgij−gtj∑j=1nwjr,ifgij−gtj>0,0,ifgij−gtj=0,−1θ∑j=1nwjrgij−gtjwjr,ifgij−gtj<0,where the parameterθ is the attenuation factor of the losses. There are three cases: (1) if gij−gtj>0, then ϕjAi,At represents a gain; (2) if gij−gtj=0, then ϕjAi,At represents a nil; and (3) if gij−gtj<0, then ϕjAi,At represents a loss.Step 5: get the overall dominance degree of the alternativeAi as follows:(10)δAi=∑t=1mϑAi,At−mini∑t=1mϑAi,Atmaxi∑t=1mϑAi,At−mini∑t=1mϑAi,At.Step 6: sort the alternatives according to the overall dominance degreeδAi.The larger the value ofδAi is, the better the alternative is.


## 4. Hybrid TODIM Method

In this section, a hybrid TODIM method with crisp number and PLTS is provided in detail.For an MCDM problem, suppose that there areAii=1,2,…,m alternatives and Cjj=1,2,…,n criteria. Based on the additive linguistic term set S=Sαα=−τ,…,−1,0,1,…,τ, the DMs assess the alternatives Ai for the criteria Cj, and the evaluation results during the probabilistic linguistic terms environment are given by Lp=Lkpkk=1,2,…,#Lp,∑k=1#Lppk≤1.The specific steps of the hybrid TODIM method with crisp number and PLTS are presented as follows:Step 1: identify the decision matrixX=xijm×n. The initial decision matrix can be constructed and identified with crisp number aij and PLTS Lijp.Step 2: obtain the normalized decision matrixG=gijm×n. The normalized decision matrix can be obtained as follows:(1)
Crisp number: the normalization of crisp numberaij˜ can be obtained as follows:(11)aij˜=aijmaxjaij,for benefit criteria,aij˜=minjaijaij,forcostcriteria.(2)
PLTS: the normalization of PLTSLijp can be obtained as follows:(12)Lij˜pijk=Lijpijk,for benefit criteria,negLijpijk,forcostcriteria,wherenegLijpijk is the negation operation for Lijpijk, which is defined in Deﬁnition 2. Besides, each criteria value of PLTS should be normalized according to Definition 6.Step 3: calculate the entropy weight for crisp number and probability linguistic entropy weight for PLTS. There are two steps:(1)
Crisp number: the entropy weight for crisp number can be determined as follows:(a)
Obtain the entropy: according to the definition of entropy, entropyEj of criteria Cj is determined as follows:(13)Ej=−1lnm∑i=1maijlnaij.(b)
Obtain the entropy weightwj′ of criteria Cj:(14)wj′=1−Ejn−∑j=1nEjj=1,2,…,n.(c)
Normalize the entropy weightwj: the entropy weight can be normalized as follows:(15)wj=wj′∑j=1nwj′,j=1,2,…,n.(2)
PLTS: based on the concepts of entropy, the probability linguistic entropy weight for PLTS can be obtained as follows:(a)
Based on the additive linguistic term setS=sα=−τ,…,−1,0,1…,τ[30,31], six methods for calculating the entropy value of a single linguistic term are proposed by Farhadinia [32]. One of the methods is as follows:(16)Esα=sinπ2α+τ2τ+sinπ2−α+τ2τ−1.(b)
LetLp=Lkpkk=1,2,…,#Lp be a PLTS, the probability linguistic entropy weight can be induced as follows:(17)EijLijpijk=∑i=1#LppijkEijSα.(c)
Normalize the probability linguistic entropy: the probability linguistic entropy can be normalized as follows:(18)Gij=Eijmax1≤i≤mEiji=1,2,…,m,j=1,2,…,n.(d)
Determine the weight: the weight of each criterion can be determined as follows:(19)wj′=1−Gj¯n−G¯j=1,2,…,n.whereGj¯=∑i=1mGij,G¯=∑i=1m∑j=1nGij.(e)
Determine the normalized weight: the normalized weight can be determined as follows:(20)wj=wj′∑j=1nwj′,j=1,2,…,n.Step 4: calculate the relative weightwjr of the criteria Cj to the reference criteria Cr as follows:(21)wjr=wjwrj=1,2,…,n,wherewj is the entropy weight for crisp number, or the probability linguistic entropy weight for PLTS of the criteria Cj and wr=maxwjj∈1,2,…,n.Step 5: obtain the dominance degreeϑ of alternative Ai over the alternative At, as follows:(22)ϑAi,At=∑j=1nϕjAi,Ati,t=1,2,…,m,whereϕjAi,At is the dominance degree of the alternative Ai over the alternative At with respect to the criterion Cj, and it can be calculated as follows:(23)ϕjAi,At=wjrdLij˜pij−Lij˜ptj∑j=1nwjr,ifLij˜pij≻Lij˜ptj,0,ifLij˜pij∼Lij˜ptj,−1θ∑j=1nwjrdLij˜pij−Lij˜ptjwjr,ifLij˜pij≺Lij˜ptj,where the parameterθ is the attenuation factor of the losses. There are three cases:(1)
IfLij˜pij≻Ltj˜ptj, then ϕjAi,At represents a gain(2)
IfLij˜pij∼Ltj˜ptj, then ϕjAi,At represents a nil(3)
IfLij˜pij≺Ltj˜ptj, then ϕjAi,At represents a lossIn this section, the distance measure can be determined as follows:(1)
Crisp number: the distance for crisp number can be determined as follows:(24)da,b=a−b.(2)
PLTS: the distance for PLTS can be determined according to the modified the Hamming distance of PLTSs based on the normalized PLTSs proposed by Lin and Xu [33].Step 6: get the overall dominance degreeδAiof the alternative Ai as follows:(25)δAi=∑t=1mϑAi,At−mini∑t=1mϑAi,Atmaxi∑t=1mϑAi,At−mini∑t=1mϑAi,At.Step 7: sort the alternatives according to the overall dominance degreeδAi.The larger the value ofδAiis, the better the alternative is.


## 5. A Case Study

### 5.1. Problem Description

The outbreak of COVID-19 epidemic has spread rapidly around the world, which has caused a great deal of property losses and a lot of casualties. It is very urgent to evaluate and respond to the urban epidemic situation involved in emergency medical management for providing scientific decision basis. Emergency medical management has a complex evolutionary process, and decision-making process has become more and more complex. In this section, a hybrid TODIM method with crisp number and PLTS is provided and presented to evaluate the severity of urban COVID-19 epidemic situation on a case study. The detailed assessment process is as follows.Five citiesAi=1,2,3,4,5i around the world and seven criteria Cjj=1,2,…,7 are selected to illustrate the point. These selected criteria by related literature and expert opinions are as follows: (1) C1: confirmed diagnosis rate per one million population; (2) C2: cure rate of confirmed cases; (3) C3: mortality rate of confirmed cases; (4) C4: growth rate of confirmed cases in three days; (5) C5: number of beds per capita of confirmed cases; (6) C6: people’s awareness of the epidemic; and (7) C7: government’s attitude toward the epidemic. Obviously, for evaluating the severity of urban epidemic situation, the criteria C1, C3, and C4 are of benefit type, and C2, C5, C6, and C7 are of cost type. The data information with crisp number and PLTS is presented, as shown in Table 1.Table 1
Data information.
C1C2C3C4C5C6C7A1162.20.050.040.463.2{S3(0.3), S2(0.5), S−1(0.2)}{S3(0.9), S1(0.1)}A2280.10.090.030.222.4{S3(0.5), S2(0.4)}{S3(0.7), S1(0.3)}A3338.00.100.010.195.3{S2(0.7), S1(0.2), S-2(0.1)}{S3(0.8), S1(0.1), S0(0.1)}A4186.70.070.010.227.3{S2(0.6), S1(0.3), S0(0.1)}{S3(0.8), S2(0.1), S1(0.1)}A5226.50.110.030.125.2{S2(0.5), S1(0.3), S-1(0.2)}{S3(0.6), S2(0.4)}Table 2
Normalized data information.
C1C2C3C4C5C6C7A10.47991.00001.00001.00000.4384{S−3(0.3), S−2(0.5), S1(0.2)}{ S−3(0), S−3(0.9), S−1(0.1)}A20.82870.55560.75000.47830.3288{ S−3(0), s−3(0.5556), S−2(0.4444)}{ S−3(0), S−3(0.7), S−1(0.3)}A31.00000.50000.25000.41300.7260{S−2(0.7), S−1(0.2), S2(0.1)}{S−3(0.8), S−1(0.1), S0(0.1)}A40.55240.71430.25000.47831.0000{S−2(0.6), S−1(0.3), S0(0.1)}{S−3(0.8), S−2(0.1), S−1(0.1)}A50.67010.45450.75000.26090.7123{S−2(0.5), S−1(0.3), S1(0.2)}{ S−3(0), S−3(0.6), S−2(0.4)}Table 3
The relative weight.
C1C2C3C4C5C6C7Weight0.21730.14520.20840.08090.17550.07570.0971Relative weight1.00000.66810.95940.37240.80770.34850.4467Table 4
Overall dominance degree for all alternatives.
AlternativeThe overall dominance degreeδAiRankingA1−8.30511.00001A2−14.76340.15114A3−13.07080.37363A4−11.96800.51852A5−15.91310.00005


### 5.2. Empirical Process

Now, the detailed evaluation steps are presented as follows:Step 1: identify the decision matrix.The decision matrix with crisp number and PLTS is constructed and identified by the following additive linguistic term set(26)S=S−3=extremely slight,S−2=very slight,S−1=slight,S0=general,S1=serious,S2=very serious,S3=extremely serious,to assess the severity of urban epidemic situation as shown in Table 1, the data of crisp number originated from the data management platform, a Wechat official account. The data of PLTS come from expert opinion, and the evaluation values during the probabilistic linguistic term environment are obtained and given according to the PLTS: Lp=Lkpkk=1,2,…,#Lp,∑k=1#Lppk≤1.Step 2: obtain the normalized decision matrix. It is obvious that the criteriaC1, C3, and C4 are of benefit type, and C2, C5, C6, and C7 are of cost type. So the normalization process is relatively simple and the normalized values can be calculated by formulas (11) and (12), Definition 2, and Definition 6, as shown in Table 2.Step 3: calculate the relative weight. The relative weight is calculated, based on the entropy weight for crisp number and probability linguistic entropy weight for PLTS according to formulas (13)–(21), as shown in Table 3. And the criteria C1 is the reference criteria, which has the greatest weight.Step 4: obtain the dominance degreeϑ of the alternative Ai over the alternative At by formulas (22)–(24), based on the dominance degree ϕj of the alternative Ai over the alternative At with respect to the criterion Cji,t=1,2,…,5,j=1,2,…,7:(27)ϕ1=0.0000−1.2671−1.5473−0.5776−0.93570.27530.0000−0.88800.24500.18560.33620.19290.00000.31190.26770.1255−1.1278−1.43540.0000−0.73620.2033−0.8544−1.23220.15990.0000,ϕ2=0.0000−0.2540−0.2694−0.2036−0.2814−1.74980.00000.0898−1.04570.1211−1.8560−0.61870.0000−1.21500.0812−1.40300.15180.17640.00000.1942−1.9385−0.8342−0.5596−1.33770.0000,ϕ3=0.00000.22830.39540.39540.2283−1.09520.00000.32280.32280.3228−1.8969−1.54880.00000.0000−1.5488−1.8969−1.54880.00000.0000−1.5488−1.09520.00000.32280.32280.0000,ϕ4=0.00000.20540.21790.20540.2445−2.53950.00000.07260.00000.1326−2.6936−0.89790.0000−0.89790.1110−2.53950.00000.07260.00000.1326−3.0226−1.6392−1.3715−1.63920.0000,ϕ5=0.00000.1387−1.2804−1.7891−1.2495−0.79030.0000−1.5046−1.9558−1.47850.22470.26400.0000−1.24950.04900.31390.34320.21930.00000.22470.21930.2594−0.2794−1.28040.0000,ϕ6=0.00000.0600−0.4691−0.3831−0.6635−0.79230.0000−0.9208−0.8800−1.03340.03550.06970.00000.0205−0.46910.02900.0666−0.27090.0000−0.54170.05020.07830.03550.04100.0000,ϕ7=0.0000−0.4785−0.4144−0.2393−0.33840.04640.00000.02320.04020.03280.0402−0.23930.00000.03280.02320.0232−0.4144−0.33840.0000−0.23930.0328−0.3384−0.23930.02320.0000,ϑ=0.0000−0.8592−2.8285−2.1845−2.4329−6.64530.0000−2.8049−3.2735−2.0397−5.8099−2.77790.0000−2.9972−1.4858−5.3477−2.5294−1.57640.0000−2.5145−5.5507−3.3285−3.3236−3.71030.0000.Step 5: get the overall dominance degreeδAi of the alternative Ai. The overall dominance degree δAi can be obtained according to formula (25) and is listed in Table 4.Step 6: rank all the alternatives. The ranking of all the alternatives can be obtained according to the overall dominance degreeδAi and is listed in Table 4. The larger the value of δAiis, the better the alternative is. We can see the best alternative is A1, and the ranking is A1≻A4≻A3≻A2≻A5. That is to say, the most serious city of the urban COVID-19 epidemic is A1. Obviously, our proposed hybrid TODIM method is feasible and can get a good decision result because our proposed hybrid method can not only involve multiple criteria or attributes about qualitative and quantitative aspects, but also express their judgments and reflect different probability degrees of all the possible linguistic information or preference information.


### 5.3. Sensitivity Analysis

In order to further verify the validity and effectiveness of the hybrid TODIM method with crisp number and PLTS, sensitivity analysis of the parameterθ is carried out for urban epidemic situation evaluation. In the hybrid TODIM method, the dominance degree ϕjAi,At of the alternative Ai over the alternative At with respect to the criterion Cj is determined based on the parameter θ, which is the attenuation factor of the losses. So, it is very necessary and important to analyze the sensitivity of the parameter θ according to the different values for each simulation, and the ranking results are presented in Table 5.Table 5
Sensitivity analysis of the parameterθ
θ = 0.1θ = 0.3θ = 0.5θ = 0.7θ = 0.9θ = 1.2θ = 1.4θ = 1.6θ = 1.8θ = 2.0δ(Ai)Rankδ(Ai)RankΔ(Ai)Rankδ(Ai)Rankδ(Ai)Rankδ(Ai)Rankδ(Ai)Rankδ(Ai)Rankδ(Ai)Rankδ(Ai)RankA11.000011.000011.000011.000011.000011.000011.000011.000011.000011.00001A20.162340.159440.156740.154340.152140.149240.147440.145740.144240.14274A30.414030.403530.394030.385330.377330.366630.360130.354130.348530.34333A40.590520.571820.554820.539320.525220.506120.494520.483920.474020.46472A50.000050.000050.000050.000050.000050.000050.000050.000050.000050.00005From Table5, the ranking orders of alternatives are consistent when the parameter θ takes different values from 0.1 to 2, which indicates the validity, effectiveness, and stability of the proposed hybrid TODIM method with the changes of the attenuation parameter θ. And our proposed hybrid TODIM method involves multiple criteria or attributes about qualitative and quantitative aspects for improving the accuracy of evaluation results; besides, our proposed hybrid TODIM method can not only express their judgments or linguistic preference with multiple linguistic terms, but also reflect different probability degrees of all the possible linguistic information or preference information. In addition, the concept of entropy and probability linguistic entropy is applied to induce hybrid criteria weight information.


## 5.1. Problem Description

The outbreak of COVID-19 epidemic has spread rapidly around the world, which has caused a great deal of property losses and a lot of casualties. It is very urgent to evaluate and respond to the urban epidemic situation involved in emergency medical management for providing scientific decision basis. Emergency medical management has a complex evolutionary process, and decision-making process has become more and more complex. In this section, a hybrid TODIM method with crisp number and PLTS is provided and presented to evaluate the severity of urban COVID-19 epidemic situation on a case study. The detailed assessment process is as follows.Five citiesAi=1,2,3,4,5i around the world and seven criteria Cjj=1,2,…,7 are selected to illustrate the point. These selected criteria by related literature and expert opinions are as follows: (1) C1: confirmed diagnosis rate per one million population; (2) C2: cure rate of confirmed cases; (3) C3: mortality rate of confirmed cases; (4) C4: growth rate of confirmed cases in three days; (5) C5: number of beds per capita of confirmed cases; (6) C6: people’s awareness of the epidemic; and (7) C7: government’s attitude toward the epidemic. Obviously, for evaluating the severity of urban epidemic situation, the criteria C1, C3, and C4 are of benefit type, and C2, C5, C6, and C7 are of cost type. The data information with crisp number and PLTS is presented, as shown in Table 1.Table 1
Data information.
C1C2C3C4C5C6C7A1162.20.050.040.463.2{S3(0.3), S2(0.5), S−1(0.2)}{S3(0.9), S1(0.1)}A2280.10.090.030.222.4{S3(0.5), S2(0.4)}{S3(0.7), S1(0.3)}A3338.00.100.010.195.3{S2(0.7), S1(0.2), S-2(0.1)}{S3(0.8), S1(0.1), S0(0.1)}A4186.70.070.010.227.3{S2(0.6), S1(0.3), S0(0.1)}{S3(0.8), S2(0.1), S1(0.1)}A5226.50.110.030.125.2{S2(0.5), S1(0.3), S-1(0.2)}{S3(0.6), S2(0.4)}Table 2
Normalized data information.
C1C2C3C4C5C6C7A10.47991.00001.00001.00000.4384{S−3(0.3), S−2(0.5), S1(0.2)}{ S−3(0), S−3(0.9), S−1(0.1)}A20.82870.55560.75000.47830.3288{ S−3(0), s−3(0.5556), S−2(0.4444)}{ S−3(0), S−3(0.7), S−1(0.3)}A31.00000.50000.25000.41300.7260{S−2(0.7), S−1(0.2), S2(0.1)}{S−3(0.8), S−1(0.1), S0(0.1)}A40.55240.71430.25000.47831.0000{S−2(0.6), S−1(0.3), S0(0.1)}{S−3(0.8), S−2(0.1), S−1(0.1)}A50.67010.45450.75000.26090.7123{S−2(0.5), S−1(0.3), S1(0.2)}{ S−3(0), S−3(0.6), S−2(0.4)}Table 3
The relative weight.
C1C2C3C4C5C6C7Weight0.21730.14520.20840.08090.17550.07570.0971Relative weight1.00000.66810.95940.37240.80770.34850.4467Table 4
Overall dominance degree for all alternatives.
AlternativeThe overall dominance degreeδAiRankingA1−8.30511.00001A2−14.76340.15114A3−13.07080.37363A4−11.96800.51852A5−15.91310.00005


## 5.2. Empirical Process

Now, the detailed evaluation steps are presented as follows:Step 1: identify the decision matrix.The decision matrix with crisp number and PLTS is constructed and identified by the following additive linguistic term set(26)S=S−3=extremely slight,S−2=very slight,S−1=slight,S0=general,S1=serious,S2=very serious,S3=extremely serious,to assess the severity of urban epidemic situation as shown in Table 1, the data of crisp number originated from the data management platform, a Wechat official account. The data of PLTS come from expert opinion, and the evaluation values during the probabilistic linguistic term environment are obtained and given according to the PLTS: Lp=Lkpkk=1,2,…,#Lp,∑k=1#Lppk≤1.Step 2: obtain the normalized decision matrix. It is obvious that the criteriaC1, C3, and C4 are of benefit type, and C2, C5, C6, and C7 are of cost type. So the normalization process is relatively simple and the normalized values can be calculated by formulas (11) and (12), Definition 2, and Definition 6, as shown in Table 2.Step 3: calculate the relative weight. The relative weight is calculated, based on the entropy weight for crisp number and probability linguistic entropy weight for PLTS according to formulas (13)–(21), as shown in Table 3. And the criteria C1 is the reference criteria, which has the greatest weight.Step 4: obtain the dominance degreeϑ of the alternative Ai over the alternative At by formulas (22)–(24), based on the dominance degree ϕj of the alternative Ai over the alternative At with respect to the criterion Cji,t=1,2,…,5,j=1,2,…,7:(27)ϕ1=0.0000−1.2671−1.5473−0.5776−0.93570.27530.0000−0.88800.24500.18560.33620.19290.00000.31190.26770.1255−1.1278−1.43540.0000−0.73620.2033−0.8544−1.23220.15990.0000,ϕ2=0.0000−0.2540−0.2694−0.2036−0.2814−1.74980.00000.0898−1.04570.1211−1.8560−0.61870.0000−1.21500.0812−1.40300.15180.17640.00000.1942−1.9385−0.8342−0.5596−1.33770.0000,ϕ3=0.00000.22830.39540.39540.2283−1.09520.00000.32280.32280.3228−1.8969−1.54880.00000.0000−1.5488−1.8969−1.54880.00000.0000−1.5488−1.09520.00000.32280.32280.0000,ϕ4=0.00000.20540.21790.20540.2445−2.53950.00000.07260.00000.1326−2.6936−0.89790.0000−0.89790.1110−2.53950.00000.07260.00000.1326−3.0226−1.6392−1.3715−1.63920.0000,ϕ5=0.00000.1387−1.2804−1.7891−1.2495−0.79030.0000−1.5046−1.9558−1.47850.22470.26400.0000−1.24950.04900.31390.34320.21930.00000.22470.21930.2594−0.2794−1.28040.0000,ϕ6=0.00000.0600−0.4691−0.3831−0.6635−0.79230.0000−0.9208−0.8800−1.03340.03550.06970.00000.0205−0.46910.02900.0666−0.27090.0000−0.54170.05020.07830.03550.04100.0000,ϕ7=0.0000−0.4785−0.4144−0.2393−0.33840.04640.00000.02320.04020.03280.0402−0.23930.00000.03280.02320.0232−0.4144−0.33840.0000−0.23930.0328−0.3384−0.23930.02320.0000,ϑ=0.0000−0.8592−2.8285−2.1845−2.4329−6.64530.0000−2.8049−3.2735−2.0397−5.8099−2.77790.0000−2.9972−1.4858−5.3477−2.5294−1.57640.0000−2.5145−5.5507−3.3285−3.3236−3.71030.0000.Step 5: get the overall dominance degreeδAi of the alternative Ai. The overall dominance degree δAi can be obtained according to formula (25) and is listed in Table 4.Step 6: rank all the alternatives. The ranking of all the alternatives can be obtained according to the overall dominance degreeδAi and is listed in Table 4. The larger the value of δAiis, the better the alternative is. We can see the best alternative is A1, and the ranking is A1≻A4≻A3≻A2≻A5. That is to say, the most serious city of the urban COVID-19 epidemic is A1. Obviously, our proposed hybrid TODIM method is feasible and can get a good decision result because our proposed hybrid method can not only involve multiple criteria or attributes about qualitative and quantitative aspects, but also express their judgments and reflect different probability degrees of all the possible linguistic information or preference information.


## 5.3. Sensitivity Analysis

In order to further verify the validity and effectiveness of the hybrid TODIM method with crisp number and PLTS, sensitivity analysis of the parameterθ is carried out for urban epidemic situation evaluation. In the hybrid TODIM method, the dominance degree ϕjAi,At of the alternative Ai over the alternative At with respect to the criterion Cj is determined based on the parameter θ, which is the attenuation factor of the losses. So, it is very necessary and important to analyze the sensitivity of the parameter θ according to the different values for each simulation, and the ranking results are presented in Table 5.Table 5
Sensitivity analysis of the parameterθ
θ = 0.1θ = 0.3θ = 0.5θ = 0.7θ = 0.9θ = 1.2θ = 1.4θ = 1.6θ = 1.8θ = 2.0δ(Ai)Rankδ(Ai)RankΔ(Ai)Rankδ(Ai)Rankδ(Ai)Rankδ(Ai)Rankδ(Ai)Rankδ(Ai)Rankδ(Ai)Rankδ(Ai)RankA11.000011.000011.000011.000011.000011.000011.000011.000011.000011.00001A20.162340.159440.156740.154340.152140.149240.147440.145740.144240.14274A30.414030.403530.394030.385330.377330.366630.360130.354130.348530.34333A40.590520.571820.554820.539320.525220.506120.494520.483920.474020.46472A50.000050.000050.000050.000050.000050.000050.000050.000050.000050.00005From Table5, the ranking orders of alternatives are consistent when the parameter θ takes different values from 0.1 to 2, which indicates the validity, effectiveness, and stability of the proposed hybrid TODIM method with the changes of the attenuation parameter θ. And our proposed hybrid TODIM method involves multiple criteria or attributes about qualitative and quantitative aspects for improving the accuracy of evaluation results; besides, our proposed hybrid TODIM method can not only express their judgments or linguistic preference with multiple linguistic terms, but also reflect different probability degrees of all the possible linguistic information or preference information. In addition, the concept of entropy and probability linguistic entropy is applied to induce hybrid criteria weight information.


## 6. Conclusions

The frequent occurrence of emergency medical incidents has caused a great deal of property losses and a lot of casualties [17]. Emergency medical management and epidemic situation evaluation has a complex evolutionary process, and emergency decision-making is very difficult to deal with the traditional decision theory. So, in this paper, based on the TODIM method considering the psychology and behavior of the DMs and PLTS expressing preference information of DMs, a hybrid TODIM method with crisp number and PLTS is proposed for aiming to assist DMs to evaluate the severity of urban COVID-19 epidemic situation involved in emergency medical management. Five main advantages of this paper are summarized as follows: firstly, the extended hybrid TODIM method is first proposed for epidemic situation evaluation during a complex humanitarian crisis environment. Secondly, in this hybrid TODIM method, multiple criteria or attributes about qualitative and quantitative aspects are involved, considering the complexity of emergency decision-making process. Thirdly, the proposed hybrid TODIM method can not only express their judgments or linguistic preference with multiple linguistic terms, but also reflect different importance degrees or probability degrees of all the possible linguistic information or preference information. Fourthly, the concept of entropy and probability linguistic entropy is applied to induce hybrid criteria weight information, which has the advantages of simple calculation and clear concept. Finally, sensitivity analysis of the parameter, which considers the psychology factors and cognitive behavior of the DMs, is further carried out on a case study to demonstrate the validity, effectiveness, and stability of the proposed hybrid method for urban epidemic situation evaluation.The limitations of this work are summarized as follows: epidemic situation evaluation has a complex evolutionary process. Through literature research, based on the complexity of emergency decision-making process, the TODIM method is still not well studied and applied for emergency medical decision-making, although it has done a lot of outreach and extension. So, in this paper, the extended hybrid TODIM method with crisp number and PLTS is first proposed for epidemic situation evaluation. However, the sample size of the data is relatively small.In future work, new and effective evaluation methods will be further researched and developed by comparative analysis to effectively respond to emergency medical emergencies during a complex humanitarian crisis environment and the crisis response strategies will also be the focus of our research. In addition, big data analysis techniques will also be further developed to reduce the impact of small sample sizes on research results.



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*Source: 4857392-2020-08-19.xml*